Las Propiedades Físicas de los Cristales L. Fuentes, CIMAV Superconductores de alta temperatura...

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)


)(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 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



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


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 de los Cristales L Fuentes CIMAV




PROPIEDADES DE ACOPLAMIENTO ELECTROMAGNEacuteTICO SENSORES EN UN




Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30


El Principio de Neumann en Cristalofiacutesica Tratamiento tensoria

Slide 33

Slide 34

Slide 35

Magnetoelectricidad el caso Cr2O3 (tema avanzado)

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45


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




















S = d middot E


μ0M = middot E

P = middot H













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



E




x

e oacute S

oacute T

Zona Elaacutestica

Zona Plaacutestica

s max

s fluencia

s YeT cS


Ley de Hooke


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




MATRIZ 3X3

S = sT T = cS


[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

]


S = sT


)(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)


)(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)








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


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









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


S = sT + dE D ( P) = dT + E

xyz




(h) = 11 h12 + 22 h2

2 + 33 h32

h1 = sen cos

h ( ) h2 = sen sen

h3 = cos

h3

h2

h1

h




ELASTICIDAD (r = 4)












taken into account







Traditional (and


collection Landolt-


(t gt 1883)






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45




















S = d middot E


μ0M = middot E

P = middot H













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



E




x

e oacute S

oacute T

Zona Elaacutestica

Zona Plaacutestica

s max

s fluencia

s YeT cS


Ley de Hooke


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




MATRIZ 3X3

S = sT T = cS


[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

]


S = sT


)(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)


)(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)








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


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









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


S = sT + dE D ( P) = dT + E

xyz




(h) = 11 h12 + 22 h2

2 + 33 h32

h1 = sen cos

h ( ) h2 = sen sen

h3 = cos

h3

h2

h1

h




ELASTICIDAD (r = 4)












taken into account







Traditional (and


collection Landolt-


(t gt 1883)






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45












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



E




x

e oacute S

oacute T

Zona Elaacutestica

Zona Plaacutestica

s max

s fluencia

s YeT cS


Ley de Hooke


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




MATRIZ 3X3

S = sT T = cS


[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

]


S = sT


)(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)


)(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)








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


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









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


S = sT + dE D ( P) = dT + E

xyz




(h) = 11 h12 + 22 h2

2 + 33 h32

h1 = sen cos

h ( ) h2 = sen sen

h3 = cos

h3

h2

h1

h




ELASTICIDAD (r = 4)












taken into account







Traditional (and


collection Landolt-


(t gt 1883)






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45




x

e oacute S

oacute T

Zona Elaacutestica

Zona Plaacutestica

s max

s fluencia

s YeT cS


Ley de Hooke


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




MATRIZ 3X3

S = sT T = cS


[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

]


S = sT


)(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)


)(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)








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


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









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


S = sT + dE D ( P) = dT + E

xyz




(h) = 11 h12 + 22 h2

2 + 33 h32

h1 = sen cos

h ( ) h2 = sen sen

h3 = cos

h3

h2

h1

h




ELASTICIDAD (r = 4)












taken into account







Traditional (and


collection Landolt-


(t gt 1883)






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45


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




MATRIZ 3X3

S = sT T = cS


[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

]


S = sT


)(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)


)(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)








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


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









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


S = sT + dE D ( P) = dT + E

xyz




(h) = 11 h12 + 22 h2

2 + 33 h32

h1 = sen cos

h ( ) h2 = sen sen

h3 = cos

h3

h2

h1

h




ELASTICIDAD (r = 4)












taken into account







Traditional (and


collection Landolt-


(t gt 1883)






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

HIPERVECTOR




MATRIZ 3X3

S = sT T = cS


[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

]


S = sT


)(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)


)(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)








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


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









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


S = sT + dE D ( P) = dT + E

xyz




(h) = 11 h12 + 22 h2

2 + 33 h32

h1 = sen cos

h ( ) h2 = sen sen

h3 = cos

h3

h2

h1

h




ELASTICIDAD (r = 4)












taken into account







Traditional (and


collection Landolt-


(t gt 1883)






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

[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

]


S = sT


)(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)


)(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)








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


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









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


S = sT + dE D ( P) = dT + E

xyz




(h) = 11 h12 + 22 h2

2 + 33 h32

h1 = sen cos

h ( ) h2 = sen sen

h3 = cos

h3

h2

h1

h




ELASTICIDAD (r = 4)












taken into account







Traditional (and


collection Landolt-


(t gt 1883)






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45


)(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)


)(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)








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


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









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


S = sT + dE D ( P) = dT + E

xyz




(h) = 11 h12 + 22 h2

2 + 33 h32

h1 = sen cos

h ( ) h2 = sen sen

h3 = cos

h3

h2

h1

h




ELASTICIDAD (r = 4)












taken into account







Traditional (and


collection Landolt-


(t gt 1883)






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45


)(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)








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


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









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


S = sT + dE D ( P) = dT + E

xyz




(h) = 11 h12 + 22 h2

2 + 33 h32

h1 = sen cos

h ( ) h2 = sen sen

h3 = cos

h3

h2

h1

h




ELASTICIDAD (r = 4)












taken into account







Traditional (and


collection Landolt-


(t gt 1883)






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45








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


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









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


S = sT + dE D ( P) = dT + E

xyz




(h) = 11 h12 + 22 h2

2 + 33 h32

h1 = sen cos

h ( ) h2 = sen sen

h3 = cos

h3

h2

h1

h




ELASTICIDAD (r = 4)












taken into account







Traditional (and


collection Landolt-


(t gt 1883)






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45


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









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


S = sT + dE D ( P) = dT + E

xyz




(h) = 11 h12 + 22 h2

2 + 33 h32

h1 = sen cos

h ( ) h2 = sen sen

h3 = cos

h3

h2

h1

h




ELASTICIDAD (r = 4)












taken into account







Traditional (and


collection Landolt-


(t gt 1883)






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45









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


S = sT + dE D ( P) = dT + E

xyz




(h) = 11 h12 + 22 h2

2 + 33 h32

h1 = sen cos

h ( ) h2 = sen sen

h3 = cos

h3

h2

h1

h




ELASTICIDAD (r = 4)












taken into account







Traditional (and


collection Landolt-


(t gt 1883)






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

xyz




(h) = 11 h12 + 22 h2

2 + 33 h32

h1 = sen cos

h ( ) h2 = sen sen

h3 = cos

h3

h2

h1

h




ELASTICIDAD (r = 4)












taken into account







Traditional (and


collection Landolt-


(t gt 1883)






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45




ELASTICIDAD (r = 4)












taken into account







Traditional (and


collection Landolt-


(t gt 1883)






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45










taken into account







Traditional (and


collection Landolt-


(t gt 1883)






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45







Traditional (and


collection Landolt-


(t gt 1883)






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

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Slide 20

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Slide 33

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Slide 43

Slide 44

Slide 45






causaCausa Efecto

Electromagnetismo

Cargas y corriente

s

Campos E y B


Propiedades


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45


2deg Rango Rango n

TENSORES POLARES


TENSORES AXIALES

B = |G|G-1 A G


n




The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

The irreps approach


6

5

4

3

2

1

3

2

1

T

T

T

T

T

T

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211







(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45






(tema avanzado)



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45



)( magnitudes

)( magnitudes

H M B j

DPE

c

i

R

RRR

Time inversion t -t












Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45








Cr2O3

cR 23 cR3 63D d

m23 m3


D3d

Space group


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45


=


















Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

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Slide 18

Slide 19

Slide 20

Slide 21

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Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30



Slide 33

Slide 34

Slide 35


Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45


















Slide 9

Slide 10

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Las Propiedades Físicas de los Cristales L. Fuentes, CIMAV Superconductores de alta temperatura...

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Transcript of Las Propiedades Físicas de los Cristales L. Fuentes, CIMAV Superconductores de alta temperatura...