International Tables for Crystallography Volume D: Physical properties of crystals

500 Pages·2003·14.01 MB·English

by  A. Authier (auth.), A. Authier (eds.)

International Tables for Crystallography (2006). Vol. D, Chapter 1.1, pp. 3–33.
1.1. Introduction to the properties of tensors
By A. Authier
1.1.1. The matrix of physical properties not the case in an anisotropic medium: a sphere cut in an
anisotropic medium becomes an ellipsoid when the temperature
1.1.1.1. Notion of extensive and intensive quantities
is varied and thermal expansion can no longer be represented by
Physical laws express in general the response of a medium to a
a single number. It is actually represented by a tensor of rank 2.
certain influence. Most physical properties may therefore be
(ii) Dielectric constant. In an isotropic medium of a perfect
defined by a relation coupling two or more measurable quantities.
dielectric we can write, in SI units,
For instance, the specific heat characterizes the relation between
a variation of temperature and a variation of entropy at a given
P ¼ "0eE
temperature in a given medium, the dielectric susceptibility the
relation between electric field and electric polarization, the D ¼ "0E þ P ¼ "0ð1 þ eÞE ¼ "E;
elastic constants the relation between an applied stress and the
resulting strain etc. These relations are between quantities of
where P is the electric polarization (= dipole moment per unit
the same nature: thermal, electrical and mechanical, respectively.
volume), "0 the permittivity of vacuum, e the dielectric
But there are also cross effects, for instance:
susceptibility, D the electric displacement and " the dielectric
(a) thermal expansion and piezocalorific effect: mechanical
constant, also called dielectric permittivity. These expressions
reaction to a thermal impetus or the reverse;
indicate that the electric field, on the one hand, and polarization
(b) pyroelectricity and electrocalorific effect: electrical response
and displacement, on the other hand, are linearly related. In the
to a thermal impetus or the reverse;
general case of an anisotropic medium, this is no longer true and
(c) piezoelectricity and electrostriction: electric response to a
one must write expressions indicating that the components of the
mechanical impetus;
displacement are linearly related to the components of the field:
(d) piezomagnetism and magnetostriction: magnetic response
to a mechanical impetus; 8
1 1 1 2 2 3
(e) photoelasticity: birefringence produced by stress; < D ¼ "1E þ "1E þ "1E
2 2 1 2 2 3
(f) acousto-optic effect: birefringence produced by an acoustic : D
3 ¼ "31E1 þ "32E2 þ "32E ð1:1:1:1Þ
wave; D ¼ "1E þ "2E þ "3E:
(g) electro-optic effect: birefringence produced by an electric
field;
The dielectric constant is now characterized by a set of nine
(h) magneto-optic effect: appearance of a rotatory polarization j
components " i; they are the components of a tensor of rank 2. It
under the influence of a magnetic field.
will be seen in Section 1.1.4.5.2.1 that this tensor is symmetric
The physical quantities that are involved in these relations can j i
(" i ¼ "j) and that the number of independent components is
be divided into two categories:
equal to six.
(i) extensive quantities, which are proportional to the volume of
(iii) Stressed rod (Hooke’s law). If one pulls a rod of length ‘
matter or to the mass, that is to the number of molecules in the
and cross section A with a force F, its length is increased by a
medium, for instance entropy, energy, quantity of electricity etc.
quantity ‘ given by ‘=‘ ¼ ð1=EÞF=A; where E is Young’s
One uses frequently specific extensive parameters, which are
modulus, or elastic stiffness (see Section 1.3.3.1). But, at the same
given per unit mass or per unit volume, such as the specific mass,
time, the radius, r, decreases by r given by r=r ¼ð =EÞF=A,
the electric polarization (dipole moment per unit volume) etc.
where is Poisson’s ratio (Section 1.3.3.4.3). It can be seen that a
(ii) intensive parameters, quantities whose product with an
scalar is not sufficient to describe the elastic deformation of a
extensive quantity is homogeneous to an energy. For instance,
material, even if it is isotropic. The number of independent
volume is an extensive quantity; the energy stored by a gas
components depends on the symmetry of the medium and it will
undergoing a change of volume dV under pressure p is p dV.
be seen that they are the components of a tensor of rank 4. It was
Pressure is therefore the intensive parameter associated with
precisely to describe the properties of elasticity by a mathema-
volume. Table 1.1.1.1 gives examples of extensive quantities and
tical expression that the notion of a tensor was introduced in
of the related intensive parameters.
physics by W. Voigt in the 19th century (Voigt, 1910) and by L.
Brillouin in the first half of the 20th century (Brillouin, 1949).
1.1.1.2. Notion of tensor in physics
Each of the quantities mentioned in the preceding section is Table 1.1.1.1. Extensive quantities and associated intensive parameters
represented by a mathematical expression. Some are direction The last four lines of the table refer to properties that are time dependent.
independent and are represented by scalars: specific mass,
Extensive quantities Intensive parameters
specific heat, volume, pressure, entropy, temperature, quantity of
electricity, electric potential. Others are direction dependent and Volume Pressure
Strain Stress
are represented by vectors: force, electric field, electric displa-
Displacement Force
cement, the gradient of a scalar quantity. Still others cannot be Entropy Temperature
represented by scalars or vectors and are represented by more Quantity of electricity Electric potential
complicated mathematical expressions. Magnetic quantities are Electric polarization Electric field
Electric displacement Electric field
represented by axial vectors (or pseudovectors), which are a
Magnetization Magnetic field
particular kind of tensor (see Section 1.1.4.5.3). A few examples
Magnetic induction Magnetic field
will show the necessity of using tensors in physics and Section Reaction rate Chemical potential
1.1.3 will present elementary mathematical properties of tensors. Heat flow Temperature gradient
(i) Thermal expansion. In an isotropic medium, thermal Diffusion of matter Concentration gradient
Electric current Potential gradient
expansion is represented by a single number, a scalar, but this is
Copyright  2006 International Union of Crystallography 3
1. TENSORIAL ASPECTS OF PHYSICAL PROPERTIES
(iv) Expansion in Taylor series of a field of vectors. Let us temperature and the associated extensive parameters: S strain, P
consider a field of vectors uðrÞ where r is a position vector. The electric polarization, B magnetic induction, entropy, respec-
Taylor expansion of its components is given by tively. Matrix equation (1.1.1.4) may then be written:
0 1 0 10 1
i 2 i T E H
i i @u j 1 @ u j k S CS CS CS CS T
u ðr þ drÞ ¼ u ðrÞ þ dx þ dx dx þ . . .
@u j 2 @u
1.1. INTRODUCTION TO THE PROPERTIES OF TENSORS
0i 0
(i) The submatrices associated with the principal properties are x ¼ x ei: ð1:1:2:2Þ
symmetric with respect to interchange of the indices related to the
j i
causes and to the effects: these properties are represented by If Ai and Bj are the transformation matrices between the bases
0
symmetric tensors. For instance, the dielectric constant and the ei and ej, the following relations hold between the two bases:
)
elastic constants are represented by symmetric tensors of rank 2
j 0 0 i
e ¼ A e ; e ¼ B e
and 4, respectively (see Section 1.1.3.4). i i j j j i
ð1:1:2:3Þ
i i 0j 0j j i
(ii) The submatrices associated with terms that are symmetric x ¼ Bjx ; x ¼ Aix
with respect to the main diagonal of matrices (C) and (R) and that
represent cross effects are transpose to one another. For instance, (summations over j and i, respectively). The matrices Aj and Bi
i j
E
matrix (C
S ) representing the converse piezoelectric effect is the are inverse matrices:
T
transpose of matrix (C ) representing the piezoelectric effect. It
P j k k
A B ¼ ð1:1:2:4Þ
will be shown in Section 1.1.3.4 that they are the components of i j i
tensors of rank 3.
k
(Kronecker symbol: ¼ 0 if i ¼6 k;¼ 1 if i ¼ k).
i
1.1.1.5. Onsager relations
Important Remark. The behaviour of the basis vectors and of the
Let us now consider systems that are in steady state and not in components of the vectors in a transformation are different. The
j i
thermodynamic equilibrium. The intensive and extensive para- roles of the matrices Ai and Bj are opposite in each case. The
meters are time dependent and relation (1.1.1.3) can be written components are said to be contravariant. Everything that trans-
forms like a basis vector is covariant and is characterized by an
J ¼ L X ;
m mn n inferior index. Everything that transforms like a component is
contravariant and is characterized by a superior index. The
where the intensive parameters X are, for instance, a tempera-
n property describing the way a mathematical body transforms
ture gradient, a concentration gradient, a gradient of electric
under a change of basis is called variance.
potential. The corresponding extensive parameters J are the
m
heat flow, the diffusion of matter and the current density. The
1.1.2.2. Metric tensor
diagonal terms of matrix L correspond to thermal conductivity
mn
(Fourier’s law), diffusion coefficients (Fick’s law) and electric We shall limit ourselves to a Euclidean space for which we have
conductivity (Ohm’s law), respectively. Non-diagonal terms defined the scalar product. The analytical expression of the scalar
correspond to cross effects such as the thermoelectric effect, product of two vectors x ¼ xie and y ¼ y je is
i j
thermal diffusion etc. All the properties corresponding to these
i j
x y ¼ x e y e :
examples are represented by tensors of rank 2. The case of i j
second-rank axial tensors where the symmetrical part of the
Let us put
tensors changes sign on time reversal was discussed by Zheludev
(1986). e e ¼ g : ð1:1:2:5Þ
i j ij
The Onsager reciprocity relations (Onsager, 1931a,b)
The nine components g are called the components of the metric
L ¼ L ij
mn nm
tensor. Its tensor nature will be shown in Section 1.1.3.6.1. Owing
to the commutativity of the scalar product, we have
express the symmetry of matrix L . They are justified by
mn
considerations of statistical thermodynamics and are not as g ¼ e e ¼ e e ¼ g :
ij i j j i ji
q
obvious as those expressing the symmetry of matrix (C ). For
p
instance, the symmetry of the tensor of rank 2 representing The table of the components g is therefore symmetrical. One
ij
thermal conductivity is associated with the fact that a circulating of the definition properties of the scalar product is that if x y ¼ 0
flow is undetectable. for all x, then y ¼ 0. This is translated as
Transport properties are described in Chapter 1.8 of this
i j i j
x y g ¼ 0 8x ¼) y g ¼ 0:
ij ij
volume.
j
In order that only the trivial solution ðy ¼ 0Þ exists, it is
1.1.2. Basic properties of vector spaces necessary that the determinant constructed from the gij’s is
different from zero:
[The reader may also refer to Section 1.1.4 of Volume B of
International Tables for Crystallography (2000).] ðgijÞ ¼6 0:
1.1.2.1. Change of basis This important property will be used in Section 1.1.2.4.1.
Let us consider a vector space spanned by the set of n basis
1.1.2.3. Orthonormal frames of coordinates – rotation matrix
vectors e , e , e ; . . . ; e . The decomposition of a vector using this
1 2 3 n
basis is written An orthonormal coordinate frame is characterized by the fact
that
i
x ¼ x e ð1:1:2:1Þ
i
g ¼ ð¼ 0 if i ¼6 j and ¼ 1 if i ¼ jÞ: ð1:1:2:6Þ
ij ij
using the Einstein convention. The interpretation of the position
One deduces from this that the scalar product is written simply as
of the indices is given below. For the present, we shall use the
simple rules: i j i i
x y ¼ x y g ¼ x y :
ij
(i) the index is a subscript when attached to basis vectors;
(ii) the index is a superscript when attached to the components.
Let us consider a change of basis between two orthonormal
The components are numerical coordinates and are therefore
systems of coordinates:
dimensionless numbers.
0 j 0
Let us now consider a second basis, e
j. The vector x is inde- ei ¼ Aiej:
pendent of the choice of basis and it can be decomposed also in
0
the second basis: Multiplying the two sides of this relation by e , it follows that
j
5
1. TENSORIAL ASPECTS OF PHYSICAL PROPERTIES
0 j 0 0 j 0 j n
e e ¼ A e e ¼ A g ¼ A (written correctly), which span the space E ðj ¼ 1; . . . ; nÞ. This set of n vectors forms
i j i k j i kj i kj
a basis since (1.1.2.12) can be written with the aid of (1.1.2.13) as
which can also be written, if one notes that variance is not j
x ¼ x e : ð1:1:2:14Þ
j
apparent in an orthonormal frame of coordinates and that the
position of indices is therefore not important, as
j
The x ’s are the components of x in the basis e . This basis is
j
0 j
e e ¼ A (written incorrectly): called the dual basis. By using (1.1.2.11) and (1.1.2.13), one can
i j i
show in the same way that
j 0
The matrix coefficients, A
i, are the direction cosines of ej with j
e ¼ g e : ð1:1:2:15Þ
j ij
respect to the e basis vectors. Similarly, we have
i
i 0 j
B
j ¼ ei ej It can be shown that the basis vectors e transform in a change
j
of basis like the components x of the physical space. They are
so that therefore contravariant. In a similar way, the components xj of a
j
j i T vector x with respect to the basis e transform in a change of basis
A ¼ B or A ¼ B ;
i j
like the basis vectors in direct space, e ; they are therefore
j
covariant:
T
where indicates transpose. It follows that )
j j 0k 0k k j
A ¼ BT and A ¼ B 1 e ¼ Bke ; e ¼ Aj e ð1:1:2:16Þ
j 0 0 i
x ¼ A x ; x ¼ B x :
i i j j j i
so that

T 1 T
A ¼ A ) A A ¼ I
ð1:1:2:7Þ
T 1 T
B ¼ B ) B B ¼ I:
1.1.2.4.2. Reciprocal space
Let us take the scalar products of a covariant vector e and a
The matrices A and B are unitary matrices or matrices of rotation i
j
contravariant vector e :
and
2 2 e e j ¼ e g jke ¼ e e g jk ¼ g g jk ¼ j
ðAÞ ¼ ðBÞ ¼ 1 ) ðAÞ ¼ 1: ð1:1:2:8Þ i i k i k ik i
[using expressions (1.1.2.5), (1.1.2.11) and (1.1.2.13)].
If ðAÞ ¼ 1 the senses of the axes are not changed – proper
j j
The relation we obtain, e e ¼ , is identical to the relations
rotation. i i
defining the reciprocal lattice in crystallography; the reciprocal
If ðAÞ ¼ 1 the senses of the axes are changed – improper
i
basis then is identical to the dual basis e .
rotation. (The right hand is transformed into a left hand.)
j
One can write for the coefficients A
i
j k k j k k
A B ¼ ; A A ¼ ;
i j i i j i
1.1.2.4.3. Properties of the metric tensor
j
giving six relations between the nine coefficients A . There are
i In a change of basis, following (1.1.2.3) and (1.1.2.5), the g ’s
ij
thus three independent coefficients of the 3 3 matrix A.
transform according to

k m 0
g ¼ A A g
1.1.2.4. Covariant coordinates – dual or reciprocal space ij i j km
0 k m ð1:1:2:17Þ
g ¼ B B g :
ij i j km
1.1.2.4.1. Covariant coordinates
Using the developments (1.1.2.1) and (1.1.2.5), the scalar i j
Let us now consider the scalar products, e e , of two contra-
products of a vector x and of the basis vectors e can be written
i variant basis vectors. Using (1.1.2.11) and (1.1.2.13), it can be
j j
x ¼ x e ¼ x e e ¼ x g : ð1:1:2:9Þ shown that
i i j i ij
i j ij
e e ¼ g : ð1:1:2:18Þ
The n quantities x are called covariant components, and we shall
i
see the reason for this a little later. The relations (1.1.2.9) can be ij
In a change of basis, following (1.1.2.16), the g ’s transform
j
considered as a system of equations of which the components x
according to
are the unknowns. One can solve it since ðg
ijÞ ¼6 0 (see the end
ij i j 0km
of Section 1.1.2.2). It follows that g ¼ BkBmg
ð1:1:2:19Þ
0ij i j km
j ij g ¼ AkAmg :
x ¼ x g ð1:1:2:10Þ
i
0 0
The volumes V and V of the cells built on the basis vectors e
with i
and e , respectively, are given by the triple scalar products of
i
ij i
g g
jk ¼ k: ð1:1:2:11Þ these two sets of basis vectors and are related by
0 0 0 0
ij V ¼ ðe1; e2; e3Þ
The table of the g ’s is the inverse of the table of the g ’s. Let us
ij
i
now take up the development of x with respect to the basis e
i: ¼ ðBjÞðe1; e2; e3Þ
i i
x ¼ x e
i: ¼ ðBjÞV; ð1:1:2:20Þ
i i
Let us replace x by the expression (1.1.2.10): where ðB Þ is the determinant associated with the transforma-
j
ij tion matrix between the two bases. From (1.1.2.17) and (1.1.2.20),
x ¼ x g e ; ð1:1:2:12Þ
j i
we can write
0 k m
and let us introduce the set of n vectors ðg Þ ¼ ðB ÞðB Þðg Þ:
ij i j km
j ij
e ¼ g e ð1:1:2:13Þ
i
6
1.1. INTRODUCTION TO THE PROPERTIES OF TENSORS

j
If the basis e
i is orthonormal, ðgkmÞ and V are equal to one, TðxÞ ¼ tix i j i j i j
0 i Pðx; yÞ ¼ pijx y ¼ tix sjy ¼ tisjx y :
ðB
jÞ is equal to the volume V of the cell built on the basis SðyÞ ¼ sjy
0
vectors e and
i
0 02 One deduces from this that
ðg Þ ¼ V :
ij
p ¼ t s :
ij i j
This relation is actually general and one can remove the prime
index:
It is a tensor of rank 2. One can equally well envisage the
2 tensor product of more than two spaces, for example,
ðg Þ ¼ V : ð1:1:2:21Þ
ij
E
F
G in npq dimensions. We shall limit ourselves in this
n p q
study to the case of affine tensors, which are defined in a space
In the same way, we have for the corresponding reciprocal
constructed from the product of the space E with itself or with its
basis n
n 3
conjugate E . Thus, a tensor product of rank 3 will have n
ij 2
ðg Þ ¼ V ; components. The tensor product can be generalized as the
product of multilinear forms. One can write, for example,

where V is the volume of the reciprocal cell. Since the tables of
ij Pðx; y; zÞ ¼ Tðx; yÞ
SðzÞ
the g ’s and of the g ’s are inverse, so are their determinants, and
ij j
i k j i k ð1:1:3:1Þ
p x y z ¼ t x y s z :
therefore the volumes of the unit cells of the direct and reciprocal ik j i j k
spaces are also inverse, which is a very well known result in
crystallography.
1.1.3.2. Behaviour under a change of basis
1.1.3. Mathematical notion of tensor
A multilinear form is, by definition, invariant under a change
1.1.3.1. Definition of a tensor of basis. Let us consider, for example, the trilinear form (1.1.3.1).
If we change the system of coordinates, the components of
For the mathematical definition of tensors, the reader may
vectors x, y, z become
consult, for instance, Lichnerowicz (1947), Schwartz (1975) or
Sands (1995). i i 0 0 k k 0
x ¼ B x ; y ¼ A y ; z ¼ B z :
j j
1.1.3.1.1. Linear forms
Let us put these expressions into the trilinear form (1.1.3.1):
A linear form in the space E is written
n
j i 0 0 k 0
i Pðx; y; zÞ ¼ pikBx Aj yB
z :
TðxÞ ¼ t x ;
i
Now we can equally well make the components of the tensor
where TðxÞ is independent of the chosen basis and the t ’s are the
i
appear in the new basis:
coordinates of T in the dual basis. Let us consider now a bilinear
form in the product space E
n
Fp of two vector spaces with n Pðx; y; zÞ ¼ p0
x0y0z0
:
and p dimensions, respectively:
i j
Tðx; yÞ ¼ t x y : As the decomposition is unique, one obtains
ij
0 j i k
p ¼ p B A B : ð1:1:3:2Þ

ik j
The np quantities t ’s are, by definition, the components of a
ij
tensor of rank 2 and the form Tðx; yÞ is invariant if one changes
One thus deduces the rule for transforming the components of
the basis in the space E
F . The tensor t is said to be twice
n p ij
a tensor q times covariant and r times contravariant: they
covariant. It is also possible to construct a bilinear form by
n transform like the product of q covariant components and r
replacing the spaces E and F by their respective conjugates E
n p
p contravariant components.
and F . Thus, one writes
This transformation rule can be taken inversely as the defini-
i j j i i j ij
Tðx; yÞ ¼ t x y ¼ t x y ¼ t x y ¼ t x y ; tion of the components of a tensor of rank n ¼ q þ r.
ij i j j i i j
Example. The operator O representing a symmetry operation has
ij
where t is the doubly contravariant form of the tensor, whereas
j i the character of a tensor. In fact, under a change of basis, O
t
i and tj are mixed, once covariant and once contravariant. transforms into O0:
We can generalize by defining in the same way tensors of rank
0 1
3 or higher by using trilinear or multilinear forms. A vector is a O ¼ AOA
tensor of rank 1, and a scalar is a tensor of rank 0.
so that
1.1.3.1.2. Tensor product
0i i k 1 l
O ¼ A O ðA Þ :
j k l j
Let us consider two vector spaces, E with n dimensions and F
n p
with p dimensions, and let there be two linear forms, TðxÞ in E
n
Now the matrices A and B are inverses of one another:
and SðyÞ in F . We shall associate with these forms a bilinear form
p
0i i k l
called a tensor product which belongs to the product space with O ¼ A O B :
j k l j
np dimensions, E
F :
n p
Pðx; yÞ ¼ TðxÞ
SðyÞ: The symmetry operator is a tensor of rank 2, once covariant and
once contravariant.
This correspondence possesses the following properties:
(i) it is distributive from the right and from the left;
1.1.3.3. Operations on tensors
(ii) it is associative for multiplication by a scalar;
1.1.3.3.1. Addition
(iii) the tensor products of the vectors with a basis E and those
n
with a basis F constitute a basis of the product space. It is necessary that the tensors are of the same nature (same
p
The analytical expression of the tensor product is then rank and same variance).
7
1. TENSORIAL ASPECTS OF PHYSICAL PROPERTIES
1.1.3.3.2. Multiplication by a scalar D ¼ "E:
This is a particular case of the tensor product.
If the medium is anisotropic, we have, for one of the compo-
nents,
1.1.3.3.3. Contracted product, contraction
1 1 1 1 2 1 3
Here we are concerned with an operation that only exists in the D ¼ " E þ " E þ " E :
1 2 3
case of tensors and that is very important because of its appli-
cations in physics. In practice, it is almost always the case that This relation and the equivalent ones for the other components
tensors enter into physics through the intermediary of a can also be written
contracted product.
i i j
D ¼ " E ð1:1:3:3Þ
(i) Contraction. Let us consider a tensor of rank 2 that is once j
covariant and once contravariant. Let us write its transformation
using the Einstein convention.
in a change of coordinate system:
The scalar product of D by an arbitrary vector x is
0j j q p
t ¼ A B t :
i p i q i i j
D x ¼ " E x :
i j i
0i
Now consider the quantity t derived by applying the Einstein
i
convention ðt0i ¼ t01 þ t02 þ t03Þ. It follows that The right-hand member of this relation is a bilinear form that is
i 1 2 3
i
invariant under a change of basis. The set of nine quantities "
0i i q p q q j
t ¼ A B t ¼ t
i p i q p q constitutes therefore the set of components of a tensor of rank 2.
t
i0i ¼ t p: Expression (1.1.3.3) is the contracted product of "ij by Ej.
A similar demonstration may be used to show the tensor
nature of the various physical properties described in Section
This is an invariant quantity and so is a scalar. This operation
1.1.1, whatever the rank of the tensor. Let us for instance
can be carried out on any tensor of rank higher than or equal to
consider the piezoelectric effect (see Section 1.1.4.4.3). The
two, provided that it is expressed in a form such that its
i
components of the electric polarization, P , which appear in a
components are (at least) once covariant and once contravariant.
medium submitted to a stress represented by the second-rank
The contraction consists therefore of equalizing a covariant
tensor T are
index and a contravariant index, and then in summing over this jk
0jk
index. Let us take, for example, the tensor t . Its contracted form i ijk
i P ¼ d T ;
0ik jk
is t , which, with a change of basis, becomes
i
0ik k qp
t
i ¼ Apt q : where the tensor nature of Tjk will be shown in Section 1.3.2. If
we take the contracted product of both sides of this equation by
The components t i
ik are those of a vector, resulting from the any vector of covariant components xi, we obtain a linear form on
jk
contraction of the tensor t . The rank of the tensor has changed the left-hand side, and a trilinear form on the right-hand side,
i
ijk
from 3 to 1. In a general manner, the contraction reduces the rank which shows that the coefficients d are the components of a
of the tensor from n to n 2. third-rank tensor. Let us now consider the piezo-optic (or
photoelastic) effect (see Sections 1.1.4.10.5 and 1.6.7). The
Example. Let us take again the operator of symmetry O. The ij
components of the variation of the dielectric impermeability
trace of the associated matrix is equal to
due to an applied stress are
1 2 3 i
O þ O þO ¼ O : ij ijkl
1 2 3 i ¼ T :
jl
It is the resultant of the contraction of the tensor O. It is a tensor
In a similar fashion, consider the contracted product of both
of rank 0, which is a scalar and is invariant under a change of
sides of this relation by two vectors of covariant components x
i
basis.
and y , respectively. We obtain a bilinear form on the left-hand
j
side, and a quadrilinear form on the right-hand side, showing that
ijkl
(ii) Contracted product. Consider the product of two tensors of the coefficients are the components of a fourth-rank tensor.
which one is contravariant at least once and the other covariant
at least once:
jk j k 1.1.3.5. Representation surface of a tensor
p ¼ t z :
i i
1.1.3.5.1. Definition
If we contract the indices i and k, it follows that Let us consider a tensor t represented in an orthonormal
ijkl...
ji j i frame where variance is not important. The value of component
p
i ¼ t iz : 0
t in an arbitrary direction is given by
1111...
0 i j k l
The contracted product is then a tensor of rank 1 and not 3. It t ¼ t B B B B . . . ;
1111... ijkl... 1 1 1 1
is an operation that is very frequent in practice.
(iii) Scalar product. Next consider the tensor product of two i j
where the B , B ; . . . are the direction cosines of that direction
1 1
vectors:
with respect to the axes of the orthonormal frame.
j j The representation surface of the tensor is the polar plot of
t ¼ x y :
i i 0
t .
1111...
After contraction, we get the scalar product:
i i
t
i ¼ xiy : 1.1.3.5.2. Representation surfaces of second-rank tensors
The representation surfaces of second-rank tensors are
quadrics. The directions of their principal axes are obtained as
1.1.3.4. Tensor nature of physical quantities
follows. Let t be a second-rank tensor and let OM ¼ r be a
ij
i j
Let us first consider the dielectric constant. In the introduction, vector with coordinates x . The doubly contracted product, t x x ,
i ij
we remarked that for an isotropic medium is a scalar. The locus of points M such that
8
1.1. INTRODUCTION TO THE PROPERTIES OF TENSORS
i j i ik j jl
t
ijx x ¼ 1 x ¼ g xk and y ¼ g yl:
It follows that
is a quadric. Its principal axes are along the directions of the
eigenvectors of the matrix with elements t . They are solutions of ij ik jl
ij t ¼ g g x y :
k l
the set of equations
i j xkyl is a tensor product of two vectors expressed in the dual
t x ¼ x ;
ij
space:
x y ¼ t :
where the associated quantities are the eigenvalues. k l kl
Let us take as axes the principal axes. The equation of the
One can thus pass from the doubly covariant form to the
quadric reduces to
doubly contravariant form of the tensor by means of the relation
1 2 2 2 3 2
t ðx Þ þ t ðx Þ þ t ðx Þ ¼ 1: ij ik jl
11 22 33 t ¼ g g t :
kl
If the eigenvalues are all of the same sign, the quadric is an This result is general: to change the variance of a tensor (in
ellipsoid; if two are positive and one is negative, the quadric is a practice, to raise or lower an index), it is necessary to make the
hyperboloid with one sheet; if one is positive and two are nega- contracted product of this tensor using gij or g , according to the
ij
tive, the quadric is a hyperboloid with two sheets (see Section case. For instance,
1.3.1).
l jl ij ijl
t ¼ g t ; t ¼ g t :
Associated quadrics are very useful for the geometric repre- k lk k kl
sentation of physical properties characterized by a tensor of rank
2, as shown by the following examples:
Remark
(i) Index of refraction of a medium. It is related to the dielectric
i ik i
1=2 g ¼ g g ¼ :
constant by n ¼ " and, like it, it is a tensor of rank 2. Its j kj j
associated quadric is an ellipsoid, the optical indicatrix, which
represents its variations with the direction in space (see Section This is a property of the metric tensor.
1.6.3.2).
(ii) Thermal expansion. If one cuts a sphere in a medium whose
1.1.3.6.3. Examples of the use in physics of different
thermal expansion is anisotropic, and if one changes the
representations of the same quantity
temperature, the sphere becomes an ellipsoid. Thermal expan-
Let us consider, for example, the force, F, which is a tensor
sion is therefore represented by a tensor of rank 2 (see Chapter
quantity (tensor of rank 1). One can define it:
1.4).
(i) by the fundamental law of dynamics:
(iii) Thermal conductivity. Let us place a drop of wax on a plate
i 2 i 2
of gypsum, and then apply a hot point at the centre. There F ¼ mC; with F ¼ m d x =dt ;
appears a halo where the wax has melted: it is elliptical, indicating
anisotropic conduction. Thermal conductivity is represented by a where m is the mass and C is the acceleration. The force appears
tensor of rank 2 and the elliptical halo of molten wax corresponds here in a contravariant form.
to the intersection of the associated ellipsoid with the plane of the (ii) as the derivative of the energy, W:
plate of gypsum.
i
F ¼ @
1. TENSORIAL ASPECTS OF PHYSICAL PROPERTIES
1.1.3.7.2. Vector product 1.1.3.8.2. Generalization
j
Consider the so-called permutation tensor of rank 3 (it is Consider a field of tensors t that are functions of space vari-
i
actually an axial tensor – see Section 1.1.4.5.3) defined by ables. In a change of coordinate system, one has
8
j j 0
< "ijk ¼ þ1 if the permutation ijk is even t i ¼ Ai B
t :
" ¼ 1 if the permutation ijk is odd
ijk
: k
"
ijk ¼ 0 if at least two of the three indices are equal Differentiate with respect to x :
j 0 0
@t
i j j @t @x
and let us form the contracted product ¼ @ t ¼ A B
k k i i 0
k
@x @x @x
1 ij i j
z ¼ " p ¼ " x y : ð1:1:3:4Þ j j
0
k 2 ijk ijk @ t ¼ A B A @ t :
k i i k
It is easy to check that j
It can be seen that the partial derivatives @ t behave under a
8 k i
< z1 ¼ x2y3 y2x3 change of axes like a tensor of rank 3 whose covar
j iance has been
3 1 3 1 increased by 1 with respect to that of the tensor t . It is therefore
z ¼ x y y x i
2
: 1 2 2 1 possible to introduce a tensor of rank 1, r (nabla), of which the
z ¼ x y y x :
3
components are the operators given by the partial derivatives
i
@
1.1. INTRODUCTION TO THE PROPERTIES OF TENSORS
1.1.4.1. Introduction – Neumann’s principle (ii) the phenomenon may exist in a medium that possesses that
symmetry or that of a subgroup of that symmetry;
We saw in Section 1.1.1 that physical properties express in
and concludes that some symmetry elements may coexist with
general the response of a medium to an impetus. It has been
the phenomenon but that their presence is not necessary. On the
known for a long time that symmetry considerations play an
contrary, what is necessary is the absence of certain symmetry
important role in the study of physical phenomena. These
elements: ‘asymmetry creates the phenomenon’ (‘C’est la dissy-
considerations are often very fruitful and have led, for instance,
me´trie qui cre´e le phe´nome`ne’; Curie, 1894, p. 400). Noting that
to the discovery of piezoelectricity by the Curie brothers in 1880
physical phenomena usually express relations between a cause
(Curie & Curie, 1880, 1881). It is not unusual for physical prop-
and an effect (an influence and a response), P. Curie restated the
erties to be related to asymmetries. This is the case in electrical
two above propositions in the following way, now known as Curie
polarization, optical activity etc. The first to codify this role was
laws, although they are not, properly speaking, laws:
the German physicist and crystallographer F. E. Neumann, who
(i) the asymmetry of the effects must pre-exist in the causes;
expressed in 1833 the symmetry principle, now called Neumann’s
(ii) the effects may be more symmetric than the causes.
principle: if a crystal is invariant with respect to certain symmetry
The application of the Curie laws enable one to determine the
elements, any of its physical properties must also be invariant with
symmetry characteristic of a phenomenon. Let us consider the
respect to the same symmetry elements (Neumann, 1885).
phenomenon first as an effect. If is the symmetry of the
This principle may be illustrated by considering the optical
phenomenon and C the symmetry of the cause that produces it,
properties of a crystal. In an anisotropic medium, the index of
refraction depends on direction. For a given wave normal, two
C :
waves may propagate, with different velocities; this is the double
refraction effect. The indices of refraction of the two waves vary
Let us now consider the phenomenon as a cause producing a
with direction and can be found by using the index ellipsoid
certain effect with symmetry E:
known as the optical indicatrix (see Section 1.6.3.2). Consider the
central section of the ellipsoid perpendicular to the direction of
E:
propagation of the wave. It is an ellipse. The indices of the two
waves that may propagate along this direction are equal to the
We can therefore conclude that
semi-axes of that ellipse. There are two directions for which the
central section is circular, and therefore two wave directions for
C E:
which there is no double refraction. These directions are called
optic axes, and the medium is said to be biaxial. If the medium is
If we choose among the various possible causes the most
invariant with respect to a threefold, a fourfold or a sixfold axis
symmetric one, and among the various possible effects the one
(as in a trigonal, tetragonal or hexagonal crystal, for instance), its
with the lowest symmetry, we can then determine the symmetry
ellipsoid must also be invariant with respect to the same axis,
that characterizes the phenomenon.
according to Neumann’s principle. As an ellipsoid can only be
As an example, let us determine the symmetry associated with
ordinary or of revolution, the indicatrix of a trigonal, tetragonal
a mechanical force. A force can be considered as the result of a
or hexagonal crystal is necessarily an ellipsoid of revolution that
traction effort, the symmetry of which is A 1M. If considered as
has only one circular central section and one optic axis. These 1
a cause, its effect may be the motion of a sphere in a given
crystals are said to be uniaxial. In a cubic crystal that has four
direction (for example, a spherical ball falling under its own
threefold axes, the indicatrix must have several axes of revolu-
weight). Again, the symmetry is A 1M. The symmetries asso-
tion, it is therefore a sphere, and cubic media behave as isotropic 1
ciated with the force considered as a cause and as an effect being
media for properties represented by a tensor of rank 2.
the same, we may conclude that A 1M is its characteristic
1
symmetry.
1.1.4.2. Curie laws
The example given above shows that the symmetry of the
1.1.4.3. Symmetries associated with an electric field and with
property may possess a higher symmetry than the medium. The
magnetic induction (flux density)
property is represented in that case by the indicatrix. The
1.1.4.3.1. Symmetry of an electric field
symmetry of an ellipsoid is
0 00 Considered as an effect, an electric field may have been
A A A
2 2 2
C ¼ mmm for any ellipsoid produced by two circular coaxial electrodes, the first one carrying
0 00
M M M
positive electric charges, the other one negative charges (Fig.
(orthorhombic symmetry)
1.1.4.1). The cause possesses an axis of revolution and an infinity
A
11A2 1 of mirrors parallel to it, A11M. Considered as a cause, the
C ¼ m for an ellipsoid of revolution
electric field induces for instance the motion of a spherical
M 1M m
electric charge parallel to itself. The associated symmetry is the
(cylindrical symmetry)
same in each case, and the symmetry of the electric field is
A 1
1 identical to that of a force, A 1M. The electric polarization or
1 C ¼ 1 for a sphere 1
M m the electric displacement have the same symmetry.
(spherical symmetry):
[Axes A are axes of revolution, or axes of isotropy, introduced
1
by Curie (1884, 1894), cf. International Tables for Crystallography
(2002), Vol. A, Table 10.1.4.2.]
The symmetry of the indicatrix is identical to that of the
medium if the crystal belongs to the orthorhombic holohedry and
is higher in all other cases.
This remark is the basis of the generalization of the symmetry
principle by P. Curie (1859–1906). He stated that (Curie, 1894):
(i) the symmetry characteristic of a phenomenon is the highest
compatible with the existence of the phenomenon; Fig. 1.1.4.1. Symmetry of an electric field.
11
1. TENSORIAL ASPECTS OF PHYSICAL PROPERTIES
1.1.4.3.2. Symmetry of magnetic induction with its symmetry on one hand and the increase of temperature,
which is isotropic, on the other. The intersection of the groups of
The determination of the symmetry of magnetic quantities is
symmetry of the two causes is in this case identical to the group of
more delicate. Considered as an effect, magnetic induction may
symmetry of the crystal. The symmetry associated with the effect
be obtained by passing an electric current in a loop (Fig. 1.1.4.2).
is that of the electric polarization that is produced, A 1M.
The corresponding symmetry is that of a cylinder rotating around 1
Since the asymmetry of the cause must pre-exist in the causes, the
its axis, ðA =MÞC. Conversely, the variation of the flux of
1
latter may not possess more than one axis of symmetry nor
magnetic induction through a loop induces an electric current in
mirrors other than those parallel to the single axis. The only
the loop. If the magnetic induction is considered as a cause, its
crystal point groups compatible with this condition are
effect has the same symmetry. The symmetry associated with the
magnetic induction is therefore ðA =MÞC.
1 1; 2; 3; 4; 6;m; 2mm; 3m; 4mm; 6mm:
This symmetry is completely different from that of the electric
field. This difference can be understood by reference to
There are therefore only ten crystallographic groups that are
Maxwell’s equations, which relate electric and magnetic quan-
compatible with the pyroelectric effect. For instance, tourmaline,
tities:
in which the effect was first observed, belongs to 3m.
@B @D
curl E ¼ r ^ E ¼ ; curl H ¼ r ^H ¼ :
@t @t
1.1.4.4.3. Piezoelectricity
It was seen in Section 1.1.3.8.3 that the curl is an axial vector Piezoelectricity, discovered by the Curie brothers (Curie &
because it is a vector product. Maxwell’s equations thus show that Curie, 1880), is the property presented by certain materials that
if the electric quantities (E, D) are polar vectors, the magnetic exhibit an electric polarization when submitted to an applied
quantities (B, H) are axial vectors and vice versa; the equations of mechanical stress such as a uniaxial compression (see, for
Maxwell are, in effect, perfectly symmetrical on this point. instance, Cady, 1964; Ikeda, 1990). Conversely, their shape
Indeed, one could have been tempted to determine the symmetry changes when they are submitted to an external electric field; this
of the magnetic field by considering interactions between is the converse piezoelectric effect. The physical interpretation of
magnets, which would have led to the symmetry A
11M for the piezoelectricity is the following: under the action of the applied
magnetic quantities. However, in the world where we live and stress, the centres of gravity of negative and positive charges
where the origin of magnetism is in the spin of the electron, the move to different positions in the unit cell, which produces an
magnetic field is an axial vector of symmetry ðA
1=MÞC while the electric polarization.
electric field is a polar vector of symmetry A
11M. From the viewpoint of symmetry, piezoelectricity can be
considered as the superposition of two causes, the crystal with its
own symmetry and the applied stress. The symmetry associated
1.1.4.4. Superposition of several causes in the same medium –
with a uniaxial compression is that of two equal and opposite
pyroelectricity and piezolectricity forces, namely A =M 1A =1MC. The effect is an electric
1 2
1.1.4.4.1. Introduction polarization, of symmetry A11M, which must be higher than or
equal to the intersection of the symmetries of the two causes:
Let us now consider a phenomenon resulting from the super-
position of several causes in the same medium. The symmetry of A 1A \
1 2
C S A 1M;
the global cause is the intersection of the groups of symmetry of crystal 1
M 1M
the various causes: the asymmetries add up (Curie, 1894). This
remark can be applied to the determination of the point groups
where S denotes the symmetry of the crystal.
crystal
where physical properties such as pyroelectricity or piezo- It may be noted that the effect does not possess a centre of
electricity are possible. symmetry. The crystal point groups compatible with the property
of piezoelectricity are therefore among the 21 noncentrosym-
metric point groups. More elaborate symmetry considerations
1.1.4.4.2. Pyroelectricity
show further that group 432 is also not compatible with piezo-
Pyroelectricity is the property presented by certain materials electricity. This will be proved in Section 1.1.4.10.4 using the
that exhibit electric polarization when the temperature is symmetry properties of tensors. There are therefore 20 point
changed uniformly. Actually, this property appears in crystals for groups compatible with piezoelectricity:
which the centres of gravity of the positive and negative charges
1; 2; m; 222; 2mm;
do not coincide in the unit cell. They present therefore a spon-
taneous polarization that varies with temperature because, owing 3; 32; 3m; 4; 4; 422; 4mm; 42m; 6; 6; 622; 6mm; 62m
to thermal expansion, the distances between these centres of
23; 43m:
gravity are temperature dependent. A very important case is that
of the ferroelectric crystals where the direction of the polariza-
tion can be changed under the application of an external electric The intersection of the symmetries of the crystal and of the
field. applied stress depend of course on the orientation of this stress
From the viewpoint of symmetry, pyroelectricity can be relative to the crystallographic axes. Let us take, for instance, a
considered as the superposition of two causes, namely the crystal crystal of quartz, which belongs to group 32 ¼ A33A2. The above
condition becomes
\
A 1A
1 2
C A 3A A 1M:
3 2 1
M 1M
If the applied compression is parallel to the threefold axis, the
intersection is identical to the symmetry of the crystal, A 3A ,
3 2
which possesses symmetry elements that do not exist in the effect,
and piezoelectricity cannot appear. This is of course obvious
because the threefold axis is not polar. For all other directions,
Fig. 1.1.4.2. Symmetry of magnetic induction. piezoelectricity may appear.
12