Crystallographic Point Groups - Springer978-1-4612-3226-1/1.pdfCrystallographic Point Groups In this...

APPENDIX A

Crystallographic Point Groups

In this appendix we examine briefly the symmetry properties of nonmagnetic crystals. According to the axiom of material invariance, the macroscopic symmetry of all nonmagnetic crystals may be described by an isotropy group {S}. Accordingly, under the transformations of the material frame of reference

X = SX, SST = STS = 1, det S = ± 1, (A.l)

the constitutive functionals must remain form-invariant for all members of the symmetry group {S}. Local properties of crystals are restricted by the point group.

The symmetry operators, that act at a fixed point 0 and leave invariant all distances and angles in a three-dimensional space, are called the point group. The symmetry operators that have these properties are rotations about axes through 0, and products (combinations) of rotations and inversions. Of course, such products include reflections in planes through o.

If the group contains only rotations, it is called a proper rotation group. This is isomorphic with the group 0+(3) of all 3 x 3 orthogonal matrices. Operators, whose matrices have determinant (-1), are called improper rotations. They are products of proper rotations and inversion. We note that the inversion commutes with all rotations.

Every subgroup of 0+(3) is a proper point group. Proper point groups of finite order are classified as: Cyclic (Cn = n); Dihedral (Dn = n22, n even, Dn = n2, n odd); Tetrahedral (T = 23); and Octahedral (0 = 432).

A crystallographic point group is restricted by a requirement that an operator must be compatible with the translational symmetry of a crystalline solid. Hence, the appropriate symmetry operations are

identity = E,

reflection in certain planes = (1,

inversion = C,

rotations = Cnr•

The rotation Cnr is an anticlockwise rotation through 2n/n radians about the axis indicated by r.

The eleven proper point groups are listed in Table A.l, together with their

374 Appendix A. Crystallographic Point Groups

Table A.t. Crystallographic pure rotation groups.

Cyclic groups

C, = 1 C2 = 2 C3 = 3 C4 =4 C6 = 6

Dihedral groups

D2 = 222 D3 = 32 D4 = 422 D6 = 622

Tetrahedral group

T= 23

Octahedral group

0=432

Symmetry elements

E E, C2z

E, C3., C3z E, C4., Ci., C2z

E, C6 ., Ci., C3., C3., C2z

E, C2x , C2y , C2z

E, C3., C3., Cl" Cl2 , Cl3 E, C4 ., Ci., C2x , C2y , C2 ., C2a , C2b

E, C6z , C6z, C3z' C3"z, C2z , C~r' Ci,

symmetry elements. In this table the first column (C1 , C2 , ••• , 0) denotes the Schonflies notation, and the second column (1, 2, ... ,432) denotes the international notation.

In addition to purely rotational symmetry, the space lattice possesses symmetries of reflections in various planes (det S = -1). In order to include such symmetry operations, we multiply the proper point group {P} by {E, C}. This produces a new set of eleven point groups that are subgroups of 0(3).

If the point group {P} has an invariant subgroup {H} of index 1 2, then

{P} = {H} + C{P - H} (A. 2)

is also a point group. This process gives ten more point groups. The possible crystallographic point groups are 32 in number, as listed in Table A.2.

By examination of the metrical properties, crystal classes are divided into seven crystal systems. Each system possesses one and the same metrical property. If hi denotes the lattice bases then the length oflattice bases \hl\ = a, \h2\ = b, \h3\ = c, and angles ex = angle(h2, h3)' p = angle(h3' hd, and y = angle(h1 , h3), for each crystal system, are the same. This is called a holohedry of the space lattices.

In Table A.l, j = 1,2,3,4; m = x, y, z; p = a, b, c, d, e,J; and r = 1,2,3; and the labels of the symmetry operations can be identified from Figures A.1-A.3. In Figures A.l and A.2 the labels of the symmetry operations are placed on the figure in the position to which the letter E is taken by that operation.

1 The index of a subgroup is the integer obtained by dividing the order of the group by that of the subgroup.

Figure A.t. Symmetry elements: triclinic, monoclinic, rhombic, and tetragonal systems,

, , , (4Z ... ,

y. -------------: __ ------1.._

Figure A.2. Symmetry elements: trigonal and hexagonal systems,

Figure A.3. Symmetry elements:

'" x

(2a

, ,

2" ~ ,

, /

\

(6, \

, , ,

(2, (2y

• , , • C ' E 21,

, , ,

, ' , ...... ', .. ' ........

(2, : ('2,

• l'

cubic system, 3 '-""--______ ---V

, , ,

~ 3"

'" • -' , 3'

ciz

2"

w

-..I

0

- :>

'"0

'"0 (1)

Tab

le A

.2. T

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2 co

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tion

al c

ryst

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Ord

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.... '<

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num

ber

Cla

ss n

ame

Sym

met

ry tr

ansf

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ricl

inic

P

edia

l C

.l

I 2

Pin

acoi

dal

ciT

I,

C

2 0'

O

Q ....

2 '" ~

2 ("

)

Mon

ocli

nic

3 S

phen

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l C

22

I,D

3 4

Dom

atic

C

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I, R

3 4

'ti 9.

4 =

... 5

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smat

ic

C2h

2/m

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C,R

3' D

3

Ort

horh

ombi

c 6

Rho

mbi

c-di

sphe

noni

dal

D22

22

I, D

., D

2, D

3 7

Rho

mbi

c-py

rom

idal

C

2v2m

m

I, R

., R

2, D

3 4

Cl

.... 8

0 s::

8 R

hom

bic-

dipy

ram

idal

D

2hm

mm

I,

C, D

., D

2, D

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., R

2, R

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0 4

'" T

etra

gona

l 9

Tet

rago

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pyra

mid

al

C4

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R. T

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10

T

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sphe

noid

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C 24

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3, D

. T3,

D2

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Tetragonal~dipyramidal

C4h

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2 T 3

, R. T

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, C, R

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12

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8 13

D

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4mm

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, D3

T3

8 14

T

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42m

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, D3

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8

15

Dit

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al

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mm

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, R.

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, R3

T 3, C

, R

., R

2, R

3, T

3, D

. T3,

16

Tri

gona

l 16

T

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3 J,

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3 17

R

hom

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6 18

T

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l D3

32

J, S

I, S

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I, D

ISI,

DIS

2 6

19

Dit

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mid

al

C3v

3m

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SI'

S2'

RI

, R

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6

20

Hex

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al-s

cale

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3v3 m

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66

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6 22

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, R

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I, D

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24

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rape

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662

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12

::s e: 25

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m

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) 24

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3S2

'<

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Cub

ic

28

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23

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D3,

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, Clj

12

0'

29

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3 J,

DI,

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D3,

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I, R

2, R

3, C

3j, C

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6j, S

6j

24

OQ

... '" 30

G

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04

32

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D ..

D2,

D3,

C2p

, C3j

, Clj

, C4m

, Cim

24

"0

31

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tetr

ahed

ral

Td4

3m

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2, D

3, (i

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3j, C

lj, S

4m, S

im

24

~ 32

H

exoc

tohe

dral

Oh

m3m

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D2,

D3,

C2P

' C3l

, Clj

, C4m

, Cim

, C, R

I, R

2, R

3, (i

,p, S

6j, S

6j,

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48

0 S4

m, S

im

5'

.... 0 ... 0 ~

"0

til

VJ

-.l

-.l

378 Appendix A. Crystallographic Point Groups

The transformation matrices are given by

I ~ (~ 0

~). rl ° ~). 1 C = 0-1 0 o 0 -1

rl

0 n R, ~(~ 0

~ ). R, ~(~ 0

~). Rl = ~ 1 -1 1 0 0 0 -1

D'~(~ 0

~). (-I 0

0) C 0

~). -1 O2 = ~ 1 o ,03 = 0 -1 0 -1 0 -1 0 0

T, ~(~ 0

!). T, ~(~ 0

~). T'~(! 1 n 0 1 0

1 0 0

M, ~(~ 1

!). M, ~(! 0

~). 0 0 (A.3)

0 1

( -1/2 ~/2 0) rl/2 -~/2 0)

Sl = -f/2 -1/2 0 , S2 = ~/2 -1/2 0, o 1 o 1

where I is the identity and C is the central inversion. R l , R2, R3 are reflections in the planes whose normals are along the Xl = X-, X2 = Y-, and X3 = zdirections, respectively. 0 1 , O2, 0 3 are rotations through n radians about the Xl -, X 2 -, and x3-axes, respectively. Tl is a reflection through a plane which bisects the X2 - and X3 -axes and contains the xl-axis. T2 and T3 are analogously defined. Ml and M2 are rotations through 2n/3 clockwise and anticlockwise, about an axis making equal acute angles with the axes Xl' x 2 , and X3' Sl and S2 are rotations through 2n/3 clockwise and anticlockwise, respectively, about the X3 = z-axis.

APPENDIX B

Crystallographic Magnetic Groups

As noted in Section 5.4, the symmetry properties of magnetic materials must include a time-inversion operator which reverses the spin of each atom. The situation is visualized simply by considering a chain of equally spaced atoms on a line (Figure B.1). Disregarding their spin, we see that the X2 -axis is a twofold symmetry axis, and in addition, the X2 X3 -plane is a reflection plane (Figure B. 1 (a)). Now if the spins are as shown in Figure B.1(b), then the situation is the same. However, if the spins are oppositely directed (Figure B.l(c)), then X 2 is no longer a twofold rotation axis. Moreover, the X 2 X 3 -

plane is not a reflection plane. Thus, the full characterization of the magnetic properties of crystals requires the incorporation of the symmetry property of the individual atoms constituting the lattice points to the symmetry of the lattice. This means the consideration of spin or, interpreted as an orbital angular momentum, time reversal. Atoms of certain materials do not possess magnetic moments and in some other materials the spin is randomly distributed. The first of these two classes of materials is called diamagnetic and the second paramagnetic. These materials may therefore be referred to as nonmagnetic, and the point group of 32 classes discussed in Appendix A constitutes their symmetry group.

However, there exist large classes of other materials which exhibit magnetic properties. These are the ferromagnetic, antiferromagnetic, and ferrimagnetic materials. In ferromagnetic materials (e.g., Fe, Zn, Co) the adjacent lattice sites possess parallel spins so that, in the absence of an external field, the material posseses net magnetization (Figure B.2(a)). In antiferromagnetic materials (e.g., CoF2, MnF2' Cr20 3 ) the spin distribution is in a periodic arrangement, alternating parallel and anti parallel motifs, that results in zero magnetization in the absence of an external field (Figure B.2(b)). The ferrimagnetic materials (e.g., MnFe20 4 , NiFe20 4 ) also contain anti parallel spin arrangements, however, the cancellation is incomplete and the body possesses magnetic dipole density. All three types of materials have highly nonlinear B-H relationships. Ferromagnetic, antiferromagnetic, and ferrimagnetic materials are called magnetic materials.

The arrangement of atomic magnetic moments can be affected in all mag-

380 Appendix B. Crystallographic Magnetic Groups

r Figure B.l. Magnetic symmetry.

(0) 0 0 -XI 0

( b) ; r + ,-XI 0

(c) + r ; -XI 0

netic classes to produce antiferromagnetism. This includes even those that exhibit ferromagnetism. For example, NiF2 in its crystallized magnetic symmetry mmm (a ferromagnetic class), exhibits antiferromagnetism. Conversely, by applying a small rotation to the spins of antiferromagnetic materials we can obtain weak ferromagnetism. This phenomenon has been observed for several substances, among which are OC-Fe203 above 250 K, NiF2, MnC03, and CoC03.

For magnetic materials, as discussed before, the spin symmetry can be incorporated into the crystal symmetry group by means of the time-reversal operator R. Alternatively, we can use a four-dimensional formalism involving 4 x 4 matrices, in Minkowski space, as the members of the symmetry group. Here, for the sake of simplicity, we briefly discuss the use of the time-reversal operator R. It is conventional to denote the time reversal by an underscore, e.g., if (E, s1, S2, ... ) denote the elements of the nonmagnetic group G. The reversal of the atomic magnetic moment for an element S" of G is denoted by ~" and is called the complement of S". If the product rule of matrices being applied to the elements of sa is S1S2 = S3, then we can easily see that the product rule for the complement group is ~1~2 = S3, ~IS2 = SI~2 = ~3. In this way, from the symmetry elements of G = {S}, we obtain complementary elements by replacing some of these symmetry operations by their complements, such that the resulting set of operations form a group under the product rule defined above. By exhausting all possibilities for the 32 elements of the

(a) (b)

Figure B.2. Magnetic materials: (a) ferromagnetic; (b) antiferromagnetic.

Appendix B. Crystallographic Magnetic Groups 381

nonmagnetic crystal group, we find that there are only 58 distinct groups which are of magnetic origin. A systematic way of determining the magnetic group is given by Tavger and Zaitsev [1956]. The 32 nonmagnetic point groups, of course, do not contain the time reversal R. The remaining 58 groups, called additional magnetic groups, contain R in combination with the spatial symmetry operators. Thus, if H is a subgroup of index 2 of the nonmagnetic group G == {S}, then the elements of the additional magnetic group are oftwo types:

(a) sa E He G; (b) RSfJ such that SfJ E (G - H).

Birss [1964J proves that sa and SfJ are disjoint, and therefore it is possible to represent a magnetic point group {M} in the form

or

{M} = {H + R(G - H)},

{M} = {H + RSfJH},

(B.1)

(B.2)

where SfJ is a particular element of the set (G - H). From (B.2) it is clear that magnetic point groups can be generated as follows:

(i) For any particular class, one group of magnetic symmetry is identical to the nonmagnetic class G.

(ii) From G select all subgroups H of index 2. (iii) Replace all elements SfJ of (G - H) (which do not belong to H) by SfJ =

RSfJ. (iv) Reject all groups {M} = {H + R( - H)} for which any element SfJ is of

odd order. This is because a magnetic group with an element RSfJ is to be rejected if SfJ is of odd order, since (RSfJt = R (n = odd) is not a magnetic symmetry group.

EXAMPLE. To illustrate, consider the prismatic class C2h = 21m = 2:m whose symmetry elements are I, D1, C, and R1.1t has three subgroups with index 2, namely,

We thus have

m = {I, Rd = C5 ,

2 = {I, Dd = C2 ,

I = {I, C} = Cj •

{~- m} = {D1' C},

{~ - 2} = {C, Rd,

{~- I} = {D1' Rd.

382 Appendix B. Crystallographic Magnetic Groups

Hence, the three magnetic groups originating from 21m are

21m: m + R {~- m} = I, R 1 , RD1 , RC,

21rJJ: 2 + R {~- 2} = I, D 1 , RC, RR1 ,

- {2-} 21rJJ: 1 + R ;;; - 1 = I, C, RD1 , RR 1 .

Note that none of the elements of these three classes are of odd order. They constitute 8 to 11 classes out of the 90 magnetic groups in Table B.1.

Table B.l. Magnetic point groups.

Classical Magnetic subgroup {H}

point No. group {M} International Schonflies G-H

1 I C1 C 2 ~ C1 D3 3 I!I 1 C1 R3 4 2/1!1 2 C2 C,R3 5 ~/m m C1h = C, C,D3 6 ~/I!I I Cj D3, R3 7 ~~2 2 C2 D1,D2 8 2mm 2 C2 R I,R2 9 ~ml!l m C, D3,RI

10 mmm 222 D2 C, R .. R 2, R3 11 I!Imm 2mm C2v C, DI, D2, R3 12 mmm 2/m C2h D1, D2, R 1, R2 13 ~ 2 C2 R2 T3, RI T3 14 4 2 C2 D2T3, DI T3 15 422 4 C4 D1, D2, CT3, R3 T3 16 ~2~ 222 D2 R2 T3, RI T3, CT3, R3 T3 17 4/1!1 4 C4 C, R 3, D2 T3, DI T3 18 ~/I!I 4 S4 C, R3, R2 T3, RI T3 19 ~/m 2/m C2h R2 T3, RI T3, D2 T3, DI T3 20 41!11!1 4 C4 R I, R2, T3, D3 T3 21 4mm 2mm C2v R2 T3, RI T3, T3, D3 T3 22 42m 4 S4 DI, D2, T3, D3 T3 23 42m 222 D2 D2 T3, DI T3, T3, D3 T3 24 42m 2mm C2v D1, D2, D2 T3, DI T3 25 4/1!I1!I1!1 422 D4 C, R I, R2, R3, D2 T3, DI T3, T3, D3 T3 26 4/l!Imm 4mm C4v C, R3, D2 T3, DI T3, D1, D2, CT3, R3 T3 27 ~/mml!l mmm D2h R2~,RI~,C~,R3~,D2~,DI~'

T3, D3 T3 28 ~/l!Iml!l 42m Dld C, R I, R2, R 3, R2 T3, RI T3, CT3, R3 T4 29 ~/ml!ll!l 4/m C4h D1, D2, R 1, R2, CT3, R3 T3, T3, D3 T3 30 3~ 3 C3 D1, D1SI, D1S2 31 3m 3 C3 R 1, R1S1, R IS2

(continued)

Appendix B. Crystallographic Magnetic Groups 383

Table B.1 (continued)

Classical Magnetic subgroup {H}

point No. group {M} International Schiinl1ies G-H

32 § 3 C3 R3, R3SI, R3S2 33 6!1JJ 6 C3h D2, D2SI, D2S2, R I, RISI, R2S2 34 6m2 3m C3v D2, D2SI, D2S2, R 3, R3SI, R3S2 35 §!1J2 32 D3 R3, R3S2, R 3SI, R I, RISI, R IS2 36 6 3 C3 D3, D3S2, D3S1

37 J 3 C3 C, CSI, CS2 38 3!1J 3 C3i DI, DISI, DIS2, R I, RISI, R IS2 39 Jm 3m C3v DI, DISI, DIS2, C, CSI, CS2 40 J!1J 32 D3 C, CSI, CS2, R I, RISI, R, S2 41 622 6 C6 DI, DISI, DIS2, D2, D2SI, D2S2 42 §2J 32 D3 D3, D3S2, D3SI, D2, D2SI , D2S2 43 6/!1J 6 C6 C, CSI, CS2, R3, R3S2, R 3S1

44 §/!1J 3 C3i D3, D3S2, D3SI, R 3, R 3S2, R3S1

45 6/m 6 C3h C, CSI, CS2, D3, D3S2, D3S1

46 6mm 6 C6 R I, RISI, R IS2, R2, R2SI, R 2S2 47 §m!1J 3m C3v D3, D3S2, D3SI, R2, R2SI, R2S2 48 §/mm!1J 62m D3h C, CSI, CS2, D3, D3S2, D3SI, D2, D2SI,

D2S2, R I, RISI, R IS2 49 §/!1Jm!1J 3m D3d D3, D3S2, D3SI, D2, D2SI, D2S2, R3, R 3SI,

R3S2, R2, R2SI, R2S2 50 6/!1J!1J!1J 622 D6 C, CSI, CS2, R3, R3SI, R3S2, R I, RISI,

R IS2, R2, R2SI, R2S2 51 6/!1Jmm 6mm C6v DI, DISI, DIS2, D2, D2SI, D2S2, C, CSI,

CS2, R 3, R3SI, R 3S2 52 6/m!1J!1J 6/m C6h DI, DISI, DIS2, D2, D2S2, R I, RISI, R IS2,

R2, R2SI, R 2S2, D2S1

53 !1J3 23 T C, S6i' S6i' R I, R2, R3 54 ~3!1J 23 T (Jdp' S4m' Sim 55 13J 23 T C2p, C4m, C4m 56 !1J3!1J 432 0 C, S6i' S6i' R I, R2, R3, (Jdp' S4m' S4m 57 !1J3m 43m ~ C, S6i' S6i' R I, R2, R3, C2P' C4m, C4m 58 m3!1J m3 T" C2P' C4m, C4m, (Jdp' S4m' Sim

APPENDIX C

Integrity Bases of Crystallographic Groups

Tables Cl.l-C1.16 give the linear combinations of the components of an absolute (polar) vector Pi. an axial vector ai. and a symmetric second-order tensor Sij. which form the carrier spaces for the irreducible representations rl •

r2 .... associated with various crystal classes. The notation r3: <p. ifJ indicates that <p and ifJ are the basic quantities associated with the representation r3 of degree one. The notation rs: (a l • a2). (bl • b2) indicates that(a l • a2) and (bl • b2) are basic quantities associated with the representation rs of degree two. and so on. Typical elements of the integrity basis for a crystal class are listed following the tables. given by Kiral [1972] and Kiral and Smith [1974]. The complete set of integrity basis would be obtained from these by using the format (5.5.18).

Table Ct. Basic quantities. For a symmetric second-order tensor Sij. a polar vector Pi. and an axial vector ai that form the carrier space for the irreducible representations r l , r 2 , ••. associated with various conventional crystal classes.

Table Ct.t

CI . r l : ai' az, a3' Su, Sn, S33' S13' SZ3' S12; r z: PI' Pz, P3; Cz ' r l : Pz, P3' ai' Su, S2Z, S33' SZ3; r z: PI' az, a3, S12' S13; Cz ' r l : PI' ai' Su, Szz, S33' SZ3; r z: Pz, P3' az, a3' SIZ' S13;

Table Ct.2

CZh ' r l : ai' Su, Szz, S33' SZ3; r z: PI; r3: Pz, P3; r 4 : az, a3' S12' S13; Cz• r l : PI' Su, Sn, S33; r z: ai' S23; r3: Pz, a3' S12; r 4 : P3' az, S13; Dz ' r l : Sl1' S22' S33; r 2: PI' ai' S23; r3: P2' az, S13; r 4 : P3' a3, S12;

Table Cl.3

Appendix C. Integrity Bases of Crystallographic Groups 385

Table C1.4

C2' r , : a3, 833 , 811 + 822; r 2: P3, 812, 811 - 822; r3: PI - iP2' a l + ia2, 813 + i823; r 4 : PI + iP2' a l - ia2, 813 - i823 ;

C4' r,: a3 , 833, 811 + 822 , P3; r 2: 812, 811 - 8zz ; r3: PI + iP2, al + iaz, 813 + i823 ; r 4: PI - iP2, al - iaz, 813 - i823;

Table Cl.S

C4~' r,: a3, 833 , 811 + 8zz ; r z: 81z, 811 - 8zz ; r3: 813 + i8z3, a, + iaz; r 4: a, - iaz, 813 - i823; r;: P3; r3: PI + ipz; r~: p, - ipz;

Table Cl.6

C4v ' r l : p" 833, 811 + 822; r z: a3 ; r3: 812; r 4: 811 - 8zz ; rs: (PI' P2)' (a2, -a,), (8'3' 8Z3 );

D4 ' r , : 833 , 811 + 822 ; r 2: P3' a3; r3: 812; r 4: 811 - 822 rs: (p" P2)' (a" a2), (823 , - 813);

D2v ' r,: 833, 8" + 822 ; r z: a3 ; r3: P3' 812; r 4: 811 - 822; rs: (PI' P2)' (ai' -a2), (8Z3 ' 813 );

Table Cl.7

D4h ' r,: 833, 8" + 8Z2 ; r 2: a3; r3: 812; r 4: 811 - 822; rs: (ai' a2), (823 , -813 ); r~: P3; r;: (PI' P2);

Table C1.8

C3' r , : P3' a3, 833 , 811 + 822 ; r z: PI - iP2' al - iaz, 813 - i823 ,

811 - 822 + 2i812 ; r3: PI + ipz, a l + ia2, 813 + i823 , 8" - 822 - 2i812;

Table Cl.l0

Table C1.9

C3V·rl:P3,833·811 +822; r 2:a3; r3: (PI' pz), (a2 , -al)' (813 , 823), (28IZ, 811 - 822 );

D3 . r,: 833, 8" + 8Z2 ; r 2: P3, a3; r3: (pz, -PI)' (a2 , -al)' (813 , 8Z3 ), (2812, 8" - 822 );

C3' r , : ai' 833, 8" + 8Z2 ; r z: a l - iaz, 813 - i823, 811 - 822 + 2i812; r3: al + ia2, 811 + i821 , 811 - 822 - 2i812; r 4 : PI; rs: PI - iP2; r6: PI + iP2;

C3h ' r ,: a3, 833 , 811 + 8Z2 ; r z: PI - iP2, 811 - 822 + 2i812; r3: PI + ip2, 8" - 822 - 2i812; r 4 : P3; rs: a l - ia2, 813 - i823; r6: a l + ia2, 813 + i823;

C6' r , : P3' a3, 833 , 8" + 822 ; r 2: 8" - 822 + 2i812; r3: 8" - 822 - 2i812; rs: PI - ipz, al - ia2, 813 - i8z3 ; r6: PI + iP2, al + ia2, 813 + i823;

386 Appendix C. Integrity Bases of Crystallographic Groups

where

Table C1.11

D3h · rl: S33, Sll + S22; r2: a3; r3: P3; r5: (a l , a2), (S23, -S13); r6: (Pt> P2), (2S12 , Sll - S22);

D3v · r l : S33, Sll + S22; r2 : a3; r4 : P3; r5: (Pl, P2); r6: (a2, -ad, (S13' S23)' (2S12 , Sl1 - S22);

D6· r l : S33, Sll + S22; r2 : a3, P3; r5: (Pl' P2), (a l , a2), (S23, -S13); r6: (2S12 , Sl1 - S22);

C6v ·rl :P3,S33,Sl1 + S22; r2:a3; r 5:(Pl,P2),(a2, -al ), (S13' S23); r6: (2S12 , S11 - S22);

Table CI.12

C6h · rl: a3' S33, Sl1 + S22; r2: Sl1 - S22 + 2iS12; r3: Sl1 - S22 - 2iS12 ; r5: al - ia2, S13 - iS23 ; r6: al + ia2, S13 + iS23 ; r;: P3; r~: Pl - iP2; r~: Pl + iP2;

Table C1.13

T·rl : Sl1 + S22 + S33; r 2: Sll + W2S22 + WS33; r3: Sll + WS22 + W2S33; r 4 : (Pl' P2, P3), (a l , a2, a3), (S23' S13' S12);

Table CI.14

T". r l : Sll + S22 + S33; r2: Sll + W2S22 + WS33; r3: Sl1 + wS22 + W2S33 ; r 4 : (a l , a2, a3), (S23, S13, S12); f 4 : (Pl' P2' P3);

Table C1.15

7;,. rl: Sll + S22 + S33; r3: (Sl1 - S33' )3/3(2S22 - Sll - S33»; r4: (S23' S13, S12), (Pl' P2' P3); r5: (a l , a2, a3);

o· r l: Sll + S22 + S33; r3: (Sll - S33' )3/3(2S22 - Sll - S33)); r4: (S23' S13, S12); r5: (Pl, P2, P3), (a l , a2' a3);

Table C1.16

0h· r l: Sll + S22 + S33; r3: (Sll - S33' )3/3(2S22 - Sll - S33»; r4: (S23, S13, S12); r5: (a l , a2' a3); f5: (Pl' P2, P3);

1 .J3 W= --+ /-

2 2'


For each crystal class there is listed a table of the form

R\rx S1 S2 SN Basic quantities

r1 T1 1 T2 1 Tf t/I, 1//, ... r2 Ti Ti ~ a, b, ...

r, T1 , T2 , TN , A,B, ...

representing the unequivalent irreducible representations r 1, r 2 , •.. , rr of the crystallographic group {S}. Tables C2.1-C2.14 display these irreducible representations for various crystal classes. These classes are identified by name and also by listing their Hermann-Mauguin, Schonfiies, and Shubnikov symbols. The basic quantities that form the carrier spaces for irreducible representations r 1 , r 2 , ••. , rr are denoted by

IjI, ,",,', 1/1",",,"', ... ,

a, b, c, d, ... ,

A = [~:], B = [!:], The irreducible representations rr are either of degree one or two. Those of degree one are either real or complex numbers, and those of degree two are expressed in terms of the matrices E, A, ... , L, listed below

-2 A- [ 1

- -y'3/2

G = [-t y'3/2] , y'3/2 t H = [-Jt/2

L = [~ ~J

-2 B- [ 1

- y'3/2

A superposed bar indicates complex conjugate. The generic elements of the integrity basis are listed following Tables C2.1-C2.14 (from Kiral and Smith [1974] and Kiral [1972]).

Pedial class. No symmetry. Hence all independent components of vectors and tensors constitute basic quantities.


Pinaeoidal class, C1 , T, 2. Domatie class, cv , m, m. Sphenoidal class, C2> 2, 2

Table C2.t

C1 I C Cv I Rl Basic C2 I Dl quantities

r 1 1 a, at, ... r2 -1 b, b', ...

Application of Theorem D.6 (Appendix D) immediately yields the result that the typical multilinear elements of the integrity bases for C1 , C., and C2

are given by 1. a;

2. bb'. (C2.1)

Prismatic class, C2h , 21m, 2:m. Rhombic-pyramidal class, C2v , mm2, 2· m. Rhombie-disphenoidal class, D2, 222, 2:2

Table C2.2

C2h I Dl Rl C C2v I Dl R3 R2 Basic D2 I Dl D2 D3 quantities

r 1 1 1 a, at, ... r 2 1 -1 -1 b, b', ... r3 -1 -1 c, c', ... r 4 -1 -1 d, d', ...

The typical multilinear elements of the integrity bases for C2h , C2v, and D2 are given by

1. a; 2. bb', ee', dd'; (C2.2) 3. bed.

Rhombie-dipyramidal class, D2h, mmm, m' 2: m. Repeated application of Theorem D.6 yields the result that the typical multilinear elements of the integrity basis for D2h are given by

1. a; 2. bb', ee', dd', AA', BB', CC', DD'; 3. bcd, bAB, bCD, eAC, eBD, dAD, dBC;

(C2.3)

4. beBC, beAD, bdBD, bdAC, edCD, edAB, ABCD.


Table C2.3

Basic D2h DI D2 D3 C RI R2 R3 quantities

r l 1 1 1 a,a', ,., r 2 -1 -1 1 -1 -1 b, b', ...

r3 -1 -1 -1 1 -1 c, c', ... r 4 -1 -1 1 -1 -1 1 d,d', '" r~ 1 1 -1 -1 -1 -1 A,A', ... r 2 -1 -1 -1 -1 1 1 B,B', ... r; -1 1 -1 -1 -1 1 C,C', ... r~ -1 -1 1 -1 1 -1 D,D', ...

Tetragonal-disphenoidal class, C2 , 4, 4. Tetragonal-pyramidal class, C4 , 4, 4.

Table C2.4

C2 I D3 DI T3 D2T3 Basic C4 D3 RI T3 R2T3 quantities

r l 1 cp, ql, ... r 2 -1 -1 t/I,t/I', ... r3 -1 -i a, b, ... r 4 -1 -i a,Ii, ...

In Table C2.4, the quantities a, b, ... denote the complex conjugates of the quantities a, b, ... , respectively. The typical multilinear elements of the integrity basis for C2 and C4 are given by

1. cp; 2. ab, '1''1''; 3. ",ab;

(C2.4)

4. abed.

Note that the presence of the complex invariants ab, 'I'ab, abed in (C2.4) indicates that both the real and imaginary parts ab ± ab, 'I'ab ± 'I'ab, abed ± abed of ab, 'I'ab, abed are typical multilinear elements of the integrity basis.

Tetragonal-dipyramidal class, C4h , 4/m, 4: m.

Table C2.5

Basic C4h D3 RI T3 R2 T3 C R3 DI T3 D2T3 quantities

r l <p, <p', ... r 2 -1 -1 1 -1 -1 '1', '1", ... r3 -1 -i -1 -i a, b, ... r 4 -1 -i 1 -1 -i ii, Ii, ... r I -1 -1 -1 -1 ~, ~', ... r 2 -1 -1 -1 -1 1 'I, ,/', ... r 3 -1 -i -1 1 -i A,B, ... r~ -1 -i -1 -i A, ii, ...


In Table C2.5, the quantities a, b, ... , A, B, ... denote the complex conjugates of a, b, ... , A, B, ... , respectively. We find upon repeated application of Theorem D6 that the typical multilinear elements of the integrity basis for C4h are given by'

1. cp; 2. ab, AB, '1"1", ee', rJl'l'; 3. 'l'ab, 'I' AB, eaA, rJaA, 'l'erJ; (C2.5) 4. abcd, abAB, abAB, ABCD, 'l'eaA, 'l'rJaA, erJab, erJAB; 5. eaABC, eAabc, rJaABC, rJAabc.

The presence of the complex invariants ab, AB, ... , rJAabc in (C2.5) indicates that both the real and imaginary parts of these invariants are typical multilinear elements of the integrity basis.

Ditetragonal-pyramidal class, C4v, 4mm, 4· m. Tetragonal-trapezohedral class, D4, 422, 4: 2. Tetragonal-scalenohedral class, D2v , 42m, 4· m

Table C2.6

C4v I D4 I Dzv

r 1 r z r3 r 4 1 rs E

Rz Dl Dl

-1 -1

1 F

Rl Dz Dz

-1 -1

1 -F

1 -E

T3 RzT3 R1T3 D1T3 RzT3 RzT3 R1T3 CT3 Basic

T3 D1T3 D zT3 D1T3 quantities

1 cp, ql, ... -1 1 1 -1 """,', ...

1 -1 -1 v, v', ... -1 -1 -1 -1 't, r', .0. K L -L -K a, b, ...

Repeated application of Theorem D.6 yields the result that the typical multilinear elements of the integrity basis for C4v, D4, and D2v, are given by

1. cp; 2. a1 b1 + a2b2' '1''1'', vv', H'; 3. 'I'(a1b2 - a2bl), v(a1b2 + a2bl)' t(a1b1 - a2b2), 'l'n; 4. alblcldl + a2b2c2d2' 'l'v(a1b1 - a2b2), 'l't(alb2 + a2b1), (C2.6)

n(al b2 + a2 bd; 5. 'I'(a1b1c1d2 + a1b1d1c2 + alc1dlb2 + b1cldla2 - a2b2c2dl - a2b2d2c1

- a2c2d2bl - b2c2d2al)·


Ditetragonal-dipyramidal class, D 4h, 4/mmm, m' 4: m

Table C2.7

Basic D4h Dl D2 D3 CT3 RIT3 R2 T3 R3T3 quantities

r 1 1 tp, !p', ...

r 2 -1 -1 -1 1 -1 '¥,'¥', ... r3 -1 -1 1 -1 -1 1 V, v', ... r 4 1 -1 -1 -1 -1 't',r', ... rs E F -F -E -K -L L K a, b, ... r; 1 1 1 1 ~, ~', ... r 2 -1 -1 -1 1 -1 1'/,1'/', ... r~ -1 -1 -1 -1 1 0,0', ... r~ 1 1 -1 -1 -1 -1 y, y',. o. r' s E F -F -E -K -L L K A,B, ...

Basic D4h C Rl R2 R3 T3 DIT3 D2T3 D3 T3 quantities

r 1 cp, cp', ... r 2 -1 -1 -1 1 -1 '¥, '¥', ... r3 -1 -1 1 -1 -1 1 v, v', ... r 4 -1 -1 -1 -1 't,r',o ..

rs E F -F -E -K -L L K a, b, .0. r' 1 -1 -1 -1 -1 -1 -1 -1 -1 ~, ~', ... r 2 -1 -1 -1 -1 '1,1'1', ... r~ -1 -1 -1 -1 0,0', ... r' 4 -1 -1 -1 -1 1 1 1 y, y', ... r~ -E -F F E K L -L -K A,B, ...

Repeated application of Theorem D.6 yields the result that the typical multilinear elements of the integrity basis for D4h are given by

1. cp; 2. albl + a2b2, AlBl + A2B2, 'P'P', vv', n', ~~', 1',,7', ee', ')1')";

3. 'P(a l b2 - a2bl ), 'P(AlB2 - A2 Bd, v(a l b2 + a2bl), V(AlB2 + A 2Bd, r(albl - a2b2), r(AlBl - A2B2), ~(alAl + a2 A2), '1(a l A2 - a2 A l), e(a l A2 + a2Ad, y(alAl - a2A2), 'Pvr, 'Pey, 'P~'1, V'1y, r~y, r'1e, v~e;

4. alblcldl + a2b2c2d2, AlBl ClDl + A2B2C2D2, (a l b2 + a2bl )(Al B2 + A2Bl ), (a l b2 - a2bd(Al B2 + A 2Bd, (albl - a2 b2)(A l Bl - A2B2), ('Pv, ey, '1e )(al bl - a2b2), ('Pr, ee, '1y)(a l b2 + a2bl ), (n, e'1, (}y)(a l b2 - a2bd, ('Pv, ey, '1(})(A l Bl - A2B2), ('Pr, e(), '1y)(AlB2 + A2Bl ), (n, ~'1, (}y)(A l B2 - A2Bl ), ('JI'1, v(), ry)(alAl + a2 A2), ('P(), V'1, re)(alA l - a2A 2), ('Py, ve, r'1)(al A 2 + a2Al), ('Pe, vy, r(})(a l A2 - a2A l ), 'Pvey, 'PV'1(}, 'Pre(}, 'Pr'1Y, vre'1, n(}y, e'1(}Y;

5. 'P(a l bl cl d2 + al bl dl c2 + al cl dl b2 + bl cl dl a2 - a2b2c2dl - a2b2d2cl - a2c2d2bl - b2c2d2al),

'P(AlBl Cl D2 + AlBlDl C2 + Al Cl Dl B2 + Bl Cl Dl A2 - A2B1C1D1 - A1B1D1Cl - A1C1D1Bl - B1C2D1A l ),


'P(a l b2 + a2bd(A I BI - A 2B2), 'P(albl - a2b2)(A I B2 + A 2Bd, v(a l b2 - a2bd(AI BI - A 2B2), v(albl + a2b2)(A I B2 - A 2Bd, r(a l b2 - a2bd(AIB2 + A 2BI ), r(a l b2 + a2 bl)(A I B2 - A 2BI ), ~(alblcIAI + a2 b2c2A 2), ~(AIBI Clal + A 2B2C2a2), t7(a l bl cI A 2 - a2 b2c2A 2), t7(AIBI Cl a2 - A 2B2C2al ), 0(a l bl c l A 2 + a2 b2c2A d, O(AIBI Cl a2 + A 2B2C2al ), y(alblclA I - a2b2c2A 2), y(AIBI Clal - A 2B2C2a2),

(C2.7)

('P~O, 'Pt7y, v~t7, vOy)(a l bl - a2b2), (a l b2 + a2bd('P~y, 'Pt70, r~t7, rOy), (v~y, Vt70, r~O, rt7y)(a l b2 - a2bl), (AIBI - A2B2)('P~0, 'Pt7y, V~t7, vOy), ('P~y, 'Pt70, r~t7, rOy)(A I B2 + A 2Bd, (AIB2 - A2BI)(V~Y, Vt70, r~O, rt7y), ('Pvy, 'PrO, nt7, t70y)(a I AI + a2A 2), (alAI - a2A2)('Pt7r, 'Pv~, nO, ~t70), ('Pvt7, 'Pr~, vry, ~t7y)(aIA2 + a2AI)' (a 1 A2 - a2A I )('PvO, 'Pry, vr~, ~Oy);

6. ~t7(albl - a2b2)(AIB2 + A 2Bd, Oy(albl - a2 b2)(A I B2 + A 2BI ), 'P~(alblcIA2 - a2b2c2A d, 'P~(AIBI Cl a2 - A 2B2C2al ), 'Pt7(a l bl cI A I + a2b2c2A2)' 'Pt7(AIBI Clal + A 2B2C2a2), 'PO(alblcIA I - a2b2c2A2)' 'PO(AIBI Clal - A 2B2C2a2), 'Py(a l bl cI A 2 + a2b2c2A I)' 'Py(AIBI Cla2 + A 2B2C2al ), (~t7,Oy)(alblcld2 + al bl d l c2 + a l c l d l b2 + bl c l d l a2 - a2b2c2dl

- a2 b2d2cI - a2 c2d2bl - b2c2d2al), (~t7, Oy)(AIBI CI D2 + AIBIDI C2 + Al CI DI B2 + BI CI DI A 2

- A 2B2C2DI - A 2B2D2CI - A 2C2D2BI - B2C2D2Ad, (a l b2 - a2bd(AI BI CI D2 + AIBIDI C2 + Al CI DI B2 + BI CI DI A 2

- A 2B2C2D I - A 2B2D2CI - A 2C2D2BI - B2C2D2Ad,

(AIB2 - A2BI)(alblcld2 + al bl d l c2 + al c l d l b2 + bl cl d l a2 - a2 b2c2dl - a2b2d2cI - a2 c2d2bl - b2c2d2ad·

Trigonal-pyramidal class, C3 , 3, 3.

Table C2.8

Basic C3 I 8 1 82 quantities

r 1 <p, <p', .•• r 2 w w2 a,b, ... r3 w2 w £1,5, ...

The quantities Q) and Q)2 in Table C2.8 are defined by

Q) = -1/2 + ifi/2, Q)2 = -1/2 - ifi/2. (C2.8)

We note that Q)3 = 1 and that a, b, ... denote the complex conjugates of the quantities a, b, ... , respectively. The typical multilinear elements of the integrity basis for C3 are given by

1. cp; 2. ab; (C2.9) 3. abc.


The presence of the complex invariants ab and abc in (C2.9) indicates that both the real and imaginary parts ab ± ab and abc ± abc of these invariants are typical multilinear elements of the integrity basis.

Ditrigonal-pyramidal class, C3v, 3m, 3· m. Trigonal-trapezohedral class, D3, 32,3:2.

Table C2.9

C3v SI S2 Rl R1S1 R1S2 Basic D3 SI S2 Dl D 1S1 D1S2 quantities

r 1 1 1 cp, cp', ... r 2 1 1 1 -1 -1 -1 'P, 'P' r3 E A B -F -G -H a, b, ...

The typical multilinear elements of the integrity basis for C3v and D3 are given by

1. <p; 2. albl + a2b2, '11\1"; 3. a2b2c2 - a1 b1 c2 - b1c1 a2 - c1a 1b2, 'P(a 1b2 - a2bl);

(C2.10)

4. 'P(a 1b1 c1 - a2b2cl - b2c2al - C2a2bl)'

Rhombohedral class, (;3' 3, 6. Trigonal-dipyramidal class, C3h, 6, 3: m. Hexagonal-pyramidal class, C6 , 6, 6.

Table C2.10

C3 SI S2 C CS1 CS2 e3• I 8 1 82 R3 R3 8 1 R382 Basic C6 I SI S2 D3 D3S1 D3S2 quantities

r 1 qJ, ql, ... r 2 w w2 w w2 a, b, ... r3 w2 w 1 w2 w a,b, ... r 4 -1 -1 -1 ~,~', ... rs w w2 -1 -w _w2 A,B, ... r6 w2 w _WI _w2 -w A, ii, ...

The quantities wand w2 appearing in Table C2.l0 are defined by (C2.8). The quantities a, b, ... , A, ii, ... denote the complex conjugates of a, b, ... , A, B, ... , respectively. Let P be a polynomial function of the quantities <p, ... , a, a, b, b, ... , ~, ... , A, A, B, ii, ... which is invariant under the first three transformations of Table C2.10. Then it is seen from the results for the group C3 that P is expressible as a polynomial in the quantities obtained from the typical multilinear quantities

<p, ab, abc, Aii, aAB (C2.ll)


and ~, aX, abA, ABC. (C2.l2)

The quantities (C2.11) remain invariant under the final three transformations of Table C2.10, and the quantities (C2.12) all change sign under any of the last three transformations of Table C2.1O. With Theorem D.6 we then see that the typical multilinear elements of the integrity basis for C3 , C3h , and C6 are given by

1. <p;

2. ab, AB, ~~'; 3. abc, aAB, ~ocA; 4. abAB, ~abA, ~ABC;

(C2.13)

5. aABCD; 6. ABCDEF.

The presence of the complex invariants ab, AB, ... , ABCDEF in (C2.13) indicates that both the real and imaginary parts ab ± ab, AB ± AB, ... , ABCDEF ± ABCi5EF of these invariants are typical multilinear elements of the integrity basis.

Ditrigonal-dipyramidal class, D3h , 6m2, m' 3: m. Hexagonal-scalenohedral class, D3v , 3m, 6· m. Hexagonal-trapezohedral class, D6 , 622, 6:2. Dihexagonalpyramidal class, C6v , 6mm, 6· m.

Table C2.11

D3h SI S2 R3 R3S1 R3S2 D 3v SI S2 C CS1 CS2

D6 SI S2 D3 D3S1 D3S2 Basic C6v I SI S2 D3 D3S1 D3S2 quantities

r 1 1 tp, cp', ... r 2 1 '1', '1", ... r3 -1 -1 -1 ~, ~', ... r 4 1 1 -1 -1 -1 1'/,1'/', .•• rs E A B -E -A -B A,B, ... r6 E A B E A B a, b, ...

D3h Rl R1S1 R 1S2 D2 D2S1 D2 S2

D 3v Dl D 1S1 D1S2 Rl R 1S1 R1S2 D6 Dl D1S1 D1S2 D2 D2S1 D2S2 Basic C6v R2 R 2 S1 R2S2 Rl R 1S1 R 1S2 quantities

r 1 cp, cp', ... r 2 -1 -1 -1 -1 -1 -1 '1', '1", ... r3 1 -1 -1 -1 ~, ~', ... r 4 -1 -1 -1 1 1 1 1'/,1'/', ••• rs F G H -F -G -H A,B, ... r6 -F -G -H -F -G -H a, b, ...


We note that the basic quantities <p, 'P, ~, ", A, a associated with Table C2.11 transform under transformations 1,2, 3, 10, 11, 12 of Table C2.11 in the same manner as do the quantities <p, 'P, 'P', <p', a, b under the transformations of Table C2.9 associated with the crystal classes C3v and D3 • Let us employ the notation

B = BI - iB2 , •.• ,

b = bi - ib2 , ....

(C2.14)

With the notation (C2.14), we see from the results for the groups C3v and D3 that the typical multilinear elements of the integrity basis for polynomial functions of the basic quantities <p, <p', ..• , 'P, 'P', ... , ~, ~', ... , ", ,,', ... , A, B, ... , a, b, ... which are invariant under the group of transformations 1,2,3, to, 11, 12 of Table C2.11 are given by

and

<p, 'P'P', ~~', AB + AB, ab + ab, 'P(AB - AB),

'P(ab - ab), ~(aA - aA), abc - abc, aAB - aAB,

'P(abc + abc), 'P(aAB + aAB),

~(abA + abA), ~(ABC + ABC)

rJ, 'P~, aA + aA, ~(ab - ab), 'P(aA - aA), ~(AB - AB),

(C2.1S)

abA - abA, ABC - ABC, ~(abc + abc), 'P(abA + abA), (C2.16)

~(aAB + aAB), 'P(ABC + ABC).

The quantities (C2.1S) remain invariant under the remaining transformations of Table C2.11 whereas the quantities (C2.16) change sign under all of the remaining transformations of Table C2.11. Application of Theorem D.6 then yields the result upon elimination of the redundant terms that the typical multilinear elements of the integrity basis for D3v , D3h , D6 , and C6v are given by

1. <Pi 2. ab + ab, AB + AB, 'P'P', ~~', rJrJ'; 3. abc - abc, aAB - aAB,

'P(ab - ab), 'P(AB - AB), ~(aA - aA), rJ(aA + aA), 'P~rJ; 4. abAB + abAB, (ab - ab)(AB - AB), 'P(abc + abc),

'P(aAB + aAB), ~(abA + abA), ~(ABC + ABC), rJ(abA - abA), rJ(ABC - ABC), 'P~(aA + aA), 'PrJ(aA - aA), (C2.17) ~rJ(ab - ab), ~rJ(AB - AB);

S. (abc + abc)(AB - AB), aABCD - aABCD, 'P(abAB - abAB), 'P~(abA - abA}, 'P~(ABC - ABC), ~~+~,~~+~~~+~~~+~;


6. ABCDEF + XBci5ifp, 'P(aABCD + iiABCD); 7. 'P(ABCDEF - XBci5ifP).

We recall that the quantities A, A, ... , F, P, a, ii, ... , c, c appearing in (C2.17) are defined as in (C2.14)

Hexagonal-dipyramidal class, C6h , 6/m, 6:m.

Table C2.12

Basic C6h I S! S2 D3 D3S! D3S2 quantities

r! tp, q/, , .. r 2 00 002 00 002 a, b, ... r3 002 00 002 00 a,b, ... r 4 -1 -1 -1 ~, ~', ... rs 00 002 -1 -00 _002 A,B, ... r6 002 00 -1 _002 -00 A.B, ... r; 1 1t, 'Tt', ...

r-2 oo 002 00 002 X, Y, ... r-3 002 00 1 002 00 X,Y, ... r 4 -1 -1 -1 <5,<5', .•. rs 00 002 -1 -00 _002 x,y, ... r 6 002 00 -1 _002 -00 x,y, ...

Basic C6 • C CSt CS2 R3 R3S1 R3S2 quantities

r! tp, q/, , .. r 2 00 002 00 002 a, b, ... r3 002 00 002 00 a, b, ... r 4 -1 -1 -1 ~, ~', ... rs 00 002 -1 -00 _002 A,B, ... r6 1 002 00 -1 _002 -00 A.B, ... r! -1 -1 -1 -1 -1 -1 1t, n', ...

r 2 -1 -00 _002 -1 -00 _002 X, Y, ... r 3 -1 _002 -00 -1 _002 -00 X,Y, ... r 4 -1 -1 -1 <5,<5', ••. rs -1 -00 _002 00 002 X, y, .0'

r 6 -1 _002 -00 002 00 x,y, ...

The quantities co and co2 appearing in Table C2.l2 are defined by (C2.8). We note that the quantities cp, ~, n, ~ and a, A, X, x associated with Table C2.12 transform under the first three transformations of Table C2.8 in the same manner as do the quantities cp and a associated with Table C2.8 (crystal class C3 ) under the transformations of Table C2.8. We see from the results for the group C3 that the typical multilinear elements of the integrity basis for polynomial functions of the basic quantities cp, ~, n, ~, a, A, X, x, which are invariant under the first three transformations of Table C2.12


are given by

<p, e, n, (j, ab, aA, aX, ax, AB, AX, Ax,

XY, Xx, xy, abc, abA, abX, abx, ABC,

ABa, ABX, ABx, XYa, XY A, XYZ, XYx, xya,

xyA, xyX, xyz, aAX, aAx, aXx, AXx.

(C2.lS)

Under any of the remaining nine transformations of Table C2.12, certain of the quantities (C2.1S) remain invariant and the others change sign. Then, repeated application of Theorem D6 will yield the result that the typical multilinear elements of the integrity basis for the crystal class C6h are given by

1. <p; 2. ab, AB, XY, xy, nn', ee', (j(j'; 3. abc, ABa, XYa, xya, AXx, eaA, exX,

(jax, (jAX, nAx, ne(j; 4. abA~, abX~!!bxy, ~BXY, ABxy, XYxy,

aAXx, aAXx, aAXx, nabX, nABX, nXlZ, nxyX, naAx, eabA, eABC, eXYA, exyA, eaXx, (jabx, (jABx, (jXYx, (jxyz, (jaAX, neax, neAX, n(jaA, e(jxX, e(jAx;

5. aABCD, aABXY, aABxy, aXlZU, aXYxy, axyzu, ABCXx, ABCXx, XlZAx, XlZAx, abxAX, abxAX, abxAX, xyzAX, xyzAX, nabAx, nABaX, nxyaX, nXYAx, eabXx, eABXx, ~XY Aa, exyAa, (C2.l9) (jabAX, (jABax, (jXYax, (jxyAX, neabx, neABx, neXYx, nexyz, neaAX, n(jabA, n(jABC, n(jXY A, n(jxyA, n(jaXx, e(jabX, e(jABX, e(jXfZ, e(jxyX, e(jaAx;

6. ABCDEF, ABCDXY, ABXYZU, XYZUVw, ABCDxy, ABxyzu, xyzuvw, XYZUxy, XYxyzu, aAxXYZ, aAXxyz, aX xABC, naABCx, naxyzA, nABCDX, nABxyX, nxyzuX, eaXfZx, ~axyzX, ~XYZUA, ~XYxyA, exyzuA, (jaABCX, (jaxYZA, (jXYZU.x, (jXY ABx, (jABCDx;

7. ABCDEXx, XYZUVAx, xyzuvAX, nAB CD Ex, nABCxyz, nAxyzuv, eXYZUVx, ~XYZxyz, ~Xxyzuv, (jABCDEX, (jABCXfZ, (jAXYZUV.

The presence of the complex invariants ab, AB, ... , (jAXYZUV in (C2.l9) indicates that both the real and imaginary parts of these invariants are typical multilinear elements of the integrity basis.


Table C2.13

T DI D2 D3 DIMI MI

r l

r 2 0) 0)

r3 0)2 0)2

r 4 I DI D2 D3 DIMI MI

T D2MI D3 M I D2M2 D3 M 2 DIM2 M2

r l

r 2 0) 0) 0)2 0)2 0)2 0)2

r3 0)2 0)2 0) 0) 0) 0)

r 4 D2MI D3M I D2M2 D3 M 2 DIM2 M2

Diploidal class, T,., m3, 6/2 (Table follows from that of T, since T,. = T x S2)

Hextetrahedral class, ]d, 43m, 3/4 Gyroidal class, 0, 432, 3/4

Table C2.14

T.J E DI D2 D3 DIT2 DIT3 D2TI D2T3 D3 TI D3 T2 TI T2 T3 0 E DI D2 D3 RIT2 RIT3 R2TI R2 T3 R3 TI R3 T2 CT1 CT2 CT3

r l 1 1 1 1 1 1 1 1 1 1 1 1 1 r 2 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1

r3 I I I I F H G H G F G F H

r 4 I DI D2 D3 D\T2 D\T3 D2T, D2T3 D3T, D3 T2 T, T2 T3 r5 I DI D2 D3 RIT2 RIT3 R2T, R2 T3 R3 TI R3T2 CT, CT2 CT3

T.J D,T, D zT2 D3T3 MI M2 DIMI D2M2 D3 M I D2M2 D3 M , D3 M 2 0 RITI R2 T3 R3 T3 MI M2 DIMI DIM2 D2M, D2M2 D3 M , D3 M 2

r l 1 1 1 1 1 1 1 1 1 1 1 r 2 -1 -1 -1 1 1 1 1 1 1 1 1

r3 G F H B A B A B A B A

r 4 D,T\ D2T2 D3 T3 M, M2 DIMI D,M2 D2MI D2M2 D3M , D3 M 2 r5 R,T, R2 T3 R3T3 MI M2 DIM, D,M2 D2MI D2M2 D3 M I D3 M 2

Hexoctahedral class, Oh' m3m, 6/4 (Table follows from that of 0, since Oh = 0 x S2)

Basic quantities

l{!, l{!', .. . l{!,l{!', .. . t, r', ...

x, y, .,.

Basic quantities

l{!,l{!', .. . l{!,l{!', .. . r, r', .. .

x, y, .,.

Basic quantities

l{!, l{!', ... )" y', ...

(::)'(:J ... X, y, ... , , ...

Basic quantities

l{!, l{!', ... y, y', ...

GJ.GD- ... x,y, ... , " ..

APPENDIX D

Some Theorems on Symmetric Polynomial Functions

Here we give some basic theorems without proof (see Weyl [1946, pp. 36, 53, 276]), that provide a systematic method for constructing the integrity basis of polynomials from the typical multilinear elements. The abbreviation L xiYj' .. Zk is understood to denote the sum of quantities obtained by permuting the subscripts in the summant cyclically, e.g.,

LX1 = LX2 = LX3 == Xl + x 2 + x 3 ,

LX 1 Y2 = L X 2Y3 = L X 3Yl == X 1 Y2 + X2Y3 + X3Yl'

Theorem 1. A set of typical multilinear elements of the integrity basis for polynomials P(x\l), X~1), ••. , x~), x~»), which are invariant under interchange of subscripts 1 and 2 on x(1), X(2), ... , x(n), is formed by the quantities

(D1)

To obtain the multilinear elements we form n sets of quantities by substituting x(1), ... , x(n) for X in (D1)1 and n(n - 1)/2 quantities by substituting xli)

for X and x(j) for Y (i, j = 1,2, ... , n; i < j) in (D1h.

Theorem 2. A set of typical multilinear elements of the integrity basis for polynomials P(x\l), X~l), x~l), ... , x~), x~), x~»), which are invariant under all permutations of the subscripts 1, 2, and 3, is formed by the quantities

(D2)

Thus the multilinear elements consist of n sets of quantities obtained by substituting x(l), X(2), .•• , x(n) in (D2)1; the n(n - 1)/2 sets are obtained by substituting xli) for x and x(j) for Y (i, j = 1, 2, ... , n; i < j) in (D2h, and the n(n - 1)(n - 2)/2 quantities are obtained by substituting xli) for x, xU) for Y and X(k) for Z (i,j, k = 1,2, ... , n; i < j < k) in (D2h. For example, for n = 3, we have

i = 1,2,3,

400 Appendix D. Some Theorems on Symmetric Polynomial Functions

LX1Y1: X~1)X~2) + X~l)X(.]l + X~1)X~2), X\2)X\3) + X~2)X~3) + X~2)X~3), X~3)X\1) + X~3)X~1) + X~3)X~1),

~> 1 Y 1 Z 1: X\1)X\2)X\3) + X~1)X~2)X~3) + X~1)X~2)X~3).

Theorem 3. A set of typical multilinear elements of the integrity basis for polynomials P(X\l), X~l), X~l), ... , x~), x~), xW»), which are invariant under cyclic permutations of the subscripts 1, 2, and 3, is formed by the quantities

LXI

LX1(Y2 - Y3)'

LX1Y1(z2 - Z3)'

(D3)

Theorem 4. A set of multilinear elements of the integrity basis for polynomials P(x~1), x~1), X~l), ... , x\n), x~), xW), L 1, ... , Lm), which are invariant under all odd permutations of the subscripts 1,2, and 3 (i.e., (12), (13), (23)) on the xl 1), •.. , xln)

(i = 1, 2, 3), with simultaneous changes of sign of the quantities L 1, L 2 , •.• , Lm, i.e.,

P(xj1), xj1), Xk1), ... , xln), xJn), Xkn), L 1, ... , Lm) _ P( (1) (1) (1) (n) (n) (n) L L ) - Xi' Xk ,Xj , ... , Xi ,Xk ,Xj ,- 1,···, - m

_ P( (1) (1) (1) (n) (n) (n) L L ) - Xk , Xi ,Xj , ... , Xk ,Xi ,Xj ' 1,· .. , m

where i, j, k is any permutation of 1, 2, 3, is formed by the quantities:

(i) LiLj (i, j = 1, ... , m; i < j); (ii) the typical multilinear integrity basis in x~), xln) (i = 1, 2, 3) which are

invariant under all permutations of the subscripts 1, 2, and 3 (Theorem 2); (iii) LiMj (i = 1, ... , m; j = 1, 2, ... ) where Mj are given by

and

Theorem 5. An integrity basis for polynomials in the variables Xl' ... , XP'

11, ... , Iq, which are invariant under a group of transformations for which 11, ... , Iq are invariants, is formed by adjoining to the quantities 11, ... , Iq an integrity basis for polynomials in the variables Xl' ... , xp which are invariant under the same group of transformations.

Theorem 6. If P is a polynomial function of the complex quantities (Xl' ... , (Xn' P1' ... , Pm, which satisfy the relation P((X1, ... , (Xn; PI' ... , Pm) = P((X1' ... , (Xn; - P1' ... , - Pm), then P is expressible as a polynomial in the quantities

(Xi (i = 1,2, ... , n,

Piik (j, k = 1, ... , m), (D4)

where Pk is the complex conjugate of Pk'

Appendix D. Some Theorems on Symmetric Polynomial Functions 401

Theorem 7. A polynomial integrity basis in the n vectors x(r) = (x~), ... , x~») (r = 1, 2, ... , n) in n-dimensional space, which is invariant under all proper orthogonal transformations, is formed by the scalar products

(D5)

and the determinant

(D6)

where i, r, s = 1,2, ... , n.

APPENDIX E

Representations of Isotropic, Scalar, Vector, and Tensor Functions

The representations for isotropic, scalar, vector, and tensor-valued functions were studied by Wang [1-4] and Smith [5, 6] using different procedures. The results given by them were not identical. After the modifications discussed by Boehler [7] both representations are made identical. For ease of reference the results are reproduced here, where Wang's notation is used.

References

[1] Wang, e.e., Arch. Rational Mech. Anal., 33,249 (1969). [2] Wang, e.C., Arch. Rational Mech. Anal., 33, 268 (1969). [3] Wang, e.C., Arch. Rational Mech. Anal., 36,166 (1970). [4] Wang, e.e., Arch. Rational Mech. Anal., 43, 392 (1971). [5] Smith, G.F., Arch. Rational Mech. Anal., 36,161 (1970). [6] Smith, G.F., Internat. J. Engng. Sci. 19,899 (1971). [7] Boehler, J.P., Z. Angew. Math. Mech. 57, 323 (1977).

Table E.1. Complete and irreducible sets of invariants of symmetric tensors A, vectors v, and skew-symmetric tensors W.

I. Invariants depending on one variable

Variable

A v W

Invariants

tr A, tr A2, tr A3

v,v trW2

II. Invariants depending on two variables when I is assumed

Variables

AloA2 A,v A,W

Invariants

tr A1A2, tr A~A2' tr A1A~, tr A~A~ v·Av, v·A2v tr AW2, tr A2W2, tr A2W 2AW v1 ·v2

v·W2 v trW1W2

(continued)

Appendix E. Representations of Isotropic Tensor Functions 403

Table E.1 (continued)

III. Invariants depending on three variables when II is assumed

Variables

A" A2, A3 A , A 2 , V

A, v,, V2 A, W" W2

A" A2,W W" W2, W3 v,, V2 , W V, W,, W2

A,v,W,

Invariants

tr A,A2A3 v'A , A2v v,' Av2, v,' A2V2 tr AW, W2, tr AW, Wf, tr AWtW2 tr A,A2 W, tr AiA2 W, tr A, W2 A2 W, tr AIA~W, trW, W2W3 V, 'WV2, V, 'W2V2 v'W, W2v, v·wtw2v, v·w,Wfv v'AWv, v'A2Wv, v·WAW2v

IV. Invariants depending on four variables when III is assumed

Variables

A" A2 , v,, v2

A, v,, v2 , W V,, v2 , W" W2

Invariants

v, '(A , A2 - A2A, )V2 v, '(AW - WA)V2 v, . (W, W2 - W2 W, )V2

Table E.2. Generators for vector-valued isotropic functions

I. Generators depending on one variable

Variable Generator

V V

II. Generators depending on two variables

Variables

A,v W,V

Generators

Av, A2v Wv, W2v

III. Generators depending on three variables

Variables

A" A2, V

WI' W2, V A,v,W,

Generators

(A,A2 - A2A,)v (WI W2 - W2 W,)v (AW - WA)v

404 Appendix E. Representations of Isotropic Tensor Functions

Table E.3. Generators for symmetric tensor-valued isotropic functions.

I. Generator depending on no variable I

II. Generators depending on one variable

Variable

A v W

Generators

A,A 2

v®v W2

III. Generators depending on two variables

Variables

A"A2

A,v A,W

Generators

A,A2 + A2A" AiA2 + A2Ai, A,Ai + AiA, v®Av + Av® v, v ®A2v + A2y ® v AW - WA, WAW,A2W - WA2, WAW2 - W2AW V,®V2+ V2®V, Wv®Wv, v®Wv + Wv®v, WV®W2V + W2V®WV W, W2 + W2 W" W, wi - WiW" WfW2 - W2 Wf

IV. Generators depending on three variables

Variables Generators

A(v, ® V2 - V2 ® v,) - (v, ® V2 - V2 ® v,)A W(v, ® V2 - V2 ® v,) + (v, ® V2 - V2 ® v,)W

Table E.4. Generators for skew-symmetric tensor-valued isotropic functions.

I. Generator depending on one variable

Variable

W

Generator

W

II. Generators depending one two variables

A, v A,W W, V

Vi' V2

W"W2

Generators

A,A2 - A2A" AiA2 - A2Ai, A,Ai - A~A" A,A2Ai - AiA2A" A2A,A~ - AiA,A2

v®Av - Av®v, v®A2v - A2v®v,Av®A2v - A2V®Av AW + WA, AW2 - W2A v®Wv - Wv®v, V®W2V - W2V®V v, ® v2 - v2 ® v, W,W2 - W2W,

III. Generators depending on three variables

Variables

A" A2, A3 A"A2 , V

Generators

A,A2A3 + A2A3A, + A3A,A2 - A2A,A3 - A,A3A2 - A3 A2A, A,v®A2v - A2v®A,v + v®(A,A2 - A2A,)v - (A,A2 - A2A,)v® V A(v, ® v2 - v2 ® v,) + (v, ® v2 - v2 ® v,)A W(v, ® v2 - v2 ® v,) - (v, ® v2 - v2 ® v,)W

APPENDIX F

Maxwell's Equations in Various Systems of Units

Three formulations of Maxwell's equations in matter (all in Lorentz-Heaviside units). After Maugin [1978a, p. 17].

(B) Three-dimensional (C) Three-dimensional (A) Four-vector Galilean formulation in a formulation in a fixed

Equation covariant formulation' co-moving frame R,(x, t)t laboratory frame RG

Gauss ~ 2 V·!2' + ~ro·J(" = q V·D = q V·D = q,

c ~ I' I I • I I aD I

Ampere (V + c- 2'6').J(" - ~q, = ~ ,I VxJ("-~!!)J=~/ VxH-~-=~J c c c c c at c

1 I • I • laB

Faraday (V + c- 2'6').$ +~:J6 = 0 Vx$+~B=O V x E + ~ at = 0 c c

Conservation ~ 2 V':J6-~ro'$=O V·B=O V·B = 0

of magnetic c

flux

Conservation • q+V'/=O aq

q + V./· = 0 -+V·J=O of charge

at

Potentials 1 21ft

iJI=V.SJ!+-ro B=VxA B=VxA c

$ = -(V + c-2'6')1ft - ~(: SJ!)1 I [dA ] $= -VIft-~ -+(VA)'v c dt

lOA E= -VIft-~-;;-

c ct

* All four-vectors (boldface type) in formulation (A) are spatial. See Chapter 15, Vol. II. t In formulation (B): B = E + c-1v X B, :f = H - c-1v X D, f = J - qv, d/dt == a/at + V· v, * D == dD/dt - (D' V)v + D(V' v) where v is the three-velocity.

Mac

rosc

opic

Max

wel

l's e

quat

ions

and

Lor

entz

forc

e in

var

ious

sys

tem

s of

uni

ts. A

fter

Jac

kson

[19

62].

The

Hea

visi

de-L

oren

tz s

yste

m is

use

d th

roug

hout

the

book

in t

heor

etic

al c

onsi

dera

tion

s.

Lor

entz

fo

rce

per

Syst

em

6 0

Jlo

D,H

M

acro

scop

ic M

axw

ell's

equ

atio

ns

unit

cha

rge

Ele

ctro

stat

ic

c-2

D =

E +

4nP

aD

aB

(e

su)

(t2/-

2)

H =

c2B

-4n

M

V·D

= 4

nq.

V x

H=

4n

J +

-V

xE

+-=

O

V·B

=O

E

+v

xB

at

at

c-2

1 E

lect

rom

agne

tic

D =

"2E

+ 4

nP

aD

aB

(em

u)

(t2/-

2)

c V

·D =

4nq

. V

x H

=4

nJ +

-V

xE

+-=

O

V·B

=O

E

+v

xB

H

= B

-4n

M

at

at

Gau

ssia

n D

= E

+ 4

nP

4n

1 aD

1

aB

v V

·D =

4nq

. V

xH

=-J+

--

Vx

E+

--=

O

V·B

=0

E

+-x

B

H =

B-4

nM

c

c at

c

at

c

Hea

visi

de-

D=

E+

P

1(

aD)

1 aB

v

Lor

entz

H

=B

-M

V·D

= q

. V

xH

=-J+

-V

xE

+--=

O

V·B

=O

E

+-x

B

c at

c

at

c

107

4n x

10-

2

Rat

iona

lize

d 4n

c2

D =

60E

+ P

aD

aB

M

KS

V

·D =

q.

Vx

H=

J+

-V

xE

+-=

O

V'B

=O

E

+v

xB

1

at

at

(q2t2

m-1

/-2 )

(m

/q-2

) H

=-B

-M

Jl

o

Whe

re n

eces

sary

the

dim

ensi

ons

of q

uant

itie

s ar

e gi

ven

in p

aren

thes

es. T

he s

ymbo

l c s

tand

s fo

r th

e ve

loci

ty o

f lig

ht in

vac

uum

with

dim

ensi

on (l

it).

~ ~ o ~ p. >;;.

~

s::: e; ~ 00

·

tTl

.c

s:: e;. o· ~ S·

~ ... o· s:: ti

l

til

~ tt

8 til o -, [ .... ti

l

References

ABLOWICZ, M.J. and SEGUR, H. [1981]: Solitons and the Inverse Scattering Transform, SIAM, Philadelphia.

ABRAHAM, M. [1909]: Zur Elektrodynamik bewegter Korper, Rend. Circ. Mat. Palermo, 28, 1-28.

ABRAHAM, M. [1910]: Sull'elettrodinamica di Minkowski, Rend. Circ. Mat. Palermo, 30,33,46. See also Theorie der Electrizitat, Vol. II, Teubner, Leipzig, 1923, p. 300.

AGRANOYICH, V.M. and Ginzburg, V.L. [1984]: Crystal Optics with Spatial Dispersion and Excitons, Springer-Verlag, New York.

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Index

Abel integral equation 197 absorption 632

resonant 634 acceleration vector 12 additive functionals 613, 679, 769 adiabatic

exponent 512 magnetization 581

admissibility axiom 144 thermodynamic 616

aerosol flow, charged 573 aether 225 Alfven

velocity 343,516 waves 515

Alfvenic flow, super 526 alkali halides 304 alleviator (influence function) 142,614 Ampere's

equation 733 law 73,732

Amperian current loops 49 amplification 306

drift-type 490 analytic function 320 angular momentum 28 anisotropic

elastic solids 163, 174, 176 fluids, magnetohydrodynamics 550

anisotropy 108 energy 108

anomalous dispersion 636 skin effect 699

antiferromagnetic configuration 495 materials 102

antiferromagnetism 110

antiferromagnets elastic 493 linear 493 magnetoelastic waves in 496 magnon-phonon couplings 494

antiplane surface waves 260, 305 antisymmetric tensors 724 apparent viscosity 598 approximation, slowly varying

amplitude 181,231 area changes 10 atomic

model 634 theory of lattices 704

attenuation 646 factor 337

attraction, intermolecular 689 Avogadro's number 96 axial

c-tensor 169 four-vectors 724

axiom of admissibility 144 of causality 133 of continuity 5 of determinism 135 of equipresence 136 of material invariance 138 of memory 143 of neighborhood 141 of objectivity 136 of time reversal 48, 138

axioms of constitutive theory 133 axisymmetric oscillations of a tube 273

balance of energy-momentum 735-736 of four-momentum 737

12 Index

balance (continued) of moment of energy-momentum

735-736 of moment offour-momentum 737 of momentum 80

balance laws 437 in continuum physics 66 in electrodynamics 72 relativistically invariant 734 resume of 85, 129 surface 67, 73 volume 66, 73

Barkausenjumps 110 Barnett effect 101, 102 barycentric velocity 55 basic equations, resume of 308 Beltrami equations 208 Beltrami-Mitchell equations 313 Benard convection problem 561,608 Bernoulli equations 556, 588 Bernoulli's theorem 514 Bianchi identities 719 biaxial crystal 120 binormal 22 birefringence 288

acoustical transverse 303 (double refraction) 120 effect 302 optical transverse linear 303

birefringent medium 640 viscoelastic materials 661

constitutive equations 657 body loads 75 Bohr's magneton 104 Boltzmann's constant 94, 585 Bose-Einstein statistics 444 bound charge 3 boundary

conditions 74 layer effect 364

Brewster angle 126 bright solitons 235 Broglie's relation 112 Brownian motion 585 buoyancy, magnetic-fluid 610

canonical decomposition 723 differentiable projection 725 isomorphism 726

canonically conjugate momentum 49 capillarity effects 609

carrier space 148 Cauchy

deformation tensor 7 relation 460 stress tensor 77,305

Cauchy's polar decomposition theorem 8

causality axiom 133 relativistic 740

Cayley-Hamilton theorem 24,161 centro symmetric cubic crystals 461 change

offrame 15 of observer 15

character of a representation 148 characteristic length, internal 675 characteristics, theory of 213 charge 2,3

conservation of 732, 734 point 696 relaxation 554

charged cylinder 192 disk 196

chocking 525 cholesteric liquids 676 Christoffel symbol 719 circulation 23 Clausius-Duhem inequality 81,452,

679, 737 Clausius-Mossoti equation 96 cnoidal wave 608 Coleman's retardation theorem 619 co-moving frame 53,721 compact operator 689 compatibility conditions 11 complex-function technique 321 complex of electric charges 27 complex variables 320 compressible flow, one-dimensional

521 conducting polarized materials 634 conduction current 32 conductors 92

ferromagnetic 491 configuration

initial 279 present 280 reference 5, 270

congruence of worldlines 722 conservation

of charge 732,734 of electric charge 73

of energy 76, 80 of magnetic flux 73, 731 of mass 56

constant Boltzmann's 94, 585 Cotton-Mouton 123 Curie 104 dielectric 95 elastoelectric 251 electromechanical coupling 357 Hall 117 Kerr 122 magnetostrictive 462 Mouton 122 piezomagnetic 472 Planck's 28, 101, 139 stress-optical 124 Verdet 123

constant magnetization in moving ferrofluid 582

constants of Y.I.G. 462 constitutive

equations 128, 156, 165, 173,437, 440

birefringent viscoelastic materials 657

nonequilibrium 579 polynomial 629,631

function 143 theory, axioms of 133

contact loads 75 continua

nonlocal 676 relativistic kinematics of 725

continuity requirement 618 continuous

memory 630 materials, thermodynamics of 613

continuum electronic spin 445 lattice 445 mechanics, relativistic 725 micromorphic 676 micropolar 676 physics 2

controllable states 265 surface loads 239

controversy (about electromagnetic stress tensor) 65

convection current 27,32 convective-time

contravariant 728 covariant 749

Index 13

derivative 16, 18 conventional crystal classes 376 conversion, energy 590 co-rotational derivative 17, 446 correlation force 58 Cotton-Mouton constant 123 Couette flow 609, 665

magnetohydrodynamic 520 Coulomb

energy 44 force 561 interaction 197

couple acting on a composite particle 42 density 59

couple-stress theory 497 coupling parameter 472 covariant

convective-time derivative 749 derivative 720

creep compliance 668-669 optical 668 test, tensile 668

critical exponent 107 cross effects 115, 688 crossover regions 479 crucial experiment 225-226 crystal

centrosymmetric cubic 461 easy-axis 463 easy-plane 463 liquid 676 systems 374 uniaxial 463

crystallographic magnetic groups 379 point group 145, 149,373 pure rotation groups 374

Curie temperature 98, 445, 586 Curie's constant 104

law 104 Curie-Weiss law 91,98,106 current 32 curvilinear coordinates 201 cylinder, charged 192

d'Alembertian inertia couple 446 operator 52

damping of magnetoelastic waves 482 of the spin precession 465

14 Index

dark solitons 236 Debye screening 696 Debye-Ioss peaks 695 decomposition, canonical 723 deformation 4

gradient 5 homogeneous 662 rates 12 relativistic 726

density gradient 603 depolarization tensor 127 depth

penetration 644 skin 644

derivative co-rotational 16-17,446 Frechet 615 instantaneous 615 Lie 728 convective-time 16, 18, 728, 749 Jaumann 16-17 material 12

descent, steepest 646 determinism, axiom 135 deviatoric part 31 diamagnetic materials 379 dielectric

constant 95 fluid 506 materials 91 moduli, frequency-dependent 695 relaxation 306 susceptibility 92 tensor 118

dielectrics 92 elastic 159, 165, 239 nonlinear 218 nonmagnetizable 218 rigid 189, 218 transparent 224

difference history 614 differentiable projection, canonical 725 dilatational heat generation 581 dipolar liquids, 695 Dirac delta function, 34, 185, 690 Dirac's relativistic quantum mechanics,

29 Dirichlet problem 183 discontinuity

line 68 surface 20

disk, charged 196 dislocation 11, 71, 132,491 dispersion 632

anomalous 338, 636 infrared 677, 704 normal 636 polariton 707 relations 370

for intersurface waves 608 pure spin waves 114

dispersive piezoelectric waves 677, 703 displacement

gradient 10 vector 9

dissipation functional 616 dissipative process 464, 745 distribution function 53 domains 97 double

layer distribution 186 normal forces 453 refraction (birefringence) 120

dragging of light 224 drift-type amplification 490 drift velocity 514 dynamic

buckling 370 magnetoelastic stability 370

dynamo theory 502

easy-axis crystal 463 easy-plane crystal 463 echoes, magnetoelastic 492 Eddy current 698-699 effect

Einstein-de Haas 101,498 electro-elastic 278 electro-optical 290 electrostrictive 162, 268, 272 Ettingshausen 116, 163 exchange-strictive 460,470 Faraday 240,297,300,500 Faraday magnetoelastic 484 Hall 163,370 Kelvin 268, 349 Kerr 240, 653, 674

transverse 293 magneto-optical 278, 297 magnetoelectric 162, 166, 168 magnetostrictive 162,470 Nemst 163 Peltier 163, 746 photoelastic 278, 287 piezoelectric resonance 257 piezomagnetic 163, 166, 169 Pockels 240, 296, 653

Poynting 162,268,349,353 pyromagnetic 166 quadratic dissipative 491 Righi-Leduc 163 Seebeck 163 skin 126,699,713 Thompson 116,746 Voigt-Cotton-Mouton 240,297,

303 effective

charge current 52 density 52

dielectric constant 256 induction 451 Lorentz force 63 magnetic field 465

eiconal equation 238 eigenfunction expansion 190 Einstein tensor 719 Einstein-Cartan

manifold 748 theory of gravitation 752

Einstein-de Haas effect 498 Einstein's equations 719 elastic

body, ferromagnetic 444 dielectrics 159, 165,239,265 ferroliquid 609 ferromagnets 437 solids

electromagnetic 441 nonlocal electrodynamics of 675

elastoelectric constants 251 electric

charge 2,27 continuum 3

conduction 114, 354 current (convection current) 27 displacement-magnetic intensity

tensor 729 moment 29

of the nth order 31 polarizability 94 quadrupole moment 31 quadrupoles 88 scalar potential 52 stress tensor 554

electrical breakdown 573 conductivity 115

electroconvective vortices 561 electrodynamics of moving media,

crucial experiment 225

Index 15

electroelastic effect 278 electrogasdynamic energy converter

569 electrohydrodynamic

convection 560 flow 567 spraying techniques 551 stability 561

electrohydrodynamics (EHD) 79,551 electromagnetic

acceleration of ionized gases 502 composite particles 40 continua, memory-dependent 611 couple density 60 elastic solids 441 energy 65 energy-momentum tensor 738,747 field, definition 439 fluids 171,503,743

linear 178 nonlinear 177

force on a point particle 35 insulators, thermoelastic 741 interactions with matter 738 loads, definitions 439 momentum 47, 62, 739 optics 117 power 60 shock waves 217 stress tensor 47, 62, 63

controversy about 65 system 405 traction 65 viscous fluids 441 wave 694 waves

in isotropic viscoelastic materials 647

in memory-dependent solids 641 electromagnetic-spin wave 500 electromagneto-optical effect 121 electromechanical coupling constant

257 electromotive intensity 54 electron 28

conduction 114 free 674

motion 637 theory 715

electron-phonon spin amplifier 490 electronic

charge 3,27 polarization 92 pressure 134

16 Index

electronic (continued) spin 103

continuum 445 electrostatic

limit 636 system of units 405

electrostriction 99, 745 electrostrictive effect 162, 268, 272 elliptic equation 204 energy

conversion 590 integrals 543 method 537

energy-momentum balance of 735-736 stress tensor 81 tensor 719, 734, 737

entropy flux 79 inequality 76

entropy-flux four-vector 734 equation of telegraphy 125 equilibrium of a free surface 589 equipresence, axiom 136 Ettingshausen effect 116, 163 Euler strain tensor, relativistic 748 Euler-Cauchyequation 465 Euler-Cauchy-Stokes decomposition

12 Eulerian

coordinates 5 strain tensor 7

event, relativistic 717 exchange

energy 456 Heisenberg 460

integral 107 interaction 107

exchange-conducting branch 491 exchange-force tensor 450 exchange-modulus tensor 108 exchange-strictive

effects 460,470 energy 457

existence theorem 183, 506 external fields 40 extraordinary waves 120

fading memory 144,615,711 Faraday effect 122,240,297,300

in ferromagnetic insulators 500 magnetoelastic 484

Faraday's equations 734 Faraday's law 73 feedback stabilization 547 ferrimagnetic deformable bodies 492 ferrimagnetism 11 0-111 ferrimagnets 501 ferroelectric

crystals 96, 304 magnet 164 materials 79

ferroelectrics 97 ferrofluid

rotation 582 viscometer 582

ferrofluids 574 interfacial stability of 603 optical properties of 609

ferro hydrodynamic approximation 585 flow 591

ferro hydrodynamics 575, 589 ferroliquid, elastic 609 ferromagnetic

bodies, hard 444 conductors 491 crystals, viscoplasticity of 491 elastic body 444 film, elastic 501 fluid 575 insulators, Faraday effect in 500 materials 79

soft 287 seals 574 walls, vibrations of 492

ferromagnetism 102,104 weak 380

ferromagnets elastic 437 magnetoelastic waves in 472 magnetostriction in 501 soft 287

field equations for incremental fields 282

Finger strain tensor 8 finite deformation 5

strain theory 13 first principle of thermodynamics 78 fission reaction 502 flexural rigidity 362 flow

Couette 609, 665 Poiseuille 665, 674 torsional 665 stabilization by magnetic fluid 609

fluid 171 simple memory-independent 172 viscous 660

fluxion space 49 force density 56 Fourier transform 690, 712 Fourier's law 746 frame

change of 15 co-moving 53,721 proper 54, 721 rest 721

Frechet derivative (functional derivative) 615, 708

free charge 3

at interface 558 electron model 114 electrons 3 energy functional 681 interface equilibrium 557 minimum 618 motion, electron 637 surface, equilibrium 589

frequency dependence of dielectric tensor 255 generation, sum and difference 655 resonance 635

frequency-dependent dielectric moduli 695

Fresnel's dragging coefficient 225 ellipsoid 119 equation 119

fringe order 670 frozen-in field 503,513 functional

basis 155 derivative (Frechet derivative) 615 isothermal static continuation of

617 representation 613

functionals 132 additive 613, 679

g-factor 29 Galilean

frame 27 group 53 invariance of Maxwell's equations

47, 52 relativity 55 transformation 52, 82, 137

Index 17

Galilean-invariant electrodynamics 47 galvanomagnetic effect 116 GASH group 99 gauge transformation 34 Gauss's

equation 733 law 73,732

Gaussian system of units 405 general

relativity 716, 718 shock solutions 219

Gosiewski's theorem 17 gradient operator 6,41 gradient-dependent materials 142 granular materials 71 gravitational potential 719 Green

deformation tensor 7 strain tensor, relativistic 748

Green-Gauss theorem 20 generalized form of 20

Green's function 186 theorem 184

group Galilean 53 index of 374 integrity basis of 384 irreducible representation of 147,

387 isotropy 139 Lorentz 741 magnetic 381

point 152, 382 space 140

orthogonal 154 point 139, 373 proper

orthogonal 139 point 373 rotation 373

Shubnikov's 140 symmetry 168 velocity 646

group crystallographic magnetic 379 point 145,149,373 pure rotation 374

gyration vector 713 gyromagnetic

effects 101 ratio 28, 501

gyroscopic couple 446

18 Index

gyroscopic (continued) nature of spin density 445 thermodynamically hidden effect

117

Hall constant 117 current 355-356 effect 116,163,370

memory-dependent 639 Hamiltonian form 45 hard

ferromagnet 316 ferromagnetic bodies 444 polarizer 207,219

harmonic function 190,210,548 Harnack theorem 190 Hartmann

electric number 568 number 518

heat conduction 163,334,498 flux vector 78 generation, dilatational 581 source 78

heated ferrofluid, stagnation point flow of 599

Heaviside unit function 339 Heaviside-Lorentz units 48 Heisenberg

exchange energy 460 model 91, 107

Helmholtz equation 55 free energy 81 solenoids 584

Helmholtz-Zorawski Criterion 25 hereditary process 491 Herglotz-Born rigid-body motion 728 Hermann-Mauguin symbols 387 high-frequency limit 637 Hilbert space 144,614, 707 history

difference 614 past 614 of the states 133

hodograph plane 208 transformation 209, 213

homogeneous deformation 239, 265, 662 polar material 94 strain in a magnet 343

hoop stress 323

Hugoniot condition 218,516, 748 hyperbolic equation 204 hyperplane, three-dimensional 722 hypersound generator 444 hyperstress 453 hysteresis

curve 97, 110 magnetic 574

image point 188 improper rotation 373 incompressible solids 346 index 747

of a group 374 of refraction 125, 219 refractive 481, 713, 751

induced optical anisotropy 120 inducement of optical anisotropy 121 inequality, Clausius-Duhem 81,452,

679, 737 inertial frame 720 inextensible strings 141 infinitesimal

perturbations, stability with respect to 539

rotation 10 strain theory 10 strains 9,460

influence function (alleviator) 142, 614 infrared dispersion 677, 704 initial configuration 279 instabilities of resistive type 502 instability

kink 547 labyrinthine 608 necking 546 sausage 546

instantaneous derivative 615 integral series, Volterra-type 715 integrity

basis 145,399 of crystallographic groups 384

interactions model of 444 spin-lattice model of 446

interface, free charges at 558 interfacial stability of ferrofluids 603 intermolecular

attraction 689 forces 71

internal characteristic length 675 field 40

constant 93

intersurface waves, dispersion relation of 608

intra-atomic contribution 44 field 40

intrinsic spin 748 invariance

requirements 82 under time reversal 82

invariants 9,402 inverse

deformation gradient 6 motion 5 optical creep function 671 scattering technique 234, 238 solution 329

inversion theorem 690 ion, migration of 573 ion-drag

anemometer 573 configuration 567

ionic crystals 95, 304 polarizability 94 polarization 92

ionized gases 502 ionosphere 630 irreducible representations 147,387 irrotational motion 4, 14 Irving-Kirkwood approximation 58 isochromatic lines 670 isomorphism, canonical 726 isothermal static continuation of a

functional 617 isotropic

elastic solids 159, 175 electromagnetic (nonlinear) 175

functions 154,402 materials 154, 170, 286 solids 241 viscoelastic materials, electromagnetic

waves in 647 isotropy group 139

Jacobi polynomial 196 Jacobian 5 Jaumann derivative 16-17 jet, magnetic fluid 610 jump

conditions 85,217,283,438,504 discontinuities 65

jumps, Barkausen 110

Kelvin effect 268, 349 force 42

Index 19

Kelvin's circulation theorem 515 Kelvin-Voigt viscoelastic solids 613 Kerr

coefficient 233 constant 122 effect 121,240,653,674

Killing's theorem 14 kinematics

ofline 19 of material cohtinua 1 of surface 19 of volume integrals 19

kink instability 547 Kleinman symmetry 655 Korteweg-Helmholtz force 554

labyrinthine instability 608 Lagrangian

coordinates 5 strain tensor 7

relativistic 727 Lame potentials 314 Landau-Lifshitz damping 466 Laplace transform 341,668 Laplace's equation 188,190,316 Larmor spin precession 451 Larmor's

precession 497 theorem 639

laser 656 technique 124

lattice continuum 445 model 304 vibrations 704

lattices, atom theory of 704 law

of balance of moment of momentum 77

of balance of momentum 77 of conservation of energy 78 of conservation of mass 77 of entropy 78

laws of balance, resume 129 Legendre

polynomial 371, 536 transformation 297, 749-750

Lie derivative 728 limit

electrostatic 636

110 Index

limit (continued) high-frequency 637 low-frequency 636

linear constitutive equations 165, 173 dielectrics 96 elastic antiferromagnets 493 functional, continuous 615 integral-operator technique 214 isotropic materials 627 momentum 27 pinch 532, 545 theory of piezoelectricity 242

linearized Eulerian strain tensor 10 liquid

cholesteric 676 crystals 676

local balance laws 73 continuum theory 71 field 93 magnetic induction 447 media, memory-dependent 441

localization 67, 70 process 76 residual 70

London's equation 701 long-range forces 44 long-wave mhgnons 113 Lorentz

condition 34 force 42, 405, 634

relativistically invariant 738 gauge condition 52, 751 group 741 invariance 739 local field 96 number 115 theory of electrons 36 transformations 720, 730

Lorentz-Heaviside system of units 95 Lorentzian signature 717 Love-Kirchhoff displacement field

359 low-frequency

limit 636 region 333

Lundquist equations 513,525

Mach number 522, 525 macroscopic

densities 55 electromagnetic theory 47

magnetic 2"-pole moment 33 anisotropy 456 behavior, nonlinear 574 dipole 33 domain 105 field

effective 465 magnetocrystalline 456

fluid flow stabilization by 609 jet 610

flux conservation of 731 tensor 729

microscopic 738 force 471 groups 381 hysteresis 574 induction, critical value 369 materials 100,380 moment 28, 32 monopole 49 point group 150,382 relaxation 583 solids, rigid 696 space group 140 spin

gyroscopic nature of 497 relaxation of 466

star, equilibrium of 533 stress tensor 318 sublattices 110 surface 547 susceptibility 100, 581

high-frequency 498 symmetry 139 two-phase flow 610

magnetic-fluid buoyancy 610 magnetically

hard material 109 saturated material 105

magnetism origin of 100 types of 102

magnetization current 52 four-vector 730 sublattice 492 vector 51

magnetized fluids, weakly 521 magnetoacoustic resonance 485 magnetocrystalline

energy 108 magnetic field 456

magnetoelastic analogue of geometrical optics 371 buckling 367 echoes 492 Faraday effect 484 resonance 473,479 waves 338

damping of 482 in antiferromagnets 496 in ferromagnets 472 in random media 338 instability of 491 surface 492

magnetoelasticity 159,307 two-dimensional 319

magnetoelectric coupling 164 effect 151, 162, 166, 168

magneto hydrodynamic approximation 507 channel generator 522 Couette flow 520 flow 518 Poiseuille flow 518 shock waves, oblique 530 simple waves 550 stability 537

magneto hydrodynamics 79,502 of anisotropic fluids 550 Bernoulli's equation 514 Kelvin's circulation theorem 515 perfect 503,512

relativistic 746 shock waves 525

magneto-optical effect 278, 297 magneto-strictive energy 457 magneto-thermoelasticity 329 magnetospheric propagation 640 magnetostriction 114,358,493, 745

constants 462 in ferromagnets 501

magnetostrictive effect 162, 470 transducers 444

magnon-phonon conversion 481

temporal 487 coupling 487

in antiferromagnets 494 interaction 472

magnons 111 mass 2

centroid 29 density 2, 55 measure 3

material continuum 3

in space-time 726 coordinates 5 derivative 12 frame indifference 15 functions 711 invariance, axiom 138 manifold 11 nonferrous 308 paramagnetic 379 stability condition 312 surface 69 symmetry 686 volume 4

Index 111

material-frame independence matrix representation 147 Matthiesen's rule 115 Maxwell-Lorentz

equations 33 theory 26

Maxwell stress tensor 48, 63 Maxwell's equations 26, 50, 438, 469,

504,509,552 covariant formulation 729 for the microscopic fields 38 four-vector formulation 733 in matter 405 in various systems of units 406 integral formulation 731

mean correlation function 58 curvature 22 field 106 life-time 114

mean value, theorem 67, 190 mechanical

balance equations 438 surface traction 86, 439

memory axiom 143 continuous 630 of strains 660

memory-dependent electromagnetic continua 611 Hall effects 639 local media 441 media, nonconducting 645 solids, electromagnetic waves in

641 metal-forming technology 329 MHD (magnetohydrodynamics) 502 MHD turbulence 502 micro-continuum 3

112 Index

micro magnetics 105 micromagnetism 105 micromorphic

continuum 71, 305 theory 676

theory 71 micropolar

continuum 676 theory 71

microscopic charge density 38 current density 38 electric polarization density 39 electromagnetic

fields 33 theory 26

magnetic polarization density 39 Maxwell's equations 36 model 458 reversibility 83 time reversal 140

axiom 620, 626 Miller's

coefficients 655 rule 655

minimal gravitational coupling 736 integrity basis 145

Minkowskian, space-time 718 mixed boundary-value problem 195 model of interactions 444 molecular field 105 molecule 93 moment

of energy-momentum, balance of 735-736

of momentum 80 momentum 62 motion 4

irrotational 4, 14 rigid"body 4, 14

Mouton constant 122 moving

discontinuity surface 68 ferrofluid, constant magnetization in

582 rigid dielectrics 224

nabla notation 6 Nanson formulas 11 nature of electromagnetic solids 158 Navier's equation 315 necking instability 546

neighborhood, axiom 141 Nernst effect 116, 163 Neumann problem 183 neutron

scattering 305 star 719

Newton's gravitational constant 537, 719

Newtonian chronology 84 nonequilibrium constitutive equations

162,579 nonferrous materials 308 nonlinear

atomic models 652 dielectrics 218 elastic dielectrics 277 electromagnetic waves 213 magnetic behavior 574 magnetization law 579 optics 230, 707 pulse propagation 233 Schrodinger equation 234 theory of rigid dielectrics 203 wave propagation 747

in magnetoelasticity 371 waves 277,370

nonlocal continua 676 field 70 media 440 moduli 693

nature of 688 rigid solids 693 theory 72

nonlocality 675 short 676

nonmagnetizable dielectrics 218 materials 141

nonpolar molecules 94 nonpolarizable materials 141 normal

dispersion 636 form 204

nuclear spin resonance 19

objective time rates 16 objectivity 15

axiom 136 principle of 741

observer, change of 15 Ohm effect 116 Ohm's law 115,634

Onsager principle 167,621 reciprocity 468

relations 116 symmetry 660

operator, compact 689 optic modes 695, 706 optical

activity, normal 712 anisotropy, induced 120 creep

function, inverse 671 modulus 668

indicatrix 119 properties of ferrofluids 609 rectification 656 relaxation moduli 667 transverse linear birefringence 303

optically isotropic body 118 optics

nonlinear 707 surface nonlinear 715

orbital motion 28 ordinary wave 120 orientational

polarizability 94 polarization 92

oscillations, radial 274

parabolic equation 204 paraelectric bodies 96 paramagnetic material 379 paramagnetism 103 parametric excitations 370 parity (in nuclear physics) 83 particle 4 past history 614 Peltier effect 116, 163,746 penetration depth 644 perfect

magnon gas 113 relativistic magnetohydrodynamic

scheme 747 permittivity 95 permutation symbol 6 perovskite structure 99 phase

function 646 space 49 transition 98 velocity 336

phonon-magnon coupling 114,444 photoelastic

Index 113

effect 278, 287 dependence on rotation 290

technique 124 photoelasticity 121, 124 photon-phonon interaction 706 photo viscoelasticity 666 physical

doublet 29 theory of dielectrics 93

piezoelectric moduli 246 powder 305 Rayleigh mode 305 resonance

effect 257 region 255

semiconductor 305 state of quiescent past 244 waves 702

dispersive 677 piezoelectrically

excited thickness vibration 253 generated electric field 263 stiffened stiffness tensor 254, 703

piezoelectricity 99, 166 piezomagnetic

constant 472 effect 163, 166, 169 energy 456

piezomagnetism 114 Piola strain tensor 8, 726 planar forces 361 Planck's constant 28, 101, 139 plane

electromagnetic waves in isotropic bodies 198

harmonic waves 299,474 wave 118

plasma frequency 640 physics 503

plasticity 491 Pockels effect 240, 296, 653 point

charge 696 group 139, 373

Poiseuille flow 665, 674 magnetohydrodynamic 518

Poisson's integral formula 189 ratio in tensile creep 669

polar decomposition 88 dielectric liquids 566

114 Index

polar (continued) molecules 94

polariton dispersion 707 polarizability 93 polarization

catastrophe 125 current 52 four-vector 730 vector 51

polarization-magnetization tensor 730 polarized matter, conducting 634 polarizer

hard 207,219 soft 207,219

polyelectrolyte 551, 573 polynomial

constitutive equations 629,631 functions 145

Volterra 631 polytropic gas 511 ponderomotive

force 49, 508 four-force 738

postulate of localization 71 potential

energy 538 in half-plane 193 theory 183

power of electromagnetic forces 43 Poynting

effect 162,268,349,353 four-vector 739 vector 47,62,201

Prandtl number 600 precessional velocity vector 446 present configuration 280 principal

axes of strain 8 section 120 stretches 8

principle of objectivity 741 of virtual power 84, 498

projection operator 289 propagation, magnetospheric 640 propagation of plane waves 329 proper

density 727 frame 54, 721 orthogonal group 139 point group 373 rotation group 373 time 721

properties of electromagnetic continua 91

pseudo-Euclidean space-time 718 pseudostress 25 pumping 487 pure Galilean transformations 82 pyroelectricity 100, 166 pyromagnetic

coefficient 587 effect 166 modulus 581

quadratic dissipative effect 491 memory dependence 621

quantum electrodynamics 29

radial motion 326 oscillation 274

radially symmetric vibration 261 radiation heat flux 81 Raman

scattering, stimulated 715 spectroscopy 305

rate of deformation tensor 13 of rotation, relativistic 727 of rotation tensor 13 of strain, relativistic 727

rate-dependent materials 143,659 rationalized MKS system of units 405 ray vector 120 Rayleigh

dissipation function 466 line diagram 530 number 564

reciprocity, Onsager 468 rectification, optical 656 reference

configuration 5, 278 state 4

reflection of electromagnetic waves 125 reflectivity 125 refraction tensor 666 refractive index 119,225,481, 713, 751 Reissner-Nordstmm solution 719 relati vistic

causality 740 continuum mechanics 725 deformation gradient 726 electrodynamics of continua 716

electromagnetic continua with intrinsic spin 752

heat conduction law 746 kinematics of continua 725 Lagrangian strain tensor 727 perfect magnetohydrodynamics 746 rate

of rotation 727 of strain 727

stress tensor 736 relativity

general 716, 718 rigid body in 728 special 716, 718

relaxation of magnetic spin 466 optical 667 property 618 time 115

representation character of 148 theorem 185

resonance 473 condition 478 frequency 256,259,262,263,635 magnetoacoustic 485 magnetoelastic 473

resonant absorption 634 response functionals 143,614 rest

frame 721 mass 28

resume of balance laws 85, 129 of basic equations 308

retardation theorem, Coleman's 619 Reynold's number 511

magnetic 511 Ricci tensor 719 Ricci's lemma 748 Riemann-Christoffel curvature 11 Riemannian

curvature 719 manifold 718 metric 717 norm 721 space-time 719

Righi-Leduc effect 116,163 rigid

body 630 in relativity 728 motions 4,14

dielectrics 189, 218 displacement 7

Index 115

electromagnetic solids 164 magnetic solids 696 materials 158 solids 14, 158

nonlocal 693 rigid-body motion, Herglotz-Born 728 Rivlin-Ericksen tensors 659 Robin problem 183 Rochelle salt 98 rotation 8

of a ferrofluid 582 of a rigid dielectric 226 rate of 13, 727 tensor 8

rotatory inertia 363

sampling function 690 saturated ferromagnetic elastic

insulators 453 sausage instability 546 scalar invariants 730 Schaudertheory 205 Schonflies symbols 387 Schrodinger, nonlinear equation 234 Schwarzschild solution 719 seals, ferromagnetic 574 second principle of thermodynamics

78 Seebeck effect 116, 163 semiconductor 92 Serret-Frenet triad 532 shear

simple 347 vertical 361

shifter 9 shock 516

compression 306 fast 527 generating function 530 slow 527 structure 530 super-Alfvenic 549 switch-off 528 switch-on 528 transverse 528 waves 277, 306

electromagnetic 217 in magnetohydrodynamics 525 magnetohydrodynamic oblique

530 shocks in soft ferroelectrics 222 short-range forces 45 Shubnikov symbols 387

116 Index

Shubnikov's group 140 simple

extension 345 material 142,437 memory-independent fluid 172 shear 29,347

of viscous ferrofluid 596 solids 142

single layer surface distribution 186 skin

depth 644 effect 126

anomalous 699, 713 slowly varying amplitude

approximation 81,231 smooth memory, axiom of 62 soft

ferromagnetic material 109,287 ferromagnets 498 polarizer 207, 219

solids, weakly magnetizable 314 solitary waves 233, 501, 610 soliton 234, 305, 608

bright 236 dark 236

sound, speed of 522 source, four-force 734 source flow, two-dimensional 593 space and time decomposition 722

canonical 738 space-time 717

Minkowskian 718 pseudo-Euclidean 718 Riemannian 719

spatial coordinates 5 four-vector 722 frame 4 isotropy 137

special relativity 716, 718 specular reflection 714 spherical

polar coordinates 202 waves 201

spherically symmetric vibrations 263 spin 3,28

amplifier, electron-phonon 490 boundary condition 453 density, gyroscopic nature of 445 electronic 103 lattice relaxation 452 precession, damping of 465 symmetry 380 system 102

tensor 736 wave 112

band 488 damping of 500 modes 495

spin-elastic surface waves 490 spin-lattice model of interactions 446 spin-orbit interaction 108 spin-precession equation 465,499, 501 spin-spin interaction 450 spinning continua 737 spontaneous magnetization 104 stability

criterion 565 electrohydrodynamic 561 interfacial (ferrofluids) 603 magnetoelastic magnetohydrodynamic 537 with respect to infinitesimal

perturbations 539 stabilization, feedback 547 stagnation-point flow 598

of heated ferrofluid 599 star

magnetic 533 neutron 719

state biased state 278 equation 511

static dielectric constant 95 magnetoelastic field 314

stationary phase method 646 statistical

average 48,55,103,738 distribution function 49 mechanics 49

steepest descent 646 Stephan-Boltzmann law 81 stiffness tensor, piezoelectrically stiffened

703 Stokes'theorem 21 Stokes-Helmholtz resolution 314 Stone-Weierstrass theorem 620, 708 strain

ellipsoid of Cauchy 9 measure 6 tensor 725

streaming birefringence 121, 124 stress

concentration 321 tensor

Cauchy 305 relativistic 736

stress-function technique 319 stress-optical constant 124 stretch tensor 8 subbodies 132 sublattice, magnetization 492 submicroscopic faults 131 super-Alfvenic

flow 526 shock 549

superconductivity 699 high-temperature 699

superconductor 699 superexchange 110 surface

balance law 67, 73 exchange contact force 447 gradient 22 nonlinear optics 715 physics 687, 603 tension 603 waves 259,371

magnetoelastic 492 spin-elastic 490

susceptibility tensor, high frequency 499

suspensions 71 switch-off shock 221 switch-on shock 221 symmetric polynomial functions 399 symmetry

breaking 287 group 168 Kleinman 655 material 139,686 operator 140

tangential discontinuity 529 Taylor number 584 Taylor's experiment 558 temperature, absolute 79 tensile creep test 668 tetrahedron 67 theorem

(Cauchy's decomposition) 8 (Cayley-Hamilton) 24, 161 (Coleman's retardation) 619 existence 183, 506 (Harnack) 190 (mean value) 190 representation 185 transport 19 uniqueness 184,204,243 (Weierstrass) 190

theory of characteristics 213 of electrons 26

Index 117

of magnetoelastic plates 359 thermal

convection 561 shock 76

thermodynamic admissibility 616, 742

thermodynamics of materials with continuous memory 613

thermoelastic electromagnetic insulators 741

thermomagnetic effect 116 thermomechanical balance laws 75,85 thermonuclear fusion experiments 502 8-pinch 532 thickness vibrations 491 Thompson effect 116,746 three-dimensional hyperplane 722 time antisymmetric tensors (c-tensors)

152 time reversal 83

axiom 138 microscopic 626 operator 140

time reversibility, microscopic 620 time symmetric tensors (i-tensors)

152 timelike coordinate 718 timelikeness 721 toroidal pinch 533 torsion 748

of a cylindrical magnet 349 torsional flow 665 traction, mechanical surface 439 translation symmetry 140 transparency, ultraviolet 637 transparent dielectric 224 transport theorems 19 transverse

isotropy 155 Kerr effect 293

true i-tensor 169 two-dimensional

magnetoelasticity 319 nonlinear problem 207 problems for special dielectrics 209 source flow 593

two-phase flow, magnetic 610 two-point

correlation function 57 probability density 57

types of magnetism 102

118 Index

ultraviolet transparency 637 uniaxial crystal 120, 463 uniformly magnetized sphere 316 unipolar

induction 229 injection 562

uniqueness theorem 184,204,243

Van Leeuwen's theorem 101 variational formulation 491 vector

group velocity 120 precessional velocity 446

velocity group 646 oflight in vacuum 28, 199 vector 12

Verdet constant 123 vertical shear 361 vibrations

extensional 257 lattice 704

virtual power principle 498 viscoelastic

materials, wave propagation 661 solids, Kelvin-Voigt 613

viscometric flow 664 viscoplasticity of ferromagnetic crystals

491 viscosity, apparent 598 viscous ferrofluid, simple shear 596 viscous fluid 660

electromagnetic 441 Voigt-Cotton-Mouton effect 122,

240,297,303 Voigt notation 247 Voigt's piezoelectricity 79 Volterra polynomial 631 Volterra-type multiple integral series

715 volume

balance laws 66, 73 changes 10

vorticity 13 four-vector 727 generation 556

walls 105 wave

conjugation 715

propag~tion, nonlinear 747 vector 119

wave-vector surface 19 waves

Alfven 515 antiplane 305

surface 260 cnoidal 608 electromagnetic 647,694

shock 217 spin 500

induced by thermal shock 338 intersurface 608 magnetoelastic 338,472,482 magnetohydrodynamic 550 nonlinear 277,370 piezoelectric 677, 702-703 plane

harmonic 299,474 propagation 329

shock 277,306 solitary 233, 501, 610 spherical 201 spin 114 surface 259,371

weak ferromagnetism 380 nonlocality 304

weakly anisotropic material 475 magnetizable solids 314

Weidemann-Franz law 115 Weierstrass theorem 190 whistlers 640 Wilson experiment 227 Wilson's equation 334 W.K.B.J. solution 489 work-hardening 491 world velocity 721 worldline 721 worldlines, congruence of 722

Young Tableaux 147 yttrium-iron-garnet (Y.I.G.) 461

Z-pinch 532 Zeeman effect 46 zero-mobility limit 571

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