Statistical approaches to the transient populations and to the macroscopic energy transfer rate for the upconversion phenomenon in monodoped amorphous solid

Statistical approaches to the transient populations and to the macroscopicenergy transfer rate for the upconversion phenomenon in monodopedamorphous solidC. Z. Hadad Citation: J. Chem. Phys. 127, 104711 (2007); doi: 10.1063/1.2766955 View online: http://dx.doi.org/10.1063/1.2766955 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v127/i10 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Statistical approaches to the transient populations and to the macroscopicenergy transfer rate for the upconversion phenomenon in monodopedamorphous solid

C. Z. Hadada�

Grupo de Química-Física Teórica, Instituto de Química, Universidad de Antioquia, A A. 1226,Medellín, Columbia

�Received 18 April 2007; accepted 3 July 2007; published online 13 September 2007�

In this article, statistical approaches to the first and the second excited state transient populations andto the temporal macroscopic energy transfer rate for the upconversion process in amorphous solidgeneric systems monodoped with trivalent lanthanide ions are reached. The plots of the expressionsshow general tendencies reported in the literature. The derivation and the analysis of the formalismallowed us to fulfill our main objective, that is, to make a theoretical study about the microscopicand statistical mechanisms present in the phenomenon and their relation with the classic kineticanalysis. The study shows that the inclusion of the minimum possible radius between two opticalcenters in a solid affects the initial slopes of the decay curves of the luminescence from theintermediate state. We also corroborate that the usual treatment of experimental data using directequations for the dynamics of the populations in laser pulsed excitation experiments falls in themistake of not considering the temporality of the macroscopic energy transfer rate. Finally, physicalexplanations are formulated about this temporal behavior and about the main factors that generatethe characteristic simple exponential decay loss of the luminescence from the intermediate state.© 2007 American Institute of Physics. �DOI: 10.1063/1.2766955�

INTRODUCTION

Because of its projections and technological uses, amongthe processes of energy transfer �ET� between luminescentcenters that have received special attention in the last yearsare those of frequency upconversion. In these processes, twoor more initially excited ions transfer their energy, eitherconsecutively �the addition de photon par transfers d’energiemechanism�1–5 or simultaneously �upconversion by coopera-tive ET�,1,2,4–7 to other ions in the ground or excited state thatfinally reach higher excited states from which they emit ra-diation of shorter wavelength than that acquired by the sys-tem during the excitation. The materials that generate thiseffect are being used, for example, in the manufacture ofrare-earth-doped solid lasers and fiber lasers,8–14 IR displaydevices and sensors,15,16 and other photonic and optoelec-tronic devices.17,18 In addition the phenomenon is taken intoaccount in the study of the higher energy states of the opti-cally active centers in solids,19 and in the design of shortlength glass waveguide amplifiers for integrated optical com-munication circuits at 1.5–1.55 �m, because ET upconver-sion is an important factor of reduction in efficiency andamplifier gain.20,21

Among the different kinds of up-conversion mechanismsand processes, the observed ones in monodoped solids withtrivalent rare earth ions are, because of their simplicity, es-pecially interesting from the theoretical point of view. Onespecific case occurs when interacting ions in an intermediate

excited state undergo an energy transfer process, with one ofthe ions reaching a second higher energy excited state, whilethe other relaxes to a lower excited state or to the groundstate.22–28 Examples of this kind of process have been re-ported, for instance, in Pr3+: glass �lead germanate glass,22

fluoroindate glass,23 fluorophosphate glass,24 tellurite glass,25

and borate glass26� in which two Pr3+ interacting ions in the1D2 intermediate state generate an ion in the 3P0 upconvertedstate, and another on the 1G4 state. This category also in-cludes the Er3+: glass systems �heavy-metal fluoride glass,29

soda-lime silicate glass,20 phosphate glass,21 and telluriteglass27� in which two ions in the 4I13/2 or 4I11/2 intermediatestate interact to generate centers in the 4I9/2 or 4I7/2 states,respectively. In these studies, the experimental record of thetime resolved luminescence from the states involved plays animportant role in the inquiry of the upconversion mecha-nisms, the kind of interactions, its magnitude, and consider-ations about the dynamics of the process.

The description of the time resolved luminescence fromthe states of interest for this type of processes is frequentlyapproached by means of system differential equations for thedynamics of the populations, where parameters are incorpo-rated such as the radiative and nonradiative decay rate con-stants and those of macroscopic �or effective� energy transferrate.3,23,30 Experimental data may be easily analyzed withthis kind of approach, which also allows the description ofseveral complex upconversion schemes.3,27 Nevertheless,this kind of analysis does not account for the detailed depen-dence on experimental and spectroscopic factors and it doesnot incorporate the statistical effects caused by the ET pro-a�Electronic mail: [email protected]

THE JOURNAL OF CHEMICAL PHYSICS 127, 104711 �2007�

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cesses in doped solids.31 One of these effects is the fact that,strictly speaking, the macroscopic energy transfer rate de-pends on time.31–33 A second oversight is the well-knownresult of exponential decay loss of time resolved lumines-cence from the intermediate state.21–23,33–38

From a conceptually more complete viewpoint, modelsfor energy transfer upconversion that have settled down frommicroscopic considerations are those of Richter and Watts,39

Philipsen and Bjarklev,40 Sergeyev and Jaskorzynska,41

Przhevuskii and Nikonorov,32 and our preliminary study inthe field.31 That of Richter and Watts corresponds to a proba-bilistic model for upconversion luminescence by two-stepenergy transfer in crystalline systems codoped withYb3+–Tm3+ or the like. The study focuses attention in steadystate experimental conditions and shows the basic influenceof the infrared pump power and the lifetime of the interme-diate excited state on the conversion efficiency, within a sim-plified scheme.39 On the other hand, Philipsen and Bjarklev40

solved the equations that characterize the microscopic behav-ior for the upconversion process in presence of migration, invitreous systems doped with Er3+, by means of Monte Carlosimulations, and focused on the study of the dependence ofthe macroscopic upconversion energy transfer rate on theintermediate state population �population inversion, in thiscase�. Following the same motivation of this last work, theapproach of Sergeyev and Jaskorzynska41 consists of an ana-lytical model for the population inversion and for the energytransfer macroscopic rate, in order to investigate the effect ofenergy migration. In the case of stationary state lumines-cence, the reached expressions are analytical and are inagreement with the predictions of Philipsen and Bjarklev40

and with experimental tendencies. In the case of �-pulsedexcitation they arrived to an integral equation for the evolu-tion of population inversion and analytical asymptotes for t→0 and t→� cases �where t is the time�. The limitations ofthe model in this last case are that the expressions are ofnumerical calculation and that the asymptote t→0 for themacroscopic rate approaching to infinite. One importantpoint of this study is that the authors demonstrate that it isnot necessary to assume the formation of clusters of ions atfixed short distances, since considering a random statisticallyuniform distribution of Er3+ ions allows correct results to beobtained for the experimental observables. On the otherhand, Przhevuskii and Nikonorov32 using Monte Carlo fordirect simulation of energy transfer upconversion, includingmigration, found that in general the average energy transferupconversion rate between pairs of optical centers is a func-tion that decreases with time and that classical kinetic analy-sis must take into account such dependence. Parallel to thepublication of this last study, we published the same conclu-sions in a paper where analytical statistical approaches werederived for the upconverted population in monodoped vitre-ous luminescent solids and transient regime.31 Regarding theresulting expressions of this study, it was found that theydepend not only on experimental variables, but that they alsoexpress the microscopic �electronic� detail of the system andof the involved transitions. The model allows a correct pre-diction of the shape of the upconversion transient curves, andalso accounts not only for the dependence of the upconver-

sion intensity on the square of the optical center concentra-tion and of the external excitation intensity at relatively lowpump power, but also allows the prediction of “saturation”effects with increasing values of these factors, a situationcomparable to what happens with steady state luminescence.

In the present article our previous work is completed,reaching statistical approaches not only to the upconvertedpopulation, but also to the first excited state �intermediatestate� and to the energy transfer macroscopic rate in mon-odoped amorphous solid generic systems and for transientexperiments. The main aim of this work is to carry out atheoretical study of the microscopic and statistical processesthat are involved in the origin of the experimental particu-larities of the phenomenon, to find its relation with the clas-sic kinetic analysis �with direct equations for the involvedpopulations� and to obtain analytical expressions to theamounts related to the upconversion experimental manifesta-tions. The expressions are reached by means of generaliza-tion and adaptation to the upconversion of the Förster-Inokuti-Hirayama statistical treatment for simple energytransfer.42,43 In the model of these authors, centers initiallyexcited �donors� transfer their energy to centers in groundstate �acceptors� by means of some electrical or magneticmultipolar interaction. The result for the luminescence inten-sity at time t, I�t�, from the state initially excited is

I�t�/I�0� = exp�− t

�0− ��1 −

3

s� c

c0� t

�0�3/s� , �1�

where I�0� is the emission intensity at the first instant, �0 isthe radiative decay time, c03/4�R0

3 is the “critical transferconcentration,” with R0 the donor-acceptor distance to whichthe transfer rate equals the radiative decay rate, c is the con-centration of acceptors, ��x� is the gamma function of x, ands=6, 8, or 10, depending on if the interaction is dipole-dipole, dipole-quadrupole, or quadrupole-quadrupole, re-spectively. Finally, it must be indicated that, given its results,the Förster-Inokuti-Hirayama formalism has been widely ac-cepted and utilized in the field of the simple ET.

STATISTICAL APPROACHES

Dynamics of the populations from the microscopiccharacterization of the system

This study considers the energy transfer upconversionprocess in one step showed in Fig. 1. For simplicity we ne-glect other competitive ET mechanisms such as cross relax-ation or energy migration which are present in many systemsand under many doping conditions, although in some cases ithas been shown that it is possible to have solely the simplecase discussed here when the concentration is low.20,21

Interacting optical centers initially are found in the inter-mediate excited state, labeled �2�, reached after pulsed selec-tive excitation of intensity Iexc, short enough not to give riseto any significant energy transfer upconverted populationwithin the excitation time. The intrinsic rates of each lumi-nescent center, �2 and �3, include contributions of intra-ionradiative and nonradiative decays to the lower states thathave the possibility of being reached, �2=states�2r

+states�2nr and �3=states�3r+states�3nr. For the sake of

104711-2 C. Z. Hadad J. Chem. Phys. 127, 104711 �2007�


simplicity, Fig. 1 only shows the intraionic transitions to theground state. The terms W�rij� and W��rij� account for themicroscopic rates for the two possible directions of energytransfer between the pair of generic centers j and i, separatedspatially by the vector rij. For them we will consider theconventional approximation1,5,31,38

W�rij� =�s

rijs , �2�

where s=6, 8, and 10 correspond to dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole interactions, respec-tively. The �s term can be expressed as a function of spec-troscopic and electronic parameters;5,31,38 in practice itcorresponds to an average among the optical centers of thesolid, so it is true that W�rij�=W��rij�.

On the other hand, the differential equations for the tem-porary evolution of the microscopic probabilities of being atthe relevant states, j

�2��t� and j�3��t�, for a generic ion j, are31

d j�2��t�dt

= − �2 j�2��t� − 2

i�j

W�rij� j�2��t�i

�2��t� , �3�

d j�3��t�dt

= − �3 j�3��t� +

i�j


�2��t� , �4�

which fulfill z�1��t�+z

�2��t�+z�3��t�=1, z j, or i

1,2 , . . . ,Nv, with Nv the total number of optical centers ina given volume, V, of the solid.

Using the following equivalence for each relevant state,est�2� , �3�:

j=1

NV

j�est��t�� V = NV

�est��t�/V = N�est��t� = j=1

N

j�est��t� , �5�

it is possible, adding member to member and reordering, toconvert the system of equations for microscopic probabilitiesin equations for macroscopic populations in the relevantstates, N�est��t�,

dN�2��t�dt

= − �2N�2��t� − 2WUPTE�t�N�2��t� , �6�

dN�3��t�dt

= WUPTE�t�N�2��t� − �3N�3��t� , �7�

which fulfills

N�1��t� + N�2��t� + N�3��t� = N , �8�

where N=NV /V is the optical density or optical center con-centration of the material. WUPTE�t� is the macroscopic up-conversion energy transfer rate, defined as

WUPTE�t� =1

NV�2��t�j=1

NV

i�j

NV


�2��t� . �9�

In this study we will demonstrate that the upconversionpopulation, N�3��t�, as well as the macroscopic energy trans-fer rate, WUPTE�t�, can be set in terms of the average prob-ability of permanence of the optical centers in the interme-diate state, �2��t�. Thus, it is sufficient to find this lastquantity starting from Eq. �3�.

Resolution for the intermediate state

The partial solution of Eq. �3� is

j�2��t� = j

�2��0�exp�− �2t − 2i�j

NV

W�rij� 0

t

i�2��t�dt� ,

�10�

where

j�2��0� = �1, centers initially excited

0, centers initially in the ground state.�

�11�

This work will consider the approximation

i�2��t� � i

�2��0�e−�2t, �12�

which is valid in the case of low or intermediate opticalcenter concentrations.31 Using this expression, the intermedi-ate state average probability is

�2��t� =1

NVj=1

NV

j�2��t�

= exp�− �2t�1

NVj=1

NV

j�2��0�

exp�− 2i�j

NV

W�rij� 0

t

i�2��0�e−�2tdt� . �13�

The temporal integral in the last expression, defined as

T�t� 0

t

e−�2tdt =1 − e−�2t

�2, �14�

does not depend on any kind of radial variable, which enableus to use the Inokuti-Hirayama method of averageresolution,43 where the statistical variable is only the dis-tance, rj. Nevertheless, in this work we correct this calcula-tion introducing the structural factor of a minimal approach

FIG. 1. Diagram of relevant energy levels of the donor-acceptor pair �i , j� ofthe same nature. The pulsed excitation transitions of intensity Iexc and theintraionic decays from the intermediate and upconversion states �with rates�2 and �3�, respectively, are shown as solid arrows. The energy transfers,with rates W��rij� or W�rij�, depending on the direction j→ i or i→ j, respec-tively, are shown by dotted arrows.

104711-3 Upconversion phenomenon in monodoped amorphous solids J. Chem. Phys. 127, 104711 �2007�


distance between two ions in the lattice, Rm. From Eq. �13�,the previous expressions for this purpose are

�2��t� = �2��0�exp�− �2t�1

NV�2��0�

j=1

NV�2��0�

exp�− i=1

NV�2��0�

2W�rij�T�t��= �2��0�exp�− �2t��

i=1

NV�2��0�

e−2W�rij�T�t��= �2��0�exp�− �2t�

� limNV

�2��0�→�

RV→�

� r=Rm

RV 4�r2

Ve−2W�r�T�t�dr�NV

�2��0�

� ,

�15�

where V=4�RV /3 is the solid volume and the condition �11�has been used for j

�2��0� and i�2��0�, the relation 1/NV

= �2��0� /NV�2��0�, and the radial distribution function

4�r2 /V.43

Calculation of the integral and the limits in Eq. �15� isdetailed in Appendix A. The final result for the average prob-ability is

�2��t� = �2��0�e−�2t exp�−4�

3N�2��0��1 − 3/s�

�2�sT�t��3/s�exp�− V�Rm�N�2��0��e−2W�Rm�T�t�

− ��1 − 3/s,2W�Rm�T�t��2W�Rm�T�t��3/s�� ,

�16�

where V�Rm�=4�Rm3 /3 and W�Rm�=�s /Rm

s correspond tothe volume and the energy transfer upconversion micro-scopic rate at Rm, respectively; ��a� and ��a ,x� are thegamma and incomplete gamma functions, respectively.

Resolution for the upconversion energy transfermacroscopic rate

Introducing the expressions �5� and �10� into Eq. �9�, weobtain

WUPTE�t� =1/NV j=1

NV i�jNV W�rij� j

�2��0�exp�− �2t − 2�0t i�j

NV W�rij�i�2��t�dt�i

�2��t�

1/NV j=1NV j

�2��0�exp�− �2t − 2�0t i�jW�rij�i

�2��t�dt�, �17�

where the numerator and the denominator have been multiplied by 1/NV. Notice that the denominator of this expressioncorresponds to the intermediate state probability average, given in Eq. �13�. The temporal derivative of the logarithm of thisaverage is

d

dtln �2��t� =

d

dtln� 1

NVj=1

NV

j�2��0�exp�− �2t − 2

0

t

i�j

W�rij�i�2��t�dt��

= − �2 −2/NV j=1

NV i�jW�rij�i�2��t� j

�2��0�exp�− �2t − 2�0t i�jW�rij�i

�2��t�dt�1/NV j=1

NV j�2��0�exp�− �2t − 2�0

t i�jW�rij�i�2��t�dt�

. �18�

Using Eq. �17� in this expression and rearranging, we obtain

WUPTE�t� = −1

2�d ln �2��t�

dt+ �2� . �19�

The explicit function is obtained by replacing �2��t� from Eq. �16�, by carrying out the derivative, and by simplifying

WUPTE�t� =2��2N�2��0��2�sT�t��3/s��1 − 3/s� − ��1 − 3/s,2W�Rm�T�t��

s�e�2t − 1�. �20�



Resolution for the upconverted state

By considering the condition N�3��0�=0, the partial solu-tion of Eq. �7� is

N�3��t� = e−�3t 0

t

e�3tWUPTE�t�N�2��t�dt

= Ne−�3t 0

t

e�3tWUPTE�t��2��t�dt . �21�

By replacing Eq. �19� in this last expression and integratingby parts,

N�3��t� = −N

2e−�3t

0

t

e�3t�d ln �2��t�dt

+ �2��2��t�dt

= −N

2e−�3t��2

0

t

e�3t�2��t�dt + 0

t

e�3td�2��t�dt

dt�=

N

2�e−�3t�2��0� − �2��t� + ��3 − �2�e−�3t

0

t

e�3t�2��t�dt� , �22�

which is the sought after function.

RESULTS AND DISCUSSIONS

Reproduction of typical curves for the intermediatestate

Figures 2 and 3 show semilogarithmic representations ofthe expression �16� to different concentrations and differentfractions of initially excited ions �and, therefore, initial in-tensities of external excitation�, respectively, for dipole-dipole interaction and for some typical values of the param-eters on which it depends. In Fig. 3, we plot the temporalpopulations according to N�2��t�=N�2��t�.

As it can be seen, the plots correctly account for thetypical effect of the luminescence exponentiality loss fromthe intermediate state and the average lifetime shorteningwith increase of the active ion concentration.20,22,33 Both

with low intensities of external excitation and low concen-trations, indicated by the solid straight lines in Figs. 2 and 3,the behavior is of simple exponential type, which indicatesabsence of ET. When there is low concentration of activeions, those ions are more mutually separated and average ETinteraction is inappreciable, being according to the N�2��0�dependence indicated in Eq. �20�.

Minimal approaching radius effect

Equation �16� corresponds to an extension to upconver-sion and a correction of the Förster-Inokuti-Hirayama42,43

theory �Eq. �1��. Correction consists of having included adistance of minimum approach between optical centers, Rm.

Since the limit of Eq. �16� when Rm→0 is �see AppendixB�

�2��t� = �2��0�e−�2t

exp�−4�

3N�2��0��1 − 3/s��2�sT�t��3/s� ,

�23�

the part of Eq. �16� represented by the third exponential

exp�− V�Rm�N�2��0��e−2W�Rm�T�t�

− ��1 − 3/s,2W�Rm�T�t��2W�Rm�T�t��3/s�� 24�

is the one that adds the effect of this structural parameter.This functionality is general and valid both for simple ET�previous replacement in Eq. �24� of N�2��0� by c, which isthe basal state acceptor concentration, and of T�t� by t� andfor upconversion. Its effect is related to the curve slope at theorigin. In fact, a problem with expression �1� is that when theluminescent response at initial times is examined, the curveof the logarithm of the intensity versus the time shows infi-nite slope in the origin. This does not correspond to what isobserved experimentally, because the curves show finite ini-tial slopes.38,44–50 The same happens in upconversionprocesses.20,22 The introduction of the functionality �24� inboth cases would correct this defect, as it is indicated in Fig.4, where Eqs. �16� and �23� are plotted. It can be noticed thatthe effect of increasing Rm from zero �solid curve� to Rm

=3.8 Å �short segment curve�, passing by Rm=3.3 Å �curve

FIG. 2. Semilogarithmic plots for the variation in time of the intermediatestate average probability, �2��t�, to several concentrations. The solid straightline corresponds to an optical density of N=11017 cm−3, while the curvesof segments correspond to concentrations N=51019 cm−3, N=1020 cm−3,and N=31020 cm−3 with decrease of the segment longitude, respectively.All these cases utilize the values �2��0�=0.2, �2=5.0 m s−1, and �s=8.86106 Å6 m s−1, where s=6 and Rm=3.5 Å.

FIG. 3. Semilogarithmic plots for variation in time of intermediate statepopulation, N�2��t�, with several initially excited ion fractions �intensities ofexternal excitation�. The solid straight line corresponds to �2��0�=0.01,while the curves of segments correspond to �2��0�=0.1, �2��0�=0.4, and�2��0�=0.6 with decrease of the segment longitude, respectively. All thesecases utilize the values N=1020 cm−3, �2=5 m s−1, and �s=8.86106 Å6 m s−1, where s=6 and Rm=3.5 Å.



of more extended segments�, is the decrease of the initialgradient.

The behavior observed in the sequence was to be ex-pected, because the microscopic ET rate increases with thedecrease of the distance between centers. The Förster-Inokuti-Hirayama case, with the possibility of Rm=0.0 Å,would match a model of punctual optical centers distributedat random and without any restriction of spatial location,thus it is an unreal case.

Energy transfer macroscopic rate and classicalanalysis for transient processes

Equations �6� and �7� constitute one of the ways to for-mulate the involved population dynamics. A more commonway consists of considering explicitly that the decrease of theintermediate state population is directly proportional to thesquare of this state population26,32 owing to the possibility ofinteraction of each one of the active optical centers with allthe others,31

dN�2��t�dt

= − �2N�2��t� − 2WUPTE� �t��N�2��t��2, �25�

dN�3��t�dt

= WUPTE� �t��N�2��t��2 − �3N�3��t� , �26�

where

WUPTE� �t� =1

�NV�2��t��2

j=1

NV

i�j

NV


�2��t� �27�

corresponds to the average energy transfer rate between allthe pairs of active optical centers of the solid. Unlike thislast one, WUPTE�t�, of Eq. �9�, corresponds to the averageenergy transfer rate among each active center and the col-lection of active ions that surrounds them. In both cases avery outstanding aspect is that these macroscopic rates �Eqs.�9�, �20�, and �27�� depend on time. This is shown in Fig. 5for the explicit expression �20� with some typical values ofits parameters.

Equation �19� indicates that the time dependence of thismacroscopic rate is related to the temporal behavior of theintermediate state probability. As the slope of the curve

ln �2��t� vs t is much greater at the initial time and thensoftens fastly, finally reaching a constant value of −�2 �ex-perimentally observed trend for several excitation powers33�,then the rate is high at the origin and then decreases rapidly.This behavior has been observed experimentally by means ofthe effective lifetime from the 4I13/2 state of Er3+ activatedwith Yb3+ in a phosphate glass matrix.51 The temporal de-pendency of this macroscopic rate has the same origin as theexponential decay loss: As each microscopic interaction istime dependent �through functions j

�2��t� and i�2��t��, the to-

tal ET interaction between a center and all centers around isalso time dependent. On the other hand, as each microscopicinteraction depends on an inverse power of the distance be-tween ions, there is a much greater probability that the near-est pairs are exhausted before the remote ones than remotest.This has the consequence that the average for these rates �Eq.�9�� is greater at the beginning than at later moments whenpairs of smaller interactions survive at every time.

On the other hand, the limit of WUPTE�t� �Eq. �20�� whent→0 is developed in Appendix C and its value correspondsto

WUPTE�0� =3N�2��0�V�Rm�W�Rm�

�s − 3�=

4�N�2��0��s

�s − 3�Rms−3 ,

�28�

which is infinite when Rm→0, as in the case of Inokuti-Hirayama model, and finite for Rm�0. This equation showsthat WUPTE�0� increases linearly with the �2��0� parameter,which agrees with the experimental results of Jaskorzynskaet al.52

Finally, it is worth to notice that starting from the dy-namics for the microscopic probabilities of relevant states�Eqs. �3� and �4�� allowed us to realize that the macroscopicrates �Eqs. �9�, �20�, and �27�� depend on time. This is animportant consideration when any population dynamicsmodel is formulated, because, strictly speaking, such tempo-ral dependency must be included,31,32 as Eqs. �6�, �7�, �25�,and �26� suggest. Nevertheless, for simplicity reasons or bynot noticing this temporal dependency, often the analysis ofthe transient population is considered without includingit.3,23,30 For example, in some analyses for simple ET �inwhich the temporal dependency also is accounted for� this

FIG. 4. Semilogarithmic plots for Eqs. �16� �segmented curves� and �23��Rm=0.0 Å, solid curve� at early moments. Both equations have been mul-tiplied by optical density, N, accounting on the temporal population in time,N�2��t�. The curve of longer segments corresponds to Rm=3.3 Å, and that ofshorter ones to Rm=3.8 Å. All cases utilize N=1.01020 cm−3, �2��0�=0.3, �2=5 m s−1, �s=8.86106 Å6 m s−1, and s=6.

FIG. 5. Examples of the variation in time of the macroscopic energy transferrate, W�t� �Eq. �20�� with the change of �2��0�: �2��0�=0.03 �shorter seg-ment curve�, �2��0�=0.1 �longer segment curve�, and �2��0�=0.3 �solidcurve�. In all the curves have been utilized the values N=6.01020 cm−3,�2=5 m s−1, �s=8.86106 Å6 m s−1, s=6, and Rm=3.5 Å.



rate is calculated as an atemporal average of the sum ofmicroscopic rates for a random distribution.38,53,54 In ourcase, such procedure would lead to the expression

�i

W�rij�� = N�2��0� Rm

RV 4�r2

V

�s

rs dr =4�N�2��0��s

�s − 3�Rms−3 ,

�29�

where the concentration of interacting optical centers is c=N�2��0� /V. But this result just matches the maximum mac-roscopic ET rate of our model �limt→0 WUPTE�t��, so it isconcluded that this way overestimates the energy transferrates.

General factors of simple exponential decay loss forthe intermediate state luminescence

Main factors by which the simple exponential decay losstakes place for a state that is deactivated by ET in dopedsolids �for crystalline as well as amorphous structures and forsimple ET or for up conversion� are the following:

�a� The random distribution of distances between interact-ing pairs of optical centers, caused either for the ran-dom spatial location of the doping centers, for theirrandom activation, or for the disorder in the structureof the host material. In fact, it consists in the greaterprobability of the earlier deactivation of the centers thatinteract more strongly, with respect to those that inter-act weakly, making the average, �2��t�, in Eq. �13� todecrease more rapidly at the beginning of the time re-lated to ulterior instants.

�b� Equation �10� notices that if the functions for the mi-croscopic probabilities of interacting ions with the ge-neric one, i

�2��t�, are different from a constant, then thefunctions for generic ions, j

�2��t�, will show, already atmicroscopic level, a nonexponential dependency thatwill be reflected later in its macroscopic function. Inthe case of simple ET generated at low pump power,the approximation i

�2��t�=i�2��0�=const can be made.

On the contrary, as Tonooka et al.38 demonstrated, athigh pump powers the temporal functionality of inter-acting ions has to be considered.

Luminescence from the upconverted state

Figure 6 shows a plot for the temporal variation of theupconverted population, given by Eq. �22� utilizing in it theexpression �16�.

It may be clearly seen how the expression �22� matchesthe kind of typical curve for the time resolved luminescencefrom the upconverted state in a process that occurs by energytransfer.22–27

CONCLUSIONS

By means of a generalization and an adaptation of theFörster-Inokuti-Hirayama statistical treatment, it has beenpossible to reach approximations to the first and second ex-cited state transient populations and to the macroscopic en-ergy transfer rate for the energy transfer frequency upconver-

sion in monodoped amorphous solid phenomenon. The plotsof these expressions show general tendencies reported in theliterature, both for the variation in time of emission intensi-ties from the intermediate and upconverted states, and alsofor the experimental trends related to the macroscopic energytransfer rate.

By introducing in the formalism the minimal radius ofapproach between two optical centers in a solid, we havedemonstrated that the equivalent one to upconversion of thesimple ET Förster-Inokuti-Hirayama equation gets multipliedby an independent term which introduces its effect. It is re-lated to the gradient of the curves in the origin: at greaterminimum possible distance of approach between two opticalcenters, the initial slope is smaller. The Förster-Inokuti-Hirayama infinite initial slope would indicate the possibilitythat two optic centers occupy the same spatial point, which iserroneous. Therefore, our generalization corrects the classicexpression of these authors.

With regard to the formulation of direct equations for thepopulation dynamics in transient experiments in solids, pro-cedure frequently used in the analysis of experimental data,this study has been able to confirm that not considering thetemporality of the energy transfer macroscopic rate consti-tutes a mistake. This conclusion is reached by deriving thedifferential equations for the macroscopic behavior startingfrom the treatment for microscopic temporal probabilities ofbeing in some state for each one of the optical centers.Therefore this last way of analysis constitutes a generalmethod to attack the problem.

In the case of transient regime experiments, the macro-scopic ET rate decreases with time, first in a rapid way, froma finite initial value and then in softer way until a null finalvalue. This behavior is due to the larger probability that thenearest couples are drained first far away, because the inter-actions are temporal and they depend on an inverse power ofthe distance between optical centers.

Similarly, the initial ET macroscopic rate under theseconditions is directly proportional to an inverse power of theminimum distance of approach between two optical centersand to the concentration of active species. This last pointagrees with the experimental tendency reported in the litera-ture.

On the other hand, the luminescence nonexponential be-havior from the intermediate state is related to the temporary

FIG. 6. Temporal variation of the upconverted state population, for the Rm

=3.5 Å, N=1,01019 cm−3, �2��0�=0,2, �2=5 m s−1, �2=60 m s−1, �s

=8.86106 Å6 m s−1, and s=6 values.



dependence of the macroscopic ET rate. This effect, of netstatistical nature, originates as much by the existence of adistribution of distances between the interacting pairs in thesolid �either for the random spatial location of the dopingcenters, for its random activation, or for a situation of disor-der in the host lattice structure of the material� and by theinteracting ion time dependence of being in the relevantstates.

The inclusion of other competitive mechanisms, as crossrelaxations or energy migration, as well as the adjustment toexperimental results in particular systems will be publishedelsewhere.

ACKNOWLEDGMENTS

I am thankful to Universidad de Antioquia, and to FUN-DACION ANDES, CONICYT, and FONDECYT of Chile.Also I am thankful to Professor Sergio Octavio Vásquez forhis invaluable observations about this work and to ProfessorAlbeiro Restrepo Cossio for the revision of this article.

APPENDIX A: DERIVATION OF �„2…„t… BY

CONSIDERING THE MINIMAL APPROXIMATIONDISTANCE EFFECT

The integral and the limit of Eq. �15� must be resolvedby the method of Förster42 and Inokuti-Hirayama,43 by intro-ducing Rm. By replacing in Eq. �15� the expression �2� forthe microscopic rate, W�r�, the resulting integral is

I = r=Rm

RV 4�r2

Ve−K/rs

dr ,

where

K = 2�sT�t� . �A1�

By making the variables change,

� =K

rs

and integrating by parts, I is

I = −4�K3/s

3V �e−�m�m−3/s − e−�V�V

−3/s + �V

�m

�−3/se−�d�� ,

�A2�

where

�m =K

Rms and �V =

K

RVs .

Given �V �m, the integral in I can be separated in

�V

�m

�−3/se−�d� = 0

�

�−3/se−�d� − 0

�V

�−3/se−�d�

− �m

�

�−3/se−�d� .

That written in the form of gamma and incomplete gammausual functions,

�V

�m

�−3/se−�d� = ��1 − 3/s� − ��1 − 3/s,�V�

− ��1 − 3/s,�m� .

Now, by introducing this last expression in Eq. �A2�, byreplacing �m=K /Rm

s , �V=K /RVs , and the value of K, given in

Eq. �A1�; also by eliminating some factors and by utilizing insome terms the expressions 4�Rm

3 /3=V�Rm� and �s /Rms

=W�Rm�, and the relation 1/V=N�2��0� /NV�2��0�, the integral,

I, becomes

I = e2�sT�t�/RVs

−V�Rm�N�2��0�

NV�2��0�

e−2W�Rm�T�t�

−V�Rm�N�2��0�

NV�2��0�

�2W�Rm�T�t��3/s��1 − 3/s�

− ��1 − 3/s,2�sT�t�/RVs � − ��1 − 3/s,2W�Rm�T�t�� .

Now, in the limit when RV→�, one has that

e−2�sT�t�/RVs

——→RV→�

1;

then,

V�Rm�N�2��0�NV

�2��0��2W�Rm�T�t��3/s��1 − 3/s,2�sT�t�/RV

s �

=V�Rm�

V�2

�s

Rms T�t��3/s

��1 − 3/s,2�sT�t�/RVs �

=Rm

3

RV3

1

Rm3 �2�sT�t��3/s��1 − 3/s,2�sT�t�/RV

s �

= �2�s

RVs T�t��3/s

n=0

��− 1�n

n!�1 − 3/s + n��2�sT�t�RV

s �1−3/s+n

= n=0

��− 1�n

n!�1 − 3/s + n��2�sT�t�RV

s �1+n

——→RV→�

0,

where it has been utilized the convergent expansion55

��a,x� = n=0

��− 1�nxa+n

n!�a + n�

valid for a�0,−1,−2,−3, . . ., which means, s� +3, +3/2 ,+1 , +3/4 , . . ., which is true. Finally, it remains to calculatethe well-known limit

limNV

�2��0�→�

INV�2��0� = lim

NV�2��0�→�

�1 −A

NV�2��0��NV

�2��0�

= e−A,

�A3�

where

A = V�Rm�N�2��0��e−2W�Rm�T�t� + �2W�Rm�T�t��3/s��1

− 3/s� − ��1 − 3/s,2W�Rm�T�t�� .

By replacing for the functions of volume and of ET rate�V�Rm�=4�Rm

3 /3, V=4�RV /3, and W�Rm�=�s /Rms � in some

terms of this last expression and reordering,



A =4�

3N�2��0��1 − 3/s��2�sT�t��3/s + V�Rm�N�2��0�

�e−2W�Rm�T�t� − ��1 − 3/s,2W�Rm�T�t��

�2W�Rm�T�t��3/s� .

So, result �A3� introduced in Eq. �15� leaves the final expres-sion for the macroscopic probability, presented in Eq. �16�.

APPENDIX B: LIMIT OF �„2…„t… WHEN THE DISTANCE

OF MINIMUM APPROACH IS ZERO

By replacing V�Rm�= �4/3��Rm3 and W�Rm�=�s /Rm

s inthe argument of the third exponential of Eq. �16� and reor-dering, it becomes

4

3�Rm

3 N�2��0��exp�− 2�s

Rms T�t��

− ��1 − 3/s,2�s

Rms T�t��2

�s

Rms T�t��3/s�

=4

3�N�2��0��Rm

3 exp�− 2�s

Rms T�t��

− ��1 − 3/s,2�s

Rms T�t��2�sT�t��3/s� .

For t�0, T�t��0, we have that

limRm→0

�Rm3 exp�− 2

�s

Rms T�t�� = 0

and

limRm→0

��1 − 3/s,2�s

Rms T�t�� = 0.

This last limit, which is not so obvious, can be easilyreached from an expansion in series of powers for the incom-plete gamma function

lim�m→�

��1 − 3/s,�m� = lim�m→�

�m

�

�−3/se−�d�

= lim�m→�

��m−3/se−�m�1 −

3

s�m

−1 + ¯ ��= 0,

in which it has been made,

�m = 2�s

Rms T�t� ——→

Rm→0� .

So, expression �24� tends to 1 and Eq. �23� is reached.Finally, when t=0, T�t�=0, and expressions �23� and

�16� are equivalent, both cases correspond to initialprobabilities.

APPENDIX C: LIMIT EXPRESSION OF THE MACROSCOPIC ENERGY TRANSFER RATE IN THE ORIGIN OF TIME

For values of t near to zero, T�t�= �1−e−�2t� /�2� t, �e�2t−1��2t. With this the expression �20� is

WUPTE�t� �2�N�2��0��2�s�3/s��1 − 3/s� − ��1 − 3/s,2W�Rm�t��

s · t1−3/s . �C1�

The limit to solve is

limt→0

��1 − 3/s� − ��1 − 3/s,2W�Rm�t��t1−3/s

= limt→0

d/dt��1 − 3/s� − ��1 − 3/s,2W�Rm�t��d�t1−3/s�/dt

= limt→0

2 2−3/se−2W�Rm�tW�Rm��W�Rm�t�−3/s

�1 − 3/s�t−3/s

=2 2−3/s�W�Rm��1−3/ss

�s − 3�.

By replacing this expression in Eq. �C1�, and by usingW�Rm�=�s /Rm

s and V�Rm�=4�Rm3 /3, result in

WUPTE�0� =4�N�2��0��s

�s − 3�Rms−3 =

3N�2��0�V�Rm�W�Rm��s − 3�

.

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Documents

Statistical approaches to the transient populations and to the macroscopic energy transfer rate for the upconversion phenomenon in monodoped amorphous solid