Chapter 8 – Symmetry in Crystal Physics – p. 1 -
9. Symmetry in Crystal Physics
9.1. Description Physical Properties of Crystals by Tensors
r v r v Isotropic material: P = χE (el. polarization, el. susceptibility, el. field); P || E physical properties, which describe the relation between vectors are generally described by scalars, implying that both vectors are parallel. r Anisotropic materials: P;E are not necessarily parallel. Suitable description?
Example:
Electric susceptibility χ
r v r v Pv = χvv Ev and Ph = χhv Ev
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:
Chapter 8 – Symmetry in Crystal Physics – p. 2 -
æ χ11 χ12 χ13 ö P1 = χ11E1 + χ12 E2 + χ13 E3 r ç ÷ v P = ç χ21 χ22 χ23 ÷E or P2 = χ21E1 + χ22 E2 + χ23 E3 ç ÷ è χ31 χ32 χ33 ø P3 = χ31E1 + χ32 E2 + χ33 E3
Abbreviated version:
3 Pi = å χijE j ; i = 1,2,3 j=1
Einstein’s Notation:
Pi = χijE j ; i, j = 1,2,3
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:
Bijk...n = aijk...npqr...u Apqr...u ; i, j,k,...,n, p,q,r,...,u = 1,2,3 with
· A: f-rank tensor
· B: g-rank tensor
Chapter 8 – Symmetry in Crystal Physics – p. 3 -
· 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 σ = dF / dA (stress, area of body element, force
acting on area), i.e. second rank tensor σ ij :
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 σ11 ,σ22 ,σ33 and 6 sheer components
σ12 ,σ13 ,σ23 ,σ 21,σ31,σ32 . 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): σ12 = σ 21,σ 23 = σ32 ,σ31 = σ13 , the stress tensor is a symmetric second rank tensor:
σij = σ ji .
(Latter relationship still holds in the case of inhomogeneous stress or body forces, see textbooks).
Chapter 8 – Symmetry in Crystal Physics – p. 4 -
The strain tensor:
Strain in 1 dimension:
dx Strain: ε = dl
In a body, along one direction, of the material in all three directions might change. Therefore:
¶xi Extension: eij = ¶li
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 εij (symmetrical part of the extension):
æ 1 1 ö ç e11 (e12 + e21 ) (e13 + e31 )÷ æ ε11 ε12 ε13 ö ç 2 2 ÷ ç ÷ ç 1 1 ÷ çε12 ε22 ε23 ÷ = (e12 + e21 ) e22 (e23 + e32 ) ç ÷ ç 2 2 ÷ èε13 ε23 ε33 ø ç 1 1 ÷ ç (e13 + e31 ) (e23 + e32 ) e33 ÷ è 2 2 ø
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:
εij = sijklσ kl
th The elastic modulus sijkl is a 4 rank tensor, in principle containing 81 components.
Chapter 8 – Symmetry in Crystal Physics – p. 5 -
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 eˆi to a new coordinate
system eˆi ', described by a transformation matrix a ji . Both sets of basis vectors are chosen
-1 orthonormal, yielding a ji = a ij .
Chapter 8 – Symmetry in Crystal Physics – p. 6 -
Normally, a vector pi in the old coordinate system would be described in the new coordinate system as:
p'i = aij p j (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:
r r r e1 e2 e3 r r r c = a ´ b = a1 a2 a3
b1 b2 b3
r (the vector product describes a vector of length c = absin Ðab , is perpendicular to a ar and b , r and which forms a right handed system ar,b,cr ). We consider a basis transformation, which changes the coordinate system from right-handed to left handed or vice versa, such as the inversion i:
Chapter 8 – Symmetry in Crystal Physics – p. 7 -
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 aij of the transformation matrix
· aij = 1: Transformation leaves hand of the axes unchanged (rotation)
· aij = -1: Transformation changes hand of the axes (inversion, reflection)
Transformation of a axial vector:
p'i = aij aij p j (note the summation over j according to the Einstein notation).
Physical examples of axial vector (1st rank axial tensor) are the angular momentum r r r ( L = mr ´ vr ) or the magnetic flux density ( F = Qvr ´ B ).
In general a tensor a connecting two properties A and B via Bijk...n = aijk...npqr...u Apqr...u 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:
T '= T (0-rank tensor)
A (polar) vector pi is described in the new coordinate system as:
T ' = a T st p'i = aij p j (or i ij j 1 -rank tensor)
For a (polar) 2nd-rank tensor
pk = Tkl ql
we obtain with p'i = aik pk and q'l = alj q j or ql = a jl q' j :
p'i = aikTkla jl q' j = T 'ij q' j or
Chapter 8 – Symmetry in Crystal Physics – p. 8 -
nd T 'ij = aika jlTkl (2 -rank tensor)
General transformation laws for tensors upon basis transformation:
Tensor Rank Polar Tensor Axial Tensor
0 T 'ij = Tkl
1 T ' = a T i ij j T 'i = aij aijTj
2 T ' = a a T ij ik jl kl T 'ij = aij aika jlTkl
3 T ' = a a a T ijk il jm kn lmn T 'ijk = aij aila jmaknTlmn n T ' = a a a ...a T ijk...n ip jq kr nu pqr...u T 'ijk...n = aij aipa jqakr ...anuTpqr...u
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:
nd Strain tensor εij : 2 -rank, symmetric second rank tensor, 6 independent components
nd Stress tensor σ ij : 2 -rank, symmetric second rank tensor, 6 independent components
th Elastic modulus sijkl : 4 rank tensor with sijkl = s jikl = sijlk = s jilk , 36 independent components.
9.5.2 Equilibrium Properties and Thermodynamic Arguments
Chapter 8 – Symmetry in Crystal Physics – p. 9 -
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):
æ ¶ε ö æ ¶ε ö æ ¶ε ö æ ¶ε ö dε = ç ij ÷dσ + ç ij ÷dE + ç ij ÷dH + ç ij ÷dT ij ç ÷ kl ç ÷ k ç ÷ k ç ÷ è ¶σ kl ø è ¶Ek ø è ¶H k ø è ¶T ø
æ ¶Pi ö æ ¶Pi ö æ ¶Pi ö æ ¶Pi ö dPi = ç ÷dσ kl + ç ÷dEk + ç ÷dH k + ç ÷dT ç ¶σ ÷ ç ¶E ÷ ç ¶H ÷ è ¶T ø è kl ø è k ø è k ø æ ¶J ö æ ¶J ö æ ¶J ö æ ¶J ö dJ = ç i ÷dσ + ç i ÷dE + ç i ÷dH + i dT i ç ÷ kl ç ÷ k ç ÷ k ç ÷ è ¶σ kl ø è ¶Ek ø è ¶H k ø è ¶T ø æ ¶S ö æ ¶S ö æ ¶S ö æ ¶S ö dS = ç ÷dσ + ç ÷dE + ç ÷dH + dT ç ÷ kl ç ÷ k ç ÷ k ç ÷ è ¶σ kl ø è ¶Ek ø è ¶H k ø è ¶T ø
According to the 1st and 2nd law of thermodynamics we obtain (reversible process):
dU = dw + dq = σijdεij + Ek dPk + H l dJl +TdS
We change the set of independent variables by introducing a Gibb’s free energy:
dG = U - σ ijεij - Ek Pk - H l J l -TS yielding
dG = -εijdσij - Pk dEk + J l dHl + SdT
From comparison with the total differential of
æ ¶G ö æ ¶G ö æ ¶G ö æ ¶G ö dG = ç ÷dσ + ç ÷dE + ç ÷dH + ç ÷dT ç ¶σ ÷ ij ç ¶E ÷ k ç ¶H ÷ l ¶T è ij ø è k ø è l ø è ø we obtain
æ ¶G ö æ ¶G ö æ ¶G ö æ ¶G ö ç ÷ = -ε ; ç ÷ = -P ; ç ÷ = -J ; ç ÷ = -S ç ¶σ ÷ ij ç ¶E ÷ k ç ¶H ÷ k ¶T è ij ø è k ø è k ø è ø
From the commutability of the second derivatives (Schwartz theorem) it follows for the
dielectric susceptibility χ jk :
Chapter 8 – Symmetry in Crystal Physics – p. 10 -
æ ¶P ö æ ¶ 2G ö æ ¶2G ö æ ¶P ö χ = ç k ÷ = -ç ÷ = -ç ÷ = ç j ÷ = χ kj ç ¶E ÷ ç ¶E ¶E ÷ ç ¶E ¶E ÷ ç ¶E ÷ jk è j ø è j k ø è k j ø è k ø
Therefore, the dielectric susceptibility tensor is symmetric. Similar arguments hold for the
magnetic susceptibility ψlk = (¶J l / ¶H k ) and the elastic modulus sijkl = (¶εij / ¶σ kl ).
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):
æ ¶P ö æ ¶2G ö æ ¶2G ö æ ¶ε ö ç k ÷ = -ç ÷ = -ç ÷ = ç ij ÷ = d ç ¶σ ÷ ç ¶σ ¶E ÷ ç ¶E ¶σ ÷ ç ¶E ÷ kij è ij ø è ij k ø è k ij ø è k ø
Similar relations are found for the piezomagnetic effect qlij = (¶J l / ¶σ ij ) = (¶εij / ¶H l ) , the
magneto-electrical polarization λlk = (¶J l / ¶Ek ) = (¶Pk / ¶Hl ), thermal dilatation and
piezocaloric-effect αlk = (¶εij / ¶T) = (¶S / ¶σ ij ) , pyroelectric and electrocaloric effect
pk = (¶Pk / ¶T) = (¶S / ¶Ek ) and pyromagnetic and magneto-caloric effect
ml = (¶J l / ¶T ) = (¶S / ¶H l ) . Thus the above set of equations can be simplified significantly:
dεij = sijkldσ kl + dkijdEk + qlij dHl + aijdT dP = d dσ + c dE + l dH + p dT k kij ij kl k lk k k dJl = qlijdσij + llk dEk + y lm dH k + ml dT
dS = aijdσij + pk dEk + ml dH k + cd ln T
9.5.3. Transport Properties and Onsager’s Principle
In irreversible thermodynamics, transport processes are described by sets of corresponding
thermodynamic forces X i and fluxes ji , chosen such that σ& = ji X i corresponds to the rate of entropy production. For the corresponding linear system of transport equations
ji = Lij X j ,
Onsager’s reciprocity relation states that
Chapter 8 – Symmetry in Crystal Physics – p. 11 -
Lij = L ji .
Example: Electric and heat transport
q 1 ¶F q · Electric transport: force X k = (el. potential); flux: electrical flux density j k T ¶xk
q q 1 ¶F q ¶F · Linear transport equations: j i = L ik (classical definition: j i = -s ik ) T ¶xk ¶xk
è Onsager relation: Lik = Lki or σik = σki (conductivity tensor is symmetric)
Q ¶(1/T) Q · Heat transport: force X k = ; flux: heat flux density j k ¶xk
· Definition of Peltier and Seebeck effect:
Q dT q q dF j i = -kik + πik j k with j k = σ kl dxk dxl
q b dF β dT j i = σik E k + σik with E k = -βkl dxk dxl
d(1/T) d(1/T) dT dT with = = T -2 it follows: dx dT dx dx
Q 2 d(1/T) 1 dF j i = -kik (-T ) + πikσklT dxk T dxl
q 2 d(1/T) 1 dF j i = -σik βkl (-T ) + σikT dxl T dxk
è Onsager relation: βklT = πkl (relation between Seebeck and Peltier effect)
9.6 Neumann’s Principle: Crystal Symmetry and Tensor Symmetry
GT Ê GC with ( Ê: subgroup or equal)
GT : symmetry group of tensor T
GC : symmetry group of crystal C
Chapter 8 – Symmetry in Crystal Physics – p. 12 -
The group of symmetry elements of a physical property includes all symmetry elements of the crystal, i.e.
· Polar tensor: Tijk ...n = aipa jq akr ...anuTpqr...u
· Axial tensor: Tijk ...n = aij aipa jq akr ...anuTpqr...u
Must be fulfilled for all transformation matrices aij 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 χij defined via Pi = χij E j (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.
æ0 -1 0ö ç ÷ aC 4 = ç1 0 0÷ . ç ÷ è0 0 1ø
We obtain:
Chapter 8 – Symmetry in Crystal Physics – p. 13 -
æ χ χ χ ö æ0 -1 0öæ χ χ χ öæ 0 1 0ö ç 11 12 13 ÷ ç ÷ç 11 12 13 ÷ç ÷ ç χ12 χ22 χ23 ÷ = ç1 0 0÷ç χ12 χ22 χ23 ÷ç -1 0 0÷ ç ÷ ç ÷ç ÷ç ÷ è χ13 χ23 χ33 ø è0 0 1øè χ13 χ23 χ33 øè 0 0 1ø æ0 -1 0öæ - χ χ χ ö ç ÷ç 12 11 13 ÷ = ç1 0 0÷ç- χ22 χ12 χ23 ÷ ç ÷ç ÷ è0 0 1øè - χ23 χ13 χ33 ø æ χ - χ - χ ö ç 22 12 23 ÷ = ç - χ12 χ11 χ13 ÷ ç ÷ è- χ23 χ13 χ33 ø
Thus:
(a) χ11 = χ22
(b) χ12 = - χ21 Þ χ12 = 0
(c) χ13 = χ23 = - χ23 Þ χ13 = χ23 = 0
Therefore, the electric polarizability tensor of a crystal belonging to class 4 has only two independent components:
æ χ χ χ ö æ χ 0 0 ö ç 11 12 13 ÷ ç 11 ÷ ç χ12 χ22 χ23 ÷ = ç 0 χ11 0 ÷ ç ÷ ç ÷ è χ13 χ23 χ33 ø è 0 0 χ33 ø
Example 1:
We consider the process of second harmonic / sum frequency generation, which is described
by the second order hyperpolarizability χijk , a polar tensor of rank 3
( Pi = χij E j + χijk E j Ek + ... ; for a fields Ew , Ew' oscillating with frequency w , w' , χijk generates frequency components with frequency 2w , 2w' , w + w' , w - w' ):
Chapter 8 – Symmetry in Crystal Physics – p. 14 -
We consider a crystal class which contains the inversion i described by the transformation matrix
æ-1 0 0 ö i ç ÷ i ì-1;i = j a = ç 0 -1 0 ÷ or a ij = -dij = í ç ÷ î 0;i ¹ j è 0 0 -1ø
Neumann’s principle requires that
i i i cijk = a il a jm a kn clmn
= å(-d il )(-d jm )(-d kn )clmn lmn
= (-1)(-1)(-1)clmn
= -clmn
Therefore, cijk = 0 , 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: εij = sijklσ kl
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:
(1,1) ® 1; (2,2) ® 2; (3,3) ® 3; (2,3) ® 4; (3,1) ® 5; (1,2) ® 6
Using the definition
ì εij ; m = 1,2,3 εm = í î2εij ; m = 4,5,6
Chapter 8 – Symmetry in Crystal Physics – p. 15 -
s n = s ij ; n = 1,2,3,4,5,6
ì sij ; m, n = 1,2,3 ï ï2sij ; m = 1,2,3;n = 4,5,6 smn = í 2s ; m = 1,2,3;n = 4,5,6 ï ij ï î4sij ; m,n = 4,5,6
Hook’s law is expressed as:
æe ö æ c c c c c c öæs ö ç 1 ÷ ç 11 12 13 14 15 16 ÷ç 1 ÷ çe 2 ÷ çc21 c22 c23 c24 c25 c26 ÷çs 2 ÷ ç ÷ ç ÷ç ÷ e 3 c31 c32 c33 c34 c35 c36 s 3 ç ÷ = ç ÷ç ÷ or e n = snms m çe 4 ÷ çc41 c42 c43 c44 c45 c46 ÷çs 4 ÷ çe ÷ çc c c c c c ÷çs ÷ ç 5 ÷ ç 51 52 53 54 55 56 ÷ç 5 ÷ ç ÷ ç ÷ç ÷ èe 6 ø èc61 c62 c63 c64 c65 c66 øès 6 ø
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
nd The value T of a physical property described by a 2 rank tensor via pi = Tij q j in the
r r r direction of q is defined as T = p|| / q , where p|| is the component of p parallel to q .
Example:
Chapter 8 – Symmetry in Crystal Physics – p. 16 -
r Electrical conductivity s , electrical field: E = Enr with nr : unit vector in field direction. r r r E E E j E j = j ×nr = j × = j i = s i E || E i E ij E E
With s nr = j|| / E it follows:
s nr = s ij ni n j
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 ji = s ij E j ), there is a simple geometrical way of representation, the so called representation quadratic defined as:
2 2 2 s ij xi x j = 1 or s 11 x1 + s 22 x2 + s 33 x3 + 2s 23 x2 x3 + 2s 31 x3 x1 + 2s 12 x1 x2 = 1
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.
Chapter 8 – Symmetry in Crystal Physics – p. 17 -
We can choose a basis transformation to a coordinate system, in which s ij is diagonal:
æs ' 0 0 ö ç 1 ÷ s '= ç 0 s '2 0 ÷ ç ÷ è 0 0 s '3 ø
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
2 2 2 s '11 x'1 +s '22 x'2 +s '33 x'3 = 1 and physical equations involving the tensor become particularly simple:
j'1 = s '1 E'1 , j'2 = s '2 E'2 , j'3 = s '3 E'3 .
The representation quadratic has two important geometrical properties (example: el.
conductivity ji = s ij E j ):
r (a) The radius r in a given direction n is related to the physical property s nr in this
2 r r r direction via s nr = 1/ r . The resulting component of j parallel to E = En is
j|| = s nr E .
2 (Proof: From s ij xi x j = 1 with xi = rni , we obtain r s ij ni n j = 1. With the result from
2 section 9.9 (s nr = s ij ni n j ), we obtain s nr = 1/ r .)
Chapter 8 – Symmetry in Crystal Physics – p. 18 -
r (b) The direction of j 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:
r r · D = e 0eE with e0e dielectrical permittivity or
r r h0hD = E with h0h dielectrical impermeability
· Maxwell equations ( m = 1):
c velocity of electromagnetic wave v = = c h e
c 1 refractive index: n = = e = v h
Optical properties of an anisotropic medium:
· Di = e 0e ij E j with e 0e ij dielectrical permittivity tensor or
nd h0hij D j = Ei with h0hij dielectrical impermeability tensor (both 2 -rank symmetric)
· Maxwell equations ( m = 1): (for proof see textbooks, e.g. Nye) In general, two plane
polarized waves with different velocity may be propagated along one direction
(double refraction).
Chapter 8 – Symmetry in Crystal Physics – p. 19 -
Graphical representation: We consider the representation quadratic of the relative dielectric
impermeability tensor hij , the so called indicatrix (note: (a) principal axes are chosen; (b) ni are called the principal refractive indices, but the refractive index is not a tensor!):
x 2 x 2 x 2 2 2 2 1 + 2 + 3 = 1 h1x1 +h2 x2 +h3 x3 = 1 or 2 2 2 n1 n2 n3
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 r two polarisation of D and the semi-axes 0A and 0B are identical to the refractive indices
n A and n A 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, m3, 432 , 432 , m3m ):
· indicatrix is a sphere (several Cn / Sn axes with n>2)
Chapter 8 – Symmetry in Crystal Physics – p. 20 -
Þ no double refraction in any direction
v (b) Optical uniaxial crystals: teragonal ( 4 , 4 , 4 m , 422 , 4mm , 42m, 4 mmm ), trigonal
( 3, 3 , 32 , 3m , 3m ), hexagonal ( 6, 6 , 6 m , 622 , 6mm , 6m2 , 6 mmm )
· indicatrix is an ellipsoid of revolution along principal symmetry axes (one Cn / Sn axes
with n>2)
Þ no double refraction along principal symmetry axis (one optical axis)
(a) Optical biaxial crystals: triclinic (1, 1 ), monoclinic ( 2 , m , 2 m ), orthorhombic ( 222 ,
mm2 , mmm ):
· indicatrix is a triaxial ellipsoid (no Cn / Sn axes with n>2)
Chapter 8 – Symmetry in Crystal Physics – p. 21 -
Þ 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:
~ GC = GC Ç GE with ( Ç greatest common subgroup)
~ GC : symmetry group of crystal C under external influence of E
GC : symmetry group of crystal C
GE : 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 hij =hij + rijk Ek + kijkl Ek El + p ijkls kl + ...
· rijk : linear electro-optical tensor ® linear electro-optical effect (Pockels effect)
· kijkl : quadratic electro-optical tensor ® quadratic electro-optical effect (Kerr effect)
· p ijkl : piezoptical tensor ® photoelastic effect
As an example, we consider the Pockels effect in ADP (ammonium-dihydrogen-phosphate).
The crystal class of ADP is 42m, belonging to the tetragonal system. The crystal is optical
Chapter 8 – Symmetry in Crystal Physics – p. 22 - 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 mm2, 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.