Thus the methods of IR spectroscopy and EPR revealed a characteristic feature of the coordination transition [BO3] ~ [BO4] in calcium-borosilicate glasses as compared with the sodium-borosilicate glasses: even with a significant excess of CaO over B203 the complete transformation of [BO~] into [BO4] does not occur. LITERATURE CITED i.

2. 3. 4. 5.

Zh. S. Tizhovka, "Synthesis and study of zirconium-containing glasses for preparing matted glazes," Author's Abstract of Candidate's Dissertation, Physical-Mathematical Sciences, Minsk (1977). N. M. Bobkova, Zh. S. Tizhovka, V. V. Tizhovka, and K. G. Chergida, Zh. Prikl. Spektrosk., 31, No. 6, 1075-1078 (1979). I. I. Plyusnina, Infrared Spectra of Silicates [in Russian], Moscow (1967). V. A. Kolesova, Izv. Akad. Nauk SSSR, Neorg. Mater., !, No. 3, 442-445 (1965). A. N. Lazarev, Vibrational Spectra and Structure of Silicates [in Russian], Leningrad

(1968). 6. 7. 8. .

i0. ii.

H. Moore and P. W. Melnilan, J. Cos. Glass. Technol., 40, No. 93, 66-161 (1956). K. Nakamoto, Infrared Spectra of Inorganic and Coordination Compounds, 2nd Ed., Wiley, New York (1970). V. A. Kolesova, Coordination of A1 and Ga Atoms in a Glass Network. The Glassy State [in Russian], Moscow (1965), pp. 219-221. G. N. Kuznetsova, V. S. Kheifets, and A. M. Shevyakov, Zh. Prikl. Spektrosk., !, No. 3, 242-247 (1964). K. A. Akhmed-Zade, V. A. Zakrevskii, and D. M. Yudin, Fiz. Khim. Stekla, 2, No. 5, 388391 (1976). G. O. Karapetyan and D. M. Yudin, Fiz. Tverd. Tela, i, No. i0, 2647-2655 (1962).

QUATERNION COUPLING EQUATIONS FOR MOVING GYROTROPIC MEDIA A. V. Berezin, E. A. Tolkachev, A. Ya. Tregubovich, and F. I. Fedorov

UDC 535.56

As is known [I], the most general coupling equation for gyrotropic media at rest takes the form D = 8E + ~ H ,

B ~ ~H-~B,

(1)

where ~, ~, E, ~ are 3 • 3 matrices. It is evident that Eq. (I) retains its form in an arbitrary inertial reference frame, but the corresponding matrices will depend nonlinearly on the velocity, generally speaking. Knowing the explicit form of this dependence would allow the influence of motion of the medium on the character of its gyrotropy to be investigated using the effective apparatus developed in [i]. A series of covariant generalization of the coupling equations is known. Thus, the case of isotropic media was considered in detail in [2] (see also the literature cited there). The corresponding equations for anisotropic media (~ = B = 0) were obtained in [3] and generalized in [4] to the case of optically active media. A geometric approach to macroscopic electrodynamics [5] was developed in [6], where the relation between the electromagnetic-field tensors in the medium and in vacuum was given a clear geometric meaning as the relation between elements of mutually dual vector spaces. Such coupling equations, which are Lorentzinvariant, satisfy the syrmnetric and antisymmetric principles of duality [7, 8], with the corresponding definitions of the gyrodual transformations. However, none of the above approaches are directly applicable to the description of elucidation of the transformational properties of ~, ~, ~, ~ relative to Lorentz-group transformations, because of "mixing" of the induction and field-strength vectors on both sides of Eq. (i). In addition, the tensor

1987.

Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 47, No. i, pp. 113-118, July, Original article submitted April 23, 1986.

0021-9037/87/4701-0747512.50

9 1988 Plenum Publishing Corporation

747

form of the equations - see [4], for example - prevents the use of the well-known advantages of the vector parametrization of the Lorentz group [9]. In [i0], a formulation of the material equations in the appropriate reference frame was given within the framework of the algebra of quaternions above a field of complex numbers, allowing, in particular, the transformational properties of ~, ~, e, p relative to the group of dual transformations to be calculated. On the basis of the results in [i0], using the well-known quaternion analog of vector parametrization of the Lorentz group [ii], a covariant generalization of quaternion coupling e q u a t i o n s i s given, and the corresponding transformational properties of the matrices ~, $, e, p are found. The dual invariance of the equations obtained is also shown. In the reference frame intrinsic to the medium, the form [i0] B--fD:=(O.H--,"~E)

sea,

the quaternion coupling equations take

a---- I, '2, 3,

(2)

where

Rep~

tt o,

Imps, = - - ~ a ,

R e % . = e~,

I r a % = ~,~.

Here the triads of the quaternions a~, ~a, ca, ~ corresponds to rows of identical matrices of Eq. (i). Remember that a complex quaternion is an element of algebra with the generatrix e i (i = 0, I, 2, 3 ) above a field of complex numbers, while cdj = - - 6 ; ? 0 + G0ej + ~J0q § eo~,,e~, where eijks is written in the and quaternion (i/2)(x - x).

(3)

the Levi-Civita symbol. This algebra is denoted by H C, and its elements are _form x = xie i = x 0 + x = x S + x V. In the given algebra, complex x* = xi*e i x = x 0 - x conjugates are defined, while x S = x0 = (i/2)(x + x), x V = x = The law of composition of two arbitrary quaternions x and y follows from Eq.

(3): .%." = x0g0 - - xy + xf~y + y0x + I.X yj,]

(4)

where • is the scalar product and [xy] t h e v e c t o r p r o d u c t o f two t h r e e - d i m e n s i o n a l vectors. Where e n e r g y d i s s i p a t i o n i n t h e medium m u s t be t a k e n i n t o a c c o u n t , aa, ~-~, ~ . ~a m u s t be r e g a r d e d a s e l e m e n t s o f some s u b a l g e b r a HG | HG o f t h e f o r m z = x + j y , w h e r e x , y 6 H G, j 2 = -1, and j commutates with all e i. However, for the purposes of the present work, this is insignificant, and a~, ~a, Ca, ~a are regarded as elements of H C. Taking this into account,

Eq. (2) evidently reduces to the system

R e F = Re [ ( ~ G + i~F)se~],

ImG =Im

[(eoF -- I'O~aO)se:~].

(5)

The q u a t e r n i o n s F=B--iE and G=H--iD are elements of the space of irreducible representations of the group SO(3.1) with a transformation law ( t h e q u a t e r n i o n L p a r a m e t r i z e s the Lorentz group) G - + G ' -- L".~G~~:, F - * F ' = L*FL~-:

(6)

a n d c o r r e s p o n d t o an a n t i s y m m e t r i c tensor in the standard approach. The r e p r e s e n t a t i o n of the group S0(3.i) with respect to which aa, ~ , e~, ~ are transformed is unknown a priori. I t i s c l e a r , h o w e v e r , t h a t t h e y m u s t a l l be t r a n s f o r m e d a c c o r d i n g t o t h e same u n a m b i g u o u s representation, i.e., according to the tensor representation in Eq. (6) or a vector representation A-,

A'

= LAU':,

(7)

which may only be realized in quaternions. Postulating the transformation law in Eq. (6), putting primes on all the quantities in Eq. (5), and adding the two equations, the result obtained is

[(e-~ - - %) F]s ea .... [(ea q- oa) F*]s-TLe,~: -~ = [(--e~ + ~)G.] s e,~ - - t(e~ § P,,3 G*] ~Te,,~,

748

(8)

where u = iu 0 + u is the four-dimensional velocity, and the identity Li.::': . . . . iu,

which is valid if L specifies s o m e (8) is equivalent to the system

is taken into account.

boost,

Gijt.gj

In index notation, Eq.

em~zF~:d,j + tzw~G~.t~7,

=:

(Sa)

Where Fij • = (i/2)eijks163 ; Ooao b = Oab; 0 a b c d = (i/2)eabnecd n, Ohm; 0 denotes one of the symbols a, ~, e, p. As is readily evident, the equations obtained do not satisfy the symmetry principle of duality. In other words, the field quantities on the left- and right-hand sides of Eq. (Sa) cannot be represented as components of some mutually dual elements of vector spaces (where the fundamental tensor is an object constructed from tensors characterizing the properties of the medium) without arriving at a contradiction with the requirement of Lorentz covariance of the equations. Thus, a single possibility remains: the quaternions ~, ~a,~a, ~a are transformed with respect to the vector representation of the Lorentz group. Calculations analogous to those used in deriving Eq. (8) give

fiTL* = (~,,,L-jL*+ i~t.--;': L*) s e~, (9)

L f L* = (Sa~i~L * - - [~aL-,~ L*) s ea,

where :F

1

(uF-

---- - -

2

j-

F'u),

I (.O--G'u), --y.

(10) ,~r =

~ = - 2 - i ( u F + F'u),

i (uG-q- G'u), 2

which, in tensor formulation, corresponds to the equations Giju i ~-- ~,yFj,~u;. + o~ijGikUl~, • o ,

F•

~

(11)

= ~i~G~uk +- ~jFjl, ul.

or in three-dimensional notation

D# + [vH #] = ~ (E # + [vB#l) + c~(H # - - [vWil), B ~ - - [vs

= t-t ( Wr - - [vD#]) + [3 (E ~r + [vBerl),

D#:=yDI-I-D'~I, v .....

(12)

V

D ! = ( 1 - - v" v) D, ,

~ .... ( 1 - - v D

1 2

,

DII = v ' v D ,

(13)

c=l,

Iv! where a.b denotes the matrix dyad (a-b)ca=acba [i]. To demonstrate dual invariance of Eq. (9) in explicit form, note that they may be represented in the form 1

(14)

where n a = LeaL* and all the quantities are written in the primed reference frame. rotations

-,

~' ,

( 7 -+- i : ~ ) v ~ , - ( 7 -+ iJ)ve---'L

For dual

(15)

749

and, therefore, Eq. (9) is dual-invariant under the condition that ~, B, ~, ~ have the transformational properties noted iN [~0] relative to the dual group. To find the form taken by Eq. (i) in an arbitrary inertial reference frame, Eq. (9) is brought to the form

(16) where n a = L*e6L*, and the quaternions ~t,T l, z., Xa, ~a in the reference frame intrinsic to the medium correspond to rows of the matrices ~-i, s - ~D-l$, _~-16, ~ - i The corresponding quaternions with a tilde differ from these by multiplication on the left by L* and on the right by L*. As follows from Eq. (16), =

"

k~a

--

~u

/S

2

(17)

and, therefore, Eq. (17) is the quaternion variant of the geometrized coupling equations introduced in [6]. Thus, it is evident that transition from one version of the material equations to another, within the framework of quaternion algebra, is accomplished by means of homomorphism of this algebra, which proves the complete equivalence of the two versions considered. Acting now on both sides of Eq. (17) with the operator Re and eliminating quantities with tildes, an equation for B is obtained. In matrix form, its solution is as follows:

B = (1 + ~H-iFvX~v x + H-t6H)-~(H-~FH-~H + H-I~HE),

(18)

H : y + (1 - - y ) ~ . ~ ,

(19)

where

6 = ~(%vX-~-VX~]),

(vX)ab =

(20)

eacbVc '

x:vX--pvX•

(21)

An e x p r e s s i o n f o r D may be o b t a i n e d f r o m Eq. ( 1 8 ) by d u a l r o t a t i o n by ~ / 2 , of the transformational properties o f ~, ~, ~, ~ f o u n d i n [ 1 0 ] . Here --4

taking

account

--]

(22) ~1 = mF -~ "-~ --f~-~,

e ~ ~.

In the case where ~ = ~ = 0 and E and D are scalar, the expressions obtained coincide with those given in [i] (p. 276). As an example, consider a uniaxial crystal where ~ = B = 0 and

=%+~'c.c,

~=~o+~'c'c,

e is the unit vector in the direction of the crystal axis (c 2 = i), and E0, ~0, constants. In this case, Eq. (18) takes the form

B = (l+~H-~vXevX)-i(H-~p,H-~H + N-~(vX--~vX~)~E).

(23) gt

, ~

;

are

(24)

Using the technique developed in [i], an explicit expression for B in terms of the vectors c, v, E, H may be obtained. In view of the unwieldiness of the corresponding formulas, only the case of transverse motion cv = 0 is considered. Then

B =/f(v)H + ~Cr(v)E,

(25)

tier (v) = e0v x + c~c. [ c v l + c2 icy]- c,

(26)

where

Co .... (1 - - ~0V~0)(l - - v 2 U'0~+ + ~'~0)) A -a,

cJ. = - - - ~ ' 4 t ' ( 1 -

750

v2(l 'w ,~"',uo,,)~'A-1 ,

C . _, = : :

'! v ' (,,. { -- e(l ~,o~

+

~r ..

p}A

'~

-.'

',

~# (v) = ,t~,,~o (v) + Wa, (v) c - c + (1~o + a.: (v)) v. v + a.~ (v)lcv]. [cv],

(27)

(28)

and ao = (1 - - v 2 (*o~+ + e ' ~ o ) ) A - h '-2, a 1 = (] - - O2S'~o) A - I ? -2,

(29) 2

a~, = --~o~o ( 1 - - v2~+~+) A - ' y -2,

a3 .... l~o2r

-2,

while

e§ ==%+e',

~+ = ~0+ ~', A = (I --V2eo~+)(l--v2~08§

(30)

I t is evident from Eq. (28) that ~ef(v) does not have the form in Eq. (23) of a uniaxial tensor. As shown in [i], any symmetric tensor may be brought to the form

T -- a-~- b(c'-C + c".c').

(31)

In the reference frame intrinsic to the medium, the tensors E and ~ for a biaxial crystal take this form. It thus follows from Eq. (28) that in a moving system the crystal takes on, as it were, a new structure, becoming effectively biaxial. This is associated with the appearance of an additional isolated direction defined by the velocity vector of the crystal. As a result of relativistic addition of the velocities of the crystal and the wave traveling in it, a specific dependence of the refractive index on the direction of propagation appears. LITERATURE C I T E D i. 2.

3. 4.

5. 6. 7.

F. I. Fedorov, Theory of Gyrotropy [in Russian], Minsk (1976). B. M. Bolotovskii and S. N. Stolyarov, Usp. Fiz. Nauk, 114, No. 4, 569-608 (1974); in: Einstein Collection 1974 [in Russian], Moscow (1976); S. N. Stolyarov, in: Einstein Collection 1975-1976 [in Russian], Moscow (1978). G. Marx, Acta Phys. Hung., !, No. 2, 75-85 (1953); A. Bressan, Relativistic Theory of Materials, Berlin (1969). B. V. Bokut', A. M. Serbyukov, and F. I. Fedorov, Electrodynamics of Optically Active Media, Preprint [in Russian], Institute of Physics, Academy of Sciences of the Belorussian SSR, Minsk (1970). I. E. Ta~n, Collection of Scientific Works, [in Russian], Vol. i, Moscow (1975). A. Z. Petrov, New Methods in General Relativity Theory [in Russian], Moscow (1966). A. E. Levashev, Motion and Duality in Relativistic Electrodynamics [in Russian], Minsk

(1979). 8. 9. i0. ii.

V. I. Vorontsov, Electromagnetic Waves in Moving Media with Weak Dispersion, Preprint ITF-47-40r [in Russian], Kiev (1974). F. I. Fedorov, Lorentz Groups [in Russian], Moscow (1979). A. V. Berezin, E. A. Tolkachev, and F. I. Fedorov, Dokl. Akad. Nauk BSSR, 29, No. 7, 595-597 (1985). A. A. Bogush, Yu. A. Kurochkin, A. K. Lapkovskii, and F. I. Fedorov, Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk, No. i, 69-75 (1976); Yu. A. Kurochkin, Quaternions and Some Applications in Physics, Preprint 109 [in Russian], Institute of Physics, Academy of Sciences of the Belorussian SSR, Minsk (1976).

751