1. Symmetry and the physical properties of crystals
Transformation of axes
Vectors and tensors
Neumann’s, Voigt’s and Curie’s principles
Symmetry constraints
Examples

2. Transformation of axes
transformation matrix α
Einstein
summation convention

3. Vectors and tensors
anisotrophy – general attribute of crystals – dependence on direction
properties of materials: link between
an independent variable – “stimulus” (vplyvy)
and
a dependent variable – “response” (odozvy)
the variables and the proportionality factors
that relate them can be tensors of various rank

4. Vectors and tensors temperature T – scalar – zero-rank tensor
intensity of electric field E, current density i
– vectors – tensors of rank one
pyroelectric coefficient pi
P – polarization vector
p – vector – tensor of rank one
electrical conductivity s
can be described
by one ”number”?
in general form
s = sij – 9 = 32 components (6 independent) – second-rank tensor

5. Transformation of tensors
axes
vectors
For transformation of a vector the same matrix
has to be used as for the axes
for a second-rank tensor
in the frame of the new axes x’
therefore

6. Transformation of tensors
in extended form
for third-rank tensor

7. Stress tensor – tenzor napätia
Force per unit area [Pa]
mechanical equilibrium
– normal components
Stress tensor is
symmetric
– shear components

8. according to definition
Strain tensor – tenzor malých deformácií l0 xi
l0 + Δl
extension per unit length
Strain tensor is
symmetric
according to definition γ x1 x2

9. Voigt notation introduced for symmetric second-rank tensors
– simplification
for stress
indices
11 → 1
22 → 2
33 → 3
23 → 4
13 → 5
12 → 6
for strain

10. Higher-rank tensors
In general, the relationship between a quantity represented by
an m-rank tensor and a quantity represented by an n-rank tensor
involves a tensor of rank m+n with 3m+n components.
Piezoelectricity
Stress applied to certain crystals may develop an electric moment
whose magnitude is proportional to the applied stress
d is the piezoelectric modulus – third-rank tensor
σ is the stress tensor

11. Hook’s law general form S – compliance tensor – tenzor poddajnosti
C – stiffness tensor – tenzor tuhosti
S and C –
4th-rank tensors
written in Voigt notation

12. Effect of symmetry pyroelectricity electrical conductivity
thermal conductivity
permeability
permittivity
piezoelectricity
elasticity
material tensors are reduced due to the crystal symmetry

13. Effect of symmetry Important questions should be answered
before starting the experiment
How many values of some physical parameter should be measured?
Which crystal directions give the simplest measurements?
Which directions should give duplicate results?
the knowledge of symmetry (point group) and
Neumann’s and Voigt’s principles
can help to answer these questions

14. Neumann’s principle
where K denotes the symmetry group of the crystal, GT is the
symmetry group of the tensor representing the physical property.
The symmetry elements of any physical property of a crystal must
include all the symmetry elements of the point group of the crystal,
The point group K of the crystal must be a subgroup
of the symmetry group GT of tensor of the physical property.
The magnitude of a particular physical property measured along a specific
direction is unchanged when the material is rotated, reflected or inverted
into a new orientation corresponding to one of the symmetry elements
of its point group.

15. Voigt’s principle
According to Neumann's principle the tensor representing any
physical property should be invariant with regard to every symmetry
operation of the given crystal class.
The condition of invariance reduces the number of the independent
tensor components, since it signifies relationships
between the tensor components.

16. Application of Neumann’s principle
property described by a second-rank tensor
monoclinic crystal with point symmetry group
symmetry elements
corresponding matrices

17. Application of Neumann’s principle
final form

18. Application of Neumann’s principle
property described by a second-rank tensor
tetragonal crystal with point symmetry group x1 x2 x3

19. Application of Neumann’s principle
final form
if s is symmetrical

20. Important remarks
Tensors have the simplest form if the coordinate axes are chosen
in the directions of the crystal axes with highest symmetry
All second-rank symmetrical tensors can be diagonalized
by an appropriate choice of coordinate axes
Property T in a given direction n
direction cosines of n
for symmetrical tensors

21. Curie’s principle
When certain causes produce certain effects, the symmetry elements
of the causes must be found in their effects.
The symmetry of the crystal subject to an external influence preserves
only the symmetry elements that are common to both the crystal
and the perturbing influence.
P is the symmetry group of the perturbed crystal, K and G are the symmetry
groups of the crystal and of the external influence, respectively.

22. Curie’s principle symmetry of symmetry of crystal
symmetry of influence
symmetry of
disturbed crystal
symmetry of crystal
uniaxial
compression σ
electric field
E(0, 0, E)
mmm
mm2

23. Examples Pyroelectricity
the effect cannot exist in a crystal possessing a centre of symmetry
if the crystal possess a unique direction, the polarization vector P lies
always along this direction
can be present only in crystals with symmetry
described by one of ten polar point groups
1, 2, 3, 4, 6, m, mm2, 3m, 4mm and 6mm
Tourmaline, trigonal crystal, symmetry group 3m
Temperature change 1° C produces a polarization 4 x 10-6 C/m2 K
that is equivalent to the intensity 740 V/cm

24. Examples Piezoelectricity of quartz – point group 32 x3 x2 x1
has no effect
is always zero

25. Examples stiffness tensor for hexagonal crystals
stiffness tensor for cubic crystals axes parallel to