the theory of the vibrations and the raman spectrum of the...

[ 105 ]

THE THEORY OF THE VIBRATIONS AND THE RAMAN SPECTRUM OF THE DIAMOND LATTICE

By H E L E N M . J . S M IT H , University o f Edinburgh

[Communicated by M . Born, F.R.S.— Received 29 December 1947)

I ntroduction

T he crystal s truc tu re o f d iam ond was first de term ined by Bragg in 1913 from X -ray p h o to graphs; the carbon atom s are a rran g ed a t the apices an d m ed ian points o f in terlinked te trahedra. Born (1914) derived expressions for the th ree elastic constants o f d iam ond in terms o f two force constants re la ted to the valency bonds betw een neighbouring atom s. But, a t th a t tim e, the only experim ental d a ta availab le w ere the com pressibility an d the Debye characteristic tem p era tu re 0 , an d precise d e term ina tion o f the valence force constants was no t possible. M eanw hile, investigation o f the op tical properties o f d iam ond h ad produced evidence for the existence o f two d istinct types, one w ith an absorp tion b an d a t 8 [iin the infra-red , the o ther tran sp aren t a t this poin t. R obertson , Fox & M artin (1934) took up this p roblem and found th a t absorp tion in the infra-red is associated w ith absorp tion in the u ltra-v io let; d iam onds tran sp aren t a t 8 ytransm it m uch fa rther in to the u ltra-v io let. Both types o f d iam ond have B ragg’s te trah ed ra l structure , the sam e refractive index, specific gravity, dielectric constant an d electron diffraction. T h e ir infra-red spectra are iden tical up to ly , and the frequency shift o f the p rincipal R am an line is the sam e. T h e derivation o f the elastic constants was again considered by N agendra N a th (1934). H e extended the theory to include central forces betw een second neighbours in the lattice. H e also suggested th a t the frequency shift o f the principal R am an line corresponds to the relative v ib ra tion o f the two carbon atom s in the u n it cell, along the line jo in in g the ir nuclei.

R am an and his collaborators have recently (1941) p u t forw ard a new theory o f la ttice dynamics according to w hich the v ibrational spectrum o f a crystal consists o f a few discrete lines. This is in d irect contradiction to the quasi-continuous v ib ra tional spectrum pred icted by classical or q u an tu m m echanics. O n this new theory there are eight fundam ental frequencies o f v ib ration for d iam o n d ; the values o f these frequencies* are deduced from the observed specific heat, u ltra-violet absorption and R am an spectrum , w hich, it is claim ed, cannot be explained by ‘o rthodox’ lattice dynam ics. R am an (1944) has suggested th a t there are, no t two, b u t four types o f d iam ond, two w ith te trahed ra l sym m etry and two w ith octahedral sym m etry depending on the electronic configurations, b u t X -ray analysis gives no indication o f this and the a ttem pts o f his school to explain the observed infra-red spectra on the basis o f their new lattice theory have been, up to now, unsuccessful.

Bhagavantam & B him asenachar (1944), have developed a new m ethod o f m easuring directly the elastic constants o f crystals and have published results for diam ond. This makes it possible for the first tim e to determ ine the force constants betw een the atom s in the lattice. T he whole problem has therefore been reconsidered here on the lines o f a general theory developed by Born (1942), assuming a rb itra ry forces betw een first-neighbour atoms and central forces

Vol. 241. A. 829. (Price gj. 6d.) [Published 6 July 1948

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106 HELEN M. J . SM ITH ON TH E TH EO RY OF TH E VIBRATIONS

betw een second-neighbour atom s. T h e values o f the force constants are ob tained from the m easured elastic constants and the frequency shift o f the principal R am an line. O ne can then derive approxim ations to the quasi-continuous v ib ra tional spectrum o f d iam ond and calculate the specific heat and D ebye’s characteristic tem pera tu re 0 . C om parison w ith the observed specific heat shows th a t first-neighbour forces alone are no t sufficient to account for the experim ental results, b u t the add ition o f second-neighbour cen tral forces produces satisfactory agreem ent. T h e second-order R am an spectrum , first observed by K rishnan in 1944, can be calculated from the v ib ra tional spectrum . T h e range and the positions o f the m axim a are obtained w ithou t any a rb itra ry assum ptions w hatever, depending only on the m easured constants. T o explain the relative intensities o f the parts o f the spectrum it is necessary to postulate certain relations betw een coupling constants. These relations form the basis for fu rther investigation of the possible differences betw een the electronic configurations in the two types o f d iam ond. I t is evident th a t R am an ’s claim , th a t lattice dynam ics is unab le to explain the R am an spectrum o f d iam pnd, is unfounded.

PART I. THE THEORY OF THE VIBRATIONS OF THE DIAMOND LATTICE

1. T heoretical basis

This section sum m arizes briefly results a lready ob tained by Born (1942) an< ®orn & Begbie (1947).

(a) Equations of motion of a crystal latticeT he lattice cell is described by three elem entary vectors a L, a 2, a 3. T h en the position vector

o f the particle a t the vertex o f any cell is

r1 = Z ^ j-f Z2a 2+Z3a 3, (1*1)where Z1, Z2, Z3 are integers.

I f there are s particles in the u n it cell w ith masses 1 ,2 , ...,s) an d rk is the positionvector o f the £th particle from the cell vertex, then

r0 =r,+r* (i-2)

defines the position of the particle in equilibrium . T he rec tangu lar com ponents o f

are xaQ (a ^ 1 ,2 ,3 ).

Now consider small a rb itra ry displacem ents u Q of the particles from equilibrium . The

potential energy O of the deform ed lattice can be expanded in powers o f the rectangular

com ponents (a = 1 ,2 ,3 ) of u Q f . T he linear terms vanish in equilibrium and the

second-order terms are

O, 2 2 2 ^w

where02<D

(a ,^ — 1> 2, 3). (1-3)

(1-4)



AND TH E RAMAN SPECTRUM OF TH E DIAM OND LATTICE 107

these second derivatives in equilibrium depend only on the difference o f cell indices ( /—/') and satisfy the condition

(1-5)n \

T he equation o f m otion of a partic le o f type k, mass mki is then

In troduce a ‘red u ced ’ d isplacem ent vector

and define the elem ents o f the dynam ical m atrix o f the lattice as

Then (1*6) becomes

( 1-8)

(1-9)

A solution o f this equation for an independen t norm al v ib ra tion o f the lattice is a p lanewave c

’.1I>II (M O )

T hen from (1*9) 0>2K ( k ) - I I D af(kqk^ ( k ' ) = O ( M l )

and

M u i r ' S(M 2 )

D(q) is the representation of the dynam ical m atrix in reciprocal space.The equations o f m otion ( I ’l l ) for a w ave-vector q in the reciprocal space o f the lattice

are a set o f 3j hom ogeneous equations in the reduced am plitudes VJJz). T he necessary and sufficient conditions th a t this set should have a non-triv ial solution is th a t

\ D ( q ) - G ) * I \ = 0 , (M 3 )

where / is the u n it m atrix o f order 3 sx 3s. For a particu la r w ave-vector q , the characteristic equation (1T3) has 3s roots <yi-. T hree o f these roots, the acoustic branches, as functions o fq , tend to zero a s q ^ O . T he rem ain ing 3^—3 roots, the optical branches, tend to finite limits asq->- 0. These 3^—3 lim iting frequencies' are the first-order R am an spectrum of the lattice. The intensities of the lines of this spectrum depend on the sym m etry o f the lattice. Born & B radburn (1947) have shown th a t if each la ttice po in t is a centre o f sym m etry then the intensities vanish and the first-order R am an spectrum does not exist.

n _i'\The elements of the m atrix Z)l , , , j are related through the sym m etry of the lattice in

the following w ay: ^A lattice poin t is defined by the vector referred to rec tangu lar axes. A sym m etry

operation of the lattice can be expressed as a transform ation m atrix T, and if is some other lattice point, //A ( l\ f

rU) = 7VU)- (M4)




( l - V k'( L - L

T he elem ents o f the m atrix Z)| , , , j will then have the transform ation law

d ( k k ' ) = t D ( J ) f > (M 5 )

w here T is the transpose o f T and the change of indices Q is obtained from (1*14).

F u rther, since the po tential energy O of the la ttice is invarian t for rigid translations and rotations, ,, *

IM kk' ) *A(i) = 1 D^(kk' ) *?(*)' (1'17)From (1*16) we can define the m atrix representing the force o f a point j . j on itself as

*P\kk (1*18)

w here the dash on the sum m ation sign indicates th a t the term s in w hich / ' = 0, = areto be om itted.

T he choice o f possible wave-vectors q is restricted by the ‘cyclic la ttic e ’ condition (Born I923)*

T he basic vectors b a o f the reciprocal la ttice are given by

11 if a = /?,

[0 if a=f=$a a-k/?

*>1 a 2 A a 3 , b .3. y a a 2

. a 2 a a 3 (1*19)

( 1*20)

a,.a2Aa3’ “ 2 a,.a2Aaj’T he vector q = 27rqlb l -f 27T 2b 2 -f 27j- 3b 3,

w here qu q2, q3 are integers, determ ines a reciprocal lattice point.

T hen if r1 = lla j + /2a 2 + /3a 3,

the scalar p roduct q .r* = 27t(^1/1 + 2/2+ ^ 3/3) (1*21)is an in tegral m ultiple o f 27r.

T he cyclic lattice condition postulates th a t the lattice deform ations have a period large com pared w ith the dimensions o f the cell, so th a t any function

f ( r L) = / M , if L* = l« + n (oc = 1 ,2 ,3 ).

H ence for to be periodic in this way, the perm itted values o f q m ust be such th a t

—, q2n * (h^ h2, /f3 — 0 ,1 ,2 , . . . , ( — I)), ( 1-22)

(b) Derivation of the elastic constantsA solution of the equation o f m otion (1-9) for long-wave acoustical vibrations is

(l\ j ilv ( h = W (/t)

\{ k ) = W (*)<««■■»>,

(1*23)

(1*24)




and (1*9) becomes

with

»2K W ~ I I 0, (1-25)

c’" 4 kk ') D“f{kk)

e-i(n(tk-rk'))

-iMi» (i-26;

Let o)1 D, then (T25) can be w ritten

Q W = C (f)W . (1-27)D ,W and C(q) can be expanded in powers o f the w ave-vector q : for we have, in rec tangular com ponents o f q ,

( 0 )

CLafi\kkf

( 2 )

*r\kk!l"AW

- | 2 I D / M** l \ j l

(7a 5/4*

(1-28)

Equating term s containing like powers o fq on either side o f (1*27), we obtain , following Born & Begbie (1947), two approxim ations to the equations o f m otion (1*27). W ritten in m atrix elem ents these are

2 2 $ 4 9 . + | f = 0;p

\ ( 2 )

1 %A / A*.' /? ^ mkmk'J

(1*29)

(1-30)

Wa (a — 1 ,2 ,3 ) are three a rb itra ry constants, trivial solutions o f (T27) corresponding to the three possible independent translations of the crystal as a whole.

(2) . . (1)Put D = o)2and elim inate Wp(k') betw een (1-29) and (1*30); the result m ay be w ritten

(1-31)1

where p — — ]£ % 1S density and va = . a 2 a a 3 is the volum e o f the u n it cell.v a k

Com parison of (1*31) w ith the equation for the am plitudes of elastic waves in elasticity theory shows th a t the D ra {q) are related to the elastic constants in the following w ay:

D'22(q)

■^23(9)

^>3,(9)

D'n (9).

51

- C16 C&

C3\

C4&

C35

i ( C4 5 + 36)

1+^55)

C16

C62

C54

J(^64 + % )

^(^56 + 14)

91

92

q l

2?2?3

29z9\ 2 9\ 92J

(1-32)

CU ^66 C55 ^65

C66 C22 C44 C2A

^55 ^44 C33 C43

C65 C24 C43 i ( C2 3 + C 44)

C46 C35 i ( ^ 4 5 ~f^36)

^(^64 + 25) ^(^56 + 14) ^(^12 + 66).

H ence the elastic constants m ay be expressed in term s of the elements of the dynam ical m atrix o f the lattice.




2. E lastic constants of diamond

T he d iam ond crystal is bu ilt up o f carbon atom s lying on two equal in te rp en etra tin g face-centred cubic lattices, relatively displaced one-quarte r o f the w ay along a cube diagonal. T he u n it cell contains two atom s, one on each la ttic e ; the distance betw een them is a */3/2, w here 2 a is the la ttice constant o f d iam ond ( = 3*56 x 10_8cm .).*

R eferred to a rec tangu lar co-ordinate system w ith origin a t a lattice po in t an d axes parallel to the edges of the cubes, the cell vectors o f the la ttice are

= a ( 0 ,1 ,1 ), a 2 = a ( l , 0 , 1), a 3 = f l ( l , l ,0 ) . (2*1)

T h e co-ordinates o f the two atom s in the u n it cell are (0 ,0 ,0 ) and ( these atom sare labelled 0 and 0 ' in figure 1. A tom s lying on the sam e cubic la ttice as 0 are labelled k — i , and those lying on the sam e cubic lattice as O' are labelled = 2. E ach atom has four nearest neighbours a rranged te trahedrally a t a distance a J3/2. W e assume as a first approx im ation th a t the elem ents o f the dynam ical m atrix are negligibly sm all except for first neighbours; then the sym bol l m ay be used to num ber these neighbours, in place of the

symbo1 ( ^ , ) .

//

Figure 1. First neighbours of the two carbon atoms in the unit cell of diamond.

T able 1 gives the co-ordinates o f the nearest neighbours o f 0 and O', the corresponding num ber / and the integers Z1/2/3 given by

r* = /1a1+/2a2+/3aj. (I*1)* To avoid introducing too many fractions, Bragg’s notation has been followed.




T able 1

X2 *3 k l Z2 Z3 kk'first neighbours \a \a \a 2 T 0 0 21

of 0 (k ' = 1) — \ a —

- \ a\a

I a - \ a

22

25

00

0- 1

— 1 0

2121

\a - \ a - \ a 2 5 - 1 0 0 21

first neighbours 0 0 0 1 1 0 0 0 12 1 Oof O' (k' = 2) a

aa0

0a

11

23

00

01

10

1L 12

0 a a 1 4 1 0 0 12

T he sym m etry operations o f the d iam ond la ttice can be described by six transform ation m atrices:

A x: T h e la ttice has a threefold axis o f ro ta tion x3, so

1 0 0 1 0 0

(2 -2)

The change of label l, in the no ta tion o f substitu tion groups, is

(1) (32) (43) (24)1

(T) (35) (is) m y

A 2: T here is a centre o f inversion a t the po in t = = this is equivalent toa centre o f inversion a t (0 ,0 ,0 ) and a translation = = \a.

Hence and (11) (22) (33) (44).

A 3: T here are three planes of reflexion. O ne is xx — x2 giving

(2-4)

"010

1 0 0 0 0 1

and (1) (2) (43) (1) (2) (43). (2-5)

The other two planes o f reflexion x2 — xs and x3 = xx can be ob tained from A j and A 3. A 4: A ro ta tion th rough abou t an axis th rough the po in t parallel to the

*3-axis followed by a ro ta tion th rough \ ti abou t an axis th rough the sam e poin t parallel to the Aq-axis, gives

(3) (21) (42) (14)(5) (2T) (52) (14).

( 2 -6)

A 5: A ro tation through \ tt abou t an axis parallel to the Aq-axis th rough the point \a)followed by a ro ta tion th rough \n abou t an axis th rough the sam e poin t parallel to the *2-axis, gives

100

0‘- 1

0and

(2) (41) (13) (34) (2) (4T) (15) (34).

(2-7)



A 6: A ro ta tion th rough \n abou t an axis parallel to the #2-axis th rough the poin t ( Jfl), followed by a ro ta tion th rough \ tt abou t an axis th rough the sam e poin t parallel

to the x3-axis, gives

° _ 1 01 (4) (31) (12) (23) (2.g)

112 HELEN M. J. SM ITH ON TH E TH EO RY OF TH E VIBRATIONS

r . 0 0 1 1 0 0

and(I) (51) (12) (25).

T o these are added the sym m etry conditions o f the dynam ical m atrix : from (1*5)

D“f{kk ) 2 ’9)

F rom A 2, $ ~ T2D lT2 =

i.e . D l = D \ D2 = D \ D 3 = Z)3, (2-10)

From (2*9) and tab le 1,

2)i = £>I« £ )\ D2— ffi— b 2, « 5 5 - (2-11)

T hus all the m atrices Dl are sym m etric.From A 1} since D l is invarian t w ith respect to Tu

D\i = DU = D J3

~cc pP * P

a.

* > l 3-

So we m ay p u t

w here m — mass of carbon atom .

D 2\

D 1Z)i

1m

From A, D 2T4D lf 4

1m

By repeated application of we find

D 31TxD2f x = —;

andD 41

TxD*Tx1

m

aP a —P

-P ~ P a

cc -P p-P a ~ P P - P a

a - / ? - / ] -P CL P~P P a -

( 2 -12)

(2*13)

(2-14)

(2-15)

(2*16)

These eight m atrices Dl and Dl [l = 1 ,2 ,3 ,4 ) satisfy all the o ther relations w hich m ay be

( 0 ) = z ) / 0 \ t eT 0 0 ‘

0 1 0 ( 2 - 1 7 )\ 1 1 / \ 2 2 / m

I----- O 0 1 __

Lastly, from (1T 8),

W e now proceed to calculate the m atrices c M , j from the relation

(1-26)



A N D T H E R A M A N S P E C T R U M O F T H E D IA M O N D L A T T IC E 113

T able 2 gives the values o f the rec tangu lar com ponents o f for the first neighboursof 0 and

T able 2

1 WfM ft mH ft MM ft 1 MjM ft

*'(21) \a —\a \o. \a

^(12) ^ **(21) \a — \a \a — \d

^(12) ~ \a ~ \d \a \a - \ a - \ a

Since Dl — Dl and r' = « •

(2-18)

Expanding in powers o fq , we find

2 ( a ) M ) . m (2-19)

so it is only necessary to calculate the expansion of *

Also

sty -*$■- o - o . f(A)-s<aT he expansion o f is

0 . (2 -20)

Hence

C

( 0 )

C

= 2 Dle - ^ tl\/— 1,2, 3,4

( 1 )c

4am

2 ia(3

1 0 01 0 1 0 0 0 1

0 9z9z 0 9 ,

9 , OJ(2) ! n \ Jig

3 |<MII^ -H “( t i + f y + f y 2 hy<ls. 2 fiqzqx 2 Hy<l “(q l+ q l+ q S .

( 0 )

C( 0 )

Cn o o' 0 1 0 1 0 0

(2 -21)

(2 -22)

(2-23)

(2*24)

(2-25)

Substituting these m atrices in the equation (1*29),

4am

T 0 0" (1) (1> / 2 iafi\771 /

‘ 0 <lz %0 1 0 (W{ 1) W {2))+Jm - <lz 0 qx

_0 0 1_ \-Qy 9x o_

W = 0,

Vol. 241. A. !5




and

Substitu ting (2*26) in (T 30),

. j 2

W { l) -W { 2 )0 <h1z 0

Via 1> ° J

W. (2-26)

2m<y2W0 1z 1y 1z 0 1,

la 1z °J

Therefore

0 <lz <ly<lz 0 <lx

Qy <lx 0_* ( t f + f y + f z )

^ k x % “ ( t f + f y + f z )

^PW x ‘Z'ky'lz

2faz<lx

^ q yqz“ ( f x + f y + f z ) .

w.

« - £ ) ( « ? + ?9 ' P(2 A ( 2~ 9 ? ^d2(o2W = ——

2 m P\(2~ i ) ^ xfy + \( * “ ) ( ? ! + ? ? ) P\(2- 9 « » « f

(2- f ) ? , ? . [2 ~ t y l y 1z “l l + [

W . (2-27)

This has the form (o2W = w here

d \M )

D ’22(q)

D I M a2

I>2,(1)~ P 2m

D’n (l)

P u b )

. tocp*

P P2oc — — oc

oc —P1

0 M 2 - ! )

° M 2 - ! )

f x

f y

d2 * , _ * ■ r'

2?y?*-

21z 1x

21x 1y

(2-28)

T he un it cell has volum e va = sl1. a 2 a a 3 = 2a3. So the density p /2a3 = m/a3.C om paring (2*28) w ith (1*32) we find the elastic constants are

1 \ ( icu = c22 — c33 — 2aa’ CAA~ c^ = c&& = 2 a \~ ~ o i) i 2a a)* (2*29)

T he rem aining elastic constants vanish.

Solving (2*29) for a and a = 2 acUift = «(^n + i2)« (2-30)

T he elastic constants satisfy the quadra tic relation

1^ 1 1— 4 4) — (^11 ~h 12)2* (2*31)




This q u ad ra tic re la tion has been ob tained already by Born (1914) using as repeating u n it a cube o f side 2 a con tain ing eight carbon atom s and considering as here only forces betw een any carbon atom and its four nearest neighbours.

N ow the atom s 0 and 0 ' in the u n it cell (figure 1) have each twelve second neighbours a t a distance a J2, lying on the sam e face-centred cubic lattice. As a second approx im ation let us consider in add ition to the first neighbour elem ents the elem ents o f the dynam ical m atrix corresponding to these second neighbours.

Figure 2. Second neighbours of the atom ‘O’ in the unit cell.

Figure 2 shows the second neighbours o f the a tom 0 ; tab le 3 specifies the co-ordinates and labelling o f these neighbours and tab le 4 the integers Z1/2/3 given by r r = Ix2lx + /2a 2 + Z3a 3.

T able 3

*1 Xg *3 k Z *1 x2 *3 k Z0 a a 1 5 la — \a — \a 2 5

0 0 a — a 1 0 6 \a — \ a \a 2 6<*-<0 a 0 a 1 7 - \ a — 2 7UQ — a 0 a 1 S 2 \a \a — \a 2 83O a a 0 1 ® g - \ a - \ a 2 9c; a — a 0 1 10 s — \a f a \a 2 10bp 0 — a — a 1 11 \a I a l a 2 11"Ca 0 — a a 1 12 . g I a I a — 2 12

— a 0 — a 1 13 l -0 i a \a f a 2 133O a 0 — a 1 14 0 - \ a \a 2a 2 14y — a — a 0 1 T5 Z | a \ a \ a 2 15

— a a 0 1 16 f a — | a \ a 2 16

T able 4

7 5 0 1 8 9 TO TT 12 13 14 15 161 l l 12 13 14 15 16 5 6 7 8 9 10Z1 1 0 0 1 0 - 1 - 1 0 0 - 1 0 1Z2 0 - 1 1 0 0 1 0 1 - 1 0 0 - 1Z3 0 1 0 - 1 1 0 0 - 1 0 1 - 1 0

11 11 11 11 11 11 11 11 11 11 11 11KK 22 22 22 22 22 22 22 22 22 22 22 22

• 15-2



T h e tw enty-four m atrices Dl and D*( / = 5 ,6 ,7 , . . . , 16) can be expressed in term s o f three constants A, /i, v by using the transform ation m atrices T ((2*2) ... (2*8)) as for the first- ne ighbour m atrices Dl. T hen

116 HELEN M. J . SMITH ON THE THEORY OF THE VIBRATIONS

Z>5Z)11

D 7 Z)1 3

D 9 D 15

Z)5Z)11

Z>13

D 5 l Z)I5|

2m

m

1m

v

0A0

0

0vM.

v0

M

00A

Z)6Z)12

Z)8Z)14

Z)ioZ)16

P 8 | Z)12j

2

2>®_1 Jt Z)14J ~ m

A 0 00 ft —v0 — v ju

H 0 -0 A

- v 0

Z)10lZ)“ )

1m

-v0

v 0

AJ

v 0 n 0 0 A

(2-32)

From (1T 8), (2T7) is replaced by

[ cl + A -f-1 0 01 0 1 0 0 0 1

(2-33)

Since Dl = Dl and r* = —r1, the equations (2*18) and (2-19) still hold and the expansion

° f is given as before by (2*22), (2*23) and (2*24).

But since 22 ) f ( *20) 4S replaced by

T able 5 gives the rec tangu lar com ponents o f or t ie secon< neighbours o f O',

from (2*34) it is necessary to calculate the expansion of only.

since

T able 5

X 5r \6 7 8 9 10 11 12 13 14 15 16

*U) ° 0 —a a — a -a 0 0 a —a a a

x*(kk) — a 0 0 — a a a a 0 0 a —a

*>{kk) a —a — a 0 0 a —a a a 0 0

kk' 22 22 22 '22 22 12 22 22 22 22 22 22

‘I 0 0"Thus

(0) / ci>2/4am 0 1 0 * (2*25)

0 0 1



AND THE RAMAN SPECTRUM OF THE DIAMOND LATTICE 117

(2-35)

( 2)c 2 a2

m

2W 2> + (?+ /*) (?* + ?*)

2

2 vqzqx

2 vqxqu

+ ( * ■ + / • ) (?!+?*) 2 v i y 9 z

2 vqzqx

2 vqaqz . (2-36)

2a?I,+ (*+ /< )

Substitu ting again in (1*29), (2*26) rem ains unaltered , b u t (1‘30) now leads to

w2W2m

[ « + 8/«] ?* a

/ ( 2_f) + 8,']?> [^(2—a) 8,<] A

+ [ a - j + 4 (^+/<)

x [ ? » + ? 9

[a + 8 /( ] jJ

+ [ a - ^ - + 4 (^ + /')

x E ?I+ ?9

[/?(2~a) + 8l’]?z?* [/?(2_a) + 8’']?J,?z [*+8a]?2A

i.e. ai2W = ^D '(q )W and

+ [^a——+ 4 (A+/i)

x [? f+ ?f]

W ,

■O'n (?)" a+8/* p2oc— —CLp2CL — — CL 0 0 0 <&

^ 2(?)P2a ——CL

+ 4(A +/«)

& 8//

+ 4(A +/0

P2CL — —oc 0 0 0 9}

■»«(?)✓»2

+ 4(A+/*)

P2oc— —ap2CL — — CL

H~4(A+/<)f

oc + 8fi 0 0 0 q2Or

= ^2m + 4(A+/*)

0

+ 4(A+/*)

0 o \ l o

crs1 0 to *cs N

0 0 0

+ 4v

0 4 r §0 Mz<lx

■ 12 (^) 0 0 0

+ 4y

0 0

+ 4y

(2-37)

(2-38)



118 HELEN M. J. SMITH ON THE THEORY OF THE VIBRATIONSC om parison w ith (1*32) gives

cn ~ c 22 = c33 =

C44 — 55 — C66 — 2Q(a ^ ^ j >

ci2 ~ c\3 ~ c23= 2(2^—a —4A — 4 //+ 8/').

(2-39)

These equations (2*39), as they stand , canno t be solved to give the atom ic constants in term s of the elastic constants, and there is no relation betw een the elastic constants analogous to (2*31).

But if it is assum ed th a t the forces betw een an atom and its second neighbours are cen tral forces, then it is a lready know n (Born 1940) th a t for a simple face-centred cubic lattice u n der cen tral forces, the elastic constants can be expressed in term s o f one atom ic constant, and

cw — 2c12 — 2c44. (2-40)

So taking the terms in (2*39) referring to second neighbour forces,

8// — 2(4A + 4 ji) = 2( — 4A — 4// + 8f),

therefore A = 0, fi — v. (2*41)

T he elastic constants in term s of a, // are then

cn ~ c22 ~ c33 ~ 2a

1 / \C4A = C55 = C66 = 2 a \ a _ T + 4 / / ) ’ “

C\2 ~ C\3 = C23 3=x 2 a a + 4 / / ) .

(2*42)

From (2*42) can be derived three quadra tic expressions for a, /?, / / in term s of cUi cl2i cu , b u t there is no relation betw een the elastic constants.

3 . F requency spectrum of diamond

T he frequencies o f v ibration of d iam ond are obtained by solving the characteristic

equation | D (q)- w 2/ | = 0 (M 3 )

for various wave-vectors q in reciprocal space.T he cell-vectors of d iam ond are

= /z(0,1,1), a 2 = //(l, 0,1), a 3 = a(l, 1,0). (^*1)

Thus from (1*19) the cell-vectors o f the reciprocal lattice are

b i = ^ ( —1 ,+ 1 ,+ 1 ) , b 2 = j ^ ( + l , — 1, + 1 ) , b 3 = 2 a( + 1> + 1’ — *)• (3<1)

T h e reciprocal lattice is a body-centred lattice.




T he wave-vector q is then

Q = g j2 jr[( — + ( (^l + 2 —^3)] (3'2)

or q = (3’3)

From the condition o f the cyclic la ttice (1*22) it is sufficient to take values o fq in reciprocal space such th a t

9.2* 3^^* (^“ )

T he elements o f D(q) are obtained from (1T2) by substitu ting the m atrices Dl given by (2*13), (2*14), (2*15), (2*16), (2*32) and (2*33), and the vectors given by (1*1) an d tables 1 and 4. They are as follows:

^ ( n ) = L>n{ ^ = ^ [ 2a+ A{2-cos27r^i_cos27rf e - ^3)}-f n{4 — cos 2Tjq2 — cos 2t — cos ( — — cos — )}],

M n ) “ M J f ) = | [ 2 a+ ^ 2 _ c o s2 ,r? 2 - cos2)r(? 3 -? i)}| + /*{4 — cos 2iTq3 — cos 2nql — cos 3) — cos 27t(^1 — q2)}],

= M j j ) = 1 [2 * + A{2 - COS 27iq3 - cos 2?f (?1 - q2) }+ y«{4 — cos 2nql — cos 2vq2 — cos 2n(q3 — q{) — cos 2ii{q2 — 3)}],

[cos 2n(ql — q2) — cos 27t 3] ,

[cos 2n( qz — q x) — cos 27 ,

[cos 27i{q2 — q3) — cos 277^].

D

D

D

13111

9

( L )

3 ( 22)

23' 11

9

m

2vm

2v

“ ( 12) - M i 2) ~ M i 2) - - M 2 ?i ) * M 21)

[ l -j-0-2mq3_j_0-2mg2-l-e-2tTigiJ

f

M 1 2 ) = M / l ) = m [ l + « - 2"i,s~ « '2’",2- ‘r 2 ”iH

M 1 2 ) = ^*3(21) = ~ m P - e - ^ + e - ^ - e - ? ’^ ] ,

PU) n _g-2niq3_g-2niq2 g~2niqi

(3*5)

where D* ^ ^ , j is the com plex conjugate of Dapj .

Since the matrices Dl are sym m etric, the dynam ical m atrix D(q) in reciprocal space is H erm itian and so all the roots (Oj o f the characteristic equation (1*13) are real.



120 H E L E N M . J . S M IT H O N T H E T H E O R Y O F T H E V IB R A T IO N S

In the qxi qyi qz space, the elem ents ofD(q) a re :

D a

cfII

(22) = ^ [ ’a + A{1 ^ COS C0S+ /*{2 — cos 7tqx cos T\q — cos 7 cos 7r^}] ,

D 22/ 1 ) _ ° 22(22) = ^ - cos n , cos nqx}

• 33!a i H « '

+ fi{2 — cos Tiqy cos nqz — cos 7 cos „}],

^22) = | [ a + ' l { l - cos)r?<cosff7J/}

• 12![ n ) = D '2'

+ fi{2 — cos irqz cos nqx — cos nqy cos 77 } ],

*22) = ^ sin,r?«sin7r?»>

-^13| ^b CO (22) = ^ sin sin

■^23|( n ) ~( q \ ±V . . [

DU\[1 2 )= ' ° 22l(1 2 )= x>33( i ^ ) = ^ ( 2 1 ) = = ^ ( 2 1 )am

_|_ e-rri(qx+qy) e~m{qz+qx) _j_ e-m(qy+qzY }

M

*2QII f % \ = —(L ("1 JL e ~ n i(q x+ q y) _ _ e - n i ( q z + q x) _ e - n i ( q y{2 1 J m L J’

■ 13! 5^. II b ^ \ — — [ 1 e~ni(-qx+qy>-\-e~niqz+q*> — e ~ +qz)~\,

21/ vfi

-^23! II 2 j \ ~ — — [ 1 — e - m (q x + q y ) — e - ir i (q z + q x) - ± . e - i r i ( q y + q z y ] '

In order to take full advantage o f the sym m etry properties o f the DaJq ) , consider the region o f allowed wave-vectors in the qx, qy,qz space. This is (see figure 3) the first Brillouin zone o f a face-centred lattice, an octahedron w ith its vertices cu t off, defined by

qx = ±1, qy = ± l , #2 = ±1, = (3*7)

Figure 3. The first Brillouin zone of a face-centred cubic lattice (Sommerfeld & Bethe 1933).



AND THE RAMAN SPECTRUM OF THE DIAMOND LATTICE 121T hen from the form o f the elem ents Dap{q) it is necessary only to consider certain points

in the positive oc tan t o f the octahedron , points such th a t

+ + (3*8)

A selection o f points in this positive oc tan t is ob ta ined by dividing the range 0 to 1 of into eighths and in troducing vector com ponents

Pi = 8& (Pi* P2>Pz in tegers). (3-9)

T hen Q = “ i t M A + A + A ) ? {Pi~ iPi+ p2 ~Pz)]

= -llP xi Py> Pzi' (3*1°)

T he integers pxi py, pz are either all odd or all even and

Q ^P z^P y^P x^S) (3*11)

T here are tw enty-nine sets o f num bers o f this type.To determ ine the frequency distribution, the roots o f (1*13) for each one o f these sets of

integers m ust be w eighted according to the num ber o f sim ilar points in the whole octahedron .

Let A — mu2, A = mD" ( l2 ) = “ M i l s ) = " M M

B = mZ)12(jQ > c — "^1 3 (^ 2 ) > ^ ~ m^ 23(x ^ )1

^ " M M G = mM /i)’ H = m D 4 n ) ’

K = m£>13( ^ ) , i = « £ )2S( / 1) >

(3-12)

then in the general case, the characteristic equation (1*13) is

> J K A B CJ C

D 1 > L B A EK L H - A C E A

A* B* c* F - A J KB* A* E* J y C

D 1 > Lc * E* A* K L s

u 1 >

0. (3*13)

A first approxim ation to the frequency distribution is ob tained by neglecting the forces between second neighbours, i.e. p u t X = ju — v = 0. T hen

j F — G — H = 4a, — = 0, (3*14)

and the equation (3T3) reduces to

L 2J3(i?2) z > * ( ] = 0 , (3-15)

where I is the un it m atrix of order 3 x 3 .

Vol. 241. A. 16



A second approxim ation to the frequency d istribution is obtained by considering the second-neighbour forces to be central forces. T hen

A — 0, n — v. (2*41)

T he equation (3*13) has then no general solution and m ust be solved num erically. H ow ever, for certain sets o f points ( px, p , pz)the de term inan t breaks dow n in to determ inants o f lower order.

(1) px — py — pzor = p2 — py These points lie on the m ain diagonal o f reciprocalspace.

T hen from (3*6) and (3T 2),

122 HELEN M. J. SMITH ON THE THEORY OF THE VIBRATIONS

F == G == Hj

T he determ inan t (3T3) reduces to

(3*16)

So

F - J - A A - B A * - B * F - J - A

F + 2 J - - A A + 2B A* + 2B* F + 2 J - A

= 0.

A 1>2= 4 (a,+ju sin2 nqx)d=2v/[a2(l + 3 cos2 71*7*) -f-/?(/?+2a) sin2frg j, 1

A 3 = 4 (a-|-4/7sin2irqx) ± 2 J[cl2{1 4*3 cos2 nqx) + —a) sin27777J .J

(3*17)

(3-18)

I f px —py —pz — 0, i.e. q = 0, then

therefore

A i>2 = 4 a ± 2 N/(4a2), A 3 =

^1,2,3 “ W4,5,6 (3-19)

This is the frequency shift o f the first-order R am an line of d iam o n d ; in w ave-num bers

v 1 /8a2ttc4 m (c — velocity o f lig h t). (3-20)

(2) px = py, p2 a rb itra ry or p x — p2i p3 a rb itra ry . This gives

C = F , F=G , (3-21)

T he determ inan t (3T3) splits into a de term inan t o f the fourth o rder and one o f the second order.

If, in addition, pz = 0, then K — L — 0, b u t no fu rther simplification o f the determ inan t results.

(3) px a rb itrary , py = pz or p x a rb itrary , p2 — T hen

B = C} G = H, ’ (3-22)

As in (2), the determ inan t splits into two sm aller determ inants, one o f the fourth o rder and one o f the second order.

If, in addition, py —pz — 0, i.e. p x — 0, p2 = p3,

T hen

F = C = 0, J = F = Z ,= 0.

G—A A + E G - A A - E F - A AA*+ E* G - A A * —E* G - A A* F - A

(3-23)

(3-24)



A N D T H E R A M A N S P E C T R U M O F T H E D IA M O N D L A T T IC E 123

Since {A + E ) ( . A*+E*) = ( A - E ) (. * ) ,

A is2 = 4a + 4//(i — cos7rqx) ± 2 j 2 J+ ^ 2 + (a2 — /?2) cos , j

A 3 = 4a + 8 //( l—c o s 7 r^ )± 2 N/ 2 a y ( l + cos7r^). J

(4) px = 8, py) pz = 0. T h en

and (3T3) becomes

[ (F —A) (G —A) ( H - A ) - E E * ( F - A ) - B B * ( H - A ) Y = 0. (3-27)

(®) Px^rPy^Pz'The roots o f (3T3) have been calcu lated num erically by a m ethod due to A itken (1937).

A = C — J = K = L — 0 ,

(3-25)

(3-26)

4 . N umerical results

T here are four quantities expressed in term s o f the three param eters a,, fi, /i; these are the elastic constants

fi 12 a (a + 8/*)> c4 4 ~ 2fl(a a + 4/*)’ a + 4A1/11 2a

and the frequency shift o f the first-order R am an line,

1 /8a277c V m '

T here is an identity betw een these four q u an titie s :

8 .47T2c2 v2 ^47t2c2 ~ + 8rx j — 16r44J

(2-42)

(3-20)

[3.4;rV2| V - 8 Cll + l<fc12] ‘(4-1)

I f the action o f second neighbours is neglected (/* == 0) (4-1) breaks up in to two identities,

4^n(gn £44) (Cn+^12)2 5

which has been obtained already in § 2, and

4tt2c22 a

8rn

(2-31)

1. (4 ‘2)

T h e frequency shift of the first-order R am an line o f d iam ond is

v=1332cm .-1, (4*3)

(see for exam ple, R obertson, Fox & M artin (1934)).T he only d irect m easurem ents o f the elastic constants o f d iam ond are those o f Bhaga-

van tam & B him asenachar (1946): they obtained the values

cn — 9*5 x 1012, c44 = 4*3 x 1012, cl2 = 3-9 x 1012 dynes/cm .2. (4*4)16-2



124 H E L E N M. J . S M IT H O N T H E T H E O R Y O F T H E V IB R A T IO N S

A dam s (1921) and W illiam son (1922) have determ ined the com pressibility o f d iam ond, bo th using the sam e apparatus. T hey found, respectively, 0T 6 and 0T8 x 10-6 per m egabar; the m ean of these results, 0T7 x 10~6 per m egabar, corresponds to a bulk-m odulus

5*9 x 1012 dynes/cm .2. (4*5)

But the bulk-m odulus - = J ( r n + 2r12), (4-6)

and substitution o f the values (4*4) gives

— = 5-8 x 1012 dynes/cm .2, (4*7)

in reasonable agreem ent w ith (4*5).Now substituting the experim ental values (4-3) and (4-4) in the identities (4T ),

and (4*2), taking /lattice constant 2 = 3*56 x 10_8cm .,

mass o f carbon atom m — 1*995 x 10_23g., -

velocity of light 3 x 1010cm ./sec.,

we find l .h .s. of (4*1) = 1*40,

l .h .s. of (2*31) = 1*10,

l .h .s. of (4*2) = 0*46.

(2-31)

(4-8)

(4-9)

(4-10)

(4-11)

T he discrepancies (4*10) and (4*11) show th a t the action o f first-neighbour forces is not sufficient to explain the physical properties o f d iam ond, b u t (4*9), w hich takes in to account second-neighbour forces, though an im provem ent on (4*11), is worse th an (4*10). As Born (1946) has already pointed out, there are three possible explanations, either there are experim ental errors, or the second-neighbour forces are not central, or m ore d istant neighbours exert appreciable forces. A cceptance o f the second or th ird a lternative would preclude, a t present, any determ ination o f num erical values o f a, /?, fi. M oreover, in view of the agreem ent betw een the two values of the bulk-m odulus (4*5) and (4*7), experim ental errors in cn and cl2 m ust be small.

Since the R am an frequency (4*3) is m easured optically and so to g reater accuracy than the elastic constants, from (3*20) and (4*3)

T hen

so from (4*4) and (4*12),

kn2c2mv28 0*157 x 106dynes/cm .

a + 8/* = 2 acu ,

(i = 0*0226 x 106 dynes/cm.,

and ~ = i(^ ii + 2^i2)

giving, from (4*7), (4*12) and (4*13),

= 0*104 x 106dynes/cm.

(4*12)

(4*13)

(4*14)




1 / B2\Now e«« = 2a{a ~ J + i/1)

= 5*0 x 1012 dynes/cm .2, (4*15)

on substitution o f the values (4-12), (4-13) and (4-14).Lastly the observed values

v = 1332 cm .-1, cu — 9-5 x 1012 dynes/cm .2, cl2 = 3*9 x 1012 dynes/cm .2,

and the calculated value c44 = 5-0 x 1012 dynes/cm .2,

give l .h .s. o f (4-1) = 1*04, l .h .s. o f (2-31) = 0-95. (4-16)

Thus a change in the value o f c44 from 4-3 x 1012 to 5-Ox 1012 dynes/cm .2 considerably improves the value o f the iden tity (4T ).

N um erical values o f elem ents o f mD(q) for the tw enty-nine sets o f integers px> py, pz given by (3T1) are ob tained from (3-6), using the values (4T 2), (4*13) and (4*14) of

T able 6. F requencies of vibration of diamond under

THE ACTION OF FIRST-NEIGHBOUR FORCES ONLY

(Units of o)j are 1014 sec.-1)Px Py P z (02 (03 (■°4 W5 oj68 4 0 2-29 2-29 1*78 1-78 103 1038 2 2 2-32 2-27 1-84 1*71 1*06 0-958 2 0 2-29 2-29 1-78 1-78 103 1038 0 0 2-29 2-29 1-78 1-78 103 1037 3 1 2-33 2-28 1-89 1-66 1-05 0-957 1 1 2-31 2-30 1-93 1-61 1-02 0-996 6 0 2-32 2-27 1-84 1*71 1-06 ' 0-956 4 2 2-36 2-33 1-92 1-62 0-94 0-856 4 0 2-35 2-29 1*94 1-59 103 0-906 2 2 2-35 2-33 1*98 1-54 0-94 0-886 2 0 2-34 2-32 204 1-46 0-97 0-916 0 0 2-33 2-33 209 1-40 0-94 0-945 5 1 2-37 2-31 1-94 1-60 0-98 0-855 3 3 2-39 2-38 1-95 1-58 0-80 0-785 3 1 2-38 2-35 2*07 1-42 0-89 0-795 1 1 2-37 2-37 2-18 . 1-25 0-83 0-824 4 4 2-40 2-40 1*91 1-63 0-73 0-734 4 2 2-40 2-38 2-02 1-49 0-81 0-734 4 0 2-40 2-34 211 1-37 0-90 0-734 2 2 2-41 2-41 217 1-26 0-71 0-694 2 0 2-43 2-38 2-25 M l 0-79 0*654 0 0 2-41 2-41 2-32 0-96 0-70 0-703 3 3 2-42 2-42 211 1-36 0-67 0-673 3 1 2-44 2-41 2-24 M 3 0-70 0-593 1 1 2-46 2-45 2-36 0-86 0*55 0-522 2 2 2-46 2-46 2-33 0-95 0-50 0-502 2 0 2-48 2-46 2*40 0-75 0-50 0-392 0 0 2-48 2*48 2-46 0-49 0-37 0-371 1 1 2-50 2-50 2*46 0-48 0-27 0-27

Table 6 gives the num erical values o f the frequencies o f v ibration o)j for these tw enty-nine points considering forces betw een first neighbours only. T ab le 7 gives num erical values of frequencies if, in addition, there are central forces betw een second neighbours in the lattice.



T h e frequencies belonging to certain wave-vectors have been p lo tted in figure 4; the wave- vectors are given in term s of ( px, py, pz). T h e separation in to optical and acoustic branches is obvious. T h e degeneracies in some of the graphs could be rem oved by a m ore accura te de term ination of the values o f a, /? and p th an was possible from the available experim ental d a ta .

T able 7. Frequencies of vibration of diamond under the

ACTION OF FIRST- AND SECOND-NEIGHBOUR FORCES

(Units of iOjare 1014 sec.-1)


Px Py Pz 0)l 0>2 (03 0)4 W5 w68 4 0 2-51 2-51 212 212 1-48 1-488 2 2 2-48 2-48 2-27 212 1*53 1-308 2 0 2-51 2-51 217 2-17 1*44 1-448 0 0 2-48 2-48 2-23 2-23 1*40 1-407 3 1 2-50 2-50 2-24 2*07 1-55 1-337 1 1 2-48 2-48 2-30 2-06 1-43 1-356 6 0 2-48 2-48 '2-27 212 1-53 1*306 4 2 2-51 2-51 2-28 2-07 1-35 1166 4 0 2-51 2-51 2-34 201 1-47 1*256 2 2 2-49 2*49 2-33 1-96 1-37 1-216 2 0 2-51 2-51 2-40 1-90 1-34 1*236 0 0 2-49 2-49 2-43 1-87 1-28 1-285 5 1 2-49 2-49 2-27 200 1-46 1155 3 3 2-49 2-49 2-32 206 113 1065 3 1 2-51 2-51 2-35 1-88 1-30 1055 1 1 2-50 2-50 2-42 1-68 117 1144 4 4 2-50 2-50 2-34 211 0-99 0-994 4 2 2-50 2-50 2-38 1-96 114 0-994 4 0 2-50 2-50 2-44 1-78 1*33 0-994 2 2 2-50 2-49 2-49 1*71 0-98 0-984 • 2 0 2-51 2-51 2-49 1-54 109 0-884 0 0 2-51 2-51 2-51 1-35 0-97 0*973 3 3 2-50 2-50 2-45 1-84 0*91 0-913 3 1 2-50 2-50 2-43 1-55 101 0-803 1 1 2*51 2-51 2-51 1-21 0-77 0*742 2 2 2-51 2-51 2-51 1-34 0-69 0-692 2 0 2-51 2-51 2-51 107 0-73 0-542 0 0 2-51 2-51 2-51 0-71 0-52 0-521 1 1 2-51 2-51 2*51 0-71 0-37 0-37

T h e frequency spectrum N(v) can be derived from the calculated frequencies by dividing the range of values o f v in to equal intervals dv and counting the num ber o f frequencies in each interval. H aving regard to the accuracy of the calculations and the to ta l num ber o f frequencies, the in terval chosen was

do) = 2irdv = 0*3 x 1014sec.-1.

Thus three histogram s were p lotted, each shifted 0T x 1014 in the o) scale from the o ther two. These histogram s were then sm oothed ou t and gave the frequency d istribution curves shown (figures 5 and 6) w hich were norm alized so th a t

(4-17)

w here N(v) is the num ber o f frequencies in the in terval v to v + dv and N is A vogadro’s num ber.



(101

4 sec

.-1)

AND THE RAMAN SPECTRUM OF TH E DIAM OND LATTICE 127

0l£___ l_____i_____i___ -S__(0.0.0) (2.2,2) (4,4.4) (6.6.6) (8.8,8) (0,0,0) (2,2.0) (4,4,0) (6,6,0) (8,8,0) (0,0,0) (2,0,0) (4,0,0) (6,0,0) (8,0,0)

(PzPyPz)(« )

(0,0,0) (22,2) (4,44) (6,6,6) (8,8,8) (2,2,0) (44,0) (6,6,0) (8,8,0) (0,0,0) (2,0,0) (4,0,0) (6,0,0) (8,0,0)iPxPyPz)

WFigure 4. (a) First-neighbour forces only, ( ) first- and second-neighbour forces.

frequency of main maximum frequency of second maximum frequency of minimum

o)x : a>2&VW3

ht. of main max. ht. of second max. ht. of main max.

T able 8

figure 5(first-neighbour

forces only)g)x = 2-35 x 1014 sec.-1 oj2 = 0*87 x 1014 sec.-1 (o3 = 1-32 x 1014 sec.-1

2*71-8

1-6

figure 6(first- and second- neighbour forces)

ojx = 2*40 x 1014 sec.-1 (o2 = 1-25 x 1014 sec.-1 o>3 s= 1*75 x 1014 sec.-1

1-91-4

4-6

7-8ht. of minimum 14-4



S eparate d istribution curves were d raw n for the six branches o f the spectra ; these are also shown in figures 5 and 6.

C om parison of tables 6 and 7 and figures 5 and 6 shows th a t the in troduction o f second- neighbour cen tral forces increases the m agnitude of the calculated frequencies for all the tw enty-nine points o f reciprocal space chosen.

T h e m ain features o f the two frequency distributions are given in table 8.T he shape o f the frequency d istribution resembles closely th a t calculated by B lackm an

(1937) for a simple cubic la ttice contain ing one type o f particle.


^ 500

w6 Ws W4 W3 Ut,U/.2nv x 10-14 sec.-1

Figure 5. Frequency distribution of diamond (first-neighbour forces only).

2000

^ 1000

2nv x 10-14 sec.- !Figure 6. Frequency distribution of diamond

(first- and second-neighbour forces).

5 . Specific heat of diamond

I f N(v) is the frequency distribution function o f a crystal then the specific heat a t constant volum e of 1 mole is ,

C„ = A;Jo N { v ) E ( ^ y v , (5T)



AND THE RAMAN SPECTRUM OF TH E DIAM OND LATTICE 129

where k = B oltzm ann’s constant, h — P lanck’s constant, T = absolute tem peratu re , and| n •

E(x) — *s t ie Einstein function.

N(v) is norm alized so th a t its in tegral over the en tire spectrum gives the to ta l num ber of frequencies per mole, i.e. in the case o f d iam ond

(4-17)

The Einstein function has been tab u la ted (Landolt-B drnstein 1927); values o f N(v) were taken from figures 5 and 6 an d the function N(v) E{hvjkT) was in teg ra ted num erically for various values o f T.

Using the d istribution function for first-neighbour forces only (figure 5),

Cv = 0-240 cal./° K a t T = 100° K , Cv = 1-159 c a l./°K a t T = 200°K . (5-2)

The results obtained using the d istribution function (figure 6) for first- and second-neighbour forces are given in tab le 9 together w ith the values o f the characteristic tem peratu re 0 derived from the specific hea t according to D ebye’s theory.

temp. (°K )

T a b l e 9

C v ca l./°K 0 (°K )100 0-0669 1910150 0-270 1800200 0-596 1810250 1-009 1840300 1-463 1870400 2-360 1910500 3-139 1920

1000 4-957 19601200 5-231 1960

Born & v. K arm an (1912) have shown th a t the characteristic tem peratu re 0 a t zero absolute tem perature is ^ / 3

where O is the atom ic volum e and v the m ean velocity of p ropagation o f waves in the crystal. v can be calculated from the elastic constants of the crystal by an approxim ation given by H opf & Lechner (1914). For d iam ond, this gives

© = 1930° K (5*4)at zero absolute tem peratu re.

Experim ental values of the specific heat o f d iam ond have been ob tained by N ernst (1911), Magnus & H odler (1926), R obertson et al. (1936) and Pitzer (1938). T he curve o f the experimental da ta for tem peratures betw een 70 and 300° K is given in figure 7. C om paring the calculated specific heats (5-2) and tab le 9 w ith experim ent it is evident th a t although first-neighbour forces alone give far from accurate results, the in troduction of second-neighbour central forces produces values of the specific heat in close agreem ent w ith experim ent.

Figure 8 shows the relation betw een experim ental and theoretical values of the characteristic tem perature 0 and illustrates the deviation from D ebye’s theory.

Vol. 241. A. 17



T here is no inform ation available on the tem peratu re varia tion o f the m easured elastic constants, b u t K rishnan (1946 a, b) has investigated the therm al expansion o f d iam ond an d the tem peratu re varia tion o f the R am an frequencies. T h e frequency shift o f the p rincipal R am an line decreases from 1333-2 cm .-1 a t 85° K to 1316 cm .-1 a t 976° K , i.e. a varia tion o f 1-3 % per 900° K of the value a t 300° K . T h e corresponding change in the atom ic constant oc over 900° K is, from (4-12), approxim ately 2 | % o f the value a t 300° K . Since the varia tion in the lattice constant is only abou t £ % per 900° K , the change in the elastic constants can be taken to be 2 \% per 900° K , i.e. % per 100° K o f the value a t room tem peratu re . This varia tion will have no appreciable effect on the calculated specific heats as it is m uch less th an the lim it o f e rro r o f the calculations.


temperature (° K)

2000

temperature (°K )Figure 7. Specific heat of diamond. Figure 8. Temperature dependence o f0 . ^

------ experimental curves, © calculated values.

PART II. THE RAMAN SPECTRUM OF DIAMOND

6. I ntroduction

T he R am an effect, the scattering of light by liquids and crystals w ith change in frequency, was first observed in 1928. In the case o f crystals, the frequency differences betw een the incident and the scattered light can be correlated w ith the v ib rational energy states. To simplify the problem , Placzek (1934) assumed th a t though it is the electrons w hich in teract w ith the incident rad iation , the changes in energy are absorbed by the nuclei w hich perform small harm onic oscillations abou t fixed positions. T he polarizability tensor o f the crystal on w hich depends the induced electric m om ent, can then be expressed as a power-series in the v ibrational am plitudes, and so the m atrix elements o f the polarizability are calculated in terms of the energy states of the nuclei and the unpertu rbed electric m om ent o f the crystal.

T he R am an spectrum o f d iam ond has been investigated since 1930, b u t before 1944 only a single intense line w ith a frequency shift o f 1332 cm .-1 had been observed (e.g. R obertson et al. 1930). This line was a ttribu ted to the triply degenerate v ibration against each o ther of the two in terpenetrating simple lattices o f carbon atoms. In 1944 K rishnan succeeded




in pho tographing the second-order R am an spectrum o f d iam ond using the 2536 A rad ia tion from a w ater-cooled q u artz m ercury arc. H e repeated his observations in 1946, obtain ing spectra o f g rea ter intensity an d h igher resolution. T his experim ental m ethod, first developed by R asetti in 1930, can be used only w ith u ltra-v io let tran sp aren t diam onds, b u t since b o th u ltra-violet tran sp aren t and u ltra-v io let opaque diam onds give the sam e frequency shift for the p rincipal R am an line, it is unlikely th a t there is any difference betw een the second-order spectra o f the two types. T h e second-order spectrum consists o f a continuous b an d w ith superim posed sm all peaks, extending over abou t 600 cm .-1 w ith its m axim um n ear the m ercury line a t 2698-9 A. T h e frequency shift of this m axim um from the exciting line is 2460 cm .-1. T h ere is a sm all peak w ith a shift o f 2176 cm .-1 a t the high-frequency side o f the band , and a t the o ther side a sharp line a t 2666 cm .-1 (the octave o f the first-order frequency) m arks the cu t-o ff o f the band .

T he R am an spectrum of any crystal consists in general o f a first-order line spectrum , a second-order line spectrum of sums and differences o f the first-order frequency shifts, and a second-order continuous spectrum o f sums and differences o f any two v ib ra tion frequencies belonging to the sam e w ave-vector. Born & B radburn (1947) first developed the theory o f the second-order continuous spectrum and applied the ir results quan tita tive ly to rocksalt. T he general theory explains satisfactorily the m ain features o f the d iam ond spectrum , the extent o f the continuous ban d and the positions o f the m axim a an d the cut-off. T h e results depend only on the m easured elastic constants, the la ttice constant and the first-order R am an frequency shift. In o rder to explain the relative intensities o f the different m axim a it w ould be necessary to consider the in teraction betw een electronic configurations a round neighbouring nuclei in the crystal.

7. G eneral theory of R aman spectra

Classically, elliptically polarized incident light

E = R( Ae-™1) = J(A (7-1)

produces an electric m om ent M in the crystal, w ith rec tangu lar com ponents

^ , - I V A (/> ,* = 1 ,2 ,3 ) , (7-2)<r

where ap(T is the polarizability tensor.I f S is a u n it vector norm al to the direction o f observation, then the intensity o f the

scattered light is/ = w4 | M s | 2 = (u4 2 2 (7’3)

per pv

From the standpoin t o f q u an tu m m echanics the intensity of light scattered in a transition from a v ibrational state v to ano ther v ib rational state v' is then obtained by replacing w4 by (w +w ^)4 and ocpa. by the m atrix elem ent since (o^^o) for all lines excited by a givenincident beam , we m ay p u t

vv' = 2 2 [ "per] w' p** ** l (7-4)

= 2 2 \}pa, nv\m/ I




For n a tu ra l inciden t light an d scattered light observed w ithou t an analyzer, averages over the polarization o f the incident an d scattered light give

(»+]*).'A , A* I0 cos 9 cos Ba

ApA* = IQsm26p

V<r = — \cos (j> (<r =#/>)»

spsp i s in

(p,(r= 1 ,2 ,3 ) , (7-5)

w here / 0 = £ | A|2 = E 2 — intensity o f the incident light, are the angles betw een the incident beam and the co-ordinate axes, and fa are the angles betw een the observed beam and the axes. In particu la r, if a n a tu ra l incident beam parallel to a co-ordinate axis is observed norm al to its d irection,

6X = 0, d2 = d3 = J tt, fa = 2'h fa

APA* sps<r = 0

(7-6)

(7-7)

/.

* A xA f = 0, A 2A * = A 3A * = I 0,

~ ^3^3 == ^2^2 = I

W ” [*23, 23L 1/ + [*31, 3 l ] t n » ' [ * 3 3 , 33]w'l* ( 7 *^)

Since the last term represents an anisotropic effect depending on the axis norm al to the p lane of the incident and observed light, we replace it, for cubic crystals, by J 2 \}pp, pp\mf'

H ence Iw** K o f t X C w L PL [iv n JConsidering generally [ip(T> = [ccpfa\vv ( 7 *10)

for all v ibrational transitions vto v',since there are m any transitions belonging to the sam e frequency, the weighted m ean of this q u an tity m ust be calculated. I f ev is the energy of the system in the state v then >£ rj ]

(i > = - — ^ ------, (7*11)\ par, pvtav. v' > V /^ e ~ev/

w here k — B oltzm ann’s constant, T is the absolute tem peratu re, and the sum m ation is over the in itial state v.

8. Expansion of the polarizability in terms of normal co-ordinates

T he equilibrium positions o f the nuclei and their small displacem ents are defined

as in § 1 above. Expanding the polarizability tensor in powers o f these displacem ents,( 0 ) ( 1) ( 2) -

V = V + V + V + - - ’ (8<1)( 0 )

w here a „ = constant,

2 v . /■(*)“*(*)> 1 (8*2)f t kl( 2)

aper 2 2 2 ^per, pv k lk 'V ( £ M K ‘)

All these quantities are symmetric in p and cr.



AND THE RAMAN SPECTRUM OF THE DIAMOND LATTICE 133T h e coefficients o f the term s in this expansion are derivatives o f the polarizability taken

in equilibrium and so satisfy certain conditions arising ou t o f the sym m etry o f the lattice. In addition the electric m om ent o f the lattice, like the po ten tia l energy, is in v arian t for a rigid translation o f the crystal as a whole.

H ence is independen t o f l, (8-3)

= At')’ (8-4)

2 v ,fW = °> (8-5)

©IIH'S (8-6)

Following Placzek we assume th a t the nuclei form a set o f 3sN coupled harm onic oscillators. (N is the n u m ber o f cells in the crystal.) T h en the equations o f m otion o f the nuclei com prise N sets, each of 3.y equations o f the type

si!'V+2 2 a '

l'k'0a' k ) 1 v r W i (1-9)

In § 1 these equations were solved by in troducing a p lane wave, b u t here we in troduce

com plex norm al co-ordinates . T h en

and (1*9) becomes '©-•(* 0,

(8-7)

( 8 -8)

where j = 1 ,2 , 3 , . . . , 3j (cf. (1T 1)). T h e choice o f possible wave-vectors q is restricted as before by the condition o f the cyclic lattice.

T h e ea{k are the com ponents o f the eigenvectors o f the la ttice corresponding to the

frequency and satisfy the orthogonality relations

5'-(‘ HS(aj3) S (8-9)

Each com plex represents two real norm al co-ordinates, b u t since

restriction <?f the values o f q to h a lf the u n it cell o f reciprocal space reduces the num ber o f co-ordinates to the correct value.

Substituting (8*7) in (8*2),

( 8-10)

( 1 )

< v = ? ^ |qj /i Ik(8-11)




T h e factor 2 is a ^-function o f q and is zero unless q = 0. H encei ?

Z3

w>6

~5bcs.

53IIb

53 1, (8-12)

w here v Q = 2 2 v , , W ^|° '| i j j M k

(8-13)

T he m atrix elem ents .]! vanish except for transitions belonging to theL \ j j

frequencies so the first-order R am an effect is a spectrum o f 35 — 3 lines. (T he th ree

acoustic branches o f the frequency spectrum o f a crystal have = 0.) This line spectrum

does no t exist if each la ttice poin t is a centre o f sym m etry, for in th a t case a^ . = 0. T he second-order polarizability becomes

V = 2 2 2 2 2 < v , / ^qj q 'j' p Ik l'k ' ? ! J(™kmk)(8-14)

Substitu ting l for ( /—/'), the factor 2 is a ^-function o f (q + q '). So using (8T 0),v

( 2)

a

w here

pa 2 2 pa'q '

l

q

Jj l * p\ j /

(!■) - —>,(* jHf mKy

(8-15)

(8-16)

9. C alculation of the intensity distribution

Consider first a set o f real oscillators characterized by q u an tu m num bers Vj. T h e m atrix elem ents \.Vj\mf°f the am plitude o f one of these oscillators are zero unless changes by ± 1 while all the o ther Vy are unchanged. T he m atrix elem ents o f the square o f theam plitude are zero unless v- changes by 0 or and the m atrix elem ents 0*4=/)o f the p roduct are zero unless v- and Vy bo th change by ± 1.

T h e non-vanishing elem ents can be arranged according to the frequencies

“V = 2 *>j (V j - V j) (9-1)

and are set ou t below in table 10, w here Cj = J ( f i / 2(t)j).

T able 10

Vj0 —

+(Oj Cj J(Vj + 1)-<0J Cj y/Vj

+2 (Oj —- 2 Q)j —

(lij+Giy ----((Oj+COy)

(Oj —(Oj> ---

VjVy Vjc](2vj +1)

cj erV[(wi + })(vr + 1)] cj fy v (*WCjCr jlVj.(Vj + l)]

c]jl(vj + l) 2)] 1)]



F urther, a p roduct o f the form {\j1j\w' [7/']t*/)av. (9-2)will be zero unless bo th factors refei to the sam e oscillator, and a p roduct

([? j (9*3)will be zero unless the first factor refers to the same p a ir o f oscillators as the second, i.e. unless• I f f I f I f f f 1 X f f f M I f fj — j i j J i J iJ J •

For com plex oscillators = rj- + the therm al average o f the am plitude p roduct

< & ] „ [S fL A v . - 2<[^]L->av (9-4)


O m itting the case v — v' (R ayleigh scattering), there are th ree possible ways o f choosing the indices j in the second-order am plitude products. T h e therm al averages corresponding to those th ree ways a r e :

i = / = j " = f , « ] 5 , > a v . = 2 (9'®)

j = j ’ j ' - r , j + r ; » ( ^ )j = f j ' = / , j * f , = 0. (9-7)

T he therm al averages for the real oscillators are ob ta ined from (7*11) and tab le 10, and are given in tab le 11 below. T h e abbreviations used are

A - 2 S v =

T able 11([Vj]vi/)av. ( [7/|]?i/)av. ( \ . V j Vj’lav.

+ (Oj C(j) r * - —- 0)j c(j) - —2<Oj — 2C(j)2 e~W —

2C(j)2 —a)j +<i)j . — — CU) C(f) r-tH*

- \<*j +*>/) — — c u )OJj — (t)jr --- --- CU) cmr*

H ence for the first-order effect, from (7*10), (8*12), (9*4) an d table 11, the intensity o f a single transition is

"per, jiv\ (9'9)'fa (anti-Stokes line).

The Stokes lines are the frequency shifts — and the anti-Stokes lines the frequency

shifts •

Sim ilarly from (7*10), (8*15), (9*5), (9*6) and tab le 11, the intensity o f a single second- order transition is

} per, pv(j') ~ 2CC')C(1') M/j)J s

Stokes

1

anti-Stokes

-4 } ) M U(9-10)

corresponding to the frequencies

( j j )[-0l+H

' } )

KSM0)(9-11)



T h en the in tensity o f a first-order line excited by an incident beam A and scattered in a direction s is, from (7*4),


T o calculate the observed second-order intensity, the single transition in tensity (9T0) has to be in tegrated over the p a rt o f i eciprocal space 0 < 1 w hich belongs to the appro-

Thus the to ta l second-order intensity produced by incident light A scattered in the

I f the lattice has m ore th an one particle in the u n it cell, then the optical branches o f the frequency spectrum tend to finite limits as q approaches zero. C onsequently the range o f in tegration in (9T3) has to be split in to two parts, one p a rt contain ing the single po in t q = 0, and the other, the rest o f the allowed range. A t the poin t q = 0 the sum m ed in tegral (9-13) reduces to a sum over j and j 'in w hich there are (3s)2 term s. T he frequency shifts corresponding to these term s are sums and differences o f the (3s—3) first-order R am an frequencies, so there will be 3^(3i'—3) second-order lines (3(3.r—3) o f these will coincide w ith the first-order lines); nine term s belong to com binations o f the acoustic branches and have zero frequency shift.

T he sum m ed in tegral (9-13) taken over the rem ainder o f the allowed p a rt o f reciprocal space has (3j )2 terms. H ence the observed spectrum will also contain (3,y)2 continuous bands, each one o f w hich has a m axim um a t a certain point. T h e result o f the superposition o f these bands is a continuous background w ith a large num ber o f peaks.

and by applying the m atrices o f the sym m etry operations given in § 2 above, we find

I ( j ) « iper,/iv

pa, /iv 0 = 1 , 2 , 3 , 1 , (a» -3 )) . (9-12)

priate frequency in terval :

= 2j j '

(9-13)

direction s, is= I (9-14)

10. Calculation of the R aman spectrum of diamond

(a) First-order line spectrumT he polarizability term s occurring in (9*9) are given explicitly by (8 T 3 ):

(8-13)

w here for d iam ond, k — 1 or 2 , j = 1,2, 3 ,4 ,5 ,6 and = m2 = m.From the invariance condition (8*5)

a w W + V , / ^ 2) = 0’ ( 10-1)

^ 1 2 ,3 (1 ) — a 2 3 , l ( l ) — ^ 3 1 ,2 ( 1 )

= — a 12, 3 ( 2 ) — — a 23, l ( 2 ) — — a 31, 2

T he rem aining a „ (k) are all zero.

1(2) = n.\

( 10-2)



T h e eigenvectors eJkjj ) a t the po in t q = 0 a re the norm alized la ten t vectors of the dynam ical m atrix D(<?) a t q = 0, viz.


e,(l 11) = 1 / A r2(l 11) = 0, *3(111) = 0 ,

« ,(11 2) = 0, «2(1 12 ) = 1 /A . e3(l | 2) = 0,

fi(l | 3) = 0, «2(1 1 3) = 0, 1 3) = 1 /A .r,(l | 4) = 1 / A e2(l | 4) = 0, «3(1| 4) = 0,

^i(l | 5) = 0, 2(i 1 = i /A , «s(l 1 5) = 0,f i(l | 6) = 0, r2(l | 6) = 0, e3(l | 6) = 1 / A

« ,(2 |1 )»---- 1 /A 2(2 11) 0, 3(2 1 1 ) =<i(2 | 2) = 0, «2(2 |2 ) = - l / A r3(2 | 2) = 0,

r,(2 | 3) = 0, «2(2 | 3) = 0, 4j(2 |3 ) = —1 / Ae,(2 | 4) = 1 / A <2(2 | 4) = 0, *s(2 | 4) = 0,e,(2 | 5) = 0, e2(2 | 5) = 1 /A . 3 (2 | 5) = 0,«,(2 | 6) = 0 , «2(2 | 6) = 0, *3(2 | 6 ) = 1/A. ■

to the frequencies

; i ) = " ( 2 ) = «(3) = //8a '

s/ m *(!)(4) = <o(5) = <u(6 ) = 0 .

in (8-13)

cT

o 11 & to ^to f ^ 0

) = “330 ) = °for a lly ,

0IIo

~5

fcf for = 1 ,2 ,4 ,5 , 6,

a23 {jj~ ® for j = 2 ,3 ,4 ,5 ,6 ,

“3I0 ) = 0for j = 1 ,3 ,4 , 5, 6,

“ 12(3) = ^ ( j ^ = *“ (§) = J m n-

(10-3)

(3-19)

(10-4)

(10*5)

( 10-6)

(10-7)

( 10-8)

But for a n a tu ra l incident beam observed norm al to its direction w ithout an analyzer,

4 t / = i ^ o [ i 2 [ V , / J + i 2 / t j w / l > ( 7 *9 )L fiv n J

so we have to form the quantities A (j) given by

= /IVFrom (10*4) to (10-8),

.4(1) = .4(2) = -4(3)

v QM

2 n2, A{4) = ^ (5 ) = ^ (6 ) = 0.

(10-9)

( 10-10)

Vol. 241. A. 18




Since <y(l) = <y( 2) = <y(3),

A = A = A ) C (l) = C(2) = C(3), (10-11)

so the intensity o f the single first-order line is

7(0) = 2 ¥ » A (j) 2C( j) .( 1 (S to k eslln e )J - 1 , 2 , 3 \e~fo (anti-Stokes line)

1 (Stokes line),

3 /°"4( 1) (anti-Stokes Une).( 10-12)

(b) Second-order spectrumT he polarizability term s in (9-10) are given by

a j ?.,] = 2 2 1 » q\e* (k ' — r , (8-16)'w\ j 117 f « > '“'•'“W / n 7 / \ K 1

w here k, fc' = 1 or 2, j , = 1 ,2 ,3 ,4 ,5 ,6 and m m.

First consider the constants the label the bonds betw een a nucleus

of the base cell and some o ther nucleus . T h e num ber o f independent can

be reduced by applying the sym m etry operations o f the lattice and the invariance condition (8-6). In the d iam ond lattice, each nucleus in the u n it cell has 4 first neighbours and 12 second neighbours; to account for the observed specific heat it is necessary to assume central forces betw een second neighbours. I f the polarizability were taken to the same

approxim ation as the dynam ical m atrix , there w ould be in all 32 different values

of j to consider. But these txp(T) pivWj^\ are derivatives o f the polarizability o f the crystal

taken in equilibrium and so depend on the in teraction betw een the electrons surrounding neighbouring nuclei in the lattice. T he determ ination o f this in teraction is a pertu rba tion problem in quan tum m echanics w hich will no t be considered here. Therefore to reduce the num ber o f a rb itra ry constants appearing in the intensity factors to a m inim um we

shall assume th a t the ap<Tf are zero f°r all neighbours except the first.

Labelling the bonds as in tab le 1, § 2, the 10 independent ap < r > are:

**11, l l( l) ^ a> a 12, l l ( l ) = a 12, 22(1) = /»a l l , 22( l ) “ a l l , 33( l ) = b,#12,33(1) =

**11,12(1) = c, #12,12(1) =

* * 1 1 , 2 3 (1 ) — d„ * * 1 2 , 2 3 ( 1 ) — j,**11,31(1) = e., #12,31(1) — k.

(10-13)

All the o ther non-zero ap<r ftv{kk') can l)e obtained from these by the use o f the sym m etry

operations o f § 2 and the invariance condition (8-6).



Now consider the eigenvectors e ^k % T here is no difficulty abou t the second-order

line spectrum for the range o f in tegra tion in (9-13) is the single po in t q = 0 and the eigenvectors required are given in (10*3) above. But for the continuous p a rt o f the spectrum , although the eigenvectors could be found for each po in t o f reciprocal space for w hich the

corresponding frequencies are a lready know n, the calculations w ould be lengthy and


tedious. T he frequencies as functions o f q, are stationary a t the po in t q = ( jW '

(cf. figure 4 ab o v e); the frequency density zj r (oj) o f 6^ , given by

zjy {(o) d(0 = JJJ dqY dqz

¥5 2>

(10-14)

(o< |&>(?.() J <(i)+dw

la,J

will be a m axim um a t this point. T hus a first approx im ation to the sum m ed in tegral (9-13) is

M = 2 v , J | / ) Z jfM , ( io -i5 )where q = ( J , | , J ) .

A t the point q = 0, (8-16) becomes

(1l34l/)-4l 5) 413)+4 ] 3 413)+41341 +£(-4I34I3)~4I34I3M iI34I3M2I34

!') 4 134 4 13 4 13)+4 1 3e*(213)+'41 3 4 1/)} ■> \ (1 1 e)

4am

J- e A l

P'12(1)= *44134 1344134 I344I34I344I34I3) 3 4 1 3) +413 4 1 3)+413 413)+4 13ef(

Substituting the eigenvectors (10-3),

All other v ( j - )

Now let

‘ 4>\ 8a1/ “ m ’ rv

I c400 1 1II ( 0 \ 8

(33/ _ m ’

“22(1°>\ 8 l j _ m 9 a22(22;( - a22| 1

00 1 11

a33(i°>\ 81/ m ’ a33(22j

^ 0 \ 8a,33/ “ m 9

a,2(ii3 = “23(2°3,) = a 130 \ 8 13/ ~ m '

(10-17)

are zero.

A(( ? v) ~ h i i m p ) i + i 4 " ( 3 ' )

(10-18)



= 1 ,2 ,3 ,

*{)?') “ * 0 + K ? ) “ 2J S ;

140 HELEN M. J . SM ITH ON T H E TH EO RY OF TH E VIBRATIONS

Since

(10-19)

( 10-20)

( 10-21)

Hence C Cl andfor j , j ' = 1 ,2 ,3 , and there will be a single second-

order R am an line for d iam ond w ith a frequency shift (o( = 2 / — and intensity, from(7-9) and (9-10), *

1 (Stokes line)or

M l( 10-22)

\e (anti-Stokes line).

A t the po in t q = (^, | , ^), (8-16) becomes

( 1 ) “ “ I {2e‘( 1 0") ■* (! I ;7 ■+: I i ) 4 (21 f ) + #,<11 i) 4 (2 l i ') + «,(a | i) 4 (1 1 / ) }

- f {2«2(1 | i ) 4 ( I I JO + 2<2(21 j ) <* (2 J j ’) + <2(1 | j ) ^ ( 2 | / ) + 4 2( 2 j i ) 4 ( l I / )

+ 2*3(1 | i ) 4 ( l | / ) + 2 ^ ( 2 | j ) e? ( 2 ] / } + e, ( l \ j ) e*(2\ f )+e3(2\j)el( l |/)}O/*

+ — W i li) 4 (2 1/) +«i(i li) (2 1 / ) +*1(2 1 j ) 4 ( i | / ) + ^ ( 2 | i ) 4 ( i 1 / ) }

o //+ - W 1 U > ? (2 |/)+ % (l l i ) 4 ( 2 |/ ) + 4 2( 2 |i ) 4 ( l | / ) + « , ( 2 |j ) 4 ( l I/)}

li)« * (2 |/)+ « 2(l | j) **(21 j') + «j(21 j) ef (1 | / ) + * 2(2 |j >*(1 |/)} ,

( j - ) = + | j K ( 2 | / ) + < 2( l 17 ) * * ( 2 1 j ”) + * , ( 2 1 j f ) « * ( 1 | j ' ) + < 2( 2 | j > f ( l I / ) }

+ 5 {« j(1 l7 > ? (2 t/)+ « 3(2 b > ? ( l I / ) }

0 h-^■{2«,(l li) 4(1 l / ) + 2c2(l li) 4(1 li') + 2«,(2 li) 4 (2 | / ) + 2e2(2 |i) « f(2 |/)

+«i(i l i ) 4 ( 2 li ')+ « 2(i l i ) 4 ( 2 li ')+ « i(2 |i )4 ( i li ')+ « 2( 2 |i ) 4 ( i li')}

+ |{ « i( i l i ) 4 ( 2 |i ')+ « 2(i li) 4 (2 l i ') + 4,(2 li) 4 ( i li ')+ * 2( 2 |i ) 4 ( i li')}2 k

+ -{ * 3(11 j) e*(2I f ) +e3(l | j)4( 2 1 f ) +^3(2 li) 4 ( 1 1/) +*3(2 li) 4 ( 1 1/)}.

(10-23)

(10-24)




T he eigenvectors e\k j j a t q = ( | , J) are the norm alized la ten t vectors o f the dynam ical

m atrix D(q)a t th a t point. F rom (3*6) and (3T8) they are as follows:

«l(l 11) = 1. «2(1 11) = 12? «s(i 11) = 0 ,

« i( l|2 ) = 1 /2 A 2(1 1 2) = 1/2 7 3 , «2(i 12 ) = -1 /7 3 ,

« i( l |3 ) = i/A <2(1 1 3) = 1/76, «j(l 1 3) = 1/76,« ,(1 |4) = i/A «2(1 1 4) = 1/76, «s(l 1 4) = 1/76,

*l(l 1 5) = l> <2(1 1 5) = —1, «j(l 1 5) = 0,

« i( l|6 ) = 1/2 7 3 . «2(1 1 6) = 1/2 7 3 , <j(l 1 6) = - 1/7 3 ,

« i(2 |l) = 1 «2(2 |1 ) = 1"“ 5 «s(2 11) = 0,«i(2 | 2) = 1/2 7 3 , «2(2 | 2) = 1/2 7 3 , *.(2 | 2) = - 1/7 3 ,« i(2 |3) = 1/76, «2(2 | 3) = 1/76, «j(2 | 3) = 1/76,

ci(2 | i) = -1 /7 6 , «2(2 | 4) = -1 /7 6 , *>(2 | 4) = -1 /7 6 ,

<i(2 1 6) = —i> e ,(2 1 5) = i , Ca(2 | 5) = 0,

e1( 2 |6 ) = - 1/2 7 3 , o2(2 1 6) = - 1/2 7 3 , <3(2 1 6) = 1/7 3 ,

(10-25)

These values are inserted in (10-23) and (10-24) and a lengthy calculation leads to the following values o f ,N 2

..................1 M 0 N ?h iflV

(10-26)

at the poin t q = (J , £):

m2A ( l l ) = ±{2(3a + 3b+.c+e)2 + ±{3b + d)2} + 2 ( f + g - j -m2A{22) = £ r{2(3a + 15b + c + 4:d+e)2 + 4:(Qa + 3b + 2 c -d + 2e)2}

+ f (5 / '+ g + I2h—j — k)2-{-%[/+%g — 3h — 2j —2k)2,m2A{33) = §{3a+ 6b-2c-2d-~2e)2+ ± {2 f+ g -6 h + 2j+ 2k)2,

m2J(4 4 ) = %(a + 2b + 2c+2d+2e)2+§{2f+g + 2h + 2j+2k)2,m2A(55) = J{2(<z+A —c—tf)2 + 4(£ —1/)2} + 2 ( / + £ —j —A)2 + 4 ( / —A)2,

m2T(66) = ^{2 (tf + 5 6 - r - 4 t / - < ) 2 + 4(2<z-F6-2r + < /-2e)2}+ i(6 /* + g —4A— A)2+ f(jf- |-2 g + A — 2j — 2A)2,

m2J(1 2 ) = }(3a — 3£ + c—2</+t?)2H -f ( / — g+ 6h+ j + k) 2,

m2A{\$) = m2T(23) = t ( 3 f l - 3 6 - 2 r + t / + < 0 H |( / - £ - 3 A + j - 2 A ) 2,

m2 4(45) = m2T(46) = f (a — r —^/ +2^)2+ f (y _ ^ + A —2j+ A )2,

m2 (56 ) = f ( a - & - c + 2 r f - * ) 2+ f ( / - £ - 2 A + j + A ) 2.

(10-27)

All other A(jj') are zero.Then the.intensity distribution of the second-order continuous spectrum is

Stokes anti-Stokes

m = v m d i ) c ( / ) ^ (l 0 '28)

with A(jj'), C(j) and /?(j) taken a t the point q = (^, ^).



11. N umerical results

142 HELEN M. J. SMITH ON THEORY OF THE VIBRATIONS

(i a)First-order line spectrum

T h e observed value o f the frequency shift of the p rincipal R am an line (1332 cm .-1) has a lready been used to determ ine the values o f the atom ic constants o f the la ttice (cf. (4T 2)). T he to ta l intensity given by (10T2) contains as a factor the a rb itra ry constant 3(1)*

T h e ra tio o f the intensity of the Stokes line to th a t o f the anti-Stokes line is w here

ho

2 n k T ‘

T aking — 2-51 x 10l4sec. *, and the tem peratu re of the crystal to be 300° K , this

ra tio is 586. T h e observed value given by K rishna^ (1946c) is 575.

(b) Second-order line spectrum

T he frequency shift of the single second-order R am an line o f d iam ond is, from (10*20), twice the frequency shift o f the first-order line, i.e. 2664 cm .-1. This agrees satisfactorily w ith K rishnan’s experim ental result o f 2666 cm .-1. T he intensity o f this line (10*22) contains three a rb itra ry constants,

a ~a n, n(l)> ^ = a n,22(l)> ^ = a i2,12(1)*

No d a ta are available on the ra tio o f the intensities o f the Stokes and anti-Stokes lines.

(c) Second-order continuous spectrum

Referring to § 4, the density functions z^.{o) can be calculated from table 7 by the sam e m ethod th a t was used to ob ta in the frequency distributions. T he range of values o f 0) is divided into equal intervals do = 0*6 x 1014sec.-1 and then the num ber o f com binations

^ o f frequencies belonging to the same wave-vector q can be counted in

each interval. T hree histogram s are p lotted, each shifted 0*2 x 1014sec.-1 in the o scale from the o ther two. These histogram s are then sm oothed out and give the frequency density curves Zjy(o).

From (10*27) we need consider only the 18 branches for w hich the A(jj') are no t zero. T he density functions Zjj>((o) for these branches are shown in figure 9 and the frequency shifts of the m axim a of the branches are given in table 12.

Com parison of figure 9 w ith K rishnan’s experim ental results shows th a t the factors A(jy') m ust be zero for the branches 4, 5, 7, 8, 9, 11, 12, 13; i.e.

A(55) = 4 (66) = 4 (45) = 4 (46) = 4 (56) = 0. (11*1)

T he branches (10), (g — o3)and (o2—o3), m ay be covered by the incident m ercury line. Further, the calculated m axim a a t 2177 and 2469 cm .-1 can be identified w ith the experim ental peaks a t 2176 and 2460 cm .-1. These calculated frequency shifts depend only on



AND RAMAN SPECTRUM OF THE DIAMOND LATTICE 143the m easured elastic constants, the la ttice constant, and the frequency shift o f the p rincipal R am an line. K rishnan claims to identify a num ber o f add itional peaks on either side o f the m ain m axim um a t 2460 cm .-1 b u t from the m icrophotom eter record there is no reason to suppose th a t these are any th ing o ther th an random fluctuations o f intensity.

2ttv x 10-14 sec.-1Figure 9 . Frequency density functions

2 tol . 2 2 (o3. 5 2 oj6. 7 (1 ) 4 + tog. 10 ftoj - t o 3 .2 ( 0 2* 3 2 (o4. 6 f(0 1 + ( 0 3. 8 0 ) 4 + tog. W 2 - t o 3 .

( “h 6t>2* 4 2 (o5. \( 0 2 + (0 3. 9 w5 + tog. 11 to4 - t o 5 .

T a b l e 12

(cm.-1) (cm.-1)2 (0 x 7 to4+to5 1805

1 2 (o 2 2602 8 to4+to6 1683to, +to. 9 to5+to6 1434

2 2to3 2469 10 toj-to3 1063 2 (0 4 2177 CO 2 ““4 2 (o5 1487 11 to4-to5 3455 2 to6 1258 12 to4-w 6 4096 toj +w3

to2+to3 2522 13 to5-w 6 133

Now from (10*26), if A (55) = 0,

a - \ - b = c-\-e, b = d, f = h, ( H ’2)

and so ^4(66) = ^4(56) = 0.

Ifd (4 5 ) = 0, then using (11*2),

a = c, b = d = e, , g = K

1 , e * a l l , l l ( l ) = a l l , 1 2 ( l ) j

®11, 2 2 ( l ) ^11 , 2 3 ( 1 ) “ ®11, 3 3 ( l ) = a l l , 3 l ( l ) >

a 12, l l ( l ) = a 12,1 2 (1 ) = a 12, 2 2 (1 ) = ^*12, 23( 1)5

a I 2 , 3 3 ( l ) = = a 1 2 , 3 l ( l ) *

(11*3)



144 H E L E N M. J . S M IT H O N T H E O R Y O F T H E V IB R A T IO N S

T hen the rem aining non-zero A (jj') becom e,

A ( n ) = A ( 2 2 ) = — 2(a* + 2 ab + 3b* + <if2),

^ (44) = p { ( 0 + 2 4 ) 2+ 3 ( 2 / + j n

QO^ ( 12) ~3m? { i(a ~ ^ ) 2 + 4/ 2}>

^ (1 3 ) = ,4(23) = 3^-2 { * (« -* )2 + ( / + 3 g ) 2}.

(11-4)

T ab le 13 gives the values o f e && and C(j) for the poin t q = (|> 2>l)> taking the tem -pera tu re o f d iam ond to be 300° K . t

T a b l e 13

j 1 2 3 4o ) ( j ) 10-14 2-50 2-50 2-34 2-11

e-fiU) 0 0 0 1 7 5 0-00175 0-00263 0-004702 C ( j ) [ h x 1014 0-401 0-401 0-429 0-476

T he frequency density functions corresponding to the non-zero A (jj') are then m ultiplied by products and squares o f these C(;) and e ~ ^ as in (10’28). Lastly, by choosing approp ria te values of ratios o f the A (jj') in (1T4) it is possible to approxim ate to the observed to ta l in tensity o f the Stokes branches; the final result is shown in figure 10, w hich contains the separate branches, the to ta l intensity, and K rishnan ’s experim ental curve. T h e ratios

betw een the A(jj') are ^ 3 3 ) ^ ( 4 4 ) :.4(13) = 4 0 :1 2 :1 :1 . '(11-5)

v (cm.-1) Stokes branches

Figure 10. Second-order Raman spectrum of diamond. Fine line is a sketch of Krishnan’s microphotometric record of the Raman spectrum of diamond (Krishnan 1946 : broken lines representtheoretical contributions to the intensity of frequency density functions Zjyioj) after multiplication by appropriate factors: thick line represents the superposition of these functions, i.e. the total theoretical intensity.



AND RAMAN SPECTRUM OF THE DIAMOND LATTICE 145

From the relations (11*3), (11*4) and (11*5) betw een the derivatives o f the po larizability tensor it m ay be possible to ob ta in inform ation ab o u t the electronic configurations in the two types o f d iam ond.

The approxim ation could be im proved by rem oving the degeneracies in the frequency spectrum , by considering elem ents o f the polarizability tensor referring to second neighbour atoms, or by calcu lating the eigenvectors for m ore th an one po in t in reciprocal space. T h e agreem ent betw een experim ent and theory is, how ever, sufficient to show th a t la ttice dynamics is able to give a satisfactory exp lanation o f the R am an spectrum of d iam ond.

I wish to thank Professor M . Born, F .R .S ., who suggested this p roblem to m e, for his continued in terest and his advice on m any occasions.

R eferences

Adams 1921 J . Wash. Acad. Sci. 11, 4 5 .Aitken 1937 Proc. Roy. Soc. Edinb. 5 7 , 2 6 9 .Bhagavantam & B him asenachar 1944 Nature, 1 5 4 , 546. Bhagavantam & B him asenachar 1946 Proc. Roy. Soc. A, 1 8 7 , 3 8 1 .Blackman 1937 Proc. Roy. Soc. A, 1 5 9 , 4 1 6 .Born 1914 Ann. P h y s . , Lpz., 44 , 6 0 5 .Born 1923 Atomtheorie desfesten Zustandes. Leipzig: Teubner.Born 1940 Proc. Camb. Phil. Soc. 3 6 , 1 6 0 .Born 1942 Rep. Progr. Phys. 9 , 2 9 4 .Born 1946 Nature, 1 5 7 , 5 8 2 .Born & Begbie 1947 Proc. Roy. Soc. A, 1 8 8 , 1 7 9 .Born & B radburn 1947 Proc. Roy. Soc. A, 1 8 8 , 1 6 1 .Born & v. K arm an 1912 Phys. Z . 1 3 , 2 9 7 .Bragg & Bragg 1913 Proc. Roy. Soc. A, 8 9 , 2 7 7 .H opf & Lechner 1914 Verb, dtsch. phys. Ges. 1 6 , 6 4 3 .K rishnan 1944 Proc. Ind. Acad. Sci. 1 9 , 2 1 6 .K rishnan 1946 a Proc. Ind. Acad. Sci. 2 4 , 33.K rishnan 1946 bProc. Ind. Acad. Sci. 2 4 , 4 5 .K rishnan 1946 c Proc. Ind. Acad. Sci. 2 4 , 2 5 .Landolt-Bornstein 1927 Phys.-Chem. Tabellen. Berlin: Springer.Magnus & H odler 1926 Ann. Phys., L pz., 8 0 , 8 0 8 .N agendra N ath 1934 Proc. Ind. Acad. Sci. 1, 333.Nernst 1911 Ann. Phys., L pz., 3 6 , 3 9 5 .Pitzer 1938 J . Chem. Phys. 6, 68.Placzek 1934 Handbuch der Radiologie, ed. M arx, 6, p a rt 2. Leipzig: Akad. Verlagsg. m .b.h. R am an 1941 Proc. Ind. Acad. Sci.14 , 4 5 9 r R am an 1944 Proc. Ind. Acad. Sci. 1 9 , 1 8 9 .Robertson & Fox 1930 Nature, 1 2 5 , 7 0 4 .Robertson, Fox & M artin 1934 Phil. Trans. A, 2 3 2 , 4 6 3 .Robertson, Fox & M artin 1936 Proc. Roy. Soc. A, 1 5 7 , 579.Sommerfeld & Bethe 1933 Handb. Phys. 2 4 , p a rt 2, 4 0 2 .Williamson 1922 J . Franklin Inst. 1 9 3 , 4 9 1 .



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