Chapter 8 – Symmetry in Crystal Physics – p. 1 - 9. Symmetry in ...

Chapter 8 – Symmetry in Crystal Physics – p. 1 -

9. Symmetry in Crystal Physics

9.1. Description Physical Properties of Crystals by Tensors

Isotropic material: EχPvr

= (el. polarization, el. susceptibility, el. field); EPvr

||

physical properties, which describe the relation between vectors are generally described by scalars, implying that

both vectors are parallel.

Anisotropic materials: EP;r

are not necessarily parallel. Suitable description?

Example:

Electric susceptibility χ

vvvv EχPvr

= and vhvh EχPvr

=

In this 2D example, two scalars are required to describe the effect of the vertical field strength, same for the

horizontal field strength.

General case in three dimensional space:


Eχχχχχχχχχ

Pvr

=

333231

232221

131211

or

3332321313

3232221212

3132121111

EχEχEχPEχEχEχPEχEχEχP

++=++=++=

Abbreviated version:

3,2,1 ; 3

1=∑=

=iEχP j

jiji

Einstein’s Notation:

3,2,1, ; == jiEχP jiji

Ultimate abbreviation for people, who don’t like summation-symbols: Summation over all

indices, which occur twice in the same term.

The dielectric susceptibility is said to be a second rank tensor.

General definition of a tensor:

A tensor is defined as a set of 3r components which describes a physical quantity. In addition

a tensor has to have special transformation properties upon a change of the basis of the

coordinate system. These transformation properties will be discussed in section 9.4.

r is called the rank of a tensor:

• Scalar: 1 = 30 component, zero-rank tensor

• Vector: 3 = 31 component, 1st-rank tensor

• Matrix: 9 = 32 components, 2nd-rank tensor

• general: 3r components, r-rank tensor

General relation between two physical properties A and B:

3,2,1,...,,,,,...,,, ; ............ == urqpnkjiAaB upqrunpqrijknijk with

• A: f-rank tensor

• B: g-rank tensor


• a: (f+g)-rank tensor

Another example: stress, strain, elastic modulus and Hooke’s law

The stress tensor:

Body acted on by external forces: state of stress.

We consider a simple case: (1) case of homogeneous stress: stress is independent of position

in the body, (2) static equilibrium, (3) no body forces or torques.

Stress is described by 9 stress components dAdFσ /= (stress, area of body element, force

acting on area), i.e. second rank tensor ijσ :

The fact that the stress components represent a tensor is to be proved. Proves can be fount in the textbooks on

crystal physics (see literature) .

Obviously, the stress has 3 normal components 332211 ,, σσσ and 6 sheer components

323121231312 ,,,,, σσσσσσ . Condition (2) (static equilibrium) immediately requires, that there are

certain relationships between the sheer components (otherwise there would be a torque acting

on the body): 133132232112 ,, σσσσσσ === , the stress tensor is a symmetric second rank tensor:

jiij σσ = .

(Latter relationship still holds in the case of inhomogeneous stress or body forces, see textbooks).


The strain tensor:

Strain in 1 dimension:

Strain: dldxε =

In a body, along one direction, of the material in all three directions might change. Therefore:

Extension: i

iij l

xe∂∂

=

The extension contains a symmetrical part and a antisymmetrical part. The antisymmetrical

part describes a rotation of the body, the symmetrical part describes the actual deformation:

Strain tensor ijε (symmetrical part of the extension):

++

++

++

=

3332233113

3223222112

3113211211

332313

232212

131211

)(21)(

21

)(21)(

21

)(21)(

21

eeeee

eeeee

eeeee

εεεεεεεεε

Thus, the strain tensor is also a symmetric second rank tensor.

With the previous definitions we can reformulate Hook’s law for a general solid:

klijklij σsε =

The elastic modulus ijkls is a 4th rank tensor, in principle containing 81 components.


9.2 Examples of Tensors Representing Physical Properties

Similar as in the case of the discussed examples many physical properties can be described in

tern of tensors of different rank. Here some examples (from E. Hartmann, Introduction to

Crystal Physics):

9.3. Polar and Axial Vectors and Tensors

We consider a basis transformation from an old coordinate system ie to a new coordinate

system 'ie , described by a transformation matrix jia . Both sets of basis vectors are chosen

orthonormal, yielding ijji aa =−1 .


Normally, a vector ip in the old coordinate system would be described in the new coordinate

system as:

jiji pap =' (note the summation over j according to the Einstein notation).

Such a (normal) vector is also called a polar vector.

In physics, there are some vectors, however, which have slightly different transformation

properties. These vectors are usually connected to a definition involving the vector product:

321

321

321

bbbaaaeee

bac

rrrrrr

=×=

(the vector product describes a vector of length ababc ∠= sin , is perpendicular to a ar and br

,

and which forms a right handed system cba rrr ,, ). We consider a basis transformation, which

changes the coordinate system from right-handed to left handed or vice versa, such as the

inversion i:


In general, a basis transformation of a polar vector generates a change of sign, if the

transformation changes the hand. A change of hands is deduced from the determinant ija of

the transformation matrix

• 1=ija : Transformation leaves hand of the axes unchanged (rotation)

• 1−=ija : Transformation changes hand of the axes (inversion, reflection)

Transformation of a axial vector:

jijiji paap =' (note the summation over j according to the Einstein notation).

Physical examples of axial vector (1st rank axial tensor) are the angular momentum

( vrmL rrr×= ) or the magnetic flux density ( BvQF

rrr×= ).

In general a tensor a connecting two properties A and B via ............ upqrunpqrijknijk AaB = is axial if

either A or B are axial. In every other case it is polar.

9.4. Transformation Properties of Tensors

Again, we consider a basis transformation to a new coordinate system. A scalar (0-rank

tensor) does not change upon this operation:

TT =' (0-rank tensor)

A (polar) vector ip is described in the new coordinate system as:

jiji pap =' (or jiji TaT =' 1st-rank tensor)

For a (polar) 2nd-rank tensor

lklk qTp =

we obtain with kiki pap =' and jljl qaq =' or jjll qaq '= :

jijjjlkliki qTqaTap '''' == or


kljlikij TaaT =' (2nd-rank tensor)

General transformation laws for tensors upon basis transformation:

Tensor Rank Polar Tensor Axial Tensor

0 klij TT ='

1 jiji TaT ='

jijiji TaaT ='

2 kljlikij TaaT ='

kljlikijij TaaaT ='

3 lmnknjmilijk TaaaT ='

lmnknjmilijijk TaaaaT ='

n upqrnukrjqipnijk TaaaaT ...... ...' =

upqrnukrjqipijnijk TaaaaaT ...... ...' =

9.5. Intrinsic Symmetry of Physical Properties

A k-rank tensor has up to 3n different components. However, the number of independent

components is much smaller in most cases, either due to intrinsic symmetries of the physical

property described (this section) or due to the crystal symmetry (section 9.6).

9.5.1 Symmetry by Definition

Some properties are defined such that the corresponding tensors exhibit an inner symmetry.

Examples:

Strain tensor ijε : 2nd-rank, symmetric second rank tensor, 6 independent components

Stress tensor ijσ : 2nd-rank, symmetric second rank tensor, 6 independent components

Elastic modulus ijkls : 4th rank tensor with jilkijlkjiklijkl ssss === , 36 independent components.

9.5.2 Equilibrium Properties and Thermodynamic Arguments


For tensors describing equilibrium properties, thermodynamic relations significantly reduce

the number of independent components.

Example:

We consider elastic, electric and magnetic work plus heat exchange in a crystal (stress, strain,

el. field, el. polarization, magn. field, magn. polarization, temperature, entropy):

dTTSdH

HSdE

ESdσ

σSdS

dTTJdH

HJdE

EJdσ

σJdJ

dTTPdH

HPdE

EPdσ

σPdP

dTTε

dHHε

dEEε

dσσε

dε

kk

kk

klkl

ik

k

ik

k

ikl

kl

ii

ik

k

ik

k

ikl

kl

ii

ijk

k

ijk

k

ijkl

kl

ijij

∂∂

+

∂∂

+

∂∂

+

∂∂

=

∂∂

+

∂∂

+

∂∂

+

∂∂

=

∂∂

+

∂∂

+

∂∂

+

∂∂

=

∂∂

+

∂∂

+

∂∂

+

∂∂

=

According to the 1st and 2nd law of thermodynamics we obtain (reversible process):

TdSdJHdPEdεσdqdwdU llkkijij +++=+=

We change the set of independent variables by introducing a Gibb’s free energy:

TSJHPEεσUdG llkkijij −−−−=

yielding

SdTdHJdEPdσεdG llkkijij ++−−=

From comparison with the total differential of

dTTGdH

HGdE

EGdσ

σGdG l

lk

kij

ij

∂∂

+

∂∂

+

∂∂

+

∂∂

=

we obtain

STGJ

HGP

EGε

σG

kk

kk

ijij

−=

∂∂

−=

∂∂

−=

∂∂

−=

∂∂ ; ; ;

From the commutability of the second derivatives (Schwartz theorem) it follows for the

dielectric susceptibility jkχ :


jkk

j

jkkjj

kkj χ

EP

EEG

EEG

EPχ =

∂∂

=

∂∂∂

−=

∂∂∂

−=

∂∂

=22

Therefore, the dielectric susceptibility tensor is symmetric. Similar arguments hold for the

magnetic susceptibility )/( kllk HJψ ∂∂= and the elastic modulus )/( klijijkl σεs ∂∂= .

Moreover it follows that the tensors describing direct and reciprocal effects are identical. A an

example we consider the piezoelectrical (stress -> el. polarization) and the reverse

piezoelectrical effect (el. field -> strain):

kijk

ij

ijkkijij

k dEε

σEG

EσG

σP

=

∂∂

=

∂∂∂

−=

∂∂∂

−=

∂∂ 22

Similar relations are found for the piezomagnetic effect )/()/( lijijllij HεσJq ∂∂=∂∂= , the

magneto-electrical polarization )/()/( lkkllk HPEJλ ∂∂=∂∂= , thermal dilatation and

piezocaloric-effect )/()/( ijijlk σSTεα ∂∂=∂∂= , pyroelectric and electrocaloric effect

)/()/( kkk ESTPp ∂∂=∂∂= and pyromagnetic and magneto-caloric effect

)/()/( lll HSTJm ∂∂=∂∂= . Thus the above set of equations can be simplified significantly:

TcddHmdEpdσdSdTmdHdEdσqdJdTpdHdEdσddPdTdHqdEddσsdε

klkkijij

lklmklkijlijl

kklkkklijkijk

ijllijkkijklijklij

ln+++=+++=+++=+++=

αψλλχ

α

9.5.3. Transport Properties and Onsager’s Principle

In irreversible thermodynamics, transport processes are described by sets of corresponding

thermodynamic forces iX and fluxes ij , chosen such that ii Xjσ =& corresponds to the rate

of entropy production. For the corresponding linear system of transport equations

jiji XLj = ,

Onsager’s reciprocity relation states that


jiij LL = .

Example: Electric and heat transport

• Electric transport: force k

kq

xTX

∂Φ∂

=1 (el. potential); flux: electrical flux density k

qj

• Linear transport equations: k

ikq

iq

xTLj

∂Φ∂

=1 (classical definition:

kiki

q

xj

∂Φ∂

−= σ )

è Onsager relation: kiik LL = or kiik σσ = (conductivity tensor is symmetric)

• Heat transport: force k

kQ

xTX

∂∂

=)/1( ; flux: heat flux density k

Qj

• Definition of Peltier and Seebeck effect:

kq

ikk

ikiQ jπ

dxdTkj +−= with

lklk

q

dxdσj Φ

=

kikkiki

q

dxdσEσj Φ

+= β withl

klkβ

dxdTβE −=

with dxdTT

dxdT

dTTd

dxTd 2)/1()/1( −== it follows:

lklik

kiki

Q

dxd

TTσπ

dxTdTkj Φ

+−−=1)/1()( 2

kik

lkliki

q

dxd

TTσ

dxTdTβσj Φ

+−−=1)/1()( 2

è Onsager relation: klkl πTβ = (relation between Seebeck and Peltier effect)

9.6 Neumann’s Principle: Crystal Symmetry and Tensor Symmetry

CT GG ⊇ with ( ⊇ : subgroup or equal)

TG : symmetry group of tensor T

CG : symmetry group of crystal C


The group of symmetry elements of a physical property includes all symmetry elements of the

crystal, i.e.

• Polar tensor: upqrnukrjqipnijk TaaaaT ...... ...=

• Axial tensor: upqrnukrjqipijnijk TaaaaaT ...... ...=

Must be fulfilled for all transformation matrices ija corresponding to symmetry operations of

the point group of the crystal (crystal class) (remark: in general, it is not necessary to test all

symmetry operations, but only the set of generating operations of the point group).

Neumann’s principle may reduce the number of independent components of a tensor or may

require that some tensor elements must vanish.

Example 1:

We consider the el. susceptibility tensor ijχ defined via jiji EχP = (polar symmetrical tensor

of rank 2). We consider a crystal which belongs to crystal class 4 (crystallographic) or C4

(Schoenfliess).

The generating element is C4.

−=

100001010

4Ca .

We obtain:


−−

−−=

−−−

−=

−

−=

331323

131112

231222

331323

231222

131112

332313

232212

131211

332313

232212

131211

100001010

100001010

100001010

χχχχχχχχχ

χχχχχχχχχ

χχχχχχχχχ

χχχχχχχχχ

Thus:

(a) 2211 χχ =

(b) 0122112 =⇒−= χχχ

(c) 02313232313 ==⇒−== χχχχχ

Therefore, the electric polarizability tensor of a crystal belonging to class 4 has only two

independent components:

=

33

11

11

332313

232212

131211

000000

χχ

χ

χχχχχχχχχ

Example 1:

We consider the process of second harmonic / sum frequency generation, which is described

by the second order hyperpolarizability ijkχ , a polar tensor of rank 3

( ...++= kjijkjiji EEχEχP ; for a fields ωE , 'ωE oscillating with frequency ω , 'ω , ijkχ

generates frequency components with frequency 2ω , 2 'ω , 'ωω + , 'ωω − ):


We consider a crystal class which contains the inversion i described by the transformation

matrix

−−

−=

100010001

ia or

≠=−

=−=jiji

a ijiji

;0;1

δ

Neumann’s principle requires that

lmn

lmn

lmnlmn

knjmil

lmnkni

jmi

ili

ijk aaa

χχ

χδδδ

χχ

−=−−−=

−−−=

=

∑)1)(1)(1(

))()((

Therefore, 0=ijkχ , i.e. second harmonic / sum frequency generation is forbidden in crystals /

media with inversion symmetry. Note: In media with inversion symmetry, all processes

described by polar tensors of odd order and axial tensors of even order are forbidden!

9.8 Contracted Matrix Notation

In literature, a special matrix notation is often used to simplify the representation of higher

rank tensors.

Example:

Hook’s law: klijklij σsε =

The elastic modulus is a 4th rank tensor with a maximum of 81 components, but symmetry

with respect to suffixes i,j and k,l reduces the number of free components to 36.

We contract the index pairs i,j to a single index m and k,l to n according to the following rule:

6)2,1( ;5)1,3( ;4)3,2( ;3)3,3( ;2)2,2( ;1)1,1( →→→→→→

Using the definition

==

=6,5,4 ;23,2,1 ;

mεmε

εij

ijm


6,5,4,3,2,1 ; == nijn σσ

=====

=

=

6,5,4, ;46,5,4;3,2,1 ;26,5,4;3,2,1 ;2

3,2,1, ;

nmsnmsnms

nms

s

ij

ij

ij

ij

mn

Hook’s law is expressed as:

=

6

5

4

3

2

1

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

6

5

4

3

2

1

σσσσσσ

εεεεεε

cccccccccccccccccccccccccccccccccccc

or mnmn s σε =

Remarks:

(1) The coefficients in the definition of the contracted notations are necessary to take into

account the reduced number of terms in the summations.

(2) This is only a notation trick! The transformation properties remain unchanged.

9.9 Value of a Physical Property in a Given Direction

The value T of a physical property described by a 2nd rank tensor via jiji qTp = in the

direction of qr is defined as qpT /||= , where ||p is the component of pr parallel to qr .

Example:


Electrical conductivity σ , electrical field: nEE rr= with nr : unit vector in field direction.

EEE

EE

EEj

EEjnjj ij

iji

i σ==⋅=⋅=r

rrr||

With Ejn /||=rσ it follows:

jiijn nnσσ =r

This equation can be used to derive tensor components from a physical measurement or vice

versa.

9.10 Geometrical Representation: The Representation Quadratic

For the important group of symmetric 2nd-rank tensors (as an example we again consider the

conductivity jiji Ej σ= ), there is a simple geometrical way of representation, the so called

representation quadratic defined as:

1=jiij xxσ or 1222 2112133132232333

2222

2111 =+++++ xxxxxxxxx σσσσσσ

This is a second degree surface, in most cases it corresponds to an ellipsoid (tensor ellipsoid):

It can be shown that upon basis transformation the representation quadratic behaves like a

symmetrical 2nd-rank tensor. Thus the transformation properties of the tensor can be derived

from a (graphical) inspection of the transformation properties of the representation quadratic.


We can choose a basis transformation to a coordinate system, in which ijσ is diagonal:

=

3

2

1

'000'000'

'σ

σσ

σ

The directions of this special set of basis vectors are referred to as the principle axes of the

tensor. In the new basis, the tensor ellipsoid points along the coordinate axes:

the representation quadratic takes a simple form

1'''''' 2333

2222

2111 =++ xxx σσσ

and physical equations involving the tensor become particularly simple:

111 ''' Ej σ= , 222 ''' Ej σ= , 333 ''' Ej σ= .

The representation quadratic has two important geometrical properties (example: el.

conductivity jiji Ej σ= ):

(a) The radius r in a given direction nr is related to the physical property nrσ in this

direction via 2/1 rn =rσ . The resulting component of jr

parallel to nEE rr= is

Ej nrσ=|| .

(Proof: From 1=jiij xxσ with ii rnx = , we obtain 12 =jiij nnr σ . With the result from

section 9.9 ( jiijn nnσσ =r ), we obtain 2/1 rn =rσ .)


(b) The direction of jr

is along the normal of the representation quadratic at the endpoint

of the radius (without proof, see textbooks).

Example 2: Optical properties of crystals

Optical properties of an isotropic medium:

• EDrr

εε0= with εε0 dielectrical permittivity or

EDrr

=ηη0 with ηη0 dielectrical impermeability

• Maxwell equations ( 1=µ ):

velocity of electromagnetic wave ηε

ccv ==

refractive index: η

ε1

===vcn

Optical properties of an anisotropic medium:

• jiji ED εε0= with ijεε 0 dielectrical permittivity tensor or

ijij ED =ηη0 with ijηη0 dielectrical impermeability tensor (both 2nd-rank symmetric)

• Maxwell equations ( 1=µ ): (for proof see textbooks, e.g. Nye) In general, two plane

polarized waves with different velocity may be propagated along one direction

(double refraction).


Graphical representation: We consider the representation quadratic of the relative dielectric

impermeability tensor ijη , the so called indicatrix (note: (a) principal axes are chosen; (b) in

are called the principal refractive indices, but the refractive index is not a tensor!):

1233

222

211 =++ xxx ηηη or 12

3

23

22

22

21

21 =++

nx

nx

nx

The indicatrix has the following important property (for lengthy proofs see textbooks):

We consider wave propagation along 0P. The central section through the indicatrix,

perpendicular to the propagation direction is an ellipse. The axes of this ellipse represent the

two polarisation of Dr

and the semi-axes 0A and 0B are identical to the refractive indices

An and An for the two waves.

From these properties of the indicatrix and Neumann’s principle we can immediately classify

all crystal classes with respect to their optical properties:

(a) Optical anaxial crystals: cubic (classes 23 , 3m , 432 , 324 , mm3 ):

• indicatrix is a sphere (several Cn / Sn axes with n>2)


⇒ no double refraction in any direction

(b) Optical uniaxial crystals: teragonal ( 4 , 4v

, m4 , 422 , mm4 , m24 , mmm4 ), trigonal

( 3 , 3 , 32 , m3 , m3 ), hexagonal ( 6 , 6 , m6 , 622 , mm6 , 26m , mmm6 )

• indicatrix is an ellipsoid of revolution along principal symmetry axes (one Cn / Sn axes

with n>2)

⇒ no double refraction along principal symmetry axis (one optical axis)

(a) Optical biaxial crystals: triclinic (1 , 1 ), monoclinic ( 2 , m , m2 ), orthorhombic ( 222 ,

2mm , mmm ):

• indicatrix is a triaxial ellipsoid (no Cn / Sn axes with n>2)


⇒ no double refraction along two axes (optical axes)

9.11 Curie’s Principle

Often, crystal properties are considered under some external influence (electrical field, strain,

etc.). Here, Curie’s principle states:

ECC GGG ∩=~ with ( ∩ greatest common subgroup)

CG ~ : symmetry group of crystal C under external influence of E

CG : symmetry group of crystal C

EG : symmetry group of external influence E

Example: electro-optical and photoelastic effects

We consider the change of the relative dielectric impermeability tensor under the influence of

an electric or stress and expand it in term of a power series:

...0 ++++= klijkllkijklkijkijij EEkEr σπηη

• :ijkr linear electro-optical tensor → linear electro-optical effect (Pockels effect)

• :ijklk quadratic electro-optical tensor → quadratic electro-optical effect (Kerr effect)

• :ijklπ piezoptical tensor → photoelastic effect

As an example, we consider the Pockels effect in ADP (ammonium-dihydrogen-phosphate).

The crystal class of ADP is m24 , belonging to the tetragonal system. The crystal is optical


uniaxial. For wave propagation along the principal symmetry axis, no double diffraction

occurs.

We apply an electric field along the principal symmetry axis. The group of the electric field

vector contains an ∞–fold rotation axis and ∞ mirror planes containing the axis (group )m∞ .

As a set of common symmetry elements two mirror planes and a C2 axis survive. We identify

the common subgroup 2mm , which belongs to the orthorhombic crystal system, i.e. an optical

biaxial group.

→ The optical uniaxial ADP crystal becomes biaxial if an electric field along z

(principal symmetry axis) is applied and double refraction occurs along z.

Electro-optical and photoelastic effects are usually employed in optical elements which

change the direction of polarization or modulate the intensity.

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