Theory of the vibrations of the sodium chloride latticersta.royalsocietypublishing.org/content/roypta/238/798/513.full.pdf · [ 513 ] THEORY OF THE VIBRATIONS OF THE SODIUM CHLORIDE

[ 513 ]

THEORY OF THE VIBRATIONS OF THE SODIUM CHLORIDE LATTICE

By E. W . K EL LER M A N N , D .P h il ., Ph .D.

University of Edinburgh

(Communicated by M . Born, F.R.S.— Received 26 October 1939)

. Introduction

T he interest in the frequency spectrum of the therm al vibrations in a crystal arose chiefly in connexion w ith the problem of the specific heat of crystals a t low tem peratures. Debye’s theory of the specific heat, however, has been so successful th a t the actual determ ination of the frequency spectrum according to Born and v. K arm an (1912) has been pushed into the background. But recent investigations, especially those of Blackman (1935,1937 a,b, 1938), have shown th a t appreciable deviations from Debye’s theory should occur according to the correct atomistic treatm ent. These deviations appear to be most pronounced near the absolute zero of tem perature. It, therefore, seemed desirable to calculate the exact frequency spectrum of a crystal.

The first a ttem pt to calculate the frequency spectrum of a crystal was m ade by Born and v. K arm an in their original paper. They assumed only quasi-elastic forces between neighbouring particles. Later calculations have been m ade for ionic lattices, for which we have a fair knowledge of the real forces which determ ine the equilibrium positions and the vibrations about them . The chief difficulty in th a t calculation has always been the long range of the Coulomb force which makes a direct sum m ation over all lattice points impossible.

Born and Thom pson (1934), using a m ethod developed by Ewald (1921), suggested a way of transform ing these sums into more rapidly convergent expressions, and Thom pson (1935) has given the final formulae for the coupling coefficients due to the Coulomb force in the equation of motion, b u t in his paper a slight mistake occurred in the definition of the coefficients, and so far no num erical results of these calculations have been published. Broch (1937) has given formulae for the case of a one-dimensional lattice making use of Epstein’s Zeta functions. Lyddane and Herzfeld (1938) have used an extension of M adelung’s m ethod (1918) and they have given some num erical results, b u t their formulae are ra ther complicated, so th a t one cannot expect to com pute the whole frequency spectrum by this method. M oreover, the problem of the therm al oscillations of an ionic lattice is not a purely electrostatic problem , and this point has not been m ade sufficiently clear by Lyddane and Herzfeld. Their treatm ent of the case of the residual rays is open to objection, and the question w hether the potential, from which the coupling coefficients are obtained, satisfies the Laplace equation or Poisson’s equation is not clear.

V ol. 238.—A 798 (Price 6s. 63 [Published 24 May 1940

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514 E. W. KELLERMANN ON THE THEORY OF THE

In this paper I have used Ew ald’s m ethod m entioned above, in a new form given by him in a recent paper (1938). By this m ethod one obtains com paratively simple and quickly convergent expressions for the coupling coefficients in the equation of m otion w hich allow a num erical calculation to an a rb itra ry degree of accuracy. Because of the good convergence it has not been too laborious to com pute num erical values for 48 different modes of vibration.

In § 2 the derivation of these expressions is given by treating the problem as an electrostatic problem , neglecting the re ta rd a tio n ; b u t the p roper way of solving the problem is to find a solution of M axw ell’s equation for the electrom agnetic field in the crystal. This will be done in § 3. From this field the force exerted on a particle and the coupling coefficients can be obtained (§4). I t will be seen th a t in this proper trea tm ent the case of infinitely long waves plays a special role and m ust be considered separately. In all the other cases this treatm ent leads to the same result as the electrostatic derivation. I f one defines a potential function from w hich the coupling coefficients are obtained as second derivatives, this potential satisfies in general L aplace’s eq u atio n ; b u t in the special case of infinitely long waves it satisfies Poisson’s equation w ith constant density.

In § 5 the coupling coefficients for the N aC l lattice are given, and in § 6 the contribution due to the repulsive forces is calculated.

In § 7 the equations for the coupling coefficients are checked by deriving from them form ulae for the elastic constants.

T he num erical values of the coefficients are given in § 8. In § 9 the corresponding frequencies are calculated and illustrated by figures, and finally (§ 10) the resulting distribution of the frequencies is discussed and the distribution curve is plotted.

I shall use the notation of Born’s Atomtheorie des festen Zustandes (1923) * cf. also Born and G oeppert-M ayer (1933).

T he lattice vectors a l5 a 2, a 3 determ ine the cell of volume The positions of the particles in the cell are represented by the basis vectors r*. (k 1 ,. . . j). T he equilibrium position of a particle (k , I) is determ ined by the vector

Z stands for the three arb itrary integers ll9. Z2, Z3. T he distance vector between two lattice points is given by

Considering small independent displacements ujj. = of each particle fromits equilibrium position, the vector between the displaced particles .is

1. T he equation of motion

— a *+ 1**, a/ — + Z2a 2+ Z3 a 3. ( 1-0)

( 1- 1)

( 1-2)



VIBRATIONS OF THE SODIUM CHLORIDE LATTICE 515

Assuming only central forces, the total potential energy of the lattice is

(1-3)kk' 11'

(The dash indicates th a t the terms I = = are om itted.) Neglecting surface effects the energy density

N v a $ (N — num ber of cells) (1-4)

(1-5)

remains finite, and the expansion of the energy in the neighbourhood of the equilibrium position when developed in powers of the displacements u* gives

F = $0 + 1 +

In equilibrium the first-order terms vanish.T he second-order terms

4>2 = -hI2 2 ( 1-6)

( f e ) ,where

with the definition (ftr')*t/

lead to the equation of m otion .

kk' 11' xy

rI_dxdy f e (f)] , >J r = r KK

- 2 2 ' ( f e f e ,k'I

mA - 2 2 2 «5z k' I' y

(1-7)

where a dot denotes differentiation with regard to time and is the mass of the particle of type k.

Considering one of the independent norm al modes of vibration w ith frequencywave-length A and wave vector k, 1 /A the displacements of the particles are

then £ 277z(k, rp ( 1-8)

(the index 0 will be om itted in the following equations) and the equation of m otion becomes

^ m Ku a + 2 2k' y

~K k x y j U K'y — 0,

where [:K Kx y.

(1-9)

..... (!-!0)

From the definition (1*6) there follows

si.*yJk=o The corresponding contributions of different forces to these bracket symbols (coeffi

cients of the equation of motion) are additive. For example,

"a:

Lxy

0 . ( M l )

[K Kx y -

cFkk'~ R\_x y _ + x y ]■

where Cmay denote the Coulomb forces and the repulsive forces.

( 1-12)

63-2



516 E. W. K E L L E R M A N N O N T H E T H E O R Y O F T H E

T he choice of the wave vectors k is restricted by the boundary condition of the “ cyclic la ttic e” (Born 1923, p. 588). I t postulates th a t the displacem ent be periodic in a volum e having the same shape as the elem entary cell and containing cells.T o form ulate this, in troduce the base vectors bj of the reciprocal lattice

(a i> b/) = $ij) b 1= —[a 2x a 3], b 2 = . . . ,V a

and the radius vector b A of this lattice,

h h = hlh l -\-h2h 2-\-h3b 3 a rb itra ry in tegers). (T14)

T h en the condition o f cyclic behaviour leads to

k = - b A = ^ b i + ^ ^ + ^ b g , (1*15)

w here kl = hlfn, k2 = h2/n3 k3 = h3/n 0, 1, ..., 1). (1*16)

2. T he coupling coefficients of the electrostatic interaction

Consider now especially the case o f electrostatic forces. H ere the series (1*10) are not absolutely convergent and have to be transform ed in to o ther series w hich correspond to the physical conditions (neutral cell) and are quickly convergent.

Following Ew ald (1921,1938) I consider a lattice sum of the form

F*(r) --- 2 J ( r - a <?2m'(k’aZ), (2*0)i

which can be w ritten jF*(r) — e2m{k>r) 2 / I r - ~a?) e2m{k>*l~r) (2*1)i

and be considered as a periodic function w hich is m odulated by a wave

exp{27n"(k, r)}.

T he sum can then be represented by a Fourier series

E J ( r _ a < ) 02m-(k,a*-r) = 2 (2 '2)I h

w ith the Fourier coefficients

n = f f / ( r ) = l y ( b ,+ k ) , (2-3)vaJ ua

where the integration extends over the whole of space and

y(b) = J /( r ) e- 2m(b,r) y, (2*4)

is the Fourier transform o f / ( r ) .I apply the transform ation (2*2) to the function




I t is then only necessary to determ ine 99(b) from (2*4) and to insert it into (2-2) and find Ew ald’s well-known formula of the T heta function transform ation (cf. Born 1923, p. 765)

Fk(r) — — ^ e-e r-a.l)2+2nUksal)__ (2‘6) J n I Va h e3 6

Applying this to the Coulomb potential

9Vk(M ) = eK'eJ ( r ), / ( r ) (2'7)

where eK is the charge of the Arth type of particle at the lattice points r(., we obtain for (1-10), since | - r ^ - a ' I = | r lK,K |,

eK,eK lim Ir-*-rKfK i °x °yr * n\_x y A

1*3( 2 -8)

As a quickly convergent representation of the sums in (2-8) can be found, it is permissible to exchange differentiation and sum m ation and to define a function

so tha t

-F*(r) = I / ( r - a * )i

= ^ ( r j ,

(2-9)

( 2-10)

T* — r A/c'/c / /ca:'and since

(cf. (1-0), (IT )) one can write for the coupling coefficients (2-8)

e2nU k,al-rKK)

r=rKK'1 * s l = ^ ? D 4 / ( r - a,)l= eKeK.e-2iri(k>r™')F ky(rj), | (2-11)

\_x y \ = ^ ^ ( r ) - -&(r )]> - 4 = *

* It must be remembered that the term (9**.) is defined by (1-6) and is not equal to

< 4 /(r)Lwhich is infinite. (9**)*„ represents the force exerted on the particle (/c, /) if it is displaced by a small amount, all other particles being kept at their equilibrium positions. It can easily be shown that for Coulomb forces and cubic symmetry this term vanishes.

k'-if Thompson (1935) has not defined the coefficients correctly. He leaves out the exponential

function exp { — 2ni(k., r KK,)}.




Now it is a t once seen th a t the lattice sum Fk (2T), (2*6) is connected by (2*5) w ith the lattice sum F k (2*9) from w hich the coefficients are derived, nam ely,

s°°F/c(r) de = F k(r).(2*12)J o

Thus im m ediate use can be m ade of (2*6) for the transform ation of the coupling coefficients. I t should be noted th a t the sum in the first representation of Fk in (2*6) converges rapidly for large values of e b u t not for small values, and the opposite holds for the second representation. Therefore, following Ewald, I divide the in tegration

z*oo f*E foo(2*12) into two parts — + , taking as in tegrand in the first integral the second

Jo Jo J Eand in the second integral the first expression (2*6).

T he in tegration yields

F k(r) = — 2 1 e- f S tt)2+2»i<b»+k.*-> + ; 1 ~ ^ ' E ' r 7 a ' ^ i!" <t.al), (2-13); | r —a11

where G is Gauss’s function

G(x) = -t- f % -P < , G( oo) -- 1. (2-14)V^J o

This expression (2*13) is very quickly convergent, as the arb itra ry param eter E can be chosen in such a way th a t the two series 2 and 2 converge rapidly, so justifying the

hIinterchange of differentiation and sum m ation in (2*10).

Finally the differentiation (2*11) gives

T* * j = I e-g(b»+k)=-2»i<b«,r„,„)Lx y J I »„» (t>»+k)2

L. r ------- . . .+ £ 2 2 [ > (E I r‘K. I) + \ E T I r‘, | )

I L I *KK, I L

T " iLx y j 4

I rL'

i^hyFjy)gj(bfc+b)2 | ^ 773 f-Vaf (bA+ k )2 ^ +

rlKK I -1

+ £22I

E f " ( E \ d \ ) - t '(E \a f | h alxaly~M J 1 q1 12 g2nUk, ai)

1 G(a) / / \ 1 r /\ !f 1 ( \ d----— = f(x ) , i (x) = f (x) =

(2-16)

For k — O the zero term in Fk (2*13) gives rise to a divergence so th a t the electrostatic derivation breaks down. The reason is th a t the problem of describing the electrom agnetic interaction in the crystal is not a purely electrostatic problem . O ne has to bear in m ind th a t because of the vibrations of the ions there will in general be an electrom agnetic field in the crystal. The proper way, therefore, of determ ining the



VIBRATIONS OF THE SODIUM CHLORIDE LATTICE 519C|-£

coupling coefficients must be to start from M axwell’s equations and to findL* y J

the proper solution of our problem. In the electrom agnetic treatm ent, then, the divergence encountered in the function Fkfor k = 0 will disappear.

3. T he electromagnetic field of the crystal

Born (1923) has calculated the electrom agnetic field of the crystal as originated by a superposition of spherical waves arising from the vibrating point charges. (This corresponds to the introduction of the "o p tic a l" instead of the electrostatic potentials (cf. Ewald 1921).) I shall give here the derivation as far as is necessary for our purposes. From the electrom agnetic field the forces acting on any particle and therefore the coupling coefficients will be obtained.

The result will be th a t the coefficients derived in this way will again be expressible as the second derivatives of a potential function to which one can apply the same transform ation as used in the preceding section. I t will be found tha t for all wave vectors k # 0 the two potential functions are identical, so tha t the results of the preceding section can be used w ithout modification. In the case of infinitely long waves (k = 0) which leads to the frequency of the residual rays, the electrom agnetic derivation will show that a modification of the potential function (2-13) is necessary. This can be applied immediately by com paring the potential function obtained by the electromagnetic derivation, with the representation (2-13) of the electrostatic values with E = co. This modification will remove the divergent term in the potential function (2-13).

I t may be surprising at first sight th a t the case of infinite waves has to be treated separately and cannot be obtained as a limiting case of long waves. The reason for this can easily be seen (cf. Born and G oeppert-M ayer 1933, p. 732). Starting from a finite crystal with n3 = N cells and a finite wave-length A, one has to deal with a double lim it: N -> co and A->- 00. Now for finite wave-lengths and a finite crystal one has in general

nro(ro lattice constant).

Here one must proceed clearly first to the lim it co. For infinite wave-lengths, on the other hand, one must first pu t A-> 00 and then proceed to the limit There is, of course, an interm ediate region where A is of the order of the linear dim ensions of the crystal. But this region is negligibly small in the scale of the numbers kl = hjn, k2 = h2ln, k3 = h3/n (cf. (1-15), (1-16)), which are the components of the wave vector k in the reciprocal lattice. For, if A is of the order I = nr0 then = 1/A is of the order k ~ l/nr0~ bjn,hence klt k2, k 3 are of the order l / » ~ 10~8 for / = 1 cm.If the accuracy of the determ ination of kl9 k2, k3 is even as high as 10-6, the region of the waves of "macroscopic dimensions" is not detectable and can be replaced by the point ki = k2 = k3 = 0. There is an apparent discontinuity at k = 0, and it is therefore no



520 E. W. KELLERM ANN ON TH E THEORY OF THE

con trad iction th a t this po in t satisfies ano ther equation th an the rest o f the spectrum . F or a detailed discussion of this discontinuity the know ledge o f the electrom agnetic field will be needed.

T he electrom agnetic field can be represented (Born 1923, p. 761) by the field of v ib ra ting dipoles plus an electrostatic field, the la tte r o f w hich does no t give rise to a force in lattices o f cubical sym m etry and for sm all displacem ents.

T he m om ent o f such a dipole is

eK u lK, w here u lK — e2ni{k> r«).

T h e field can be described by the H ertz vector

Z = S e-i<ote2ni(k,t)9(Z.o)

w hich is the sum of H ertz ’s solutions for the various v ib ra ting dipoles. T h e vector S is a function o f space. I t has been determ ined by Born (1923); he finds

S = 2 P « S (r-r„ ), P„ = «,U„ (3-1)

1 e 2nUbh)r)w here Sis the F ourier series — — 2 te:— £2»

k0 A0 27 '

(3'2)

A0 is the w ave-length in vacuum corresponding to the frequency <y0.Now separate the zero term of the series (3*2) w hich represents the m ean value o f the

electrom agnetic fieldS = ~5+S,x '

^ 1 1 1irvak2—kl > (3'3)

w 1 pr)

5 = ^ ? ' ( V f k Y - k y ]

w here the refractive index n = k/k0 — A0/A has been in troduced.T he H ertz vector corresponding to the m ean electrom agnetic field is,

(3T ), (3-0), given byaccording to

(3-4)

where p _ p ^ g—i(ot r)

1o - - 2 p «-

ua k

(3-5)

P m ay be in terpreted as the m om ent per un it volume.




From the H ertz vector (3-4) the m ean electrom agnetic field vectors are obtained:_ _ i ,927 1E = grad d i v Z - ^ - p - = 4)7;^—y { P - n 2s(P ,s)} ,

1 ,dZ —curl-^- c ot 4* r f r r i [ s x p ]>

k I 8.

(3-6)

H ere s is a unit vector in the direction of propagation of the wave.These are the same formulae as those which one obtains as a solution of M axwell’s

equations for a plane wave by putting B = H and splitting up D into E + 4ttP as has been pointed out by Born and G oeppert-M ayer (1933, p. 776).

In order to discuss the formulae (3-6) one has to consider the m agnitude of

n — k/kQ — A0/A

which will make it possible to distinguish between the two different cases 0 and k =(= 0.

The frequency of the fastest vibrations occurring in crystals is of the order 1013 sec.-1, and the corresponding smallest wave-length A0 = l/k0 ~ 10-3 cm. This is very large com pared with the lattice distance ( ~ 10-8 cm.). For the therm al vibrations of the crystal one can consider the wave-length A as very small com pared with the length A0 of the light wave of the same frequency in vacuo and can therefore put n — co. The only exception is k = 0; in this case n vanishes since =(= 0 (cf. (3-2): 4= 0).

Therefore, in the case A — 0 (3-6) becomes

E = — 4ttP , H = 0; k = 0, not neglected, (3-7)

and in the case k4= 0

E — — 4?rs(P, s ), H = 0; 4= 0, neglected. (3-8)

The value (3-0) for the H ertz vector developed here can be used for the solution of the dynam ical problem of the proper vibrations of the crystal only under the condition th a t the system can be considered as closed, i.e. tha t there is no emission of radiation. I t will be seen, however, tha t this is the case for all values of except 0. In these cases {kH= 0) the total m om ent of the crystal must vanish, since all the dipoles in the crystal vibrate with a difference of phase which is not invariant against a translation of the lattice vector. T h a t can be seen a t once (cf. Born and Goeppert-M ayer 1933, p. 643) from the expression for the total m om ent itself

l l P i = H > X = e~i(0tI e Ku y ™ (k>r<> 2 (3-9)I k I k k I

This sum is zero for all values of k except k — 0.I shall now show th a t our solution for the H ertz vector corresponds to this fact.

The optical properties of crystals have been carefully investigated by Ewald (cf. Born

V ol. 238.—A 64




1923, p. 774). Consider a crystal which is bounded on one side by a p lane perpendicular to the vector b 3, so th a t

(r, b 3) > 0 , inside the crystal, 1

(r, b 3) < 0 , outside the crystal.]

T he H ertz vector for such a “ half-crystal” has been found by Ewald. I t is

(3-10)

Z (in)

2 (out)Z (0)+ Z (1).z<2), (3-11)

where Z (in) and Z (out) are the H ertz vectors inside and outside the crystal. H ere the vectors Z (j) ( j — 0, 1, 2) are determ ined with the help of equations (3*0), (3T) by the functions S{j)given by

Sm

Sa)

§<2>

e2ni(bh,r)

OTa h(b ^ + k )2—2i +00 g2nilli(*9bj) +fe(r5 2) +7/i(r> s)

Vah?h^ ( 7 i - 72) ( l - ^ 2m>0 :

2? +G0 277Z*[Zi(r5 b!> +Z2<r> b2) +?/2(r> b3)l

(3-12)

S,(0) is identical w ith the expression (3-2) for the infinite lattice.T he constants rjl 3 rj2 in (3-12) are the two solutions of the equation

(/ib i + /2b 2+ 7 b 3+ k ) 2 —These are given by

71,72 = ^2 1-(b3> <i) ±V((b3> q)2+£§ (ô- 72)}]5|

q = /1b 1 + /2b 2+ k . jI consider first the case when either lxor l2 or both are different from zero. W riting k as

k = kxbj d- k2\i2-f- A3b 3,

where — (cf. § 5), we see th a t q cannot be parallel to b 3, so th a t

(Kq)2< 7 2£|-

Since k0<| b j| it follows th a t k0in (3*14) m ay be neglected, so th a t the square rootis im aginary and ijl32 are conjugate complex. In this case for the solution w ith positive im aginary p a rt must be used, and for rj2 the solution with negative im aginary part. I f such a complex solution is inserted in (3-12) it follows with (3-10) th a t S,(1) and S(2)decrease exponentially w ith the distance from the surface. They represent only small surface effects and m ay be neglected.

(3-13)

(3-14)




Turning to the zero terms of the series S{1) and (i.e. = /2 = 0) it is found tha tthe solution (3T4) reduces to

7 i,2 — (^3> k)±V{(b3, k )2 + £§(A$ —A:2)}], for 0. (3*15)

I f k#= 0, so th a t k0< k, kQ m ay be neglected again and the solutions 2 are again conjugate complex, if k is not parallel to b 3. In this case exponentially decreasing expressions are again obtained. The only exception is when k is parallel to b 3. Here I get from (3T5)

7i»2 = "V*0; for (b„ k) = |b , | I k | (3-16)y3

and tjl9 2b 3 = — k ± k 0. (3-17)

Inserting these values in (3-12) one obtains with (3-0), (3-1) the H ertz vectors Z (1), Z (2) corresponding to the zero terms of Sil), S{2).

Z (1) ? i(otg2ni(r9 k0)

va b3kQ( 1-~**»«*-*wj-J2 p g2iri(rK9 k —ko)^

— 6—i(ot _g-2ni(r9 k0)

va b3k0(l — e2ni{k+kg2iti(rK,k

(3-18)

Since k0< k, k0< b3 and ( rK, b^) ~ 1 it follows th a t (r*, k 0)<^ 1. To the first order in k0 the result is

i p-i(ot fi2m(r,ko) V n ) yd) _ _ _______ _ ^ k V k

va b3k0 1 — e2nik/b3'V 0—io)t n—2m it ko) V n7(2) — _____________Z/cr/c

vab3k0

(3-19)

In this case there is indeed an outgoing radiation field. But the effect of this field is negligible. This may be seen either by calculating the force from Z (1) or by calculating the energy flow of the outgoing radiation from Z (2) and com paring it with the energy of the oscillator. I f this is calculated by means of (3-6) only values of the m agnitude k0/b3 are obtained which can be neglected.

The field of the crystal for k 4= 0 is, therefore, completely determ ined by 6,(0) (3-12) or (3-2), which gives the right solution for our problem for k =# 0.

Now only the case k = 0 remains to be considered. For the zero term of the Hertz vectors Z (1), Z <2) one obtains once more the expression (3-18) and need only pu t 0. But in this case k0/b3 in the exponential function of the denom inator cannot be neglected. Expanding this exponential we find, instead of (3-19),

Z (1) =j[ g2ni(t9 ko)

27TVa MI 0—io)t g—2m(r, ko)

27TVa MZ (2) =

(3-20)




Z (2) represents again an outgoing wave, b u t the am plitude contains Ijkl and therefore it cannot be neglected.

Therefore the crystal cannot be considered as being in a stationary state for k = 0; in order to render it stationary rad ia tion has to be sent to and absorbed by the crystal. Indeed , residual rays can only be observed by absorption or reflexion m easurem ents, as the loss o f energy o f the incident beam . Now it is well known th a t the boundary conditions for electrom agnetic waves are such th a t the incident wave is continued in the in terior by the diffracted wave, b u t this is identical w ith the m ean wave of the dipoles. This la tte r has, therefore, to be considered as produced by the incident wave, i.e. as a given external wave w hich acts on the particles o f the lattice. T he in teraction forces of the v ibration p roper o f the lattice are therefore described only by the periodic term s of the field strength. For this reason one has to cancel the constant term in the expression (3*2), (3*1), (3*0) of the Hertz vector.

In this way is found the na tu re of the ap p aren t discontinuity in the spectrum a t k = 0 which has been in troduced a t the beginning o f this paragraph .

4. T he coupling coefficients of the electrodynamic interaction

I t is now possible to w rite dow n the H ertz vector for all states of vibrations of the crystal (3*0), (3*1), (3*2). Since k 0 is small com pared to the vectors of the reciprocal lattice, k 0 m ay be neglected in (3*2) except for the zero term bA = (0, 0, 0). As long as k 4= 0, k 0 m ay be neglected also in the zero term , as has ju s t been shown. For k ------- 0,on the other hand, the zero term m ust be om itted. Therefore one finds for :

X e 2ni(bh, t )

s = m J ' ( V F k p ’ (4-0)

where the dash indicates th a t for k — 0 the zero term b A = 0 m ust be om itted .fH aving now determ ined the H ertz vector one arrives in the same way as in § 1 a t

the equation of m otion by w riting down the expression for the force which is determ ined by the H ertz vector

Z ‘* -- Z - z i , (4-1)

where z lK is the H ertz vector of the dipole (a:, /) itself which has to be subtracted. From (4-1) one obtains by means of (3*6) the electrom agnetic field which is given by (cf. (3-0), (3-1). (3-2))

EL = g rad , div Zj* ~ f l*

= <r“ e2-rilfc.d | { |A y^ # ‘ ( r - r i ) + | (4-2)k'¥=k

f While following closely and in formal identity the derivation of Born and Goeppert-Mayer (1933, p. 776) it must be emphasized that the dash in Born’s function means complete omission of the zero term, irrespective of the value of k.




whereI p2m(hh+k,t) \

f k(r) = Sir)e2<"',t>'1 = — I ' , . , , , 17 K ' w m>at (t>«+k)2’

p-t(r) = ^ ( r ) - A -

(4-3)

T he terms due to grad div Z** contain factors {bh-\-k)x (bh+ k)y3 whereas the term due 1 •* (iP*to has the factor — 4tt2 and can therefore be neglected. The m agnetic field

vector, too, can be neglected as it gives only a second order contribution to the force.From (4*2) one obtains easily the force exerted on the particle (x, I) and the equation

of m otion

* æ + i z E ; ' ] ^ = o, (4-4)

with [x y ] =e^ [ d x d y r { r ) \ „ J 2" ’(4-5)

Com paring these electrodynamic expressions with the corresponding electrostatic ones (2*11) it follows th a t they are identical, if the function r) is replaced by ). In order to com pare these two functions I take the param eter in (2T3) equal to oo. Then Fkis given by

Fk( r)277z(b +k,r)

^ 1 ( 1 >4+ k ) 2(4-6)

This is identical w ith the expression (4-3) for f k(r), w ith the exception of the case k = 0. In this case the divergent zero term in is om itted in O ne finds

f k{r) =Fr ) ; k 4= 0,- 1 p2m(k9r)-i

# 0 ( r ) = l im ; k = 0./c-H) L 7tV0 K J(4-7)

Thus for k 4= 0 the electrostatic derivation is justified which is only natural since the retardation represented by k0can be neglected. Fk (2-9) is defined as a sum of potentials of point charges. Therefore it satisfies Laplace’s equation

AFk = A fk = 0; k 4= 0. (4-8)

W ith (4-5) the following relation follows:

cM + r c + ? C = : f M = ° -Lx x J Ly y A LzzJ 7 Lx x J (4-9)




For k = 0 (cf. Born 1923, p. 728), on the o ther hand, direct differentiation gives

and

— limkÔ L

AFk+ — enmzi r »2L-»O L TlVa K J

477 ,— : k = 0Va

z T * 4 7 7 ^ 'X LX x Ak=0 Va

(4-10)

(4-11)

This holds also for k' = kif one substitutes ijr0for ijr0 in (4-10). T he potential i/r° satisfies, therefore, Poisson’s equation, which m ay be in terpreted as the existence o f a uniform charge distribution of am ount — 1 per un it cell.

In the case of cubical sym m etry it follows th a t

M\_x x J4* eKeK, 3 Lx xJk=o~

f * K'~] = T * f | = cycl = 0.Lx y J k=0 LxyJk=0

47t

"3 V(4-12)

T hen one gets for the coupling force in the equation o f m otion (4-4)

Fa = Y [ * Kx \ U s , = f v = f

which is the well-known expression for the Lorenz-Lorentz force.

T he question w hether the sum 2 P" 1 is equal to zero or to 4tt i.e. w hether

the potential from which the coupling coefficients are derived satisfies Poisson’s or Laplace’s equation has been discussed before, b u t has never been cleared in a satisfactory way. T he definite result of the foregoing derivation is th a t it proves w ithout artificial assumptions the apparen t discontinuity (cf. p. 519) of this potential as function of the wave vector k . Lyddane and Herzfeld (1938) agree with Born th a t for k =» 0

_ r*% I . £ ,the sum 2 is really equal to 477-^ and thus derived from a potential corre-

X LX xJk=o v a

spending to a uniform charge distribution — 1 per cell, b u t in order to explain the different properties of the potential for k 4= 0 they assume th a t for all o ther wavelengths one has to perform a displacem ent of the uniform charge distribution, which gives rise to surface charges which ju st compensate the term kn\va so th a t the above sum is equal to zero. This argum ent is not only ra ther artificial, since the displaced uniform charge — 1 has no physical significance, bu t leads also to consequences which are only partly correct. They obtain two different frequencies for the residual rays, one for “ transverse” waves which corresponds to the value hitherto found— and also in this paper— and one for “ longitudinal” waves.




5. T he coupling coefficients of the N aCl lattice

In the case of N aCl there are two different particles in the cell 2); but, as far as symmetry is concerned, it m ay be regarded as a simple lattice, i.e. the N a sites are entirely equivalent to the Cl sites. Therefore, to any lattice vector there corresponds another one of the same m agnitude and in the opposite direction. The same is true for the reciprocal lattice. For this reason the coupling coefficients are all real, as may be seen from (1 T 0 ); in this sum all im aginary terms cancel so tha t

[* y ] = ? (#«■)«» cos 2” (k > r ^)>

G y] “ ? cos 2”(k’ a');(5-0)

and since = - r A , [ * * ] = [* ^ 1 )

From (4-5) it can be seen th a t the coefficients are entirely symmetrical in and y so tha t

M S p f l . (5-2)Lx yJLy x j

Furtherm ore, the N aCl lattice is entirely symmetrical in N a and Cl (if the repulsive forces between all bu t nearest neighbours are neglected); therefore

G 3 H J 3 - <*-3>

The cell vectors of the NaCl lattice are

a i == r0(0, 1, 1), 1*21 = 1*12 = (I) I) I), ja 2 = r0 ( l , 0 , 1), va = 2 1 (5-4)

a 3 = ?o(l, U0) , J

where r0 is the distance between nearest neighbours or the lattice constant and thevolume of the cell. A lattice vector is therefore of the form

a* ~ ro( 2 + 3» 3 + l» ^1+^2) mro(4> (s)> 2 4 even,X

r 21 = ro( 2 + 3 + lj 3 + l + l 5 l + 2+l ) — ro{m x> m y> m z ) ’ odd.« X

Here lu l2, /3 cover all integer numbers. Therefore, ly, lz cover all sets of integers for which lx I ly I 1/2 = 2 4 is even and mx,my, mz cover all sets of integers for which 2 mx is

X Xodd.




T he reciprocal vectors are given by (1*13). W ith (5*4) they are

b i = 2 7 o ( - l , 1 ,1 ) .

b 2 = ^ r ( l , - 1 . 1 ) .

1, - 1),

so th a t a vector in the reciprocal lattice is given by

^ h==2 r h l +h2—h3)

(5-6)

12r0

(hx, h ,hz), hx, h , h2 all odd or all even, (S'?)

where hx, hy, hz cover all sets of integers w hich are either all odd or all even, i.e. thereciprocal lattice is body-centred, as is well known.

T he sums (2T5) representing the coupling coefficients m ay now be w ritten in dim ensionless form by w riting e/r0 for the param eter Eand substituting for the wave vector k :

k — klb l + k2h 2+ k3h 3 — g~j <7J ,2r0

qx — k2 + k3—kl} qy — k3-\-kl — k + —k^,

IE i P 1_] = _ Gn + H + - ^ - e 3# 'e2 L* y j ’* + , xll + 3 j ir *«’

—Gl2e* l x yj «'

(5-8)

(5-9)

where

Gil = 2 e~ 5 <h+,)2 cos n(!y¥kt +hz).

m

x yP cos 7r(q, 1]

2_ e~‘v i[d )Jne P + P ’4 „ . 6 e-«2'2 , 3Wei)g(l) = - f e3e- s«r-+ e

f(el) = l - ^ J o e"p 4 ,

< - W I - i V W + 5 + ® fThese equations only hold for A: 4= (0, 0, 0),

/3

(5-10)




For some wave vectors k the calculation can be reduced appreciably by considering the cubic symmetry.

(1) qx — qy = qz. This case is symm etrical in z. Therefore from (4-9) it follows th a t

cpc k' I cVkk'~\ cPa: ■L X x j L y y \ Lz J — 5 9 z-(5-11)

(2) qz — 0 (or qz = 1)*. In this case in Gxthe two terms A J and hy, —hz)ju st cancel and Gxz — 0. Putting e = co,Hxz also vanishes (the coupling coefficients are, of course, independent of e). Therefore

T li=T121=T H -T 2l=o.[_x z j [_xzj L z j L z j (5-12)

(3) Consider two wave vectors which differ only in the sign of the z-component. In this case pu t e — 0 so th a t only a contribution from is obtained (cf. (5-10)st); it follows tha t

P P = _ P P P 2i = _ P PLx z j k Lxzj k.’ z j kp ii PI 11 ri 2i PI 2iLy zJk “ L yzjk,’ zJk ~ L

(5-13)

(7*5 (7 k/ 1/5 (7z ?Z5

whereas all other coefficients are the same for both wave vectors.(4) Consider two wave vectors k , k ' which are identical apart from an interchange

of the components of qx,qy. Taking again e — 0, then

PI i t _ r i i i r i 2 i _ r i 21Lx x jk ~ LyyJk'5 Lxd k " L y y

c a - c a . c a l c a ,(5-14)

all the other coefficients being equal.

* This can also be seen by considering the function only (e = 0). For =(= we find developingcos 7r(q, 1) = cos n(qxlx + qyly+ ),

Hxi22^(7) sinn qxlx sinn qzlz cosn qyl. (5-10)sl

All other terms arising from the development of cos 7r(q, 1) cancel because of symmetry lz assume the same positive and negative values).

We see at once from (5-10)a that for qz = 1 (or qz — 0), Hxz — 0 and therefore

It is not surprising that these bracket coefficients vanish, for in the reciprocal lattice the point given by the position vector qx — qy = 0, qz = 1 is symmetrically situated with respect to all lattice points. If qx, qy =t= 0 the symmetry in the z-direction is not destroyed, and it will be seen from (e — co) thatthe coefficients in question vanish on account of this symmetry.

V ol. 238.—A 65




From the condition of the cyclic lattice (IT S ), ( I T 6) it follows th a t kl} k3 range from 0 to 1. In order to obtain a fair survey of the coefficients as functions of the wave vector, I divide this range into tenths and consider the vector com ponents

A = {Pi>p2>Psin te g e r s ) . (S T 5)

In order to m ake full use of the sym m etric properties of the coefficients it is m ore convenient to consider the region of allowed wave vectors in the qz space. Since the reciprocal lattice is of the body-centred type this region is of the form of an octahedron w ith its vertices cut off (cf. for exam ple, Sommerfeld and Lethe 1933, figure 27). T he boundaries are given by the equations

qx± qy± q z = ±f ; <7* = ± 1 , < = ± 1 , <?z = ± 1 . (5T6)

In view of the properties (5T3) ju s t considered one m ay restrict the calculation to positive values of qx,qy, qz. All other coefficients m ay then be obtained from (5T3). Furtherm ore, in view of (5T4) the calculation can be restricted to sets of num bers such th a t qx^ q y^ q z- Thus only those values of q such th a t

(£H 7)

?x+?ir+7z f> Jneed be considered.

I t will be convenient to introduce as in (5T5) whole num bers p v p z given by (5-7)

A — P2 ~Srp3~Pl> Py — Py~Pi ~Pzi ( 6 T 8 )

the sets of whole num bers p x) p y1 p z are either all odd or all even and satisfy the conditions

P,+P,+P*<1B- JT here are forty-eight sets of num bers of this type.

I could, of course, also divide the range in the qx) qz space into tenths. But this would give more sets of num bers p x,py , p z i i.e. a closer division of our range which is not necessary for our purposes. If, on the other hand, I divided it into fifths, it would give too small a choice.

6. T he repulsive forges

A part from the Coulomb forces there are other forces present in an ionic lattice, m ainly the repulsive forces which prevent the lattice from collapsing, and also the van der W aals forces. Let us collect all those other forces in a potential These forces decrease very rapidly with the distance, w ith the exception of the V an der W aals forces (cf. (Lennard-)Jones and Ingham 1925). But, as Born and M ayer (1932) have shown, these forces form only a very small percentage of the potential so th a t it is sufficient




to consider only the interaction between nearest neighbours. In the N aCl lattice each ion is surrounded by six nearest neighbours so tha t the energy per cell is given by

= — a —+ 6y(r0), (6*0)'o

where a is the M adelung constant and r0the distance between nearest neighbours.The first two derivatives of v(r)can be obtained from the condition of equilibrium

and the compressibility; putting for abbreviation

the condition of equilibrium is

so tha t

dr0e2 e2

IT 65,

( 6- 1)

( 6 ' 2)

using M adelung’s constant — 1-7476. The compressibility is given by

1K

o i < P j$0. i drl 18r0 drl 18r0 Va = 2r|,

so tha t (6-3)

A depends, of course, on the particular crystal considered. Here N aCl has been chosen where k = 4-16 x 10-12 cm .2/dyne, r0 = 2-814 x 10~8 cm. W ith e = 4-8 x 10-10 e.s.u., I get

A — 10-18 (for NaCl). (6-4)

Consider now th a t p a rt of the coupling coefficients which is due to the repulsive forces. This part is given by (1-10), (1-12),

T* ] = 2 W A , (6-5)

In the case k' =1= kthe sum (6-5) extends over the six nearest neighbours given by the vectors

r0(zh I? 0, 0), j

r0( 0, d= 1, 0), I (6-6)

r0( 0, 0, ±1).J65-2




T he differentiation in (6*5) yields w ith the definition (6T)

T l 21 1*2Lx y\%va i

y l nil X12!/12W 2)2

+ b (s —+ T V W 2) 2 / J

g2m(k} r12) ( 6 -7 )

T he sum over the vectors (6-6) gives

* r i 21 n* n 21 vU y j = ° ’ * + y '\_x x J ? ==A008 27r^r0+ Jg(cos 277^r0+ co s 2irkzr0).

( 6-8)

In the case k ' — k only the zero term in (6-5) rem ains, all o ther term s representing interaction between d istant ions:

\_x y \ = (6-9)

This expression is independent of k and can therefore be calculated from (1-11)

T1 il+T121 = 0, lx y l Lx

s° th a t i * y = ° ’ x * r > i * 3 ? = - ^ + 2 £ )- c«-l0)

I f for k the dimensionless vector q given by (5-8) is introduced, (6-8) and (6-10) m ay be w ritten

V I - T I - 0 .L x y j L x y j

[ * = + l3 2 = ^ 008 7Tqx + B(cOS 77 + COS lTqz ) .

(6-11)

7. T he elastic constants

Born (1923) has shown th a t the coupling coefficients in the equation of m otion are connected with the elastic constants.

T he la tter are defined by Born (1923, p. 547)

[xyxy] =~I2(<!>LKK) Xy xlKK-yLKK'-aKK* I (7-0)

The elastic constants in the usual notation are related to the bracket symbols (7-0) for a crystal of cubic symmetry as follows:

[xmr] —

[xxyy] =[xyxy] = 544.

C12~C44

(7-1)

The Cauchy relation (7-2)




holds; this is a consequence of the assumption of central forces in a simple lattice with central symmetry about every particle.

O ne finds exactly the expressions (7-0) if the coupling coefficients (T10) are developed in a power series of the inverse w ave-length; w ith 2?rk — rs, where s is a unit vector one finds

p c k'1 _ p c /c '1 <0> , p c Ar," |(1)T + p : k ' 1 (2),Lx y J Lx y J + Lx y J lx y J (7-3)

where pc yc,-](0)Lx y A

MLx y j

ru iLx y j

yv-id)

y"k Ar'n(2)

y

2 k'k) xy i I

« 2 ( (§i-k'k J xy y

Ii 2 2 % % x ^ y ^ ( ^ J .

I x , y

(7-4)

Com paring (7-3) with (7-0) one notices tha tHe *■'i (2)

= 2 , „ •X, § KK l# y J (7-5)

Cr-KIn order to calculate J , develop the expressions (5-9), (5-10) for the coupling

coefficients of the N aCl lattice. The result is

i T * *1 = afT1 H ^ + T 1 2TLx x j I Lx xj Lx x j

r0 PITT h

'2+ 4

[ I | X.W + 2 [(/(/) +g(l)I) li- 2 [(/(rn) +«(m) H < ] ]

2 x » t o + 2

; 2 x « t o + 2L" h I

l l /2l2)

nr2

2m

- 2 m

2! K H

(/W + g (0 - 2 ( / M +g(rn)

where

X # ) = p 'h2h2niiMy A2

h - i hl\A4 ‘ A2H e2 1 hy 4 / e4 8

A2 7T2 L * _ _ A2h2

(7-6)

(7-7)

Cr*- y^'q(2)Similarly one obtains for Lx yJ

Cr> *-/- | (2) rcT l T V 2) e n o-i

$ c ;] H C ] -H 3cn i i (2) . cn 2i (2)

y.z?2 pi /2 /2 772 Z72"l

<7-8)




where4 _ fr2/*2 r /?2 L2

4e- 8 -2-Jh? L

2 ~ 1 + 2Â2 e2 1 A2 + Â |A 2-|

e* 4 J(7-9)

Com paring (7-6), (7-8) w ith (7-5) one finds th a t p a rt o f the elastic constants which is due to the Coulom b forces:

<?2 i 2 n% i = c[ » « ] = " l E l X x W + I M + s m

v a r0 1" A L I J

cc,jS*J.2Vo***

2x„to+2 ~ m + g ( i ) lj ?

l l l l

11-2m

l l - 2

f ( m ) + g ( m ) ^ m 2

f(m )+ g(m )^ f ]m 2}, - (7-10)

% 4 = c[xyxy] = 2 x W + 1 §(l) ljj r - } •var 0v" a I 1 m m )

T h a t p a rt of the elastic constants which is due to the repulsive forces can be calculated in the same way. From (6-8), (6-10) one has

Skk '

1skk '

pc k'i {2)_ j Rr i r i (2) La; a; J 1 La; a:J

pi 9~i(2)) p 2 ^G3

'x '-c;

II /

c>

I

so th a t ? e2 AR _ R11 — o ) 12 — q ) 44 — O.

VJo2

(7-11)

(7-12)

T he C auchy relation need not be satisfied for the individual contributions o f the Coulom b and repulsive forces, b u t only for the sum, if the condition of equilibrium is taken into account.*

T he num erical calculation gives

^ „ 62% i = - 2*562^; X = 5 -09^ ; C\\ —

e2 — e2 62% 2= 1*28^; % 2 = -0 -5 8 3 ^ ; r12 = 0-695^;

%4 = o -6 9 e2 ; RC\\ — O, r44 = 0 -696^ .

* The reason is seen from the direct derivation of the elastic constants; they are given by Born (1923, p. 548)

\xyxy\ = jr—■ 2 2 \PxyPKK' KK'V KK'“ va kk' I

where

The equilibrium condition leads to the disappearance of the term PlKK,, if is the total potential, and

\xyxy\ — ~n~ 2 2 QK * Va kk' I

is symmetric in all the four indices x,y, x, y and can be written \xyxy\ — \xxyy\. But this does not hold for the Coulomb part of the potential separately.




The value ccncalculated by M adelung’s m ethod has been given by Born (1920);e2 .

he finds the value — 2-55 7-7. Since the num erical calculations are confined to an2 r%

accuracy of 1 % , the two values agree with each other.In table 1 the results are com pared with the experim ental values.* The units are

dynes/cm .2 x 1011.T a b le 1

Exp. Theor.cll 4 96 4-65C12 14 1-28( 11 + 2 12) 2-581 2-401/k 2-40/

C44 1-28 1-28

The theoretical value of the reciprocal compressibility (2-40 x lO 11 dynes/cm.2)3satisfies, of course, exactly the relation - — r11 + 2cl2 w ith the theoretical values of

and c12, and coincides with the experim ental value since this value has been used for the determ ination of the constant A.But it differs from the experim ental elasticity constants. This is an incongruity not of the theory, b u t of the experim ental results. In fact, the measurements of cl2are ra ther inaccurate (it has been measured only within 10 % erro r). Therefore the experim ental values agree with the theoretical values w ithin the limits of error.

The perfect agreem ent of the experim ental and theoretical values of are a strong confirmation of the theory ; for, the theoretical value of does not depend on A (or , bu t only on the lattice constant.

8. N umerical results for th e coupling coefficients

For the purpose of num erical calculation of the coupling coefficients the adjustable param eter ein equation (5TO) has been chosen equal to 1. All terms smaller than 1 % of the largest term in each series were neglected. Since the series in question converge rapidly this gives an accuracy of 1 % —2 % .

All coefficients have been calculated independently from each other so tha t the equations (4-9) could be used to check the results. These equations were satisfied in each case within 1 % . The only exceptions are p = (6, 6, 0) and p — (6, 6, 2). In these cases the final coefficients are the differences between two nearly equal quantities and therefore the error is larger than 1 % , bu t as in this case the electrical p a rt is much smaller than the repulsive part of the coefficients, the total error is again not larger than 1 % so tha t it is not worth while to increase the accuracy of the electric part.

* The elastic constants have been calculated from the elastic moduli given in 1935) 3 Erg.Bd. 1, 74.




A part from this, each coefficient has been checked independently either by calculating the coefficients belonging to the wave vector —qz) w hich involves a differentorder in the evaluation of the terms of the series or by repeating the calculation w ith a different value of e.Each coefficient has therefore been calculated twice by means of num erically different series. T he agreem ent in each case was w ithin 1 % .

T he results are given in tables 2, 2a,which contain th a t p a rt o f the coupling coefficients which is due to the Coulom b force. T he values are given in units of e2/va and depend therefore only on the lattice structure, b u t not on the volume. In the first

colum n the wave vector of the m ode of v ibration is given in units o f — ; the num bers

describing the wave vector are identical w ith the com ponents px, py, pz in troduced in (5T9). T ab le 3 contains the to tal coupling coefficients for N aCl, obtained by sum m ing the contributions o f the Coulom b force and the repulsive force (cf. ( I T 2)). T he la tte r contribution, of course, depends on the properties of the N a and Cl ions. T he coefficients are given again in units of e2jva.

Since the coefficientstr*£

J = 0; x4= y(cf. (6-8)), the bracket symbols

cpc k'~\_ pc k'~ [_x y A lx y A

are already the total coefficients and therefore table 2 and table 3 represent the whole set of the coefficients for the equation of motion.

9. Evaluation of the frequencies

In the case of N aC l (s----- 2) the equations of m otion for any given wave vector k constitute a system of six homogeneous equations for the am plitudes U* of the vibration. For a non-trivial solution the determ inant of the system m ust vanish. This leads to the secular equation for the frequencies:

1 1

2x1 1y x2 1y x1 iZ X

2 1 Z

1 2

2 2

1 2 )

y2 2y x1 2 z2 2 z x

1 1 x y2 1 x y1 i

y y2 1

y y1 i z y2 1 z y

1 2 x y 2 2 x y1 2

y y 2 2 y y 1 2 z y2 2 z y

1 1 x z

- X

0. (9-0)




T a b l e 2

p x P y P z

10 5 0*10 4 010 2 210 2 010 0 09 5 19 3 39 3 19 1 18 6 08 4 28 4 08 2 28 2 08 0 07 5 37 5 17 3 37 3 17 1 16 6 26 6 06 4 46 4 26 4 06 2 26 2 06 0 05 5 55 5 35 5 15 3 35 3 15 1 14 4 44 4 24 4 04 2 24 2 04 0 03 3 33 3 13 1 12 2 22 2 02 0 01 1 10 0 0

° n 1~ K

-0*785 -1*812 -2*938 -3*606 -4*330 -0*714 -1*583 -2*699 -4*047 + 0*059 -1*471 -1*975 -3*184 -3*911 -4*738 -0*286 -0*894 -1*859 -3*049 -4*699 + 0*140 + 0*017 -0*552 -1*594 -2*090 -3*694 -4-563 -5-782

0-0*228-0*600-1*659-3*050-5*526

0-0*751- 1*220-3*115-4*642-6*987

0-1*362-5*377

0-1*893-8*013

0

T H sI JUyj e+ 1*594 + 1*640 + 1*468 + 1*961 + 2*160 + 1*453 + 0-795 + 1*663 + 2*022 + 1*195 + 1*142 + 1*450 + 1*595 + 2*031 + 2*366 + 0*571 + 0*846 + 0*932 + 1*527 + 2*350 + 0*140 +0*017 + 0*279 + 0*621 + 0*684 + 1*803 + 2*066 + 2*891

0-0*228 -0*600 + 0*836 + 0*840 + 2*764

0-0*751 - 1*220 + 1*561 + 1*414 + 3*512

0-1*352 + 2*684

0-1*893 + 3*997

0

-0*785 + 0*181 + 1*468 + 1*638 + 2*160 -0*714 + 0*795 + 0*987 + 2*022 -1*298 + 0*339 + 0*535 + 1*595 + 1*882 + 2*366 -0*286 + 0*062 + 0*932 + 1*526 + 2*350 -0*274 - 0*022 + 0*279 + 0*976 + 1*428 + 1*803 + 2*500 + 2*891

0+ 0*450 + 1*214 + 0*836 + 2*224 + 2*764

0+ 1*502 + 2*455 + 1*561 + 3*228 + 3*512

0+ 2*706 + 2*684

0+ 3*796 + 3*997

0

V[_x el

+ 10*981 + 12*166 + 13*448 + 14*214 + 15*043 +10*530 + 11*513 + 12*860 + 14*430 + 8-549 + 10*375 + 11*079 + 12*524 +13*402 +14*386 + 6*838 + 8*148 + 9*281 + 10*970 + 13*021 + 4*312 + 5*004 + 4*933 + 7*120 + 8-025 + 9*933 + 11*200 + 12*683

0+ 2*379 + 3*929 + 5*252 + 7-467 + 10*591

0+ 2*351 + 3*406 + 5-892 + 7-850 + 10*623

0+ 2*407 + 7*282

0+ 2*421 + 8*994

0

T 1 2>U y je 20

-2*313-6*725-6*071-7*520

0-5*758 -4*439 -7*214 + 3*003 -2*375 -1*692 -6*263 -5*633 -7*193

0+ 1*342 -4*642 -3*368 -6*512 + 4*312 + 5*004 -2*470 -0-548 + 0*192 -4*964 -4*360 -6*344

0+ 2*379 + 3*929 -2*628 -1*198 -5*299

0+ 2*351 + 3*406 -2*945 -2*137 -5*313

0+ 2*407 -3*641

0+ 2*421 -4*491

0

T 1 2U «

-10*981- 9-856- 6*725- 8*142- 7*520 -10*530- 5-758- 8*473- 7*214 -11*548- 8*003- 9*391- 6*263- 7*779- 7*193- 6-838- 9*479- 4*642- 7*605- 6*512- 8*615- 9-997- 2*470- 6*566- 8*201- 4*964- 6-844- 6*344

0- 4*761- 7*845- 2-628- 6*267- 5*299

0- 4*702- 6-799- 2*945- 5*717- 5*313

0- 4*813- 3*641

0- 4*820- 4*491

0

* The vector (10, 5, 0) is really not a vector within our choice. I have, however, considered it, since it is one of the corner points of our phase space.

V o l . 238.—A 66




Px Py Pz pL* yJ

P H i\_x z j e210 5 0 0 010 4 0 0 010 2 2 0 010 2 0 0 010 0 0 0 09 5 1 -1-009 — 0-3229 3 3 -0856 - 0 - 8 6 69 3 1 -0-883 -0-3369 1 1 -0366 -0-3568 6 0 -1-796 08 4 2 -1-946 -1 -2 0 18 4 0 -1-985 08 2 2 -1-288 -1-2888 2 0 -1-344 08 0 0 0 07 5 3 -2-810 -2-3037 5 1 - 2 - 8 6 8 -0-8707 3 3 -2-422 -2-4227 3 1 -2-620 -0-9747 1 1 -1-093 -1-0936 6 2 -3177 -1-9376 6 0 -3-234 06 4 4 -3-299 -3-2996 4 2 - 3-606 -2-1456 4 0 -3-806 06 2 2 -2-546 - 2-5466 2 0 -2-772 06 0 0 0 05 6 5 -3-615 -3-6155 5 3 - 3-833 -2-9805 5 1 -4202 -1-1745 3 3 - 3-582 -3-5825 3 1 -4-273 -1-5465 1 1 -1-963 -1-9634 4 4 -3-668 -3-6684 4 2 -4-560 -2-5904 4 0 -5-088 04 2 2 -3-720 - 3-7204 2 0 -4464 04 0 0 0 03 3 3 -3-810 -3-8103 3 1 -5363 -1-8723 1 1 -3-243 -3-2432 2 2 -3-986 -3-9862 2 0 -6-038 02 0 0 0 01 1 1 -4-132 -4-1320 0 0 0 0

P * ] u\_x Z/J e2 [ i a s E ®

0 0 00 0 00 0 -0 3 9 30 0 00 0 0

+ 0-438 0 -0-438+ 0-251 + 0-251 -0-856+ 0-450 + 0-120 -0 2 6 7+ 0-197 + 0-197 -0 0 8 3+ 0-803 0 0+ 0-810 + 0-238 -0-670+ 1-050 0 0+ 0-634 + 0-634 -0-244+ 0-793 0 0

0 0 0+ 0-742 0 -0-742+ 1-423 + 0-075 -0-191+ 0-918 + 0-918 -0-308+ 1-522 + 0-434 -0-051+ 0-688 + 0-688 + 0-006+ 1-293 -0-172 -0-172+ 1-697 0 0+ 0-694 + 0-694 -0-203+ 1-866 + 0-693 + 0-116+ 2-346 0 0+ 1-532 + 1-532 + 0-242+ 1-911 0 0

0 0 00 0 0

+ 1-506 + 0-436 + 0-436+ 2-510 + 0-276 + 0-276+ 1-853 + 1-853 + 0-794+ 2-974 + 0-911 + 0-439+ 1-483 + 1-483 -0 2 3 6+ 1-330 + 1-330 + 1-330+ 2-932 + 1-259 + 1-259+ 3-695 0 0+ 2-760 + 2-760 + 1-270+ 3-637 0 0

0 0 0+ 2-511 + 2-511 + 2-511+ 4-359 + 1-398 + 1-398+ 2-869 + 2-869 + 0-937+ 3-421 + 3-421 + 3-421+ 5-531 0 0

0 0 0+ 3-993 + 3-993 + 3-993

0 0 0

T able 2

o0

-1-03800

-1-009-2-099-0-762-0-281

0-1-815

0-1-077

00

-2-810-1-030-2-190-0-829-0-320-1-937

0-3-205-1-588

0-1-291

00

-3-615— 2-980-1-174-2-642-1-098-0-481- 3-668- 2-690

0-2-034

00

-3-810-1-872- M i l-3-986

00

-4-1320




Table 3

P x A P z P 1-hl_X X j P H sU i /J e 2 [ ‘ D P|_x2> xJ e1 p 2" h

u y je P 2>[_z e2

10 5 0 - 8 635 — 6-256 -8-635 — 0-364 0 + 0-36410 4 0 - 9-662 - 6-210 - 7-669 + 0-461 + 0-833 + 1-12910 2 2 -10-788 -6 3 8 2 -6-382 + 1-383 + 1-734 + 1-73410 2 0 -11-456 -6-889 - 6-212 + 1-927 + 2-165 + 2-26110 0 0 -12-180 - 6-690 - 5-690 + 2-633 + 2-660 + 2-6609 5 1 - 8-564 -6-397 -8-664 — 0-260 0 + 0-2609 3 3 - 9-433 -7-055 - 7-055 + 0-461 + 0-649 + 0-6499 3 1 -10-549 -6-187 -6-863 + 1-385 + 1-545 + 1-6329 1 1 - 11-897 -5-828 -5-828 + 2-532 + 2-468 + 2-4688 6 0 - 7-791 — 6-655 -9-148 — 0-492 -0 3 6 6 -0 0 6 68 4 2 - 9-321 -6-708 -7-511 + 0-837 + 0-771 + 0-8168 4 0 - 9-825 -6 4 0 0 - 7-315 + 1-318 + 1-232 + 1-3728 2 2 -11-034 -6 2 6 6 -6-255 + 2-403 + 1-973 + 1-9738 2 0 -11-761 -5-189 -5-968 + 3-069 + 2-380 + 2-4018 0 0 -12-588 -6-484 -5-484 + 3-820 + 2-764 + 2-7647 5 3 - 8-136 -7-279 -8-136 + 0-169 0 -0-1697 5 1 - 8-744 -7-004 -7-788 + 1-056 + 0-919 + 0-8887 3 3 - 9-709 -6-918 - 6-918 + 1-927 + 1-342 + 1-3427 3 1 -10-899 -6 3 2 3 -6-324 + 3-193 + 2-193 + 2-0777 1 1 -12-549 -5-500 -5-500 + 4-821 + 2-747 + 2-7476 6 2 - 7-710 -7-710 -8-124 + 0-583 + 0-583 + 0-3416 6 0 - 7-833 -7-833 -7-872 + 1-053 + 1-053 + 0-9036 4 4 - 8-402 - 7-571 - 7-671 + 1-067 + 0-676 + 0-6766 4 2 - 9-444 - 7-229 -6-874 + 2-672 + 2-015 + 1-6706 4 0 - 9-940 -7-166 -6-422 + 3-354 + 2-533 + 1-9796 2 2 -11-444 -6-047 -6-047 + 4-902 + 2-689 + 2-6896 2 0 -12-413 -5-784 -5-350 + 5-947 + 3-071 + 2-7536 0 0 -13-632 -4 9 5 9 -4 9 5 9 + 7-207 + 3-031 + 3-0315 5 5 - 7-860 -7-850 -7-850 0 0 05 5 3 - 8-078 -8-078 -7-400 + 1-694 + 1-694 + 1-2235 5 1 - 8-450 -8-450 - 6-636 + 2-821 + 2-821 + 1-8375 3 3 - 9-509 -7-014 -7-014 + 3-882 + 2-671 + 2-6715 3 1 -10-900 - 7-010 -5-626 + 6-683 + 3-678 + 2-7305 1 1 -13-376 -6 0 8 6 - 5-086 + 8-375 + 3-275 + 3-2754 4 4 - 7-860 -7-850 -7-850 + 2-426 + 2-426 + 2-4264 4 2 - 8-601 -8-601 -6-348 + 4-195 + 4-195 + 2-8144 4 0 - 9-070 -9-070 - 5-395 + 5-027 + 5-027 + 2-6614 2 2 -10-965 -6-289 -6-289 + 7-153 + 3-989 + 3-9894 2 0 -12-492 -6-436 -4-622 + 8-889 + 4-574 + 3-1614 0 0 -14-837 -4-338 -4-338 + 11-439 + 3-342 + 3-3423 3 3 - 7-850 -7-850 -7-850 + 4-614 + 4-614 + 4-6143 3 1 - 9-202 -9-202 -5-144 + 6-598 + 6-598 + 3-4993 1 1 -13-227 -5-166 -5-166 + 11-050 + 4-248 + 4-2482 2 2 - 7-860 -7-850 -7-850 + 6-351 + 6-351 + 6-3512 2 0 - 9-743 - 9-743 -4-054 + 8-550 + 8-550 + 3-4752 0 0 -15-863 - 3-863 - 3-853 +14-900 + 3-582 + 3-5821 1 1 - 7-850 -7-850 -7-850 + 7-466 + 7-466 + 7-4660 0 0 — — — — — —

66-2




H ere

is introduced so th a tT he determ inant is sym m etric since

y J

A = « 2.

(9-1)

(9-2)

Uc

as follows from (5-1), (5-2). '* V ' ^ *For the calculation o f the roots o f the equation (9*0) which gives the frequencies

of the crystal I have used a m ethod given by A itken (1937). (I wish to thank D r A. C. A itken for advising me to use this m ethod, w hich was a very great help in these calculations.)

I t is, however, not always necessary to consider determ inants of the sixth order. In our choice of wave vectors (p x, p y > fiz) the determ inant (9*0) splits up very frequently into determ inants o f a lower order. Considering all these cases one obtains a satisfactory survey over the frequency spectrum if the wave vector is described, not w ith the help of (A, PyiP)ibu t by its components [pX)(cf. (5*18)).

In a large num ber of cases the determ inant (9*0) can easily be split up into a p roduct of three determ inants.

(I) P = (A) 0, 0) or p = (0, p2,A t); P2 = PvVariation o f A, i.e. of/?2, therefore,means going along the diagonal of the ground plane of the phase space. W e have w ith (5*12), (6*11), (6*11)

{ * * ' ) - ( * * ' )I y y f \z z ) , [x y j z j \z x )

I introduce for abbreviation the notation

*=n> Hr*)- H * 2k

The determ inant can then be w ritten as the product

a—X c a '-Xc b - X c' b ' - X

with the solutions

(9'3)

(9*4)

(9-5)

By inspection of the equation of m otion it is easily seen th a t the first two of these frequencies correspond to longitudinal waves. The second two give transverse waves. They have to be counted twice since the corresponding determ inant in (9*4) is squared, giving the two independent directions of polarization of the transverse waves.




(2) For a wave vector (10, A,, 0) it follows from (5T2), (6T1) th a t the determ inantsplits up into three. The vibrations are in the x, y and z directions. In the case (10 0)one has in general three different determ inants. O nly in the case (10, 5, 0) one has, because of the symmetry

[* K'~\ = P ^ 1 (cf. table 3), (9-6)LX -1(10, 5, 0) L z zj(10, 5, 0)

two different determ inants, b u t the frequencies 2 have to be counted twice.The frequencies for the cases (10, 5, 0), (10, 0, 0), (5, 5, 5) and (0, 0, 0) have already

been calculated by Lyddane and Herzfeld. T heir values for the electric p a rt of the coefficients agree w ith ours, bu t there is a difference in the constants A and B which is due to the repulsive forces. These authors obtain for 10-606, 1-073 (cf. (6-4)).This is due to the fact tha t they use the repulsive potential of Born and M ayer (1932) for the calculation of these constants, thus neglecting the V an der W aals (London) forces, which are roughly included in our general expression:

(3) P « ( A ,Py,Pz); Px == Py == Pzor p =

Clearly, variation of the pimeans moving along the m ain diagonal of the phase space. The following identities hold:

6 3 - 6 9 7 6 3 - H I K H D - l

The secular determ inant gives

a-3r2d—X c-\-2f a—d—X c—f

1OQ++<0 c—f b —e—X(9'8)

Again two longitudinal and twice two transverse waves are obtained with the frequencies :

A h

A h (9-9)a = a-\-2d, a! = a — d,P = b + 2e, [T = b — e,7 = c+2f, Y = c - f .




(4) ( px, pyi0). T he determ inan t splits up into two, of the fourth and the second order respectively. T he la tte r characterises a transverse v ibration in the z direction.

(5) {px — PyiPz)' T he determ inant splits up into one o f the fourth and one of the second order. T he transverse v ibration is perpendicular to the diagonal x

(6) ( Px~Pyj O). H ere the determ inant splits up into three. T he following coefficients are e q u a l:

n n ni\ d1 1

2 2

1 2

1 1

y y 2 2 y y 1 2

y y

e —

f -

1 1 x y

2 2 x y 1 2 x y

(1 1]1Iz ZJ1 *

I22]

d\z z\>'

f1 2]iiz zji-j

(9-10)

All other coefficients vanish. T he secular equation is

tt~\~d—A -}- a—d—X c—f a—X cc + f c

0. (9-11)

Two longitudinal and four transverse waves are obtained w ith the frequencies

a = a+d, ,ft — b-\-e, /?' =y = c+ f9 —

(9-12)

T he direction of polarization of the transverse waves is the axis (cots 4) and one of the diagonals in the xyplane (<yf 2) respectively.

In general, two of the vibrations will be quasi-longitudinal and four quasi-transverse. The num erical values of all the frequencies are collected in table 5; I have also plotted (figures 1—8) the frequencies belonging to the vectors

{PvPvPs) - from (0, 0, 0) to (0, 5, 5),„ (0, 0, 0) „ (5, 5, S),

from (0, 1, 1) to ( 0 ,1, 5), from (2, 3, 3) to (2, 3, 6),

>> (1, 1, 1) » (1, 1, 5), „ ( 3, 3, 3) „ (3, 3, 6),„ (2, 2, 2) „ (2, 2, 6), „ (3 ,4 ,4 ) „ (3 ,4 ,7 ) .



-01

X

CO E

I-OI X

xi-O

I X CO


(-1,-2,-3)(A, Py )

- - longitudinal ------ transverseFigure 1. p: (0.0.0) to (0.5.5)

(A, Py, Ar)------longitudinal transverse

Figure 3. p: (0.1.1) to (0.1.5)

60

5:5

50

4-54 0

3-5302-52-0

1*510

0*5

2.2.22 .2.2

2.2.33.3.1

2.2.44.4.0

2.255.5.1

2.2.6 ( - 1 , - 2 , - 3 )6-6-2 (A, A, A)

longitudinal transverse

( - 1 , - 2 , - 3 )

- - lon gitu d in al------transverseFigure 2. p : (0.0.0) to (5.5.5)

---- longitudinal ------- transverseFigure 4. p: (1.1.1) to (1.1.5)

7oX3

5-5

5-0

4-5

4-0

3-5 3-0

2-5

2 0 1-5 1-0

0-5

2.3.3 2.3.44.2.2 5.3.1

2.3.5 2.3.6 (p i y 2, A )6.4.0 U Hpx>

— transverseFigure 5. p : (2.2.2) to (2.2.6)

---- longitudinal -Figure 6. p : (2.3.3) to (2.3.6)




In the figures the vector com ponents are described both in terms of the A, Pz and of the Pi>p2>PyT he units are 1013 sec.-1. For the longitudinal as well as for the transversevibrations the figures show the well-known picture of the “ o p tica l” and “ acoustic” branches. I t can be seen th a t the branches of the transverse frequencies in figures 1 and 2 (where those frequencies have to be counted twice (cf. pp. 540, 541)) split up in the m ore general case into two branches each.

This degeneration, as shown in figures 1 and 2, is, o f course, due to neglecting higher order terms in the developm ent o f the energy (1-6). For the same reason it m ust be understood th a t in those cases where two branches of the frequencies intersect each other, say for instance two transverse frequency branches, this degeneration would be rem oved by taking account of higher terms in the developm ent of the energy. T he cross-point would be dissolved in such a way th a t the two lower parts of the original branches would be connected, and the two higher parts as well.

0 5 - O-5-

3.3.3 3 3 4 3 3 5 33.63.3.3 4.4 2 5.5.1 6.6.0

------ lo n g i tu d in a l ------- transverse

3 .4.4 34.5 346 34.75 .3 3 6 4 .2 7.5.1 8 .6.0

------ longitudinal ------- transverse'Figure 7. p : (3.3.3) to (3.3.6) Figure 8 . p: (3.4.4) to (3.4.7)

For the case (0, 0, 0) of the residual ray we obtain uniquely the frequency

so) — J e~+ 2B — — 2-86 x 1013 sec.-1.

Lyddane and Herzfeld (1938) obtain the same lim iting frequency for transverse waves and beside this a lim iting frequency

(o = y ^ v 4 + 2 £ + y J = 6-02 x 1013 sec.-1

for “ longitudinal” waves. T he num erical values here are slightly smaller than those of Lyddane and Herzfeld due to the different choice of B (cf. p. 541). But, as shown in § 4, one cannot accept this second frequency as one of residual rays which is, therefore, given in square brackets in table 5.




Table 4. Frequencies of the NaCl lattice

P x P y P z (02 (04 W5 w610 5 0 343 2-91* 2-76 2-35* — —

10 4 0 3-63 3-28 2-95 2-91 2-52 2-2810 2 2 3-88 3-25 3-01 2-94 2-45 1-8210 2 0 4-03 3-13 3-06 3-05 2-02 1-9610 0 0 4-20 3-10 3-09* 1-77* — —

9 5 1 3-60 3-36 2-89 2-89 2-59 2-239 3 3 3-79 3-32 3-02 2-75 2-66 1-889 3 1 3-90 3-21 3-02 2-89 2-41 2-019 1 1 4-17 3-13 3-05 3-05 1-96 1-768 6 0 3-52 3-62 2-87 2-84 2-75 2-038 4 2 3-97 3-17 3-06 2-76 2-43 1-978 4 0 3-93 3-24 2-98 2-69 2-42 2-168 2 2 4-17 3-09 2-96 2-92 2-30 1-758 2 0 4-25 3-11 3-00 2-81 1-92 1-708 0 0 4-38 3-07* 2-99 1-69* — —

7 5 3 4-23 3-24 2-79 2-78 2-40 1-757 5 1 4-04 3-23 3-00 2-63 2-40 2-037 3 3 4-33 3-17 2-84 2-72 2-28 1-787 3 1 4-39 3-08 2-91 2-72 2-16 1-837 1 1 4-67 3-02 3-00 2-83 1-76 1-606 6 2 4-17 3-33 2-96 2-48 2-30 1-926 6 0 4-06 3-30 2-89 2-59 2-54 1-966 4 4 4-47 3-51 2-63 2-51 2-05 1-866 4 2 4-62 3-05 2-92 2-51 2-07 1-936 4 0 4-66 3-13 2-78 2-48 2-13 1-936 2 2 4-76 2-87 2-83 2-76 1-94 1-566 2 0 4-81 3-04 2-83 2-62 1-65 1-626 0 0 4-87 3-01* 2-59 1-42* — —

5 5 5 4-52 3-64 2-40* 1-93* — —

5 5 3 4-69 3-36 2-67 2-40 1-93 1-925 5 1 4-60 3-08 2-84 2-41 2-08 1-925 3 3 4-91 2-95 2-68 2-67 1-86 1-615 3 1 5-01 2-97 2-60 2-52 1-75 1-675 1 1 5-14 3-02 2-83 2-34 1-57 1-084 4 4 4-86 3-22 2-50* 1-77* — —

4 4 2 6-06 2-83 2-71 2-49 1-69 1-624 4 0 5-09 3-03 2-49 2-39 1-69 1-654 2 2 5-34 2-81 2-70 2-26 1-46 1-274 2 0 5-39 2-96 2-73 2-05 1-27 1-254 0 0 6-44 2-94* 1-90 1-03* — —

3 3 3 5-33 2-65* 2-48 1-42* — —

3 3 1 5-48 2-90 2-63 2-01 1-30 1-253 1 1 6-67 2-88 2-83 1-69 0-96 0-892 2 2 5-71 2-76* 1-67 1-00* — —

2 2 0 6-78 2-91 2-76 1-35 0-86 0-792 0 0 5-87 2-89* 1-02 0-64* — —

1 1 1 5-94 2-84* 0-84 0-51* — —

(0 0 0 [6-02] 2-86 0-00 — — —

The frequencies belonging to a wave vector are ordered with regard to their absolute magnitude. The units are 1013 sec.-1.

* To be counted twice.

V ol . 238.—A 67




10. T he frequency spectrum

From the frequencies calculated, it is possible to obtain the frequency spectrum N(v), w hich is required for m any applications. I divide the scale into a num ber of equal intervals Avand count the num ber of calculated frequencies in each interval. In order to obtain the spectrum very accurately it is, of course, desirable to choose as small as possible. O n the other hand, there is a lower lim it for the size o f First, there should be a fair num ber o f frequencies in each interval; secondly, it is obviously pointless to decrease the interval beyond the accuracy w ith w hich the frequencies are

calculated. I have chosen Aco — 2itAv = 0*3 x 1013 sec.-1 which corresponds to the accuracy of our calculation. In th a t case the average num ber of frequencies in each interval is about 14. Therefore, the spectrum obtained is accurate w ithin a possible shift of some frequencies of not more than 0-3 x 1013 sec.-1. O nly a t the ends of the spectrum the average num ber of frequencies per interval is too small as to give this accuracy. In this way three step-curves have been plotted, each of which is shifted against the other by a th ird of an interval Ao) in the scale of the s. The distribution curve (figure 9) is the approxim ation of these graphs by a smooth curve which is norm alized so th a t the total num ber of frequencies is 3 N,

where N(v)dv is the num ber of frequencies between v and and N is Avogadro’snum ber.

15

2irv x 10-1303 0-6 0-9 1-2 1-5 1-8 21 2-4 2 7 13-0 33 3 6 3 9 4-2 4 5 4-8 51 54 57 6 02 nvR

vR: residual rays frequency Figure 9. Frequency distribution




O ne recognizes three m axim a a t 1-8, 2-85, and 4-2 x 1013 sec."1. These m axim a are due to the different types of norm al modes. I have analysed the frequency spectrum by drawing separate distribution curves for longitudinal optical longitudinalacoustic (La.),transverse optical (t.o.), and transverse acoustic (t.a.) waves respectively (although a strict classification into longitudinal and transverse waves is not always possible).

The corresponding graph is given in figure 10. The different m axim a in the distribution curve (figure 9) correspond to these four kinds of v ibrations; the m axim a due to the longitudinal acoustic and the transverse optical frequencies coincide and give rise to the large m axim um in figure 9 near the frequency of the residual rays.

0 3 0-6 0-9 1-2 1-5 1-8 2-1 2-4 27| 3-0 3 3 3 6 3-9 4-2 4-5 4-8 5-1 5-4 57 6 0

vR: residual rays frequency

Figure 10. Frequency distribution N(v), analysed.

A more careful consideration of the frequency spectrum will be made in connexion with its application to the problem of the specific heat, which will follow shortly.*

I am very much indebted to Professor M. Born, who suggested this problem to me, for his advice on many occasions.

I am also very grateful to Dr K. Fuchs for many suggestions.The numerical results were obtained with the help of a Muldivo calculating machine.

I wish to thank Professor W. Oliver who kindly allowed me the use of the machine.

* Note added in proof. A slight numerical error has occurred in the computation of the figures 9 and 10.This, however, does not alter the result appreciably. The correct curves will be published in the paper on the specific heat.

67-2



548 E. W. KELLERMANN

Summary

According to Born’s trea tm ent of polar crystals, the frequency equation for a v ibrating crystal contains in its coefficients lattice sums w hich are due to long range Coulom b forces. Using a m ethod developed by Ew ald it has been possible to find a quickly convergent form of those sums. T he general form ulae for the coefficients have been developed and a special application has been m ade to the case of sodium chloride. T he coefficients and also the frequencies themselves have been calculated for 48 different states of vibration of the crystal which are chosen in such a way as to m ake possible a fair survey over the whole frequency spectrum of the crystal. I t appears th a t the purely electrostatic derivation of the general form ulae for the coefficients does no t give inform ation about the case of the residual rays. This can only be obtained by taking account of the electrodynam ic boundary conditions, nam ely th a t the crystal as a whole m ust not em it rad iation , w hich leads to the correct solution for the frequency of the residual rays. T he form ulae for the coefficients have also been used for the calculation of the elastic constants of sodium chloride.

R eferences

Aitken, A. C. 1937 Proc. R. Soc. Edinb. 57, 269.Blackman, M. 1935 Proc.Roy. Soc. A, 148, 365.

— 193Z Proc. Roy. Soc. A, 149, 117.— 1937 Proc.Roy. Soc. A, 159, 416.— 1937 Proc. Camb. Phil. Soc. 33, 94.— 1938 Proc.Roy. Soc. A, 164, 62.

Born, M. 1920 Ann.Phys., Lpz., (4), 61, 87.— 1923 Atomtheorie Des Festen Zustandes, 2nd ed. Leipzig and Berlin.

Born, M. and Goeppert-M ayer, M. 1933 Handbuch der 2nd ed. 24,part 2 , p. 623. Berlin.

Born, M. and v. K arm an, Th. 1912 PhyZ. 13, 297. Born, M. and M ayer, J . 1932 Z. Phys.75, 1. Born, M. and Thompson, J . H. C. 1934 Roy. Soc. A, 147, 594.Broch, E. 1937 Proc. Camb. Phil. Soc. 33, 486.Ewald, P. P. 1921 Ann.Phys., Lpz., (4), 64, 253.

— 1938 Nachr. Ges. Wiss. Gottingen, N.F. 11, 3, 56.Jones, J . E. and Ingham , A. E. 1925 Proc. Roy. Soc. A, 107, 636.Lyddane, R. H. and Herzfeld, K. F. 1938 Rev. 54, 846.M adelung, E. 1918 Phys.Z. 19, 524.Sommerfeld, A. and Bethe, H . 1933 Handbuch der Physik, 24, part 2 , p. 402. Thompson, J . H . G. 1935 Proc.Roy. Soc. A, 149, 487.



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Theory of the vibrations of the sodium chloride latticersta.royalsocietypublishing.org/content/roypta/238/798/513.full.pdf · [ 513 ] THEORY OF THE VIBRATIONS OF THE SODIUM CHLORIDE