I L NUOVO CIMENTO

VOL. 9 B, ~ . 1

11 Maggie 1972

Phonons in a Strong Static Magnetic Field (*). A. ItOLZ (*') (?enter /or the Application o] Mathematics, Lehigh University - Bethlehem, Pa. (ricevnto il 17 Dicembre 1970;

manoscritto revisionato ricevuto il 25 Maggie 1971)

Summary. - - The Hamiltonian of an ionic crystal lattice in the harmonic approximation and in the presence of a static magnetic field is diagonalized in terms of creation and annihilation operators. Phonon energies and polarizations are determined in this t h e o r y b y means of the solutions of an eigenvalue problem of the first kind with twice the dimension of the corresponding problem without magnetic field. The energy eigenvalues of the Hamiltonian are linear combinations of the phonon energies ~%(k) with integers as coefficients, just as in the case without a magnetic field. F o r crystals in which every ion is at a center of inversion symm e t r y the phor~o~, spectrum will conserve its inversion s y m m e t r y in k-space. F o r crystals without inversion s y m m e t r y this will no longer be true. In every case the polarizations of the phonons will be elliptical.

1.

-

Introduction.

T h e p r o b l e m of p h o n o n s in s t r o n g s t a t i c m a g n e t i c fields h a s b e e n c o n s i d e r e d for m e t a l s in a n u m b e r of p u b l i c a t i o n s (1). T h e effect of t h e m a g n e t i c field on t h e p h o n o n s w a s c a l c u l a t e d v i a i t s i n f l u e n c e on t h e f r e e c h a r g e c a r r i e r s w h e r e a s t h e d i r e c t i n f l u e n c e of t h e m a g n e t i c field on t h e m o t i o n of t h e i o n s h a s n o t b e e n t a k e n i n t o a c c o u n t . I n t h e f o l l o w i n g , h o w e v e r , we c o n s i d e r t h e p r o b l e m f r o m t h e l ~ t t e r p o i n t of v i e w , w h i c h is a p p r o p r i a t e for ionic c r y s t a l s .

(*) The research reported in this p a p e r was carried out with the support of a grant from the National Science Foundation to Lehigh University. (**) Present address: D e p a r t a m e n t o de Fisica, Univcrsidad de Chile, Casilla 5487, Santiago. (1) See for example: S. J. ~[IYAKE and IV[. YOKOTA: Journ. Phys. Soc. Japan, 28, 1369 (1970); D. P. CgOCK and Y. C. LEE: Phys. Lett., 3 2 A , 53 (1970). 83

84

A. HOLZ

The dynamics of the system in this case is completely determined by the motion of the ions. Accordingly, one is faced with a less difficult problem than in a metal where the collective motion of the electrons has also to be taken into account. The fact t h a t the problem stated above has not been considered so far is partly due to the assumption t hat even strong magnetic fields will have a small influence on ionic motion, and partly because the solution to the anisotropic oscillator problem has only recently (5) been given. We can make a rough estimate of the magnitude of the effect by comparing the cyclotron frequency vc of an ion in a typical ionic crystal like LiF with the phonon frequency v~. For Li + one obtains % ~ 2 . 1 0 2 H s -1 where H is in Oerstedt and for the frequency of an optical phonon we have %~1013 s -1. Accordingly, fields of the order 1011 Oe are needed to bring both frequencies to the same order of magnitude. However, these strong fields are not necessary to observe for example the Faraday effect due to optical phonons or the change of thermodynamic properties of crystals at low temperatures. In the latter case, only low-frequency modes contribute and therefore fields as low as 10 ~ Oe could induce appreciable effects. In the following we show t h a t the phonon spectrum will be displaced by the application of a magnetic field. This displacement will be roughly proportional to re. Furthermore, the phonons in a magnetic field will in general be elliptically polarized. Therefore, the concept of longitudinal and transversal modes no longer applies. In the following a procedure is developed to diagonalize the Hamiltonian of a crystal lattice (in the harmonic approximation) in a uniform magnetic field in terms of creation and annihilation operators b*(k) and b(k), respectively. The Hamiltonian of this problem is a quadratic form of the Fourier transformed single-particle operators q(k) and p(k). The effect of the magnetic field on the force constants has not been taken into account because it is probably p r e t t y small. After having expressed b*(k) and b(k) as linear combinations of q(k) and p(k) the diagonalization of the Hamiltonian proceeds in the same way as outlined in (5) and thus will not be given in detail. The eigenvalue problem which determines the expansion coefficients of b*(k) and b(k) will be transformed into an eigenvalue problem of the first kind whose eigenvalues give the phonon spectrum. The polarizations of the phonons are obtained from the eigenvectors of this problem by a simple transformation. In Sect. 4 the ease of a crystal is considered where every ion is at a center of inversion symmetry. We point out t hat the diagonalization procedure applied in this paper can be modified in the following way: instead of expressing the Hamiltonian in terms of q(k) and p(k) the Hamiltonian can be written in terms of (2) E. E. Bn~GMA~N and A. HOLZ: Nuovo Cimento, 7 B, 265 (1972). The notation used in this paper is slightly different.

PltONONS IN A STRONG STATIC MAGNETIC F I E L D

85

creation and annihilation operators b*o(k) and be(k) of the u n p e r t u r b e d problem. To the resulting quadratic form in b*o(k) and be(k) a Bogoliubov transformation can be applied. Hamiltonians of this kind arise in the spin-wave t h e o r y of ferrites (3). No numerical estimates have been made so far. Thus, only v e r y general implications of the effect of a magnetic field on the phonon spectrum of a crystal can be made. I n particular, a t r e a t m e n t of this problem b y group-theoretical methods as was carried out in (4.5) for phonons w i t h o u t magnetic field seems to be promising. A detailed t r e a t m e n t of lattice vibrations without magnetic field can be found in ref. (e,7).

2. - The Hamilton operator of a crystal lattice in a uniform magnetic field.

The H a m i l t o n operator of a crystal lattice in the presence of ~ magnetic field is, according to the principle of minimal substitution, given b y

)(

(2.1)

)

H e r e M , and e, are the mass and charge respectively, of the ~-th ion in the primitive unit cell, which is assumed to have r ions, u~(ln) is the a-th Cartesian component oi the displacement of the n-th atom in the /-th unit cell from its equilibrium position x~(ln), io~(ln) is the corresponding linear m o m e n t u m operator, q)~(/n; l'~') are the atomic force constants of the crystal, c is the velocity of light, and B is the vector potential which is subject to the Lorentz gauge

(2.2)

~ ~B~(~)/~u~(~) = 0 ct

p~(l~) and u,(ln) satisfy the c o m m u t a t i o n relation

(2.3)

(s) See for example, L. R. WALKER: in Magnetism, edited by G. T. RADO and H. SUIaL, Vol. 1 (New York and London, 1963), p. 311. (4) A. A. ~r and S. H. Vosr~o: Rev. Mod. Phys., 40, 1 (1968). (~) J . L . WARREN: Rev. Mod. Phys., 40, 38 (1968). (e) ~. BORN and K. HUANG: Dynamical Theory of Crystals (Oxford, 1954). (7) G. LEIBFRIED: in Handbueh der Physik, edited by S. FL/3GGE, Vol. 7, Part I (Berlin, 1955).

86

A. l ~ o L z

where ~ is Planck's constant and (~=~is Kronecker's delta. Due to the gauge invariance of the magnetic field the vector potential B can be referred to the equilibrium position of each ion separately. This implies

(2.4)

B~(lu) = ~ C~u~(lg), fl

where C is an a n t i s y m m e t r i c 3 • 3 matrix. Inserting eq. (2.4) into (2.1) yields, with the use of eqs. (2.2), and (2.3),

(2.5)

H = 1 ~. (M~lp~(lz)p~(lu) + 2(e,,/cM,,)u:(lu) ~ C~pa(lu)) + I.~t,ot

fl

+ 12 2

2

M:, X c=,

It' ug' ~tfl

The translational s y m m e t r y of the crystal implies t h a t the force constants obey the relation (7) (2.6)

qS~(1 + m, u; l' + m, u') = qi~(lu; l'u'),

where m is an arbitrary integer. The Hamiltonian (2.5) is, therefore, invariant under the s y m m e t r y operations of the translational group of the crystal. Accordingly, the following Fourier representations are possible:

u=(~) = (_,VM,,)-~ ~: q=,,(k) exp [-- ik.x(Zu)], (2.7)

k

p~(l~) = (M,,/N') 89~.p~,(k) exp [ik .x(lu)]. Ir

Here the number of unit cells per volume unit of the crystal is denoted by N. The N allowed values of the wave vector k are determined by the cyclic boundary conditions. The Fourier coefficients in eq. (2.7) can be considered as 3r-component column vectors in a 3r-dimensional space. This suggests the notation

(2.s)

q(k)=--{q~,(k)},

p(k)=--{p,,(k)}.

F r o m the Hermitian properties of the single-particle operators it follows by means of eq. (2.7) t h a t (2.9)

q+(k) = q ( - - k) ,

p+(k) = p ( - - k)

holds, where q+(k) is the Hermitian conjugate of q(k). we obtain (2.10)

[q~,,(k), p~.,(k')]_ =- il~O,,~O,o,,A(k-- k')

By means of eq. (2.3)

87

Pt{ON01~S IN A STRONG STATIC MAGNETIC FIELD

with 1,

A(k)=

k~-0~

0,

k#0.

L e t us note n e x t the relations

~ e x p [ik.x(/~)] = (2.11) exp [ik.x(/~)] = l

iV,

x(lz) ~= O, x(l~) = 0 ,

0

k#g,

0

'

'

N,

k=g,

where g is a reciprocal lattice vector. Inserting (2.7) into (2.5) and observing eq. (2.11), we obtain

(2.12)

H = 89~ (p(k).p*(k) § q(k)K(k)q+(k) + 2q(k)Ap(k)). It

H e r e the following abbreviations have been introduced: (2.13a)

(2.13b)

K(k) = {K~3,,(k)}

=

-

where K(k) is the Fourier t r a n s f o r m e d dynamical m a t r i x of the problem with an additional field-dependent t e r m . I t has the s y m m e t r y properties (')

(2.14)

K*(k) = K ( - - k) ,

K+(k) = K ( k ) ,

where K*(k) means the complex conjugate of K(k). I n eq. (2.12) and in all the following equations vectors appearing on the left-hand side of other vectors or matrices are understood to be transposed.

3. - Creation and annihilation operators for propagating waves.

Creation and annihilation operators for phonons with a wave vector k are assumed in the form (3.1,)

bt~(k) = v~(k)-qt(k) -~- e~(k).p(k),

(3.1b)

bj(k) = vj(k).q(k) § ej(k).p+(k) ,

88

A, HOLZ

where j ranges from 1 to 3r, and thus labels the different phonon branches. vj(k) and er are 3r-component vectors which are defined by

v,(a) - {v~.~(k)},

(3.2)

e~(k) -= {~,. (k)}.

In addition we introduce the 3r • (3.3)

matrices

V(k) = {V~.~(k)} ~- {v~.j(k)},

E(k) = {E~.~(k)} ~ {e~.;(k)},

which will be used later. The vectors vj(k) and e~(k) in eqs. {3.1a) and (3.1b) are now determined by the following conditions: (3.4a)

Ibm(k), H]_ = -- ~o)j(k)b~(k),

(3.4b)

[bj(k), HI_ = ~wj(k) bj(k),

where ]/e)~(k) is a positive number and represents the energy of the phonon. B y means of eq. (2.10) we obtain for the commutator on the left-hand side of eq. (3.4a)

(3.5)

[b~r

H]_ = i?i{v*j(k).p(k) -- 89[e*~(k)K(k)q+(k) +

+ q+(k)g(-- k)ej(k) + 2q+(k)Av*(k) --2e*j(k)Ap(k)]}. B y comparison of the right- and left-hand sides of eq. (3.4a) and eq. (3.5) we obtain the set of equations

{ ~(v:(k) - ze~(k)) = - ~j(k) e*(k), (3.6) i(Av~(k)

-- Kt(k)

e*(k))

= -- r

.

Equations (3.6) represent a homogeneous system of 6r equations whose solutions belonging to positive eigenvalues wj(k) determine the creation operator (3.]a). The complex conjugate of these solutions determines the annihilation operator (3.1b) in accordance with eq. (3.4b). The system of equations determining bj(--k) can be derived similarly and is given by

i ( v j ( - - k ) - - A e j ( - - k ) ) : cor

,

(3.7) i(Avj(-- k) -- Kt(k) ej(-- k)) = ~ ( - - k) vj(-- k) . Equations (3.7) differ from eqs. (3.6) only in the sign in front of o~(--k). Therefore, the solutions of eqs. (3.6) with positive eigenvalues o~(k) will determine b~(k) and the one with negative eigenvalues (or positive ogj(--k)) determine b~(-- k).

PHONONS

IN A STRONG STATIC MAGNETIC

89

FIELD

By taking the complex conjugate of eq. (3.6) we obtain eq. (3.7) only when the condition

K*(k) = K(k)

(3.8)

holds. Consequently, we have in this ease vj(-- k)

----

exp [i~j] vj(k) ,

(3.9) e j ( - - k) = exp [i%] e j ( k ) ,

where ~v~ is an a r b i t r a r y phase factor which, for convenience, we set equal to zero. Another ease where a similar relation exists will be discussed in Sect. 4. Only in these two cases does the equation

eo~(k) = r

(3.10)

k)

hold. In all other cases eqs. (3.6) do not necessarily have 3r positive eigenvalues r and cqs. (3.7) do not necessarily have 3r positive eigenvalues ~ j ( - - k ) . We will not consider these cases as t h e y are most likely to occur only when the magnetic field is v e r y strong. I n the same way as outlined in (2) we p u t the eqs. (3.6), (3.7) into the form ~ l j ( k ) = .Q,l(k)O(k) ~b(k),

(3.11)

where E and O(k) are 6r x 6r matrices which are given b y

(3.12)

iI

'

K*(k

'

where I is a unit m a t r i x and ~j(k) is a 6r-component vector which is given b y

t-

(3.13)

~j(k) = \y~(k)

/ "

The solutions of cq. (3.11) are related to the solutions of eqs. (3.6) ~md (3.7) as follows. For Y2j(k)> 0 we obtain (3.14a)

toj(k) = zQj(k),

v*(k) = xj(k),

e*~(k)= y j ( k ) ,

and for ~9j(k)< 0 we obt,~in (3.14b)

~,~j(--k)----.Q~(k),

v~(--k) = xj(k),

e~(--k)=y~(k).

A. HOLZ

90

The matrices ~ and O(k) are Hermitian. In addition it was shown in (2) t h a t O(k) is positive definite when the m a t r i x (3.15)

K*(k) -- AA ~

is positive definite. However, this m a t r i x is positive definite whenever the dynamical m a t r i x of the unperturbed problem is positive definite, i.e. for stable lattices. This can be seen by substituting K * ( k ) by eq. (2.13a). If -~ and O(k) are H e r m i t i a n and O(k) is positive definite, then eq. (3.11) can be transformed into the following eigenvalue problem of the first kind (s):

(3.16)

T ( k ) xr

-= .Q71(k)xr

where v2(k) = T * N T ,

x,(k) = A v b ( k ) , (3.17) O(k)

= At A ,

T

= A -~ .

The m a t r i x T ( k ) is Hermitian and hence all eigenvectors of eq. (3.16) can be chosen such t h a t (3.18a)

x~(k).x,(k)

= ~j(k)~,j

for

sign ~Q~(k) = sign .@~(k)

and (3.18b)

x ~ ( k ) , x,(k) = 0

for

sign tg~(k) :/: sign tgj(k),

where ~(k) is an arbitrary positive real number. As in (~) we take ~j(k) = ~-l%(k)

for zQ~(k)> 0 ,

(3.19) ~(k) = ~-1r

k)

for ~j(k) < 0 .

]~y assuming the normalization of the eigenveetors in this form it can be shown, as in (3), t h a t the following equations hold: (3.20)

I i ~ ( v ~ ( k ) . e ~ ( k ) - - v*(k).e~(k)) = 6~j ,

[

v~(-- k ) . e r

- - v j ( k ) " e i ( - - k) = O .

(s) R. ZUR~0L: Matrizeu (Berlin, GSttingen, Heidelberg, 1961), p. 193.

PtlONONS

91

IN A STRONG STATIC MAGNETIC FIELD

B y means of eq. (2.10) it is easy to show t h a t the left-hand side of eq. (3.20) is proportional to the c o m m u t a t o r between the creation and annihilation operators. Hence, we obtain

[bs(k), b~,(k')]_

5~r

[bj(k), b~,(k)]_

0.

(3.21)

This holds for all wave vectors k. F u r t h e r m o r e , the Hamiltonian can now be expressed in the form (3.22)

H = Z

+

k

89

t

I t is obvious from eq. (3.21) and eq. (3.22) t h a t the formalism for phonons in uniform magnetic fields is the same as for phonons without external fields. For example, the eigenvalues, which we denote b y e, of H are given b y ~ = ~ ~h~or

(3.23)

k

89

i

where the nj(k) are integers. The characteristic frequencies e~j(k) and the polarizations of the phonons will be modified b y the presence of a magnetic field. I n order to obtain the p h o n o n polarizations we have to express q(k) in terms of b~(k) and bj(k). B y means of the 3r-component quantities

b*(k) = {b*j(k)},

(3.24)

b(k) ~- {be(k)},

we can write eqs. (3.1a), (3.1b) in the form

(3.25)

{

b+(- k)

b(k)

V*(-- k) q(k) q- E*(-- k)p+(k), -= V(k) q(k) ~- E(k)p+(k) .

To solve this equation for q(k) we write eq. (3.20) as follows:

4h(V(k)E+(k) -- E(k) V+(k)) = I , (3.26)

V(-- k)Et(k) -- E(-- k) Vt(k) = 0 . Using eqs. (3.25), and ( 3 . 2 6 ) a n d some simple algebraic operations of (3), we obtain (3.27)

q(k) = -- ihEt(k) b(k) ~- i~Et( - k) b*(-- k).

92

x. HOLZ

Finally, we can express the displacement u~(lu) of each ion in terms of and b(k) as follows:

b+(--k)

(3.28)

u~,(l~) = i(hrM,) -89h ~ (-- e,~*~z(k) b~(k) q- e:,,,z(-- k) b~(-- k)) . kd

9*exp [--

ik .x(/n)] .

F r o m this expression it is evident t h a t b~(k) creates a phonon with wave vector k with a polarization which is proportional to er In general, ej(k) will have complex values and, consequently, the phonons are elliptically polarized. We m a y use eq. (3.28) to calculate the interaction between phonons and other modes 9

4. - Crystal w i t h i n v e r s i o n s y m m e t r y .

I n this Section we consider the case of a crystal whose point group contains the inversion. I t is f u r t h e r assumed t h a t e v e r y ion is a center of inversion 9 According to MARADUDIN (4) the dynamical m a t r i x can t h e n be t r a n s f o r m e d into a real m a t r i x of the form (4.1)

K,(k) = UK(k) U* ,

where the u n i t a r y m a t r i x U is defined by

(4.2)

~r = { G,,.~,,} ~ { ~ , . ,

exp [ - i k . x ( , ~ ) ] } .

L e t us introduce n e x t a u n i t a r y m a t r i x F (in the 6r-dimensional space) defined b y

(4.3)

P =

. 0

B y means of

F, eq. (3.11) can be t r a n s f o r m e d into

(4.4) where (4.5a)

~,,(k) = F~j(k),

(4.5b)

O,(k) =

(4.5c)

A~

=

\A,g,(k)/

UA U*.

93

P I t O N O N S I N A S T R O N G S T A T I C NIAGNETIC FI]~LD

In the case where U is given by cq. (4.2), we have (4.6)

A t -- A .

:By means of eq. (4.5a) we obtain

[ V:(k) (4.7)

/

= VV*(k),

E*(k)

Vt(--k ) = U V ( - - k ) ,

= UE*(k),

E,(--k) = U.E(--k).

B y taking the complex conjugate of eq. (4.4) and using eq. (4.6) it is seen that, because Ot(k ) is real and s y m m e t r i c the positive and negative eigenvalues of eq. (4.4) form pairs of equal magnitude. Hence, we obtain the following result: (4.8a)

~oj(k) = to~(-- k),

(4.8b)

v~j(-- k) = exp [isvr

,

%{-- k) = exp [i~v~]%(k),

where q~ is an a r b i t r a r y phase factor which, for convenience, we set equal to zero. The solutions of eq. (4.4) satisfy eqs. (3.18a), (3.18b). Consequently, eqs. (3.26) hold in the form

(4.9)

I /

ih(V,(k)E:(k)--Et(k)V*t(k))

= O,

Vt(-- k)Ett( - k) -- Et(-- k) V~(k)

O.

If eq. {4.7) ~md its complex conjugate are inserted into eq. (4.9) and use is made of (4.10)

U # U = 1,

eq. (3.26) results. F r o m this it follows t h a t once the solutions of eq. (4.4) are obtained and t r a n s f o r m e d into the original co-ordinates the formalism developed in Sect. 3 applies. I n addition it follows from eqs. (4.7) and (4.8b) t h a t the equation (4.11)

Uej(-- k) = U*ej(k)

holds. I t should be pointed out t h a t exactly the same equation holds for phonons w i t h o u t applied field. In this case, however, Uej(--k) is a real vector. We note t h a t there will always be a unitary transformation which transforms K(k) into a real matrix. The m a t r i x At, however, will in general not be real. Therefore, O*(k) will be different from •t(k) and hence eq. (4.8a) does not follow. I n particular, for crystals without inversion s y m m e t r y U will have nondiagonal elements. Hence, only for v e r y special cases can A and K(k) be transformed simultaneously into real matrices. The consequences t h a t eq. (4.8a) does not hold for crystals without inversion s y m m e t r y will be discussed in the n e x t Section.

94

5. -

A. HOLZ

Discussion

and results.

The Hamiltonian H of a crystal lattice in the harmonic approximation and in the presence of a magnetic field has been diagonalized exactly. I t follows that H in terms of the transformed creation and annihilation operators has the same form as without field. There are, however, a number of differences with respect to unperturbed phonons. Let us consider first the case of a real dynamical matrix. Then according to the eqs. (3.9) and (3.10) phonons with opposite wave vectors k will have the same energy and parallel polarizations. Therefore, standing waves can be formed. In contrast to the case without magnetic field the polarization e~(k) are not real vectors. Hence, phonons in a magnetic field will be, in general, ellipticMly polarized. This follows from the fact t h a t a physical system in the presence of a magnetic field does not have time-inversion symmetry. In addition to a change of the phonon energy, the degeneracy of the phonon spectrum at points or lines of high symmetry in the Brillouin zone will be decreased. This is due to a lowering of tile point symmetry of the crystal in a magnetic field. To what extent this will occur can only be decided by methods similar to (~.~), and has not been investigated so far. Another special case represents a crystal where every ion is a center of inversion symmetry. Here the dynamicM matrix can be transformed into a real one by means of a unitary transformation. The same type of eigenvalue problem then results as for the anisotropic harmonic oscillator (2). Phonons with opposite wave vectors will have the same energies. In the general case of a crystal without inversion symmetry the magnetic field will destroy the inversion symmetry of the phonon spectrum in k-space. For certain special orientations of the magnetic field and for certain k-vectors one might find that phonons belonging to opposite wave vectors have the same energy. This will be the case whenever a unitary transformation exists which transforms A and K(k) simultaneously into real matrices. A physical explanation of the effect follows already from the classical equations of motion of the problem, where in a problem without space- and time-inversion symmetry no relation between lattice waves belonging to opposite wave vectors can be found. As a consequence of ~(k)=/= ~o~(--k) extremM points of the phonon spectrum originally located at points of high symmetry in the Brillouin zone will be displaced by the magnetic field. The periodicity of wj(k) in k-space, however, will be conserved since K(k) = K ( k ~- g) . Another point of interest is t hat the numbers of positive and negative eigenvalues of eq. (3.11) are not necessarily equal. Therefore, one phonon branch could disappear at a certain point ko and at --ko an extra branch would appear.

P]~IONONS IN A STRONG STATIC MAGNETIC FIELD

9

RIASSUNT0

95

(*)

Si diagonalizz~ l ' h u m i l t o n i a n ~ di u n rcticolo crist~llino ionieo n e l l ' a p p r o s s i m ~ z i o n e a r m o n i c a ed i n p r e s e n z a di u n e ~ m p o m ~ g n e t i c o st~tieo i n t e r m i n i degli o p e r a t o r i di creazione ed a n n i e h i l a z i o n e . Si d e t e r m i n a n o i n q u e s t a t e o r i a le energie e le polarizzazioni dei f o n o n i p e r mezzo delle soluzioni di u n p r o b l e n m di a u t o v a l o r i di p r i m a specie con d i m e n s i o n e d o p p i a del e o r r i s p o n d e n t e p r o b l e m a s e n z a c a m p o m a g n e t i e o . Gli autov a l o r i d e l l ' e n e r g i a d e l l ' h a m i l t o n i a n o sono e o m b i n a z i o n i delle energie dei f o t o n i h%(k) con i n t e r i come coefficienti, p r o p r i o c o m e nel caso s e n z a c a m p o m a g n e t i c o . P e r i cristalli in eui ciaseun lone 6 al c e n t r o della s i m m e t r i a di i n v e r s i o n e lo s p e t t r o dei f o n o n i cons e r v a la s u a s i m m e t r i a di i n v e r s i o n c hello spazio k. P e r eristalli s e n z a s i m m e t r i a di i n v e r s i o n e q u e s t o n o n 5 pif~ vero. I n ogni easo le p o l a r i z z a z i o n i dei f o n o n i sono ellittiehc.

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Traduzione a cura della Redazione.

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