Upload
rosa-maria-moya-de-la-cruz
View
238
Download
0
Tags:
Embed Size (px)
Citation preview
Las Propiedades Fiacutesicas de los CristalesL Fuentes CIMAV
Superconductores de alta temperatura
Aleaciones con memoria de forma
magneacuteticaKUllako at all Helsinki
University of Technology
NGlavatska at al Institute for Metal Physics Kiev
Ukraine
(animacioacuten)
(animacioacuten)
Las propiedades fiacutesicas como tensores
P E
P E
Introductorio P = o P E
Cristalofiacutesica P = o P∙ E
P es un tensor de 2ordm rango
En general las propiedades se asocian a relaciones
constitutivas
Y = K middot XSi los rangos respectivos de (X Y) son (mn) entonces la propiedad K es un tensor
de rango m+n
Property Related magnitudes Tensor
Heat capacity C Entropy (P0) Temperature (P0) P0
Elasticity s Strain (P2) Stress (P2) P4
Electr susceptibility χP Polarization (P1) Elec Intensity (P1) P2
Magn susceptibility χM Magnetization (A1) Magn Intensity (A1) P2
Thermal expansion Strain (P2) Temperature (P0) P2
Pyroelectricity p Polarization (P1) Temperature (P0) P1
Pyromagnetism i Magnetization (A1) Temperature (P0) A1
Piezoelectricity d Polarization (P1) Stress (P2) P3
Piezomagnetism b Magnetization (A1) Stress (P2) A3
Magnetoelectricity Magnetization (A1) Elec Intensity (P1) A2
P POLAR A AXIAL N = Tensor rank
THERMO-ELASTO-ELECTRO-MAGNETIC PROPERTIES
Algunos efectos y relaciones constitutivas
Paraelectricidad P = ε0χPmiddotE
Paramagnetismo μ0M = μ0χMmiddotH
Elasticidad S = s middot T
Dilatacioacuten S = middot Δ
Piezoelectricidad P = d middot T
S = d middot E
Efecto magnetoeleacutectrico
μ0M = middot E
P = middot H
INTERACCIONES PRINCIPALES Y DE ACOPLAMIENTO
PROPIEDADES DE ACOPLAMIENTO ELECTROMAGNEacuteTICO
SENSORES EN UN COCHE
Sensores de temperaturaSensores de condicioacuten del aceite y de humedad
Sensores de presioacuten
X - BY WIRE (X FRENOS DIRECCIOacuteN)
El servomecanismo de direccioacuten hidraacuteulica requiere presurizacioacuten constante del fluido hidraacuteulico Al sustituiacutersele por motores eleacutectricos se economizaraacute
aproximadamente 5 ndash 10 de combustible
ldquoSTEERING BY WIRErdquo A NIVEL DE LABORATORIO
httpwwwdelphicomautomotivenextechproductsxbywirehttp42voltdupontcomenSystemsbywire_mainhtml
httpwwwvisteoncomtechnologyautomotivepowertrain_controlshtml
EXPLICACIOacuteN CUALITATIVA DE ALGUNOS ACOPLAMIENTOS
B = mE m = eb
(T = eE)
e = dc
(c = s-1)
m = ds-1b
E E
T
H
D
S
B
p d
m
C
s
b
i
T = eE
Piezoelectricidad Piezomagnetismo
Magnetoelectricidad1 2
E
ESFUERZO (T oacute ) Y DEFORMACIOacuteN (S oacute )
EN UN ENSAYO DE TRACCIOacuteN
S = sTs = ldquocompliancerdquo
x
e oacute S
oacute T
Zona Elaacutestica
Zona Plaacutestica
s max
s fluencia
s YeT cS
T = cS c∙s = 1c = ldquostiffnessrdquo
Ley de Hooke
TENSORES ESFUERZO (T oacute ) y DEFORMACIOacuteN (S oacute )
333231
232221
131211
TTT
TTT
TTT
T
333231
232221
131211
SSS
SSS
SSS
S
i
j
j
iij x
u
x
uS
2
1
T FA (Nm2 = Pa)
S LL
HIPERVECTOR
ELASTICIDAD Y NOTACIOacuteN MATRICIAL
Aij puede ser por ejemplo el tensor esfuerzo
o el tensor deformacioacuten
MATRIZ 3X3
S = sT T = cS
s = compliance c = stiffness
[1198781
1198782
1198783
1198784
1198785
1198786
]=[11990411
11990421
11990431
11990441
11990451
11990461
11990412
11990422
11990432
11990442
11990452
11990462
11990413
11990423
11990433
11990443
11990453
11990463
11990414
11990424
11990434
11990444
11990454
11990464
11990415
11990425
11990435
11990445
11990455
11990465
11990416
11990426
11990436
11990446
11990456
11990466
] ∙[119879 1
119879 2
119879 3
119879 4
119879 5
119879 6
]
La elasticidad a traveacutes del tensor compliance en notacioacuten matricial
S = sT
La Elasticidad en Materiales Isotroacutepicos
)(200000
0)(20000
00)(2000
000
000
000
1211
1211
1211
111212
121112
121211
ss
ss
ss
sss
sss
sss
s
s11 = 1E s12 = -E (MEI)
La Elasticidad en Materiales Isotroacutepicos
)(2
100000
0)(2
10000
00)(2
1000
000
000
000
1211
1211
1211
111212
121112
121211
cc
cc
cc
ccc
ccc
ccc
c
211
111
Ec
21112
Ec (MEI)
(Soacutelido ideal incompresible = 05 frecuentemente ~ 03)
Moacutedulo de Young ldquoErdquo T11 = ES11
allongitudinndeformacioacute
lateralndeformacioacute
Coeficiente de Poisson ldquordquo
La Elasticidad en Materiales Isotroacutepicos
La Elasticidad en Materiales Cuacutebicos
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
441211441211 22
1ccccssss aa
441211441211 22
1ccccssss aa
Anisotropiacutea
Inverso
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Las propiedades fiacutesicas como tensores
P E
P E
Introductorio P = o P E
Cristalofiacutesica P = o P∙ E
P es un tensor de 2ordm rango
En general las propiedades se asocian a relaciones
constitutivas
Y = K middot XSi los rangos respectivos de (X Y) son (mn) entonces la propiedad K es un tensor
de rango m+n
Property Related magnitudes Tensor
Heat capacity C Entropy (P0) Temperature (P0) P0
Elasticity s Strain (P2) Stress (P2) P4
Electr susceptibility χP Polarization (P1) Elec Intensity (P1) P2
Magn susceptibility χM Magnetization (A1) Magn Intensity (A1) P2
Thermal expansion Strain (P2) Temperature (P0) P2
Pyroelectricity p Polarization (P1) Temperature (P0) P1
Pyromagnetism i Magnetization (A1) Temperature (P0) A1
Piezoelectricity d Polarization (P1) Stress (P2) P3
Piezomagnetism b Magnetization (A1) Stress (P2) A3
Magnetoelectricity Magnetization (A1) Elec Intensity (P1) A2
P POLAR A AXIAL N = Tensor rank
THERMO-ELASTO-ELECTRO-MAGNETIC PROPERTIES
Algunos efectos y relaciones constitutivas
Paraelectricidad P = ε0χPmiddotE
Paramagnetismo μ0M = μ0χMmiddotH
Elasticidad S = s middot T
Dilatacioacuten S = middot Δ
Piezoelectricidad P = d middot T
S = d middot E
Efecto magnetoeleacutectrico
μ0M = middot E
P = middot H
INTERACCIONES PRINCIPALES Y DE ACOPLAMIENTO
PROPIEDADES DE ACOPLAMIENTO ELECTROMAGNEacuteTICO
SENSORES EN UN COCHE
Sensores de temperaturaSensores de condicioacuten del aceite y de humedad
Sensores de presioacuten
X - BY WIRE (X FRENOS DIRECCIOacuteN)
El servomecanismo de direccioacuten hidraacuteulica requiere presurizacioacuten constante del fluido hidraacuteulico Al sustituiacutersele por motores eleacutectricos se economizaraacute
aproximadamente 5 ndash 10 de combustible
ldquoSTEERING BY WIRErdquo A NIVEL DE LABORATORIO
httpwwwdelphicomautomotivenextechproductsxbywirehttp42voltdupontcomenSystemsbywire_mainhtml
httpwwwvisteoncomtechnologyautomotivepowertrain_controlshtml
EXPLICACIOacuteN CUALITATIVA DE ALGUNOS ACOPLAMIENTOS
B = mE m = eb
(T = eE)
e = dc
(c = s-1)
m = ds-1b
E E
T
H
D
S
B
p d
m
C
s
b
i
T = eE
Piezoelectricidad Piezomagnetismo
Magnetoelectricidad1 2
E
ESFUERZO (T oacute ) Y DEFORMACIOacuteN (S oacute )
EN UN ENSAYO DE TRACCIOacuteN
S = sTs = ldquocompliancerdquo
x
e oacute S
oacute T
Zona Elaacutestica
Zona Plaacutestica
s max
s fluencia
s YeT cS
T = cS c∙s = 1c = ldquostiffnessrdquo
Ley de Hooke
TENSORES ESFUERZO (T oacute ) y DEFORMACIOacuteN (S oacute )
333231
232221
131211
TTT
TTT
TTT
T
333231
232221
131211
SSS
SSS
SSS
S
i
j
j
iij x
u
x
uS
2
1
T FA (Nm2 = Pa)
S LL
HIPERVECTOR
ELASTICIDAD Y NOTACIOacuteN MATRICIAL
Aij puede ser por ejemplo el tensor esfuerzo
o el tensor deformacioacuten
MATRIZ 3X3
S = sT T = cS
s = compliance c = stiffness
[1198781
1198782
1198783
1198784
1198785
1198786
]=[11990411
11990421
11990431
11990441
11990451
11990461
11990412
11990422
11990432
11990442
11990452
11990462
11990413
11990423
11990433
11990443
11990453
11990463
11990414
11990424
11990434
11990444
11990454
11990464
11990415
11990425
11990435
11990445
11990455
11990465
11990416
11990426
11990436
11990446
11990456
11990466
] ∙[119879 1
119879 2
119879 3
119879 4
119879 5
119879 6
]
La elasticidad a traveacutes del tensor compliance en notacioacuten matricial
S = sT
La Elasticidad en Materiales Isotroacutepicos
)(200000
0)(20000
00)(2000
000
000
000
1211
1211
1211
111212
121112
121211
ss
ss
ss
sss
sss
sss
s
s11 = 1E s12 = -E (MEI)
La Elasticidad en Materiales Isotroacutepicos
)(2
100000
0)(2
10000
00)(2
1000
000
000
000
1211
1211
1211
111212
121112
121211
cc
cc
cc
ccc
ccc
ccc
c
211
111
Ec
21112
Ec (MEI)
(Soacutelido ideal incompresible = 05 frecuentemente ~ 03)
Moacutedulo de Young ldquoErdquo T11 = ES11
allongitudinndeformacioacute
lateralndeformacioacute
Coeficiente de Poisson ldquordquo
La Elasticidad en Materiales Isotroacutepicos
La Elasticidad en Materiales Cuacutebicos
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
441211441211 22
1ccccssss aa
441211441211 22
1ccccssss aa
Anisotropiacutea
Inverso
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Property Related magnitudes Tensor
Heat capacity C Entropy (P0) Temperature (P0) P0
Elasticity s Strain (P2) Stress (P2) P4
Electr susceptibility χP Polarization (P1) Elec Intensity (P1) P2
Magn susceptibility χM Magnetization (A1) Magn Intensity (A1) P2
Thermal expansion Strain (P2) Temperature (P0) P2
Pyroelectricity p Polarization (P1) Temperature (P0) P1
Pyromagnetism i Magnetization (A1) Temperature (P0) A1
Piezoelectricity d Polarization (P1) Stress (P2) P3
Piezomagnetism b Magnetization (A1) Stress (P2) A3
Magnetoelectricity Magnetization (A1) Elec Intensity (P1) A2
P POLAR A AXIAL N = Tensor rank
THERMO-ELASTO-ELECTRO-MAGNETIC PROPERTIES
Algunos efectos y relaciones constitutivas
Paraelectricidad P = ε0χPmiddotE
Paramagnetismo μ0M = μ0χMmiddotH
Elasticidad S = s middot T
Dilatacioacuten S = middot Δ
Piezoelectricidad P = d middot T
S = d middot E
Efecto magnetoeleacutectrico
μ0M = middot E
P = middot H
INTERACCIONES PRINCIPALES Y DE ACOPLAMIENTO
PROPIEDADES DE ACOPLAMIENTO ELECTROMAGNEacuteTICO
SENSORES EN UN COCHE
Sensores de temperaturaSensores de condicioacuten del aceite y de humedad
Sensores de presioacuten
X - BY WIRE (X FRENOS DIRECCIOacuteN)
El servomecanismo de direccioacuten hidraacuteulica requiere presurizacioacuten constante del fluido hidraacuteulico Al sustituiacutersele por motores eleacutectricos se economizaraacute
aproximadamente 5 ndash 10 de combustible
ldquoSTEERING BY WIRErdquo A NIVEL DE LABORATORIO
httpwwwdelphicomautomotivenextechproductsxbywirehttp42voltdupontcomenSystemsbywire_mainhtml
httpwwwvisteoncomtechnologyautomotivepowertrain_controlshtml
EXPLICACIOacuteN CUALITATIVA DE ALGUNOS ACOPLAMIENTOS
B = mE m = eb
(T = eE)
e = dc
(c = s-1)
m = ds-1b
E E
T
H
D
S
B
p d
m
C
s
b
i
T = eE
Piezoelectricidad Piezomagnetismo
Magnetoelectricidad1 2
E
ESFUERZO (T oacute ) Y DEFORMACIOacuteN (S oacute )
EN UN ENSAYO DE TRACCIOacuteN
S = sTs = ldquocompliancerdquo
x
e oacute S
oacute T
Zona Elaacutestica
Zona Plaacutestica
s max
s fluencia
s YeT cS
T = cS c∙s = 1c = ldquostiffnessrdquo
Ley de Hooke
TENSORES ESFUERZO (T oacute ) y DEFORMACIOacuteN (S oacute )
333231
232221
131211
TTT
TTT
TTT
T
333231
232221
131211
SSS
SSS
SSS
S
i
j
j
iij x
u
x
uS
2
1
T FA (Nm2 = Pa)
S LL
HIPERVECTOR
ELASTICIDAD Y NOTACIOacuteN MATRICIAL
Aij puede ser por ejemplo el tensor esfuerzo
o el tensor deformacioacuten
MATRIZ 3X3
S = sT T = cS
s = compliance c = stiffness
[1198781
1198782
1198783
1198784
1198785
1198786
]=[11990411
11990421
11990431
11990441
11990451
11990461
11990412
11990422
11990432
11990442
11990452
11990462
11990413
11990423
11990433
11990443
11990453
11990463
11990414
11990424
11990434
11990444
11990454
11990464
11990415
11990425
11990435
11990445
11990455
11990465
11990416
11990426
11990436
11990446
11990456
11990466
] ∙[119879 1
119879 2
119879 3
119879 4
119879 5
119879 6
]
La elasticidad a traveacutes del tensor compliance en notacioacuten matricial
S = sT
La Elasticidad en Materiales Isotroacutepicos
)(200000
0)(20000
00)(2000
000
000
000
1211
1211
1211
111212
121112
121211
ss
ss
ss
sss
sss
sss
s
s11 = 1E s12 = -E (MEI)
La Elasticidad en Materiales Isotroacutepicos
)(2
100000
0)(2
10000
00)(2
1000
000
000
000
1211
1211
1211
111212
121112
121211
cc
cc
cc
ccc
ccc
ccc
c
211
111
Ec
21112
Ec (MEI)
(Soacutelido ideal incompresible = 05 frecuentemente ~ 03)
Moacutedulo de Young ldquoErdquo T11 = ES11
allongitudinndeformacioacute
lateralndeformacioacute
Coeficiente de Poisson ldquordquo
La Elasticidad en Materiales Isotroacutepicos
La Elasticidad en Materiales Cuacutebicos
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
441211441211 22
1ccccssss aa
441211441211 22
1ccccssss aa
Anisotropiacutea
Inverso
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Algunos efectos y relaciones constitutivas
Paraelectricidad P = ε0χPmiddotE
Paramagnetismo μ0M = μ0χMmiddotH
Elasticidad S = s middot T
Dilatacioacuten S = middot Δ
Piezoelectricidad P = d middot T
S = d middot E
Efecto magnetoeleacutectrico
μ0M = middot E
P = middot H
INTERACCIONES PRINCIPALES Y DE ACOPLAMIENTO
PROPIEDADES DE ACOPLAMIENTO ELECTROMAGNEacuteTICO
SENSORES EN UN COCHE
Sensores de temperaturaSensores de condicioacuten del aceite y de humedad
Sensores de presioacuten
X - BY WIRE (X FRENOS DIRECCIOacuteN)
El servomecanismo de direccioacuten hidraacuteulica requiere presurizacioacuten constante del fluido hidraacuteulico Al sustituiacutersele por motores eleacutectricos se economizaraacute
aproximadamente 5 ndash 10 de combustible
ldquoSTEERING BY WIRErdquo A NIVEL DE LABORATORIO
httpwwwdelphicomautomotivenextechproductsxbywirehttp42voltdupontcomenSystemsbywire_mainhtml
httpwwwvisteoncomtechnologyautomotivepowertrain_controlshtml
EXPLICACIOacuteN CUALITATIVA DE ALGUNOS ACOPLAMIENTOS
B = mE m = eb
(T = eE)
e = dc
(c = s-1)
m = ds-1b
E E
T
H
D
S
B
p d
m
C
s
b
i
T = eE
Piezoelectricidad Piezomagnetismo
Magnetoelectricidad1 2
E
ESFUERZO (T oacute ) Y DEFORMACIOacuteN (S oacute )
EN UN ENSAYO DE TRACCIOacuteN
S = sTs = ldquocompliancerdquo
x
e oacute S
oacute T
Zona Elaacutestica
Zona Plaacutestica
s max
s fluencia
s YeT cS
T = cS c∙s = 1c = ldquostiffnessrdquo
Ley de Hooke
TENSORES ESFUERZO (T oacute ) y DEFORMACIOacuteN (S oacute )
333231
232221
131211
TTT
TTT
TTT
T
333231
232221
131211
SSS
SSS
SSS
S
i
j
j
iij x
u
x
uS
2
1
T FA (Nm2 = Pa)
S LL
HIPERVECTOR
ELASTICIDAD Y NOTACIOacuteN MATRICIAL
Aij puede ser por ejemplo el tensor esfuerzo
o el tensor deformacioacuten
MATRIZ 3X3
S = sT T = cS
s = compliance c = stiffness
[1198781
1198782
1198783
1198784
1198785
1198786
]=[11990411
11990421
11990431
11990441
11990451
11990461
11990412
11990422
11990432
11990442
11990452
11990462
11990413
11990423
11990433
11990443
11990453
11990463
11990414
11990424
11990434
11990444
11990454
11990464
11990415
11990425
11990435
11990445
11990455
11990465
11990416
11990426
11990436
11990446
11990456
11990466
] ∙[119879 1
119879 2
119879 3
119879 4
119879 5
119879 6
]
La elasticidad a traveacutes del tensor compliance en notacioacuten matricial
S = sT
La Elasticidad en Materiales Isotroacutepicos
)(200000
0)(20000
00)(2000
000
000
000
1211
1211
1211
111212
121112
121211
ss
ss
ss
sss
sss
sss
s
s11 = 1E s12 = -E (MEI)
La Elasticidad en Materiales Isotroacutepicos
)(2
100000
0)(2
10000
00)(2
1000
000
000
000
1211
1211
1211
111212
121112
121211
cc
cc
cc
ccc
ccc
ccc
c
211
111
Ec
21112
Ec (MEI)
(Soacutelido ideal incompresible = 05 frecuentemente ~ 03)
Moacutedulo de Young ldquoErdquo T11 = ES11
allongitudinndeformacioacute
lateralndeformacioacute
Coeficiente de Poisson ldquordquo
La Elasticidad en Materiales Isotroacutepicos
La Elasticidad en Materiales Cuacutebicos
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
441211441211 22
1ccccssss aa
441211441211 22
1ccccssss aa
Anisotropiacutea
Inverso
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
PROPIEDADES DE ACOPLAMIENTO ELECTROMAGNEacuteTICO
SENSORES EN UN COCHE
Sensores de temperaturaSensores de condicioacuten del aceite y de humedad
Sensores de presioacuten
X - BY WIRE (X FRENOS DIRECCIOacuteN)
El servomecanismo de direccioacuten hidraacuteulica requiere presurizacioacuten constante del fluido hidraacuteulico Al sustituiacutersele por motores eleacutectricos se economizaraacute
aproximadamente 5 ndash 10 de combustible
ldquoSTEERING BY WIRErdquo A NIVEL DE LABORATORIO
httpwwwdelphicomautomotivenextechproductsxbywirehttp42voltdupontcomenSystemsbywire_mainhtml
httpwwwvisteoncomtechnologyautomotivepowertrain_controlshtml
EXPLICACIOacuteN CUALITATIVA DE ALGUNOS ACOPLAMIENTOS
B = mE m = eb
(T = eE)
e = dc
(c = s-1)
m = ds-1b
E E
T
H
D
S
B
p d
m
C
s
b
i
T = eE
Piezoelectricidad Piezomagnetismo
Magnetoelectricidad1 2
E
ESFUERZO (T oacute ) Y DEFORMACIOacuteN (S oacute )
EN UN ENSAYO DE TRACCIOacuteN
S = sTs = ldquocompliancerdquo
x
e oacute S
oacute T
Zona Elaacutestica
Zona Plaacutestica
s max
s fluencia
s YeT cS
T = cS c∙s = 1c = ldquostiffnessrdquo
Ley de Hooke
TENSORES ESFUERZO (T oacute ) y DEFORMACIOacuteN (S oacute )
333231
232221
131211
TTT
TTT
TTT
T
333231
232221
131211
SSS
SSS
SSS
S
i
j
j
iij x
u
x
uS
2
1
T FA (Nm2 = Pa)
S LL
HIPERVECTOR
ELASTICIDAD Y NOTACIOacuteN MATRICIAL
Aij puede ser por ejemplo el tensor esfuerzo
o el tensor deformacioacuten
MATRIZ 3X3
S = sT T = cS
s = compliance c = stiffness
[1198781
1198782
1198783
1198784
1198785
1198786
]=[11990411
11990421
11990431
11990441
11990451
11990461
11990412
11990422
11990432
11990442
11990452
11990462
11990413
11990423
11990433
11990443
11990453
11990463
11990414
11990424
11990434
11990444
11990454
11990464
11990415
11990425
11990435
11990445
11990455
11990465
11990416
11990426
11990436
11990446
11990456
11990466
] ∙[119879 1
119879 2
119879 3
119879 4
119879 5
119879 6
]
La elasticidad a traveacutes del tensor compliance en notacioacuten matricial
S = sT
La Elasticidad en Materiales Isotroacutepicos
)(200000
0)(20000
00)(2000
000
000
000
1211
1211
1211
111212
121112
121211
ss
ss
ss
sss
sss
sss
s
s11 = 1E s12 = -E (MEI)
La Elasticidad en Materiales Isotroacutepicos
)(2
100000
0)(2
10000
00)(2
1000
000
000
000
1211
1211
1211
111212
121112
121211
cc
cc
cc
ccc
ccc
ccc
c
211
111
Ec
21112
Ec (MEI)
(Soacutelido ideal incompresible = 05 frecuentemente ~ 03)
Moacutedulo de Young ldquoErdquo T11 = ES11
allongitudinndeformacioacute
lateralndeformacioacute
Coeficiente de Poisson ldquordquo
La Elasticidad en Materiales Isotroacutepicos
La Elasticidad en Materiales Cuacutebicos
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
441211441211 22
1ccccssss aa
441211441211 22
1ccccssss aa
Anisotropiacutea
Inverso
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
X - BY WIRE (X FRENOS DIRECCIOacuteN)
El servomecanismo de direccioacuten hidraacuteulica requiere presurizacioacuten constante del fluido hidraacuteulico Al sustituiacutersele por motores eleacutectricos se economizaraacute
aproximadamente 5 ndash 10 de combustible
ldquoSTEERING BY WIRErdquo A NIVEL DE LABORATORIO
httpwwwdelphicomautomotivenextechproductsxbywirehttp42voltdupontcomenSystemsbywire_mainhtml
httpwwwvisteoncomtechnologyautomotivepowertrain_controlshtml
EXPLICACIOacuteN CUALITATIVA DE ALGUNOS ACOPLAMIENTOS
B = mE m = eb
(T = eE)
e = dc
(c = s-1)
m = ds-1b
E E
T
H
D
S
B
p d
m
C
s
b
i
T = eE
Piezoelectricidad Piezomagnetismo
Magnetoelectricidad1 2
E
ESFUERZO (T oacute ) Y DEFORMACIOacuteN (S oacute )
EN UN ENSAYO DE TRACCIOacuteN
S = sTs = ldquocompliancerdquo
x
e oacute S
oacute T
Zona Elaacutestica
Zona Plaacutestica
s max
s fluencia
s YeT cS
T = cS c∙s = 1c = ldquostiffnessrdquo
Ley de Hooke
TENSORES ESFUERZO (T oacute ) y DEFORMACIOacuteN (S oacute )
333231
232221
131211
TTT
TTT
TTT
T
333231
232221
131211
SSS
SSS
SSS
S
i
j
j
iij x
u
x
uS
2
1
T FA (Nm2 = Pa)
S LL
HIPERVECTOR
ELASTICIDAD Y NOTACIOacuteN MATRICIAL
Aij puede ser por ejemplo el tensor esfuerzo
o el tensor deformacioacuten
MATRIZ 3X3
S = sT T = cS
s = compliance c = stiffness
[1198781
1198782
1198783
1198784
1198785
1198786
]=[11990411
11990421
11990431
11990441
11990451
11990461
11990412
11990422
11990432
11990442
11990452
11990462
11990413
11990423
11990433
11990443
11990453
11990463
11990414
11990424
11990434
11990444
11990454
11990464
11990415
11990425
11990435
11990445
11990455
11990465
11990416
11990426
11990436
11990446
11990456
11990466
] ∙[119879 1
119879 2
119879 3
119879 4
119879 5
119879 6
]
La elasticidad a traveacutes del tensor compliance en notacioacuten matricial
S = sT
La Elasticidad en Materiales Isotroacutepicos
)(200000
0)(20000
00)(2000
000
000
000
1211
1211
1211
111212
121112
121211
ss
ss
ss
sss
sss
sss
s
s11 = 1E s12 = -E (MEI)
La Elasticidad en Materiales Isotroacutepicos
)(2
100000
0)(2
10000
00)(2
1000
000
000
000
1211
1211
1211
111212
121112
121211
cc
cc
cc
ccc
ccc
ccc
c
211
111
Ec
21112
Ec (MEI)
(Soacutelido ideal incompresible = 05 frecuentemente ~ 03)
Moacutedulo de Young ldquoErdquo T11 = ES11
allongitudinndeformacioacute
lateralndeformacioacute
Coeficiente de Poisson ldquordquo
La Elasticidad en Materiales Isotroacutepicos
La Elasticidad en Materiales Cuacutebicos
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
441211441211 22
1ccccssss aa
441211441211 22
1ccccssss aa
Anisotropiacutea
Inverso
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
ldquoSTEERING BY WIRErdquo A NIVEL DE LABORATORIO
httpwwwdelphicomautomotivenextechproductsxbywirehttp42voltdupontcomenSystemsbywire_mainhtml
httpwwwvisteoncomtechnologyautomotivepowertrain_controlshtml
EXPLICACIOacuteN CUALITATIVA DE ALGUNOS ACOPLAMIENTOS
B = mE m = eb
(T = eE)
e = dc
(c = s-1)
m = ds-1b
E E
T
H
D
S
B
p d
m
C
s
b
i
T = eE
Piezoelectricidad Piezomagnetismo
Magnetoelectricidad1 2
E
ESFUERZO (T oacute ) Y DEFORMACIOacuteN (S oacute )
EN UN ENSAYO DE TRACCIOacuteN
S = sTs = ldquocompliancerdquo
x
e oacute S
oacute T
Zona Elaacutestica
Zona Plaacutestica
s max
s fluencia
s YeT cS
T = cS c∙s = 1c = ldquostiffnessrdquo
Ley de Hooke
TENSORES ESFUERZO (T oacute ) y DEFORMACIOacuteN (S oacute )
333231
232221
131211
TTT
TTT
TTT
T
333231
232221
131211
SSS
SSS
SSS
S
i
j
j
iij x
u
x
uS
2
1
T FA (Nm2 = Pa)
S LL
HIPERVECTOR
ELASTICIDAD Y NOTACIOacuteN MATRICIAL
Aij puede ser por ejemplo el tensor esfuerzo
o el tensor deformacioacuten
MATRIZ 3X3
S = sT T = cS
s = compliance c = stiffness
[1198781
1198782
1198783
1198784
1198785
1198786
]=[11990411
11990421
11990431
11990441
11990451
11990461
11990412
11990422
11990432
11990442
11990452
11990462
11990413
11990423
11990433
11990443
11990453
11990463
11990414
11990424
11990434
11990444
11990454
11990464
11990415
11990425
11990435
11990445
11990455
11990465
11990416
11990426
11990436
11990446
11990456
11990466
] ∙[119879 1
119879 2
119879 3
119879 4
119879 5
119879 6
]
La elasticidad a traveacutes del tensor compliance en notacioacuten matricial
S = sT
La Elasticidad en Materiales Isotroacutepicos
)(200000
0)(20000
00)(2000
000
000
000
1211
1211
1211
111212
121112
121211
ss
ss
ss
sss
sss
sss
s
s11 = 1E s12 = -E (MEI)
La Elasticidad en Materiales Isotroacutepicos
)(2
100000
0)(2
10000
00)(2
1000
000
000
000
1211
1211
1211
111212
121112
121211
cc
cc
cc
ccc
ccc
ccc
c
211
111
Ec
21112
Ec (MEI)
(Soacutelido ideal incompresible = 05 frecuentemente ~ 03)
Moacutedulo de Young ldquoErdquo T11 = ES11
allongitudinndeformacioacute
lateralndeformacioacute
Coeficiente de Poisson ldquordquo
La Elasticidad en Materiales Isotroacutepicos
La Elasticidad en Materiales Cuacutebicos
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
441211441211 22
1ccccssss aa
441211441211 22
1ccccssss aa
Anisotropiacutea
Inverso
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
EXPLICACIOacuteN CUALITATIVA DE ALGUNOS ACOPLAMIENTOS
B = mE m = eb
(T = eE)
e = dc
(c = s-1)
m = ds-1b
E E
T
H
D
S
B
p d
m
C
s
b
i
T = eE
Piezoelectricidad Piezomagnetismo
Magnetoelectricidad1 2
E
ESFUERZO (T oacute ) Y DEFORMACIOacuteN (S oacute )
EN UN ENSAYO DE TRACCIOacuteN
S = sTs = ldquocompliancerdquo
x
e oacute S
oacute T
Zona Elaacutestica
Zona Plaacutestica
s max
s fluencia
s YeT cS
T = cS c∙s = 1c = ldquostiffnessrdquo
Ley de Hooke
TENSORES ESFUERZO (T oacute ) y DEFORMACIOacuteN (S oacute )
333231
232221
131211
TTT
TTT
TTT
T
333231
232221
131211
SSS
SSS
SSS
S
i
j
j
iij x
u
x
uS
2
1
T FA (Nm2 = Pa)
S LL
HIPERVECTOR
ELASTICIDAD Y NOTACIOacuteN MATRICIAL
Aij puede ser por ejemplo el tensor esfuerzo
o el tensor deformacioacuten
MATRIZ 3X3
S = sT T = cS
s = compliance c = stiffness
[1198781
1198782
1198783
1198784
1198785
1198786
]=[11990411
11990421
11990431
11990441
11990451
11990461
11990412
11990422
11990432
11990442
11990452
11990462
11990413
11990423
11990433
11990443
11990453
11990463
11990414
11990424
11990434
11990444
11990454
11990464
11990415
11990425
11990435
11990445
11990455
11990465
11990416
11990426
11990436
11990446
11990456
11990466
] ∙[119879 1
119879 2
119879 3
119879 4
119879 5
119879 6
]
La elasticidad a traveacutes del tensor compliance en notacioacuten matricial
S = sT
La Elasticidad en Materiales Isotroacutepicos
)(200000
0)(20000
00)(2000
000
000
000
1211
1211
1211
111212
121112
121211
ss
ss
ss
sss
sss
sss
s
s11 = 1E s12 = -E (MEI)
La Elasticidad en Materiales Isotroacutepicos
)(2
100000
0)(2
10000
00)(2
1000
000
000
000
1211
1211
1211
111212
121112
121211
cc
cc
cc
ccc
ccc
ccc
c
211
111
Ec
21112
Ec (MEI)
(Soacutelido ideal incompresible = 05 frecuentemente ~ 03)
Moacutedulo de Young ldquoErdquo T11 = ES11
allongitudinndeformacioacute
lateralndeformacioacute
Coeficiente de Poisson ldquordquo
La Elasticidad en Materiales Isotroacutepicos
La Elasticidad en Materiales Cuacutebicos
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
441211441211 22
1ccccssss aa
441211441211 22
1ccccssss aa
Anisotropiacutea
Inverso
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
ESFUERZO (T oacute ) Y DEFORMACIOacuteN (S oacute )
EN UN ENSAYO DE TRACCIOacuteN
S = sTs = ldquocompliancerdquo
x
e oacute S
oacute T
Zona Elaacutestica
Zona Plaacutestica
s max
s fluencia
s YeT cS
T = cS c∙s = 1c = ldquostiffnessrdquo
Ley de Hooke
TENSORES ESFUERZO (T oacute ) y DEFORMACIOacuteN (S oacute )
333231
232221
131211
TTT
TTT
TTT
T
333231
232221
131211
SSS
SSS
SSS
S
i
j
j
iij x
u
x
uS
2
1
T FA (Nm2 = Pa)
S LL
HIPERVECTOR
ELASTICIDAD Y NOTACIOacuteN MATRICIAL
Aij puede ser por ejemplo el tensor esfuerzo
o el tensor deformacioacuten
MATRIZ 3X3
S = sT T = cS
s = compliance c = stiffness
[1198781
1198782
1198783
1198784
1198785
1198786
]=[11990411
11990421
11990431
11990441
11990451
11990461
11990412
11990422
11990432
11990442
11990452
11990462
11990413
11990423
11990433
11990443
11990453
11990463
11990414
11990424
11990434
11990444
11990454
11990464
11990415
11990425
11990435
11990445
11990455
11990465
11990416
11990426
11990436
11990446
11990456
11990466
] ∙[119879 1
119879 2
119879 3
119879 4
119879 5
119879 6
]
La elasticidad a traveacutes del tensor compliance en notacioacuten matricial
S = sT
La Elasticidad en Materiales Isotroacutepicos
)(200000
0)(20000
00)(2000
000
000
000
1211
1211
1211
111212
121112
121211
ss
ss
ss
sss
sss
sss
s
s11 = 1E s12 = -E (MEI)
La Elasticidad en Materiales Isotroacutepicos
)(2
100000
0)(2
10000
00)(2
1000
000
000
000
1211
1211
1211
111212
121112
121211
cc
cc
cc
ccc
ccc
ccc
c
211
111
Ec
21112
Ec (MEI)
(Soacutelido ideal incompresible = 05 frecuentemente ~ 03)
Moacutedulo de Young ldquoErdquo T11 = ES11
allongitudinndeformacioacute
lateralndeformacioacute
Coeficiente de Poisson ldquordquo
La Elasticidad en Materiales Isotroacutepicos
La Elasticidad en Materiales Cuacutebicos
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
441211441211 22
1ccccssss aa
441211441211 22
1ccccssss aa
Anisotropiacutea
Inverso
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
TENSORES ESFUERZO (T oacute ) y DEFORMACIOacuteN (S oacute )
333231
232221
131211
TTT
TTT
TTT
T
333231
232221
131211
SSS
SSS
SSS
S
i
j
j
iij x
u
x
uS
2
1
T FA (Nm2 = Pa)
S LL
HIPERVECTOR
ELASTICIDAD Y NOTACIOacuteN MATRICIAL
Aij puede ser por ejemplo el tensor esfuerzo
o el tensor deformacioacuten
MATRIZ 3X3
S = sT T = cS
s = compliance c = stiffness
[1198781
1198782
1198783
1198784
1198785
1198786
]=[11990411
11990421
11990431
11990441
11990451
11990461
11990412
11990422
11990432
11990442
11990452
11990462
11990413
11990423
11990433
11990443
11990453
11990463
11990414
11990424
11990434
11990444
11990454
11990464
11990415
11990425
11990435
11990445
11990455
11990465
11990416
11990426
11990436
11990446
11990456
11990466
] ∙[119879 1
119879 2
119879 3
119879 4
119879 5
119879 6
]
La elasticidad a traveacutes del tensor compliance en notacioacuten matricial
S = sT
La Elasticidad en Materiales Isotroacutepicos
)(200000
0)(20000
00)(2000
000
000
000
1211
1211
1211
111212
121112
121211
ss
ss
ss
sss
sss
sss
s
s11 = 1E s12 = -E (MEI)
La Elasticidad en Materiales Isotroacutepicos
)(2
100000
0)(2
10000
00)(2
1000
000
000
000
1211
1211
1211
111212
121112
121211
cc
cc
cc
ccc
ccc
ccc
c
211
111
Ec
21112
Ec (MEI)
(Soacutelido ideal incompresible = 05 frecuentemente ~ 03)
Moacutedulo de Young ldquoErdquo T11 = ES11
allongitudinndeformacioacute
lateralndeformacioacute
Coeficiente de Poisson ldquordquo
La Elasticidad en Materiales Isotroacutepicos
La Elasticidad en Materiales Cuacutebicos
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
441211441211 22
1ccccssss aa
441211441211 22
1ccccssss aa
Anisotropiacutea
Inverso
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
HIPERVECTOR
ELASTICIDAD Y NOTACIOacuteN MATRICIAL
Aij puede ser por ejemplo el tensor esfuerzo
o el tensor deformacioacuten
MATRIZ 3X3
S = sT T = cS
s = compliance c = stiffness
[1198781
1198782
1198783
1198784
1198785
1198786
]=[11990411
11990421
11990431
11990441
11990451
11990461
11990412
11990422
11990432
11990442
11990452
11990462
11990413
11990423
11990433
11990443
11990453
11990463
11990414
11990424
11990434
11990444
11990454
11990464
11990415
11990425
11990435
11990445
11990455
11990465
11990416
11990426
11990436
11990446
11990456
11990466
] ∙[119879 1
119879 2
119879 3
119879 4
119879 5
119879 6
]
La elasticidad a traveacutes del tensor compliance en notacioacuten matricial
S = sT
La Elasticidad en Materiales Isotroacutepicos
)(200000
0)(20000
00)(2000
000
000
000
1211
1211
1211
111212
121112
121211
ss
ss
ss
sss
sss
sss
s
s11 = 1E s12 = -E (MEI)
La Elasticidad en Materiales Isotroacutepicos
)(2
100000
0)(2
10000
00)(2
1000
000
000
000
1211
1211
1211
111212
121112
121211
cc
cc
cc
ccc
ccc
ccc
c
211
111
Ec
21112
Ec (MEI)
(Soacutelido ideal incompresible = 05 frecuentemente ~ 03)
Moacutedulo de Young ldquoErdquo T11 = ES11
allongitudinndeformacioacute
lateralndeformacioacute
Coeficiente de Poisson ldquordquo
La Elasticidad en Materiales Isotroacutepicos
La Elasticidad en Materiales Cuacutebicos
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
441211441211 22
1ccccssss aa
441211441211 22
1ccccssss aa
Anisotropiacutea
Inverso
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
[1198781
1198782
1198783
1198784
1198785
1198786
]=[11990411
11990421
11990431
11990441
11990451
11990461
11990412
11990422
11990432
11990442
11990452
11990462
11990413
11990423
11990433
11990443
11990453
11990463
11990414
11990424
11990434
11990444
11990454
11990464
11990415
11990425
11990435
11990445
11990455
11990465
11990416
11990426
11990436
11990446
11990456
11990466
] ∙[119879 1
119879 2
119879 3
119879 4
119879 5
119879 6
]
La elasticidad a traveacutes del tensor compliance en notacioacuten matricial
S = sT
La Elasticidad en Materiales Isotroacutepicos
)(200000
0)(20000
00)(2000
000
000
000
1211
1211
1211
111212
121112
121211
ss
ss
ss
sss
sss
sss
s
s11 = 1E s12 = -E (MEI)
La Elasticidad en Materiales Isotroacutepicos
)(2
100000
0)(2
10000
00)(2
1000
000
000
000
1211
1211
1211
111212
121112
121211
cc
cc
cc
ccc
ccc
ccc
c
211
111
Ec
21112
Ec (MEI)
(Soacutelido ideal incompresible = 05 frecuentemente ~ 03)
Moacutedulo de Young ldquoErdquo T11 = ES11
allongitudinndeformacioacute
lateralndeformacioacute
Coeficiente de Poisson ldquordquo
La Elasticidad en Materiales Isotroacutepicos
La Elasticidad en Materiales Cuacutebicos
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
441211441211 22
1ccccssss aa
441211441211 22
1ccccssss aa
Anisotropiacutea
Inverso
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
La Elasticidad en Materiales Isotroacutepicos
)(200000
0)(20000
00)(2000
000
000
000
1211
1211
1211
111212
121112
121211
ss
ss
ss
sss
sss
sss
s
s11 = 1E s12 = -E (MEI)
La Elasticidad en Materiales Isotroacutepicos
)(2
100000
0)(2
10000
00)(2
1000
000
000
000
1211
1211
1211
111212
121112
121211
cc
cc
cc
ccc
ccc
ccc
c
211
111
Ec
21112
Ec (MEI)
(Soacutelido ideal incompresible = 05 frecuentemente ~ 03)
Moacutedulo de Young ldquoErdquo T11 = ES11
allongitudinndeformacioacute
lateralndeformacioacute
Coeficiente de Poisson ldquordquo
La Elasticidad en Materiales Isotroacutepicos
La Elasticidad en Materiales Cuacutebicos
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
441211441211 22
1ccccssss aa
441211441211 22
1ccccssss aa
Anisotropiacutea
Inverso
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
La Elasticidad en Materiales Isotroacutepicos
)(2
100000
0)(2
10000
00)(2
1000
000
000
000
1211
1211
1211
111212
121112
121211
cc
cc
cc
ccc
ccc
ccc
c
211
111
Ec
21112
Ec (MEI)
(Soacutelido ideal incompresible = 05 frecuentemente ~ 03)
Moacutedulo de Young ldquoErdquo T11 = ES11
allongitudinndeformacioacute
lateralndeformacioacute
Coeficiente de Poisson ldquordquo
La Elasticidad en Materiales Isotroacutepicos
La Elasticidad en Materiales Cuacutebicos
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
441211441211 22
1ccccssss aa
441211441211 22
1ccccssss aa
Anisotropiacutea
Inverso
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
(Soacutelido ideal incompresible = 05 frecuentemente ~ 03)
Moacutedulo de Young ldquoErdquo T11 = ES11
allongitudinndeformacioacute
lateralndeformacioacute
Coeficiente de Poisson ldquordquo
La Elasticidad en Materiales Isotroacutepicos
La Elasticidad en Materiales Cuacutebicos
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
441211441211 22
1ccccssss aa
441211441211 22
1ccccssss aa
Anisotropiacutea
Inverso
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
La Elasticidad en Materiales Cuacutebicos
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
4444
12111211
1212
12111211
121111
1
)2)(()2)(( cs
cccc
cs
cccc
ccs
441211441211 22
1ccccssss aa
441211441211 22
1ccccssss aa
Anisotropiacutea
Inverso
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
PIEZOELECTRICIDAD EN NOTACIOacuteN MATRICIAL
3
2
1
6
5
4
3
2
1
P
P
P
T
T
T
T
T
T
dddddd
dddddd
dddddd
363534333231
262524232221
161514131211
oacute
d ∙ T = P
diα para el cuarzo
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
CUARZO PIEZOELECTRICIDAD LONGITUDINAL Y TRANSVERSAL
httpwwwmodalshopcomcalibrationaspID=176
P1 = d11T1 + d12T2 +d14T4 (Px = 23 Txx - 23 Tyy - 067 Tyz)
P2 = d25T5 + d26T6 (Py = 067 Txz - 46 Txy) P3 = Pz = 0
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
TABLAS ELASTO-PIEZO-
DIELEacuteCTRICAS IEEE
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
S
S
S
S
S
S
D
D
D
=
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s s s s s d d d
s s
1
2
3
4
5
6
1
2
3
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
41 42 43 44 45 46 41 42 43
51 62 53 54 55 56 51 52 53
61 62 63
s s s s d d d
d d d d d d
d d d d d d
d d d d d d
T
T
T
T
T
T
E
E
E
64 65 66 61 62 63
11 12 13 14 15 16 11 12 13
21 22 23 24 25 26 21 22 23
31 32 33 34 35 36 31 32 33
1
2
3
4
5
6
1
2
3
MATRIZ ELASTO-PIEZO-DIELEacuteCTRICA
S = sT + dE D ( P) = dT + E
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
xyz
REPRESENTACIOacuteN SUPERFICIAL DE LAS PROPIEDADES (r = 2)
Constante dieleacutectrica logitudinal del cristalPbBi4Ti4O15 11 = 22 = 3000 33 = 426
Grupo puntual estructural 4m m mGrupo puntual de la propiedad
(h) = 11 h12 + 22 h2
2 + 33 h32
h1 = sen cos
h ( ) h2 = sen sen
h3 = cos
h3
h2
h1
h
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
PIEZOELECTRICIDAD LONGITUDINAL (r = 3)
CUARZO 321 BaTiO3 4mm
-cuarzo d( ) = d11 sin3 cos 3- BaTiO3 d() = (d33 - d15 - d31) cos3 + (d15 + d31) cos -iquestdoacutende estaacute el origen
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
ELASTICIDAD (r = 4)
CINC - HEXAGONAL COBRE - CUacuteBICO- cuacutebicos11 = s11 ‑ 2 (s11 ‑ s12 ‑ frac12 s44) (h1
2 h22 + h22 h32 + h12 h32) - hexagonals11 = (1 ‑ h32)s11 + h34 s33 + h32 (1 ‑ h32 )(2s13 + s44)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Cryst Syst Intern Sch type PRE PZE PRM PZM ME
triclinic 1 C1 e + + + + + -1 Ci c + + monoclinic 2 C2 e + + + + + m Cs nc-ne + + + + + 2m C2h c + + orthorhombic 222 D2 e + + + 2mm C2v nc-ne + + + + mmm D2h c + tetragonal 4 C4 e + + + + + -4 S4 nc-ne + + + + 4m C4h c + + 422 D4 e + + + 4mm C4v nc-ne + + + + -42m D2d nc-ne + + + 4mmm D4h c + trigonal 3 C3 e + + + + + -3 S6 c + + 32 D3 e + + + 3m C3v nc-ne + + + + -3m D3d c + hexagonal 6 C6 e + + + + + -6 C3h nc-ne + + + 6m C6h c + + 622 D6 e + + + 6mm C6v nc-ne + + + + -6m2 D3h nc-ne + + 6mmm D6h c + cubic 23 T e + + + m3 Th c + 432 O e + -43m Td nc-ne + m3m Oh c
Leyend Int International notationSch Schoenfliese enantiomorphic c centric nc-ne non centric-non enantiomorfic + possible PRE piroelectricity PZE piezoelectricityPRM piromagnetism PZM piezomagnetism ME magnetoelectricity
CRYSTALLOGRAPHIC POINT GROUPS AND
ELECTROMAGNETIC COUPLING EFFECTS
L Fuentes (1998) Textures and Microstructures 30 167-189
To find out if a magnetic property is possible in a
givencrystal the ordinary
symmetry elements of its color point group must be
taken into account
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
ldquoSAMZrdquo y ldquoMPODrdquo
httpblogscimavedumxluisfuentesarchivos
httpcrystalcimavedumxsamz
httpwwwmaterialpropertiesorg
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Application for surface representation of propertiesDownloadable from a CIMAV blog
httpblogscimavedumxluisfuentesarchivos
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Traditional (and
great) tool for data
collection Landolt-
Boumlrnstein Springer
(t gt 1883)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
An emerging option with modern ldquoopenrdquo approach httpmpodcimavedumx
Daniel Saulius Giancarlo
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Today MPOD and SAMZ are linked and can be run from any place with internet even from a cell phone (wi-fi)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
EL PRINCIPIO DE NEUMANN
La simetriacutea del efecto siempre es al menos igual a la simetriacutea de la
causaCausa Efecto
Electromagnetismo
Cargas y corriente
s
Campos E y B
Cristalofiacutesica Estructura
Propiedades
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
El Principio de Neumann en Cristalofiacutesica Tratamiento tensorial
2deg Rango Rango n
TENSORES POLARES
A = G-1 A G Aijhellipm = GikGjlhellipGmnAklhellipn
TENSORES AXIALES
B = |G|G-1 A G
Bijhellipm = | G | GikGjlhellipGmnBklhellip
n
Las operaciones de simetriacutea generalizada de cualquier propiedad fiacutesica macroscoacutepica forman un grupo de color que contiene como subgrupo al grupo puntual de simetriacutea
de color de la estructura considerada
A = tensor de propiedad G = matriz de rotacioacutenldquo+rdquo simetriacutea ldquordquo antisimetriacutea
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
The irreps approach
Piezoelectricity in C2v
6
5
4
3
2
1
3
2
1
T
T
T
T
T
T
dddddd
dddddd
dddddd
P
P
P
363534333231
262524232221
161514131211
IEEE Standards for Piezoelectricity
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
MAGNETOELECTRIC POINT GROUPS
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Magnetoelectric longitudinal surfaces
Cr2O3 (-3rsquomrsquo)
Aurivillius Phase Bi5FeTi3O15 (2mm)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Magnetoelectricidad el caso Cr2O3
(tema avanzado)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Time-reversal and spatial-inversion symmetry in ferroics
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Effect of time inversion
)( magnitudes
)( magnitudes
H M B j
DPE
c
i
R
RRR
Time inversion t -t
E P D ldquoirdquo polar vectors
j ldquocrdquo polar vectors
B H M ldquocrdquo axial vectors
L Fuentes-Cobas J Matutes-Aquino ME Fuentes Montero KHJ Buschow (ed) Handbook Magnetic Materials Vol 19 Chap 3 Magnetoelectricity 129-229 Elsevier Amsterdam (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Ordinary or single-color groups (the 32 classical crystallographic point groups)
2 Gray symmetry Coincidence of white and black (spin-up and spin-down) conditions Para- and
dia-magnetic materials (32 ldquograyrdquo groups) 3 ldquoBlack and whiterdquo point groups (58 BampW
groups) BampW point groups are formed as follows G is an ordinary group of order g H is a sub group of G of order g2 Multiply (G minus H ) by the time inversion operation and build the ldquoblack
and whiterdquo group M = H +(G minus H ) In the Schoenflies notation it is denoted as GH
Magnetic (BampW) point groups
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Simetriacutea del -Fe
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Cr2O3
cR 23 cR3 63D d
m23 m3
Structural point group
D3d
Space group
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
m23Irreps of point group D3d =
=
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Grupo puntual magneacutetico Co-irreps
m3 = D3dD3 = D3 + (D3d ndashD3)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Tensor magnetoeleacutectrico y mediciones
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)
Bibliografiacutea- Nye J F The Physical Properties of Crystals Clarendon Press Oxford
(1957) - Hartmann E An Introduction to Crystal Physics International Union of
Crystallography Teaching Pamphlet 18 (2005) httpwwwiucrorg - Fuentes L Fuentes ME ldquoLa Relacioacuten Estructura-Simetriacutea-Propiedades
en Cristales y Policristalesrdquo Reverteacute Madrid (2008)- IEEE Standards on Piezoelectricity ANSIIEEE Std 176 (1987)- Chateigner D et al Material Properties Open Database MPOD- httpmpodcimavedumx- Muntildeoz A Aquino G Templeton I Saacutenchez D Fuentes L Programa
SAMZ para representacioacuten superficial de propiedades httpcrystalcimavedumxsamz
- L Fuentes J Matutes Ma E Fuentes ldquoMagnetoelectricityrdquo Capiacutetulo 3 del Volumen 19 del ldquoHandbook of Magnetic Materialsrdquo Editor KHJ Buschow Elsevier (2011)