REVISTA MEXICANA DE FÍSICASome of those designs include periodic band gap filters like defected ground structure (DGS), that is actually a transmis-sion line along with a well-defined

REVISTA MEXICANA DE FISICA

Director:Francisco Ramos GomezFacultad de Ciencias, UNAM, Mexico

Consejeros EmeritosLeopoldo Garcıa-ColınUniversidad Autonoma Metropolitana – Iztapalapa, Mexico

Manuel PeimbertInstituto de Astronomıa UNAM, Mexico

Fernando AlbaInstituto de Fısica, UNAM, Mexico

Consejo Editorial

Materia Condensada:Carlos BalseiroCentro Atomico de Bariloche, Argentina

Alipio G. CallesFacultad de Ciencias, UNAM, Mexico

Manuel CardonaInstitute Max Planck, Stuttgart, Alemania

Robert CavaUniversity of Princeton, USA

Roberto EscuderoInstituto de Investigaciones en Materiales, UNAM, Mexico

Francisco JaqueUniversidad Autonoma de Madrid, Espana

Harold KrotoFlorida State University

Fısica Atomica y Molecular:Gerardo Delgado-BarrioConsejo Superior de Investigacion Cientıfica, Espana

James McGuireTulane University, USA

Alfred SchlachterAdvanced Light Source, LBL Berkeley California, USA

Fısica Nuclear:Alejandro FrankInstituto de Ciencias Nucleares, UNAM, Mexico

Arturo MenchacaInstituto de Fısica, UNAM, Mexico

Andres SandovalGSI, Alemania & CERN, Suiza

Termodinamica y Fısica Estadıstica:Eugenio E. VogelUniversidad de la Frontera, Chile

Ivan L’heureuxUniversity of Ottawa, Canada

Vıctor RomeroInstituto de Fısica, UNAM, Mexico

Optica:Alejandro CornejoInstituto Nacional de Astrofısica, Optica y Electronica, Mexico

Eugenio MendezCICESE, Mexico

Jumpei TsujiuchiInstitute of Technology, Tokio, Japon

Fernando MendozaCentro de Investigaciones en Optica, Mexico

Gravitacion y Fısica Matematica:Octavio ObregonInstituto de Fısica, Universidad de Guanajuato, Mexico

Fernando QuevedoUniversity of Cambridge, Inglaterra

Instrumentacion:Victor CastanoCentro de Fısica Aplicada y Tecnologıa Avanzada, UNAM, Mexico

Daniele FinotelloKent State University, USA

Partıculas Elementales y Teorıa de Campo:Gerardo Herrera CorralCINVESTAV, IPN, Mexico

Fısica Medica:Marıa Ester BrandanInstituto de Fısica, UNAM, Mexico

Secretaria:Marıa Magdalena Lopez ReynosoSociedad Mexicana de Fısica

Edicion Tecnica:Raul A. Espejel MoralesFacultad de Ciencias, UNAM

Asistentes Tecnicos:Efraın R. Garrido RomanSociedad Mexicana de Fısica

Paris M. Sanchez CarreonSociedad Mexicana de Fısica

La Revista Mexicana de Fısica es una publicacion bimestral de la Sociedad Mexicana de Fısica, A.C., Apartado Postal 70-348, Coyoacan,04511 Mexico, D.F., MEXICO. Se publica con el patrocinio de: Consejo Nacional de Ciencia y Tecnologıa, Instituto Nacional de AstrofısicaOptica y Electronica, de la UNAM: Rectorıa, Coordinacion de la Investigacion Cientıfica, Instituto de Astronomıa, Instituto de CienciasNucleares, Instituto de Investigaciones en Materiales, Instituto de Fısica y Facultad de Ciencias.

Indizada en: Actualidad Iberoamericana, Astron. & Astrophys. Abstr., Bull. Signal., Chem. Abstr., Curr. Cont., Curr. Math. Pub., Curr.Pap. Phys., Electr. & Electron. Abstr., INIS Atomind., Math. Sci., LatIndex, Math. Rev., Nucl. Sci. Abstr., PERIODICA, Phys. Abstr.,Phys. Ber., Res. Alert, Sci. Abstr., Sci. Cit. Ind., y SciSearch. Incluida en el Indice de Revistas Mexicanas de Investigacion Cientıfica yTecnologica del Consejo Nacional de Ciencia y Tecnologıa (CONACyT).

Las instrucciones para autores aparecen publicadas en el numero 6 de cada volumen y en la pagina de internet http://rmf.smf.mx. Elcosto de la suscripcion anual es de $1000.00 pesos para la Republica Mexicana, $130 USD para America Central y del Norte y $160 USDpara el resto del mundo. Precio del ejemplar $170.00 pesos.

Revista Mexicana de Fısica ISSN 0035-00IX

La Revista Mexicana de Fısica es publicada y distribuida semestralmente por la SociedadMexicana de Fısica A.C., con domicilio en: 2 piso, Departamento de Fısica, Facultad de Cien-cias, Ciudad Universitaria, 04510 Mexico, D.F. Tel.: (+52 55) 5622-4946; fax: (+52 55) 5622-4848.

Director : Francisco Ramos Gomez

Domicilio de la Revista Mexicana de Fısica: 2 piso, Departamento de Fısica, Facultadde Ciencias, Ciudad Universitaria, 04510 Mexico, D.F., Apartado postal: 70-348, delegacionCoyoacan, 04510 Mexico, D.F. Telefono: (+52 55) 5622-4946; fax: (+52 55) 5622-4848. Correoelectronico: [email protected], Pagina en Internet: http://rmf.smf.mx/

Impresion: Reproducciones Graficas del Sur, S.A. de C.V., Amatl No. 20, Col. Santo Domingo,Delegacion Coyoacan, 04369 Mexico, D.F., Tel.: 5619-4088.

Certificado de licitud numero 13482 y de contenido numero 11055 otorgado por la ComisionCalificadora de Publicaciones y Revistas Ilustradas de la Secretarıa de Gobernacion. Reservadel tıtulo numero 04–2005–062911592100–102 de la Direccion General de Derechos de Autor.Publicacion periodica: Registro numero 038 0488, caracterısticas 210241109, otorgado por laoficina del Servicio Postal Mexicano.

El volumen 57, numero 3, junio de 2011, se termino de imprimir en junio de 2011; se tiraron1000 ejemplares.

Se autoriza la reproduccion parcial o total de su contenido citando la fuente: Revista Mexicanade Fısica o Rev. Mex. Fis. Los artıculos firmados son responsabilidad de los autores.

Diseno de portada: Arte Grafico, Sur 71 No. 501, Col. Justo Sierra, 09460 Mexico, D.F.

Impreso en Mexico–Printed in Mexico

CARTA REVISTA MEXICANA DE FISICA 57 (3) 184–187 JUNIO 2011

Spurline structures and its application on microwave coupled line filter

J.R. Loo-Yau, O.I. Gomez-Pichardo, and F. Sandoval-IbarraCentro de Investigacion y de Estudios Avanzados-Guadalajara Unit,

Av. Del Bosque 1145, Col. El Bajıo, 45015 Zapopan, Jal.Tel: +52 (33) 3777-3600,

e-mail: [email protected]

M.C. Maya-Sanchez and J.A. Reynoso-HernandezCentro de Investigacion Cientıfica y de Educacion Superior de Ensenada,

Carretera Ensenada-Tijuana 2918, Zona Playitas, 28860, Ensenada, Baja California,Tel: +52 (646) 175-0500

Recibido el 28 de febrero de 2011; aceptado el 18 de mayo de 2011

We propose and demonstrate experimentally that spurline structures enhance the rejection bandwidth of microwave bandstop coupled linefilters. We have investigated the influence of spurlines structures on the rejection bandwidth of a typical microwave coupled line filters (witha notch frequency at 3.0 GHz). Momentum simulations and experimental results show that using spurline structures (designed to present anotch frequency at 2.4 GHz and 3.2 GHz) enhance in high percentage the performance of microwave coupled line filters (CLF).

Keywords: Filters; microwave circuits.

En este trabajo se propone y demuestra experimentalmente que las estructuras “spurlines” son capaces de mejorar el ancho de banda de losfiltros de microondas de lıneas acopladas. Se ha investigado la influencia de las estructuras “spurlines” sobre la banda de rechazo de unfiltro de microondas de lıneas acopladas tıpico (con una frecuencia de supresion de 3.0 GHz). Simulaciones con el metodo de Momentos yresultados experimentales muestran que utilizando las estructuras “spurlines” (disenadas para rechazar frecuencias de 2.4 y 3.2 GHz) mejoranen un alto porcentaje el ancho de banda del filtros de microondas de lıneas acopladas (FLA).

Descriptores: Filtros; circuitos de microonda.

PACS: 84.30.Vn; 84.40.Dc

1. Introduction

Since several decades ago, both active and passive filters arewidely used to suppress unwanted signals. At microwave fre-quencies, passive components like transmission lines havebeen used for designing passive filters oriented to wirelesscommunications applications [1], where open, short stubs,and coupled lined are some common proposals. In particular,bandstop and bandpass filters can be designed by using cou-pled line structures as shown in Fig. 1a-b, respectively. How-ever, these filters are narrowband designs. In recent years, theso-called metamaterials have been developed for the fabrica-tion of special substrates to design microwave filters [2,3].Some of those designs include periodic band gap filters likedefected ground structure (DGS), that is actually a transmis-sion line along with a well-defined etching in the backsideground plane (see Fig. 1c) [4]. In the practice, the advantageof DGS filters lies in the sharp cut-off frequency response,presenting a low pass behavior. An alternative periodic bandgap filter is based on spurline structure [5]. The frequencyresponse of a spurline filter is a notch type, with narrow re-jection bandwidth. How to enhance the rejection bandwidthin spurline filters is an open research field, from which Liuet al have proposed the design of a bandstop filter (from 2.3to 5.6 GHz) based on a meander spurline [6]. In that pro-posal, the simulation-based in optimization process impliesoptimize five design parameters, representing a difficult work

because the time interval between calculated points is long,i.e. one need to run the simulation for a long time in order toget optimized values.

In this paper, we propose to use spurline structures to en-hance the rejection bandwidth of a microwave coupled linefilter. The proposed design, fabricated on Rogers substrateRT/D 5880, was theoretically verified using EM simulations,based on the method of momentum from ADS.

2. Experiment

Figure 2a shows the basic structure of a spurline, whichworks as a narrow bandstop filter, where the frequency re-sponse is determined by optimizing three parameters: length(a), height (b), and gap (s) [7]. The lengtha and heightb dictate the frequency of the notch characteristic. How-ever, it is more properly, in a non-reflective transmission line(Z0=50Ω), tune the notch frequency by modifying the lengtha instead of the heightb. Moreover, Fig. 2b shows simula-tions results of the frequency response for the spurline struc-ture as a function of the internal gap,s. It is evident that thebandwidth of the rejection band is directly related with thisparameter. Another characteristic of the filter, not shown inthe figure, is that the frequency response at which the notchoccurs increases when the lengtha decreases. According tothat, the main idea to design a CLF with an enhanced rejec-tion bandwidth is to get a total frequency response based on

SPURLINESTRUCTURES AND ITS APPLICATION ON MICROWAVE COUPLED LINE FILTER 185

FIGURE 1. Microwave couple line filters. Bandstop (a); bandpass(b); Defected Ground Structure (c).

FIGURE 2. Spurline structure. Transmission line with an etchingprocess describing an L shape (a); frequency response at differentinternal gap,s=0.2 mm (b).

FIGURE 3. Bandstop filter. Coupled line filter on RT/D 5880 sub-strate at 3 GHz (a); S21 frequency response (b).

the response of three narrow bandstop filters (a CLF embed-ded into spurlines structures), where each notch frequencywill represent the lowest (fL), central (f0) and highest (fH )frequency of the rejection bandwidth. To validate our hypoth-esis the proposed filter will be compared with the response ofa typical 3 GHz coupled transmission line filter (see Fig. 3a),which correspond to an electrical length of 210 with a char-acteristic impedanceZ0 > 100 Ω.

Figure 3b compares theS21 between Momentum resultsand experimental data of the bandstop coupled line filter;EM simulation was performed without SMA connectors, andthe experimental data were collected using a VNA (Anritsu,37347D) that was calibrated according to the SOLT tech-nique. Note that this design is composed by three sections,one of them is the gapg between transmission lines; the elec-trical characteristics of the microstrips used in each sectionare depicted in Table I. Moreover, Fig. 3b shows that exper-imental data have a good agreement with simulation. Theerror between both results, around 3.5 GHz, is attributed to apoor coplanar to microstrip transition. Defining 10 dB of in-sertion loss, the resulting rejection bandwidth of the bandpassis of the order of 1.042 GHz.

2.1. Proposed filter

Figure 4a shows the proposed bandstop filter, where the cou-pled lines section was also designed according to the physicaldimension reported in the Table I. As we can see in Fig. 4b

Rev. Mex. Fıs. 57 (3) (2011) 184–187

186 J.R.LOO-YAU, O.I. GOMEZ-PICHARDO, F. SANDOVAL-IBARRA, M.C. MAYA-SANCHEZ, AND J.A. REYNOSO-HERNANDEZ

FIGURE4. Layout of the proposed filter (a); coupled line filter withspurline structure (b); S21 frequency response (c).

the length “a” of the spurline 1 is slightly larger than thelength “a” of the spurline 2, because the first one hasto present the notch at the lowest frequency (2.4 GHz);the spurline 2 was designed at a notch frequency of3.2 GHz. Table II reports the physical dimensionsof the spurline structures. Figure 4c shows, a com-parison between EM simulation results and experimen-tal data. The experimental response of each section,not shown in the figure, presents approximately a valueS21=-40 dB at the notch frequency.

This characteristic is well reproduced also by measuringthe response of the proposed filter, as shown in Fig. 4c.However, from experimental point-of-view, even when arepresented the expected notch frequencies, one of them is ap-proximately at 2.7 GHz. This value can be explained by cal-culating the central frequency of the bandwidth given by

f0 =√

fLfH ≈ 2.7713 GHz (1)

TABLE I. Characteristics of the transmission line of the bandstopcoupled line filter.

Section1 Section 2 Section 3

Z0 = 50Ω Z0= 128.8Ω Z0 = 50Ω

L = 90.0 L = 192.0 L = 90.0

g = 0.3 mm

TABLE II. Spurlines structure characteristics.

Spurline 1 Spurline 2

a 23.6 mm 17.3 mm

b 2.0 mm 2.0 mm

s 1.0 mm 1.0 mm

wherefL= 2.4 GHz andfH= 3.2 GHz. This result, by onehand, affects the symmetry of the bandwidth. Such a fre-quency shift, respect to the notch frequency of the gap be-tween coupled lines, also affects the value ofS21(≈-7 dB)around 3.0 GHz. On the other hand, the rejection bandwidthin general is increased in a 50% compared with the responseof bandpass coupled line filter.

3. Conclusions

We have used a technique based on spurline structure toenhance the rejection bandwidth of coupled line filtersfor microwave applications. This technique was used toalso demonstrate that the notch frequency of an individualspurline structure could be tuned by optimizing just three ba-sic parameters: length (a), height (b), and internal gap (s).Hence, the proposed filter is to superimpose the frequencyresponse of three narrow bandstop filters. Simulated results,using Momentum, have a high correlation with experimentaldata. The rejection bandwidth of the filter (up to this worknot investigated) increased in a 50% respect to the responseof typical bandpass coupled line filters. Finally, we presentthe origin of the central frequency (2.7 GHz) as a function ofthe notch frequency of both spurline structures.

Acknowledgments

The authors thank to Agilent Technologies Mexico for dona-tion of ADS, to Rogers Corporation for providing the sub-strate used in this work, and J.L. Urbina-Martınez for assist-ing us on the use of Momentum.

Rev. Mex. Fıs. 57 (3) (2011) 184–187

SPURLINESTRUCTURES AND ITS APPLICATION ON MICROWAVE COUPLED LINE FILTER 187

1. D.M. Pozar,Microwave Engineering, Second Edition(John Wi-ley & Sons, Inc., 1998) p. 474.

2. J. Bonache, I. Gil, J. Garcia-Garcia, and F. Martin,IEEE Trans.on Microwave Theory Tech. 54 (2006) 265.

3. H. Lobato-Morales, A. Corona-Chavez, and J. Rodriguez-Asomoza,Microwave Optical Tech. Lett51 (2009) 1155.

4. Dal Ahn, Jun-Seok Park, Chul-Soo Kim, Juno Kim, YongxiQian, and Tatsuo Itoh,IEEE Trans. on Microwave Theory andTechniques49 (2001) 86

5. F.C. Nguyen and K. Chang,IEEE Trans. on Microwave Theoryand Techniques33 (1985) 1416.

6. H. Liu, R.H. Knoechel, and K.F. Schuenemann,ETRI Journal29 (2007) 614.

7. H. Liu, L. Sun, and Z. Shi,Microwave and Optical TechnologyLetters49 (2007) 2805.

Rev. Mex. Fıs. 57 (3) (2011) 184–187

INVESTIGACION REVISTA MEXICANA DE FISICA 57 (3) 188–192 JUNIO 2011

Entrelazamiento cuantico espurio con matrices seudopuras extendidas 4 por 4

J.D. Bulnesa y L.A. Pecheb,†aUniversidade Estadual do Norte Fluminense,

Av. Alberto Lamego 2000, Campos dos Goytacazes - CEP: 28013-602, RJ, Brasil.bUniversidade do Estado do Rio de Janeiro,

Rua Sao Francisco Xavier, 524, Maracana, Rio de Janeiro - CEP: 20550-900, RJ, Brasil.

Recibido el 26 de julio de 2010; aceptado el 26 de abril de 2011

En este artıculo consideramos una extension matematica para el concepto de matriz seudopura RMN, en el caso particular de matrices4× 4,a partir de matrices no fısicas. Se presentan ejemplos de matrices densidad entrelazadas son presentados. Aquel entrelazamiento, identificadopor el criterio de Peres-Horodecki, es espurio debido a que no es posible asignar estas matrices a cualesquiera estados cuanticos de un sistemade espines nucleares RMN.

Descriptores:Entrelazamiento cuantico; criterio de Peres-Horodecki; estado seudopuro.

In this paper, we consider one mathematical extension for the concept NMR pseudo-pure matrix in the particular case of4 × 4 matrices,from non-physical matrices. Examples of entangled density matrices are presented. Those entanglement, identified by the Peres-Horodeckicriterion, is spurious because it is not possible to assign these matrices to any physical state of a system of NMR nuclear spins.

Keywords: Quantum entanglement; Peres-Horodecki criterion; pseudo-pure state.

PACS: 03.65.Ud; 03.67.Mn; 03.65.-w

1. Introduccion

A pesar de la abundante y diversa produccion en el campode la computacion cuantica [1,2], en la cual se vienen con-siderando distintos sistemas fısicos y el uso de varias tecni-cas, inclusive hıbridas, lo que sabemos del entrelazamientocuantico (quantum entanglement) [3,4] en lo que se refiere asu significado fısico, ası como a su generacion, preservaciony medida, lo que en buena parte depende de la interpretacionde los resultados experimentales, permanece claramente in-completo. La identificacion teorica de dicho entrelazamientoes un asunto que, en el caso mas general de mezclas estadısti-cas, aun no ha sido aclarado, mientras que su representacionmatematica esta formalmente incorporada en la estructura dediversos algoritmos y protocolos. Del lado de las implemen-taciones experimentales de computacion cuantica contamosvarios casos destacables, algunos de ellos habiendo hecho usode sistemas fısicos (moleculas) con espın nuclear sensible alfenomeno de la resonancia magnetica nuclear (RMN), comoen las Refs. 5, 6 y 7, correspondientes al caso de espın nuclearI = 1/2, los que rapidamente llamaron la atencion y gene-raron un intenso debate sobre la capacidad de los sistemasRMN para implementar o simular el entrelazamiento [8-11];otras implementaciones han sido realizadas usando nucleoscuadrupolares deI = 3/2, como en [12-14]. En todos esosexperimentos fueron manipulados (indirectamente) un nume-ro macroscopico de partıculas cuanticas, pero solo un numeropequeno debits cuanticos, los mismos que fueron realizadosgracias a la preparacion del sistema RMN en un estado ini-cial especial, a la evolucion unitaria reflejada en los estadoscuanticos y a los tiempos de coherencia relativamente largos.Dicho estado especial, denominado seudopuro (pseudopurestate) [15,16] y que fue propuesto dentro del modelo de la

computacion cuantica RMN de soluciones lıquidas a tempe-ratura ambiente para lidiar con la limitacion impuesta por laimposibilidad de describir a la totalidad de los espines RMNpor un estado puro, se puede expresar de la siguiente manera:

ρε =1− ε

2NI + ερ1, (1)

dondeρε es una matriz densidad que representa el estadoRMN de una molecula,ρ1 es una matriz densidad corres-pondiente al elemento observable en un experimento RMN yε ≈ µH/2NkBT , siendoT la temperatura,kB la constantede Boltzmann yN el numero debitscuanticos, es un parame-tro que mide el grado de polarizacion del sistema de espines(en el campo magneticoH).

Por otro lado, en la computacion cuantica hay la necesi-dad de identificar el caracter separable o entrelazado de losestados cuanticos; es por ello que fueron establecidos algu-nos criterios, como el de Peres-Horodecki [17,18], que esta-blece, en el caso particular de matrices densidad4 × 4, unacondicion necesaria y suficiente de separabilidad. Ademas, alconsiderar la naturaleza de los asuntos que aquı son aborda-dos, conviene tener presente las siguientes informaciones:

(i) una matriz no solo por ser del tipo matriz densidad ten-dra asegurada una representacion de (correspondenciacon) un estado fısico accesible a un sistema cuantico;

(ii) no todo estado que se pueda escribir con la forma en-trelazada correspondera, necesariamente, a un entrela-zamiento verdadero, es decir, fısico [19,20]; y

(iii) el concepto de entrelazamiento matematico, que hasido introducido para diferenciarlo del verdadero en-trelazamiento y que fue discutido primero por Val-qui [20], sera usado para designar matrices que, tenien-do el aspecto de entrelazadas, no puedan ser colocadas

ENTRELAZAMIENTO CUANTICO ESPURIO CON MATRICES SEUDOPURAS EXTENDIDAS 4 POR 4 189

en correspondencia con cualquier estado cuantico ac-cesible al sistema considerado. Finalmente, convieneindicar que en las Refs. 21 y 22 se presentan situacio-nes concretas de entrelazamiento matematico en el ca-so de estados puros.

A continuacion presentamos la estructura de este artıcu-lo: en la Sec. 2 definimos matematicamente la extension paralos estados seudopuros4 × 4, la que incluye una subseccionsobre la estrategia usada para generar las matrices extendidasentrelazadas. En la Sec. 3 presentamos los ejemplos numeri-cos concretos de matrices con entrelazamiento. En la Sec. 4presentamos una caracterizacion parcial del conjunto de es-tas matrices entrelazadas y, finalmente, presentamos nuestrasconclusiones.

2. Matriz seudopura extendida

Volvamos a la Ec. (1) con la intencion de identificar la con-dicion (apenas matematica) mas general que garantice queρε sea una matriz densidad. Se puede percibir que tal condi-cion existe, tratandose, en verdad, de 2 condiciones, siendoque una de ellas establece queρ1 no necesita ser una matrizdensidad.

Denominamos matriz seudopura extendida (ρEε ) a una

matriz densidad que tiene la forma (1), pero que, a diferenciadeesta, se obtiene a partir de matricesρ1 extendidas (ρE

1 ) queno son matrices densidad: ellas son hermiteanas, tienen traza1 y al menos un autovalor negativo. Entonces escribimos,

ρEε = (1− ε)

I4

4+ ερE

1 , (2)

dondeI4 esla matriz identidad4 × 4. ComoρE1 no es una

matriz densidad, la matrizρEε , de acuerdo con (2), tambien

podrıa tener algun autovalor negativo; por ello, para dar con-sistencia a nuestra extension, debemos imponer una segundaexigencia o condicion, asegurando ası queρE

ε sea una matrizdensidad. De la Ec. (2) es evidente queρE

ε y ρE1 conmutan,

entonces, si representamos porλε y λ1 a los respectivos auto-valores, definidos para el conjunto de autovectores en comun,la siguiente relacion es valida:

λε = (1− ε)/4 + ελ1. (3)

Imponiendo queλε ≥ 0, obtenemos la siguiente relacion:

λ1 ≥ −(1− ε)/4ε. (4)

Esta condicion sobre los autovalores deρE1 garantiza que las

ρEε sean matrices densidad. La extension matematica ası de-

finida nos conducira a los primeros ejemplos numericos de

entrelazamientoespurio para mezclas estadısticas4 × 4 y alhacerlo estaremos confrontados con el problema acerca delas condiciones bajo las cuales un criterio matematico (el dePeres-Horodecki) llega a tener validez fısica para el sistemacuantico considerado.

2.1. Metodo numerico

El esquema que hemos implementado para buscar matricesρE

ε que sean entrelazadas consiste en generar (repetidamen-te y dentro de ciertos intervalos) un conjunto de 15 numerosaleatorios que consideramos como los coeficientes indepen-dientes de la expansion de una matrizρE

1 individual en la basede productos de matrices de Pauli. Cuando todos los autova-lores de la matrizρE

1 ası construıda satisfagan la Ec. (4) en-tonces, a traves de la Ec. (2) y para un valor del parametroε previamente elegido, construimos la matrizρE

ε y la matriztranspuesta parcialmente,PρE

ε, ası como sus autovalores. Si

verificamos que (al menos) un autovalor dePρEε

es negativo,entonces, de acuerdo con el criterio de Peres-Horodecki, lacorrespondiente matrizρE

ε sera entrelazada. A continuaciondesarrollamos explıcitamente este metodo para los siguientesvalores del parametroε: 10−2 y 3.3× 10−5.

3. Ejemplos de matrices4 × 4 con entrelaza-miento espurio

En la base de productos de matrices de Pauli, construida apartir deσ1, σ2, σ3, σ4, conσ1 = I2, σ2 ≡ σx, σ3 ≡ σy,σ4 ≡ σz, una matriz densidadρ, de tamano 4 × 4, se puedeexpresar de la siguiente manera:

ρ =14

4∑

i,j=1

Ci,j σi ⊗ σj , (5)

dondeC1,1 = 1 y −1 ≤ Ci,j ≤ 1. Pero si necesitamos re-presentar matrices que sonunicamente hermiteanas y tienentraza igual a 1, entonces el intervalo de posibles valores paralos 15 coeficientes independientesCi,j es mucho mas amplio.En particular, para el conjunto de coeficientesCi,j ordenadosen la matrizC,

C=

+1.0000 −0.6429 −29.4845 −7.2208+4.1172 −8.5393 +5.5682 −23.6294−22.7868 −18.9588 +4.6408 −6.2254−28.7033 −15.2617 −19.8301 −16.7560

, (6)

resulta, usando la Ec. (5), la siguiente matriz:

ρE1 =

−12.9200 −3.9762 + 12.3287i −4.8780 + 7.2530i −3.2950 + 3.3476i−3.9762− 12.3287i −0.9316 −0.9746 + 6.1318i +6.9367 + 4.1403i−4.8780− 7.2530i −0.9746− 6.1318i +9.8096 +3.6547 + 2.4136i−3.2950− 3.3476i +6.9367− 4.1403i +3.6547− 2.4136i +5.0420

(7)

Rev. Mex. Fıs. 57 (3) (2011) 188–192

190 J.D.BULNES Y L.A. PECHE

verificando que es hermiteana, tiene Tr(ρE1 ) = 1 y dos autovalores negativos,

λ1 = −23.1909, λ2 = −4.4928, λ3 = +7.3209, λ4 = +21.3628,

siendo, por lo tanto, una matriz no fısica. Al considerar la matrizρE1 anterior y el valorε = 10−2 en la Ec. (2) obtenemos la

siguiente matriz:ρEε ,

ρEε =

+0.1183 −0.0398 + 0.1233i −0.0488 + 0.0725i −0.0330 + 0.0335i−0.0398− 0.1233i +0.2382 −0.0097 + 0.0613i +0.0694 + 0.0414i−0.0488− 0.0725i −0.0097− 0.0613i +0.3456 +0.0365 + 0.0241i−0.0330− 0.0335i +0.0694− 0.0414i +0.0365− 0.0241i +0.2979

, (8)

verificando que es hermiteana, tiene Tr(ρEε ) = 1 y autovalo-

res no negativos:

λ1 = +0.0156, λ2 = +0.2026,

λ3 = +0.3207, λ4 = +0.4611;

por lo tanto,ρEε es una matriz densidad. Ahora aplicaremos

de manera formal el criterio de Peres-Horodecki a la matrizρE

ε ; para ello, primeramente, debemos definir una base dematrices densidad:ρ1, ρ2, ρ3, ρ4. Entonces consideramoslas siguientes matrices:

ρ1 =12

(1 00 1

), ρ2 =

12

(1 −ii 1

),

ρ3 =12

(1 11 1

), ρ4 =

(1 00 0

). (9)

Al reescribir la matrizρEε , en (8), en la base de matrices den-

sidad, dada en (9), tenemos

ρEε =

14

4∑

i,j=1

Di,j ρi ⊗ ρj ,

dondelos valores de los coeficientesDi,j han sido ordenadosen la siguiente matriz:

D=

+2.3456 +0.0576 +0.6592 +2.6488+0.4288 +0.1336 −0.4280 −1.0416+1.1232 −0.2536 −0.2216 −0.7248+1.6032 −0.9848 −0.8000 −0.5456

, (10)

obtenida por una transformacion de cambio de base a partirde los coeficientes deρE

ε en la base de productos de matri-ces de PauliBi,j = tr(ρE

ε σi ⊗ σj). La matriz transpuestaparcial deρE

ε , representada aquı porP (ρEε ) y definida (indis-

tintamente) por la transposicion de cualquiera de sus partes,como a continuacion,

P (ρEε ) =

14

4∑

i,j=1

Di,j ρi ⊗ (ρj)T

nosconduce a la siguiente matriz:

P (ρEε ) =

+0.1183 −0.0398− 0.1233i −0.0488 + 0.0725i −0.0097 + 0.0613i−0.0398 + 0.1233i +0.2382 −0.0330 + 0.0335i +0.0694 + 0.0414i−0.0488− 0.0725i −0.0330− 0.0335i +0.3456 +0.0365− 0.0241i−0.0097− 0.0613i +0.0694− 0.0414i +0.0365 + 0.0241i +0.2979

, (11)

la cual tiene los siguientes autovalores:

λ1 = −0.0080, λ2 = +0.2379, λ3 = +0.3425, λ4 = +0.4275,

conλ1 < 0. Por lo tanto, de cuerdo con el criterio de Peres-Horodechi, la matrizρEε es entrelazada.

Un segundo ejemplo de esta situacion pero usando un valor del parametro igual aε = 3.3× 10−5, que es del mismo ordende magnitud como en las implementaciones experimentales de CC-RMN de soluciones lıquidas a temperatura ambiente, y lasiguiente matriz de coeficientes deρE

1 en la base de productos tensoriales de matrices de Pauli:

C = 103 ×

+0.0010 +9.1000 +8.9909 +8.8544+9.7067 +8.9003 +9.9779 +9.5595+9.9593 +9.6036 +9.7759 +9.3788+9.8103 +9.6321 +9.2731 +9.9868

. (12)

Entonces, usando la Ec. (2), obtenemos la siguiente matriz:

ρEε =

+0.4864 +0.1545− 0.1507i +0.1589− 0.1595i −0.0072− 0.1615i+0.1545 + 0.1507i +0.1755 +0.1541 + 0.0031i +0.0012− 0.0048i+0.1589 + 0.1595i +0.1541− 0.0031i 0.1597 −0.0044 + 0.0023i−0.0072 + 0.1615i +0.0012 + 0.0048i −0.0044− 0.0023i +0.1784

, (13)

Rev. Mex. Fıs. 57 (3) (2011) 188–192

ENTRELAZAMIENTO CUANTICO ESPURIO CON MATRICES SEUDOPURAS EXTENDIDAS 4 POR 4 191

la cual es hermiteana, tiene Tr(ρEε ) = 1 y autovalores

λ1 = +0.0008, λ2 = +0.0256, λ3 = +0.2179, λ4 = +0.7557,

y la siguiente matriz transpuesta parcialmente:

P (ρEε ) =

+0.4864 +0.1545 + 0.1507i +0.1589− 0.1595i +0.1541 + 0.0031i+0.1545− 0.1507i +0.1755 −0.0072− 0.1615i +0.0012− 0.0048i+0.1589 + 0.1595i −0.0072 + 0.1615i +0.1597 −0.0044− 0.0023i+0.1541− 0.0031i +0.0012 + 0.0048i −0.0044 + 0.0023i +0.1784

(14)

con autovaloresλ1 = −0.0024, λ2 = +0.0288, λ3 = +0.2181, λ4 = +0.7555,

siendo un autovalor negativo, como en el primer ejemplo. Por lo tanto, ambas matricesρEε , dadas en (8) y (13), corresponden

a situaciones de entrelazamiento de acuerdo con el criterio de Peres-Horodechi.

4. Caracterizando el conjunto de matricesρEε

entrelazadas

Los ejemplos anteriores de matricesρEε entrelazadas, gene-

radas a partir de matricesρE1 , no constituyen casos aislados.

En esta seccion vamos a caracterizar parcialmente el conjun-to de las matrices de ese tipo. La Fig. 1 muestra un mapadonde se presenta la fraccion de matricesρE

ε entrelazadas enfuncion de los intervalos de definicion de las matricesρE

1 ,las mismas que, a traves de la Ec. (2), producen las matricesentrelazadas. La Fig. 2 muestra un histograma del numerode matrices entrelazadas que tienen una distancia de Hilbert-Schmidt dada al denominado “estado gato”, el estado con elmaximo grado de entrelazamiento, donde las matrices con-sideradas fueron definidas a partir de matricesρE

1 dentro deun intervalo< Cmin, Cmax > fijo (ver sub-Sec. 2.1); estafigura nos da una idea del grado de entrelazamiento asociadocon las matrices extendidas entrelazadas.

FIGURA 1. Fraccion de matricesρEε entrelazadas (en escala de co-

lores) encontradas en 1000 matricesρE1 generadas aleatoriamente

(como es descrito en la sub-seccion 2.1) dentro de cada interva-lo < Cmin, Cmax >; fueron considerados 130 valores tanto paraCmin como paraCmax, siendo el valor del parametro usado iguala ε = 5× 10−3.

FIGURA 2. Histograma del numero medio de matrices seudopu-ras extendidas y entrelazadas que tienen una distancia de Hilbert-Schmidt dada al “estado gato”, considerandoε = 5× 10−4. Se ge-neraron (aleatoriamente) 500,000 matrices en cada ciclo de calcu-los y se realizo un promedio sobre 10 ciclos.

5. Conclusiones

En los ejemplos de matricesρEε dados en (8) y (13) las corres-

pondientes matricesρE1 , u otras de esa misma naturaleza, no

pueden ser implementadas experimentalmente a traves de laaplicacion de pulsos de radiofrecuencia (representados pormatrices unitarias), al contrario de lo que corresponde paramatrices densidadρ1 de estados seudopuros. En consecuen-cia, y a pesar de que las matricesρE

ε anteriormente mencio-nadas son entrelazadas de acuerdo con el criterio de Peres-Horodecki, no es posible establecer ninguna correspondenciafısica entre cualquiera de las matricesρE

ε y un estado fısi-co accesible a un sistema de espines nucleares RMN; de es-ta manera, el entrelazamiento de las matricesρE

ε mostradasen (8) y (13) no es fısico, sino espurio o matematico. Parale-lamente, los ejemplos construidos han sidoutiles para mos-trar que la sola aplicacion del criterio matematico de Peres-Horodecki, a pesar de usado con matrices 4×4, no es sufi-ciente para identificar el entrelazamiento fısico; aquel debe

Rev. Mex. Fıs. 57 (3) (2011) 188–192

192 J.D.BULNES Y L.A. PECHE

ser complementado con la evaluacion teorica sobre la posibi-lidad de establecer una correspondencia entre matrices den-sidad y estados cuanticos accesibles al sistema considerado.Por otro lado, aparte de los procedimientos de implementa-cion de un estado cuantico existen los de medida de estadocuantico. En la CC-RMN es clara la razon por la cual lasmatricesρε no pueden ser medidas experimentalmente; encambio, las matricesρ1 sı son medibles, lo que se consiguea traves del procedimiento experimental conocido como “to-mografıa de estado cuantico”. En ese procedimiento se asu-me que la matriz a ser tomografiada es una matriz densidad;es decir, no se verifica en laboratorio que las matrices imple-mentables son matrices densidad, sino que se parte de ellopara construir experimentalmente tal matriz. De manera que,para conseguir medir alguna matriz que no sea del tipo matrizdensidad, sino solamente hermiteana y con traza igual a uno,como se ha considerado para las matrices extendidasρE

1 , de-berıan tomarse esas consideraciones como exigencias en elprocedimiento de tomografıa de estado cuantico. Aun mas,a partir de la matriz tomografiada (que resulta de conside-rar no el procedimiento formal de tomografıa, sino aquel queincorpora la modificacion mencionada anteriormente) podrıaobtenerse, a partir de otras medidas, el valor de la denomina-

da “fidelidad de entrelazamiento”; en ese caso, no podrıamosestar seguros de que a traves de los valores de tal funcion seestarıa midiendo alguna cantidad asociada con el entrelaza-miento fısico en el sistema considerado. Finalmente, el con-texto en el que hemos definido la extension matematica, losejemplos mostrados y la discusion ofrecida nos han servidopara aclarar el significado de la siguiente pregunta: siρ es unamatriz densidad4× 4, cuya matriz transpuesta parcialmentetiene (al menos) un autovalor negativo, entonces ¿la matrizρcorrespondera (necesariamente) a una situacion de entrelaza-miento? Tal pregunta tiene respuesta negativa en el caso delentrelazamiento fısico.

Agradecimientos

JDB agradece al prof. Holger Valqui (UNI, Lima) por lasaclaraciones sobre distintos aspectos del entrelazamientocuantico, ası como a los profesores Ivan S. Oliveira (CBPF,Rio de Janeiro) y Henrique Saitovitch (CBPF, Rio de Janeiro)por los comentarios; tambien agradece al Programa Nacio-nal de Pos-Doutorado - PNPD, de la Fundacao Coordenacaode Aperfeicoamento de Pessoal de Nıvel Superior - CAPES,Brasil, por el apoyo financiero.

†. Luis Alberto Peche fallecio la manana del dıa 6 de diciembredel 2010 en la ciudad de Rio de Janeiro a los 44 anos de edad.Fue un fısico que trabajo en asuntos de caos cuantico, sistemasfuertemente correlacionados y en el modelamiento fısico condatos de radar y sısmicos. Paso por la UNI, en Lima, ası comopor el CBPF, PUC, ON y UERJ, en Rio de Janeiro.

1. M.A. Nielsen y I.L. Chaung,Quantum Computation and Quan-tum Information (Cambridge University Press, Cambridge,2002).

2. I.S. Oliveiraet al., NMR Quantum Information Processing(El-sevier, 2007).

3. A. Peres,Quantum Theory: Concepts and methods(Kluwer,1993).

4. A. Aspect,Gazeta de Fısica22(1999) 16; H.G. Valqui,Revciu-ni 7 (2003) 115. http://fc.uni.edu.pe/publicaciones/rev07-03/pdf/hgvalqui.pdf

5. M.A. Nielsen, E. Knill y R. Laflamme,Nature396(1998) 52.

6. L.M.K. Vandersypenet al , Nature414(2001) 883.

7. M. Mehring, J. Mende y W. Scherer,Phys. Rev. Lett.90 (2003)153001.

8. S. Braunstein, C. Caves, Jozsa, Linden, S. Popescu y Schack,Phys. Rev. Lett.83 (1999) 1054.

9. N. Menicucci y C. Caves,Phys. Rev. Lett.88 (2002) 167901.

10. G. Longet al., Commum. Theor. Phys.38 (2000) 306.

11. A. Kessel y V. Ermakov,quantu-ph, ArXiv Los Alamos(2000)0011002.

12. J.D. Bulneset al., Braz. J. Phys.35 (2005) 617.

13. F.A. Bonket al., J. Mag. Res.175(2005) 226.

14. F.A. Bonket al., Phys. Rev. A69 (2004) 042322.

15. N.A. Gershenfeld y I.L. Chuang,Science275(1997) 350.

16. D.G. Cory, A.F. Fahmy y T.F. Havel,Proc. Natl. Acad. Sci. USA94 (1997) 1634.

17. A. Peres,Phys. Rev. Lett. 77 (1996) 1413.

18. M. Horodechi, P. Horodechi, R. Horodechi,Phys. Lett. A223(1996) 1.

19. A. Wojcik, Science301(2003) 1183.

20. H.G. Valqui,Revciuni11 (2007) 67. Disponible en:http://www.bibliotecacentral.uni.edu.pe/pdfs/REVCIUNI/1,2007/art- 010.pdf

21. H.G. Valqui, Proceedings of the Fifth Latin American Sympo-sium High Energy Physics, Lima, Peru, 12-17 July 2004(WorldScientific Publishing, 2006). p. 272.

22. A.M. Basharov y E.A. Manykin,Optics and Spectroscopy96(2004) 81.

Rev. Mex. Fıs. 57 (3) (2011) 188–192


Electrostatic models of charged hydrogenic chains in a strong magnetic field

A. EscobarInstituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico,

Circuito Exterior s/n Ciudad Universitaria, Mexico, D.F., 04510, Mexico.

Recibido el 20 de septiembre de 2010; aceptado el 10 de febrero de 2011

Simple one-dimensional electrostatic models of one-(two) electron molecular systems H+2 , H2+

3 , H3+4 and H2, H+

3 , H2+4 in a strong magnetic

field are proposed to estimate the binding-(ionization) energy of the corresponding ground states. The study is carried out in the range ofmagnetic fiedsB = 102 − 106 a.u. The models are inspired (and based) on the quasi one-dimensional form of the ground state electronicdistribution function which is obtained by precise variational calculations in the Born-Oppenheimer approximation in a non-relativisticframework. It is shown that the models give, for all magnetic fields considered, a very good description of the binding-(ionization) energy ofthe one-electron molecular systems H+

2 , H2+3 , H3+

4 , being accurate in2.5%, 5% and5% respectively, and15% for the two-electron systemsH+

3 , H2+4 (30% for H2) as compared with the corresponding variational calculations.

Keywords: Hydrogenic chains; strong magnetic field; electrostatic models.

Modelos electrostaticos unidimensionales simples de los sistemas moleculares de 1-(2) electronesH+2 , H2+

3 , H3+4 y H2, H+

3 , H2+4 en campos

magneticos intensos son propuestos para estimar la energıa de amarre-(ionizacion) del correspondiente estado base. El estudio se lleva acabo en el rango de campos magneticosB = 102 − 106 a.u. Los modelos estan inspirados (y basados) en la forma cuasi-unidimensionalde la funcion de distribucion electronica, del estado base, que se obtiene mediante calculos variacionales muy precisos realizados en laaproximacion de Born-Oppenheimer en un tratamiento no relativista. Se muestra que los modelos brindan, para los campos magneticosconsiderados, una muy buena aproximacion a la energıa de amarre-(ionizacion) de los sistemas moleculares de un electronH+

2 , H2+3 y H3+

4 ,con una precision relativa del2.5%, 5% y 5% respectivamente, y con una precision relativa del15% para los sistemas moleculares de 2electronesH+

3 , H2+4 (30% paraH2) comparadas con los calculos variacionales correspondientes.

Descriptores: Cadenas hidrogenoides; campos magneticos intensos; modelos electrostaticos.

PACS: 31.15.Pf; 31.10.+z; 97.10.Ld

1. Introduction

Strong magnetic fields are present in the surfaces ofneutron stars, where typically magnetic field varies inB≈1012−1013 G, and can reach extraordinary values ofB ≈ 1014 − 1016 G in the surface of the recently discoveredmagnetarsi. Since the discovery of the existence of strongmagnetic fields several questions arose about the stability andstructure of atoms and molecules exposed to such extrememagnetic fields [1,2].

First investigations about the structure of matter in strongmagnetic fields [1,2] gave qualitative indications that dueto the large quadrupole moment of the elongated electronicdensity cloud, new molecular systems, in the form of linearchains, could exist in an aligned configuration parallel to themagnetic field direction.

An accurate description of atomic and molecular systemsin strong magnetic fields (even the most simple ones) has re-quired the development of non-perturbative techniques whichcan give reliable results. An important step in the searchfor new exotic molecules in magnetic fields was achievedin 1999 with the theoretical discovery of the molecular ionH2+

3 [3] which can exist as a bound state for magnetic fieldsB & 1011 G. Later, it was shown that starting at differentthresholds in the domain of magnetic fieldsB ∈ [0, 1016] Gnew chains of one and two electron systems composed of hy-drogen and helium can exist in linear geometry (for a list ofspecific compounds and a review see [4] and also [6]).

In particular, the fact that the electronic cloud in astrong magnetic field acquires a cigarette-like form has sug-gested to use quasi-one dimensional approaches to solvethe Schroedinger equation [7,8]. It is known that the one-dimensional Coulomb problem [9-11] describes a quantumsystem with many uses in atomic, molecular and condensedmatter physics despite its apparent simplicity. For example,in the theory of a Mott exciton in a strong magnetic field [12]or in describing the problem of electrons over a pool of liq-uid helium. In this last case, given the charge and its imageis hence clear that the electron is acted by a Coulomb in-teraction [13]. Additionally, an essentially two-dimentionalset of electrons trapped in the levels of the one-dimentionalCoulomb problem has been suggested as a possible realiza-tion of a quantum computing device [14].

Recently, some heuristic one-dimensional electrostaticmodels for one-electron molecular systems in a strong mag-netic field have been introduced in Ref. 4. Altough verysimple, their accuracy is enough to gain a certain qualitativeinsight about the structure of molecules in magnetic fields.For example, a model for a one-electron diatomic molecu-lar ion (Z,Z, e) like H+

2 (Z=1) in a strong magnetic fieldis described in Ref. 4; guided by the evolution of the elec-tronic distribution of H+2 as the magnetic field is increased,i.e. evolving from a two peak configuration, for small mag-netic fields, to a one (centered) peak configurationii, at strongmagnetic fields, this model assumes that at equilibrium theelectronic cloud can be mimicked by a point-like charge sit-

194 A. ESCOBAR

uated exactly in the middle between the charged centers. Allthree charges (two heavy centers of chargeZ and one elec-tron) are confined by the magnetic field inside a narrow cylin-drical channel whose radius is limited to a domain defined bythe Larmor radius. The electrostatic Coulomb energyEcoul

(in a.u.) of such linear configuration of point charges is eas-ily calculated, being

Ecoul = −Z(4− Z)Req

, (1)

whereReq is the equilibrium distance between protons.Ecoul

is negative forZ < 4 predicting that the system can be boundeven forZ = 3, the case of Li5+2 which actually is predictedto exist for magnetic fieldsB & 104 a.u. [15].

We should mention that the model presented, as all elec-trostatic models, assumes that point charges are fixed for eachvalue of the magnetic field, then a configuration is calledequilibrium configuration due to the fact that accurate vari-ational results for the equilibrium distance(s)Req and thecorresponding binding energies were used to develop themodeliii.

Of course, the validity of this model relies on the phe-nomenological assumption that the binding energy can bewell approximated by the Coulomb electrostatic energy ofsuch linear configuration of point chargesiv. There mightexist a domain of magnetic field strength where this picturemakes sense. This is, indeed, the case for very strong mag-netic fields. For example, forZ = 1 (hydrogen molecularion) the binding energy obtained from the above relation (1)is slightly overestimated being larger in 10% for a magneticfield B = 104 a.u. while it is larger in 5% forB = 106 a.u.(see [4]). It indicates that the accuracy of the model increasesas a magnetic field grows. In any case, the approximate re-sults given by this simple model are very surprising. Modelsof pointlike charges were proposed in Ref. 4 also for one-electron systems with 3 and 4 charged centers. Similar mod-els have been also used for two-electron systems in a mag-netic field [5].

Our goal in the present paper, is to develop further theabove mentioned simple electrostatic models of the groundstate of the one-electron hydrogenic chains H+

2 , H2+3 , H3+

4

and the two-electron molecular systems H2, H+3 , H2+

4 in astrong magnetic field. The development consists of incorpo-rating into the models the information contained in the elec-tronic distribution which is obtained by precise variationalcalculations. Atomic unitsme = −e = = a0 = 1 are usedthroughout although the energy is given in Rydbergs.

2. One-electron molecular systems in a strongmagnetic field: electrostatic model for theground state

For the non-relativistic description of the groud state1σg ofthe one-electron molecular systems in a strong magnetic fieldB (parallel configurationv), we can develop an electrostatic

model as follows: In the first instance, we consider that theelectronic charge distribution can be modeled by a linear den-sity of chargeλ(z) situated along the magnetic field direction(z-direction), whereλ(z) is defined through thez-profile ofthe ground state electronic distributionΨvar obtained by vari-ational calculations [4] being

λ(z) = e

∫ |Ψvar(ρ, φ, z)|2ρ dρdφ∫ |Ψvar(ρ, φ, z)|2 ρ dρdφdz, (2)

where(ρ, φ, z) are the cylindrical coordinates of the positionvector of the electron, and (e= −1 in atomic units) is theelectron charge. In a second step, the linear charge distribu-tion (2) is approximated by a linear superposition of a finitenumber of standard Gaussian functions situated symmetri-cally with respect to the center of the molecular axis. Af-terwards, each Gaussian curve is replaced by a point charge,located at the center of the Gaussian and whose value is equalto the corresponding linear coefficient amplitude (see Figs. 1to 3). The number of Gaussian functions used in the modeldepends on the shape of the profileλ(z). Roughly, for eachlocal maximum of the profile a Gaussian function is intro-duced. However, this number is rather arbitrary. For exam-ple, for the system H+2 whose profile in the range of magneticfield considered presents a single maximum, three Gaussianfunctions were used (although, considering only one Gaus-sian function placed in the mid point between the line con-necting the protons, does not alter the results significantly inthis case). Let us take, for example, such configuration ofthree point charges used to construct a model for the molec-ular ion H+

2 (see Fig. 1)

λ(z) =q1√2πσ2

1

e− z2

2σ21

+q2√2πσ2

2

(e− (z−z2)2

2σ22 + e

− (z+z2)2

2σ22

), (3)

whereqi, σi, zi (i = 1, 2) are interpolation parameters. Suchparameters are not all independent. In addition to the normal-ization condition, which allows us to writeq2 in terms ofq1,we can impose the condition

Ecoul = Eb , (4)

Eb being the variational binding energy andEcoul the elec-trostatic Coulomb energy. The Coulomb energyEcoul (in Ry-dbergs) is obtained after the replacement of these Gaussiansby point charges (Fig. 1) and is given byvi

Ecoul(B)=2

(1

Req+

4q1

Req+

2q2

|Req

2 −z2|+

2q2

|Req

2 +z2|

), (5)

where the values of the equilibrium interproton distanceReq(B) as a function of the magnetic field strengthB aretaken from the variational calculations [4], which then allowsus to express one parameter amongqi, σi, zi (i = 1, 2) interms of the others (for exampleq1=q1(Eb, Req; z1, q2, z2) ).

Rev. Mex. Fıs. 57 (3) (2011) 193–203

ELECTROSTATIC MODELS OF CHARGED HYDROGENIC CHAINS IN A STRONG MAGNETIC FIELD 195

FIGURE 1. Electrostatic model for the ground state1σg of the one-electron molecular ionH+

2 in parallel configuration with a uniformand constant strong magnetic fieldB = (0, 0, B). The electroniccloud is replaced by three point like charges with fractional chargeq2, q1, q2 at the positions−z2, 0, z2 respectively.

The condition (4) is not as strong as it migth seem. Mak-ing use only of the normalization condition the relative dif-ference betweenEcoul andEb is less than 15% for all oneelectron systems and for all magnetic fields considered. Onthe other hand, imposing only (4) the resulting linear densityof chargeλ(z) breaks the normalization condition, but in lessthan2% for all systems and for all magnetic fields consid-ered.

Condition (4) can be or not satisfied, but normalizationcondition must hold anyway. Then, according to this physicalrequirement we impose (4) and afterwards the normalizationonλ(z) is restored multiplying by an overall factor onλ(z).

Suitable interpolation functions of the parametersReq, qi,zi (i = 1, 2) as functions of the magnetic fieldB were found,which allows us to write the Coulomb energyEcoul=Ecoul(B)as an explicit analytical approximation as function of themagnetic fieldB.

2.1. Calculations

In this section we show, for each one-electron system(H+

2 , H2+3 , H3+

4 ), the interpolating functions of the param-etersReq, qi andzi (i = 1, 2, 3) as a function of the magneticfield strengthB. Hereafter, magnetic field is defined in di-mensionless units (a.u.) asB/B0, whereB0 = 2.35×109 G,which we continue to denote asB.

2.2. Ion H+2

The study is carried out in the range of magnetic fields4255 ≤ B ≤ 105 a.u. For this system a superposition ofthree Gaussian functions as in (3) was used to model the lin-ear charge densityλ(z) (see 2). This charge density is de-fined by thez-profile of the corresponding ground state elec-tronic distributionΨvar taken from [4]. The parameters of themodel (3) which approximateλ(z) for each magnetic fieldstudied are summarized in the Table I.

The equilibrium interproton distanceReq(B) as a func-tion of the magnetic field strengthB is taken from the varia-tional calculations [4] and fitted by the formula given by

Req(B)=1.7288

1+1.1365 log1+(0.0232B)2+(0.0018B)4 , (6)

TABLE I. H+2 state1σg. Parameters which approximate the elec-

tronic charge densityλ(z) given as the sum of three (normalized)Gaussian functions. The electrostatic model is obtained replacingeach Gaussian curve by a point charge, located at the center of theGaussian and whose value is equal to the corresponding linear co-efficient amplitude (see Fig. 1).

B (a.u.) q1 (a.u.) q2 (a.u.) z2 (a.u.)

4255 -0.89403 -0.05298 0.28201

10000 -0.92383 -0.03808 0.26209

18782 -0.95033 -0.02483 0.24239

42553 -0.95773 -0.02113 0.22279

100000 -0.95988 -0.02005 0.20276

Thefunctional form of the dependence ofReq(B) on themagnetic field, as well as for the interpolation forz2(B)(see below), is taken from the physics-inspired approxima-tions based on the assumption that the dynamics of the one-electron Coulomb system in a strong magnetic field is gov-erned by the ratio of transverse to longitudinal size of theelectronic cloud (for details see [16]). The protons of thesystemH+

2 are thus located atz = ±(Req/2) with respectto the origin situated in the mid point on the line connectingprotons.

As a function of the magnetic fieldB, the interpolatingformula for the point chargeq1 located in the center (z= 0)is given by

q1(B) = −0.0059 + 22.0211√

B + B

9.7408 + 31.9754√

B + B. (7)

This formula describes the approximated dependence ofq1(B) on the magnetic fieldB. Notice that forB → ∞,q1(B) → -1 implying that as the magnetic fields increases,the electrostatic model is reduced to that with a one singlepoint charge situated in the middle between the two protonsof the moleculeH+

2 as it was proposed in Ref 4. The chargeconservation conditionq1 +2q2 = −1 gives the interpolationfor the two symmetric point chargesq2(B) (see Fig. 1). Fi-nally, the interpolation for the position of these two chargesz2 is given by

z2=2.1117

1+0.2523 log1+(1.0075 B)2+(0.1090 B)4 , (8)

where z2, as Req, is given in atomic units of distance(a0=1 a.u.).

Notice that the lateral chargesq2 are situated outside ofthe equilibrium distance for all values of the magnetic fieldconsidered and that forB → ∞ the lateral charges decreasetheir relative importance.

Table II shows, as a function of the magnetic fieldB, thevalues of the Coulomb energyEcoul of the electrostatic modelof point charges and the corresponding variational binding

Rev. Mex. Fıs. 57 (3) (2011) 193–203

196 A. ESCOBAR

TABLE II. H+2 state1σg. Comparison ofEcoul vs Eb. The relative

difference∆E = (|Ecoul−|Eb||/|Eb|) is less than2.5% for all therange of magnetic fields considered.

B(a.u.) Ecoul(Ry) Eb(Ry) ∆E(%)

4255 -36.5454 -35.7538 2.2

10000 -46.8141 -45.7970 2.2

18782 -55.8387 -54.5016 2.4

42553 -68.7737 -67.5826 1.7

100000 -82.7359 -83.5814 1.0

TABLE III. H2+3 state1σg. Parameters which approximate the

electronic charge densityλ(z) given as the sum of four (normal-ized) Gaussian functions situated symmetrically with respect to thecenter of the molecular axis.

B (a.u.) q2 (a.u.) z2 (a.u.) q3 (a.u.) z3 (a.u.)

425 -0.04188 0.04429 -0.45811 0.19696

1000 -0.03950 0.03061 -0.46049 0.15537

4255 -0.02081 0.00931 -0.47918 0.09948

10000 -0.01390 0.00443 -0.48609 0.07793

18782 -0.01042 0.00258 -0.48957 0.06562

energy Eb, where it is shown that the relative differencevaries∆E varies from 1 to 2.5 in all the range of magneticfields studied.

2.3. IonH2+3

The study is carried out in the range of magnetic fields425 ≤ B ≤ 18782 a.u. For this system a superposition offour Gaussian functions were used to fit thez-profile of theelectronic distribution,i.e. λ(z) (see Fig.2). The parametersof the model which approximateλ(z) for each magnetic fieldstudied are summarized in the Table III.

The Coulomb energyEcoul (in Rydbergs) of such config-uration of point charges is given by

Ecoul(B) = 2

(5

Req+

2q2

|Req

2 − z2|+

2q2

|Req

2 + z2|

+2q3

|Req

2 − z3|+

2q3

|Req

2 + z3|+

2q2

z2+

2q3

z3

), (9)

wherethe equilibrium distanceReq(B) as a function of themagnetic field strengthB is taken from the variational calcu-lations [4] and fitted by the formula

Req(B)

=0.8761

1+0.3308 log1+(0.0026B)2+(0.00047B)4 , (10)

The protons of the systemH2+3 are thus located at

z=0,±(Req/2) with the origin situated in the mid point onthe line connecting protons.

FIGURE 2. H2+3 state1σg: Electrostatic model for the ground

state1σg of the one-electron molecular ionH2+3 in parallel con-

figuration with a uniform and constant strong magnetic fieldB=(0, 0, B). The electronic cloud is replaced by four point likecharges with fractional chargeq3, q2, q2, q3 located at the positions−z3,−z2, z2, z3 respectively.

As a function of the magnetic fieldB, the interpolatingformula for the point chargeq3 located atz = ±z3 is given by

q3(B) = − 1.1634 + 0.00044 B

2.57928 + 0.00089 B. (11)

This formula describes the approximated dependenceof q3(B) on the magnetic fieldB. Notice that forB→∞, q3(B) → −0.494 implying that as the magneticfields increases, the electrostatic model is reduced to (al-most) that with a two negative point charges situated at±z3, z3<(Req/2)∀B. The charge conservation condition2q1 + 2q2 = −1 gives the interpolation for the two symmet-ric point chargesq2(B) (see Fig. 2). The interpolation for thepositionzi (i = 2, 3) of the two point chargeqi is given by

z2(B) =7.1659

1 + 3.0835 log1 + (0.00024 B)2

×(

1 + 0.00405 B

1 + B

), (12)

z3(B)=1.6333

1+1.3342 log1+(0.0345B)2+(0.004B)4 . (13)

Notice thatz2 < z3 for all values of the magnetic fieldB.Table IV shows, as a function of the magnetic fieldB,

the values of the Coulomb energyEcoul of the electrostaticmodel of point charges and the corresponding variationalbinding energyEb.

TABLE IV. H2+3 state1σg. Comparison ofEcoul vsEb. In general,

the relative difference∆E = (|Ecoul− |Eb||/|Eb|) is less than5%for all the range of magnetic fields considered.

B(a.u.) Ecoul(Ry) Eb(Ry) ∆E(%)

425 -15.3869 -15.1580 1.5

1000 -20.5693 -20.7829 1.0

4255 -35.5729 -34.3905 3.4

10000 -47.7120 -45.4081 5.0

18782 -57.9764 -55.2311 4.9

Rev. Mex. Fıs. 57 (3) (2011) 193–203


TABLE V. H3+4 state1σg. Parameters which approximate the elec-

tronic charge densityλ(z) given as the sum of four (normalized)Gaussian functions situated symmetrically with respect to the cen-ter of the molecular axis.

B (a.u.) q2 (a.u.) z2 (a.u.) q3 (a.u.) z3 (a.u.)

105 -0.04792 0.03037 -0.45207 0.05755

2× 105 -0.03431 0.02583 -0.46568 0.04756

3× 105 -0.03086 0.02370 -0.46913 0.04284

4× 105 -0.02749 0.02242 -0.47250 0.03995

5× 105 -0.02628 0.02113 -0.47371 0.03759

106 -0.02232 0.01849 -0.47767 0.03210

2.4. Ion H3+4

The study is carried out in the range of magnetic fields1 × 105 ≤ B ≤ 106 a.u. For this system a superpositionof four Gaussian functions were used to fit thez-profile ofthe electronic distribution,i.e λ(z) (see2). The parametersof the model which approximateλ(z) for each magnetic fieldstudied are summarized in the Table V.

The equilibrium interproton distancesR1, R2 (see Fig. 3)as a function of the magnetic field strengthB are taken fromthe variational calculations [4] and fitted by the respectiveformula

R1(B)=1.0919

1+1.3482 log1+(0.0024B)2+(0.00015B)4

×(

1 + 1.0908 B

1 + 0.9593 B

), (14)

R2(B)=1.1206

1+1.2352 log1+(0.00105B)2+(0.000096B)4

×(

1 + 1.11996 B

1 + 0.9466 B

). (15)

The protons of the systemH3+4 are thus located at

z = ±R1

2, ± (R1 + 2R2)

2with the origin situated in the mid point on the line connect-ing protons. Notice thatR1 < R2 ∀B.


q3(B)=− 0.04144+0.7888 log1+(0.00004B)21.00636+1.60047 log1+(0.000034B)2 , (16)

This formula describes the approximated dependence ofq3(B) on the magnetic field. Notice that forB → ∞,q3(B) → −0.492 implying that as the magnetic fields in-creases, the electrostatic model is reduced to (almost) thatwith two negative point charges situated at

±z3 ,R1

2< z3 <

R1 + 2R2

2∀B.

FIGURE 3. H3+4 state1σg: Electrostatic model for the ground

state1σg of the one-electron molecular ionH3+4 in parallel con-

figuration with a uniform and constant strong magnetic fieldB=(0, 0, B). The electronic cloud is replaced by four point likecharges with fractional chargeq3, q2, q2, q3 located at the positions−z3,−z2, z2, z3 respectively.

TABLE VI. H3+4 state1σg. Comparison ofEcoul vsEb. In general,

the relative difference∆E = (|Ecoul − |Eb||/|Eb|) is less than2.5% for all the range of magnetic fields considered.

B(a.u.) Ecoul(Ry) Eb(Ry) ∆E (%)

1× 105 -77.5705 -74.0368 4.7

2× 105 -93.0368 -91.1208 2.1

3× 105 -104.8920 -102.4702 2.3

4× 105 -113.7690 -111.1897 2.3

5× 105 -120.7930 -118.3093 2.0

106 -143.0300 -142.7426 0.2

Thecharge conservation condition2q1 +2q2 = −1 givesthe interpolation for the two symmetric point chargesq2(B)(see Fig. 3). The interpolation for the positionzi (i = 2, 3)of the two point chargeqi is given by

z2(B) =0.1781

1 + 0.1897 log1 + (0.000018 B)2

×(

1 + 0.3070 B

1 + 1.3992 B

), (17)

z3(B) =0.3150

1 + 0.2434 log1 + (0.000017 B)2

×(

1 + 0.3369 B

1 + 1.3785 B

). (18)

Notice thatz2 < (R1/2)∀B.Table VI shows the values of the Coulomb energyEcoul of

the electrostatic model of point charges and the correspond-ing variational binding energyEb.

2.5. Results

In the last sections simple one-dimensional electrostatic mod-els of the one electron molecular systemsH+

2 , H2+3 , H3+

4 in astrong magnetic field were proposed to estimate the binding-

Rev. Mex. Fıs. 57 (3) (2011) 193–203

198 A. ESCOBAR

(ionization) energy of the corresponding ground states, giv-ing the following main results:

• Compared with the binding energy obtained by precisevariational calculation [4], the Coulomb energyEcoul

has a relative error of less than2.5%, 5% and5% forthe systemsH+

2 , H2+3 , H3+

4 respectively for any valueof the magnetic field considered.

• The model predicts the existence of the systemH+2 for

any value of magnetic field in agreement with varia-tional results. In the range0 6 B < ∞, the corre-sponding Coulomb energyEcoul(B) is a smooth, neg-ative and monotonously decreasing function.

• For the systemH2+3 the electrostatic model predicts

binding energies of -169.523 Ry atB = 106 a.u. and-262.108 Ry atB = 107 a.u. which implies a rela-tive error (compared with variational calculations [4])of less than6%, 0.07% respectively even for thesemagnetic fields (two orders of magnitud larger that thevalues considered in the present study). For magneticfieldsB & 200 a.u. the Coulomb energyEcoul(B) isa smooth, negative and decreasing function, for smallvalues (B < 30) a.u. it becomes positive, which maybe an indication that the systemH2+

3 is not bound inthis region of magnetic fields. According to variationalresults [4]H2+

3 has bound states fromB & 102 a.u.

• For the system H3+4 and for magnetic fields

B&7×104 a.u., the Coulomb energyEcoul(B) is a neg-ative, smooth and decreasing function. In the mag-netic field interval0 6 B 6 7 × 104 a.u., the en-ergy Ecoul(B) is negative but not monotone. On theother side, using the variational method, it is found thatthe moleculeH3+

4 is bound only for magnetic fieldsB & 104 a.u., therefore extrapolation of the functionEcoul(B) for magnetic fields0 6 B 6 104 a.u. is notphysical.

3. Two-electron molecular systems: electro-static model of point charges and a chargedcylinder

For the non-relativistic description of the ground state3Πu

(the notation used in Ref. 4 is adopted) of the two-electronmolecular systems in a strong magnetic fieldB (parallelconfigurationvii), we follow a treatment similar to that forthe case of one-electron systems. In the first instance, weconsider that the electronic total charge can be modeled bya linear density of chargeΛ(z) situated along the magneticfield direction (z-direction), whereΛ(z) is defined throughthe z-profile of the ground state electronic distributionΨvar

obtained by variational calculations [6]

Λ(z) = 2e

∫Ψ∗varΨvarρ1ρ2dρ1dz2dρ2dϕ∫

Ψ∗varΨvardr1dr2;

z = z1(z2), (19)

where(ρi, φi, zi), i = 1, 2 are the cylindrical coordinates ofthe position vectorri of the electroni, ϕ is the angle betweenr1 andr2 ande (= −1 in atomic units) is the electron charge.

In a second step, in analogy to the case of one-electronmolecular systems, the linear charge distribution (17) is ap-proximated by a linear superposition of a finite number ofstandard Gaussian functions situated symmetrically with re-spect to the center of the molecular axis. For simplicity,we assume equal contribution of each electron to this linearcharge distributionΛ(z)

Λ(z) = λ1(z) + λ2(z) ; λ1 = λ2 (20)

where λ1(z) is the linear charge density associated withelectrone1 and similary fore2. Both charge distributionλi(z), i = 1, 2, normalized by construction to−1 (electroncharge in a.u.), are given by the same linear superpositionof Gaussian functions asΛ(z), except that the linear coef-ficient amplitudes of the Gaussian curves are reduced by afactor of 2. However in a state3Πu (a state withM = 1)we consider that one electron, let us say electrone1, is in itsground state (m1 = 0) and the other electron is in an ex-cited state (m2 = 1) wheremi (i = 1, 2) denotes the angularmomentum projection of thei-electron on the magnetic fielddirection, andM = m1 + m2 represents the total magneticquantum number.

This suggest reduce the linear charge distributionλ1(z),associated withe1 (m1 = 0), to a configuration of pointcharges replacing each Gaussian curve by a point charge lo-cated at the center of the Gaussian and whose value is equal tothe corresponding linear coefficient amplitude, whileλ2(z)associated withe2 (m2 = 1) is projected onto the surfaceof an infinite cylindrical shell of radiusρ0, thus the electron2 is described by a surface charge distributionσ2(z). Weassumeviii ρ0 ≈ 〈ρ〉. The number of Gaussian functionsused in the model depends on the shape of the profileΛ(z).Roughly, for each local maximum of thez-profile a Gaussianfunction is introduced.

Explicitly the resulting surface charge distributionσ2(z)(normalized) has the form

σ2(z) =λ2(z)2πρ

;∫

σ2dV = −1. (21)

Is important to emphasize that the ground state (m= 0)wave function of an electron placed in a uniform and constantmagnetic fieldB = (0, 0, B) has not nodes.

Using this electrostatic modelix, we then obtain theCoulomb interaction energy

Ecoul = Ecoul(R, ρ0; σi, qi, zi),whereR denotes the equilibrium internuclear distance(s) ofthe system. Normalization condition allows us to express oneparameter amongqi, σi, zi (i = 1, 2) in terms of the others.

Rev. Mex. Fıs. 57 (3) (2011) 193–203


However, unlike the previous case of one electron systems,the Coulomb energyEcoul can not be written in a closed formin terms of elementary functions.

Suitable interpolation functions of the parametersqi, zi,σi (i = 1, 2, 3) as a functions of the magnetic field strengthBwhere found, such functions allows us to write the CoulombenergyEcoul = Ecoul(B) as an explicit analytical approxima-tion as function of the magnetic fieldB, with the distance (s)R andρ0 playing the roles of external parameters. If we usethe values ofR andρ0 obtained in variational calculations,we see that the electrostatic energyEcoul is not a good esti-mate for the variational energy -EI

x. However we can findintervals[ρ0−, ρ0+], [R−, R+] for which the relative differ-ence ofρ0, R andEcoul from their variational counterparts〈ρ〉, Req and -EI respectively are less than15% (30% for H2).

3.1. Calculations

In this section we show, for each two-electron system(H2, H+

3 , H2+4 ), the interpolating function of the parame-

tersqi, zi andσi (i = 1, 2, 3) as a function of the magneticfield strengthB and the corresponding intervals[ρ0−, ρ0+],[R−, R+]. Hereafter, magnetic field is defined in dimension-less units (a.u.) asB/B0, whereB0 = 2.35× 109 G, whichwe continue to denote asB.

3.2. Molecule H2

As a basic system, the moleculeH2 has been studied in pres-ence of strong magnetic fields. It was found that the moststable configuration of the system is parallel to the magneticfield [6]. A relevant fact is that the state with the lowest totalenergy (ground state) depends on the magnetic field strengthB. It evolves from spin-singlet1Σg state for small mag-netic fields0 . B . 0.18 a.u. to spin-triplet3Πu statefor 12.3 a.u.. B The moleculeH2 does not exist in therange of magnetic fields 0.18. B . 12.3 a.u., in which thelowest energy state corresponds to a repulsive (unbounded)3Σu state. The electrostatic study is carried out in the rangeof magnetic fields100 ≤ B ≤ 10000 a.u. For this sys-tem a superposition of three Gaussian functions was used tomodel the linear charge densityΛ(z) (see 17) defined by thez-profile of the corresponding ground state electronic distri-bution. The electrostatic model consists of (i) a set of pointchargesqi (m1 = 0) obtained replacing each Gaussiancurve ofλ1(= Λ/2) by a point charge, located at the centerof the Gaussian and whose value is equal to the correspond-ing linear coefficient amplitude and (ii) an infinite chargedcylindrical shellσ2(z) = (λ2)/(2πρ) (m2 = 1) of radiusρ0,(see Fig. 4). The parameters of the model which approximateΛ(z) for each magnetic field studied are summarized in theTable VIII.

The Coulomb energyEcoul (in Rydbergs) is given by

Ecoul = 2(Ep−p + Ep−e1,e2 + Ee1−e2) , (22)

TABLE VIII. H2 state3Πu. Parameters which approximate theelectronic charge densityΛ(z) given as the sum of three Gaussianfunctions situated symmetrically with respect to the center of themolecular axis.

B (a.u.) q1 (a.u.) σ1 (a.u.) q2 (a.u.) z2 (a.u.) σ2 (a.u.)

100 0.63435 0.42474 0.18282 0.14724 0.15957

425 0.64121 0.31403 0.17939 0.10007 0.10580

1000 0.64873 0.26154 0.17563 0.07969 0.08446

4255 0.66495 0.19581 0.16752 0.05617 0.05912

10000 0.67017 0.16740 0.16491 0.04567 0.04930

FIGURE 4. H2 state3Πu: Electrostatic model for the ground state3Πu of the two-electron molecular ionH2 in parallel configurationwith a uniform and constant strong magnetic fieldB = (0, 0, B).The electronic cloud is replaced by three point like charges withfractional chargeq2, q1, q2 (m1 = 0) located at the positions−z2, z1, z2 respectively and an infinite charged cylindrical shell(m2 = 1) of radiusρ0. The protons of the systemH2 are located atz = ± (R/2) with the origin situated in the mid point on the lineconnecting protons.

where

E(p−p) =1R

, (23)

E(p−e1,e2) =4q1

R+

2q2

|R2 − z2|+

2q2

|R2 + z2|

+ Φ(−R

2

)+ Φ

(R

2

), (24)

E(e1−e2) = q1Φ(z1) + q2Φ(−z2) + q2Φ(z2) , (25)

where the subscriptsp ande1, e2 denote the proton and elec-trons respectively (using this notation,E(p−p) is the repulsiveinteraction between protons, and similary forE(p−e1,e2) andE(e1−e2)). As usual the electrostatic potentialΦ(z; ρ0), pro-duced by the cylindrical shellσ2(z; ρ0), is given by

Φ(z; ρ0) =∫

σ2(z′; ρ0)√ρ20 + (z − z′)2

dz′dρ′dφ′ , (26)

As a function of the magnetic fieldB, the interpolatingformula for the point chargeq1 located in the center (z= 0)is given by

q1 = −1.6757 + 0.001137 B

2.6531 + 0.001682 B, (27)

Rev. Mex. Fıs. 57 (3) (2011) 193–203

200 A. ESCOBAR

TABLE VIII. H2 state 3Πu: Intervals ∆ρ0=[ρ0−, ρ0+] and∆R=[R−, R+]. The relative difference∆E=(|Ecoul−|EI ||

/|EI |)is less than30%.

B(a.u.) ∆R (a.u.) ∆ρ0 (a.u.) Ecoul

100 [1.26Req,1.3Req] [1.26〈ρ〉,1.3〈ρ〉] [-1.296EI ,-1.238EI ]

425 [1.27Req,1.3Req] [1.27〈ρ〉,1.3〈ρ〉] [-1.293EI ,-1.246EI ]

1000 [1.27Req,1.3Req] [1.27〈ρ〉,1.3〈ρ〉] [-1.295EI ,-1.247EI ]

4255 [1.27Req,1.3Req] [1.27〈ρ〉,1.3〈ρ〉] [-1.285EI ,-1.236EI ]

10000 [1.29Req,1.3Req] [1.29〈ρ〉,1.3〈ρ〉] [-1.283EI ,-1.266EI ]

This formula describes the approximated dependence ofq1(B) on the magnetic fieldB. Notice that forB→∞,q1(B) → -0.67 . The charge conservation conditionq1 + 2q2 = −1 gives the interpolation for the two symmetricpoint chargesq2(B) (see Fig. 4).

The interpolation function for the positionz2 of the twochargesq2 is given by

z2(B)

=1.0152

1+1.0553 log1+(0.16148B)2+(0.01403B)4 , (28)

For all the range of magnetic fields considered,z2<(Req/2) whereReq is the equilibrium distance obtainedwith variational calculations [6].

Finally the corresponding variancesσ1, σ2 are fitted by

σ1(B)

=2.68367

1+0.62449 log1+(0.69290B)2+(0.03944B)4 , (29)

σ2(B)

=3.70302

1+3.4712 log1+(0.24225B)2+(0.02064B)4 . (30)

Notice thatσ1 > σ2 ∀B. In relative units (Req = 1), ingeneral both variances decrease as functions of the magneticfield B.

Table VIII shows the intervals[ρ0−, ρ0+], [R−, R+] forwhich the relative difference ofρ0, R andEcoul from theirvariational counterparts〈ρ〉, Req and -EI respectively are lessthan30%.

3.3. Ion H+3

The moleculeH+3 has been studied in presence of strong

magnetic fields [17]. It was found that the most stable con-figuration of the system is parallel to the field. A relevant factis that the state with the lowest total energy (ground state)depends on the intensity of the fieldB. It evolves from spin-singlet 1Σg state for small magnetic fieldsB <0.2 a.u. toweakly-bound spin-triplet3Σu state for intermediate fields0.2 . B . 20 a.u. and eventually to spin-triplet3Πu statefor B > 20 a.u. The electrostatic study is carried out in the

TABLE IX. H+3 state3Πu. Parameters which approximate the elec-

tronic charge densityΛ(z) given as the sum of four Gaussian func-tions situated symmetrically with respect to the center of the molec-ular axis.

B (a.u.) q3 (a.u.) z3 (a.u.) σ3 (a.u.) q2 (a.u.) z2 (a.u.) σ2 (a.u.)

100 0.32950 0.20773 0.29793 0.17049 0.19735 0.27051

1000 0.32376 0.10027 0.12499 0.17624 0.09147 0.26435

10000 0.28318 0.04655 0.05925 0.21682 0.04649 0.14583

18782 0.25980 0.04575 0.05794 0.24019 0.03901 0.13892

FIGURE 5. H+3 state3Πu: Electrostatic model for the ground state

3Πu of the two-electron molecular ionH+3 in parallel configuration

with a uniform and constant strong magnetic fieldB = (0, 0, B).

range of magnetic fields100 ≤ B ≤ 18782 a.u. For thissystem a superposition of four Gaussian functions was usedto model the linear charge densityΛ(z) (see 17) defined bythez-profile of the corresponding ground state electronic dis-tribution. The parameters of the model which approximateΛ(z) for each magnetic field studied are summarized in theTable X.

The Coulomb interaction energyEcoul (in Rydbergs) isgiven by


where

Ep−p =5R

, (32)

Ep−e1,e2 =2q3

|R2 − z3|+

2q3

|R2 + z3|

+2q2

|R2 − z2|+

2q2

|R2 + z2|+

2q3

|z3|

+2q2

|z2| + Φ(−R

2

)+ Φ(0) + Φ

(R

2

), (33)

Ee1−e2 = q3Φ(−z3)

+ q3Φ(z3) + q2Φ(−z2) + q2Φ(z2). (34)


q3(B) =1.88093 + 0.000179 B

5.66932 + 0.000728 B. (35)

Rev. Mex. Fıs. 57 (3) (2011) 193–203


This formula describes the approximated dependence ofq3(B) on the magnetic field. Notice that forB→∞,q3(B)→ − 0.24, z2 < z3 < (R/2)∀B. The charge conservation condition2q1 + 2q2 = −1 gives the interpolation forthe two symmetric point chargesq2(B) (see Fig. 5). The interpolation for the positionzi (i = 2, 3) of the two point chargesqi

is given by

z3(B) =1.39441

1 + 1.38206 log1 + (0.077667 B)2 + (0.009420 B)4 , (36)

z2(B)=0.19845

1+0.20969 log1+(8.83244× 10−5B)2+(0.004052B)4 , (37)

z3 > z2 ∀B, finally the corresponding variancesσ3, σ2 are fitted by

σ3(B) =1.55132

1 + 1.1988 log1 + (0.055867 B)2 + (0.010451 B)4 , (38)

σ2(B) =1

1 + 0.95415 log1 + (0.019994 B)2

(1 + 1.76429 B

1 + B

). (39)

TABLE X. H+3 state3Πu: Intervals∆ρ0 = [ρ0−, ρ0+] and∆R

=[R−, R+]. The relative difference∆E=(|Ecoul−|EI ||)/(|EI |)is less than15% for all the range of magnetic fields considered.

B(a.u.) ∆R (a.u.) ∆ρ0 (a.u.) Ecoul(Ry)

100 [1.1Req,1.15Req] [1.1〈ρ〉,1.15〈ρ〉] [-1.115EI ,-1.075EI ]

1000 [1.1Req,1.15Req] [1.1〈ρ〉,1.15〈ρ〉] [-1.083EI ,-1.044EI ]

10000 [1.1Req,1.15Req] [1.1〈ρ〉,1.15〈ρ〉] [-1.085EI ,-1.048EI ]

18782 [1.1Req,1.15Req] [1.1〈ρ〉,1.15〈ρ〉] [-1.071EI ,-1.034EI ]

Table X shows the intervals[ρ0−, ρ0+], [R−, R+] forwhich the relative difference ofρ0, R andEcoul from theirvariational counterparts〈ρ〉, Req and -EI respectively are lessthan15%.

3.4. Ion H2+4

It is well known that in the absence of a magnetic field, theexotic molecular systemH2+

4 does not exist. However, forB & 2000 a.u. the system becomes bound in the linear con-figuration aligned along the magnetic line, the state with thelowest total energy (ground state) being realized by the spin-triplet 3Πu state [6]. For. B . 2000 a.u. the ground statecorresponds to the repulsive spin-triplet3Σu state. As themoleculeH+

3 , the electrostatic study is carried out in therange of magnetic fields100 ≤ B ≤ 18782 a.u. For thissystem a superposition of four Gaussian functions was usedto model the linear charge densityΛ(z) (see 17) defined bythez-profile of the corresponding ground state electronic dis-tribution. The parameters of the model which approximateΛ(z) for each magnetic field studied are summarized in theTable XI.

The Coulomb energyEcoul (in Rydbergs) is given by


where

Ep−p =1

R1+

2R2

+2

R1 + R2+

1R1 + 2R2

, (41)

Ep−e1,e2 =2q2

|R12 + R2 − z2|

+2q2

|R12 + R2 + z2|

+2q2

|R12 − z2|

+2q2

|R12 + z2|

+2q3

|R12 + R2 − z3|

+2q3

|R12 + R2 + z3|

+2q3

|R12 − z3|

+2q3

|R12 + z3|

+Φ(−R1

2

)+Φ

(− (R1 + 2R2)

2

)+Φ

(R1

2

)

+ Φ(

(R1 + 2R2)2

), (42)

Ee1−e2 = q2Φ(−z2)

+ q2Φ(z2) + q3Φ(−z3) + q3Φ(z3). (43)


FIGURE 6. H2+2 state3Πu: Electrostatic model for the ground

state 3Πu of the two-electron molecular ionH2+4 in parallel

configuration with a uniform and constant strong magnetic fieldB=(0, 0, B).

Rev. Mex. Fıs. 57 (3) (2011) 193–203

202 A. ESCOBAR

TABLE XI. H2+4 ground state. Parameters which approximate the electronic charge densityΛ(z) given as the sum of four Gaussian functions

situated symmetrically with respect to the center of the molecular axis.

B (a.u.) q2 (a.u.) z2 (a.u.) σ2 (a.u.) q3 (a.u.) z3 (a.u.) σ3 (a.u.)

100 0.470132 0.098422 0.412104 0.029867 0.448526 0.30471

1000 0.462995 0.043426 0.212223 0.037004 0.192696 0.16069

10000 0.362742 0.021508 0.109725 0.137257 0.094341 0.12014

18782 0.111161 0.018676 0.058242 0.388838 0.083406 0.08205

q2(B)=0.46730

1+2.85481 log1+(3.2607× 10−5B)2 . (44)

This formula describes the approximated dependence ofq2(B) on the magnetic field. The charge conservation condition2q1 + 2q2= − 1 gives the interpolation for the two symmetric point chargesq3(B) (see Fig. 6). The interpolation for thepositionzi (i = 2, 3) of the two point chargesqi is given by

z2(B) =0.59035

1 + 1.36931 log1 + (0.060622 B)2 + (0.009125 B4 , (45)

z3(B) =3.06313

1 + 1.60496 log(1 + 0.059972 B)2 + (0.009441 B)4), (46)

z3 > z2 ∀B, finally the corresponding variancesσ3, σ2 are fitted by

σ2(B) =5.65161

1 + 2.76837 log1 + (5.01591× 10−6 B)2 + (0.010304 B)4 , (47)

σ3(B) =1.67862

1 + 1.65664 log1 + (0.098154 B)2

(1 + 1.56427 B

1 + B

). (48)

TABLE XII. H2+4 state 3Πu: Intervals ∆ρ0=[ρ0−, ρ0+] and

∆R=[R−, R+]. The relative difference∆E=(|Ecoul−|EI ||/|EI |)is less than15% for all the range of magnetic fields considered.

B(a.u.) ∆R (a.u.) ∆ρ0 (a.u.) Ecoul(Ry)

100 [Req,1.05Req] [〈ρ〉,1.05〈ρ〉] [-1.038EI ,-1.126EI ]

1000 [Req,1.05Req] [〈ρ〉,1.05〈ρ〉] [-1.064EI ,-0.974EI ]

10000 [1.1Req,1.15Req] [1.1〈ρ〉,1.15〈ρ〉] [-1.115EI ,-1.060EI ]

18782 [1.1Req,1.15Req] [1.1〈ρ〉,1.15〈ρ〉] [-1.081EI ,-1.053EI ]

Table XII shows the intervals[ρ0−, ρ0+], [R−, R+] forwhich the relative difference ofρ0, R andEcoul from theirvariational counterparts〈ρ〉, Req and -EI respectively are lessthan15%.

3.5. Results

In the last sections simple electrostatic models of two elec-tron molecular systemsH2, H+

3 , H2+4 in a strong magnetic

field were proposed to estimate the double ionization energyof the corresponding ground states, giving the following mainresults:

• Two electron molecular systemsH2, H+3 , H2+

4 ina strong magnetic field are described by a heuris-tic model (electrostatic model of point charges and acharged cylinder) inspired in precise variational calcu-lations.

• Compared with the double ionization energy obtainedby precise variational calculations [4], the CoulombenergyEcoul has a relative error of less than30%, 15%and15% for the systemsH2, H+

3 , H2+4 respectively for

any value of the magnetic field considered.

4. Conclusions

Simple one-dimensional electrostatic models of one-(two)electron molecular systemsH+

2 , H2+3 , H3+

4 and H2, H+3 ,

H2+4 in a strong magnetic field are presented to estimate

the binding-(ionization) energy of the corresponding groundstates, being accurate for the systemsH+

2 , H2+3 , H3+

4 in 2.5%,5% and5% respectively, and15% for the two-electron sys-temsH+

3 , H2+4 (30% for H2) compared with the correspond-

ing variational calculations.

Rev. Mex. Fıs. 57 (3) (2011) 193–203


Acknowledgments

I am grateful to A.V.Turbiner and J.C.L.Vieyra for providingmany remarkable insights on the subject of matter in strong

magnetic fields and especially to JCLV for his constant inter-est in this work. AER was supported in part by a CONACyTgrant for PhD studies (Mexico).

i. The conversion factor1 a.u.= 2.35 × 109 G is used in thepresent work.

ii. The electronic distribution of H+2 is symmetric with respect tothe plane whose normal is parallel to the magnetic field direc-tion. For strong magnetic field the peak is situated in the middlebetween the charged centers.

iii. From a classical point of view and considering only electro-magnetic interactions, a closed system of charged particles inthe vacuum can not be in a stable equilibrium. However, inquantum mechanics such equilibrium configuration can be re-alized.

iv. As some confusion may arise at this point, we must empha-size that the present approach is a pure electrostatic modeland not an effective one-dimensional approximation of theSchroedinger equation. In particular we do not define an ef-fective Coulomb interaction.

v. The infinitely-heavy charged centers located on a line parallelto the magnetic field direction.

vi. The interactions between the electronic point chargesqi arenot taken into account due to the fact that the electron does notinteract with itself.

vii. The infinitely-heavy charged centers located on a line parallelto the magnetic field direction.

viii. Where〈ρ〉 ≡∫

ρ2|Ψvar|2dr1dr2∫ |Ψvar|2 dr1dr2.

ix. The interactions between the point chargesqi are not takeninto account due to the fact that the electrone1 does not interactwith itself, but the interaction between the point chargesqiand the cylindrical surface charge distributionσ2 (associatedwith e2) is taken into account.

x. EI denotes the double ionization energy obtained by variationalcalculations [6].

1. B.B. Kadomtsev and V.S. Kudryavtsev,Pis’ma ZhETF 15(1971) 61;Sov. Phys. JETP Lett. 13, 9 (1971) 42.

2. M. Ruderman,Phys. Rev. Lett.27 (1971) 1306.

3. A. Turbiner, J.C. Lopez, and U. Solis H.,Pis’ma v ZhETF69(1999) 800;JETP Lett.69 (1999) 844.

4. A.V. Turbiner and J.C. Lopez Vieyra,Phys. Repts.424 (2006)309.

5. A.V. Turbiner,Astrophysics and Space Science308(2007) 267.

6. A.V. Turbiner, N.L. Guevara, and J.C. Lopez Vieyra,Phys. Rev.A 81 (2010) 042503.

7. R.D. Benguria, R. Brummelhuis, P. Duclos, and S. Perez-Oyarzun,J. Phys. B37 (2004) 23112320.

8. L.D. Landau, and E.M. Lifshitz,Quantum Mechanics (Non-relativistic theory)3rd Ed. p. 462.

9. R.P. Martınez-y-Romeroet al., Rev. Mex. Fis.35 (1989) 617.

10. Nunez-Yepezet al., Phys. Rev. A.39 (1989) 4307.

11. L.J. Boya, M. Kmiecik, and A. Bohm,Physical Review A37(1988) 3567.

12. Elliot y Loudon,J. Phys. Chem. Solids15 (1960) 196.

13. M.W. Cole and M.H. Cohen,Phys. Rev. Lett.23 (1969) 1238.

14. P.M. Platzman and M.J. Dykman,Science284(1999) 1967.

15. H. Olivares-Pilon, D. Baye, A.V. Turbiner, and J.C. LopezVieyra,J. Phys. B: At. Mol. Opt. Phys.43 (2010) 065702.

16. A.V. Turbiner, A.B.Kaidalov, and J.C. Lopez Vieyra,[arXiv:0506019v1] [physics.atom-ph] .

17. A.V. Turbiner, N.L. Guevara, and J.C. Lopez Vieyra,Astro-physics and Space Science308(2007) 499.

Rev. Mex. Fıs. 57 (3) (2011) 193–203


Estimador estocastico para un sistema tipo caja negraJ. de J. Medel Juarez

Centro de Investigacion en Computacion del Instituto Politecnico Nacional,Av. Juan de Dios Batiz s/n casi esq. Miguel Othon de Mendizabal,

Unidad Profesional Adolfo Lopez Mateos, Mexico D.F. 07738, Mexico.

R.U. ParrazalesCentro de Investigacion en Computacion del Instituto Politecnico Nacional,

Av. Juan de Dios Batiz s/n casi esq. Miguel Othon de Mendizabal,Unidad Profesional Adolfo Lopez Mateos, Mexico, D.F. 07738, Mexico.

R.P. OrozcoCentro de Investigacion en Ciencia Aplicada y Tecnologıa Avanzada del Instituto Politecnico Nacional,

Legaria No. 694, Mexico D.F. 11500, Mexico.

Recibido el 3 de noviembre de 2010; aceptado el 29 de abril de 2011

Este artıculo considera a un sistema tipo caja negra con dinamica interna desconocida. Para describirla se requiere de un estimador basadoen la variable instrumental, de la matriz de transicion y del identificador que es resultado de un modelo simplificado. El modelo propuesto demanera recursiva esta en espacio de estados y tiene explıcitamente la ganancia interna, como elunico elemento desconocido por describir. Elestimador se aproxima y en el mejor de los casos, converge a una vecindad de la referencia, lo que permite ser una herramienta del identifi-cador al usar a la matriz de transiciones de una manera analıtica resolviendo el problema de convergencia del filtro. La convergencia puedeobservarse por el funcional recursivo del error de identificacion. Como ejemplo, se desarrollo la simulacion del modelo en diferencias finitasde un motor de CD requiriendo conocer que dinamica interna de operacion tiene. El estimador con la variable instrumental logro describir alparametro para diferentes condiciones de operacion y se dio seguimiento a la senal. El funcional de error para diferentes ganancias dentro dela region de estabilidad discreta, es convergente. Y la funcion de distribucion del identificador se aproxima a la corriente directa del modelo.

Descriptores:Procesos estocasticos; estimacion; filtrado; identificacion.

This paper considers a black box system with unknown internal dynamics. The estimator based on instrumental variable requires, the transi-tion matrix used in the identifier which results in a simplified model. The recursive space state model allows an explicit internal gain which isunknown and undescribed. The recursive estimator allows knowing the internal dynamics of the black box system in an analytic manner andin the best cases, converges to a reference neighborhood, becoming a necessary identification tool solving the convergence filter problem. Theconvergence estimator and the identifier are seen from the recursive functional identification error. An example was developed to simulatethe DC motor in a finite differences model that requires knowing the operation of internal dynamics. The instrumental variable estimatordescribes the different operating condition parameters and monitors the direct current signal in finite differences. The functional error todifferent gains in the stability discrete region converges, and approximates the distribution of the direct current model.

Keywords: Stochastic processes; estimation; filtering; identification.

PACS: 02.30.Yy; 07.50.-x; 02.70.Br

1. Introduccion

Un filtro es un dispositivo que elimina, extrae, predice, re-construye y describe parte de la informacion de un sistema,de acuerdo a un criterio previamente establecido [15].

El proceso de filtrado sin alterar la dinamica del siste-ma de referencia, requiere de una planeacion experimental,seleccionando un modelo y validandolo para describir deuna manera aproximada la senal observableyk Consideran-do su excitacion νk, el proceso de filtrado esta integrado porel estimadorak y el identificadormk; que para aproximar-se a la senal deseada requiere de la excitacion νk, que per-mite obtener una senal identificada observableyk; el errorde identificacion ek se establece por la diferencia entre lasenal observableyk y su identificada observableyk, de acuer-do con la Fig. 1.

En la practica, los modelos son de naturaleza recursiva,como ejemplo de filtros digitales recursivos se tienen los fil-tros de Kalman [19] y de Medel-Poznyak [12].

Con base en el problema de presuponer conocidos losparametros, se desarrollaron aplicaciones en fısica [16-18]con aportaciones a vision artificial para el seguimiento detrayectorias de movimiento de partıculas, que en condicio-nes dinamicas se requerıan secuencias de matrices de transi-cion, lo cual llevo al desarrollo de estimadores de parametrosdinamicos recursivos [13].

Dentro de la teorıa de filtrado existen dos procesos basi-cos segun Haykin [15] y Medel [10], que son laestimaciony la identificacion. La estimacion describe la dinamica delos parametros del sistema representados porak y la iden-tificacion, la dinamica de los estados representados pormk

todo ello con respecto al sistema de referencia, de acuer-do con la Fig. 1.

ESTIMADORESTOCASTICO PARA UN SISTEMA TIPO CAJA NEGRA 205

FIGURA 1. Esquema general de un filtro digital funcionando comoestimador de parametros e identificador de estados.

Cualificar la eficiencia de un filtro, significa describir suspropiedades de convergencia a traves del funcional del errorJ(k) dado por el segundo momento de probabilidad del errorde estimacion o de identificacionek [10,15].

El sistema de estudio es un modelo tipo caja negra en elque solo se observayk ∈ R, y νk ∈ R, ambas conk ∈ Z+.La relacion en el sistema entre la salidayk y la excitacionexternaνk = νi : i = 1, . . . , k, k ∈ Z+ esta descrita porf(νk, k) ∈ R. Ası, el modelo se representa en la forma (1):

yk = f(νk, k) + νk, (1)

dondef(νk, k) es la funcion objetivo del filtro [8,9] yνk es elproceso de innovacion dado por la variable aleatoria del errorde medicion.

El identificador es descrito poryk con errorek formadopor la diferencia entre el filtro y la salida de referencia.

La estimacion es descrita a traves de un proceso de in-novacion involucrado en la formulacion y validacion del mo-delo (1) con respecto a la senal de referencia, permitiendoconstruir la matriz de transiciones necesaria para el identifi-cador [1,2,12]. En el caso de queesta no sea estacionaria nihomogenea no es posible utilizar el metodo de estimacion demınimos cuadrados.

En este artıculo se aplica el metodo de estimacion re-cursiva mediante el segundo momento de probabilidad comoen [2,3] y se desarrolla un filtro con la variable instrumen-tal como proceso de innovacion para expresarlo de manerarecursiva y observar su evolucion en el tiempo.

2. Estimador con proceso de innovacion

El modelo en espacio de estados con ruidos acotados por unadistribucion gaussiana segun la Ref. 4 es descrito en (1). Lafuncion f(νk, k) esta definida como el producto interior en-tre el estado observable con un retardoyk−1, y el parametrodesconocidoak.

Teorema 2.1 Sea el modelo del sistema tipo caja ne-gra con entradaxk y salida acotadayk, con dominios en

N(µ, σ2 < ∞). Existe un modelo del proceso tipo caja ne-gra simplificado dado por (2).

yk = ayk−1 + νk, (2)

dondea y νk son el parametro y el ruido del proceso, respec-tivamente.

Demostracion 2.1Ver anexo, demostracion 5.1.Segun la demostracion del teorema 2.1, y basados en los

conceptos sobre sistemas de secuencias de estados [9] al con-siderar la varianzaQk = Ey2

k−1 − d(νk−1yk−1), la cova-rianzaPk = Eykyk−1, la mediaµ, ası como sus formasrecursivas, se determina el parametro estocasticoak, basadosen los conceptos sobre sistemas de secuencia de estados [9].

Teorema 2.2 Existe un estimador estocastico recursivo3 paraak con respecto a un sistema tipo caja negra 2.

ak =(ykyk−1) + (k − 1)Pk−1

y2k−1 − d(νk−1yk−1) + (k − 1)Qk−1

(3)

Demostracion 2.2Ver anexo, demostracion 5.2.Se define el filtro estocastico recursivo covariante com-

pleto por el teorema 2.3 y por el teorema 2.4.Teorema 2.3 Sea el modelo del sistema tipo caja negra

con entrada y salida estocasticaxk e yk, respectivamente yque tienen las propiedades de invarianza observadas en sussegundos momentos. Existe un filtro estocastico recursivo co-variante definido por (5).

yk = akyk−1 + νk (4)

Demostracion 2.3Ver anexo, demostracion 5.3.Y la evaluacion de la convergencia del filtro mediante su

error queda dado por (5).Teorema 2.4Sea el error de identificacionek = yk−yk.

El funcional del errorJk esta dado de manera recursivapor (5).

Jk =1k

(e2k + (k − 1)Jk−1

)(5)

Demostracion 2.4Ver anexo, demostracion 5.4.

3. Aplicacion y resultados

3.1. Aplicacion

El modelo matematico de un motor de corriente directa (CD)tipo serie se representa por la ecuacion del circuito electri-co (6) y la ecuacion mecanica (7):

u = Ri + Ldi

dt+ λ0ω, (6)

Jω = τm − τL. (7)

Rev. Mex. Fıs. 57 (3) (2011) 204–210

206 R.P. OROZCO, R.U. PARRAZALES Y J. DE J. MEDEL JUAREZ

FIGURA 2. Simulacion del comportamiento deik en un motor de

CD. A) Comparacion de senalesik, ik. B) Estimacion del parame-tro internoak.

El modelo en diferencias finitas queda expresado en (8).

ik = ak ik−1 + wk. (8)

Corresponde a la forma descrita en (2) requieriendo de (3)para conocer sus parametros y de (4) para observar la evolu-cion de su estado, ası como de (5) para observar en el filtradosu nivel de convergencia.

En la Fig. 2A) se muestra una comparacion entre la senal

de referenciaik y la senal identificadaik. Como se puedeapreciar ambas se encuentran acotadas en2A y son estocasti-cas. En la Fig. 2B), se muestra la evolucion del parametrointernoak.

En la Fig. 3A), se presenta el funcional del errorJk aso-ciado a la estimacion del parametro internoak, convergiendoa cero. En la Fig. 3B), se muestra el histograma del compor-tamiento deik en un motor de CD, esto para verificar que lasenal de referencia es estocastica y normal.

3.2. Resultados

Se analizan dos aspectos importantes, i) las salidas delas Ecs. (10) y (15), ii) el parametro obtenido por la Ec. (27)

FIGURA 3. Simulacion del comportamiento de un motor de CD.A) Funcional del errorJk, B) Histograma del comportamiento deik en un motor de CD.

y el recursivo por la Ec. (31), convergiendo hacia el parame-tro ak del sistema original (10), para valores de|ak| ≤ 1.

Se desarrollo el proceso de estimacion para (32) usan-do como herramienta de desarrollo MatLab(R) con diferentesvalores dentro del intervalo de(−1, 1).

La Fig. 4A), muestra los resultados de la estimacion paradiferentes valores deak.

La Fig. 4B), muestra los resultados del funcional del errorpara las diferentes condiciones deak.

Observando que en cualquiera de los casos converge auna region acotada y es menor a 0.01 unidades.

Para el caso de que el sistema descrito en (15) tenga unparametro de 0.7 unidades, el estimador de acuerdo con (32)dio como resultado lo mostrado en la Fig. 4C).

El resultado de la identificacion al sustituir (32) en (15)queda esquematizado en la Fig. 4D).

En la Fig. 5A), se describe la respuesta del modelo delsistema (15).

Rev. Mex. Fıs. 57 (3) (2011) 204–210


FIGURA 4. A) Estimacion de parametros para diferentes condicio-nes dea, de acuerdo con (15) y (32).B) Funcional del error paradiferentes condiciones deak. C) Convergencia paraak = 0.7, delos parametros estimados de acuerdo con el modelo recursivo (32)(ak). D) Identificacion estocastica de la salida simplificadayk y lasalida covariante del modelo (15) paraak = 0.7.

FIGURA 5. A) Amplitud en decibeles deyk. B) Amplitud en deci-

beles deyk. C) Funcional del error de estimacion (5).D) Conver-gencia en funcion de distribucion del identificador (4) hacia senalde referencia (2).

Rev. Mex. Fıs. 57 (3) (2011) 204–210


La Fig. 5B), presenta el resultado de la simulacion del sis-tema descrito por (32) sustituida en (15). La Fig. 5C), presen-ta el funcional del error de la convergencia del identificadorcon respecto al sistema (15). La Fig. 5D), presenta las distri-buciones de la respuesta del sistema de referencia (2) y la delidentificador.

4. Conclusion

El parametro estimadoak de forma recursiva (3) se acoplo almodelo simplificado (4), construyendo el identificador. El ni-vel de convergencia de la respuesta del identificador con res-pecto a la respuesta del sistema tipo caja negra (1) se logro atraves del funcional de error recursivo (5).

La simulacion del estimador (3), del identificador (4), delfuncional del error (5), y de las distribuciones de (1) y de (4),se desarrollaron en 1400 muestras para diferentes condicio-nes de operacion. La Fig. 4a) presenta el parametro estocasti-co concentrado del sistema (1) en sus dos versiones: la cova-riante y la recursiva, convergiendo estaultima a una condi-cion estacionaria. En la Fig. 4B) se presento el seguimientodel identificador (4) y la salida del sistema (1).

La expresion del filtro recursivo estocastico (3) con elidentificador (4) permitio dar el seguimiento de la respuestadel sistema tipo caja negra, con una precision de convergen-cia de acuerdo con (5) de1× 10−4 unidades.

Anexo

En esta seccion se detallan las demostraciones de los teore-mas de la Sec. 2.

Demostracion 5.1 (Teorema 2.1). El sistema esta repre-sentado por las Ecs. (10) y (10).

xk+1 = axk + bwk (9)

yk = cxk + dνk (10)

donde (10) es la ecuacion de transicion de estado,xk es elvector de estados,wk ⊆ N(µw, σ2

w < ∞) es la senal deruido adherida a la ecuacion de estado,yk es la salida obser-vada,νk ⊆ N(µν , σ2

ν < ∞) es la senal de ruido agregada ala salida,a y c son parametros del sistema,b y d son parame-tros del ruido de medicion de los estados interno y externo,respectivamente [6-8].

Considerando que (10) es retardada en el tiempo:

xk = axk−1 + bwk−1 (11)

Al sustituir (11) en (10) se obtiene (12):

yk = c(axk−1 + bwk−1) + dνk

= caxk−1 + cbwk−1 + dνk. (12)

De (10) se obtiene el valor del estado interno con un re-tardo (13).

yk = cxk + dνk,

yk − dνk = cxk,

xk = c−1yk − c−1dνk,

xk−1 = c−1yk−1 − c−1dνk−1. (13)

Sustituyendo (13) en (12) se obtienen (14) y (15).

yk = ca(c−1yk−1 − c−1dνk−1) + cbwk−1 + dνk

= cac−1yk−1 − cac−1dνk−1 + cbwk−1 + dνk (14)

= akyk−1 + νk,

∴ yk = akyk−1 + νk. (15)

Se observa en (15) que la salida del sistema de maneraexplıcita cuenta con dos tipos de terminos: la senal retardadade la salidayk−1 y los ruidos−cac−1dνk−1 +cbwk−1 +dνk.QED

Demostracion 5.2 (Teorema 2.2). La normalizacion delas senales, tanto de entrada como de salida del sistema ti-po caja negra hace un uso adecuado del espacio filtrado. Elconjunto muestreado de estados internos

X(ω) := x(ω, k) : ω ∈ Ω, k ∈ T,de estados externos

Y (ω) := y(ω, k) : ω ∈ Ω, k ∈ T,con F (X(ω), Y (ω)) ⊆ Y (ω) ⊆ N(µy, σ2

y<∞). El se-gundo momento de probabilidad de (15) respecto deyk−1

esta determinado por (16):

Eykyk−1 = Eakyk−1yk−1+ Eνkyk−1= akEyk−1yk−1+ Eνkyk−1= akEy2

k−1+ Eνkyk−1. (16)

Sustituyendoνk de (14) en (16) se obtiene (18).

Eykyk−1 = akEyk−1yk−1+ E(−cac−1dνk−1 + cbwk−1 + dνk)yk−1= akEyk−1yk−1+ E(−cac−1dνk−1)yk−1+ E(cbwk−1)yk−1+ E(dνk)yk−1= akEy2

k−1+ (−cac−1d)Eνk−1yk−1+ (cb)Ewk−1yk−1+ (d)Eνkyk−1= akEy2

k−1+ akdEνk−1yk−1+ cbEwk−1yk−1+ dEνkyk−1= akEy2

k−1+ akdEνk−1yk−1. (17)

Rev. Mex. Fıs. 57 (3) (2011) 204–210


∴ Eykyk−1 = ak(Ey2k−1+ dEνk−1yk−1). (18)

Ya que no hay correlacion entre el ruido y el estado pasadodel sistema,Ewk−1yk−1 = Eνkyk−1 = 0 , como pue-de considerarse a (14). El sistema (16) tiene la forma (18) yse tiene el parametro desconocido:

ak =Eykyk−1

Ey2k−1 − d(νk−1yk−1) =

Pk

Qk, (19)

conretardo en el tiempo:

ak−1 =Pk−1

Qk−1. (20)

Considerandola ergodicidad de (19), con respecto aPk

se obtiene (21) y su forma recursiva (22):

Pk =1k

k∑

i=1

yiyi−1, (21)

Pk =1k

[(ykyk−1) +

k−1∑

i=1

yiyi−1

]. (22)

Tomando (21) con un retardo de tiempo y aplicadoen (22) se obtiene (23):

Pk =1k

[(ykyk−1) + (k − 1)Pk−1] . (23)

Ahora, considerando la ergodicidad de (19) con respectoaQk su forma recursiva se presenta en (24):

Qk =1k

[k∑

i=1

y2i−1 − d

k∑

i=1

νi−1yi−1

]. (24)

Retardadaen el tiempo (24) y manteniendo las condicio-nes de invarianza, se tiene (25):

Qk−1 =1

k − 1

[k−1∑

i=1

y2i−1 − d

k−1∑

i=1

νi−1yi−1

]. (25)

Al considerar a (25) en (24), la forma recursiva deQk sepresenta en (26):

Qk =1k

[y2

k−1 − dνk−1yk−1 + (k − 1)Qk−1

]. (26)

Entonces el parametro para el filtro esta dado por la esti-macion (27), y al sustituir (23) y (26) en (19).

ak =1/k [(ykyk−1) + (k − 1)Pk−1]

1/k[y2

k−1 − dνk−1yk−1 + (k − 1)Qk−1

]

=ykyk−1 + (k − 1)Pk−1

y2k−1 − dνk−1yk−1 + (k − 1)Qk−1

(27)

Demostracion 5.3(Teorema 2.3). De (19):

ak =Pk

Qk(28)

Retardadaen el tiempo se obtiene (29):

ak−1 =Pk−1

Qk−1,

Pk−1 = ak−1Qk−1 (29)

Sustituyendo(29) en (23) la covarianza recursiva se pre-senta en (30):

Pk =1k

[(ykyk−1) + (k − 1)ak−1Qk−1

](30)

Sustituyendo (30) en (28) el estimador tiene la forma(31):

ak =1k

[(ykyk−1) + (k − 1)ak−1Qk−1

]

Qk(31)

Finalmente,el parametro recursivo queda dado por (32):

ak =[(k − 1)Qk−1

kQk

]ak−1 +

ykyk−1

kQk(32)

Al sustituir (32) en (15) se obtiene el filtro identificador(4).

yk = akyk−1 + νk

Demostracion 5.4 (Teorema 2.4). El funcional delerror (33).

Jk = E(yk − yk)2 = Ee2k (33)

=1k

k∑

j=1

(yj − yj)2

=1k

(yk − yk)2 +

k−1∑

j=1

(yj − yj)2

=1k

(yk − yk)2 +

k − 1k − 1

k−1∑

j=1

(yj − yj)2

=1k

(yk − yk)2 + (k − 1)

1k − 1

k−1∑

j=1

(yj − yj)2

El funcional del error retardado (34):

Jk−1 =1

k − 1

k−1∑

j=1

(yj − yj)2 (34)

Jk =1k

[(yk − yk)2 + (k − 1)Jk−1

]

La forma recursiva es (35):

∴ Jk =1k

[e2k + (k − 1)Jk−1

](35)

Rev. Mex. Fıs. 57 (3) (2011) 204–210


1. J.Abonyi, Fuzzy model identification for control1st ed (Birk-hauser Boston 2003) p. 19.

2. P. Englezos y N. Kalogerakis,Applied Paramter Estimation forChemical Engineers1st ed (Marcel Dekker 2001) p. 218.

3. A. Aguado Behar y M. Martınez Iranzo,Identificacion y Con-trol Adaptivo1st ed (Pearson 2003). p. 50.

4. A. Sinha,Linear Systems: Optimal and Robust Control1st ed(CRC Press. 2007). p. 150.

5. W.S. Levine,The Control Handbook1st ed (CRC Press. 1996).p. 575.

6. R.E. Curry,Estimation and Control with Quantized Measure-ments1st ed (MIT Press. 1970). p. 14-15.

7. Hidden Markov Models Estimation and Control1st ed (Sprin-ger USA 1995). p. 19.

8. M. Godin,Identification and Estimation for Models Describedby Differential Algebraic Equations1st ed (Linkoping 2006) p.12-13.

9. T. Sodertrom y P. Stoica,System Identification1st ed (PrenticeHall 1989). p. 402.

10. J. J. Medel,Computacion y Sistemas(2004).

11. J.J. Medel y C.V. Garcıa,Rev. Mex. Fıs. (2010).

12. J.J. Medel y M.T. Zagaceta,Rev. Mex. Fıs. (2010).

13. E.M. Gutierrez-Ariaset al., Rev. Mex. Fıs. (2011).

14. I. Estrada y J.J. Medel,Filtrado adaptivo para sistemas AR de1er. orden(Tesis Maestrıa, CIC-IPN 2010).

15. S. Haykin,Kalman filtering and neural networks1st ed (JohnWiley and Sons Inc., 2001). p. 3-10.

16. E. Wellset al., J. Phys. B: At. Mol. Opt. Phys.43 (2010) DOI:10.1088/0953-4075/43/1/015101.

17. I. Rodrigueset al., J. Phys. B: At. Mol. Opt. Phys.43 (2010)DOI: 10.1088/0953-4075/43/12/125505.

18. W. Wang, J. Shen y X.X. Yi,J. Phys. B: At. Mol. Opt. Phys.42(2009) DOI: 10.1088/0953-4075/42/21/215504.

19. R.E. Kalman,A New Approach to Linear Filtering and Pre-diction Problems(Transactions of the ASME–Journal of BasicEngineering, 82, Series D, 1960). p. 35.

Rev. Mex. Fıs. 57 (3) (2011) 204–210


Ferromagnetismo en manganitas sustituidas con plata de estructura perovskita

N. Hernandez, T. Hernandez, I. Dzul y Y. PenaLaboratorio de Materiales I, Centro de Laboratorios Especializados,

Universidad Autonoma de Nuevo Leon, Facultad de Ciencias Quımicas,Ciudad Universitaria, Av. Pedro de Alba S/N, 66450, San Nicolas de los Garza, Nuevo Leon, Mexico,

e-mail: [email protected]

Recibido el 17 de diciembre de 2010; aceptado el 4 de abril de 2011

Se prepararon por primera vez una serie deoxidos mixtos de formula general Sm1−xAgxMnO3 con estructura perovskita en un intervalo decomposicion 0.1≤ x ≤ 0.5 por reaccion convencional en estado solido. Se estudia la estructura, morfologıa y magnetismo de las muestrassintetizadas. Los patrones de difraccion de rayos-X muestran que para x = 0.1 se tiene la presencia de una sola fase con estructura perovskita,mientras que para x≥ 0.2 las muestras consisten de una fase perovskita ferromagnetica y dos fases no magneticas correspondientes a Agmetalica y Ag1.8Mn8O16. El analisis de x= 0.1 por SEM revela que la morfologıa y tamano de las partıculas es aleatorio, resultado del metodode preparacion. Las muestras de Sm1−xAgxMnO3 de x entre 0.1 y 0.5 muestran, que aplicando campos de 10 Teslas,estas no alcanzan unvalor de saturacion. El comportamiento ferromagnetico de Sm1−xAgxMnO3 se ve disminuido por el aumento de la composicion de Ag.

Descriptores:Perovskita; manganita; reaccion en estado solido; ferromagnetismo.

A series of mixed oxides of general formula Sm1−xAgxMnO3 with perovskite structure were prepared by first time by conventional solid-state reaction processing. The structure, morphology and magnetism of the samples are investigated. The X-ray diffraction patterns show thatthe x = 0.1 sample is a single perovskite structure, while x≥ 0.2, samples consist of a ferromagnetic perovskite phase and two nonmagneticphases, Ag and Ag1.8Mn8O16. The ferromagnetic behavior of Sm1−xAgxMnO3 decrease with increase of Ag composition. The SEManalysis when x = 0.1 revealed that the random distribution of morphology and size of particles result of preparation method. The samples ofSm1−xAgxMnO3 by x between 0.1 and 0.5 show that applying 10 T fields these cannot reach a saturation value.

Keywords: Perovskite; manganite; solid state reaction; ferromagnetism.

PACS: 81.20.Ev; 61.66.Fn; 75.47.Lx; 75.60.Ej

1. Introduccion

Los oxidos de metales de transicion (OMT’s) con estructu-ra perovskita tienen una gran historia en la investigacion yhan sido conocidos como materiales con una interesante va-riedad de propiedades tales como electricas [1,2], magneti-cas [3-5], dielectricas [6,7] yopticas [8,9] que aun no sondel todo comprendidas. Caracterısticas como valencia mixtay cambios estructurales conducen a que estos materiales pre-senten fenomenos de ordenamiento de carga (OC), diversasestructuras magneticas, transiciones de fase metal-aislante yotros fenomenos de fundamental y potencial importancia tec-nologica [10,11].

La primera renovacion de interes cientıfica en los OMT’socurrio cuando fue descubierta la superconductividad a altatemperatura en los cupratos con estructura perovskita lami-nar. El segundo auge en investigacion fue atraıdo, principal-mente, hacia la magnetorresistencia, propiedad inicialmenteobservada en sistemas de multicapas Fe/Cr [12]. La magne-torresistencia gigante (MRG) se ha observado en solidos gra-nulares ası como enoxidos de manganeso con estructura tipoperovskita [13], el cambio en la resistividad observada en losoxidos de manganeso fue tan grande que no era comparablecon otra forma de magnetorresistencia, este efecto observa-do en las manganitas de formula Ln1−xMxMnO3 donde Lnes una tierra rara y M un cation divalente fue llamado mag-netorresistencia colosal (MRC). Una cantidad enorme de es-tudios sobre la MRC ha sido llevada a cabo en losoxidos

con estructura tipo perovskita que han sido preparados pordiferentes metodos de sıntesis en la busqueda de un modelocorrecto que explique sus propiedades magneticas, electricasy de magnetotransporte; ası como de la posible aplicacion deestos materiales como sensores magneticos. La sustitucion deAg en manganitas es causante de peculiares propiedades re-sultantes [14,15]. En la bibliografıa se pueden ubicar diversostrabajos de investigacion donde se indica que la sustitucion deAg en manganitas causa una mejora en la temperatura de Cu-rie (Tc) [3,16-18]. Tao y colaboradores [18] encontraron quemuestras policristalinas de La0.7Ag0.3MnO3 preparadas porreaccion en estado solido consistıan de una fase perovskitaferromagnetica y una fase metalica no magnetica.

En la presente investigacion se informa de una serie demuestras policristalinas Sm1−xAgxMnO3 con estructura pe-rovskita en un intervalo de composicion 0.1≤ x ≤ 0.5 sinte-tizadas por primera vez por reaccion convencional en estadosolido. El proposito de este trabajo consiste en estudiar las ca-racterısticas estructurales, morfologicas y magneticas de loscompuestos sintetizados.

2. Procedimiento experimental

Se preparo por primera vez una serie de muestras policris-talinas de Sm1−xAgxMnO3 por reaccion de estado solidocon un intervalo de sustitucion de 0.1≤x≤0.5; para ello semezclaron cantidades estequiometricas de Sm(NO3)3 6H2O

212 N. HERNANDEZ, T. HERNANDEZ, I. DZUL Y Y. PENA

FIGURA 1. Patrones de difraccion de rayos X en polvos de lasmuestras Sm1−xAgxMnO3 cuando x= 0.1–0.5.

FIGURA 2. Micrografıas SEM y microanalisis EDX deSm0.9Ag0.1MnO3.

(99.9 % Alfa Aesar), AgNO3 (99.99 % Aldrich) y Mn(NO3)24H2O (97.0 % Fluka), se molieron hasta obtener una pastahomogenea y luego se calcinaron durante 10 h a 1000C. Lospolvos obtenidos se molieron y se dispusieron para ser ca-racterizados. La estructura de los compuestos obtenidos fueidentificada por difraccion de rayos X en polvos a temperatu-ra ambiente utilizando un Difractometro Philips X Pert MPDcon radiacion de Cu-Kα en intervalo de 7 a 90 en 2θy conun tamano de paso de 0.025. Se analizo la microestructuray morfologıa para el compuesto Sm0.9Ag0.1MnO3 medianteun microscopio electronico marca JEOL modelo JSM-6510LV acoplado a un detector para espectroscopıa por dispersion

FIGURA 3. Medidas de magnetizacion de las muestrasSm1−xAgxMnO3 cuando x= 0.1, 0.2 y 0.5. Enfriadas a campoaplicado de 500 Oe (FC) y en ausencia de campo (ZFC). En elinserto se muestra el inverso de la magnetizacion vs temperatura.

FIGURA 4. Curvas de magnetizacion contra campo de las mues-tras Sm1−xAgxMnO3 cuando x= 0.1, 0.2 y 0.5, realizadas a unatemperatura de 20 K.

de energıa de rayos X (EDS). Las medidas magneticas sellevaron a cabo con un magnetometro de muestra vibrante(VSM) que cuenta con un criostato de flujo continuo de he-lio l ıquido (permite realizar mediciones desde 4.2 K hasta300 K). Se realizaron medidas de magnetizacion frente a latemperatura en un intervalo de 5–100 K para un campo apli-cado de 500 Oe. Las medidas de magnetizacion frente al cam-po se realizaron a una temperatura de 20 K y en ciclos de10 Teslas.

3. Resultados y discusion

La Fig. 1 presenta los patrones de difraccion de rayos X(DRX) de las muestras. La estructura cristalina correspon-de a una sola fase con estructura tipo perovskita ortorrombi-ca cuando x = 0.1, para las muestras con 0.5≥ x ≥ 0.2, seobservan picos de difraccion correspondientes a Ag metalicay al compuesto Ag1.8Mn8O16. Este resultado sugiere que lasolubilidad de Ag+ esta limitada en un compuesto perovski-ta monofasico [18]. Hay un lımite maximo de solubilidad deAg+ en Sm1−xAgxMnO3, que hace que la Ag+ en exceso no

Rev. Mex. Fıs. 57 (3) (2011) 211–214

FERROMAGNETISMO EN MANGANITAS SUSTITUIDAS CON PLATA DE ESTRUCTURA PEROVSKITA 213

pueda entrar en la estructura perovskita; es decir, los iones deAg+ y Mn3+/4+ residuales en Sm1−xAgxMnO3 traeran co-mo consecuencia Ag metalica y Ag1.8Mn8O16 para formaruna mezcla de fases. La ubicacion de los picos correspon-dientes a la estructura perovskita se realizo en base a las refe-rencias 13 y 18. La Fig. 2 muestra la microscopıa electronicade barrido (SEM) de la muestra cuando x = 0.1. Se puedecorroborar que la morfologıa de las partıculas es el resultadodel metodo de preparacion, no se distingue una morfologıacomun en ellas, el tamano de partıcula es aleatorio y van des-de 0.3 hasta 0.6µm (Fig. 2a); el aumento de la imagen a25,000X corrobora lo antes mencionado, se observa que laspartıculas estan sinterizadas, es evidente el contacto cohesi-vo entre ellas, por lo que la micrografıa indica la etapa finalde sinterizacion del solido policristalino en la que el creci-miento de granos implica el encogimiento de otros (Fig. 2b).El analisis de rayos X por energıa dispersiva (EDX) para lamuestra cuando x = 0.1 se presenta en la Fig. 2c. En el espec-tro se observa que no hay senal de impurezas en la muestraexaminada. Partiendo de los datos del analisis semicuantita-tivo se puede decir que los resultados para cuando x = 0.1se aproximan a la composicion esperada para la muestra. Lasmedidas del analisis por EDX se realizaron para dos regio-nes diferentes de la muestra, mostrando alta reproducibilidadde los resultados; es decir, la composicion de la muestra paracada region fue la misma.Las curvas de ZFC (Enfriamien-to a campo cero)-FC (Enfriamiento con campo aplicado) pa-ra SmxAg1−xMnO3 cuando x = 0.1, 0.2 y 0.5 se presentanen la Fig. 3. Dentro de la misma figura se muestra el grafi-co del inverso de la magnetizacion en funcion de la tempe-ratura. Ambas curvas, ZCF y FC, presentan una transicionferromagnetica-paramagnetica (FM-PM), con una disminu-cion en la temperatura de Curie para las fases analizadas amayor contenido de Ag; ademas, para el grafico del inversode la susceptibilidad cuando x = 0.1, 0.2 y 0.5 se sigue la leyde Curie-Weiss,χ = C/(T-θp), por encima de la Tc y en elintervalo de temperaturas sobre los cuales los datos fueronmedidos.

La temperatura de Curie (Tc) definida como el mınimo dedM/dT se midio a campos bajos para x= 0.1, 0.2 y 0.5 resul-tando los valores de 57±1, 56±1 y 42±1 K, respectivamente.

Las temperaturas de Curie de las muestras son muy cer-canas entre sı. La temperatura de Curie cuando x = 0.1 esligeramente mas grande que para x= 0.2 y 0.3. Dado que la fa-ses de Ag metalica y Ag1.8Mn8O16 presentes en las muestrasson no magneticas, las propiedades magneticas de las mues-tras deben tener su origen principalmente a la fase perovskitaferromagnetica. El aumento del contenido de Ag solo aumen-tara ligeramente la relacion Mn4+/Mn+3 de la fase perovski-ta, lo que aumenta la interaccion de doble canje, dando lu-gar a una pequena disminucion de Tc de las muestras. Dado

que la solubilidad de Ag+ es limitada, cuando x≥ 0.2, losiones de Ag+ excedentes ya no entran en la fase perovskitade Sm1−xAgxMnO3. Ya que la Tc es constante, simplemen-te sugiere que, cuando x≥ 0.1, las muestras tienen la mismacomposicion de la fase perovskita ferromagnetica.

En la Fig. 3, se observa una discrepancia entre las curvasZFC y FC a bajas temperaturas en todas las muestras. Estoindica una gran heterogeneidad con caracterısticas semejan-tes a un spin-glass o cluster-glass, comportamientos similaresya han sido observados [19,20].

De la evolucion de la magnetizacion frente al campoaplicado a 20 K, se observa que las curvas para el siste-ma SmxAg1−xMnO3 cuando x = 0.1, 0.2 y 0.5 no alcan-zan un valor de saturacion (ver Fig. 4), comportamiento quepresentan perovskitas con samario cuando se realizan medi-das de magnetizacion vs campo a temperaturas inferiores a115 K [21], por bibliografıa estos materiales se conocen co-mo ferromagnetos insaturados [22,23]. La magnetizacion es-pecıfica de las muestras disminuye al aumentar el contenidode Ag, lo cual se atribuye al aumento de las fases no magneti-cas de Ag metalica y Ag1.8Mn8O16 en las correspondientesfases.

4. Conclusiones

Se sintetizo por primera vez Sm1−xAgxMnO3 (x= 0.1, 0.2y 0.5); ademas, se investigo la estructura, morfologıa y mag-netismo de las muestras. La difraccion de rayos X mostro queel compuesto para x=0.1 consta de una sola fase perovski-ta, mientras que para muestras con contenido de plata mayora 0.1 se presenta una mezcla de fases; una fase perovskitaferromagnetica y otra fase de plata yoxido de manganesono magneticos. Ademas, se sabe que el material sin dopar(SmMnO3) posee un caracter antiferromagnetico, lo cual noslleva a la conclusion que la insercion de plata provoca uncambio en el estado de valencia del manganeso, que a su vezcambia las propiedades magneticas de las muestras dopadascon plata [24]. A mayores contenidos de plata se presenta unadisminucion en la propiedad ferromagnetica de los materialesdebido a la presencia de fases no magneticas en el sistema.

Agradecimientos

Los autores agradecen al Dr. Jose Manuel Barandiaran jefedel grupo de magnetismo en el departamento de Electricidady Electronica de la UPV/EHU por su colaboracion en las me-didas magneticas, a la Dra. Patricia Zambrano por su colabo-racion en la realizacion de las micrografıas. Y especialmentea la Universidad Autonoma de Nuevo Leon por prestar susinstalaciones y recursos para realizar este proyecto.

Rev. Mex. Fıs. 57 (3) (2011) 211–214

214 N. HERNANDEZ, T. HERNANDEZ, I. DZUL Y Y. PENA

1. S.M. Bukhari y J.B. Giorgi,Solid State Ionics180(2009) 198.

2. J.M. Michalik, J.M. De Teresa, J. Blasco, C. Ritter, P.A. Alga-rabel, M.R. Ibarra and Cz. Kapusta,Solid State Sci. 12 (2010)1121.

3. T. Tang, C. Tien and B.Y. Hou,Physica B. 403(2008) 2111.

4. A. Levstiket al, Solid State Commun150(2010) 1249.

5. T. Hernandez, F. Plazaola, J.M. Barandiaran y J.M. Greneche,Hyperfine Interact. 161(2005) 113.

6. C. Bharti and T.P. Sinha,Solid State Sci.12 (2010) 498.

7. R.J. Boothet al, Mater. Res. Bull. 44 (2009) 1559.

8. B. Ghebouli, M.A. Ghebouli, M. Fatmi y A. Bouhemadou,So-lid State Commun.150(2010) 1896.

9. Chang-Yeoul Kim, T. Sekino y K. Niihara,J. Sol-Gel Sci. Te-chn.55 (2010) 306.

10. M. Matsuda, S. Katano, T. Uefuji, M. Fujita y K. Yamada,Phys.Rev. B66 (2002) 172509.

11. K. Ibrahimet al, Phys. Rev. B.70 (2004) 224433.

12. M.N. Baibichet al, Phys. Rev. Lett. 61 (1988) 2472.

13. R. Von Helmolt, J. Wecker, B. Holzapfel, L. Schutz y K. Sam-wer,Phys. Rev. Lett.71 (1993) 2331.

14. S. Ifrah, A. Kaddouri, P. Gelin y D. Leonard,C. R. Chimie, 10(2007) 1216.

15. T. Tanget al, J. Magn. Magn. Mater.222(2000) 110.

16. R. Shreekalaet al, Appl. Phys. Lett.74 (1999) 2857.

17. M. Koubaa, W. Cheikhrouhou-Koubaa y A. Cheikhrouhou,J.Alloy Compd.473(2009) 5.

18. T. Taoet al, Appl. Phys. Lett.77 (2000) 723.

19. L. Pi, X.J. Xiao y Y.H. Zhang,Phys. Rev. B.62 (2000) 5667.

20. X.M. Liu, X.J. Xiao y Y.H. Zhang,Phys. Rev. B.62 (2000)15112.

21. V.V. Runov, G.P. Kopitsa, A.I. Okorokov, M.K. Runova y H.Glattli, Letters to J. Exp. Theor. Phys.4 (1999) 353.

22. A.K. Kundu, Md.M. Seikh, A. Srivastava, S. Mahajan y R.Chatterjee,Arch. Condens. Matter1-18(2011) 268827.

23. P. Raychaudhuriyet al., J. Phys.: Condens. Matter9 (1997)10919.

24. D. O’Flynn, C.V. Tomy, M.R. Lees y G. Balakrishnan,J. Phys.Conf. Ser.200(2010) 012149.

Rev. Mex. Fıs. 57 (3) (2011) 211–214


Effects on the quantum not and controlled-not gates of a modular magnetic fieldin the z-direction in a chain of nuclear spin system

G.V. LopezDepartamento de Fısica, Universidad de Guadalajara,

Blvd. Marcelino Garcıa Barragan 1421, esq. Calzada Olımpica, Guadalajara, Jalisco, Mexico,e-mail: [email protected]

M. AvilaCentro Universitario UAEM Valle de Chalco, UAEMex,

Marıa Isabel, 56615, Valle de Chalco, Estado de Mexico, Mexico,e-mail: [email protected]

Recibido el 7 de enero de 2011; aceptado el 9 de marzo de 2011

We study the simulation of a single qubit rotation and Controlled-Not gate in a solid state one-dimensional chain of nuclear spins systeminteracting weakly through an Ising type of interaction with a modular component of the magnetic field in the z-direction, characterizedby Bz(z, t) = Bo(z) cos δt. These qubits are subjected to electromagnetic pulses which determine the transition in the one or two qubitssystem. We use the fidelity parameter to determine the performance of the Not (N) gate and Controlled-Not (CNOT) gate as a function of thefrequency parameterδ. We found that for|δ| ≤ 10−3 MHz, these gates still have good fidelity.

Keywords: Modular magnetic field; quantum gates; chain of nuclear spins.

Estudiamos la simulacion de una rotacion de un solo qubit y una compuerta Control-Not en un sistema uni-dimensional compuesto poruna cadena de espınes nucleares interactuando debilmente a traves de una interaccion tipo Ising con una componente modular del campomagnetico en la direccion z, caracterizada porBz(z, t) = B0(z) cos δt. Estos qubits estan sujetos a pulsos electromagneticos los cualesdeterminan la transicion en los sistemas de uno y dos qubits. Usamos el parametro fidelidad para determinar la actuacion de la compuertaNOT (N) y Contol-not (CNOT) como una funcion del parametro frecuenciaδ. Hallamos que para| δ |≤ 10−3 MHz, estas compuertastienen buena fidelidad.

Descriptores: Campo magnetico modular; compuertas cuanticas; cadena de espines nucleares.

PACS: 03.65.-w; 03.67.-a; 03.67.Ac; 03.67.Hk

1. Introduction

Almost any quantum system with at least two quantum lev-els may be used, in principle, for quantum computation. Thisone uses qubits (quantum bits) instead of bits to process in-formation. A qubit is the superposition of any two levels ofthe system, called|0〉 and |1〉 states,Ψ = C0|0〉 + C1|1〉with |C0|2 + |C1|2 = 1. The tensorial product of L-qubitsmakes up a register of lengthL, say |x〉 = |iL−1, ..., i0〉,with ij = 0, 1, and a quantum computer with L-qubits worksin a 2L dimensional Hilbert space, where an element of thisspace is of the formΨ =

∑Cx|x〉, with

∑ |Cx|2 = 1. Anyoperation with registers is done through a unitary transfor-mation which defines a quantum gate, and one of the mostimportant result about quantum gates and quantum logicaloperation is that any quantum computation can be done interms of a single qubit unitary operation and a Controlled-Not (CNOT) gate or a single qubit unitary operation and aControlled-Controlled-Not (CCNOT) gate since CNOT andCCNOT are universal gates [1,2].

Although quantum computers of few qubits [3-9] havebeen done in operations for some time and they have beenused successfully so far, to make serious computer calcu-lations one may requires a quantum computer with at leastof 100-qubits registers, and hopefully this will be achieved

in a future not so far away. One solid state quantum com-puter model that has been explored for physical realizationand which allows to make analytical and numerical studiesof quantum gates and protocols [10] is the one made of one-dimensional chain of nuclear spins systems [11,12] insidea strong magnetic field in the z-direction (with very stronggradient in that direction) and an RF-field in the transversedirection. Such a model physically is unlikely to be con-structed, however this represents a good approximation forsimulation of quantum algorithms and gates whose respec-tive results could be applied in more realistic quantum com-puters. Furthermore, the approach relies in the universal char-acter of Quantum Mechanics. In this model, the Ising inter-action is considered among first and second neighbor spinswhich allows to implement ideally this type of computer upto 1000-qubits or more [13,14]. Among other gates and algo-rithms [15], one qubit rotation and CNOT gates were studywith this quantum computer model [16]. One of the impor-tant statement of this model is that one keeps constant themagnetic field in the z-direction at the location of each qubit.However, this statement may be not so realistic in practicefor this model or other solid state quantum computer basedon spin system with very strong axial magnetic field. Themain reason is that the strong magnetic field must be donewith superconducting magnets which, in turns, are made of

216 M. AVILA AND G.V. L OPEZ

superconducting cables, and on the wires of these cables eddycurrents are induced which can last for some time [21] andcan produce modulation on the magnetic field, and then wewonder: if there is a magnetic field modulation where thisfield change slowly with time, how these basic elements,one qubit rotation and CNOT gates, would be affected. Ofcourse, in this case, the usual analytical approximation with-out field modulation is not valid anymore, and a full numer-ical calculation is required to see the possible effect of thismodulation on 1-qubit rotations and CNOT gates. In ourstudy we will assume that the system is completely insulatedfrom the environment, such that the important decoherenceeffects [24,22,25] which normally would appear in this quan-tum system is not considered.

In this paper, we want to study this modulation effect ofthe magnetic field on the Not (particular case of 1-qubit rota-tion, or unitary operation) and CNOT quantum gates. To dothis, we will assume an additional cosine time dependenceon the normal z-direction of the magnetic field and will de-termine, using the fidelity [5,18,23] parameter, the minimumvariation in the frequency of this modulation to keep thesequantum gates elements still well defined. The paper is struc-tured as follows: In Sec. 2 the Quantum-not gate in presenceof the modular magnetic field is studied. The modularity ef-fects on the CNOT gate are determined in Sec. 3. The paperis concluded with a discussion of the obtained results.

2. Quantum Not-gate

Consider a single paramagnetic particle with spin one-half ina magnetic field given by

B = (Ba cos(ωt), Ba sin(ωt), B0(z) cos δt) (1)

where the first two components represent the RF-field, andthe third component represents the strong magnetic field inthis direction. The interaction between this particle and themagnetic field is given by the HamiltonianH = −~µ · B,where~µ is the magnetic moment of the particle which is re-lated with the nuclear spinS = ~I as~µ = γ~I, with γ thegyromagnetic ratio of the particle. So, the Hamiltonian is

H=−~µ ·B=−~ωo cos δt Iz − ~Ω2 (I+e−iωt+I−eiωt), (2)

whereωo = γB0(zo) (zo is the location of the particle) is theLarmor frequency,Ω = γBa is the Rabi frequency, andI±represents the ascent (descent) operator,I± = Ix±iIy. If |0〉and|1〉 are the two states of the spin one-half, one has that

Iz|i〉 =(−1)i

2|i〉 , I+|0〉 = |1〉 , I−|1〉 = |0〉 . (3)

The ground state of the system is represented by|0〉, whichrepresents the spin of the particle in the direction of the thirdcomponent of the magnetic field. To solve the Schrodingerequation,

i~∂|Ψ〉∂t

= H|Ψ〉 , (4)

one proposes a solution of the form

|Ψ〉 = co(t)|0〉+ c1(t)|1〉 (5)

such that|co|2 + |c1|2 = 1 at any time. Doing this, one getsthe following ordinary differential equations

ico = −ωo cos δt

2co − Ω

2c1e

iωt (6a)

andic1 = +

ωo cos δt

2c1 − Ω

2coe

−iωt . (6b)

Choosingc0(t) = eiωt/2d0(t) andc1(t) = e−iωt/2d1(t) inabove equations, one has

id0 = +ω − ωo cos δt

2d0 − Ω

2d1 (7a)

andid1 = −ω − ωo cos δt

2d1 − Ω

2d0 (7b)

which, in turns, can be written as the following uncoupledsimilar Mathieu equation [19],

d0 + α(t)d0 = 0 (8a)

where the complex functionα(t) is given by

α(t)=14

[Ω2+ω2

(1−ωo

ωcos δt

)2]

+iωoδ

2sin δt , (8b)

andd1 is obtained from (7a),

d1 =ω − ωo cos δt

Ωd0 − i

2Ω

d0 . (9)

For δ = 0 and on resonance (ω= ωo), one has thatα = Ω2/4, and the system oscillates between the states|0〉and|1〉with and angular frequency corresponding to the RabifrequencyΩ, as one expected [16]. Forδ 6= 0 the solutionof this equation is far to be trivial, and instead of solvingthe Eq. (8a), we will find directly the numerical solution ofthe system (7) with the given initial conditions. By takingω = ωo (resonant case), one expects to obtain the transition|0〉 ←→ |1〉 and to get the quantum Not-gate with a phase.

To study the performance of the quantum Not-gate as afunction of the modulation frequencyδ , we will calculate thefidelity parameter at the end of aπ-pulse and make the com-parison of the ideal wave function,Ψexpected, with the wavefunction resulting from our simulation,Ψsim.

F = 〈Ψsim|Ψexpected〉 , (10)

where |Ψsim〉 is the state obtained from numerical simula-tions, and|Ψexpected〉 is the ideal expected state. for the initialcondition|Ψo〉 = |0〉, of course, the fidelity coincide with thecoefficient|c1|2. At this point we want to stress that we de-fine |F |2 in this way due that any quantum gate or algorithmis represented by the final wave function of the quantum sys-tem. Ideally, if the quantum gate is fully realizable this wave

Rev. Mex. Fıs. 57 (3) (2011) 215–219

EFFECTSON THE QUANTUM NOT AND CONTROLLED-NOT GATES OF A MODULAR MAGNETIC FIELD IN THE Z-DIRECTION. . . 217

FIGURE 1. Quantum Not-gate: (a) Global behavior (b) Local be-havior with respect toδ.

function is represented by|Ψexpected〉. However, the non res-onant transitions and the error systems (modulation) makethat the resulting wave function of the complete simulationis given by|Ψsim〉. In this way, the fidelity is a measure ofthe good operation of gates and algorithms. On the otherhand, there is another measurement for the calculation ofthe the distance between two states and this is the so calledUhlmann-Josza fidelity [17]. However, in Ref. 18 it has beenshown that Eq. (10) is a lower bound for the Uhlmann-Joszafidelity. Such a result favors the present results. on the otherhand, it is worth to point out that in Refs. 20 and 23, a dis-cussion on the effects of the noise on the fidelity associatedto the quantum gates is given.

Figure 1a and 1b show the behavior of the fidelity andthe probabilities as a function of the parameterδ at the endof a π-pulse,τ = π/Ω. We have used the parameters (units2π MHz) Ω = 0.1 andωo = 200. The RF-frequency hasbeen chosen equal to the resonant frequencyω = ωo. As onecan see, forδ ≤ 0.2 × 10−3 MHz we can have a very welldefined quantum Not-gate.

3. Two qubits model and quantum CNOT gate

Figure 2 shows two paramagnetic nuclear particles of spinone-half (qubits) subjected to a magnetic field of Eq. (1),making and anglecos θ =

√3/2 to eliminate the dipole-

dipole interaction between them. The interaction of the mag-netic field with the qubits is carried out through the couplingwith their dipole magnetic moment~µi = γSi(i = 1, 2),whereγ is the gyromagnetic ratio andSi is the spin of theith-nucleon (S = ~I). The interaction energy is given by

H = −~µ1 ·B1 − ~µ2 ·B2 + ~JI(1)z I(2)

z = H0 − ~Ω2×

(I(1)+ e−iωt + I

(1)− eiωt + I

(2)+ e−iωt + I

(2)− eiωt

), (11)

whereJ is the coupling constant of interaction between near-est neighboring spins,Ω = γBa is the Rabi frequency,H0

is the part of Hamiltonian which is diagonal in the basis|i1io〉ij=0,1 and is given by

H0 = −~(ω1I

(1)z + ω2I

(2)z

)cos δt + ~JI(1)

z I(2)z . (12)

whereωi are the Larmor’s frequencies which are defined as

ωi = γB0(zi) i = 1, 2 (13)

with zi being the z-location of the ith-qubit. The eigenvaluesof H0 on the above basis forδ = 0 are

E00 = −12

ω1 + ω2 − 1

2J

E01 = −12

ω1 − ω2 +

12J

E10 = −12

−ω1 + ω2 +

12J

E11 = −12

−ω1 − ω2 − 1

2J

(14)

Theground state of the system is denoted by|00〉 whichcorresponds to the case of having both spins parallel in thedirection of the third component of the magnetic field. Bydoingω = (E11−E10)/~ = ω2−J/2, one gets the resonanttransition which defines the CNOT operation|10〉 ←→ |11〉with a phase involved (eiπ/2), where the left qubits is the con-trol and the right one is the target. To solve the Schrodingerequation,

i~∂|Ψ〉∂t

= H|Ψ〉 , (15)

we can assume that the wave function can be written as

Ψ=C00(t)|00〉+C01(t)|01〉+C10(t)|10〉+C11(t)|11〉 (16)

FIGURE 2. Two qubits configuration.

Rev. Mex. Fıs. 57 (3) (2011) 215–219

218 M. AVILA AND G.V. L OPEZ

such that∑ |Cij |2 = 1. Thus, we arrive to the following

system of complex-couple ordinary differential equations

iC00 = −12

((ω1 + ω2) cos δt− 1

2J

)C00

− Ω2

(C01 + C10) eiωt (17a)

iC01 = −12

((ω1 − ω2) cos δt +

12J

)C01

− Ω2

(C00e

−iωt + C11eiωt

)(17b)

iC10 = −12

((ω2 − ω1) cos δt +

12J

)C10

− Ω2

(C00e

−iωt + C11eiωt

)(17c)

iC11 = −12

(−(ω1 + ω2) cos δt− 1

2J

)C11

− Ω2

(C01 + C10) e−iωt. (17d)

Doing the transformation

C00 = eiωt/2D00, C01 = e−iωt/2D01,

C10 = e−iωt/2D10, and C11 = e−i3ωt/2D11,

one gets rid of the fast oscillations and gets the followingequations for the coefficientsD′s:

iD00 = −12

((ω1 + ω2) cos δt− 1

2J − ω

)D00

− Ω2

(D01 + D10) (18a)

iD01 = −12

((ω1 − ω2) cos δt +

12J + ω

)D01

− Ω2

(D00 + D11) (18b)

iD10 = −12

((ω2 − ω1) cos δt +

12J + ω

)D10

− Ω2

(D00 + D11) (18c)

iD11 = −12

(−(ω1 + ω2) cos δt− 1

2J + 3ω

)D11

− Ω2

(D01 + D10) . (18d)

We solve numerically these equation, and forδ = 0 andω = ω2 − J/2, a full transition will occur between thestates|10〉 and|11〉. Note that one hasCij(0) = Dij(0) and|Cij(t)|2 = |Dij(t)|2. For δ 6= 0, we consider two initiallyconditions cases: Digital case, where the initial condition isgiven by

|Ψo〉 = |10〉 , (19a)

FIGURE 3. CNOT behavior, digital case.

FIGURE 4. CNOT behavior, superposition case.

that isC00(0) = 0, C01(0) = 0, C10(0) = 1, C11(0) = 0.Superposition case, where the initial condition is

|Ψo〉=√

210|00〉+ 1√

10|01〉+

√610|10〉+ 1√

10|11〉 . (19b)

Thesetwo initial states can be gotten from our ground state|00〉 by applying it Hadamard and/or CNOT gates which isnot the point in our study. So, we are assuming that these ini-tial states are given, and we use a simple state and a superpo-sition state to cover a general situation. For our simulation,we use the following parameters (units2π MHz) Ω = 0.1,ω1 = 100, ω2 = 110, andJ = 10. The RF-frequency cho-sen is the resonant frequencyω = ω2 − J/2, and applyinga π-pulse,τ = π/Ω, we should get the respective CNOTtransition|10〉 ←→ |11〉. Figure 3 shows the behavior of theprobabilities and the fidelity as a function of the parameterδat the end of theπ-pulse and for the digital case. Figure 4shows the same as before but for the superposition case. Thiscase is more stable (the fidelity decays more slowly than thedigital case) due to non zero contribution to the termsC00 and

Rev. Mex. Fıs. 57 (3) (2011) 215–219

EFFECTSON THE QUANTUM NOT AND CONTROLLED-NOT GATES OF A MODULAR MAGNETIC FIELD IN THE Z-DIRECTION. . . 219

C01 which always contribute with the same constant proba-bility 3/10.

4. Conclusions

For a quantum computer model of a chain of qubits in a mag-netic field where its z-component varies with respect the time,we have studied the Not and Controlled-Not gate behavior asa function of the frequencyδ of variation of this component.In general, one can say that forδ ≤ 10−3 MHz these quan-tum gates remain well defined with a fidelity very close toone. This small value inδ means that it is enough to consider

up to the next to leading order term in Taylor expansion of thecosine function in Eq. (1). We have seen that the fidelity forthe superposition case is more stable than the digital case dueto the contribution to the fidelity parameter of the other nozero states involved in the dynamics. Of course, this safetyregion, defined byδ, for these quantum gates does not meansafety for a full quantum algorithm, which is under studied.

Acknowledgments

We want to thank UAEMex for the grant 2594/2008U.

1. D. Deutsch, A. Barenco, and A. Ekert,Proc. R. Soc. London A449(1995) 669.

2. S. Loyd,Phys. Rev. Lett.75 (1995) 346.

3. D. Boshi, S. Branca, F.D. Martini, L. Hardy, and S. Popescu,Phys. Rev. Lett.80 (1998) 1121.

4. C.H. Bennett and G. Brassard,Proc. IEEE international Con-ference on Computers, Systems, and Signal Processing, N.Y.(1984) 175.

5. I.L. Chuang, N.Gershenfeld, M.G. Kubinec, and D.W. Lung,Proc. R. Soc. London A454(1998) 447.

6. I.L. Chuang, N. Gershenfeld, and M.G. Kubinec,Phys. Rev.Lett.18 (1998) 3408.

7. I.L. Chuang, L.M.K. Vandersypen, X.L. Zhou, D.W. Leung,and S. Lloyd,Nature393(1998) 143.

8. P.Domokos, J.M. Raimond, M. Brune, and S. Haroche,Phys.Rev. Lett.52 (1995) 3554.

9. J.Q. You, Y. Nakamura, F.Nori,Phys. Rev. Lett.91 (2002)197902.

10. G.P. Berman, D.I. Kamenev, G.D. Doolen, G.v. Lopez, and V.I.Tsifrinovich,Contemp. Math.305(2002) 13.

11. S. Lloyd,Science261(1993) 1569.

12. G.P. Berman, G.D. Doolen, D.D. Holm, and V.I TsifrinovichPhys. Lett. A1993(1994) 444.

13. G.P. Berman, G.D. Doolen, D.I. Kamenev, G.V. Lopez, and V.I.Tsifrinovich Phys. Rev. A6106(2000) 2305.

14. G.V. Lopez, T. Gorin, and L. Lara,Int. Jou. Theo. Phys.47(2008) 1641.

15. G.V. Lopez and L. Lara,J. Phys. B: At. Mol. Phys.39 (2006)3897.

16. G.P. Berman, G.D. Doolen, G.V. Lopez, and V.I. Tsifrinovich,Phys. Rev. A61 (2000) 062305.

17. A. Uhlmann,Rep. Math. Phys.9 (1976) 273.

18. P.E.M. Mendonca, R.d.J. Napolitano, M.A. Marchiolli, C.JFoster, and Y-Ch. Liang,Phys. Rev. A78 (2008) 052330.

19. I.S. Grandshteyn and I.M. Ryzhik,Table of Integrals, Series,and Products(Academic Press, Inc., 1980, Sec. 8.6).

20. P.E.M. Mendonca, M.A. Marchiolli, and R.d.J. Napolitano,J.Phys. A. Math. Gen.38 (2005) L95.

21. E.A: Badea and O. Craiu,IEEE Transations on Magnetics33(1997) 1330.

22. P.E.M.F. Mendonca, M.A. Marchiolli, and R.d.J. Napolitano,J.Phys. A: Math. Gen.38 (2005) L95.

23. R. Cabrera, O.M. Shir, R. Wu, and H. Rabitz,J. Phys. A: Math.Theor.44 (2011) 095302.

24. I. Sinaysky, F. Petruccione, and D. Burgarth,Phys. Rev. A78(2008) 092301.

25. W.H. Zurek,Rev. Mod. Phys.75 (2003) 715.

Rev. Mex. Fıs. 57 (3) (2011) 215–219


Spherical MoS2 micro particles and their surface dispersion due toaddition of cobalt promoters

M.A. Ramosa,c, V. Correab, B. Torresc, S. Floresa, J.R. Farias Mancillaa, and R.R. Chianellic

aDepartamento de Fısica y Matematicas, UACJ-Instituto de Ingenierıa y Tecnologıa,#610 Avenida del Charro, Cuidad Juarez, 32310, Mexico,

e-mail: [email protected] Department, Metropolitan University,

San Juan, Puerto Rico.cMaterials Research and Technology Institute,

500 W. Univesity Ave, Burges Hall #303, El Paso, Texas 79902, U.S.A.


We present here a hydrothermal synthesis on spherical shape molybdenum di-sulfide (MoS2) micro-particles using thiomolybdate salts andsodium silicate as reducing agent. To understand the role of cobalt promoters on this particular MoS2 spherical shape a second reactionwas carried out using same precursors plus addition of Co following same pressure and temperature conditions. Both products (before andafter Co promoter) were characterized using scanning electron and transmission electron microscopic analysis. From SEM measurements aspherical average size diameter of∼ 2.855µm on pure MoS2 is observed and disperse surface once cobalt is incorporated into the reaction.From TEM observations an interlayer average distance of∼ 0.63 nm is obtained for MoS2-MoS2 slabs on samples with Co content. X-raydiffraction indicated principal crystallographic planes to be (002), (100), (101), (102), (103), (006), (105), and (110) for both MoS2 andMoS2/Co samples.

Keywords: Molybdenum sulfide; cobalt; X-ray; TEM.

Presentamos aquı una sıntesis quımica de micropartıculas esfericas de sulfuro de molibdeno (MoS2) utilizando sales de tiomolibdato ysilicato de sodio como agente reductivo. Para comprender el rol de los promotores de cobalto (Co) en estas particulares micro-esferasde MoS2, una segunda reaccion fue realizada utilizando los mismos precursores y la adicion de Cobalto bajo las mismas condiciones depresion y temperatura. Ambos productos (antes y despues del Co) fueron caracterizados utilizando microscopios de barrido y transmisionelectronicos (SEM y TEM). Los resultados del SEM indican un diametro promedio de∼2.855µm en esferas de puro MoS2, ası como unadispersion cuando el cobalto es incorporado en la reaccion. Observaciones en TEM indican una distancia promedio de∼0.63 nm en laslaminas de MoS2para muestras que contienen cobalto. Los resultados de rayos-X indican que los principales planos de difraccion son: (002),(100), (101), (102), (103), (006), (105), y (110) para ambas muestras las de MoS2 and MoS2/Co.

Descriptores: Sulfuro de molibdeno; cobalto; rayos-X; TEM.

PACS: 81.16.Be; 81.07.-b; 87.64.Ee; 87.64.kd

1. Introduction

Molybdenum di-sulfide has several applications, in whichmost important found are hydrodesulphurization of diben-zothiophene also known as HDS [1], one can find also pub-lished in the literature several research articles indicating itspotent properties when used as lubricant in high vacuumconditions [2]. From the literature molybdenum di-sulfideunit cell is found to have a coordinated tetragonal S-Mo-Sbonding array usually called slabs, which are stack in an ar-ray due to Van der Waals bonding fact that make it relativeeasy to glide when is used as lubricant [3]. But also it hasbeen discovered that MoS2 can have a different final arrayin the structure, depending on the synthesis method [4,5],that structure is due to the array of MoS2 slabs which couldbe spherical shape, nano-tubes, flakes and nano-rods [4-6].Also the structure/function relation tells us that properties inthose MoS2-based nanostructures will depend strongly on fi-nal shape formed after synthesis [7]. Based on that an el-egant three-dimensional MoS2 micro-flowers were recently

synthesized by heating a precursor MoO2 thin film in a vaporsulfur atmosphere and used because it appeared to be excel-lent field emitters [8], that investigation lead to a conclusionwhich is the role of sodium silicate on forming of those MoS2

flower-like structures [9]. Now, when using MoS2 as a cat-alyst on a HDS reaction, previous investigations prove thatcatalytic activity almost doubled when using a promoter suchas nickel (Ni), cobalt (Co) or tungsten (W), leading also anew phase first discovered by meaning of Mossbauer spec-troscopy [10], and later under x-ray synchrotron [11], andx-ray photoelectron spectroscopy [12], after those investiga-tions a new term was coined called “CoMoS” in the case ofCo-promoter and NiMoS in case of Ni-promoter, reason whywhen MoS2/Co are in contact its mean to say a CoMoS phasehas been formed [10]. One simple technique to understandfinal structure/function after synthesis is by observing finalproducts under microscopy techniques, which can be scan-ning electron microscope (SEM) or transmission electronmicroscopy (TEM), the comparison between techniques isthe way electrons are accelerated and also sample thickness

SPHERICALMoS2 MICRO PARTICLES AND THEIR SURFACE DISPERSION DUE TO ADDITION OF COBALT PROMOTERS 221

FIGURE1. A) SEM image of MoS2 mico-spheres with average size diameter of 2.86µm. B) Surface dispersion due to Co-promoter addition.

which in most cases thick specimens won’t let electrons topenetrate through the sample, provoking almost zero trans-mission. This research report is divided into three sectionswhich is 1) Synthesis of spherical shape MoS2 and synthesisof MoS2 with the addition Co by meaning of Hydrothermalmethods; 2) TEM and SEM observations in final productsand 3) X-ray powder diffraction for both spherical MoS2 andMoS2/Co cases.

2. Hydrothermal synthesis of spherical shapeMoS2 and MoS2/Co

The experimental procedure is following previous investi-gations found in the literature [2-4] as follows: 3 mmol ofsodium molybdate (Na2MoO4.2H2O) and 9 mmol of thioac-etamide (CH3CSNH2) were dissolved in 30 mL of deionized

water, and then 0.5 g of sodium silicate (Na2SiO3.9H2O) wasadded into the solution under violent stirring. The pH valueof the solution was adjusted to 6.0 by dropping 12 M hy-drochloric acid (HCl) solution while violent stirring. 0.50g of cobalt chloride (CoCl2) was added to the solution be-fore the hydrothermal reaction. Once cobalt is added the newsolution became magenta color-like. The resulting magentasolution was transferred to a 50 mL Teflon-lined and placedinside the hydrothermal reactor rising temperature value to220C for 24 h and allowed to cool down naturally. Theblack resulting precipitates were collected and washed firstwith 1M of Sodium Hydroxide (NaOH) solution for severaltimes to remove possible residues specially from silicic acidand later with deionized water and absolute ethanol, finallyboth products were dried separate at 60C for 6 h in openflow furnace.

The reaction could be described as follows:

1) 6CoCl26H2O + 12Na2MoO4 + Na2SiO3 + 26HCl−→ H4SiCo6Mo12O40+26NaCl + 47H2O + 6Cl2

2) CH3CSNH2 + 2H2O−→ CH3COOH + NH3+H2S

3) H4SiCo6Mo12O40 + 27H2S−→ 12Co0.5MoS2+H2SiO3 + 3H2SO4 +25H2O

FIGURE 2. Left: TEM image of spherical shape MoS2. Right: TEM image of spherical shape surface dispersion due to addition of Cobalt.

Rev. Mex. Fıs. 57 (3) (2011) 220–223

222 M.A. RAMOS, V. CORREA, B. TORRES, SERGIO FLORES, J.R. FARIAS MANCILLA, AND R.R. CHIANELLI

FIGURE 3. High Resolution TEM image of disperse sphericalshape MoS2/Co indicating a interlayer distance of 0.52 nm in MoS2

structure.

3. Characterization of MoS2 samples by XRD,SEM and TEM

Each individual MoS2 and MoS2/Co product were placedsubjected to XRD analysis using a Rigaku XRD diffractionsystem Miniflex goniometry at room temperature with a stepsize of 0.05 and Cu-kα radiation (λ – 1.5418 nm).

SEM was done on a field emission gun model Hitachi S-4800, each individual product were stick directly to a carbondouble sided tape and placed on the high vacuum chamberwith an accelerating voltage of 12 kV and current value of 8-10 A to avoid electron charge in the surface. Finally TEMwere obtained using a Hitachi with an operational voltageof 200 kV, both products were dispersed individually in iso-propanol using ultrasonic bath for 15 min, and then one dropof the resulted solution placed onto a Cu/C 200 mesh TEMgrid allowing them to dry naturally.

4. Results and discussion

SEM images presented in Fig. 1 show a spherical shape madeof MoS2 slabs stacking naturally, the particle average size di-ameter is found to be∼ 2.86µm as measured using DigitalMicrograph software. Bending on the layers is also observedon the MoS2 slabs could be due to high energetic while athydrothermal reaction. Figure 2 presents a spherical surfacedispersion in samples with cobalt content, this could be at-

tributed that Co-promoters have a principal(10−1 0) planar

nucleation site as described by theoretical and experimentalmethods found in the literature [16-18].

Results from Transmission Electron Microscope indicatefringes-like structure which is characteristic of MoS2 as de-scribed by others [15]; one can observed also the presence ofMoS2 and MoS2/Co phases, as it is presented Fig. 2. Us-ing Digital Micrograph Software (precision of 0.01 +/- nm)images of 2 nm in resolution an interlayer distance value of

FIGURE 4. XRD data of MoS2 and MoS2/Co as compared withMoS2 simulated for rhombohedra MoS2 ideal crystallographicstructure.

0.52 nm is obtained for MoS2; this value seems to be smallerwhen comparing to 0.62 nm of an ideal MoS2 crystal struc-ture [15], this latter can be attributed to engineered nano-scaled stress cause from the interaction of Cobalt and MoS2

as observed in Fig. 3.Principal planar directions obtained from XRD were

(002), (100), (101), (102), (103), (006), (105), and (110) ascompared with simulated XRD on rhombohedra MoS2 us-ing Cerius2 molecular modeling package, Fig. 4 presents ob-tained results. Broadening is observed from XRD peaks (blueline) corresponding to Co addition onto the MoS2 could bedue to engineered induced nano-scaled stress from dispers-ing MoS2 spherical surface, slabs have form nano-crystals asobserved in SEM images.

5. Conclusions

We present here a successful synthesis of spherical shapeMoS2, using a hydrothermal synthesis method. In order tostudy the effect that cobalt can cause on the structure a sec-ond reaction was carried out; addition of Co created a disper-sion of spherical MoS2 surface as confirmed by XRD, SEMand TEM characterization techniques. The dispersion effectcan be interpreted due to nucleation affinity between princi-

pal(10−1 0)-MoS2 plane to Co atoms as it is described by oth-

ers [18]. Since MoS2 when is promoted with cobalt (additionof cobalt) enhances its catalytic properties authors proposedas future work to measure the catalytic activity on MoS2/Cosamples.

Acknowledgements

The Universidad Metropolitana (UMET) Puerto Rico for re-search funds support and the Materials Research and Tech-nology Institute of University of Texas at El Paso for the us-age of their equipment and facilities. Authors thank PhD.Candidate: Sara Gaytan for TEM measurements.

Rev. Mex. Fıs. 57 (3) (2011) 220–223

SPHERICALMoS2 MICRO PARTICLES AND THEIR SURFACE DISPERSION DUE TO ADDITION OF COBALT PROMOTERS 223

1. R.R.Chianelli, G. Berhault, and B. TorresCatalysis Today147(2009) 275.

2. J. Xu, M.H. Zhu, Z.R. Zhou, Ph. Kapsa, and L. Vincent,Wear255(2003) 253.

3. N. Hiraoka,Wear249(2001) 1014.

4. D. Vollath, and D.V. Szabo, Materials Letters35 (1998) 236.

5. H.A. Therese, N. Zink, and U. Kolb,Wolfgang Tremel SolidState Sciences8 (2006) 1133.

6. N. Elizondo-Villarreal, R. Velazquez-Castillo, D.H. Galvan, A.Camacho, and M.J. Yacaman, Applied Catalysis A: General328(2007) 88.

7. V.C. Fox, N. Renevier, D.G. Teer, J. Hampshire, and V. Rigato,Surface and Coatings Technology116-119(1999) 492.

8. Lin Ma, Wei-Xiang Chen, Hui Li, Yi-Fan Zheng, and Zhu-DeXu, Materials Letters62 (2008) 797.

9. F.L. Deepak, A. Mayoral, and M.J. Yacaman,Materials Chem-istry and Physics118(2009) 392.

10. H. Topsoe, B.S. Clausen, R. Candia, C. Wivel, and S. Mørup,Journal of Catalysis68 (1981) 433.

11. G. Berhault, M. Perez De la Rosa, A. Mehta, M.J. Yacaman,and R.R. Chianelli,Applied Catalysis A: General345 (2008)80.

12. J. Iranmahboob, D.O. Hill, and H. Toghiani,Applied SurfaceScience, 185(2001) 72.

13. M.H. Siadati, G. Alonso, B. Torres, and R.R. Chianelli,AppliedCatalysis A: General305(2006) 160.

14. K. Edaet al., Journal of Solid State Chemistry179(2006) 1453.

15. N. Elizondo-Villarreal, R. Velazquez-Castillo, D.H. Galvan, A.Camacho, and M.J. Yacaman, Applied Catalysis A: General328(2007) 88.

16. L.S. Byskov, J.K. Nørskov, B.S. Clausen, and H. Topsøe,Jour-nal of Catalysis187(1999) 109.

17. M. Daage and R.R. Chianelli,Journal of Catalysis149 (1994)414.

18. J.V. Lauritsenet al., Journal of Catalysis249(2007) 220.

Rev. Mex. Fıs. 57 (3) (2011) 220–223


Estructura y morfolog ıa de pelıculas de pm-Si:H crecidas por PECVD variandola diluci on de diclorosilano con hidrogeno y la presion de trabajo

C. Alvarez-Macıasa, J. Santoyo-Salazarb, B.M. Monroya, M.F. Garcıa-Sancheza, M. Picquartc,A. Ponced, G. Contreras-Puentee y G. Santanaa,e,∗.

aInstituto de Investigaciones en Materiales, Universidad Nacional Autonoma de Mexico,Av. Universidad No. 3000, Col. Ciudad Universitaria, Apartado Postal 70-360, Coyoacan, Mexico, D.F., 04510, Mexico,

e-mail: [email protected] Departamento de Fısica, Centro de Investigacion y de Estudios Avanzados del Instituto Politecnico Nacional,

Apartado Postal 14-740, Mexico, D.F. 07000, Mexico.c Departamento de Fısica, Universidad Autonoma Metropolitana Iztapalapa,

Apartado Postal 55-534, Mexico, D.F. 09340, Mexico.dKleberg Advanced Electron Microscopy Center Department Physics & Astronomy

University of Texas at San Antonio One UTSA, San Antonio, TX. 78249.eEscuela Superior de Fısica y Matematicas del Instituto Politecnico Nacional,

Edif. 9, UPALM, Col. Lindavista, 07738.

Recibido el 20 de enero de 2011; aceptado el 1 de abril de 2011

El silicio polimorfo hidrogenado (pm-Si:H) es un material atractivo para la industria fotovoltaica al tener propiedades optoelectronicassimilares al silicio amorfo hidrogenado (a-Si:H), actualmente utilizado, y mejor estabilidad ante exposicion prolongada a la radiacion solar.En este trabajo se reportan los resultados experimentales de caracterizaciones estructurales y morfologicas de pelıculas de pm-Si:H crecidascon diferentes condiciones de deposito en terminos de tamano de grano y fraccion cristalina, obtenidos por espectroscopıa Raman, y el analisisde la rugosidad superficial por microscopıa de fuerza atomica. Las pelıculas fueron obtenidas con la tecnica PECVD bajo condiciones degeneracion de nanocristales de silicio embebidos en una matriz de silicio amorfo. El analisis estructural confirma que el tamano y la densidadde los nanocristales es sensible tanto a cambios en la presion de la camara del reactor (250 y 500 mTorr) como a variaciones en la diluciondel gas precursor de silicio (SiH2Cl2) en hidrogeno (H2) (tazas de flujo de 25, 50, 75 y 100 sccm). El analisis morfologico mostro quepara 25 sccm de H2 hay una diferencia de rugosidad RMS de 8 nm entre ambas presiones, mientras que para los demas flujos de H2 seobtuvo la misma rugosidad RMS (entre 2 y 3 nm). Los resultados muestran que mientras el tamano y la densidad de los nanocristales semodifican considerablemente con las condiciones de deposito, la rugosidad no se modifica considerablemente a diluciones de diclorosilanomayores a 25 sccm, lo que permite obtener materiales diferentes con interfaces de buena calidad para la aplicacion del material en dispositivosfotovoltaicos.

Descriptores:Silicio polimorfo hidrogenado; nanocristales; AFM; Raman.

Hydrogenated polymorphous silicon (pm-Si:H) is an attractive material for applications in the photovoltaic industry as it has optoelectronicproperties similar to amorphous silicon (a-Si:H), which is the material currently used, and better stability to prolonged exposure to solarradiation. In this work we report experimental results of structural and morphological characterization of pm-Si:H films in terms of grainsize and crystalline fraction obtained by Raman spectroscopy studies and the analysis of surface roughness with atomic force microscopy.The films were obtained by PECVD under conditions of generation of nano-crystalline silicon inclusions embedded in an amorphous siliconmatrix. Changes in pressure of the reactor chamber (250 and 500 mTorr) and variations in the dilution of the silicon precursor gas (SiH2Cl2)with hydrogen (H2) (flow rates of 25, 50, 75 and 100 sccm) modified the nanocrystals size and the crystalline fraction. The morphologicalanalysis showed that for 25 sccm of H2 there is a difference of 8 nm in the RMS roughness between both pressures, while for the other flowsof H2 the same RMS roughness (2 to 3 nm) was obtained. These results are very important because the optoelectronic properties of materialsdepend on the size and density of nanocrystals and the roughness analysis helps determine the growth conditions to produce high qualityinterfaces for application of the material in photovoltaic devices.

Keywords: Hydrogenated polymorphous silicon; nanocrystals AFM; Raman.

PACS: 68.35.Ct; 68.37.Ps; 73.61.Jc; 78.67.Bf; 81.07.Bc; 81.15.Gh

1. Introduccion

Las pelıculas delgadas de silicio amorfo hidrogenado (a-Si:H) producidas por deposito quımico en fase vapor asistidopor plasma (PECVD por sus siglas en ingles) han sido utiliza-das en la industria fotovoltaica debido a su mejor coeficientede absorcion y bajo costo de produccion, en comparacioncon los materiales cristalinos [1-3]. En general, estos mate-

riales se obtienen descomponiendo el silano (SiH4) en unplasma rico en hidrogeno. Sin embargo, a pesar de los es-fuerzos que han permitido optimizar los procesos de plasmay el diseno de las celdas solares, la degradacion inducida porla luz sigue siendo uno de los principales obstaculos de estatecnologıa [1-4]. Por otra parte, las pelıculas de silicio micro-cristalino (µc-Si) obtenidas tambien por PECVD presentanmejores propiedades de transporte de portadores de carga y

ESTRUCTURA Y MORFOLOGIA DE PELICULAS DE PM-SI:H CRECIDAS POR PECVD VARIANDO LA DILUCION. . . 225

TABLA I.

Potencia= 150 W, F(Ar) = 50 sccm, Flujo de SiH2Cl2 = 5 sccm

Tsustrato= 200C, Tiempo de crecimiento = 30 min

Presionde la Flujo de Razon Dilucion

camara [mTorr] hidrogeno [sccm] SiH2Cl2/H2 H2/(SiH2Cl2+H2)

250

25 0.2 83.3

50 0.1 90.9

75 0.067 93.8

100 0.05 95.24

500

25 0.2 83.3

50 0.1 90.9

75 0.067 93.8

100 0.05 95.24

estabilidadante exposicion prolongada a la radiacion so-lar [2-3]. Sin embargo, este material no tiene la capacidad deabsorcion del a-Si:H y su costo de produccion es mayor [2-4].Idealmente, se busca un material que tenga las propiedadesopticas del a-Si:H, con propiedades de transporte y una esta-bilidad similar a la delµc-Si. Este objetivo ha llevado a mu-chos grupos en busca de nuevas condiciones de crecimientoque favorezcan la transicion de la red amorfa hacia una masrelajada en los dominios delµc-Si [2-4]. Recientemente, porPECVD y utilizando silano (SiH4) como gas precursor de si-licio en altas diluciones en hidrogeno se han obtenido nuevosmateriales formados por una matriz de silicio amorfo con in-

clusiones nanometricas de silicio cristalino embebidas. Talesmateriales han sido denominados indistintamente como si-licio nanocristalino (nc-Si:H) o silicio polimorfo (pm-Si:H)[2-10], esteultimo debido al hecho de que consisten en unamezcla de tres fases: amorfa, nanocristalina y cristalina. Seha reportado que esta nueva estructura mejora las propieda-des de transporte con respecto al silicio amorfo, aun despuesde exposicion prolongada a la radiacion solar [4-9]. Ademas,resultados recientes han mostrado mejoras en los dispositi-vos fabricados con combinaciones de estos nuevos materia-les [3-6].

La tecnica de PECVD permite modificar la microestruc-tura de las pelıculas depositadas, y en consecuencia sus pro-piedades optoelectronicas, con la variacion de los parametrosdel proceso de deposito (la presion en la camara, la poten-cia RF, la distancia entre electrodos, la temperatura del subs-trato, etc.) [5-10]. En particular se ha visto que existe unatransicion de silicio amorfo a microcristalino con la varia-cion en la relacion H2/SiH4 [3,7,10-12]. Recientemente se haobservado que utilizando diclorosilano (SiH2Cl2) como gasprecursor de silicio, en lugar de SiH4, disminuye la cantidadde hidrogeno en la matriz y se tiene un mejor control de lamicroestructura del material obtenido [4,12-15]. Sin embar-go no hay reportes de como estos cambios en las condicionesde deposito afectan la superficie de la pelıcula crecida.

El presente trabajo muestra la influencia de la presion enla camara del reactor y la dilucion de diclorosilano en hi-drogeno en las propiedades estructurales y morfologicas delpm-Si:H. La hipotesis del trabajo es que los cambios en las

FIGURA 1. Espectros de Raman de las pelıculas delgadas crecidas.

Rev. Mex. Fıs. 57 (3) (2011) 224–231

226 C. ALVAREZ-MACIAS et al.

condiciones de deposito, tanto en la presion como en la di-solucion del gas precursor de Si (SiH2Cl2), influyen en laestructura y morfologıa de las pelıculas pm-Si:H. La micro-estructura del material determinara sus propiedades optoelec-tronicas y la morfologıa superficial es esencial para garantizarinterfaces de calidad en dispositivos fotovoltaicos a pelıculasdelgadas.

2. Procedimiento experimental

Las pelıculas delgadas de pm-Si:H se crecieron en un siste-ma de PECVD convencional con placas paralelas de 150 cm2

de superficie y 1.5 cm de separacion, activado por una senalde 13.56 MHz de RF [13-15]. Los crecimientos se realizarona presiones de 250 y 500 mTorr y con tazas de flujo masicode H2 de 25, 50, 75 y 100 sccm, manteniendo el resto de losparametros constantes. Las condiciones de crecimiento se re-sumen en la Tabla I. Las pelıculas se depositaron al mismotiempo sobre sustratos de silicio cristalino (100) de tipo n dealta resistividad y superficies epitaxiales de rugosidad RMSde 0.1 nm para los estudios de AFM, substratos de cuarzopara las caracterizaciones de Raman y cristales de cloruro desodio para microscopıa electronica de transmision de alta re-solucion. Antes del deposito, los sustratos fueron sometidosa un proceso de limpieza estandar, que en el caso del silicioincluye 1 min de ataque conacido fluorhıdrico diluido (5 %)para remover eloxido nativo de la superficie.

El analisis estructural se realizo con un equipo RamanT64000 Jobin-Yvon Horiba con triple monocromador. Lafuente de excitacion fue la lınea 514.5 nm de un laser de Ar+

Lexcel. Todas las mediciones fueron realizadas a tempera-tura ambiente en aire. Las muestras fueron irradiadas a unapotencia de 20 mW. El rango de medicion fue entre 400 y600 cm−1, el tiempo de integracion de las mediciones fue de1 min y la senal Raman fue adquirida por un detector CCDenfriado. Los estudios de microscopıa electronica de trans-mision de alta resolucion (HRTEM por sus siglas en ingles)se realizaron en un FEI-Titan de 80-300 KV. Las imagenes seobtuvieron en la condicion focal de Scherzer y se grabaronen tiempo real con una camara CCD. Se utilizo el softwareGatan Digital Micrograph para el analisis de las imagenes deHRTEM.

Para la morfologıa superficial se utilizo la tecnica de con-tacto intermitente (Tapping), en un microscopio de fuerzaatomica (AFM por sus siglas en ingles), JEOL JSPM 4210.Una vez obtenidas las imagenes se utilizo el software deanalisis WinSPM Processing, version 2.00 para determinarla rugosidad cuadrada media (RMS) de las pelıculas.

3. Resultados experimentales y discusion

3.1. Raman

En la Fig. 1 se muestran los espectros obtenidos por Ramande las pelıculas crecidas. En general, se ha demostrado quepara la mezcla de fases amorfa/cristalina de silicio, el espec-

tro de Raman consta de dos contribuciones correspondien-tes al modooptico transversal (TO): un pico ancho centra-do a 480 cm−1 (fase amorfa) y un pico estrecho centrado en520 cm−1 (fase cristalina) [16-18]. Estos numeros de onda semuestras como referencia en la Fig. 1 mediante lıneas verti-cales continuas.

En la Fig. 1 es posible verificar como la variacion de losparametros de crecimiento modifica la forma general de losespectros, lo cual esta relacionado a la existencia de diferen-tes fases en las pelıculas de silicio polimorfo. Por ejemplo, a500 mTorr y a 25 sccm de H2 se observa una distribucion an-cha centrada a 480 cm−1, lo que indica que la pelıcula es pre-dominantemente amorfa. Con una dilucion en H2 de 50 sccmla distribucion es de forma aguda y centrada en 520 cm−1,indicando que la estructura predominante en la pelıcula esla cristalina a pesar que aun se puede observar la influenciade la fase amorfa de la matriz, representada por el hombro ala izquierda del pico fundamental. Sin embargo, al continuaraumentando la dilucion en hidrogeno a 75 y 100 sccm res-pectivamente, los espectros muestran, ademas del pico agu-do cercano a la lınea de 520 cm−1 y la distribucion anchacentrada en 480 cm−1, una asimetrıa de estas distribucionesentre ambos picos. Este tipo de asimetrıas localizadas entrelos 500 y los 519 cm−1 se asocian a la existencia de unafase nanocristalina [11,16-22]. Lo anterior, de acuerdo a loreportado por la literatura, sugiere que bajo esas condicionesespecificas de crecimiento coexistan tres fases del silicio: lafase amorfa formada por la matriz, una fase nanocristalina yotra cristalina. Se puede comprobar que la presion de deposi-to juega un papel importante en la estructura de las pelıculas,pues se observan espectros de Raman con estructura predo-minantemente amorfa (50 sccm), predominantemente crista-lina (75 sccm) y la existencia de fases nanocristalinas (25 y100 sccm), cuando se disminuye la presion a 250 mTorr.

En general, para ambas presiones no se observa un corri-miento monotono del pico desde una posicion en la faseamorfa a otra en la fase cristalina con el aumento de H2,como es descrito por otros autores al usar silano como gasprecursor [3,21-23]. En nuestro caso se observa que a la pre-sion de 500 mTorr hubo un primer corrimiento brusco en elpico al aumentar el flujo de 25 a 50 sccm de H2 (es decir, laestructura de la muestra paso de una fase amorfa a otra poli-morfa). Posteriormente, al aumentar el flujo de 50 a 75 sccmde H2, ocurre un nuevo corrimiento del pico de Raman co-mo consecuencia del paso de la fase polimorfa a una fasepracticamente nanocristalina. Al seguir aumentando el flujode H2 a 75 y 100 sccm las posiciones de los picos indicanque las pelıculas presentan estructuras representadas por las3 fases. Respecto a la presion de 250 mTorr, al aumentar de25 a 50 sccm de H2 el desplazamiento en la posicion del picoinicia desde una fase predominantemente nanocristalina has-ta otra practicamente amorfa. Posteriormente, al aumentar a75 sccm se observa un cambio brusco de la posicion desdela fase amorfa hasta una cercana a la cristalina. Sin embargo,al incrementar el flujo a 100 sccm el pico regresa a la fasenanocristalina.

Rev. Mex. Fıs. 57 (3) (2011) 224–231


FIGURA 2. Posiciones de los picos intermedios obtenidos de la de-convolucion de los espectros Raman en funcion del flujo de H2.

Otra de las observaciones de la Fig. 1 es que en la mayorıade los espectros hay un ensanchamiento del pico asociado ala fase amorfa al cambiar la presion de trabajo al doble (de250 a 500 mTorr), siendo el caso de 50 sccm de H2 la unicaexcepcion. Este comportamiento indica que con el aumentode la presion aumenta el desorden en la red amorfa. A pre-siones bajas (250 mTorr), donde es menor la intensidad de lafase amorfa, se observa mas claramente la presencia de pi-cos localizados entre 480 y 520 cm−1. N. Budini y col. [23]mostraron que las posiciones del pico intermedio por deba-jo de 500 cm−1 obedecen a efectos en las fronteras de granoentre los nanocristales y la matriz amorfa, mientras que lospicos encontrados en la region entre 500-519 cm−1 se asig-nan a los modos TO generados por confinamiento cuanticoen nanocristales de diferentes tamanos [16-19].

Con el proposito de cuantificar el analisis anterior, a lascurvas de la Fig. 1 se les realizaron deconvoluciones bajo lossiguientes criterios. Dado que en el pm-Si:H se puede tener lacoexistencia de tres fases: amorfa, nanocristalina y cristalina,los espectros son ajustados por tres curvas de distribucion ti-po gaussiana correspondientes a cada una de ellas. Los picosde las fases amorfa y cristalina, se ajustan en posiciones fijasde 480 cm−1 y 520 cm−1, respectivamente [17-23]. Para to-dos los casos, la posicion del pico intermedio se dejo libre almejor ajuste entre 500 y 520 cm−1. Con este analisis es po-sible monitorear la posicion e intensidad del pico intermedio,o nanocristalino, en funcion del flujo de H2.

La Fig. 2 muestra la evolucion en la posicion del pico in-termedio con el aumento del flujo de H2 para las dos presio-nes de trabajo. En la Fig. 2 se observa que para la presion de500 mTorr hay un corrimiento monotono en la posicion delpico intermedio, el cual va aumentando de 500 a 512 cm−1

con el incremento en el flujo de H2 de 25 a 75 sccm. Estecorrimiento de la posicion del pico hacia mayores valores conel aumento en el flujo de H2 ya ha sido observado por otrosautores [21-23]. Para la presion de 250 mTorr, el corrimientoen la posicion del pico al incrementar el flujo de H2 no esmonotono, primero disminuye y despues aumenta. Por otro

FIGURA 3. Micrografıas de HRTEM mostrando la diferencia enestructuras de las pelıculas de pm-Si:H. a) Muestra crecida a250 mTorr de presion y 50 sccm de flujo de H2. b) Muestra cre-cida a 500 mTorr de presion y 50 sccm de flujo de H2

TABLA II.

Presionde Flujo de Corrimiento Tamano promedio

la camara hidrogeno de frecuencia de los

[mTorr] [sccm] ∆ω [cm−1] nanocristales [nm]

250

25 13.3 2.58

50 19.6 2.12

75 18.6 2.18

100 7.56 3.42

500

25 19.3 2.14

50 13.4 2.57

75 8.4 3.24

100 9 3.13

lado,es posible observar que para ambas presiones todos lospicos intermedios se localizan en posiciones dentro del inter-valo 500-514 cm−1. Por lo tanto, las posiciones de los picosintermedios observadas en la Fig. 2 indican la existencia deuna fase nanocristalina en las pelıculas crecidas.

A partir de la posicion del pico asociado a la fase na-nocristalina se puede obtener el tamano promedio de losnanocristales que componen dicha fase en cada una de lasmuestras. En este caso se usa el modelo de ConfinamientoCuantico que relaciona el tamano de grano,DR, con el corri-miento en frecuencia∆ν de los picos de Raman respecto a520 cm−1, y es dado por la relacion 1 [17,21,24].

DR = 2π√

2.24/∆ν (1)

La Tabla II muestra el tamano promedio de los nanocris-tales obtenido de cada muestra, calculados con la relacion 1,como una funcion del flujo de H2 para las dos presiones detrabajo utilizadas. En la Tabla II se observa que en todas lasmuestras el tamano promedio de los nanocristales no superalos 5 nm de diametro, de lo cual se infiere que las propieda-desopticas de las pelıculas deberan estar influenciadas porel comportamiento cuantico [25]. Tambien es posible obser-var que en general a mayores presiones y diluciones de H2 eltamano de los nanocristales es mayor, aunque este comporta-miento no es monotono.

Rev. Mex. Fıs. 57 (3) (2011) 224–231


FIGURA 4. Comportamiento de la fraccion volumetrica cristalinaen funcion del flujo de H2 para las dos presiones de trabajo.

La Fig. 3 corresponde a dos imagenes obtenidas por mi-croscopıa electronica de transmision de alta resolucion (HR-TEM). Con el objetivo de confirmar visualmente la diferen-cia entre la estructura predominantemente amorfa con la pre-dominantemente cristalina, las micrografıas corresponden ala muestra crecida a 250 mTorr de presion con 50 sccm deflujo de H2 y a 500 mTorr de presion con 50 sccm de flu-jo de H2, respectivamente. En la micrografıa a) es notablela estructura amorfa que compone a la matriz de la pelıculacon la existencia de algunos nanocristales. En la micrografıab) se pueden apreciar muy pocas zonas donde se observa lamatriz amorfa, pues la muestra presenta una gran densidadde nanocristales de silicio facetados, con diferentes orienta-ciones y de muy diferentes tamanos, los cuales se senala concırculos para ayuda del lector. El tamano promedio de los na-nocristales obtenidos por esta tecnica es de 2.2 nm, lo queconfirma los resultados obtenidos por Raman. Esta corres-pondencia entre ambas tecnicas ya ha sido reportada en estosmateriales [9,18].

De los resultados de Raman se puede calcular tambienla fraccion cristalina. Para esto se recurre a la relacion 2[16,17,19,21,23], la cual proporciona este parametro a partirde lasareas bajo las curvas gaussianas de las deconvolucionesrealizadas a los espectros Raman de cada muestra.

xc =IN + IC

IN + IC + yIA(2)

donde IA, IN , e IC , representan elarea de la gaussianacorrespondiente a la fase amorfa, nanocristalina y cristalinarespectivamente. El factory = 0.9 considera la mayor excita-cion fononica de la seccion transversal de la matriz de silicioamorfo, con respecto al cristalino [16].

La Fig. 4 muestra la fraccion cristalinaXc, obtenida de larelacion 2, en funcion del flujo de H2 para las dos presionesde trabajo. En la Fig. 4 se observa que la fraccion cristalinade las pelıculas crecidas oscila con el aumento del flujo deH2 para las dos presiones de trabajo. Por ejemplo, para laspelıculas crecidas a 250 mTorr primero hay una disminucionde la fraccion cristalina del 26 al 5 %, al cambiar el flujo de

H2 de 25 a 50 sccm. Al aumentar el flujo a 75 sccm hay unincremento de la fraccion cristalina hasta 70 %, pero al in-crementar el flujo de H2 a 100 sccm vuelve a disminuir a un60 %. Por su parte, en las pelıculas crecidas a 500 mTorr, pri-mero hay un aumento muy pronunciado de la fraccion cris-talina, el cual va del 21 al 82 % al aumentar el flujo de 25 a50 sccm. Posteriormente, disminuye al 50 % cuando se incre-menta el flujo a 75 sccm y finalmente hay un leve aumentoal 55 % cuando se aumenta el flujo a 100 sccm. Este tipode comportamiento oscilante ha sido observado anteriormen-te en otros parametros medidos en pelıculas crecidas a partirdel diclorosilano como gas precursor de silicio [13-15]. Di-chos comportamientos se relacionan a la presencia del cloro ysu efecto en la quımica del plasma, en donde bajo determina-das condiciones puede crearse un plasma atacante y en otrasocasiones se favorece la creacion de una gran cantidad deespecies condensables, contribuyendo al aumento de la cris-talinidad y al espesor de la pelıcula [14,15].

Es importante notar que para una misma dilucion de di-clorosilano en hidrogeno e identicos parametros de creci-miento, con el solo hecho de duplicar la presion dentro dela camara de crecimiento la fraccion cristalina se puede in-crementar hasta en un 80 %. Esto permite controlar la micro-estructura, y de este modo las propiedades optoelectronicasdel material, solamente modificando alguno de los parame-tros de crecimiento. Pero para la aplicacion de este materialen dispositivos optoelectronicas otro parametro importante esla morfologıa superficial de las pelıculas crecidas.

4. Analisis por microscopıa de fuerza atomica(AFM)

La morfologıa superficial de las pelıculas crecidas dependede las condiciones de deposito y su control es importante,ya que las inhomogeneidades degradan las propiedades elec-tronicas del material [12,26-28]. Ademas, la alta rugosidadsuperficial afecta la utilizacion de estas pelıculas como in-terfaces en dispositivos, aspecto importante en la fabricacionde estructuras fotovoltaicas [27]. La obtencion de las pelıcu-las bajo las diferentes condiciones de crecimiento, como elaumento de la presion en la camara de reaccion y la alta di-lucion de diclorosilano en H2, pueden variar la rugosidad su-perficial debido al aumento en la cantidad de especies queinteraccionan con la superficie de crecimiento y su concen-tracion [26]. Por lo tanto, es importante observar como se veafectada la morfologıa superficial en funcion de los cambiosen los parametros de crecimiento utilizados en el presentetrabajo.

La Fig. 5 muestra las imagenes bi- y tridimensionales delas superficies observadas con AFM en modotappingde lasmuestras crecidas bajo los parametros mostrados en la Ta-bla I. Las micrografıas bidimensionales cubren unarea de250 nm2.

En la Fig. 5 se observa como se modifica la morfologıasuperficial de las pelıculas con la variacion de los parametros

Rev. Mex. Fıs. 57 (3) (2011) 224–231


FIGURA 5. Micrografıas superficiales de las pelıculas de pm-Si:H crecidas.

de crecimiento. Por ejemplo, para las muestras crecidas a unapresion de 250 mTorr la morfologıa es mas de tipo granularque las crecidas a 500 mTorr. Estoultimo se refiere a que a250 mTorr la superficie de las pelıculas esta formada por es-tructuras alargadas de mas de 200 nm compuestas de granospequenos con tamanos promedio de 10 nm. Estas estructurasvan cambiando su forma, ası como la prominencia de los gra-nos con la variacion del flujo de H2. Por ejemplo, al aumentarel flujo de H2 de 25 a 50 sccm hay cambios en la cantidad re-lativa y la forma de las estructuras grandes con un aumento enel tamano y la prominencia de los granos. Al seguir aumen-tando los flujos de H2 a 75 y 100 sccm se dejan de observarlos granos y solo hay cambios en la forma y tamano de las es-tructuras grandes. A diferencia de las muestras anteriores, enlas muestras crecidas a 500 mTorr no se observan estructurascon granos pequenos. En este caso se observa que al aumen-tar el flujo de H2 de 25 a 50 y hasta 75 sccm, las estructurasgrandes van cambiando de tamano, pero en general son massuaves. Sin embargo, al aumentar el flujo de H2 a 100 sccm,la morfologıa de las estructuras exhibe superficies porosas.Estructuras similares fueron observadas por AFM por P. Dut-ta y Col. [26] en muestras de pm-Si:H crecida por PECVDutilizando silano como gas precursor. Dutta y Col. observa-ron que las estructuras grandes son separadas en los granos

pequenos al realizarle a sus muestras un tratamiento termicoen ambiente de hidrogeno.

La Fig. 6 muestra las rugosidades RMS en funcion delaumento de flujo de H2. El area de analisis fue de 1µm2. Dela Fig. 6 el aspecto mas notable es la diferencia en las rugo-sidades RMS de las pelıculas crecidas cuando el flujo de H2

es de 25 sccm para ambas presiones de trabajo: Se observaque para la presion de 250 mTorr la rugosidad RMS da unvalor menor a un nanometro, mientras que para la presion de500 mTorr la rugosidad esta por arriba de 8 nanometros.

Otro resultado importante que se observa en la Fig. 6 esla interseccion de los intervalos de error de la rugosidad RMScuando los flujos son mayores a 25 sccm de H2 para ambaspresiones de trabajo. Se observa que aunque hay una ligeratendencia a que la rugosidad RMS crezca para 250 mTorr ydecrezca para 500 mTorr, al incrementar el flujo de H2 de50 y hasta 100 sccm las rugosidades de las pelıculas caenen el mismo intervalo, aproximadamente entre 2 y 3 nm. Elanalisis anterior permite observar que bajo estas condicionesde crecimiento, es posible obtener pelıculas de pm-Si:H conpropiedades estructurales muy diferentes pero caracterısticasmorfologicas parecidas.

La variacion de la rugosidad RMS con los cambios en elflujo de H2 y la presion de la camara se basa en la competen-

Rev. Mex. Fıs. 57 (3) (2011) 224–231


FIGURA 6. Rugosidad RMS superficial en funcion del flujo de H2para las dos presiones de trabajo.

cia entre estas condiciones de crecimiento, las cuales actuancomo un regulador tanto de incorporacion de cloro como dehidrogeno metaestable [14,15]. Por un lado, con el incremen-to de la presion en la camara se disminuye el camino libremedio de las especies en fase gaseosa y se incrementan lasreacciones en el plasma estimulando el crecimiento de losnanocristales [19-21]. Por el otro lado, al aumentar el flujode H2, se extrae mas cloro, ya que los precursores SiHxClyson inestables y se combinan con el hidrogeno atomico enforma gaseosa produciendo HCl [12]. Esto provoca una dis-minucion de enlaces Si-Cl permitiendo la formacion de mascentros de nucleacion sobre la superficie. [14,15]. De estamanera, cuando el flujo es de 25 sccm y la presion es ma-yor (500 mTorr) (en la Fig. 6) se tienen muchas reaccionesen el plasma y un exceso de Cl incorporado a la red. Esto ha-ce que baje la estabilidad quımica de las pelıculas, haciendoque la rugosidad RMS se vea incrementada en casi un ordende magnitud respecto a las pelıculas crecidas a presion masbaja (250 mTorr), en donde hay una menor cantidad de reac-ciones en el plasma [12,14]. Por su parte, la alta rugosidad esconformada por una gran cantidad de defectos estructurales ycon ello enlaces meta-estables. Esta situacion induce mayordegradacion de las pelıculas ante exposiciones prolongadas aradiacion solar [26,28].

5. Conclusiones

Se han mostrado resultados de caracterizacion estructural ob-tenidos en pelıculas de pm-Si crecidas con diferentes con-diciones de deposito. El analisis por espectroscopıa Ramanmostro la existencia de tres fases: amorfa, nanocristalina ycristalina. Se pudo constatar que tanto el aumento del flujode H2 como el aumento de la presion de trabajo modifican eltamano promedio de los nanocristales embebidos en la ma-triz amorfa, ası como la fraccion cristalina de las pelıculascrecidas y la relacion entre las tres fases presentes.

El analisis de AFM mostro que para un flujo de 25 sccmde H2 la rugosidad RMS se aumenta en casi un orden de mag-nitud al aumentar al doble la presion de trabajo (de 250 a500 mTorr). Mientras que para las demas flujos de H2, mayo-res a 50 sccm y para ambas presiones, las rugosidades caenen el mismo intervalo y menor a 5 nm. Este resultado sugiereque estas muestras deberan presentar una menor degradacionen comparacion con las crecidas a 25 sccm de flujo de H2.

Por tanto es posible obtener pelıculas con diferentes ca-racterısticas estructurales y rugosidades parecidas. Las dife-rentes caracterısticas estructurales dan la posibilidad de quelas pelıculas tengan diferentes propiedades optoelectronicas.Mientras que las rugosidades parecidas y menores de 5 nmpermiten que estas pelıculas puedan ser utilizadas como in-terfaces en dispositivos fotovoltaicos.

6. Agradecimientos

Agradecemos el apoyo financiero parcial de esta obraa DGAPA-UNAM PAPIIT: Proyectos IN116409-2 yIN115909-2, a CONACyT Mexico con el proyecto 48970y al ICyTDF con el proyecto PIFUTP08-143. Los autoresagradecen al Dr. JC Alonso y el Dr. A. Ortiz por el uso delaboratorio durante la preparacion de muestras y al Dr. A.Remolina Millan por su contribucion en la preparacion de lasmuestras. C.Alvarez y M. F. Garcıa agradecen los apoyosfinancieros por CONACyT a traves de la beca doctoral CVU165872 y de la beca posdoctoral ICyTDF, respectivamente.

1. K. Ohkawaet al., Sol. Energy Mater. Sol. Cells66 (2001) 297.

2. P. Roca i Cabarrocas, A. Fontcuberta i Morral y Y. Poissant,Thin Solid Films403–404(2002) 39.

3. S. Guha, J. Yang, A. Banerjee, B. Yan y K. Lord,Sol. EnergyMater. Sol. Cells78 (2003) 329.

4. C.R. Wronskiet al., NCPV and Solar Program Review MeetingNREL/CD-520-33586(2003) 789.

5. W. Bronneret al., J. Non-Cryst. Solids299–302(2002) 551.

6. C. Longeaudet al., J. Non-Cryst. Solids227-230(1998) 96.

7. P. Roca I. Cobarrocas,J. Non-Cryst. Solids266-269(2000) 31.

8. P. Kleideraet al., Thin Solid Films403 –404(2002) 188.

9. R. Butteet al., J. Non-Cryst. Solids266-269(2000) 263.

10. S. Thomson, C.R. Perrey y T.J. Belich,J. Appl. Phys.97 (2005)034310.

11. B.P. Swain,S. Afr. J. Sci.105(2009) 77.

12. H. Matsui, T. Saito, J.K. Saha y H. Shirai,J. Non-Cryst. Solids354(2008) 2483.

13. G. Santana, J. Fandino, A. Ortiz y J.C. Alonso,J. Non-Cryst.Solids351(2005) 922.

Rev. Mex. Fıs. 57 (3) (2011) 224–231


14. B.M. Monroyet al., J. Lumin.121(2006) 349.

15. A. Remolinaet al., Nanotechnology20 (2009) 245604.

16. G. Viera, S. Huet y L. Boufendi,J. App. Phys.90 (2001) 4175.

17. J. Gopeet al., J. Non-Cryst. Solids355(2009) 2228.

18. J.H. Shim, S. Im y N.-H. Cho,Appl. Surf. Sci.234(2004) 268.

19. S. Halindintwaliet al., S. Afr. J. Sci.105(2009) 290.

20. S. Lebib y P. Roca i Cabarrocas,J. App. Phys.97 (2005)104334.

21. S. Ray, S. Mukhopadhyay y T. Jana,Sol. Energy Mater. Sol.Cells90 (2006) 631.

22. A. Mossad Ali,J. Lumin.126(2007) 3126.

23. N. Budini, P.A. Rinaldi, J.A. Schmidt, R.D. Arce y R.H. Bui-trago,Thin Solid Films518(2010) 5349.

24. J. Sancho-Parramon, D. Gracin, M. Modreanu y A. Gajovic,Sol. Energy Mater. Sol. Cells93 (2009) 1768.

25. P.F. Trwoga, A.J. Kenyon y C.W. Pitt,J. Appl. Phys.83 (1998)3789.

26. P. Dutta, S. Paul, D. Galipeau y V. Bommisetty,Thin SolidFilms 518(2010) 6811.

27. P. Roca i Cabarrocas, N Chabane, A V Kharchenko, y S Tcha-karov,Plasma Phys. Control. Fusion46 (2004) B235.

28. D. Franta, I. Ohlıdal, P. Klapetek, y P. Roca i Cabarrocas,ThinSolid Films455–456(2004) 399.

Rev. Mex. Fıs. 57 (3) (2011) 224–231


Single-electron Faraday generator

G. GonzalezDepartamento de Matematicas y Fısica, Instituto Tecnologico y de Estudios Superiores de Occidente,

Periferico Sur Manuel Gomez Morın 8585 Tlaquepaque, Jal., 45604, Mexico,e-mail: [email protected]


In this paper I study the posibility of inducing a single-electron current by rotating a non-magnetic conducting rod with a small tunneljunction immerse in a uniform magnetic field perpendicular to the plane of motion. I show first, by using a thermodynamic approach, theconditions needed to pump electrons around the mechanical device in the Coulomb blockade regime. I then use a density matrix approach todescribe the dynamics of the single-charge transport including many-body effects. The theory shows that it is possible to have single-electrontunneling (SET) oscillations at low temperatures by satisfying conditions similar to the Coulomb blockade systems.

Keywords: Coulomb blockade; SET oscillations; electromotive force; tunneling Hamiltonian.

Se demuestra la posibilidad de inducir una corriente de tunelaje electronico al rotar una varilla conductora con una junta tunel inmersa enun campo magnetico homogeneo perpendicular al plano de rotacion. Utilizando la energıa libre de Helmholtz se obtienen las condicionesnecesarias para generar una fuerza electromotriz (fem) que induce la corriente de tunelaje electronico en el regimen de bloqueo Coulombiano.Utilizando la matriz de densidad se demuestra que es posible tener oscilaciones en el transporte electronico de carga.

Descriptores: Bloqueo Coulombiano; oscilaciones SET; fuerza electromotriz; Hamiltoniano de tunelaje.

PACS: 73.23.Hk

1. Introduction

The field of single-electronics started when new effects dueto the quantization of charge in ultrasmall tunnel junctions,both in the superconducting and the normal state, where pre-dicted by Averin and Likharev [1]. The theory of Averin andLikharev considers a tunnel junction which is biased by anexternally fixed currenti and whose voltageV is measuredby a very high impedance voltmeter with metallic shunt con-ductanceGS . A tunnel junction consists of two conductingelectrodes separated by a thin layer of insulating material andis characterized by its capacitanceC and tunnel resistanceRT . When a voltage is applied to the small capacitance tun-nel junction, the charge will flow continuously through theconductor and it will accumulate on the surface of the elec-trode against the insulating layer of the junction (the adjacentelectrode will have equal but opposite surface charge). Onthe other hand, the insulating layer is thin enough for elec-trons to tunnel through. The state of the junction is describedby the surface chargeQ (which is a continuous variable) andthe electronsn that have tunnel through the insulating layer(which is a discrete variable). Averin and Likharev predictedthat if the chargeQ at the junction is greater than|e|/2, anelectron can tunnel through the junction in a particular di-rection, subtracting|e| from Q. Likewise, if Q is less than−|e|/2, an electron can tunnel through the junction in oppo-site direction, adding|e| to Q. But if Q is less than|e|/2and greater than−|e|/2, tunneling in any direction wouldincrease the energy of the system, hence tunneling will notoccur. This suppression of tunneling is known today as theCoulomb blockade [2]. The physical origin of the Coulombblockade of single-electron tunneling (SET) is quite simple.In a current-biased junction, each tunneling event leads to

a change of the electrostatic energy of the system given by∆E = e(Q ± e/2)/C. If the initial chargeQ is within therange−|e|/2 < Q < |e|/2, the energy change∆E is positiveand at low temperatures tunneling events are impossible. Onthe other hand, if|Q| > |e|/2, tunneling is possible becausethis process reduces the electrostatic energy. An interestingprediction of Averin and Likharev was the SET oscillationsin the voltage across the junction [1,2]. Due to the Coulombblockade of tunneling the chargeQ on the junction accumu-lates until its threshold valuee/2 is reached and then the junc-tion is recharged by the externally fixed current. This wholeprocess repeats itself with a frequencyν = i/e [3].

The purpose of this article is to show that there are SEToscillations without an external applied current in a small tun-nel junction. In this case, the SET oscillations are driven bythe Lorentz force due to the rotation of a conductor with asmall tunnel junction and an applied external magnetic field.In addition, this mechanically driven device is proposed asa transducer of motion into electricity. The paper is orga-nized as follows. First I will start by giving a thermodynamicformulation of the problem and the basic relations of the the-ory. Then I will analyze the system using a density matrixapproach to include many-body effects and show the SET os-cillations in the system. The conclusions are summarized inthe last section.

2. Thermodynamic Formulation

Consider the Faraday generator shown in Fig. (1), where aconducting rod of length rotates with constant angular ve-locity ω in a constant magnetic field that is perpendicular tothe plane of motion. The rod completes the circuit, with one

SINGLE-ELECTRON FARADAY GENERATOR 233

FIGURE 1. Rotational motion of the conducting rod in theXYplane when the switch is (a) off or (b) on. The crosses indicate thata uniform magnetic field is pointing into the page.

contact point on one end of the rod and the other on the circu-lar rim. The circuit containing the galvanometer is completedby an open wire structure with a switch.

For the case when the switch is open there will be no elec-tromotive force, however we know from elementary electro-dynamic courses that charge will pile up at the two ends ofthe rod and will produce an electric field that balances theLorentz force felt by the moving charges inside the conduc-tor (See Fig. 1a).

The free energy of the system is given byF = E−~µ · ~B,whereE is the total energy of the rotating rod,~µ is the mag-netic dipole moment of the circulating charge at the end ofthe rod and~B is the constant magnetic field perpendicular tothe plane of motion. Taking the magnetic field as~B = −Bz,the free energy is given by

F = Ein +Iω2

2− QBω`2

2, (1)

whereEin is the internal energy of the rod,I denotes the mo-ment of inertia for the rod with respect to the axis of rotationand we have taken the current asi = Qω/2π. It should beremembered that rotation in general changes the distributionof mass in the body, and so the moment of inertia and internalenergy of the body are in general functions ofω [4].

When the switch is turned on there is an electromotiveforce E in the circuit and hence charge will be circulatingaround the circuit. The free energy of the system in this caseis given by

F = Ein +Iω2

2− (Q−∆Q)Bω`2

2+ E∆Q, (2)

where∆Q is the amount of charge circulating around the cir-cuit andE∆Q is the work done by the system. The change infree energy is obtained by subtracting Eq. (2) from Eq. (1),which gives us

∆F =(

Bω`2

2+ E

)∆Q, (3)

if the system is in thermodynamic equilibrium then∆F = 0and we obtainE = −Bω`2/2, as we know from elementaryelectrodynamic courses [5]. Note thatE < 0.

Now I will consider the case when there is a tunnel junc-tion at positionr with thicknessδr in the conducting rod asshown in Fig. 2.

FIGURE 2. Schematic diagram for the rotation of the conductingrod with a tunnel junction of thicknessδr and a uniform magneticfield pointing into the page. Note how the charge accumulates onthe surface of the electrode against the insulating layer. For thiscase current will only flow when a tunnel event occurs.

For this case, even if the switch is turned on there willbe no current flowing through the circuit because the chargeQ will accumulate on the surface of the electrode againstthe insulating layer as depicted in Fig. 2. Nevertheless,quantum mechanically speaking there is a probability forthe charge to tunnel through the junction. The free en-ergy of the system before quantum tunneling is given byF = E − ~µ · ~B − ~µ1 · ~B − ~µ2 · ~B, where~µ and ~µ1(2)

corresponds to the magnetic dipole moment of the charge ac-cumulated at the end of the rod and against both sides of theinsulating layer, andE = Iω2/2 + Q2/2C, whereC is thecapacitance of the tunnel junction. Therefore, the free energyof the system before quantum tunneling is given by

F =Iω2

2+

Q2

2C−QBω`2

2−QBωr2

2+

QBω

2(r+δr)2. (4)

Equation (4) can be written in the following form

F =I ′

2ω2 +

Q′2

2C, (5)

where

I ′ = I − C

[B`2

2− Bδr(2r + δr)

2

]2

Q′ = Q− Cω

[B`2

2− Bδr(2r + δr)

2

]. (6)

Whenthere is quantum tunneling there is a change in chargeby ±|e| and an electromotive forceE in the circuit whichcauses a change in the free energy given by

F =I ′

2ω2 +

(Q′ ± |e|)22C

+ E|e|. (7)

The change in free energy is obtained by subtracting Eq. (7)from Eq. (6), which gives us

∆F =|e|22C

(1± 2Q′

|e|)

+ |e|E . (8)

Rev. Mex. Fıs. 57 (3) (2011) 232–235

234 G. GONZALEZ

AssumingE < 0 from our previous result, we see fromEq. (8) that a tunnel event becomes energetically favorableand a currenti flows throughout the circuit whenQ′>(|e|/2).When a tunnel event takes place the chargeQ′ will changeby −|e| and after a time|e|/i the rotational motion of theconducting rod immerse in the constant magnetic field willrecharged the junction and another tunnel event will takeplace. As a result the tunneling events will occur periodicallywith frequencyν = i/|e|.

For the particular case in which the tunnel junction ofthicknessδ lies exactly in the middle of the conducting rod,i.e. r = `/2 − δ/2 andr + δr = `/2 + δ/2, then a tunnelevent becomes energetically favorable when

Q >|e|2

+CωB`(`− δ)

2. (9)

Equation(9) can be expressed in terms of the electrostaticvoltage in the following way

V >|e|2C

+Bω`(`− δ)

2. (10)

Sincethe maximum voltage allowed for the system is givenby Bω`2/2, then Eq. (10) reads

Bω`2

2> V >

|e|2C

+Bω`(`− δ)

2. (11)

Equation (11) gives us the following restrictionBω`δ>|e|/C. Using typical values of the tunnel junc-tion capacitanceC ≈ 3 × 10−15 F and tunnel thicknessδ = 10 A [2], we needBω` > 105 V/m, to satisfy the re-striction condition. If we have a magnetic field ofB = 1 Tand` = 1 cm, then we will need a rotational frequency ofaroundνr ≥ 10 MHz. The current delivered by the single-electron Faraday generator for this rotational frequency isaroundi ≈ 1 pA.

3. Many-body effects

To study in more detail the dynamics of the charge trans-port we need the total Hamiltonian of the system depictedin Fig. (2), which is given by:

H = F(Q′) + HT + H1 + H2 + HS − iΦ. (12)

The first term in Eq. (12) represents the free energy of thesystem which is given by Eq. (5). The charge operatorQ′

can be expressed via Fermion operators

Q′ = −e

2

(∑

k1

c†k1ck1 −

∑

k2

c†k2ck2

)−Q0, (13)

wherec†k and ck are the electron creation and annihilationoperators and

Q0 = Cω

[B`2

2− Bδr(2r + δr)

2

]

is a constant term. The second term in Eq. (12) representsthe tunneling Hamiltonian which is given by

HT =∑

k1,k2

Tk1k2c†k2

ck1 +∑

k1,k2

Tk2k1c†k1

ck2 , (14)

where the summation is carried out over all statesk withinthe electrodes 1 and 2 andTk1k2 is the tunneling rate acrossthe junction. The HamiltoniansH1, H2 andHS describe theenergy of the internal degrees of freedomk1, k2 andkS ofthe two electrodes of the junction and of the shuntGS , re-spectively. The last term in Eq. (12) is the operator of themagnetic flux defined as

Φ = −∫Edt, (15)

whereE is the electromotive force (emf) around the circuit.Note that Eq. (12) corresponds exactly to the basic Hamilto-nian given by Averin and Likharev. The only new feature isthe shift of the charge operatorQ′ = Q − Q0 arising fromthe magnetic dipole interaction between the spinning chargeand the external magnetic field. One should remember thatQ′ is essentially the surface charge of the junction formingthe capacitor and is a quasi-continuous variable as expressedby Eq. (13), where the tunneling, (as a discrete process), canchangeQ′ only by integer numbers and the induced currentflowing through the Faraday generator, (as a continuous pro-cess), changesQ′ by any amount on the scale ofe dependingon the rotational frequency. Therefore, the carrier motion inthe usual conductor is virtually not quantized, at least on thescale ofe, and what is quantized is the charge on the junction.

Restricting ourselves to the case when the current throughthe junction and shunt are not too large we can consider themas perturbations and one can write an explicit time evolutionequation for the density matrix to describe the equation ofmotion governing the charge on the junction. Following thepioneering work of Averin and Likharev [1], and assumingthatGS , GT ¿ 4e2/h, the resulting master equation is givenby

∂f

∂t= FT + FS (16)

wheref(Q′, t) is the classical probability distribution andFT

andFS are contributions due to the tunneling and shunt cur-rent, respectively, and are given by

FT (Q′)=Γ+(Q′−e)f(Q′−e, t)+Γ−(Q′+e)

×f(Q′+e, t)−[Γ+(Q′)+Γ−(Q′)]f(Q′, t) (17)

FS=GS

C

∂

∂Q′

(CkBT

∂f

∂Q′+fQ′)

, (18)

where Γ± are the tunneling rates for forward (plus sign)and backward (minus sign) single electron tunneling over thejunction and can be expressed as

Γ±(Q′) =1ei(∆F±/e)

[1− exp

(−∆F±

kBT

)]−1

, (19)

Rev. Mex. Fıs. 57 (3) (2011) 232–235

SINGLE-ELECTRON FARADAY GENERATOR 235

wherei is the d.c. induced current and

∆F± = ± e

C(e/2±Q′) . (20)

Note that the master equation given in Eq. (16) correspondsexactly to the master equation given in Averin and Likharevpaper [1], the only difference is that there is no external cur-rent or bias.

If one looks at the regime−e/2 < Q′ < e/2, the tunnel-ing contributions may be neglected,i.e. FT = 0, and Eq. (16)can be expressed as

∂f

∂t= GSkBT

∂2

∂Q′2

[f +

V (t)kBT

], (21)

whereV (t) is the time dependent voltage across the tunneljunction and is given by [6]

V (t) =1C

∫Q′f(Q′, t)dQ′. (22)

Making the substitutionF (Q′, t) = f(Q′, t) + V (t)/kBT ,we end up with a reaction-diffusion equation given by

∂F

∂t= GSkBT

∂2F

∂Q′2 +1

kBT

dV

dt. (23)

For the system with constant electrostatic potentialV0, i.e. notunneling, the solution to Eq. (23) is [7]

F =1√texp

[ −Q′2

4kBTGSt

]+

V0

kBT, (24)

wherethe first term in Eq. (24) is the solution to the mas-ter equation forf(Q′, t), which properly normalized can beexpressed as

f(Q′, t) =1√

4πkBTGStexp

[ −Q′2

4kBTGSt

]. (25)

Equation (25) represents a Gaussian probability packet de-scribing the distribution of chargeQ that will move due to the

Lorentz force untilQ′ > e/2, at this point the rateΓ−(Q′)becomes nonvanishing and this leads to a rapid decay of thepacket,i.e. f(Q′, t = tT ) = 0, wheretT = C/GT is the timewhen a tunneling event occurs. In this regime the tunnelingevent leads to a noticeable change in the voltage across thejunction, i.e. ∆V = ±e/C, which is described in Eq. (23)by the last term,i.e.

∂F

∂t= GSkBT

∂2F

∂Q′2 ±e

CkBTδ(t− tT ), (26)

The general solution to Eq. (26) is just a shift of the solutiongiven in Eq. (25) to the starting timet = tT , i.e.

f(Q′, t) =Θ(t− tT )√

4πkBTGS(t− tT )

× exp[ −Q′2

4kBTGS(t− tT )

], (27)

whereΘ(t − tT ) is the Heaviside funtion. It is evident fromEq. (27) that the whole process of the Gaussian packet forma-tion repeats periodically with every tunneling event showingthe periodic SET oscillations in the voltage across the junc-tion.

4. Conclusions

The main contribution of this article is to show that there canbe SET oscillations across a tunnel junction without an exter-nally applied current source. This result is in contrast to thesystem analyzed by Averin and Likharev where an externalfixed current is always present. The thermodynamic and mi-croscopic derivation shows how a single-electron current canbe induced by rotating a conducting rod with a small tunneljunction in the presence of a uniform magnetic field perpen-dicular to the plane of motion. An estimate of the currentdelivered by the single-electron Faraday generator for rota-tional frequencies ofνr ≈ 10 MHz is i ≈ 1 pA. Thus, thisdevice could serve as a fundamental standard of d.c. current.

1. D.V. Averin and K.K. Likharev,J. of Low Temperature Physics62 (1986).

2. H. Grabert and M.H. Devoret,NATO ASI SeriesVol. 294(Plenum Press, New York, 1992). Chap. 2.

3. S. SelberherrComputational Microelectronics(Springer, Wien,NewYork, 2001) Chap. 2.

4. L.D. Landau and E.M. Lifshitz,Statistical PhysicsThird Edi-tion Part 1, (Butterworth-Heinemann 1980).

5. H.C. Ohanian and J.T. Market,Physics for Engineers and Sci-entistsVol. 2, Third Edition, (McGrawHill, page 1005 in span-ish).

6. D.K. Ferry and S.M. Goodnick,Transport in Nanostructures(Cambridge University Press, 1997).

7. I. Sneddon,Elements of Partial Differential Equations(Inter-national Student Edition, 1957).

Rev. Mex. Fıs. 57 (3) (2011) 232–235


γ-Fe2O3/ZnO composite particles prepared by a two step chemical soft method

S. Lopez-Romero and F. Morales LealInstituto de Investigaciones en Materiales, Universidad Nacional Autonoma de Mexico,

Apartado Postal 70-360, Mexico D.F., 04510, Mexico,e-mails: [email protected]; [email protected]

Recibido el 3 de febrero de 2011; aceptado el 8 de abril de 2011

Composite iron oxide-Zinc oxide (γ- Fe2O3/ZnO) was synthesized by two step method: in the first one stepγ-Fe2O3 particles were obtainedby a cetyltrimethylammonium hydroxide (CTAOH) assisted hydrothermal method at low temperature (60C). In the second step, theγ-Fe2O3

particles were included in the ZnO particles synthesis, which were obtained by a hexamethylenetetramine (HMTA) assisted hydrothermalmethod at low temperature (90C). SEM study of the samples revealed that theγ-Fe2O3/ZnO composites present a compact morphology. Theγ-Fe2O3 and ZnO phases were identified by XRD, energy dispersive X-ray analysis (EDX) and analysis of the IR spectrum. The compositeexhibit the characteristic emissions of ZnO under UV radiation and ferromagnetic behavior ofγ-Fe2O3 under an external magnetic field.

Keywords: Iron oxides; zinc oxides; hydrothermal method; composites.

En este trabajo el compuesto formado entre eloxido de hierro y eloxido de zinc (γ-Fe2O3/ZnO) fue sintetizado por el metodo sol-gel endos pasos: en el primer paso partıculas deloxido de hierroγ-Fe2O3 fueron obtenidas por el metodo hydrotermico asistido con hidroxidode cetiltrimetilamonio (CTAOH) a baja temperatura (60C). En el segundo paso, las partıculas deloxido γ-Fe2O3 fueron incluidas en lasıntesis de las partıculas de ZnO las cuales fueron obtenidas por el metodo hydrotermico asistido con hexametilentetramina HMTA) a unatemperatura de 90C. El analisis SEM de las muestra revelo que las partıculas del compuestoγ-Fe2O3/ZnO presentan una morfologıaredonda compacta. Las fasesγ-Fe2O3 y ZnO fueron identificadas por XRD, analisis de rayos x con energıa dispersiva (EDX), y analisis deespectro IR. El compuesto exibe emision caracterıstica del ZnO bajo radiacion UV y comportamiento ferromagnetico delγ-Fe2O3 bajo uncampo magnetico.

Descriptores: Oxido de hierro;oxido de zinc; metodo hidrotermico; materiales compuestos.

PACS: 81.05Mh

1. Introduction

The synthesis of composite particles consisting of magneticcores and luminescent cells such asγ-Fe2O3/ZnO has gainedacceptance in few years due to its magnetic, photolumi-nescence and catalytic properties [1]. Also, as active ele-ment in gas sensors [2]. This type of composite particleshas biological and biomedical potential applications such asdetection of cancer cells, bacteria and viruses, and mag-netic separation [3]. Recently Ruipeng Fuet al. [4] pre-paredγ-Fe2O3/ZnO composite particles via a simple solutionmethod, and investigated its morphology; indicating that theγ-Fe2O3/ZnO composite particles are of typical sphere-likemorphology with diameter in the range of 300-400 nm. Also,the γ-Fe2O3/ZnO composites exhibit magnetic response toan external magnetic field and efficient characteristic emis-sions of ZnO under UV excitation. Dong Kee Yiet al. [5],by a two step synthesis obtained silica-coated nanocompos-ites of γ-Fe2O3 magnetic nanoparticles (MPs) and CdSephotoluminescent quantum dots (QDs), its analysis showedthat the presence of CdSe increased the effective magneticanisotropy of theγ-Fe2O3 containing particles, indicatingthat the QDs were closely connected to MPs, concludingthat the SiO2/MPs-QDs nanocomposite particles preservedthe magnetic properties ofγ-Fe2O3 and optical properties ofCdSe QDs. Hongwei Gu [1] reported on a one-pot chemicalsynthesis method for generating heterodimers of nanoparti-cles by taking advantage of lattice Mismatch and selective an-

nealing at a relatively low temperature. They deposited amor-phous CdS on the surface of FePt nanoparticles to form ametastable core-cell structure, in which the CdS transformedinto a crystalline state upon heating. Moreover, the core cellstructures which have sizes less tan 10 nm, also exhibit bothsuperparamagnetism and florescence.

In this work, we report on a two –step surfactant as-sisted hydrothermal method to synthesizeγ-Fe2O3/ZnOcomposite particles. In the first stepγ-Fe2O3 nanoparti-cles were synthesized by a sol-gel method using ferric nitrate[Fe(NO3)3] as sourse material and cetyltrimetilamonium hy-droxide (CTAOH) as surfactant and catalyst. In the secondstep zinc nitrate [Zn(NO3)2.6H2O] and hexamethylenete-tramine, also called methenamina were taken as starting ma-terials.

2. Experimental details

2.1. Materials

All chemicals (Sigma Aldrich) used in this study were of an-alytical reagent grade and used without further purification.

2.2. Preparation ofγ-Fe2O3 nanoparticles

The process to obtain a colloidal solution to synthesizer theγ-Fe2O3 nanoparticles is based on the sol-gel method [6].Briefly: ferric nitrate (Fe(NO)3) and cetyltrimethylammo-

γ-Fe2O3/ZnOCOMPOSITE PARTICLES PREPARED BY A TWO STEP CHEMICAL SOFT METHOD 237

FIGURE 1. Scanning electron microscopy image of the seed mag-neticγ-Fe2O3 particles.

nioum hydroxide (CTAOH) were taken as starting materials.Initially 0.01 M of ferric nitrate was dissolved in ethyl alcoholand magnetically stirred at 60C for 1 h. Then, cetyltrimethy-lammonioun hydroxide was mixed into solution with a molarratio of Fe+3/CTAOH of 1/1.6 and later refluxed at 60C for2 h. After few minutes the solution became dark-red, whichindicated theγ-Fe2O3 nanoparticles formation. Finally, afterrefluxing for 2 h the solution was cooled to room tempera-ture, and a black-red precipitate was obtained upon addingalcohol and centrifuging.

2.3. Preparation ofγ-Fe2O3/ZnO composites particles

The synthesis method to obtain theγ-Fe2O3/ZnO compos-ites particles is as follow: Zinc nitrate (Zn(NO3)2.6H2O)was taken as the source material and hexamethylenetetramine((CH3)6N4) as the surfactant and catalyst. The precursor wasprepared by dissolving 3.0 g of Zinc nitrate and 8.0 g ofmethenamina in deionized water under vigorous stirring at30C for 1h to form a 0.01 M equimolar solution. Then, theγ-Fe2O3 nanoparticles prepared in the before step, were de-posited into the solution and heated at 90C for 10 h. It wasobserved that a pink powder precipitated at the flask bottom,indicating theγ-Fe2O3/ZnO composites particles formation.

3. Characterization

The x-ray diffraction patterns ofγ-Fe2O3/ZnO compositeswere obtained with an x-ray diffractometer (SIEMENS D500) using the CuKα (λ=1.5406A) radiation, with a scan-ning speed of 1 per min, 35 kV and 30 mA, the scan-ning in 2θ was from 2 to 70. Morphology of the com-posite samples and analysis X-ray EDX were obtained us-ing a JEM600-Lv scanning electron microscopy. IR spectraof theγ-Fe2O3/ZnO composite particles in KBr pellets were

FIGURE 2. SEM image ofγ-Fe2O3/ZnO composite particles.

recorded in the range of 4000-500 cm−1 on an AmericanNicolet AVANTAR 380 FT-IR spectrometer. The photo-luminescence spectrum was obtained using a fluorescencespectrometer (Hitachi.H-4600). Magnetization (M) measure-ments were performed with a SQUID based magnetometer(Quantum Design). The temperature (T) range of measure-ments was between room temperature and 2 K, and the rangeof applied magnetic field (H) in these experiments was be-tween± 10 kOe. M(T) measurements were performed in thezero field cooling (ZFC) and field cooling (FC) mode.


Figure 1 is a SEM image of the magneticγ-Fe2O3 nanopar-ticles prepared by the first step of the method. The nanopar-ticles are non uniform monodispersed and have a grain sizein the range of 100-200 nm, and can be considered as sphere-like particles. Figure 2 is a SEM image of as-synthesizedγ-Fe2O3/ZnO composite particles; they present a compactmorphology and are bigger than theγ-Fe2O3 seed particles.The grain size of the composite particles is in the range of400-600 nm.

Consistent with the reaction mechanisms proposed inhydrothermal syntheses [7], which has established that thegrowth unit are the anions [Zn(OH)4]2−, the following chem-ical reactions are proposed:

(CH3)6N4, 90C

Zn(NO3)2 + 2H2O→ Zn(OH)2 + 2HNO3 (1)

Zn(OH)2 ↔ Zn2+ + 2HO− (2)

Zn2+ + 2HO− ↔ ZnO+ H2O (3)

Zn(OH)2 + 2OH− ↔ [Zn(OH)4]2− (4)

Rev. Mex. Fıs. 57 (3) (2011) 236–240

238 S.LOPEZ-ROMERO AND F. MORALES LEAL

FIGURE3. Composition and EDX of theγ-Fe2O3/ZnO composite.

[Zn(OH)2] ↔ Zn2+ + 4OH− (5)

Zn2+ + 4OH− ↔ ZnO+ H2O (6)

In reaction (1) Zn2+ ions are first combined with OH−

radicals in the aqueous solution to form a Zn(OH)2 colloidvia the reaction Zn2+ + 2OH− → Zn(OH)2 . Later, in the hy-drothermal process a portion of the Zn(OH)2 is separated intoZn2+ ions and OH− radicals according to reaction (2). Then,ZnO nuclei are formed according to the reaction (3), whenthe concentration of Zn2+ ions and OH− radicals reaches thesupersaturation degree of ZnO. Finally, the growth units of[Zn(OH)4]2− radicals are replicated by means of reaction (4).Continuing thus with the dissolution-nucleation cycle accord-ing to reactions (5) and (6), respectively.

Figure 3 shows the composition and the image of EDXof theγ-Fe2O3/ZnO composite. Zn, Fe and O elements be-longedγ-Fe2O3/ZnO composite.

The Fe to Zn ratio initial in wt% was of 0.8 and the sameratio taken from the EDX measurement was of 0.15.

Figure 4 shows the XRD patterns of theγ-Fe2O3/ZnOcomposites obtained at room temperature. The diffractionpeaks of ZnO are marked by diamonds, corresponding tothe hexagonal phase with lattice constants;a = 3.25A andc=5.21 A (JCPD file No 36-1451). The lattice parametersobtained for this samples area=3.24880A andc=5.20540A.The peaks marked by dots were indexed to the cubic phaseof γ-Fe2O3 with a=4.822A (JCPDS file No 39-1346). Thelattice parameter obtained in this case isa=4.682A, conclud-ing that the x-ray pattern showed that in the compositeγ-Fe2O3/ZnO particles each compound conserve its crystallinestructures.

Figure 5 shows the FT IR spectrum of theγ-Fe2O3/ZnOcomposite. The broad band centered at 3433 cm−1 can beassigned to the stretching vibration of OH−. The peaks at1747 and 1702 cm−1 are due at CO− groups. The peak at1643 cm−1 is assigned to COOFe groups. The absorptionsat 1389, 1149 and 1064 cm−1 can be ascribed to the COO-

FIGURE4. XRD pattern ofγ-Fe2O3/ZnO composite particles. Thediffraction peaks of ZnO are indicated by diamonds and the diffrac-tions peaks ofγ-Fe2O3 are indicated by dots.

FIGURE 5. FT IR spectrum ofγ-Fe2O3/ZnO composite particles.

FIGURE 6. Spectrum of photoluminescence ofγ-Fe2O3/ZnO com-posite particles under 358 nm excitation.

groups. The peak at 684 cm−1 is a characteristic absorptionof γ-Fe2O3. The peak at 554 cm−1 is a characteristic absorp-

Rev. Mex. Fıs. 57 (3) (2011) 236–240

γ-Fe2O3/ZnOCOMPOSITE PARTICLES PREPARED BY A TWO STEP CHEMICAL SOFT METHOD 239

tion of ZnO. All these IR spectrum data are in consistencewith the reported values [8-11].

Figure 6 shows the room temperature photoluminescencespectrum of theγ-Fe2O3/ZnO composite particles, obtainedwith a laser radiation of 358 nm. It is well knew that the emis-sion spectra of a ZnO particle consist of two emission bands:the first is an exciton emission band in the UV region witha maximum in∼ 380 nm, this band is caused by the radia-tive annihilation of excitons, and the second, an intense bandin the green region of the visible spectrum with a maximumin between 500 and 530 nm due to the radiative recombina-tion of an electron from a level in the conduction band andan deeply trapped hole in the bulk (V¨0) of an ZnO parti-cle [12,13]. In Fig. 6 it is observed that the exciton emissionoccurs near ultraviolet (380-400 nm) and the visible emissionoccur in the green light (480 nm) regions. In addition to, thefigure 6 shows that the intensity of the emission visible is lessthan the intensity of exciton emission. This is due accord-ing to Van Dijken [13] to an effect of particle size in whichto point out that as the size of the ZnO particles increasesthe intensity of visible emission decreases and the intensityof exciton emission increases. In our case, from Fig. 2 theγ-Fe2O3/ZnO particles size (∼300nm) is bigger than Dijkenparticles decreasing thus the intensity of visible emission re-spect to that of exciton emission. The occurrence of the exci-ton and visible peaks in photoluminescence spectrum confirmthe presence of ZnO in theγ-Fe2O3/ZnO composite.

The temperature dependence of magnetic behavior M(T)of theγ-Fe2O3/ZnO composite shows a difference betweenthe ZFC and FC measurements as is shown in Fig. 7. TheZFC and FC curves show an irreversible temperature regionthat finish about a maximum in the ZFC curve. This max-imum is about 12 K and it is independent of magnetic fieldas shown by the curves measured a magnetic field of 10 Oeand 1 kOe. Similar M(T) behavior has been observed inγ-Fe2O3 nanoparticles with size around 6.4 nm [14], however,

FIGURE 7. Magnetization as a function of temperature of theγ-Fe2O3/ZnO composite. Measurements correspond to the samesample under magnetic field of 10 Oe and 1 kOe, in the ZFC andFC mode. The arrows indicate the maximum in the ZFC curve thatoccurs at T=12 K.

FIGURE 8. Magnetization as a function of applied magnetic fieldof a sample ofγ-Fe2O3/ZnO composite, the curves were measuredat 2 K (circles) and 12 K (squares). Note that the curve measured at12 K does not show hysteresis. The inset shows the coercive fieldas a function of temperature.

the temperature where the maximum occurs in the ZFC curvedepends from applied magnetic field. This temperature hasbeen associated whit the blocking temperature and changefrom 101 K, when the applied field was a 200 Oe, to 68 Kunder a field of 50 Oe [15].

Magnetization as a function of magnetic field at low tem-peratures shows hysteretic behavior. Figure 8 shows M(H)cycles obtained at 5 and 12 K. The hysteretic behavior dis-appears when the temperature is increased, it occurs at about12 K. From this temperature and above M(H) shows a su-perparamagnetic behavior, as can be observed in the curvemeasured at 12 K (Fig. 8). Measurements of M(H) at tem-peratures 12 K and above do not show hysteresis. This behav-ior is in agreement with the temperature at which the M(T)curve shows difference between the ZFC and FC measure-ments. The inset in Fig. 8 shows the temperature behaviorof the coercive field (HC) from 2 K until 100 K. The co-ercive field observed at 5 K is about 300 Oe, higher than20 Oe reported for nanoparticles of 6.8 nm [15] and lowerthan 1 kOe reported for nanoparticles of 8.5 nm [14]. It isnoted that HC(T) decreases almost exponentially. This be-havior is in disagreement to the HC ∝ T1/2 behavior ob-served inγ-Fe2O3 nanoparticles with diameter distributionfrom 3 to 5 nm [16,17]. As mentioned above, theγ-Fe2O3

nanoparticles size is between 100-200 nm, it could be inter-esting to know if the exponential behavior of HC(T) is relatedto the particles size.

5. Conclusions

In this work were synthesized compactγ-Fe2O3/ZnO com-posite particles by a two step assisted-hydrothermal method.From studious made by SEM microscopy, the morphologyof the composite particles resulted be compact with a grainsize in the range of 400-600 nm. The phases and purity ofthe γ-Fe2O3/ZnO composite particles were investigated by

Rev. Mex. Fıs. 57 (3) (2011) 236–240

240 S.LOPEZ-ROMERO AND F. MORALES LEAL

XRD , EDX and IR analysis and reveled that theγ-Fe2O3andthe ZnO nanoparticles conserved its phases respective. Fromphotoluminescence and magnetic behavior M(T) studious oftheγ-Fe2O/ZnO composite particles can be conclude that theZnO andγ-Fe2O3 particles preserved the unique optic prop-erty of the first and magnetic property of the second.

Acknowledgements

The author gratefully acknowledge useful contribution withL. Banos and Adriana Tejeda for his support in carrying outx-ray study, Omar Novelo for their support in electron mi-croscopy characterization and Miguel A. Canseco for his sup-port in carrying out IR study.

1. HongweiGu et al, J. Am. Chem. Soc.126(2004) 5664.

2. Teixeiraet al, J. Bra. Chem. Soc.11 (2000) 27

3. Deseng Wanget al., Nano Lett.3 (2004) 409.

4. Ruipeng Fuet al., Mater. Lett.62 (2008) 4066.

5. Don Kee Yi,et al., J. Am. Chem. Soc.127(2005) 499.

6. Chin-Hsien Hung and Wha-Tzong Wang,Mater. Chem. Phys.82 (2003) 705.

7. Yongshen Zanget al., Appl. Sur. Sci.242(2005) 212.

8. Xiang Yang Kong and Zhong Lin Wang,Nano Lett.3 (2003)1625.

9. Sihui Zhanget al, Coll. Interface Sci308(2007) 265.

10. M. Andres-Verges and C.J. Cerna,J. Mater Sci. Let.7 (1988)970.

11. Zhijon Jing,J. Mater. Lett.60 (2006) 2217.

12. Svetozar Musicet al, J. Alloys Comp.448(20008) 277.

13. S. Monticoneet al, J. Phys. Chem. B.102(1998) 2854.

14. A. Van Dijkenet al, J. Luminescence87-89(2000) 454.

15. P. Duitaet al, Phys. Rev. B70 (2004) 174428.

16. K. Jhon,et al, J. Appl. Phys.73 (1993) 5109.

17. Jong-Ryul Jeong,et al, Phys. Status Solidi B241(2004) 1593.

Rev. Mex. Fıs. 57 (3) (2011) 236–240


Room temperature thermal properties of Pb(Fe1/2Nb1/2)O3ferroelectromagnetic ceramics

R. Fonta,b, E. Marınc, A. Lara-Bernalc, O. Raymonda, A. Calderonc, J. Portellesa,b, and J.M. SiqueirosaaCentro de Nanociencias y Nanotecnologıa, Universidad Nacional Autonoma de Mexico,

Km 107 Carretera Tijuana-Ensenada, Ensenada, 22860, Baja California, Mexico.bFacultad de Fısica, Universidad de La Habana,

San Lazaro y L, Vedado, La Habana, 10400, Cuba.cCentro de Investigacion en Ciencia Aplicada y Tecnologıa Avanzada-Instituto Politecnico Nacional,

Legaria 694, Col. Irrigacion, 11500, Mexico, D.F., Mexico.


The thermal properties of ferroelectromagnetic Pb(Fe1/2Nb1/2)O3 ceramics obtained using the conventional ceramic method at differentsintering temperatures between 850C and 1000C by stoichiometric mixing of the corresponding oxides and using different kinds ofprecursors, have been investigated for the first time. In particular the thermal conductivity was calculated from the measured values ofthermal diffusivity and specific (volume) heat capacity using the photoacoustic technique and the temperature relaxation method, respectively.Whereas no influence of the kind of precursor used for sample preparation on the thermal conductivity (k) was observed, we have found thatthe value ofk depends on sintering temperature and has a maximum for samples synthesized at 900C, regardless of the use of precursors ornot. This paper shows that such feature is determined by the competition of the thermal conductivity mechanisms inside the grains and thoseat the grain boundaries in combination with the morphologic features.

Keywords: Thermal diffusivity; thermal conductivity; specific heat capacity; ferroelectromagnetic; multiferroic; lead iron niobate (PFN).

Se determinan por primera vez las propiedades termicas de la ceramica ferroelectromagnetica Pb(Fe1/2Nb1/2)O3 obtenida utilizando elmetodo ceramico convencional a diferentes temperaturas de sinterizacion, entre 850C y 1000C, mezclando estequiometricamente losoxidos correspondientes y utilizando diferentes tipos de precursores. En particular, la conductividad termica se calculo a partir de los valoresmedidos de la difusividad termica y el calor especıfico (a volumen constante) utilizando la tecnica fotoacustica y el metodo de relajaciontermica, respectivamente. Aunque no se observo ninguna influencia del tipo de precursor utilizado en la preparacion de la muestra sobrela conductividad termica (k), se encontro que el valor dek depende de la temperatura de sinterizacion y alcanza su valor maximo para lasmuestras sinterizadas a 900C, sin importar si se utilizaron precursores o no. Este artıculo muestra que esta propiedad esta determinadapor una combinacion de los mecanismos de conduccion dentro del grano con los que suceden en la frontera de grano y las caracterısticasmorfologicas.

Descriptores: Difusividad termica; conductividad termica; calor especıfico; ferroelectromagnetico; multiferroico; niobato de plomo y nio-bio (PFN).

PACS: 77.84.Dy; 78.20.Hp; 81.40.Rs

1. Introduction

Multiferroics are materials of great scientific and technolog-ical interest because they show coexisting features such asferro- or antiferromagnetism, ferroelectricity, or ferroelastic-ity/shape memory effects [1-5]. In the last few years the fer-roelectric and antiferromagnetic single phase compound withperovskite structure lead iron niobate [Pb(Fe1/2Nb1/2)O3,PFN for short], in which electric and magnetic order coex-ist, has been widely investigated [6-16]. Although detailed,systematic studies about their physical properties have beenreported before, to the best of the authors’ knowledge, re-ports on their room temperature thermal properties are notyet available in the literature. Thus the main objective of thiswork is the evaluation of the thermal properties of PFN ce-ramics and we focus our attention on the influence of the sin-tering temperature and the effects of different kinds of precur-sors used for sample preparation on the thermal conductivity.

2. Experimental

The PFN ceramic samples studied here were obtained us-ing the conventional ceramic method. One group of sam-ples, labeled PFNoxides, were produced from calcined pow-ders synthesized by solid state reaction of reagent grade iron,niobium and Pb oxides (Fe2O3, Nb2O5 and PbO respec-tively) in stoichiometric amounts. The other two groupswere obtained by the B-site precursor method using the fer-rocolumbite (FeNbO4) as precursor, which has been recog-nized as an effective way to obtain a pure perovskite phase inlead-based systems. They were labeled PFN1075 (when themonoclinic phase FeNbO4 precursor synthesized at 1075Cwas used) and PFN1200 (the orthorhombic phase FeNbO4

precursor synthesized at 1200C was used). The sampleswere sintered at different temperatures between 850C and1000C. More details about the fabrication process are givenin previous reports [7,9].

242 R. FONT, E. MARIN, A. LARA-BERNAL, O. RAYMOND, A. CALDERON, J. PORTELLES, AND J.M. SIQUEIROS

FIGURE 1. Average density values in percent of the theoreticalvalue of the PFN1075 samples as function of the sintering temper-ature.

Thermal conductivity (k) was calculated from the mea-sured values of thermal diffusivity (α), and specific (volume)heat capacity (C), by means of the well known relationship

k = αC. (1)

The thermal diffusivity was obtained using the photoa-coustic (PA) technique in its open cell configuration [17-19],in which the sample is mounted directly on top of an electretmicrophone where the PA signal is detected, while the spe-cific heat capacity was measured using a calorimetric tech-nique, namely the temperature relaxation method [20,21].This methodology has been successfully employed beforeby several authors for the characterization of a great vari-ety of materials, such as semiconductors [22], foods [23],wood [24], among others, as well as other kinds of ferro-electric ceramics [25,26], therefore we will not give specificdetails here.


The thermal diffusivity, studied as a function of the sinteringtemperatures for three groups of samples, exhibits a maxi-mum value for the samples sintered at 900C. On the otherhand, we observe that the specific heat capacity, measuredwith the technique described in detail by E. Marın and H.Valiente [27], does not show an appreciable variation withrespect to the sintering temperature, having an average valueof (1.50±0.10)×106J/cm3K. This constant value ofC can beexplained by taking into account its definition as the productof the density (ρ) and the specific heat (c). The specific heatis defined as the change in the internal energy per unit of tem-perature change; thus, if the density of a solid increases (ordecreases) the solid can store less (or more) energy. There-fore, as the density increases with sintering temperature (seeFig. 1), the specific heat must decrease and then the productC=ρc stays constant for all sintering temperature values. Asa consequence ofC being constant, and according to Eq. (1),the behavior of the thermal conductivity is similar to that of

FIGURE 2. Thermal conductivity as function of the sintering tem-perature. Solid line links the average value for each temperature.

TABLE I. Values of average density and grain size, Fe3+/Fe2+ con-centration ratio, maximum dielectric constant and diffuseness ex-ponent for the PFN ceramics obtained by three different kinds ofprecursor at a sintering temperature of 900C.

Sample

PFNoxides PFN1075 PFN1200

Average

Density (g/cm3)* 7.43(88) 6.77(80) 6.54(77)

Average

Grain Size (µm) 1.9 1.8 2.1

Fe3+/Fe2+

Concentration Ratio 0.66 0.57 0.56

Dielectric Constant

Maximum, ε′ ωim♣ 695 641 6790

Diffuseness Exponentα 1.65 1.86 1.73

∗Values in parenthesis are average densities in percent of the theoretical

value. ♣Values at 100 kHz

the diffusivity, as is illustrated in Fig. 2, showing a maximumat 900C.

As can be seen in Fig. 2, the samples prepared with differ-ent precursors exhibit very similar values of thermal conduc-tivity, regardless of the temperature at which they were sin-tered, although their dielectric and ferroelectric properties aresignificantly different according to previous studies. This isan understandable result if we take into account the structuraland morphological characteristics of the different samples.As an example, let us examine the samples sintered at 900C,previously studied in detail [9-11,14,16], which are thosethat also show the lowest dispersion in thek values com-pared with those obtained at different temperatures. Also,these samples obtained with different precursors, showed nostructural differences, a uniform grain distribution, the sameferroelectric-paraelectric transition temperature of 110C,

Rev. Mex. Fıs. 57 (3) (2011) 241–244

ROOM TEMPERATURE THERMAL PROPERTIES OF Pb(Fe1/2Nb1/2)O3 FERROELECTROMAGNETIC CERAMICS 243

FIGURE 3. Dielectric loss as function of the sintering temperaturefor PFNoxides and PFN1200 samples at a frequency of 100 Hz.

and a normal diffuse phase transition (non–relaxor behav-ior), as summarized in Table I. We also see that among thesamples sintered at 900C, PFN1200 shows the best electri-cal behavior, meaning higher values of dielectric permittivityand remanent polarization, and the lowest dielectric losses[9-11,16]. Thus the following question arises: What is theorigin of the maximum of thermal conductivity at the sinter-ing temperature of 900C, at which also a lower dispersionin thek values is observed?

Making a simple inspection of Fig. 2, we note that fortemperatures below 900C the thermal conductivity is higherthan for temperatures above that value. From this fact, one islead to believe, according to the above discussion, that possi-ble causes of this behavior may be attributed to the relation-ship between the morphologic features of the samples andthe strong competition between the thermal transport mech-anisms inside the grains and those at the grain boundaries,as functions of the sintering temperature. It is known that,in general, near room temperature, the two principal thermalconduction mechanisms in solids are due to conduction elec-trons (especially in metals) and lattice vibration (phonons)(especially in insulators, such as ceramics) [28]. From pre-vious dc and ac conductivity analyses in the samples herestudied in the temperature range from room temperature to300C [7,9-11], we found that all observed conduction andrelaxation processes were assumed to take place inside thegrains, conditioned by the grain and ferroelectric domainsizes, the degree of deformation of the lattice and the crystal-lites, as well as the potential barriers in the grain boundariesdue to space charge accumulated at these interfaces. Fourcontributions of electrical conduction mechanisms were iden-tified in the studied temperature range for the three kinds ofsamples:n and/orp type hopping charge, small polarons,oxygen vacancy conduction, and the intrinsic ionic conduc-tion which occurs at higher temperatures. Then − p hop-ping charge and small polarons mechanisms were associatedto Fe2+ presence and the other two to oxygen vacancies, bothgenerated during the sintering processes. The small polarons,

defined as a coupling of lattice deformations withn or p-type charges (electron-phonon interaction), is a characteristicmechanism of these ferroelectric materials under electricaland thermal fields, in correspondence with their piezo- andpyroelectric properties [9-11]. Consequently, polarons resultin suitable transport carriers for thermal energy.

With this picture in mind, the behavior of the thermal con-ductivity in Fig. 2 can be explained using the following ar-guments: At low sintering temperatures the samples look likepressed powders, with smaller grain size and very high poros-ity that hinders the intergrain heat transfer rate by the effectsof interfacial thermal resistance; moreover, this phenomenonis favored in a similar way, by the very small crystallite sizeinside the grains which decreases the intragrain (intercrystal-lite) long range electrical conductivity. The latter situationis illustrated in Fig. 3 where, for PFNoxides samples (withsmaller crystallite size), dielectric loss is more pronouncedcompared to that of PFN1200, due to a higher chaotic con-duction along crystallite boundaries [11].

When the sintering temperature increases and approaches900C, the density of the sample increases (see Fig. 1); how-ever, the density is not high enough to make the electricalconduction along grain boundaries the preponderant mecha-nism. On the contrary, in these cases the dielectric behavioris determined by the polarization and short range conductionprocesses taking place inside the grains. At 900C the sam-ples exhibit the best dielectric properties, as is illustrated inFig. 3, showing the lowest values of the dielectric loss. Froma previous study it was demonstrated that the higher contri-butions come from small polarons, so that heat transfer andtherefore thermal conductivity are enhanced, thus justifyingthe maximum in Fig. 2.

For samples sintered at temperatures above 900C we ob-served that the density approaches the theoretical value due toa higher intergrain fusion. Additionally, the ratio Fe2+/Fe3+

and the number of oxygen vacancies increase; consequently,a higher concentration of free charge carriers develops andthe long range conduction mechanisms become predominantboth inside the grains and along the grain boundaries with aconsequent increase in the dielectric losses as can be seen inFig. 3 for the PFN1200 and PFNoxides samples. However,this high mobility of charges at the grain boundaries leadsto a higher chaotic thermal conduction in detriment of direc-tional energy transport, reducing thus the thermal conductiv-ity. This last effect is more pronounced for samples obtainedat 950C, where a minimum ofk is observed in correspon-dence with the maximum of the dielectric loss at this temper-ature (see Fig. 3); moreover, this is enhanced for PFNoxidessamples due to their small crystallite sizes, as was discussedabove.

4. Conclusions

In summary, the thermal conductivity behavior in PFN ce-ramics is not affected by the kind of precursor used for theirpreparation; it is determined instead by the morphological

Rev. Mex. Fıs. 57 (3) (2011) 241–244

244 R. FONT, E. MARIN, A. LARA-BERNAL, O. RAYMOND, A. CALDERON, J. PORTELLES, AND J.M. SIQUEIROS

features and the competition between the thermal and elec-trical transport mechanisms (preponderantly small polarons)inside the grains and across the grain boundaries, and theconduction processes along the grain boundaries. The sam-ples sintered at 900C show the highest value of the thermalconductivity in correspondence with its best dielectric prop-erties facts that are explained in terms of conduction and po-larization processes by small polarons and short range con-duction mechanisms taking place inside the grains, and notthe long range conduction mechanisms occurring along thegrain boundaries. This work shows the potential of the PAtechnique, aided with specific heat capacity measurements,to study the heat transport mechanisms in PFN ceramics and

similar materials. For a more comprehensive analysis ofthese mechanisms, temperature dependent measurements ofthermal properties are necessary. Work in this direction isunder way.

Acknowledgements

This work was partially supported by projects SIP-IPN20080032 and SEP-CONACyT 83289, 82503, and 49986-F, as well as DGAPA-UNAM projects N IN105711 andIN107811. The support of COFAA-IPN through the PIFI andSIBE programs is also greatly acknowledged. Thanks are dueto I. Gradilla for technical support.

1. R.Ramesh and N.A. Spaldin,Nature Mater.6 (2007) 21.

2. J.F. Scott,Nature Mater6 (2007) 256.

3. M. Gajeket al., Nature Mater.6 (2007) 296.

4. W. Eerenstein, M. Wiora, J.L. Prieto, J.F. Scott, and N.D.Mathur,Nature Mater.6 (2007) 348.

5. Y.H. Chu, L.W. Martin, M.B. Holcomb, and R. Ramesh,Mater.Today10 (2007) 16.

6. X.S. Gao, X.Y. Chen, J. Yin, J. Wu, Z.G. Liu, and M. Wang,J.Mater. Sci.35 (2000) 5421.

7. O. Raymond, R. Font, N. Suarez-Almodovar, J. Portelles, andJ.M. SiqueirosFerroelectrics294(2003) 141.

8. Y. Yang, J.M. Liu, H.B. Huang, W.Q. Zou, P. Bao, and Z.G.Liu, Phys. Rev. B70 (2004) 132101.

9. O. Raymond, R. Font, N. Suarez-Almodovar, J. Portelles, andJ.M. Siqueiros,J. Appl. Phys.97, 084107 (2005).

10. O. Raymond, R. Font, N. Suarez-Almodovar, J. Portelles, andJ.M. Siqueiros,J. Appl. Phys.97 (2005) 084108.

11. O. Raymond, R. Font, N. Suarez-Almodovar, J. Portelles, andJ.M. Siqueiros,J. Appl. Phys.99 (2006) 124101.

12. L. Yan, J. Li, C. Suchicital, and D. Viehland,Appl. Phys. Lett.89 (2006) 132913.

13. D. Varshney, R.N.P. Choudhary, and R.S. Katiyar,Appl. Phys.Lett.89 (2006) 172901.

14. R. Font, G. Alvarez, O. Raymond, J. Portelles, and J.M.Siqueiros,Appl. Phys. Lett.93 (2008) 172902.

15. M.H. Lenteet al., Phys. Rev. B78 (2008) 054109.

16. R. Font, O. Raymond, E. Martinez, J. Portelles, and J.M.Siqueiros,J. Appl. Phys.105(2009) 114110.

17. D.P. Almond and P.M. Patel,Photothermal Science and Tech-niques in Physics and its Applications(Chapman and Hall,London 1996).

18. L.F. Perondi and L.C.M. Miranda,J. Appl. Phys.62 (1987)2955.

19. M.V. Marquezini, M.N. Cella, A.M. Mansanares, H. Vargas,and L.C.M. Miranda,Meas. Sc. Technol.2 (1991) 396.

20. A.M. Mansanares, A.C. Bento, H. Vargas, N.F. Leite, andL.C.M. Miranda,Phys. Rev. B42 (1990) 4477.

21. H Valiente, O. Delgado-Vasallo, R Galarraga, A. Calderon, andE. Marin, International Journal of Heat Transfer276 (2006)1859.

22. J.L. Pichardoet al., Appl. Phys. A65 (1997) 69.

23. M.E. Rodriguezet al., Z. Lebensm. Unters. Forsch.200(1995)100.

24. J.A. Balderas-Lopezet al., Forest Products Journal46 (1996)84.

25. S. Martınezet al., Journal of Materials Science39(2004) 1233.

26. S. Garcıaet al., Journal de Physique IV125(2005) 309.

27. E. Marın and H. Valiente,Journal de Physique IV125 (2005)305.

28. R.E. Newnham,Properties of Materials: Anisotropy, Symme-try, Structure(Oxford University Press Inc., New York 2005).

Rev. Mex. Fıs. 57 (3) (2011) 241–244


Applications and extensions of the Liouville theorem on constants of motion

G.F. Torres del CastilloInstituto de Ciencias de la Universidad Autonoma de Puebla,

Puebla, Pue., 72570 Mexico.


We give an elementary proof of the Liouville theorem, which allows us to obtainn constants of motion in addition ton given constants ofmotion in involution, for a mechanical system withn degrees of freedom, and we give some examples of its application. For a given setof n constants of motion that are not in involution with respect to the standard symplectic structure, there exist symplectic structures withrespect to which these constants will be in involution and the Liouville theorem can then be applied. Using the fact that any second-orderordinary differential equation (not necessarily related to a mechanical problem) can be expressed in the form of the Hamilton equations, theknowledge of a first integral of the equation allows us to find its general solution.

Keywords: Hamilton–Jacobi equation; constants of motion; symplectic structures.

Se da una prueba elemental del teorema de Liouville, el cual permite obtenern constantes de movimiento adicionales an constantes demovimiento en involucion dadas, para un sistema mecanico conn grados de libertad, y se dan algunos ejemplos de su aplicacion. Para unconjunto dado den constantes de movimiento que no estan en involucion con respecto a la estructura simplectica estandar, existen estructurassimplecticas con respecto a las cuales estas constantes estaran en involucion y puede aplicarse entonces el teorema de Liouville. Usando elhecho de que cualquier ecuacion diferencial ordinaria de segundo orden (no necesariamente relacionada con un problema mecanico) puedeexpresarse en la forma de las ecuaciones de Hamilton, el conocer una primera integral de la ecuacion permite hallar su solucion general.

Descriptores: Ecuacion de Hamilton–Jacobi; constantes de movimiento; estructuras simplecticas.

PACS: 45.20.Jj; 02.30.Jr; 02.30.Hq

1. Introduction

The Hamilton–Jacobi (HJ) equation provides a powerfulmethod to solve various problems of classical mechanics(see,e.g., Refs. 1 and 2) as well as to relate different prob-lems [3]. As is well known, given a Hamiltonian for a me-chanical system withn degrees of freedom, the knowledgeof a complete solution (that is, a solution containingn non-additive arbitrary constants) of the corresponding HJ equa-tion allows one to obtain the solution of the equations of mo-tion. Then arbitrary parameters contained in a complete so-lution of the HJ equation are then constants of motion, thatare identified with half of a new set of canonical coordinates.

Usually, the complete solutions of the HJ equation areobtained by means of separation of variables, which requiresexpressing this equation in a suitable coordinate system (see,e.g., Refs. 1 and 2). Actually, in most textbooks on classicalmechanics the method of separation of variables is the onlyone employed to solve the HJ equation. (Similarly, in mosttextbooks on quantum mechanics, the only method employedin the solution of the Schrodinger equation is that of separa-tion of variables.) Nevertheless, there exist some other meth-ods for solvingfirst-order partial differential equations (see,e.g., Ref. 4) such as the HJ equation. In one of these less-known methods, when applied to the HJ equation, one hasto express the canonical momenta in terms of the coordinatesand n constants of motion; a complete solution,S, of theHJ equation can then be obtained fromdS = pidqi − Hdt.However, it turns out that the expression on the right-handside is an exact differential if and only if the constants of mo-tion employed in this process are in involution, that is, their

Poisson brackets are all equal to zero, as Liouville found by1855 [2]. In the case where there is only one degree of free-dom, the Liouville theorem can be applied making use of anarbitrary constant of motion, since the Poisson bracket of afunction with itself is trivially equal to zero.

The aim of this paper is to give an elementary proof of Li-ouville’s theorem, with some illustrative examples of its ap-plication finding complete solutions of the HJ equation, with-out relying on an specific coordinate system (by contrast withthe method of separation of variables). When one hasn con-stants of motion that are not in involution, it is still possibleto find a different symplectic structure (that is, another defini-tion of the Poisson bracket) so that these constants of motionbe in involution. Since any second-order ordinary differentialequation (ODE), or any pair of first-order ODEs, can be ex-pressed in the form of the Hamilton equations (in an infinitenumber of different ways) [5], with the aid of Liouville’s the-orem, making use of a first integral one can find the completesolution.

In Sec. 2 we state the Liouville theorem stressing its anal-ogy with the procedure followed in the use of a complete so-lution of the HJ equation in the solution of the equations ofmotion. Section 3 contains four examples of the applicationof the Liouville theorem and in Sec. 4 an elementary proof ofthe Theorem is given.

2. The Liouville theorem

In this section we begin by recalling some basic facts relatedwith the application of the complete solutions of the HJ equa-

246 G.F. TORRES DEL CASTILLO

tion and we show that, in a certain sense, the same steps ap-pear in the Liouville theorem but in the opposite order.

For a given HamiltonianH(qi, pi, t) of a system withndegrees of freedom, the corresponding HJ equation, in itsstandard form, is the first-order partial differential equation

H

(qi,

∂S

∂qi, t

)+

∂S

∂t= 0. (1)

A complete solution of this equation is a functionS(qi, t, Qi)containingn non-additive arbitrary parametersQ1, . . . , Qn

that satisfies Eq. (1). Under the appropriate regularity con-ditions, such a function generates a canonical transformationrelating the original canonical coordinatesqi, pi with a newset of canonical coordinatesQi, Pi, which are constants ofmotion (since the new Hamiltonian is equal to zero), accord-ing to

dS = pidqi −Hdt− PidQi, (2)

that is,

pi =∂S

∂qi, Pi = − ∂S

∂Qi, i = 1, 2, . . . , n (3)

(see,e.g., Refs. 1 and 2).Since theQi, Pi are canonical coordinates, the Poisson

brackets among theQi are all equal to zero

Qi, Qj = 0, i, j = 1, 2, . . . , n. (4)

As pointed out in the Introduction, the complete solutionsof the HJ equation are usually obtained by separation of vari-ables, in which case the separation constants can be takenas theQi, but the application of this method requires an ap-propriate choice of the coordinatesqi, pi. As we shall show,given a set ofn functionally independent constants of motionQ1, . . . , Qn, satisfying Eq. (4) (that is, theQi are in invo-lution) one can find a complete solution of the HJ equation(and, therefore, the solution of the equations of motion) with-out having to use some special coordinate system. Indeed, ifwe haven constants of motion

Qi = Qi(qj , pj , t), i = 1, 2, . . . , n (5)

(which may depend explicitly on the time), assuming thatthese relations can be inverted to express thepi in terms ofQj , qj , andt, we obtainn functionsFi such that

pi = Fi(qj , t, Qj). (6)

Substituting these expressions into the Hamiltonian we ob-tain a function

H(qi, t, Qi) ≡ H(qi, Fi(qj , t, Qj), t). (7)

Then, treating theQi as constants, the linear differential formFidqi− Hdt is exact, that is,Fidqi − Hdt is the differentialof some functionS (which depends parametrically on theQi)

dS = Fidqi − Hdt (8)

[cf. Eq. (2)] andS(qi, t, Qi) is therefore a complete solutionof the HJ equation. In the next section we give some exam-ples, deferring the proof of the exactness ofFidqi − Hdt toSec. 4.

3. Examples

In this section we give some illustrative examples of the pro-cedure outlined above to find complete solutions of the HJequation.

3.1. The Kepler problem in two dimensions

As is well known, the HJ equation for the Kepler problem intwo dimensions, which corresponds to the Hamiltonian

H =1

2m(px

2 + py2)− k√

x2 + y2,

expressed in Cartesian coordinatesx, y, wherem is the massof the particle andk is a positive constant, is separable inpolar and parabolic coordinates (see,e.g., [1,6]) but itis notseparable in Cartesian coordinates.

SinceH is time-independent and invariant under rota-tions about the origin,

Q1 ≡ H, Q2 ≡ xpy − ypx

(the total energy and the angular momentum about the ori-gin) are constants of motion, which are in involution (as canbe seen from the fact that the angular momentum is a constantof motion). Inverting these expressions one finds

px =−Q2y ± x

√2mQ1r2 + 2mkr − (Q2)2

r2,

py =Q2x± y

√2mQ1r2 + 2mkr − (Q2)2

r2,

wherer2 ≡ x2 + y2, which gives the functionsF defined byEq. (6). Thus, the right-hand side of Eq. (8) becomes

Q2 (−ydx + xdy)r2

±√

2mQ1r2 + 2mkr − (Q2)2

r2(xdx + ydy)

or, equivalently,

Q2d(arctan

y

x

)

±√

2mQ1 +2mk

r− (Q2)2

r2dr.

This last expression is indeed the differential of a function,which must be a complete solution of the HJ equation. It maybe noticed that this function is the sum of separate functionsof the polar coordinatesθ, r (which is a consequence of usingthe angular momentum as one of the constants of motionQi).

Rev. Mex. Fıs. 57 (3) (2011) 245–249

APPLICATIONS AND EXTENSIONS OF THE LIOUVILLE THEOREM ON CONSTANTS OF MOTION 247

3.2. A time-dependent Hamiltonian

Now we shall consider the time-dependent Hamiltonian

H =p2

2m− ktx, (9)

wherek is a constant. The corresponding HJ equation is notseparable, but one can readily verify that

Q ≡ p− kt2/2

is a constant of motion (note thatQ depends explicitly on thetime). Hence

F (x, t,Q) = Q + kt2/2

and

H(x, t, Q) = (1/2m)(Q + kt2/2)2 − ktx.

Then, it can readily be verified that the differential form

Fdx− Hdt =(

Q +kt2

2

)dx

−[

12m

(Q +

kt2

2

)2

− ktx

]dt

is exact and that it is the differential of the function

S = Qx +12kt2x− 1

2m

(Q2t + Qk

t3

3+ k2 t5

20

), (10)

which is, therefore, a complete solution of the HJ equation.It may be noticed that this function is not the sum of separatefunctions ofx andt.

From Eqs. (3) and (10) one finds a second constant ofmotion

−P =∂S

∂Q= x− Qt

m− kt3

6m.

As usual, the values ofQ andP are determined,e.g., by theinitial conditions, and the formulas above give the solution ofthe equations of motion.

This example is also interesting because the Schrodingerequation corresponding to the Hamiltonian (9) cannot besolved by separation of variables, but it turns out thatψ=exp(iS/~), with S given by Eq. (10), is a solution of thisequation.

3.3. Two constants of motion that are not in involution

The Hamiltonian

H =px

2 + py2

2m+ mgy (11)

corresponds to a particle of massm in a uniform gravitationalfield in Cartesian coordinates and one can readily verify thatthe two functions

Q1 ≡ px, Q2 ≡ pxpy

m+ mgx (12)

are constants of motion (as well as the Hamiltonian itself).Making use of the standard definition of the Poisson bracket(x, px = 1 = y, py) one finds thatQ1, Q2 is differentfrom zero and therefore these two constants of motion do notseem suitable to find a complete solution of the HJ equation.

However, bydefiningthe Poisson bracket in such a waythat

x, py = 1 = y, px, (13)

Q1 andQ2 are in involution. But, then, taking into accountthat px is the momentum conjugate toy andpy is the mo-mentum conjugate tox, in order to reproduce the equationsof motion

x = px/m, y = py/m, px = 0, py = −mg,

in place of (11), we have to useQ2 as the Hamiltonian; infact,

x =∂Q2

∂py, y =

∂Q2

∂px, px = −∂Q2

∂y, py = −∂Q2

∂x

(see Ref. 7 for details).Thus

px = Q1, py =m(Q2 −mgx)

Q1

and [performing the appropriate changes on the right-handside of Eq. (8)]

dS =m(Q2 −mgx)

Q1dx + Q1dy −Q2dt

must yield a complete solution of the HJ equation (corre-sponding to the new HamiltonianQ2).

3.4. Application to second-order ODEs

Following the algorithm given in Ref. 5, any second-orderODE, or any system of two first-order ODEs, can be ex-pressed in the form of the Hamilton equations in an infinitenumber of different ways. Then, any constant of motion isuseful to apply Liouville’s Theorem, and one can make useof the complete solution of the HJ equation thus obtained tofind a second constant of motion and, therefore, the generalsolution of the original ODE or ODEs.

For instance, in Ref. 5 the Emden–Fowler equation(which arises in the study of a self-gravitating gas)

x +2x

t+ xk = 0,

wherek is a constant, has been considered, showing that it isequivalent to the Hamilton equations with

H =p2

2t2+

t2qk+1

k + 1

Rev. Mex. Fıs. 57 (3) (2011) 245–249

248 G.F. TORRES DEL CASTILLO

andq = x, p = t2x, as can be readily verified. It was alsoshown that in the particular case wherek = 5,

Q ≡ 3p2

t+ 3pq + t3q6

is a constant of motion. Inverting this last expression, onefinds

p = F (q, t,Q) =t

6(− 3x±

√9q2 − 12t2q6 + 12Q/t

)

and the right-hand side of Eq. (8) takes the form

d(−u

4− Q

6ln |t|

)± 1

12

√9− 12u2 + 12Q/u du,

with u ≡ q2t. Clearly, this differential form is the differen-tial of a function, which must be a complete solution of theHJ equation (it may be noticed that this function is the sumof separate functions ofq2t andt).

A second constant of motion, which together withQ,yields the complete solution of the original equation, is givenby

−P =∂S

∂Q= −1

6ln |t| ± 1

2

∫du√

9u2 − 12u4 + 12Qu.

4. Proof of the Liouville theorem

We shall prove that, ifQ1, . . . , Qn aren functionally inde-pendent constants of motion, and the original momentapi

can be expressed in the formpi = Fi(qj , t, Qj), the differ-ential formFidqi − Hdt is exact if and only if theQi are ininvolution. According to the standard criterion, this differen-tial form is exact if and only if

∂Fi

∂qj=

∂Fj

∂qi, i, j = 1, 2, . . . , n (14)

and

∂Fi

∂t= −∂H

∂qi, i = 1, 2, . . . , n (15)

(these last equationslook likehalf of the Hamilton equations,but, as we shall see, they hold as a consequence of the con-stancy of theQi).

Substitution of the relationspi = Fi(qj , t, Qj) into theexpressions for the constants of motionQi give the equations

Qi = Qi(qj , Fj(qk, t, Qk), t)

which have to hold identically; hence, making use of thechain rule, we obtain

0 =∂Qi

∂qm+

∂Qi

∂pk

∂Fk

∂qm, (16)

and

0 =∂Qi

∂t+

∂Qi

∂pk

∂Fk

∂t. (17)

Making use of Eq. (16) we find that

Qi, Qj =∂Qi

∂qm

∂Qj

∂pm− ∂Qj

∂qm

∂Qi

∂pm

= −∂Qi

∂pk

∂Fk

∂qm

∂Qj

∂pm+

∂Qj

∂pk

∂Fk

∂qm

∂Qi

∂pm

= −∂Qi

∂pk

∂Fk

∂qm

∂Qj

∂pm+

∂Qj

∂pm

∂Fm

∂qk

∂Qi

∂pk

=∂Qi

∂pk

∂Qj

∂pm

(∂Fm

∂qk− ∂Fk

∂qm

),

so thatQi, Qj = 0 if and only if Eqs. (14) hold.Similarly, from the definition ofH we have

∂H

∂qi=

∂H

∂qi+

∂H

∂pj

∂Fj

∂qi, (18)

and the fact thatQi is a constant of motion amounts to

0 =∂Qi

∂t+

∂Qi

∂qm

∂H

∂pm− ∂Qi

∂pm

∂H

∂qm.

Substituting Eqs. (16), (17), and (18) into this last equationwe find that

0 = −∂Qi

∂pk

∂Fk

∂t− ∂Qi

∂pk

∂Fk

∂qm

∂H

∂pm

− ∂Qi

∂pm

(∂H

∂qm− ∂H

∂pk

∂Fk

∂qm

)

= −∂Qi

∂pk

(∂Fk

∂t+

∂H

∂qk

)

− ∂Qi

∂pk

∂H

∂pm

(∂Fk

∂qm− ∂Fm

∂qk

),

thus completing the proof.

5. Conclusions

The Liouville theorem allows one to get then complementaryconstants of motion to a given set ofn constants of motionof a system withn degrees of freedom, without having to usesome particular coordinate system. Among other things, theresults presented here show the usefulness of having varioussymplectic structures (and Hamiltonians) for a given mechan-ical system and of expressing an arbitrary second-order ODEin the form of the Hamilton equations.

Rev. Mex. Fıs. 57 (3) (2011) 245–249

APPLICATIONS AND EXTENSIONS OF THE LIOUVILLE THEOREM ON CONSTANTS OF MOTION 249

1. M.G. Calkin,Lagrangian and Hamiltonian Mechanics(WorldScientific, Singapore, 1996),

2. E.T. Whittaker,A Treatise on the Analytical Dynamics of Par-ticles and Rigid Bodies, 4th ed. (Cambrige University Press,Cambridge, 1993). Chap. XII.

3. G.F. Torres del Castillo,Rev. Mex. Fıs.44 (1998) 540.

4. I.N. Sneddon, Elements of Partial Differential Equations(Dover, New York, 2006). Chap. 2.

5. G.F. Torres del Castillo,J. Phys. A: Math. Theor.42 (2009)265202.

6. G.F. Torres del Castillo and J.L. Calvario Acocal,Rev. Mex. Fıs.44 (1998) 344.

7. G.F. Torres del Castillo and G. Mendoza Torres,Rev. Mex. Fıs.49 (2003) 445.

Rev. Mex. Fıs. 57 (3) (2011) 245–249


Controlling a laser output through an active saturable absorber

M. Wilson, V. Aboites, A. Pisarchik, V. Pinto, and Y. BarmenkovPhotonics Department Center for Research in Optics,

Loma del Bosque 115 Col. Valle del Campestre 37150 Leon.

Recibido el 3 de marzo de 2011; aceptado el 15 de abril de 2011

Using the modified Statz - De Mars equations to describe a two laser optical system, it is shown how a saturable absorber can be made into anactive control device when an external Electro Optic Modulator modulated low-intensity laser pumps directly into a saturable absorber insidea dye laser cavity. The direct modulation enables to control when and how the pulse train coming from the saturable absorber is released.The results here presented show that the pulse characteristics such as width, intensity and pulse frequency coming from the dye laser cavity,depend on the absorber characteristics and the modulation frequency.

Keywords: Laser; optical resonator; laser dynamics.

Usando las ecuaciones de Statz - De Mars modificadas para describir un sistemaoptico de dos laseres se muestra como un absorbedorsaturable puede convertirse en un dispositivo de control activo cuando una senal externa de baja intensidad, modulada por un ModuladorElectroOptico, es inyectada directamente enel cuando se encuentra dentro de una cavidad de laser de colorante. La modulacion directapermite controlar cuando y como se libera el tren de pulsos que proviene del absorbedor saturable. Los resultados aquı presentados muestranque las caracterısticas de intensidad, anchura y frecuencia de pulso proveniente de la cavidad laser de colorante dependen de las caracterısticasfısicas del absorbedor y de la frecuencia de modulacion aplicada al laser de control.

Descriptores: Laser; resonadoresopticos; dinamica de laseres.

PACS: 42.55.-f; 42.60.Da; 42.65.Sf

1. Introduction

The Statz - De Mars equations’ system has been broadlyused to model laser systems for almost 50 years. Originallydesigned to describe the oscillations in a maser [1], it hassince undergone many modifications to describe more com-plex systems. Particularly in this work, terms that take intoaccount on one hand a saturable absorber (SA) and on theother a control beam are added to these equations.

Fluorescent dye lasers have been widely used as an am-plifier media for incident laser signal beams at several wave-lengths [2,3]. Moreover it has also been shown [4] that thesteady-state transmission of a pump beam through a laseramplifier can be controlled by the intensity of a signal beamfrom an auxiliary laser, suggesting that the signal laser beamcan be used as an optical switch. While the authors [4] useda dye cell as an active media and switching device, in thepresent paper a dye laser beam will be used as a signal beamand a SA controlled by an external modulated beam (controlbeam) will replace the dye cell, in this way the saturable ab-sorber is in fact turned into an active device.

A SA is a non-linear optical component with a certainoptical loss, which is reduced at high optical intensities [5].Depending on the parameters of the SA they are usually usedfor passive mode-locking and Q-switching [6]. It is wellknown that continuous wave dye lasers with SAs provideshort pulses due to self-stabilization of the pulse shaping pro-cess [7]. A SA inserted into a laser cavity increases the non-linearity of such system and enriches the laser operation dy-namics [8]. Mode-locking laser dynamics has been an impor-tant study subject [9], traditionally mode-locking is activelyobtained with an optical modulator inside the cavity or pas-

sively with a SA. In this work, we believe for the first time,the two are combined by reallocating the optical modulatoroutside the cavity and injecting its signal into an intracavitySA; the modulated signal comes from a low-intensity contin-uous wave laser.

While, the important characteristics of a SA are: modu-lation depth (maximum possible change in optical loss), un-saturable losses (unwanted losses which cannot be saturated),recovery time, saturation fluence, saturation energy and dam-age threshold (given in terms of intensity) [5], the modifiedStatz - De Mars equations require only to take into accountfor the SA, its geometry, its active absorbent nuclei densityand its relaxation time [10].

For a long time SAs have been considered passive de-vices [11-13]. As shown in Ref. 15 a dye cell with an externalsignal beam may be used as a switching device. If now an ex-ternal modulated low intensity beam is injected directly intothe SA, the optical losses and the saturation fluence are mod-ified according to the frequency injected transforming the de-vice into an active one. Thus the saturable absorber becomesan important control tool.

This work presents a numerical simulation of a dye laseroutput with an intracavity SA behaving as an active devicedue to an external modulation. The simulation considers anauxiliary beam coming out of the low-energy CW laser andmodulated by an electro-optical modulator (EOM) that is in-jected transversally into the SA; so that the control of themain laser is obtained through the absorber. The dye laseris described by the Statz - De Mars equations, these will bemodified to take into account both the SA and an externalmodulation. The parameters where chosen for a typical dye

CONTROLLING A LASER OUTPUT THROUGH AN ACTIVE SATURABLE ABSORBER 251

laser [16], while we played with the modulation frequencyand the SA’s absorption parameter. The results will showthat with such device the laser output goes from a continu-ous wave regime to a narrow train of high peaks with con-stant amplitude (comb like) passing through a modulationfrequency window where the overall shape of the wave isvery smooth even though it may present a collection of un-damped undulations at different pulse locations depending onthe working parameters.

2. Theoretical model

Using the Statz-De Mars equations for a three level laser witha SA [10] the derivatives of the emitted-photon densityS, thepopulation in the active mediumN , and the population inver-sion in the SAka, are given as:

dS

dt= ΓvσNS − Γv

lal

kaS − 1T

S (1)

dN

dt= −β

σ

~wNS +

N0 −N

τ(2)

dka

dt= −2σakaS

~w+

k0a − ka

τa(3)

wherethe parameters

Γ = (l/ν)/(l/ν + la/νa + [L/c− (l + la)]/c),

T = Γν[η1 + (la/l)η1a + ηa]−1,

ν, and σ stand for cavity-filling coefficient, photon cavitylife-time, optical frequency and active medium cross-sectionrespectively,β is an integer used to describe the variation ofthe difference in population inversion between the transitionlevels during a photon emission,l is the cavity length, whilela is the SA length,k0a is theka steady-state value,σa rep-resents the SA cross-section,N0 is the total initial populationin the active medium,τ andτa stand for relaxation time inthe active medium and the SA respectively,η, ηa represent,respectively, the total absorption coefficient for the activemedium and for the SA, finally,~w is the product betweenthe Planck’s constant and the optical frequency, representingthe photon energy.

In order to simplify the manipulation of this equa-tions system it is a standard procedure [10,14] to trans-form it into an adimensional one, to do so the follow-ing parameters are defined as t’=t/τ , G= τ /T, δ = τ /τa,ρ=2σa/βσ, α = ΓνσTN, αa=-ΓνTk0a(la/l), and the equa-tion variables as: n(t’)=ΓνσTN(t’), na(t’)=-Γν(la/l)Tka(t′)and m(t’)=βστS(t’)/~w.

The adimensional Statz - De Mars equations system canbe written as:

dm

dt= Gm(n + na − 1) (4)

dn

dt= α− n(m + 1) (5)

FIGURE 1. Laser scheme with an active intracavity saturable ab-sorber.

dna

dt= δαa − na(ρm + δ) (6)

On the other hand it has already been demon-strated [15,16] that an external signal injected directly intothe cavity can be used as an optical switch to modify theshape of the output signal,i.e. an optical system formed bya high-power laser and a low-power control one. If these re-sults are generalized for a laser with a SA and a modulatedcontrol signal is injected into the SA one obtains anactivedevice; the laser output depends directly of the SA behav-ior. As soon as a modulated signal is injected into the ab-sorber the output signal shows a periodic behavior, its periodis clearly proportional to the control signal one. In this workthe Statz – De Mars equations will be modified to numeri-cally describe the behavior of a laser made up by a dye activemedium (AM), two mirrors (total reflexion, M1, and semi-transparent mirror, M2) and an intracavity saturable absorber(SA) coupled with a low-power continuous wave (CW) laserthrough an Electro-Optical Modulator (EOM) as shown inFig. 1.

The modified adimensional Statz – De Mars equationsthat take into account the modulated control signal,cos(ωt)into the system stand as follows:

dm

dt= Gm(n + na − 1) (7)

dn

dt= α− n(m + 1) (8)

dna

dt= δαa(

1 + cos(ωt)2

)− na(ρm + δ) (9)

whereω standsfor the modulation frequency applied to theEOM.

Rev. Mex. Fıs. 57 (3) (2011) 250–254

252 M. WILSON, V. ABOITES, A. PISARCHIK, V. PINTO, AND Y. BARMENKOV

FIGURE 2. Dye laser output intensity. a) without modulation; b, c, d, e, f) with modulation and anαa of 0.3, 1, 3, 30 and 60 respectively.

FIGURE 3. Phase diagramm vsna. a) withαa= 1.4 b)withαa= 30.

3. Numerical results

To find the solutions of the Eqs. (7-9) a Mathematica codewas build using the typical [16] Statz - De Mars parametersfor a Dye laserG= 200,α= 4, δ= 1, ρ = 0.001 and as initialvaluesm0= 0.25, n0= 0 andna0= 0.152. It was observed

that the system actually tends to a fixed point in a short com-putational time and to a periodic behavior when the exter-nal modulation is on. With a fixedω the only parameter wecan actually play with isαa, a measure of the active centersabsorbent density. Since the geometrical parameters that areinvolved in the definition ofαa will be fixed, the parameter

Rev. Mex. Fıs. 57 (3) (2011) 250–254

CONTROLLING A LASER OUTPUT THROUGH AN ACTIVE SATURABLE ABSORBER 253

FIGURE 4. Pulse shape atαa= 15 with frequenciesω. a) 1/4, b) 1/2, c) 2, d) 5, e)10, f) 35.

αa will be identified as absorption ratio and therefore willdepend on the chosen SA (in a dye SA its dependence varyaccording to the dye concentration), the values for this nu-merical experiment were chosen in the practical range (0.3,63). When the Statz - De Mars equations were solved with-out modulation,m reached a fixed point in few iterations(Fig. 2a), as soon as the modulation is injected into the ab-sorber a periodic pulse train begins to appear (Fig. 2b, c, d,e, f) withαa as small as 0.3, asαa is increased the maximumintensity reached increases and a region of different frequen-cies coexistence begins to appear at a certain location in thepulse, this location depends on the absorption ratio.

It can be observed from Fig. 2 that the laser output tendsto a fixed point when the SA is in passive configuration,i.e.without external modulation; as soon as the SA is turned intoan active device, different packages of pulses are obtaineddepending on the SA’s absorption ratio (αa), the bigger theconcentration the higher the reached intensity. The laser out-put, in this case, goes from a continuous wave regime to ahigh-intensity peaks train passing, through an undamped un-dulations window moving across the pulse’s body.

When the absorption ratio is larger than a threshold thatdepends on the modulation frequency, the phase diagram be-

tweenm andna exhibits instead of one, two critical points,as shown in Fig. 3. When the frequency is 1 this phenomenonappears for anαa value near 1.4. Ifαa= 30 the two criticalpoints in the phase diagrams are far enough from one anotherto emphasize the phenomenon (Fig. 3b).

The laser output signal is very important, if the geomet-rical variables are fixed the only variables that have a greatimpact on the laser dynamics areαa and ω. Therefore, acomprehensive study on the output signal in terms ofω wasdone for anαa= 15, the value was chosen first of all becausethe undamped undulations are obvious for smallω and arenowhere near the maximum intensity reached by the laseroutput; its behavior is representative of what happens at anyfrequency even if there are some differences in detail. Fig-ure 4 shows how the signal frequency increases following thechange in the modulation frequency. As the frequency rises, athreshold is obtained where the narrow pulse train is reached,its value is closely related withαa, a change in the absorp-tion ratio only moves the undamped undulations window;the window is shifted towards the right (both thresholds getlarger) asαa is increased. The second threshold correspondsto the appearance of a narrow pulse train, whenαa= 15 thenarrow high intensity pulse train is obtained at aroundω=35.

Rev. Mex. Fıs. 57 (3) (2011) 250–254

254 M. WILSON, V. ABOITES, A. PISARCHIK, V. PINTO, AND Y. BARMENKOV

FIGURE 5. Pulse width against control frequency. It is shown howthe pulses width become narrower as the control frequency is in-creased.

When a SA is used in a conventional manner, once theSA operational parameters are fixed, the laser repetition rateand the peak intensity achievable are also fixed due to the SAproperties. However, in the proposed scheme these proper-ties can be externally controlled by modifying the injectedpower frequency, given as a result, a wider range of pulsesgeneration. This is the main advantage of our proposal.

The dependence between the output pulse width and themodulation frequency, shown in Fig. 5, is clearly a very goodapproximation to an exponential decay. It must be noted thatthe modulation frequencyω enters the equation as an inte-ger multiple of the laser relaxation frequencyw, so that thepulse duration is calculated from the relaxation time. Fromthe above considerations one can observe a width lower limitaround 19 ns achieved approximately at 80 KHz, which is 35times the relaxation frequency.

4. Conclusions

This paper effectively shows how a saturable absorber can bemade in to anactivedevice to control the output of a Dyelaser. The insertion of a modulated signal directly into thesaturable absorber modifies the continuous output driving itinto a periodic one. The larger theαa the higher the laserintensity obtained, and the more comb-like it becomes. Fora givenαa there are two important thresholds: the modula-tion frequency where the undamped undulations appear andthe one where they disappear, as the absorption ratio is in-creased, the undulations window is shifted towards higherfrequencies. For a givenαa, as the modulation frequencyis increased, the output signal changes from a smooth pe-riodic function to a clear comb-like pulse train whose widthdecreases exponentially, thus the saturable absorber is behav-ing as anactivedevice.

1. H. Statz, and G. De Mars,Quantum Electronics(1960) 530.

2. P.P. Sorokin, and J.R. Lankard,IBM J. Res. Develop.10 (1966)162.

3. L.W. Hilman,Dye laser principles(Academic press, 1990).

4. M.J. Damzen, K.J. Baldwin, and P.J. Soan,J. Opt. Soc. Am. B11 (1994) 313.

5. R. Paschotta,Encyclopedia of laser physics and technology(Wiley, 2008).

6. B.K. Garside and T.K. Lim,J. Appl. Phys.44 (1973) 2335.

7. W. Dietel, E. Dopel, and D. Kuhlke,Sov. J. Quantum Electron.12 (1982) 668.

8. V.V. Nevdakh, O.L. Gaiko, and L.N. Orlov,Opt. Comm.127(1996) 303.

9. A.N. Pisarchik, A.V. Kir’yanov, Y.O. Barmenkov, and R.Jaimes-Reategui,J. Opt. Soc. Am. B22 (2005) 2107.

10. L. Tarassov,Physique des processus dans les generateurs derayonnement optique coherent(Editons MIR, 1981).

11. M.E. Fermann,Opt. Lett.18 (1993) 894.

12. A. Schmidtet al., Opt. Lett.33 (2008) 729.

13. W. Dietelet al., Sov. J. Quantum Electron12 (1982) 668.

14. M. Braun,Differential equations and their applications: An in-troduction to applied mathematics(Springer, 1992).

15. V. Aboites, K.J. Baldwin, G.J. Crofts, and M.J. Damzen,Opt.Comm.98 (1993) 298.

16. V. Aboites, K.J. Baldwin, G.J. Crofts, and M.J. Damzen,Rev.Mex. Fis.39 (1993) 581.

Rev. Mex. Fıs. 57 (3) (2011) 250–254


Simultaneous phase-shifting cyclic interferometer for generation oflateral and radial shear

D.I. Serrano-Garcıaa, N.I. Toto-Arellanoa, A. Martınez Garcıaa, J.A. RayasAlvareza,A. Tellez-Quinonesa, and G. Rodrıguez-Zuritab

aCentro de Investigaciones enOptica,A.C. Loma del Bosque 115 Col. Lomas del Campestre, 37150 Apartado Postal 1-948, Leon, Gto. Mexico,

e-mails: [email protected]; [email protected] de Ciencias Fısico-Matematicas de la Benemerita Universidad Autonoma de Puebla,

Apartado Postal 1152, Puebla, 72001, Pue., Mexico.

Recibido el 4 de marzo de 2011; aceptado el 15 de abril de 2011

We present experimental results obtained by a phase-shifting interferometer employing polarization capable of retrieve directional derivativesin x-direction (lateral shear). The system was adapted to obtain radial derivative (radial shear) and implemented with a cyclic interferometerwith phase grid to multiplex the interference patterns. Using phase shifting by polarization, the interferometer is capable of processing theoptical phase data with n-interferograms captured in a single shot. Experimental results are presented.

Keywords: Diffraction; phase shifting; lateral shear; radial shear; interferometry; polarization.

En este trabajo se presentan los resultados experimentales obtenidos con un sistema interferometrico de corrimiento de fase por polarizacion,que puede obtener la derivada direccional en direccion x (desplazamiento lateral) y adaptando al arreglo para obtener la derivada radial(desplazamiento radial). El interferometro que se presenta consiste de un interferometro cıclico con rejilla de fase para multiplexar lospatrones de interferencia. Usando las tecnicas de corrimiento de fase por polarizacion, el interferometro es capaz de procesar la faseoptica apartir de n-interferogramas capturados en una sola toma. Se presentan los resultados experimentales obtenidos.

Descriptores: Difraccion; corrimiento de fase; desplazamiento lateral; desplazamiento radial; interferometrıa; polarizacion.

PACS: 42.87.Bg; 42.79.Ci; 42.79.Dj; 42.15.Eq; 42.25.Hz; 07.05.Pj

1. Introduction

In the lateral shearing interferometers the same wavefront issuperposed with its copy but displaced a distance∆x. When∆x is sufficiently small, the phase difference can be approx-imated as the directional derivative of the wavefront in thedisplacementx-direction [1-2]. Due to this reason, it is sen-sitivity against high phase changes and the resultant inter-ferograms are known asshearograms. Lateral shear in-terferograms have different fields of applications like opticaltests, wavefront aberrations [3], phase singularities detection(optical vortex) [4], mechanical stress [5] among others. Inthe case of radial shear interferometers the superposition isagainst the same wavefront at different scales (contracted orexpanded) with no displacement. In optical testing, the sys-tem is particulary sensitive at the astigmatism and coma aber-rations and insensitive to defocus. The resultant patterns canbe directly related to a Twyman-Green interferometer. Theproposed system presents the advantage of obtain lateral andradial shear by using of the adequate components, both casesare studied and presented in this work.

2. Cyclic-Path interferometer

Figure 1 shows the interferometers proposed for lateralshear [6-8] and radial shear [9-13], the two optical systemsare capable of obtain in a single shotn-shearograms with in-dependent phase shifts [14-16]. Figure 1(a) shows the two

possible interferometers that can be coupled to the 4-fsys-tem, each of them is used separately to generate a specifiedshear, the system-I is a Cyclic Shear Interferometer (CSI),where the shear is generated by moving the mirrorM by asmall distance4s. The system-IIis a cyclic radial shearinterferometer (CRI), the optical setup is similar to the pre-viously proposed, adding a pair of lensL1 andL2 to obtainthe two beams with different diameter in the transversal sec-tion. In this case also lateral shear can be obtained only witha small displacement inM , resulting in a combination of alateral and radial shear interferometer. Both systems are cou-pled to the 4-fsystem separately, each replica of the inter-ference pattern obtained in the image plane, can be used togenerate independent phase shift by placing linear polarizerfilters in each replica. Figure 1(b) shows the plots represent-ing the superposition of amplitude spectra in the output of thetwo systems.

2.1. Lateral shear interferometer

The system uses a laser source of He-Ne operating atλ = 632.8 nm. The collimated beam has a transversal sec-tion of a = 8.6 mm and linear polarization oriented to 45

generated by a quarter-wave retarder (Q0) and a linear polar-izer (P0). The optical setup is a combination of a cyclic pathinterferometer and a 4-f system. In Fig. 1(a) the system-Irepresents the cyclic shear interferometer (CSI). By the useof a polarizing beam splitter (PBS) the resulting beams have

256 D.I. SERRANO-GARCIA, et al.

FIGURE 1. Shearing interferometers with phase grid and modulation of polarization. (a) ConfigurationI is a variable lateral-shear interfer-ometer. The configurationII is a cyclic radial interferometer (CRI), which comprises a polarizing beam splitter (PBS), two lenses (L1, L2)and two mirrors (M, M ′). Q1: quarter-wave retarder;Pi: linear polarizers;ψi: transmission angle of polarization;∆s: linear shear;x0:beam separation;F0: order separation. (b) Upper row: diffraction orders of the same diffraction numerical order superimpose for lateralshear. Lower row: diffraction orders of different numerical order superimpose for radial shear.

cross linear polarization and after passing the quarter waveretarder (Q1), right and left circular polarization is obtained.The mirrors used in the system are defined asM andM ′.The 4-f system coupled to the cyclic interferometer usestwo equals lenses (f= 20 cm) and a phase grid defined asG(µ, ν) placed in the posterior focal plane of the first lens asthe system’s pupil with a spatial periodd, whereµ = u/λfandν = v/λf are the frequency coordinates (u, v) scaled tothe wavelengthλ and the focal lengthf . Replicated beamsof the interference pattern are obtained from this configura-tion. Placing a linear polarizer on each replica, an indepen-dent phase-shift can be obtained. No extra corrections wereused in the angle of the linear polarizers used because thequarter wave retarders (Q0, Q1 ) operate at the wavelength ofthe source [14-15].

2.2. Replicated interference patterns and modulation bypolarization

Defining the resultant vectorial beam amplitud of the CSIwhich enters to the 4-f system as

~t(x, y) = ~JLw (x, y) + ~JRw′ (x + x0, y) . (1)

wherex0 is the mutual separation of the beams alongx, w isaperture of the reference beam andw′ is the aperture of thebeam sheared by distancex0. ~JR and ~JL represents Jonesvectors for right and left circular polarization as

~JL =(

1i

), ~JR =

(1−i

). (2)

The lateral shear of the wavefronts are adjusted by mov-ing the mirror M of the CSI. The diagram in Fig. 1(a),system-Ican be adjusted in such way that the lateral shear∆s = x0 must be smaller than the transversal sectiona ofthe beams, this implies also that the lateral displacement issmaller than the diffraction order separation defined asF0,see upper row of Fig. 1(b). The two mutually separatedbeams by a distance∆s enters to the 4-fsystem with crosslinear polarization, after passing the quarter-wave retarder(Q1) circular cross polarization are obtained. The phase gridis used to obtain replicated shearograms generated by the in-terferometric system (CSI). The replicated interference pat-terns can be modulated by polarization to obtain phase-shifts.The fringe pattern are defined as [15]:

I = 2J2q J2

r 1 + cos[2ψ −∆φ(xq, yr)] (3)

whereψ representing the angle of the linear polarizer,

xq = x− qF0, yr = x− rF0,

∆φ(x, y) = φ(x, y)− φ(x− x0, y),

Jq andJr are the Bessel function of orderq andr respec-tively. The interference patterns presents unitary fringe mod-ulation.

Rev. Mex. Fıs. 57 (3) (2011) 255–258

SIMULTANEOUS PHASE-SHIFTING CYCLIC INTERFEROMETER FOR GENERATION OF LATERAL AND RADIAL SHEAR 257

FIGURE 2. Typical interferograms for lateral shear obtained in asingle shot (a) Interference patterns, (b) Unwrapped phase distribu-tions of the interferograms sheared inx-direction.

FIGURE 3. Typical interferograms for Radial shear obtained in asingle shot. (a) Sheared interferograms of an aberrated wavefront.(b) Unwrapped phase distributions of the interferograms sheared inradial direction.

3. Radial shear interferometer

In Fig. 1(a), the system-IIshows the optical arrangementused, where polarized light at 45 entering the interferom-eter generated by a quarter wave retarder (Q0) and a linearpolarizer (P0). The cyclic radial interferometer (CRI) usesa polarizer beam splitter (PBS), two lens (L1, L2) and twomirrors (M, M′). Transversal sections of the beams can bedescribed as:

w(x, y) = circ

(ρ

Ma

)· eiφ(x/Ma,y/Ma),

w′(x, y) = circ(ρ) · eiφ(x,y) (4)

whereρ =√

x2 + y2 andMa = f2/f1 denotesthe relativemagnification of the pupils as the focal lengths of both lenses(L1,L2). In the image plane of the 4-f system the fringe pat-tern obtained is modulated by the Bessel function as (3) butwith radial symmetry, where

∆φ(x, y) = φ(x, y)− φ(x/Ma, y/Ma).

As before in the lateral shear interferometer, at the imageplane of the 4-f system replicated interference patterns thatcan be modulated by polarization are obtained with indepen-dent phase shifts.

4. Phase data processing

In general, the interference pattern can be described as [17]:

Ii(x, y) = A(x, y) + B(x, y) cos[2ψi − ∂φ(x, y)

∂x

], (5)

for the case of lateral shear inx direction. Ii(x, y) repre-sents thei = 1...4 intensity distribution captured by the CCDcamera in a single shot, the polarization filters angles are:ψ1 = 0, ψ2 = 45, ψ3 = 90 andψ4 = 135, each of themrepresent phase shifts of0, π/2, π and3/2π respectively. Byconsidering thatA andB are constant[15] the relative phasecan be calculated as [18-19]:

∂φ(x, y)∂x

= arctan[I1(x, y)− I3(x, y)I2(x, y)− I4(x, y)

](6)

for the case of radial shear, Eq. (6) can be used only with theconsideration of the radial dependency.

5. Experimental results with phase grid shear-ing interferometers

The experimental results presented in Fig. 2 and Fig. 3 wereobtained by lens misalignments at defocus and paraxial fo-cus respectively. Lateral shear interferograms representingspherical aberration with defocusing are shown in Fig. 2. TheFig. 2(a) presents the four patterns obtained simultaneouslywith relative shifts ofπ/2. Phase derivative inx-direction ispresented in Fig. 2(b). A sets ofn=4 typical experimentalinterferograms for radial shear are shown in Fig. 3(a) and theresulting unwrapped phase data in Fig. 3(b), representing awavefront affected by spherical aberration in paraxial focus.Experimental results are presented for both cases of shear foran oil drop collocated on a microscope slide in Fig. 4 andFig. 5, showing the capability of this system for measure-ment of surface deformations in fluids and also measurementof the concentration gradient profile of liquids. Figure 4(a)shows a typical sequence of four shearograms obtained in sin-gle shot, and Fig. 4(b) shows the resulting unwrapped phase.Figure 5(a) shows sets of four experimental interferograms

FIGURE 4. Static oil drop for the case of lateral shear. (a) Set offour shearograms obtained in single shot. (b) Unwrapped phaseshowing the directional derivative.

Rev. Mex. Fıs. 57 (3) (2011) 255–258

258 D.I. SERRANO-GARCIA, et al.

FIGURE 5. Static oil drop for the case of lateral and radial shear-ing combined. (a) Interference patterns with phase shifts ofπ/2.(b) Unwrapped phase distributions of the interferograms sheared inradial direction.

obtained combining linear shear and radial shear and 5(b)the resulting unwrapped phase. Experimental results are pre-sented using four interference patterns to retrieve the opticalphase data using the four steps algorithm [20-21]. Each ofthe interferograms was subjected to the same scaling (0-255gray levels) and low-pass filtering process before phase cal-culation.

6. Conclusions

A cyclic path shearing interferometer coupled to a 4-fsystemwith a phase grid, to achieve simultaneouslyn-shearograms

for phase measurement by phase-shifting techniques has beenimplemented for the case of lateral and radial shear. The sys-tem presented can use other grating types but the irradianceratios, fringe modulation values and polarization distributionchanges. The system is mechanically stable against exter-nal vibration and can be used in beam characterization, mi-croscopy, tomography, holography, phase slope measurementor optical testing.

Acknowledgements

One of the authors (NITA) occupies a second year to post-doctoral research position at CIO and expresses sincere ap-preciation to Luisa, Miguel, and Damian for the support pro-vided, and to CONACYT for grant 102137/43055. Partialsupport from El Consejo Nacional de Ciencia y Tecnolo-gia (CONACYT) and Centro de Investigaciones enOpticaA.C. (CIO) through projects 290597 (CONACYT-CIO) and124145 (CONACYT-BUAP) is also acknowledged. The au-thor DISG (Grant:227470/31458) is very grateful to CONA-CyT for the graduate scholarship granted and expresses sin-cere appreciation to Geliztle. ATQ is grateful to CONACYTfor financial support through scholarship 227470/31458.

1. M.V. Mantravadi,Lateral shearing interferometers2nd ed. (D.Malacara, ed. Wiley, New York, 1992).

2. M.V. Murty, Appl. Opt.12 (1973) 2765.

3. K. Mastuda, Y. Minami and T. Eiju,Appl. Opt.31 (1992) 6603.

4. D. Pal Ghai, S. Vyas, P. Senthilkumaran, and R.S. Sirohi,Op-tics and Lasers in Engineering46 (2008) 419.

5. K. Patorski,Appl. Opt.27 (1988) 3567 .

6. D. Malacara,c.4 in Optical Shop Testing3nd ed. (D. Malacara,ed. Wiley, New York, 2007).

7. T. Kreis,J. Opt. Soc. Am. A3 (1986) 847.

8. A. Cornejo-Rodriguez,Ronchi test, c.9 in Optical Shop Testing3nd ed., (D. Malacara, ed. Wiley,New York, 2007).

9. P. Hariharan and D. Sen,J. Sci. Instrum.37 (1960) 374.

10. R.F. Horton,Opt. Engineer.27 (1988) 1063.

11. D. Malacara,Appl. Opt.13 (1974) 1781.

12. M.V. Murty, Appl. Opt.3 (1964) 853.

13. R.F. Horton,Opt. Engineer.27 (1988) 1063.

14. G. Rodrıguez-Zurita, C. Meneses-Fabian, N. Toto-Arellano,J.F. Vazquez-Castillo, and C. Robledo-Sanchez,Opt. Express16 (2008) 7806.

15. N. Toto-Arellano, G. Rodrıguez-Zurita, C. Meneses-Fabian,and J.F. Vazquez-Castillo,Opt. Express16 (2008) 19330.

16. G. Rodrıguez-Zurita, N. Toto Arellano, C. Meneses-Fabian,and J. Vazquez-Castillo,Opt. Letters33 (2008) 2788.

17. N.I. Toto-Arellano, A. Martınez-Garcıa, G. Rodrıguez-Zurita,J.A. Rayas-Alvarez, and A. Montes-Perez,Appl. Opt.49(2010)6402.

18. D.K. Sharma, R.S. Sirohi, and M.P. Kothiyal,Appl. Opt.23(1984) 1542.

19. B. Barrientos-Garcıa, A.J. Moore, C. Perez-Lopez, L. Wang,and T. Tschudi,Appl. Opt.38 (1999) 5944.

20. B. Barrientos-Garcıa, A.J. Moore, C. Perez-Lopez, L. Wang,and T. Tschudi,Opt. Eng.38 (1999) 2069.

21. D. Malacara, M. Servin, and Z. Malacara,c.6 in Phase detec-tion algorithms in Interferogram Analysis for Optical Testing(Marcel Dekker, New York 1998).

Rev. Mex. Fıs. 57 (3) (2011) 255–258

INSTRUMENTACION REVISTA MEXICANA DE FISICA 57 (3) 259–265 JUNIO 2011

Sistema automatizado para la medicion de la conductividad termica de lıquidosmediante el metodo del alambre caliente

S. Alvarado, E. Marın∗, A.G. Juarez, y A. CalderonCentro de Investigacion en Ciencia Aplicada y Tecnologıa Avanzada, Instituto Politecnico Nacional,

Legaria 694, Colonia Irrigacion, Mexico D.F., 11500 Mexico.

R. IvanovFacultad de Fısica, Universidad Autonoma de Zacatecas,

Calz. Solidaridad Esquina Paseo de la Bufa s/n, Zacatecas, Zac., 98060, Mexico.


Se presenta el montaje y puesta a punto de un sistema experimental automatizado basado en la tecnica del alambre caliente, para medir laconductividad termica de lıquidos, empleando equipos de alta precision (fuente de corriente y medidor de voltaje) que permiten prescindir deun arreglo de resistencias conocido como puente de Wheatstone, comunmente implementado en instrumentos de este tipo. Tambien se utilizaun criterio para verificar que se trabaja sobre la region lineal adecuada de la curva∆T versus ln(t), que es la utilizada para el procesamientode los datos experimentales. Finalmente se valida el funcionamiento del montaje experimental mediante mediciones de conductividad termicaen lıquidos de propiedades termicas bien conocidas.

Descriptores:Tecnica hot-wire; conductividad termica; nanofluidos.

We present the implementation of an automated system based on the hot-wire technique for measurement of the thermal conductivity ofliquids using high precision equipment (current source and voltage meter) that allow to work without an array of resistances known as aWheatstone bridge, commonly used in such equipments. We also use a criterion to verify that we are working on the correct linear region ofthe curve∆T versus ln(t), which is used for experimental data processing. Finally we validate the functionality of the hot-wire experimentalarray by measuring the thermal conductivity in samples of liquids with well-known thermal properties.

Keywords: Thermal conductivity; hotwire technique; nanofluids.

PACS: 65.20.-w; 66.25.+g; 66.30.Xj; 66.30.Xj; 81.70.Pb

1. Introduccion

Sistemas refrigerantes mas eficientes son indispensables endiferentes industrias, como la electronica y la automotriz [1],por lo que aumentar la eficiencia de la transferencia de calores importante ya que el calor generado durante la operacionde dispositivos mecanicos y electronicos debe ser extraıdoeficazmente para evitar su rotura o dano permanente.

Suspensiones estables de partıculas solidas de dimensio-nes nanometricas en solventes apropiados, los llamados na-nofluidos, han demostrado tener valores elevados de conduc-tividad termica,k, con respecto al fluido base, por lo cual es-tos compuestos representan una ruta atractiva para ser usadospara lograr la disipacion eficiente y efectiva del calor que segenera en diferentes sistemas. Diferentes autores han repor-tado incrementos significativos de la conductividad, algunosde los cuales sobrepasan lo que predicen las teorıas existen-tes (de medios efectivos) [2-4], lo que ha motivado muchostrabajos tanto para proponer nuevos mecanismos de transfe-rencia de calor [5], como para proponer correcciones a losmodelos asociados a las variantes experimentales usadas paralas mediciones [6], pasando por la propuesta de nuevas tecni-cas de medicion [7-9]. Sin embargo se ha venido trabajandopara lograr un consenso en cuanto a cuales son las causas delmencionado aumento en la conductividad termica, aunque re-cientemente un grupo internacional de autores de alrededorde 30 instituciones reportaron [10] el resultado del llamado

INPBE (International Nanofluid Property Benchmark Exer-cise), en el cual midieron ese parametro en muestras identicasde varios tipos de nanofluidos con diferentes tecnicas, llegan-do a la conclusion de que los incrementos en la conductividadtermica en general coinciden con los predichos por las teorıasde medios efectivos [11].

De cualquier manera, como la mayorıa de los trabajos re-portan la medicion de la conductividad termica de estos siste-mas con la tecnica del alambre o hilo caliente (HW del inglesHot-wire), un primer paso en cualquier investigacion enfoca-da en esta direccion es el disenar, fabricar y poner a punto unsistema basado en esta tecnica.

La tecnica del alambre caliente consiste en mantener in-merso un alambre conductor en el lıquido a investigar, y ha-cer pasar a traves deel una corriente electrica constante, demanera que se caliente mediante efecto Joule. Debido a ladisipacion del calor generado hacia la muestra mediante con-duccion, la temperatura (T) del alambre varıa en el tiempo (t)y la cinetica de esta variacion depende de las propiedadestermicas del lıquido. La conductividad termica se puede de-terminar a partir de la curva∆T versus ln(t) con ayuda deuna formula que es obtenida mediante la resolucion de laecuacion de difusion de calor con las condiciones de fronteraapropiadas. Debido a la complejidad del calculo matemati-co y de las ecuaciones que resultan, la mayorıa de los autoressimplifican la solucion haciendo muchas aproximaciones, en-

260 S.ALVARADO, E. MARIN, A.G. JUAREZ, A. CALDERON Y R. IVANOV

tre ellas la de despreciar las dimensiones laterales del alam-bre y del contenedor, las perdidas de calor por convecciony radiacion, entre otras. De esa manera coinciden en utilizarcomo rango de medicion aquel donde se tiene una relacionlineal entre la variacion de temperatura y el logaritmo natu-ral del tiempo de medicion, para cuya seleccion es necesariocontar con un criterio que permita garantizar que se ha ele-gido la region lineal correcta. Por otra parte, la medicion delas variaciones de temperatura en el alambre generalmente selleva a cabo midiendo los cambios en su resistividad usan-do un puente de resistencias tipo Wheatstone, y despreciandoen muchos casos las variaciones con la temperatura de suscomponentes.

Por todo ello el objetivo de este trabajo es implemen-tar la tecnica del alambre caliente sin necesidad de utilizarun puente de resistencias, y emplear un criterio de suficien-cia para seleccionar el rango de medicion donde se cumplala condicion de linealidad en la curva de∆T versus ln(t).La metodologıa propuesta y la funcionalidad del aparato ex-perimental disenado y puesto a punto fueron validadas conmediciones en varios lıquidos de propiedades termicas bienconocidas.

2. Marco teorico

La tecnica del alambre caliente es parte de un grupo de meto-dos que emplean un flujo transitorio de calor para determinarla conductividad termica. En particular se considera un buenmetodo para determinar la conductividad termica de materia-les que pueden acomodarse alrededor de un alambre delga-do, lo que la hace ideal para medir ese parametro en lıquidos.Otra ventaja es que se pueden despreciar los efectos de trans-ferencia de calor por conveccion, ya que cuandoesta existese aprecian desviaciones en la linealidad de la grafica de∆Tversus ln(t). Por otra parte, el mismo alambre funciona comofuente de calor y sensor de temperatura en las mediciones,como veremos mas adelante.

El modelo matematico desarrollado para el metodo delalambre caliente [12], considera una fuente lineal de calor,infinitamente larga y delgada, con una distribucion de tem-peratura uniforme, que disipa un flujo de calor por unidad delongitud en un medio homogeneo e infinito. La suposiciongeneral es que la transferencia de calor hacia el medio essolamente por conduccion y por lo tanto se incrementan en el

FIGURA 1. (a) Esquema que muestra las conexiones entre los componentes involucrados en la implementacion de la tecnica Hot-wire.(b) Fotografıa del montaje de la tecnica del alambre caliente, donde: A) celda de medicion, B) control de temperatura, C) caja de conexioneselectricas, D) nanovoltımetro, E) fuente (Sourcemeter) y F) computadora con software Hot-wire. (c) Imagen en la que se pueden apreciar loscomponentes de la celda de medicion y su ensamble.

Rev. Mex. Fıs. 57 (3) (2011) 259–265

SISTEMAAUTOMATIZADO PARA LA MEDICI ON DE LA CONDUCTIVIDAD TERMICA DE LIQUIDOS MEDIANTE. . . 261

tiempo la temperatura de la fuente y la del medio de prueba.La ecuacion que gobierna esta tecnica se deriva de la ecua-cion general de difusion de calor de Fourier y esta dada por

∆T =q

4πk

[−γ + ln

4αt

r2

](1)

dondeq es la potencia disipada por unidad de longitud delalambre,k es la conductividad termica del medio,γ es lacontante de Euler,α es la difusividad termica,t es el tiempode medicion y r es el radio del alambre utilizado.

Por lo tanto despues de desarrollar la parte logaritmica dela Ec. (1), derivarla con respecto aln(t) y despejar la conduc-tividad termica (k) se obtiene:

k =q

4π

[d∆T

d(ln t)

]−1

(2)

Porlo tanto, si la temperatura del alambre se mide en fun-cion del tiempo, la conductividad termica del medio (k) en elque esta inmerso es proporcional al flujo de calor por uni-dad de longitud del alambre e inversamente proporcional a lapendiente de la region lineal de la curva∆T versus ln(t).

Es importante senalar que en la practica el alambre tienedimensiones finitas. Si consideramos que su radio es igual aay su longitud2b, que esta inmerso en un medio cilındrico deradior, y que la temperatura es medida en su centro, enton-ces, la solucion de la ecuacion de difusion de calor serıa [13]:

∆T =q

4πk

∞∫

r2/4αt

u−1e−ue−(q/r)2uI0

× 2au

rerf

(b

r

√u

)du (3)

dondeI0 es la funcion de Bessel modificada de orden 0, yerf (x) es la funcion error. Suponiendo que el medio que ro-dea al alambre tiene dimensiones mucho mas grandes queeste, entonces se cumple que:

exp[(−a

r

)2

u

]I0

2au

r→ 1 cuando

a

r→ 0 (4)

erf

[b

r

√u

]→ 1 cuando

b

r→∞ (5)

y por consiguiente, de la Ec. (3) se obtiene la Ec.(1). De-bemos senalar que en la deduccion de la Ec. (3) y en la dela Ec. (1) se continua suponiendo que el medio en el quese encuentra el alambre es homogeneo e isotropico, que laconductividad termica del alambre es infinitamente grande,se desprecia la conveccion termica en el medio y las perdidasde calor por radiacion, entre otras. Aunque estas suposicionesno son estrictamente satisfechas en la practica, las aproxima-ciones que se hacen son adecuadas para obtener medicionesprecisas de la conductividad termica, si el experimento se rea-liza convenientemente y los datos experimentales se procesande manera apropiada, como veremos en la proxima seccion.

3. Experimental

3.1. Sistema de medicion

En la Fig. 1(c) se muestra el sistema de medicion propues-to. La parte (a) representa el diagrama de conexiones electri-cas mientras que en la (b) aparece una fotografıa del equipocon las partes que lo constituyen. Este esta conformado poruna celda de medicion, una fuente de corriente (Sourceme-ter, Keithley 2400), un nanovoltımetro (Keithley 2182A), yuna computadora que comunica y controla a la fuente y elnanovoltımetro a traves de una interface GPIB (General Pur-pose Interface Bus) y de un software hecho en el ambientede programacion LabViewR©(National Instruments), el cualpermite tambien llevar a cabo la adquisicion, almacenamien-to y procesamiento de los datos obtenidos. Los parametroscontrolados a traves del software son: el voltaje y la corrien-te que suministra la fuente, la cantidad de mediciones y lavelocidad con que las realiza el nanovoltımetro, ası como elnombre asignado al archivo de datos para su almacenamien-to.

En Fig. 1(c) se muestra la fotografıa de la celda de me-dicion, la cual esta conformada por un recipiente cilındricode cobre, y un soporte de NylamidR©M que permite suje-tar y mantener tenso al alambre de platino (Alfa Aesar) quetiene 0.12 m de longitud y 76.2µm de diametro, cumplien-do ası con las condiciones (4) y (5) para ser considerado unalambre de radio cero y longitud infinita. El volumen de lıqui-do que se requiere para cubrir completamente al alambre deplatino es de 162 ml.

Como parte del sistema tambien se diseno un sistema de“bano Marıa” automatico para mantener la celda de medi-cion a una temperatura inicial deseada y estable. Este con-trol de temperatura emplea como calefactor una resistenciaelectrica, la cual es encendida y apagada por un interrup-tor electromecanico (Tianbo HJR-3FF-S-Z), el cual a su vezes controlado por un control digital de temperatura (YuyaoXMTG818) que permite verificar constantemente la tempe-ratura del bano a traves de un termopar tipo K. En el controldigital de temperatura se programa la temperatura deseada.El rango de desactivacion o activacion del sistema de calen-tamiento es de± 1 grado respectivamente. Todos estos ele-mentos estan colocados en una hielera (Coleman) de 15.1 L.

3.2. Metodologıa para la determinacion de la conducti-vidad termica

La conductividad termica de un lıquido puede ser calculada apartir de la Ec. (2), conociendo la potencia disipada por uni-dad de longitud del alambre de platino (q), y sus variacionesde temperatura con respecto al logaritmo natural del tiempode medicion. Pero debido a que en nuestro casoq es gene-rado por efecto Joule al hacer pasar una corriente electricade intensidadI a traves del alambre de resistenciaR0, y las

Rev. Mex. Fıs. 57 (3) (2011) 259–265


TABLA I. Resultados de la conductividad termica de la glicerina a 20C para diferentes valores de corriente electrica.

Glicerina(99.90 % J.T. Baker)

I [mA] 60 70 80 90 100

k [W/mk] 0.29± 0.01 0.29± 0.01 0.29± 0.01 0.29± 0.01 0.29± 0.01

variaciones de temperatura (∆T) las medimos de manera in-directa a traves de las variaciones de voltaje (∆V), resultapertinente reescribir la Ec. (2) en terminos de esas variables.Luego, considerando que la resistencia electrica del platinodepende de la temperatura segun:

R(t) = R0(1 + σ∆T (t)) (6)

donde σ es el coeficiente termico resistivo del platino,σ=(0.00335±0.00003) K−1 [12], usando la Ley de Ohm po-demos escribir:

V (t) = R(t)I = R0(1 + σ∆T (t))I (7)

Sustituyendo (1) en (7) se obtiene

V (t) = IR0

1 + σ

q

4πk

[ln

(4αt

r2

)− γ

](8)

donde:

q =I2R0

L(9)

Simplificandoy derivando la Ec. (8) con respecto aln(t), ob-tenemos:

d∆V (t)d ln(t)

= m =I3R2

0σ

4πLk(10)

donded∆V (t)/d ln(t) es el valor de la pendiente (m) quese obtiene al hacer un ajuste de mınimos cuadrados sobre laregion lineal de la curva∆V versusln(t)

Para obtener la conductividad termica se sigue el siguien-te procedimiento: Primero se suministra un pulso de corrienteelectrica, durante el cual se mide el incremento de la diferen-cia de potencial (∆V) con respecto al tiempo. Segundo sehace una grafica de∆V versusln(t), en la que se hace unajuste de mınimos cuadrados sobre la region lineal obtenien-do ası su pendiente y finalmente empleando la Ec. (10) secalcula la conductividad termica del lıquido analizado. Estees basicamente el procedimiento empleado por la mayorıa delos autores, sin embargo debemos verificar que se haya ele-gido la region lineal correcta, ya que podrıa suceder que nosubicaramos en otra region lo suficientemente pequena de lacurva obtenida y apreciasemos un comportamiento lineal, loque nos llevarıa a obtener resultados errados.

Por lo anterior se sigue el procedimiento que se describea continuacion [14].

3.2.1. Verificacion de la linealidad de la curva∆T versusln(t)

La Ec. (10) se puede rescribir de la siguiente manera:

m = BI3 (11)

donde:

B =R2

0σ

4πLk(12)

Al aplicar logaritmo natural sobre la Ec. (11) obtenemos:

ln(m) = 3 ln(I) + ln(B) (13)

Supongamos que se realizan mediciones dem usando di-ferentes valores de corriente electrica segun el procedimientoindicado en la seccion anterior y graficamosm versusI endoble escala logarıtmica. Notese que la Ec. (13) representauna recta de pendientem′= 3 en una grafica deln(m) en fun-cion deln(I). Si las regiones elegidas en las curvas de∆Vversusln(t) como lineales son correctas al hacer un ajuste demınimos cuadrados el valor de la pendientem′ debe ser iguala tres.

3.2.2. Determinacion de la conductividad termica

Por otra parte, vemos de la Ec. (11) que al hacer una grafi-ca dem en funcion deI3 se obtiene una recta de pendienteigual aB, a partir de la cual podemos calcular la conducti-vidad termica usando la Ec. (12). ParaR se utilizara el valorpromedio obtenido de los valores medidos antes de cada ex-perimento.

4. Resultados experimentales

Para realizar la calibracion y verificar el correcto funciona-miento de nuestro instrumento se llevo a cabo la medicionde la conductividad termica de los siguientes lıquidos: glice-rina, etilenglicol, agua, metanol y aceite de motor SAE 50,comunmente utilizados como patrones [14-17], por poseerpropiedades termicas bien conocidas. La temperatura de lasmuestras fue mantenida a (290±1) K.

En la Fig. 2 se muestra la grafica de∆V versusln(t)medida en una muestra de Glicerina paraI=80 mA. ∆Vrepresenta la diferencia de potencial entre el potencial trans-currido un tiempo determinado y el potencial al instanteinicial. La lınea continua representa el resultado del me-jor ajuste lineal mediante mınimos cuadrados del cual secalculo la pendiente y con su valor, y usando la Ec. (10) con

Rev. Mex. Fıs. 57 (3) (2011) 259–265


FIGURA 2. Grafica del aumento de la diferencia de potencial ver-sus logaritmo natural del tiempo con ajuste de mınimos cuadradossobre la region lineal.

FIGURA 3. Grafica del aumento de la diferencia de potencial enfuncion del tiempo, a diferentes valores de corriente, para unamuestra de glicerina.

R0=(2.5816±0.0001)Ω, se obtiene para la conductividadtermica el resultado

kmed = (0.29± 0.01) W/mk (14)

que coincide con el valor reportado en la literatura(krep=0.286 W/mk).

Con el objetivo de aplicar el criterio de verificacion pro-puesto en 3.2 se realizaron una serie de mediciones con es-te mismo fluido pero aplicando distintos valores de corrienteelectrica (I), que van de los 50 a los 110 mA. En la Fig. 3se presentan las curvas de∆V versust obtenidas para estosvalores de corriente.

Como ya se menciono anteriormente el modelo teorico dela tecnica del alambre caliente solo contempla transferenciade calor por conduccion y una gran ventaja de esta tecnicaes que permite identificar facilmente cuando se presenta con-veccion ya queesta provoca desviaciones en la curva de∆Vversust. Como se puede apreciar en el grafico de la Fig. 3

FIGURA 4. Grafica del aumento de la diferencia de potencial ver-sus logaritmo natural del tiempo con diferentes valores de corrientey su respectivo ajuste de mınimos cuadrados sobre la region lineal.

FIGURA 5. Grafica con doble escala logarıtmica de las pendien-tes (m) en funcion de los valores de corriente empleados. La lıneacontinua representa el ajuste de mınimos cuadrados para los puntosobtenidos.

al aumentar el valor de la corriente electrica, la temperaturase eleva mas rapidamente y en la medicion realizada con 110mA, a tiempos cortos se observan desviaciones en la trayecto-ria de la curva que revelan la presencia de conveccion, por loqueesta medicion no se emplea para determinar la conduc-tividad termica (k) de la glicerina. Otra curva que tambiense descarta es la de 50 mA, debido a que con esta corrienteelectrica la variacion de temperatura es muy pequena.

En la Fig. 4 se presenta la grafica con las curvas de∆V versusln(t), empleando los distintos valores de corrienteelectrica, y su respectivo ajuste de mınimos cuadrados sobrela region lineal de las curvas. Una vez obtenidos los valoresde las pendientesm, estos se emplean en la Ec. (10) junto conlos demas parametros involucrados y se obtienen los resulta-dos que se muestran en la Tabla I. Es importante senalar quelos ajustes realizados en las Figs. 2 y 4 solamente se reali-zan en las regiones mostradas, porque al extenderlos hasta el

Rev. Mex. Fıs. 57 (3) (2011) 259–265


FIGURA 6. Grafica de las pendientes (m) en funcion de los valoresde corriente elevados al cubo (I3). La lınea continua representa elajuste de mınimos cuadrados para los puntos obtenidos.

TABLA II. Resultados obtenidos para la conductividad termica dedistintos lıquidos a 20C a partir de la metodologıa propuesta yvalores de la pendiente de la curvaln(m) vs. ln(I)

Lıquido Pendiente (b) k (medido) k (literatura)

(ln(m) vs ln(I) W/mK W/mK

Etilenglicol

(J.T. Baker 99.90 %) 3.01± 0.01 0.252± 0.009 0.256

Agua destilada Q.P. 3.010± 0.007 0.59± 0.03 0.609

Metanol

(J.T. Baker 99.90 %) 3.03± 0.02 0.208± 0.009 0.202

Aceite SAE 50

(Quaker State) 3.02± 0.03 0.141± 0.006 0.145

Glicerina

(J. T. Baker 99.90 %) 3.03± 0.01 0.288± 0.006 0.286

final de las curvas se observo que los valores de la pendien-te se hacen ligeramente mayores posiblemente debido a quecomienzan a establecerse efectos convectivos, resultando va-lores de conductividad termica que pueden diferir hasta en un8 % del valor reportado.

A partir de la grafica de la Fig. 5 podemos decir que lasregiones lineales elegidas en las curvas de∆V versusln(t)(Fig. 4) son las correctas debido a que la pendiente del ajus-te de mınimos cuadrados de la graficaln(m) versusln(I) esaproximadamente igual a tres.

Porultimo, de la grafica dem versusI3 de la Fig. 6, ob-tuvimos el valor del coeficienteB mediante ajuste lineal porel metodo de mınimos cuadrados, a partir del cual empleandola Ec. (12) se calcula la conductividad termica, obteniendose:

k = (0.288± 0.006)W/mk (15)

valor que coincide con el obtenido en las mediciones con dis-tintos valores de corriente (ver Tabla I) y el reportado en laliteratura.

Aplicando el mismo procediendo para el resto de loslıquidos medidos se obtuvieron los resultados que se mues-tran en la Tabla II.

5. Conclusiones

En este trabajo se monto y puso a punto un sistema experi-mental para la medicion de la conductividad termica de lıqui-dos segun el metodo del alambre caliente. A diferencia detrabajos previos que emplean un puente de Wheatstone paramedir las variaciones de temperatura inducidas en el alambre,en este trabajo disenamos un sistema de medicion automati-zado que hace uso de una fuente de corriente y de un me-didor de voltaje de alta precision, y se utilizo un criterio desuficiencia para verificar que el ajuste de mınimos cuadradosse realice sobre la region lineal correcta en las curvas de∆Tversusln(t). El sistema y la metodologıa de medicion pro-puesta fueron probados satisfactoriamente midiendo la con-ductividad termica de lıquidos, de propiedades termicas bienconocidas, como la glicerina, etilenglicol, agua, metanol yaceite para motor SAE 50. En los resultados obtenidos a par-tir de las ecuaciones del criterio de verificacion se tiene unaincertidumbre que varıa entre 1.5 y 2 %, que es casi la mitadque la obtenida en los resultados determinados directamentea partir de las curvas de∆V versusln(t) con distintos valo-res de corriente. Estos errores son muy bien aceptados paracualquier tecnica de medicion y en particular para las de ca-racterizacion termica. Hay que mencionar que el instrumentodisenado ofrece ventajas respecto a los pocos que se ofertanen el mercado [18] fundamentalmente en cuanto al costo ya que aquellos funcionan como “cajas negras” que arrojanel resultado de la medicion y no ofrecen al investigador laposibilidad de manipular los datos y establecer criterios decomprobacion como los aquı utilizados.

Agradecimientos

A CONACyT por su apoyo economico mediante el proyectoNo. 83289, a la SIP-IPN por su apoyo a traves de los proyec-tos, 220090160, 20100780 y 20100622 y a COFAA-IPN porel soporte a nuestro trabajo a traves de los programas SIBEy PIFI. Tambien deseamos agradecer al Dr. C. Falcony, delDepartamento de Fısica del CINVESTAV, Mexico D.F., porpermitirnos el uso temporal de parte del equipamiento de suLaboratorio. Porultimo se agradece sinceramente alarbitroanonimo por susutiles comentarios.

Rev. Mex. Fıs. 57 (3) (2011) 259–265


∗. e-mails: [email protected]; [email protected]

1. A.R.A. Khaled y K. Vafai.,Intern. J. Heat Mass Transf.48(2005) 2172.

2. U.S. Choi,Nanofluid Technology: Current status and future re-search(U.S. Department of Energy, Office of Scientific & Te-chnical Information, 1999).

3. J.A. Eastmanet al., Materials science forum312(1999) 629.

4. X.Q. Wang y A R Mujumdar,Int. J. Th. Sci. 46 (2007) 1.

5. D.G. Cahillet al., J. Appl. Phys. 93 (2003) 793.

6. J.J. Vadaszet al., Int. J. Heat and Mass Transfer48 (2005)2673.

7. E. Marın, Internet Electron. J. Nanoc. Moletron5 (2007) 1007.

8. E. Marın, A. Calderon y D. Dıaz,Analytical Sciences25 (2009)705.

9. R. Ivanov, E. Marin, I. Moreno y C. Araujo,J. Phys. D: Appl.Phys.,43 (2010) 225501.

10. J. Buongiornoet al., J. of Appl. Phys.106(2009) 094312.

11. Ce-Wen Nan, R. Birringer, D.R. Clarke y H. Gleiter,J. Appl.Phys.81 (1997) 6692.

12. H.S. Carlslaw y J.C. Jaeger,Conduction of Heat in Solids(Ox-ford University Press, London, 1959).

13. G.J. Kluitenberg, J.M. Ham, y K.L. Bristow,Soil Sci. Soc. Am.J. 57 (1993) 1444.

14. M. Khayet y J.M. Ortiz de Zarate,Int. J. Thermophysics26(2005) 3.

15. R.A. Perkins, M.L.V. Ramires y C.A. Nieto de Castro,J. Res.Natl. Inst. Stand. Technol.105(2000) 255.

16. W.A. Wakeham y M. Zalaf,Fluid Phase Equilib. 36(1987) 183.

17. J.P. Garnieret al., Int. J. of Thermophysics, 29 (2008) 468.

18. KD2 user manual(Decagon Incorporated, 2006).

Rev. Mex. Fıs. 57 (3) (2011) 259–265

INSTRUMENTACION REVISTA MEXICANA DE FISICA 57 (3) 266–275 JUNIO 2011

The shadowgraph imaging technique and its modern applicationto fluid jets and drops

R. Castrejon-Garcıaa, J.R. Castrejon-Pitab, G.D. Martinb, and I.M. HutchingsbaCentro de Investigacion en Energıa, Universidad Nacional Autonoma de Mexico,

Priv. Xochicalco s/n, Temixco, Mor. 62580, Mexico.bInstitute for Manufacturing, University of Cambridge, 17 Charles Babbage Road,

Cambridge, CB3 0FS, United Kingdom.


The shadowgraph technique is discussed in terms of some modern application to fluid visualization and the characterization of free-surfaceflows. A brief description of shadowgraph photography is presented which emphasizes the parameters which need to be controlled to obtainuseful images for digital processing and analysis. Several examples of shadowgraph images are presented together with their analysis toexemplify the variety of quantitative measurements that are possible with modern imaging tools. Three experimental setups were developedfor different novel applications at different length scales: centimeter-long jets, millimeter-size sprays and micrometer-size droplets. Thecomponents used for these shadowgraph imaging systems range from standard photographic flash sources to specialized spark flash lampsand from single lens reflex (SLR) camera lenses to purpose-built systems with research-grade optical elements. Image analysis was developedand used to derive various types of information from the different systems. For jets it was used to determine the dynamic surface tension ofthe jetted fluid, for sprays it was employed to determine the ligament-droplet size distribution along the jetting axis and in inkjet printing itwas used to determine the angular deviation and terminal speed of the jets and droplets.

Keywords: Sprays; droplets; jets; shadowgraph.

La tecnica de fotografıa de sombra se discute en terminos de algunas de sus aplicaciones modernas en la visualizacion de fluidos y lacaracterizacion de flujos con superficie libre. La fotografıa de sombra se describe brevemente, haciendo hincapie en los parametros que sedeben controlar para obtener imagenes aptas para el procesamiento digital y analisis. Se presentan tambien algunos ejemplos de imagenesde fotografıa de sombra con su correspondiente analisis, para ejemplificar la variedad de mediciones cuantitativas que pueden hacerse conlas herramientas modernas de imagen. Se desarrollaron tres montajes experimentales para algunas aplicaciones novedosas: chorros decentımetros de largo, aerosoles de milımetros de tamano y gotas de micras de tamano. Los componentes que se utilizan en estos sistemas defotografıa de sombra van desde fuentes de destello estandar, hasta lamparas especializadas de destello con chispa; y desde camaras reflex deun solo lente (SLR), hasta sistemas construidos expresamente con elementosopticos de alta calidad. El metodo de analisis de las imagenes fuedesarrollado para obtener diversos tipos de informacion de los diferentes sistemas; en los chorros se usa para encontrar la tension superficialdinamica del fluido eyectado; en los aerosoles se usa para determinar la distribucion de tamano de los ligamentos y las gotas a lo largo deleje del aerosol; y en los chorros de impresion de tinta se usa para determinar su desviacion angular y la velocidad terminal de los chorros ylas gotas.

Descriptores: Aerosoles; gotas; chorros; fotografıa de sombra.

PACS: 47.80.Jk; 47.55.db; 47.80.Jk; 81.15.Rs; 82.70.Rr

1. Introduction

Among the different optical techniques used in the study ofparticles, liquids or gases in motion, shadowgraph imagingstands out as a usually inexpensive but powerful technique.Traditionally, the shadowgraph technique has been often usedto qualitatively study the dynamics of fluids through the vi-sualization of flows or the recording of flow streamlines. Theaim of this paper is to provide some examples where shad-owgraphy can be used in conjunction with digital image pro-cesses to quantitatively determine fluid and flow properties,and to show how shadowgraphy can be used as a quantitativetechnique.

In general terms, shadowgraph photography highlightsthe difference in refractive index at the interface between abody and its surroundings (medium), or between media, [1].With back-illumination, the light that does not interact withthe object produces a bright background, whereas the light

refracted at the interface is dispersed and thus the interfaceappears dark. As a consequence, shadowgraph images con-sist of a bright background and the shadows of the interfacesbetween regions with different refractive indexes. Generallyspeaking, the use of shadowgraphy is appropriate when themedium being studied is transparent and for media with largedifferences in refractive index (e.g. air and water). Mostshadowgraph systems consist of two main components: theillumination source and the recording element. In brief, the

FIGURE 1. Generic diagram of a shadowgraph system.

THE SHADOWGRAPH IMAGING TECHNIQUE AND ITS MODERN APPLICATION TO FLUID JETS AND DROPS 267

FIGURE 2. (Colour online) Schematic diagrams of the optical andillumination setups used for the different applications. The top il-lustration shows the system used for jets, the middle one for spraysand the bottom for droplets. All the Nikon DSLR cameras had a23.6×15.8 mm 10.4 Megapixel CCD sensor. Dimensions not toscale.

aim of the illumination arrangement is to obtain a homoge-neous background at the recording sensor and to provide thelight which is refracted by the object or media under study.The recording system contains the optical elements requiredto form and record an image of the object’s silhouette orshadow. Usually, short-duration illumination is provided by aflash-lamp, spark generator or LED, and film or a CCD sen-sor are used as recording elements.

Although the optical principles behind this technique aresimple, it does not always lend itself to easy laboratory im-plementation. Figure 1 shows a typical schematic view ofa shadowgraph system and exemplifies the simplicity of thedesign. However, there are several variables that must becontrolled and adjusted to obtain useful images for analysis.Generically, all shadowgraph systems are similar althoughthey differ in their detailed components. The specificationsof these components are important as the contrast, sharpnessand brightness of shadowgraph images depend on parame-ters such as the magnification, light sensitivity, field of view,depth of field and focal length of the optical system. Ide-ally, a shadowgraph system should be designed to fulfil somedesignated conditions of field of view, sharpness and bright-ness but in practice, some systems are designed more prag-matically, based on the (local or commercial) availability ofoptical components. Primarily, the size of the feature to beidentified on the images determines the optical magnifica-tion which is the dominant specification of the optical com-ponents. Other variables such as the depth of field and thesensor size and sensitivity are often chosen and adjusted toimprove the quality of the image formed on the sensor. Addi-tional variables such as the optical distortion and the workingdistance are usually chosen based on component availabilitybut with the aim to enhance the image quality or to facili-tate the operation of the system. To visualize centimeter ormillimeter-scale objects, conventional photography offers awide variety of optical elements such as lenses, flash light

sources and shutters. However, for micrometer-scale experi-ments, the ability to choose the depth of field and the work-ing distance independently is restricted by the availability ofmicroscope objectives and optical zoom assemblies. Theserestrictions may cause difficulties in the use of shadowgra-phy for a particular application and need to be known andcontrolled as part of the design of the system.

Shadowgraph photography has advantages over other vi-sualization techniques because it is non-invasive and allowsthe recording and in many cases measurement of several char-acteristics such as interface speed and direction of motion,and object sizes [2,3]. In this paper, it is shown how theprocessing of shadowgraph images can be used to quanti-tatively determine important properties of jets, sprays anddroplets. It is also shown that this visualization techniquecombined with digital image analysis can be employed to in-vestigate and quantitatively determine:i) the dynamic sur-face tension and the the surface profile of modulated jets inthe Rayleigh regime,ii) the ligament and droplet size distri-bution of sprays,iii) the straightness or angle of deviationof jetted droplets andiv) the effects of nozzle defects on thedirectionality of jetted droplets. The image analysis methodused in these studies is simple and generic and can be repro-duced in several numerical computing environments and pro-gramming languages (e.g.Matlab, LabView, or Visual C).These analyzes are based on tracing the outlines of objectsin shadowgraph images through the detection of local colouror intensity level gradients. Once the feature boundaries arefound and recorded, this information is then used to measurethe sizes of the objects and their positions. This data is thenutilized to measure different properties according to the par-ticular application.

2. Shadowgraph imaging

The three imaging systems described below were conceivedand built to cover three different regimes found in the ejectionof liquids through nozzles: single droplets, jets and sprays.These regimes are ultimately determined by the properties ofthe fluid, the geometry of the nozzle and the pressure historyapplied to the liquid. Generally, a single short pressure pulseproduces a single droplet, a sustained high pressure producesa jet and a long pressure pulse of higher amplitude creates anspray [5,10]. These three modes of fluid ejection are impor-tant as they are extensively used in industry to deliver and de-posit fluids in a variety of applications. In addition, the phys-ical dimensions of the experimental setups were chosen todemonstrate the use of the shadowgraph technique and digitalimage analysis over a wide range of conditions, ranging fromcentimeter-long jets down to micrometer-sized droplets. As aconsequence, most of the optical elements were different be-tween these three systems. The only component common toall systems was the 23.6×15.8 mm 10 Megapixel CCD cam-era sensor (in Nikon D40x and D80 DSLR cameras) whichwas used to capture all the shadowgraph images in this work.Fig. 2 illustrates the various components used.

Rev. Mex. Fıs. 57 (3) (2011) 266–275

268 R. CASTREJON-GARCIA, J.R. CASTREJON-PITA, G.D. MARTIN, AND I.M. HUTCHINGS,

FIGURE 3. (Colour online) Sectional view of the millimeter-scaleprinthead. Relative sizes are in correct proportion. The nozzleshape consists of a 45 degree (half-angle) conical inlet, followedby a parallel cylindrical section 2.2 mm in diameter and 2.2 mmlong.

2.1. Jets

The first setup was a shadowgraph system built to observeand record instantaneous images of a harmonically disturbedcontinuous liquid jet. The interest in these observations liesin the fact that the dynamics of such a system were theoret-ically described by Rayleigh (for inviscid liquids) in 1878and by Weber (for viscous fluids) in 1931, [11,12] and [2].The Rayleigh and Weber models predict the surface profileof a weakly modulated liquid jet emerging from a cylindri-cal nozzle. These models can be used in conjunction withshadowgraph imaging and analysis of centimeter-long jets toestimate the dynamic surface tension of the jetted fluid anddetermine the speed and size of the droplets formed after jetbreak-up. These studies are relevant to the printing industryas modulated jets form the basis of continuous ink-jet print-ers.

The model printhead used to generate and harmonicallydisturb the jets is a modified version of a design describedelsewhere [10]. Briefly, it consists of a sealed, pressur-ized liquid-filled chamber with the outlet nozzle in its lowersurface and a rubber membrane forming the upper face, asshown schematically in Fig. 3. The membrane is in contactwith the liquid on one surface and with an electromagneticvibrator on the other, and is used to transmit the oscillatorymotion from the actuator to the fluid. The vibrator (LDS Testand Measurement Ltd, model V201) is driven by an amplifier(Nikkai 250, total harmonic distortion<0.025%) connectedto a function generator (TTi model TG550). The liquid isfed into the chamber by a regulated pump (Promotec modelBL58) and emerges from the nozzle as a jet. The jet velocityis adjusted by varying the power applied to the pump and thepressure in the chamber is measured with a dynamic pressuretransducer (Entran Sensors & Electronics, EPX-N12-1B) in

FIGURE 4. Examples of shadowgraph pictures of jets formed fromglycerol/water mixtures. The jet is modulated with a 333 Hz si-nusoidal waveform and breaks up into droplets with a speed of3.9 m/s. Pressure modulation amplitudes are 4.8%, 3.5%and 2.6%respectively.

contact with the liquid. This transducer is used to monitorthe amplitude and shape of the pressure waveform producedby the function generator. In these experiments, a solutionof 74.7% (by mass) pure glycerol in triple-distilled waterwas jetted through a 2.2 mm diameter nozzle with a 45 de-gree (included angle) conical inlet and a cylindrical section2.2 mm long. The static pressure used to jet the fluid at3.50± 0.03 m/s was 11.4 kPa. The vibrator was set up toproduce a pressure modulation amplitude of 1% of that value

Rev. Mex. Fıs. 57 (3) (2011) 266–275


FIGURE 5. (Colour online) Shadowgraph images of quenching oil at 38C emerging from a 1.2 mm atomizer nozzle. The average terminalspeed of the spray droplets is 15 m/s.

FIGURE 6. (Colour online) Cutaway view of the nozzle used forthe production of oil sprays. The nozzle has a 45 degrees conicalinlet. Diagram is to scale.

(0.1 kPa) at a frequency of 333 Hz. The modulation pres-sure amplitude used in the experiments presented in this pa-per (1%) corresponds to the minimum modulation requiredto cause stable break-up of the jet. Under this conditions, the

jet breaks up at a distance of 27.5 cm from the nozzle plane(commonly called the break-up length).

The optical system used to visualize the centimeter-scalejets utilizes a vibration reduction Nikkor 18-55 mm f/3.5-5.6 lens mounted on a Nikon D80 DSLR camera body. Thisarrangement is set to produce a field of view of 34×23 cm,with a working distance of 45 cm and a depth of field of 5 cm.To obtain the shadowgraph illumination, single flashes wereproduced by phase-lock triggering of a SB-800 SpeedlightNikon Flash in the 1/128 option. This setting produces lightpulses of≈2 ms duration that are short enough to freeze themotion of the jet in the image. The flash was placed 2 maway from the jet and a generic projector screen was usedas an optical diffuser, at a distance of 30 cm from the jet, toproduce an even background illumination. The camera shut-ter was opened for 2 seconds in darkness so that the camerasensor only captured light from the flash. The triggering cir-cuitry was locked to the phase of the sinusoidal modulationused to break up the jet. No illumination lens was used in thissetup as the raw intensity of the flash perceived by the cam-era sensor at a sensitivity of 600 ISO was enough to producehigh quality images with good contrast, as shown in Fig. 4.The uniformity of the background was adjusted by varyingthe relative position of the diffuser, the flash and the jet. Theexposure time in this and the other two shadowgraph systemswas controlled by the flash duration and not by the camerashutter, and only single-flash photography was used.

2.2. Sprays

The second shadowgraph system discussed here was devel-oped for the study of sprays. Sprays and aerosols are used asdispensing systems in a wide range of applications, from thedelivery of materials from pressurized canisters (e.g.paints,deodorants, insecticides, etc.) to the delivery of combustiblesfor burning (e.g. internal combustion engines and oil com-bustion for power generation). Typically, a spray or an aerosol

Rev. Mex. Fıs. 57 (3) (2011) 266–275


FIGURE 7. (Colour online) Schematic and cutaway view of part ofthe nozzle plate from a commercial ink-jet printhead (Xaar XJ126).The nozzle profile presents a conical inlet with a final diameter ofca. 50µm. Relative sizes are in proportion.

is produced by the continuous flow of over-pressurized liquidthrough a nozzle. In a spray, the fluid emerges as a continuousjet until surface tension forces stretch and eventually breakup the liquid into droplets, Fig. 5. The distribution of thedroplets along the spray is determined by the fluid propertiesand also by aerodynamic effects. The analysis of shadow-graph images from sprays can be used to quantitatively ob-tain the droplet size distribution at different locations alongthe spray. This analysis can be used in practical applicationsto determine the optimum position of a substrate for spraydeposition or the best position of an ignition element in acombustion chamber.

Sprays were formed by forcing oil through a genericswirler nozzle with an oil pump at a measured pressure of460 kPa and a flow rate of 0.45 l/min; the geometry of thenozzle is shown in Fig. 6. After ejection, the fluid is col-lected and pumped back so that the process can be run con-tinuously. The nozzle outlet diameter is 1.2 mm. Under theseconditions the break-up process for quenching oil at 38Cstarts to occur at approximately 1.2 cm from the nozzle plane,and spray droplets are first obtained at a distance of approxi-mately 2.0 cm, as shown in Fig. 5.

The optical system used to visualize these sprays consistsof an Apo Nikkor 305 F/9 lens attached to a 610 mm exten-sion tube and mounted on a Nikon D80 DSLR camera bodyto produce a field of view of 23×31 mm, a working distanceof 40 cm and a depth of field of approximately 17 mm. A sin-gle generic biconvex lens of 110 mm diameter and 320 mmfocal length was used in the illumination system. This lensfocused the light pulses produced by an argon jet stabilizedspark generator (Pulse, Instrumentation and Controllers Ltd)on to a glass diffuser to produce the uniform background il-lumination. The nominal light pulse duration for this gener-ator is 300 ns with an energy of 2.5 J. The spray generatorwas positioned 610 mm away from the camera lens and at732 mm from the spark generator, resulting in uniform il-lumination. The triggering time of the spark generator wasexternally controlled by a series of SN74121 integrated cir-cuits interconnected to produce time delays within the rangeof 5 to 150 ms. The camera sensor was set at a sensitivityof 400 ISO. As in the previous setup, the camera shutter wasopened for 2 seconds in darkness and the exposure time con-trolled by the the spark duration. Examples of shadowgraphimages of these sprays are shown in Fig. 5.

FIGURE 8. Figure caption.(Colour online) Shadowgraph picture ofdroplets jetted from a Xaar XJ126 printhead. The droplets travel atan speed of 5 m/s and have a volume of 80 picoliters.

2.3. Droplets

The third shadowgraph system was designed to visualizemicrometer-size droplets from a commercially available ink-jet printhead. The aim of this study was to develop a tech-nique to quantify the directionality (straightness) of individ-ual droplets jetted from an ink-jet nozzle. The method usesdouble flash shadowgraph images of detached droplets gener-ated by a row of nozzles to calculate the flight direction. Thisstudy is of interest to the ink-jet industry as it can be used toquantify the effect of defects on the behavior and reliabilityof printing nozzles or alternatively, to characterize the naturalangular variability within an array of nozzles. The techniquewas tested with a commercially available printhead.

The droplets were produced with a Xaar XJ126 drop-on-demand (DoD) printhead which contains a linear arrayof 50 µm diameter nozzles, [4]. The fluid is ejected bythe distortion of piezoelectric elements located between theink chambers behind the nozzle plate. Typically, this print-head operates at droplet speeds of 5-6 m/s with droplet vol-umes of 80 picoliters. The nozzle has a conical inlet with a30 degree included angle and has a depth of approximately20 µm as shown in Fig. 7. The fluid used in these exper-iments is a transparent oil with a Newtonian viscosity of20 mPa·s [4]. The optical system used to visualize the 50µmdroplets consists of a 12X zoom Navitar microscope lens witha 2x lens attachment mounted on a Nikon D40x DSLR cam-era body. This arrangement provides a working distance of32 mm, a field of view of 3.5×2.4 mm and a depth of field of500µm. Under these conditions, the resolution of the imageis 2.3µm/pixel.

The system used to back-illuminate the droplets consistedof a dual-spark flash generator (High-Speed Photo-Systeme),a multimode fiber optic, a condenser lens, an array of twoplano-convex lenses, an optical diffuser, a microscope lensand the camera sensor, as illustrated in Fig. 2. Each of thetwo spark flash generators, which can be independently trig-gered, produces flashes of 20 ns of duration with 25 mJ of

Rev. Mex. Fıs. 57 (3) (2011) 266–275


FIGURE 9. (Colour online) Steps in image processing. Raw image(above), processed image with edge detected (middle) and jet edgeonly (below).

energy. The light from both spark sources is focused by acondenser lens on to the end of the optical fiber. The light atthe output of the fiber encountered an engineered optical dif-fuser (Thorlabs and RPC Photonics) and was then focused bythe pair of plano-convex lenses. The first lens, with a 60 mmfocal length, was positioned 35 mm from the diffuser. Thesecond lens, with a 45 mm focal length, was placed 10 mmin front of the first lens and 150 mm from the front of themicroscope lens. The droplets were generated in front of thearray of lenses and at 35 mm from the microscope lens. Apulse generator (TTi) was used to trigger the spark flashes in-dividually, separated by a time delay. The camera sensor wasset at 600 ISO. An example of a single flash shadowgraph ofan array of 38 droplets is shown in Fig. 8.

3. Image Analysis

The process of image analysis is based on the detectionof fluid boundaries in shadowgraph pictures. As discussedabove, the identification of features in images is nowadaysa common function in image processing programs; amongmany examples, Matlab has a built-in routine called “bw-traceboundary” to outline object boundaries in black andwhite images, Labview has the “Threshold” tool on the NIVision Assistant and HarFA and PEJET have a thresholdingprocess for colour images, [13,14,4] and [15]. In this work,both HarFA and PEJET were used to analyze colour RGBshadowgraph images. Briefly, one way to identify the bound-aries of the fluid regions is to compare the intensity levels ofthe RGB channels to a pre-established threshold level; thisis actually the method used by HarFA, Matlab and Labview.Another way to detect the boundary, used by PEJET, is byidentifying local changes of RGB intensity, which are at theirlargest at the fluid boundaries. This approach produces betterresults when the image background brightness is not perfectlyeven, [4]. In any case, the result of both approaches is highlydependent of the evenness of the background and the sharp-ness of the boundary, and as a consequence the correct designof the shadowgraph imaging setup is essential.

Once the fluid boundaries are detected and their coor-dinates recorded, several studies can be carried out. In thiswork, the boundary profiles obtained from the different sys-tems were analyzed in different ways depending on the appli-cation.

3.1. Analysis of Jets

As noted above, continuous ink-jet (CIJ) printheads producestreams of droplets by harmonically disturbing a liquid (ink)jet. Most commercial CIJ printers recirculate and reuse thefluid that has been jetted but not used for printing. As thisprocess is repeated, the concentration of the solvent or the inkcomponents may change as a result of various factors suchas evaporation, sedimentation and/or temperature variations.These processes modify the fluid properties, the jet break-up behavior and ultimately the behavior and spacing of thedroplets, which can lead to reductions in print quality, [9].Modern commercial printers have built-in rheometers andsensors that constantly monitor the viscosity and temperatureof the ink, but other properties such as the surface tension re-main unmeasured. This section reports a method where fluidproperties (e.g.the surface tension) can be continuously mon-itored through the analysis of shadowgraph images of jets.This approach does not affect the jetting or the break-up pro-cess, and although it was tested on centimeter-long jets gener-ated by a large nozzle, could readily be adapted to a nozzle attypical commercial CIJ scale (ca. 100 micrometer diameter).

The experimental setup described in Sec. 2A was usedto jet a viscous Newtonian liquid consisting of a solutionof 74.7% glycerol (99.9% pure) in water with the followingmeasured properties: densityρ =1250 kg/m3, surface tensionσ = 81.0± 0.5 mN/m and viscosityµ = 36.7± 0.2 mPa·s.Surface tension and viscosity of the fluid were measured us-ing a bubble pressure tensiometer (SITA Messtechnik, Proline t15, at a bubble lifetime of 100 ms) and a portable vis-cometer (Hidramotion, Viscolite 700). The liquid was jettedthrough a2.2±0.05 mm diameter nozzle and the driving pres-sure was modulated with a sinusoidal waveform with an am-plitude of 100 Pa. The resulting jet was imaged as describedabove and the images were analyzed as follows. First, the im-age was cropped from its original size (3872×2592 pixels) toa smaller format (3144×151 pixels) to show only the field ofview corresponding to the modulated jet and the first few sep-arate droplets (the resolution of the image was of 114±1 pix-els/cm). This step is necessary to speed up the analysis andavoid the contour detection of features extraneous to the jet.Next, the image was analyzed, the jet profile detected and itscontour recorded. This process is illustrated in Fig. 9. Boththe shadowgraph image and the surface profile can be stud-ied to quantitatively determine jet properties. The jet axialspeed (v) can be obtained directly by measuring the dropletseparation using a relationship obtained from mass and flowconservation:

v = fλ (1)

whereλ is the wavelength of the surface disturbance or thedroplet separation after break-up andf is the imposed mod-ulation frequency [9]. The droplet separation (wavelength)can be accurately measured by calculating the center of massof the droplets from the contour analysis or via a spatial fastFourier transform (FFT) of the radial jet profile. These meth-ods produce a value ofλ = 10.5 ± 0.2 mm. In addition to

Rev. Mex. Fıs. 57 (3) (2011) 266–275


FIGURE 10. (Colour online) Experimental and theoretical instan-taneous jet profiles in terms of the axial position of a modulatedviscous jet. The experimental profile was obtain by analysis ofshadowgraph images.

this simple analysis, the growth of the disturbance observedon the jet profile can provide information about the fluid dy-namic properties. Rayleigh and Weber derived expressionsthat describe the exponential grow of surface perturbationsin harmonically disturbed cylindrical jets [2]. In cylindri-cal coordinates, the surface profile of a weakly harmonically-disturbed jet is given by

r = a + ξ0eikz+(αr+i2πf)z/v, (2)

where a constant axial speed (v) is assumed,a is the initialradius, andξ0, k, andαr are the initial amplitude, the wavenumber and the growth rate of the disturbance respectively;k = 2π/λ. The growth rate depends on the fluid propertiesand for Newtonian viscous fluids is given by

α2r

kaI0(ka)2I1(ka)

+ αrk2 µ

ρ

[2ka

I0(ka)I1(ka)

l2

l2 − k2− 1

− 2laI0(la)I1(la)

k2

l2 − k2

]=

σk2

2ρa(1− k2a2), (3)

wherel = k2 + ρ/µ. A detailed derivation of these expres-sions can be found in Ref. 2. It is through Eq. 3 and theprofile obtained by the image analysis that jet properties canbe determined as shown in Fig. 10. The initial radius (a) iscalculated by averaging the section of the radial profile cor-responding to the first few surface oscillations (the first 5 os-cillations were used in this example although this number isnot critical as the disturbance is only detectable after about10 oscillations). The growth rate (αr) is obtained from a lin-ear regression analysis, as the slope of the logarithmic posi-tion of the local maxima of the surface disturbance (Figs. 10and 11). As the value of the growth rate is theoretically de-termined by Eq. (3) and the fluid properties, this analysis canbe utilized to study the variation of the fluid properties overtime.

FIGURE 11. (Colour online) Logarithm of the position of the in-stantaneous jet surface local maxima (from the initial radius) froman instantaneous shadowgraph image of a continuous modulatedjet. The slope of the data corresponds to the growth rate of the jetdisturbance.

To test this method, shadowgraph images of a modulatedjet were analyzed and the jet profile compared with the profilepredicted from Rayleigh’s model. The theoretical expressionfor the jet profile with the measured fluid properties producesa growth rate value ofαr = 60.2±2.0 s−1. In contrast, fromthe slope value shown in Fig. 11, the image analysis givesa value ofαr = 61.5 ± 3.3 s−1, which is consistent withthe prediction within experimental errors. The shadowgraphjet profile is shown in Fig. 10 together with the theoreticalcurve predicted from the measured fluid properties, the noz-zle diameter and an initial disturbance amplitude calculatedby recursive fitting; the close agreement is evident. This anal-ysis is of interest to the inkjet community as is can be used tomonitor fluid properties that tend to vary during the operationof their printhead systems [16].

3.2. Analysis of sprays

The delivery of liquid through sprays and aerosols is a tech-nique widely used in manufacturing and other industrial pro-cesses. The atomization process occurs as a result of theinteraction between the liquid and the surrounding air, andinvolves several stages through which the oil eventually be-comes an aerosol. Figure 12 shows a transparent oil sprayproduced by the pressure atomizer described above. Spraygeneration is similar to other types of jetting such as CIJ orDoD ink-jet printing as the liquid passes generally throughthree stages. In the first, the liquid is discharged as a jet; thisstep is easily recognized as the jetted liquid can still be iden-tified as a single continuous region (i). In the second stage,the liquid is extended and stretched to form liquid threads orligaments (ii ). The last stage is identified by the collapse orbreak-up of these ligaments into droplets by surface tension

Rev. Mex. Fıs. 57 (3) (2011) 266–275


FIGURE 12. (Colour online) Steps in image processing applied to a shadowgraph image of an oil spray. Raw image (far left), processedimage after contour detection, (middle-left), liquid boundary (middle right) and individual features (far right). Three characteristic stages areobserved as indicated:i single-body jet,ii ligament regime andiii droplet domain.

FIGURE 13. (Colour online). Liquid feature size distribution in theligament (in red) and droplet (in blue) domains of a liquid spray.Please note the logarithmic scale on the vertical axis.

forces (iii). Downstream, these droplets can achieve, undersome conditions, a terminal homogeneous velocity of sev-eral meters per second. In a similar way to the CIJ case, thejetting process of sprays depends on several variables suchas the nozzle geometry, the liquid properties and the drivingpressure. These properties govern the droplet size and speeddistributions and may change in time depending on the condi-tions under which the dispensing system is operated. UnlikeCIJ, sprays are not forced or modulated to break up at a well-determined frequency to produce droplets periodically. As aconsequence the break-up of liquid sprays is random and soare the droplet separation and volume making their character-ization difficult. This section describes an imagining methodto monitor the size distribution of droplets generated by aspray dispenser. The method is non intrusive and does notaffect the operation or the spray mechanisms.

The experimental setup described in Sec. 2B was utilizedto discharge quenching oil through a 1.2 mm diameter noz-zle into air. The image analysis of the captured sprays con-sists of the study and identification of the fluid edges withina selected area of interest. This process requires the detailedexamination of these boundaries to determine which side ofthe edges correspond to the liquid and which side does not;a similar method is described for droplets in Ref. 4. Thisis necessary when shadowgraph images of transparent liq-uids are analyzed as some of the flash light can be refractedthrough the liquid onto the camera sensor and not away fromit. This effect is visible in the left-hand image in Fig. 12where the liquid in the upper part of the spray is transmittinglight from the background and the internal structure of theliquid is visible. This internal structure needs to be excludedfrom the image analysis as it can be wrongly recognized aspart of the fluid boundary; as seen on the middle-left imageof Fig. 12. With the fluid boundaries detected, an algorithmthat identifies the outermost closed boundary is used to filterout all the unwanted features, [4]. The results of this processare seen in the third image of Fig. 12.

Once the real features are detected and labeled, their pix-els positions are recorded. To estimate the size of individ-ual liquid features, the transverse section (or the projectedarea) of the objects is calculated as the number of pixels con-tained inside their boundary. This process is illustrated onthe far-right image on Fig. 12 where fluid edges have beenfilled to reconstruct the transverse section of features. Al-though the precise volume of the features recorded in regionsi and ii (given the lack of cylindrical or spherical symme-try) cannot be calculated with this technique, the measuredtransverse sections can be utilized to estimate the feature sizedistribution along the spray. The histogram in Fig. 13 showsthe number of features or objects in terms of their projectedarea within a spray section. Figure 13 indicates that the pre-dominant size in both domains corresponds to a feature sizeof around 0.03 mm2. The histogram also shows that there are

Rev. Mex. Fıs. 57 (3) (2011) 266–275


FIGURE 14. (Colour online) Raw image (left) and processedversion (right) of a double-flash shadowgraph picture of a set of15 droplets. The droplets on the top are recorded during the firstflash event and the ones at the bottom during the second fash.The droplet deviation angle is determined from the position of thedroplet at the first flash and its position after 480µs.

twice as many objects of this size in the droplet domain asin the filament domain, showing that filaments are broken upinto droplets of that size.

Although this technique is not adequate to measure thevolumetric distribution, it can be used in industrial applica-tions to monitor the break-up behavior of sprays by monitor-ing changes on the size distribution. However, in the dropletdomain, as most features have a circular cross section, theresults of the image analysis could also be used to estimatethe volume distribution by simple geometry as described inRef. 4.

3.3. Analysis of droplets

Droplets provide an excellent method to deposit liquid mate-rials. One of the most successful examples of their practicaluse in industry is in ink-jet printing. As in jets and sprays, thebehavior of printed droplets is governed by the liquid proper-ties, the jet driving pressure and the dimensions of the nozzle,and is affected by other factors such as nozzle imperfections,the presence of dried ink around the nozzle, inhomogeneitiesin the piezoelectric driving elements and in the nozzle platematerial, and the presence of gas bubbles in the printheadchambers, [9] and [17]. Jet directionality is a key parameteras it influences the quality of printing. Although jet direction-ality is a key parameter as it influences the quality of printing,the effect on jet directionality of nozzle properties is a topicwhich has not been extensively studied because most theo-retical models and numerical simulations assume symmetri-cal geometries or are valid only for specific liquid properties(i.e. Newtonian liquids, [18,19] and [20]). This section de-scribes a method for characterizing the directionality of jetsby shadowgraph imagining of droplets.

Jets and droplets were produced from a Xaar XJ126 print-head and recorded with the shadowgraph system described inSec. 2. Briefly, the image analysis calculates the jetting an-gle by detecting the droplet positions at two times definedby two flash events; this analysis is shown schematically inFig. 14. First, an image is analyzed to detect the edges of theliquid droplets. Next, the positions of the droplet boundariesare used to calculate its center of mass (the center of massposition being determined using the method described in

FIGURE 15. (Colour online) Results from the image analysis ofdouble-flash shadowgraph images. The error bars are the resultof 5 consecutive experiments.

Ref. 4. Finally, the positions of the centers of masses arecorrected for lens distortion and the angle formed betweenthe droplet position at the two flashes events is then calcu-lated and recorded. During the analysis, the original imageis divided into sections where each portion contains only theimages of one single droplet recorded by the two flashes. Forthese experiments, the field of view of the imaging systempermitted the visualization of 36 or 37 jets in a single im-age, as shown in Fig. 8. During the analysis, these imageswere divided into slices with widths equal to the nozzle pitchof the printhead (140µm). Pincushion distortion is removedby a third-order polynomial correction algorithm whose co-efficients were obtained through the imaging and correctionof an accurate square mesh object pattern from a microscopegrating. This analysis was applied to images correspondingto all 126 nozzles of the Xaar printhead; some of the resultsare shown in Fig. 15. As a result, these experiments con-firmed the reported native jet variability stated by the print-head manufacturer (reported to be<1 degree).

This non-intrusive imaging technique could be used inindustrial applications to assess the jetting directionality ofindividual nozzles in a printhead.

4. Conclusions

Image analysis methods have been developed and appliedto shadowgraph images obtained from three different ex-perimental setups to study three industrial methods for thedelivery of liquids: continuous inkjets, continuous sprays,and droplets jetted from a drop-on-demand inkjet printhead.These methods produce quantitative data that can be used tomonitor processes such as printing and spraying.

It has been shown that shadowgraph imaging can be uti-lized to monitor variations on the liquid properties of contin-uous modulated jets by the analysis of their profile. In thecase of sprays, these experiments have shown that the analy-sis of shadowgraph images can be used to estimate the dropsize distribution and its variation with distance from the noz-

Rev. Mex. Fıs. 57 (3) (2011) 266–275


zle. This method could be used to monitor and/or control thebehavior of a spray in order to deliver a certain distributionof droplet size at some distance from the nozzle. In the fieldof inkjet printing, these experiments were used to asses thenozzle directionality and could be used on a production lineto test the properties of fabricated printheads.

All these imaging techniques are non-intrusive and canbe applied continuously during the jetting process.

Acknowledgments

This work was supported by the Engineering and PhysicalSciences Research Council (UK) and industrial partners inthe Next Generation Ink-jet Printing project.

1. G.S.Settles,Schlieren and Shadowgraph Techniques: Visual-izing Phenomena in Transparent Media(Springer, Heidelberg2001).

2. G. Brenn, Z. Liu, and F. Durst,International Journal of Multi-phase Flow26 (2000) 1621.

3. L. Le Moyne, V. Freire, and D.Q. Conde,Chaos, Solitons andFractals38 (2008) 696.

4. I.M. Hutchings, G.D. Martin, and S.D. Hoath,Journal of Imag-ing Science and Technology51 (2007) 438.

5. A. Frohn and N. Roth,Dynamics of Droplets(Springer Verlag,Berlin 2000).

6. D. Gonzalez-Mendizabal, C. Olivera-Fuentes, and J.M.Guzman,Chem. Eng. Comm56 (1987) 117.

7. M. Levanoni,IBM. J. Res. Develop21 (1977) 56.

8. S.A. Curry and H. Portig,IBM. J. Res. Develop21 (1977) 10.

9. C.A. Bruce,IBM. J. Res. Develop20 (1976) 258.

10. J.R. Castrejon-Pita, G. Martin, S. Hoath and I. Hutchings,Rev.Sci. Instrm.79 (2008) 075108.

11. L. Rayleigh,Proc. Lond. Math. Soc.10 (1878) 71.

12. C. Weber,Z. Angew Math. Mech11 (1931) 136.

13. M. Nezadal and O. Zmeskal,Harmonic and Fractal Im-age Analyzer(code) (2001), e-Archive: http://www.fch.vutbr.cz/lectures/imagesci

14. K. Tomankova, P. Jerabkova, O. Zmeskal, M. Vesela, and J.Haderka,Journal of Imaging Science and Technology, 50 583(2006).

15. J.C. Russ,The imaging Processing Handbook4th ed. (CRCPress, Boca Raton, 2002).

16. V.P. Janule,Pigment and Resin Technology25 (1996) 10.

17. J. de Jonget al., J. Acoust. Soc. Am.120(2006) 1257.

18. N.F. Morrison and O.G. Harlen, Rheol. Acta DOI:10.1007/s00397-009-0419-z (2010).

19. C.S. Kimet al., Computers & Fluids38 (2009) 602.

20. A.S. Yang, C.H. Cheng and C.T. Lin,J. Mechanical Engineer-ing Science: Proc. IMechE 2006220(2006) 435.

Rev. Mex. Fıs. 57 (3) (2011) 266–275

continuacion/continued

contenido/contents Rev. Mex. Fıs.57 (3) (2011) junio 2011

Room temperature thermal properties of Pb(Fe1/2Nb1/2)O3

ferroelectromagnetic ceramics,R. FONT, E. MARIN , A. LARA-BERNAL, O. RAYMOND , A. CALDERON,J. PORTELLES, AND J.M. SIQUEIROS 241–244

Applications and extensions of the Liouville theorem on constants of motion,G.F. TORRES DELCASTILLO 245–249

Controlling a laser output through an active saturable absorber,M. W ILSON, V. ABOITES, A. PISARCHIK, V. PINTO, AND Y. BARMENKOV 250–254

Simultaneous phase-shifting cyclic interferometer for generation of lateraland radial shear,

D.I. SERRANO-GARCIA , N.I. TOTO-ARELLANO, A. MARTINEZ GARCIA ,J.A. RAYAS ALVAREZ , A. TELLEZ-QUINONES, AND G. RODRIGUEZ-ZURITA 255–258

Instrumentacion/InstrumentationSistema automatizado para la medicion de la conductividad termica delıquidos mediante el metodo del alambre caliente,

S. ALVARADO , E. MARIN , A.G. JUAREZ, A. CALDERON Y R. IVANOV 259–265The shadowgraph imaging technique and its modern application to fluidjets and drops,

R. CASTREJON-GARCIA , J.R. CASTREJON-PITA , G.D. MARTIN ,AND I.M. HUTCHINGS 266–275

REVISTAMEXICANA DE

FISICAVOLUMEN 57NUMERO 3

JUNIO2011

CONTENIDO/CONTENTS

Cartas/LettersSpurline structures and its application on microwave coupled line filter,

J.R. LOO-YAU , O.I. GOMEZ-PICHARDO, F. SANDOVAL -IBARRA,M.C. MAYA -SANCHEZ, AND J.A. REYNOSO-HERNANDEZ 184–187

Investigacion/ResearchEntrelazamiento cuantico espurio con matrices seudopuras extendidas 4 por 4,

J.D. BULNES Y L.A. PECHE 188–192Electrostatic models of charged hydrogenic chains in a strong magnetic field,

A. ESCOBAR 193–203Estimador estocastico para un sistema tipo caja negra,

R.P. OROZCO, R.U. PARRAZALES Y J. DE J. MEDEL JUAREZ 204–210Ferromagnetismo en manganitas sustituidas con plata de estructura perovskita,

N. HERNANDEZ, T. HERNANDEZ, I. DZUL Y Y. PENA 211–214Effects on the quantum not and controlled-not gates of a modular magneticfield in the z-direction in a chain of nuclear spin system,

M. AVILA AND G.V. LOPEZ 215–219Spherical MoS2 micro particles and their surface dispersion due to additionof cobalt promoters,

M.A. RAMOS, V. CORREA, B. TORRES, SERGIO FLORES,J.R. FARIAS MANCILLA , AND R.R. CHIANELLI 220–223

Estructura y morfologıa de pelıculas de pm-Si:H crecidas por PECVDvariando la dilucion de diclorosilano con hidrogeno y la presion de trabajo,

C. ALVAREZ-MACIAS, J. SANTOYO-SALAZAR , B.M. MONROY,M.F. GARCIA -SANCHEZ, M. PICQUART, A. PONCE, G. CONTRERAS-PUENTE

Y G. SANTANA 224–231Single-electron Faraday generator,

G. GONZALEZ 232–235γ-Fe2O3/ZnO composite particles prepared by a two step chemical softmethod,

S. LOPEZ-ROMERO AND F. MORALES LEAL 236–240

continuacion/continued

Documents

REVISTA MEXICANA DE FÍSICASome of those designs include periodic band gap filters like defected ground structure (DGS), that is actually a transmis-sion line along with a well-defined