Electron-ion interactions in close packed metals

Pram~n.a, Vol. 24, No. 5, May 1985, pp. 757-771. © Printed in India.

Electron-ion interactions in close packed metals

V R A M A M U R T H Y

Department of Physics, Indian Institute of Technology, New Delhi 110016, lnOla MS received 1 December 1984; revised 5 February 1985

Abstract. The electron-ion interactions are evaluated exactly over the actual shape of the atomic polyhedron by making use of simple co-ordinate axes transformations and lattice symmetry in the case of hcp and ccp structures. It is shown that the expressions for the interference factor, S(q, t) of hcp structure are complex while those of cop structure are real, even when both atomic arrangements are referred to the same orthorhombic co-ordinate axes, and in each case, lattice atom contributions could be distinguished from basis atom contributions to S(q, t). By comparing these expressions with each other as well as with those obtained by approximating their atomic polyhedra by an ellipsoid of equivalent volume, apparent differences between interference factors of hop and cop structures, validity of Wigner-Seitz approximation for a diatomic lattice and the manner in which the electron-ion interactions contribute to the different modes of vibration in'a hexagonal lattice are discussed.

Keywords. Electron ion interaction; interference factor; coordinate axes transformation;

umklapp process; hop lattice; ccp lattice.

PACS No. 63"20

1. I n t r o d u c t i o n

T h e e l e c t r o n - i o n i n t e r a c t i o n s m a n i f e s t t h e m s e l v e s w h e n e v e r the t h e r m a l m o t i o n o f the ion p e r t u r b s the energies o f c o n d u c t i o n electrons. It is necessary to e v a l u a t e the f o r m e r by a v e r a g i n g the l a t t e r over the a c t u a l s h a p e o f the a t o m i c p o l y h e d r o n in o r d e r t o e n s u r e t h a t the f o r m factors a n d the i n t e r f e r e n c e factor, S(q) so o b t a i n e d a r e sensitive to the s y m m e t r y o f the lattice. R a m a m u r t h y (1978, h e r e i n a f t e r referred to as I) has s h o w n t h a t this s u m c o u l d be e v a l u a t e d exactly in the case o f c u b i c s t r u c t u r e s as well as t e t r a g o n a l s t r u c t u r e s ( R a m a m u r t h y 1979) b y e x p l o i t i n g their lattice s y m m e t r y a n d in each case there a r e several a l t e r n a t i v e ( b u t e q u i v a l e n t ) w a y s o f w r i t i n g d o w n the i n t e r f e r e n c e factor, S(q). Besides, o n e o r the o t h e r o f these e x p r e s s i o n s was o b t a i n e d b y Bross a n d B o h n (1967), S h a r a n et al (1972, 1973) a n d A s h o k k u m a r (1973). O n the c o n t r a r y , n o a t t e m p t has been m a d e so far to d e d u c e e q u i v a l e n t e x p r e s s i o n s for S(q) in the case o f h e x a g o n a l c l o s e - p a c k e d (hcp) s t r u c t u r e s a n d to c o m p a r e t h e m with the c o r r e s p o n d i n g expressions, p u b l i s h e d elsewhere, in the case o f c u b i c c l o s e - p a c k e d (ccp) structures. Since the a m p l i t u d e o f t h e r m a l m o t i o n o f a lattice a t o m differs f r o m t h a t o f a basis a t o m in a d i a t o m i c lattice, it is all the m o r e i m p o r t a n t to d i s t i n g u i s h b e t w e e n t h e i r c o n t r i b u t i o n s to S(q) as well as to d e t e r m i n e the m a n n e r in which the u m k l a p p p r o c e s s e s c o n t r i b u t e a n d restore the t r a n s l a t i o n a l s y m m e t r y to the o p t i c a l m o d e s o f v i b r a t i o n . H e n c e , the p r e s e n t p a p e r d e s c r i b e s an exact e v a l u a t i o n o f the i n t e r f e r e n c e f a c t o r s a s s o c i a t e d with the close p a c k e d s t r u c t u r e s which m a k e s use o f t h e i r lattice 757

758 V R a m a m u r t h y

symmetry. To facilitate the comparison between these structures and their interference factors, the atomic arrangements in both eases are referred to orthorhombic co- ordinate axes.

2. Theory

When the effect o f thermal motion o f the ion, represented by the wave vector, q is averaged over the conduction electrons present in the atomic polyhedron without approximating it by a sphere or an ellipsoid o f equivalent volume, but treating the electrons as free (the band structure effects and the exchange as well as the correlation effects being taken into account through appropriate effective mass and screening function (Ramamurthy and Singh 1978), respectively the interference factor, S(q) is given by

S(q) =

foexp

(iq. r)dn/D. (1)

where f2 is the volume of the atomic polyhedron. This expression could be reduced, by making use o f Gauss" divergence theorem and some vector identities involving V (see equation (2) in It, to the following forms:

(crx + ay + try)

S2(q) = i)(qx + qy + q:)' (2at

(q~trx + qy% + q, crz)

S 3 ( q ) = n(q~ + qy + q~) ' (2b)

Here a~, .y and a. are the Cartesian components of e defined by

lls

~ = ~- exp (iq.r)dS, (3)

where the integration is over the surface of the atomic polyhedron and hence has to be evaluated separately for each atomic arrangement. This integral is evaluated by adopting the procedure described in I in the ease of diatomic hcp and triatomic ccp structures in the next section.

3. Evaluation of the interference factors 3.1 hcp structure

The atomic polyhedron ofhcp arrangement is shown in figure 1 (at. It is a dodecahedron consisting of three pairs o f isosceles trapezoidal faces, denoted by (200), (110)and (1 TO), parallel to the z axis and three pairs of rhombic faces, denoted by (1 ] 1 ), (] 11 ) and (021), which intersect the z axis at + (3c 2 + 4a2)/6c where 2a and 2c are the lattice constants along x and z directions. The former has one side of length 2a(l + 3t2) ~/2/3t with t = c/a common with the latter. All faces, except a pair of (021) rhombic faces contribute to try, a pair of (200) trapezoidal faces do not contribute to ~y whereas none of the trapezoidal faces contribute to try. Nevertheless, the symmetry associated with the polyhedron reduces the evaluation of the integral (3) to that of contributions from trapezoidal and rhombic faces to ax and try, respectively. Since the (200) trapezoidal faces are

Electron-ion interactions in close packed metals

759 perpendicular to the x axis, their contribution to a~ could be expressed as

o : 0 ,

⁼

f°"'

^exp

^{(iq, y)}

^exp

(iq=z)dS~,,

(4) L i J-* J-~t~,

where tl = x/3 and d = (3c 2 -

2aZ)/6c.

Besides, the co-ordinate axes transformation X = cos ~b (x + ti y), Y = sin e p ( y l q -

x)

and Z = z, (5) which rotates x and y axes through an angle ~ = t a n - ~ (tl) = 60 ° about the Z axis and orients the (110) trapezoidal faces perpendicular to the X axis, is made use of to write down their contribution to ax in the form

o=(llO, ₌ [ e x p { i ( q . + t i q , , X } - ~ / i

r./2 f~

_exp

{i(qylh - qx) Y}

_]-a12 d -al2 d

exp

(iq=Z ) dS x.

(6)

The evaluation o f these integrals yields

ax(200) = 4 sin

(q~a)

{sin [(q, +

q=/s)a']

exp

(iq=d)/(qy + qJs) iq=

-- sin

[ (q, -- q=ls)a' ]

exp ( -

iq=d)l(q, - q=ls) }

(7) and

4 sin

[ (qx + t lqy)a/2] ~

sin

[ (h qx - qy + qUy)a'/E]exp (iq=d)

ox(110) =

zq, ( (t~qx ^- q, ⁺ q=h)

sin [ (ti

q: -_ q, - q=h)a'12]

^{exp ( -}

iq.d) ~ ,

(8)

it,q. - q, - q=lT) J

with a ' =

a/h, s = tit

and 7 =

t~tl2.

The corresponding contribution from (1i0) trapezoidal faces could easily be written, by substituting - q y for qy in the complex conjugate o f (8), as

4 sin [(q~

- tt qy)al2]

~sin [(tl

q: + qr - q=l~,)a'12]

exp (/q=d)

ax(l-lO)

7~ [ (tlq. + q,

^- q=/7)

sin

[ (ttq=

^÷

q, + q=lT)a'12 ]

exp (-iq=d)

~

(9)

(tiq= + q, + q=lT) j"

Further, the co-ordinate axes transformation

tl

[x + t i y ] and

Z

⁼

( 4 + 3 t z ) t l 2 [ t z + x - y / t , ] , X

⁼

3 t

r / = [2(4 + 3tz)] ill [ }zlt ^- x +

y/t,],

(10) orients the ( l i d rhombic face perpendicular to the Z axis and transforms it into a square. Hence its contribution to as could be expressed as

° = ( l l l ) = [ e x p { i ( t q = + q ~ - q ' / t i ) Z } ~ / 2 i

~_.12

Io/2 exp{i(q~,+tiq,)z}

exp{i(5lq=lt-q=+q,lt,)~l} dSz.

(11)

all d - a l l

760 V R a m a m u r t h y

Making use o f the co-ordinate axes transformation

Z = Z , X = ½ ( ~ + i ) a n d Y = ½ ( i f - X ) , (12) which rotates g and t/axes through 45 ° a b o u t the Z axis, the integral is evaluated to obtain

~z (1]1) = 8tl sin [ (qJ~ - q q x - qy)a'/2] sin [ (qJy + 2qy)a'/2]

exp

[i(tq.

+ qx - q , / t , ) a / 2 ] / i ( q / ) , - t , qx - q , ) ( q . h ' +

2q,,). (l 3)

The corresponding contributions from (1 H), (i [1) and ( i ] [ ) rhombic faces could be written as follows by replacing q~ by - q ~ , q~ by - q ~ and qz, q~ by - q z , - q x respectively in (13):

~(1 ] 1) = 8q sin [ (q J 7 + tl qx + qy)a'/2] sin [ (q J 7 - 2qy)a'/2]

exp [ - i(tqz - qx + qyltl)al2]/i(q:/? + t l qx + qy) ( q . l ~ - 2q,), (14)

(~o)

"021) ~ 0 ) "

),...

! ,j'.. • /

'"0

:~o;:

(021)

/

/ l

o---- - . ~ . __t.__. __J

--~",,

^.x

~TO,

• i

//

R I (a)

E l e c t r o n - i o n interactions in close p a c k e d m e t a l s

Cv

761

l l l l l l I l

(I11)

e ( 0 2 1 )

/ /

, .

1 1 / I l /

N x

%xxl

,~i~o, I

• I

\

- - - - I 0

,'.021:

(111)

/ / /

• 1 / ~ r ' " /

(b)

and

Filgate 1. Atomic polyhedra o f close packed metals referred to orthorhombic coordinate axes: (a) hcp structure (b) c~p structure. O, O and triple integers denote the centre and Miller indices o f each face. The cubic coordinate axes pass through P(a, a/~/3, c/2), Q ( - a, a/~/3, c/2) and R (0, - 2 a / x / 3 , c/2).

a=(I]'l) = 8tt sin [ ( q J y + t~qx - - q , ) a ' / 2 ] s i n [(q=/7 + 2q,)a'/2]

exp [i(tq= -- q~ -- qy/tl)a/2]/i(q=/y + ttq~ -- q,) (q=/7 + 2q,) (15)

a=(ii 1-)

= 8tx sin [(q=/7 - tlqx + qy)a'/'2] sin [ ( q J T - 2qy)a'/2]

exp [ -

i(tq= + qx + qy/t~)a/2 ] / i ( q J 7 - t xqx + qy) (q=/T - 2qy). (16) O n the other hand, the co-ordinate axes transformation

Z = sin O(Tz + y), x = x and r / = cos O(y - z/y), (17) rotates the z and y axes through an angle 0 = t a n - ~ (1/7) about the x axis and orients the (021) rhombic face perpendicular to the Z axis. Its contribution to a= is therefore given

762 V R a m a m u r t h y by

exp {i(q, - } dSz. (18)

This integral is evaluated with the help of the co-ordinate axes transformation X = ½(x + tit/), Y = ½(~I/tl - x ) and Z = Z, (19)

• which transforms the rhombic face into a square face and rotates x and v/axes through 45 ° about the Z axis, to arrive at the expression

az(021) = 8tt sin [ (q:/y - qy + t l qx)a'/2]sin [ (qffy - qy - t l qx)a'/2 ]

exp [i(yqz + qy)a']/i{(qffy.- q y ) 2 _ (tlqx)2 }. (20) The corresponding contribution from (02 i) rhombic face could be written down, by substituting - q z for q~ in (20), as

a,(02i) = 8tl sin [ (qffy + qy - t~qx)a'/2] sin [ (qffy + qy + t~qx)a'/2]

exp [ - i(yq~ - qr)a']/i{ (qffy + qy)2 _ (t lqx)2 }. (21) Similar expressions for the contributions from ( 11 l) rhombic faces and from all faces, except (200) trapezoidal faces, to tr~ and ay respectively, are obtained by taking into account the corresponding components of area of these faces. For instance, x and y components are t - t and s - ~ times the z component o f area of (111) rhombic faces and hence their contributions to the former and the latter are related by

o~(111) + o~(lii) + ox(| iI) + tr~(f ii-)

[ a,(1 l l ) - , , ( 1 i f ) - a=(i f l ) + a,(fiT) ] . (22)

L

^t

d

and

a,(l i1) + %(1I f) + a,(I fl) + ~r,(li i')

= [-a,(lil)+o,(l-lI)-oAiil)+tr,(IIi)]s " (23)

On the contrary, y component is y- t times the z component ofarea of (021) rhombic faces while it is tt times the x component of area of (110) trapezoidal faces and their contributions to a r are therefore given by

try(021) + oy(02i) = [oz(021) - o,(02i)]/y (24)

and

ay(110) + a,(l I0) = tt [o~(110) -- a~(l i-0)]. (25)

3.2 ccp structure

The atomic polyhedron ofccp arrangement is shown in figure 1 (b). It is a dodecahedron consisting of three pairs of parallelogram faces, denoted by (200), (110) and Cli0), parallel to the z axis and three pairs of rhombic faces, denoted by (111), (1 I1) and (021) which intersect the z axis at -t- (3c 2 + 4a2)/6c where 2a and 3c are the lattice constants along the x and z directions. The former has one side of length 2a(1 + 3t2) 1/2/3t with

Electron-ion interactions m close packed metals

763

t = c/a,

common with the latter. All faces, except a pair of (021) rhombic faces contribute to try, a pair of (200) parallelogram faces do not contribute to try while only the rhombic faces contribute to try. However, the symmetry associated with the polyhedron reduces the evaluation of the integral (3) to that of contributions to a~ and a, from parallelogram and rhombic faces, respectively. The contribution to tr~ from the (200) parallelogram faces which are perpendicular to the x axis, could be written as i- 1 - . 3- o' a exp

(iqry)

^exp

(iqzz) dSx

⁽²⁶⁾

whereas that from the (110) parallelogram faces could be expressed using the co- ordinate axes transformation (5) in the form

l exp

{i(q~ +_ t,q,)X } 1 "/2

trx( l 10)

L

^l _ l - a / 2 I

a / 2 ~ exp{i(qp/tl-qx)Y}exp(iqzZ)dSx"

- , / 2 - ~ (27)

The evaluation of these integrals yields

a~(200) = 8 sin

(q~a)

sin [(qy +

q,/s)a']sin (q,d)

⁽²⁸⁾ q, (q, + q,/s)

and

8 sin [(qx +

t~q,)/a/2]

^{sin [(tt}

q~ - q, + q,/?)a'/2]

^sin

(q,d)

a~( l 1 O) = q, (ttq~ - qy +

q=/7) (29)

The corresponding contribution from (1 i-0) parallelogram faces could be written as follows by replacing

qy, q,

by - q y , - q , in (29):

ax(l ]0) = 8 sin [(q~ -

tlqy)a/2]sin [(tlq~ + q, - qz/y)a'/2]

sin

(q,d)

(30) q , ( t i qx + qy - q J ~ )

In addition, since the co-ordinate axes transformation (I0) transforms the ( l l l ) rhombic faces into square faces and orients them perpendicular to the Z axis, their contribution to tr, could be written in the form

a~(l~l) f e x p { i ( t q z + q ' - q ' / t ' ) Z } ~_

| a[2

f ~/2

~a/2 exp

{ i(qx + t l qy) X

}exp {i(~[q,/t

- q~ + q / t l )~! } dS z.

a12 3 - a / 2

(31)

The evaluation of this integral, by making use o f the co-ordinate axes transformation (12) which rotates ~f and r/axes through 45 ° about the Z axis, yields

o,(1il) = 16q

sin[(tq,+q~-q/t~)a/2]sin[(q~/y-t~q~-q,)a'/2]sin[(qd),+2qy)a'/2]

(32)

(q,/~ - t, q,~ - q,,) (q,/~, +

~,)

The corresponding contribution from ( i i l ) rhombic faces could be written, by substituting - q x for qx in (32), as

a , ( | i l ) =

16ttsin[(tq~-qx-q~/tt)a/2]sin[(qJ~+ttq~-qy)a'/2]sin[(qz/~+2q')a'/2]

(33) (q,/r ^{+ t~q,, - #,)} (q,/~ ^{+ ~ , )}

764 V Ramamurthy

On the contrary, the co-ordinate axes transformation (17) rotates the z and y axes through an angle 0 = t a n - ~ (ID) about the x axis and orients the (021) rhombic faces perpendicular to the Z axis. Hence their contribution to az is given by

a~(O21)=[exp{i('q~+q')Z}~j,.f:.f~iexp(iq,x)

exp

{ i( q, - q,/r)tl } dS z.

(34)

By means o f the co-ordinate axes transformation (19) which transforms these rhombic faces into square faces and rotates the x and ~/axes through 45 ° abbut the Z axis, the integral is evaluated to obtain

16tt sin [(~q~ + q,)a'] sin [ ( q , / y - qy + t l q x ) a ' / 2 ] sin [ (qz/~ - q , - t l q x ) a ' / 2 ]

oz(021) = [ (q,/? _ q,)2 _ (ttq~)2] (35)

Similar expressions for the contributions to or, from (I 1 I) rhombic faces and to oy from all faces except the (200) parallelogram faces are obtained by taking into consideration the differences in the corresponding components of area of these faces.

For example, z component is t times and s times the x and y components of area of (I I I) rhombic faces, respectively and hence their contributions to the latter are given by

o~(lIl) + a ~ 0 i l ) = [ a , ( l i l ) - a , ( | I 1 ) ] / t (36) and

tr,(1 i l ) + a , ( i D) = - [a,(lI1) +

o,(Iil)]/s.

(37) On the other hand, the z component of area o f (021) rhombic faces is y times its y component while the y component o f area o f (110) parallelogram faces is t~ times its x component and hence their contributions to try are given by

a,(021) -- a,(O21)D (38)

and

#,(110) + o,(1T0)

=

t,[oA11o)-

o.(I-I-0)]. (39) 3.3

Expressions for

S(q)

It is possible to write down the interference factor for each of these structures in two different forms by substituting the expressions for a~, a~ and o, in 2(a) and (b). Since the expressions so obtained for S2(q) and S3(q) consist o f scalar terms, they could be reduced to convenient forms. For instance, the expression for S2(q) in the case of hcp structure is reduced to (A1) when the terms with common denominators are coUected together while that for Sa(q) is reduced to (A2) when the products of the trigonometric functions are transformed into their sums. On the other hand, the expression for S~(q) in the case o f ccp structure goes over to (A3) when the sums of the trigonometric functions are transformed into their products whereas the reverse transformation reduces that for Sa(q) to (A4). Hence these expressions which are collected together in appendix A are just two of the several alternative (but equivalent) ways of writing clown the interference factor. However, the contributions from the rhombic faces which bisect the basis vectors in both structures and those from the trapezoidal or parallelogram faces of the atomic polyhedron are not shown here separately as the former could easily

Electron-ion interactions in close packed metals 765 be distinguished from the latter in all expressions for S(q). Besides, these expressions for ccp structure with t = 1.633 are reduced to the corresponding expressions for fcc structure (viz A 1 and A2 in I) by a co-ordinate axes transformation which orients the z, x and y axes of the former along the [ 111], [ i10] and [i i2] directions of the latter. On the contrary, there is no way of transforming the former, at any value of t, to those of hcp structure which are invariably complex, mainly because of the differences in their atomic arrangements.

It could, however, be shown that the apparently different expressions for S2(q, t) and Sa(q, t) reduce to the same expressions along the principal symmetry directions of the crystal, provided their singularity is overcome by means of L'Hospital's rule. These expressions for the interference factor along [~00], [0~0] and [00~] directions denoted respectively by (B1), (B2) and (B3) in the case ofhcp structure and by (B4), (BS) and (B6) in the case ofccp structure, where ( is the appropriate reduced wavevector, are included in appendix B. It may be observed that the corresponding expressions for the two structures are identical except for the imaginary part of S((, t) in (B2) and in the limit of t ---. 1.633, they go over respectively, to those of the fcc structure along [~(0], [~2~] and [(~(] directions. In addition, each of these expressions tends to unity (while the imaginary part in B2 tends to zero) in the limit of ~ --* 0 and becomes zero when corresponds to a reciprocal lattice vector, g # 0. They have been plotted as a function of

in the case of hcp and cop structures with t = 1.633 in figures 2(a) and (b) respectively along [~00], [00(] and [0~0] directions as well as along a non-symmetry direction, [((~]. Numerical values of the interference factor, G(qr,) which approximates the atomic polyhedron by an ellipsoid of equivalent volume, calculated from the expression (CI) in appendix C have also been plotted in these figures to facilitate their comparison with the corresponding values of S((, t).

4. Discussion

It is obvious from figures 2(a) and (b) that the interference factor, S(¢, t) varies significantly with the direction at any value oft and its real as well as imaginary parts go through zero at values of¢ corresponding to g # 0 whereas G(qr,) does not. Further, the numerical values of G(qr,) for hcp and ccp structures which are insensitive to the shapes of their atomic polyhedra, are identical at all wavevectors. On the other hand, the basis atoms ofhcp structure destroy the centre of symmetry of its polyhedron and hence the expressions for S(~, t) become complex while those of ccp structure, referred to orthorhombic co-ordinate axes, do not. Nevertheless, the imaginary parts of the |ormer are reduced to zero and the corresponding expressions for S((, t) become equal along [~00], [00(] and [((~] directions mainly because the atomic arrangements of both structures are identical along these directions, whereas the differences along other directions manifest themselves as imaginary parts of S(~, t). Since the surface of the ellipsoid of equivalent volume does not match with that of the atomic polyhedron in any direction and the former lies outside the latter along [~00], [0~0], [00~] and [ ~ ] directions, the values of G(qr,) are higher than those of S((, t), but the proximity of these surfaces at t = 1.633 gives rise to least differences between the two along [(00] and [ ( ~ ] directions. However, the numerical values of t associated with the hcp arrangement invariably deviate from its unique value, 1-633 associated with the ccp arrangement. As a consequence, interference factors of the former differ from those of

l ' w 9

766 V Ramamurthy

1.01

0.~

0.6

0.4

ffl

0.~

[~001 [oo~J

_(-)

0.0

-0.2 L

0.0 1.0 2.0 3.0 4.0

1.0

0.8

0.6

I

^0.4

0.2

0.0

-0,2 r

0.0

_/[0 ~

o]

xXNN I

(b)

I I I I

0.5 I. 0 1.5 2.0

Figure 2. C-dependence of interference factors in the case of hcp and cep structures along (u) [C00], [00~] directions and (b) [0~0], [CC~] directions (see appendix for appropriate expressions): ( ), S(O; ( - - - ) G(q, rs); (" " "), Imaginary part of S(O for hcp structure. Other values are identical for both structures. The arrow in (a) indicates that S(C) passes through zero whereas G(qr~) does not, at a nonzero reciprocal lattice vector.

Electron-ion interactions in close packed metals 767 the latter at all wavevectors and the differences between S(~, t) and G(qrs) of hcp structure manifest themselves even at small values of ~ in all directions.

It was shown in I that the Wigner-Seitz approximation leads to an erroneous evaluation of the contributions from umklapp processes to the thermal and electrical properties of metals and is not consistent with the symmetry requirements of the lattice.

Besides, the acoustical and optical modes of vibration of a diatomic lattice are associated, respectively with the 'in-phase' and 'out-of-phase' motion of the basis atoms. Different amplitudes of thermal motion of lattice and basis atoms make it all the more important to separate the latter contributions to the interference factor from the former contributions. Since there was no means of distinguishing these different contributions to G(qr~), several authors (Gupta and Dayal 1965; Sharan and Bajpai 1970; Bose et a11973, Upadhyaya and Verma 1973; Saxena and Rathore 1984) assumed wrongly that the contributions from electron-ion interactions to the optical modes are the same as those to the acoustical modes and neither of these branches conforms to the translational symmetry of the lattice. It should therefore be apparent from this discussion that the correct evaluation of the contributions from normal and umklapp processes to different modes of vibration in a hexagonal lattice requires the exact evaluation of the interference factor over the actual shape of its atomic polyhedron so that it is expressed as a sum of the contributions from rhombic planes (bisecting the basis vectors) and trapezoidal planes (bisecting the lattice vectors). Further, the co-ordinate axes transformation which transforms the triatomic ~'p structure, referred to the orthorhombic co-ordinate axes, to the monoatomic cop structure referred to cubic co-ordinate axes (see §3.3) has been exploited to identify the correct set of reciprocal lattice vectors as well as to ensure that the method of evaluation of the electron-ion interactions in a polyatomic structure, described elsewhere (Ramamurthy 1985) is free from any mathematical or numerical errors.

5. Conclusions

The symmetry of the atomic polyhedra of hcp and ccp structures simplifies considerably the exact evaluation of the electron-ion matrix elements over their actual shape and facilitates the separation of the lattice atom contributions from the basis atom contributions to the interference factors, S(q, t). The imaginary part of S(q, t) for hcp structure reduces to zero and hence the two close packed structures have the same expressions for S(q, t) in certain directions, but they differ from each other in other directions as well as at different values of t. Since the lattice and basis atom contributions are inseparable in G(qrs), Wigner-Seitz approximation is not valid for a diatomic lattice and exact expressions for S(q, t) are required for a proper evaluation of thermal and electrical properties of hcp metals and to satisfy the symmetry requirements of the lattice.

Acknowledgements

The author is grateful to Dr S B Rajendraprasad and Mr A Thayumanavan for their assistance in computation and to Dr D Ranganathan for valuable discussions.

768

V Ramamurthy

Appendix A. Expressions for S(q,/): general direction (a)

hcp structure

S d q , t) = 2<i

(I-exp (ill09) fsin u sin ~ + a09)

(u + v + 09) L i09 [. (v' + otto) (1

+ q ) s i n ( ~ - ) s i n ( u-v'+2~09)2

(u - v' + 2a09)

(1- t i ) s i n l t ~ ) ~

}

(u + v' - 2a09)

+ a term with 0 9 a n d - 0 9 i n t e r c h a n g e d / + / 2 ( 2 + s ) J [

t09)12]sin( u+d-2~09. )-2. ^sin

( u - v' + 2 a 0 9 ) 2 }

{

exp [i(2v' +

i[u 2 - (v' -

20t09) 2 ]

+ 2(2 - s) {a term with to and - 09 interchanged}

, - [u+v'-2¢t09xl

~exp[i(u-v

+ t09)/2J s i n t - - ~ _ ~ - - - ) s i n ( v ' +09)

t

+ (s - t I - 1) {a term with u and - u interchanged}

+ (t i - 1 - s ) {a term with 09 and -09 interchanged}

" 1 / \

- ( t t + 1 + s ) { a term with u, 0 9 a n d - u , - 09 interchanged} / / s ),

_111

^(AI)

S3 (q, t) = (u s + v~ + 09~) \ L i09 u (v' + ~09)

+ (u + tiv)(c°s(2v'-°t09)-c°s(u + v' + ct09) v' +

2ct09)

,{cos(2v'-ot09)-cos(u-v'-ct09))}

+(u-t:'t,

+ a term with co and - to interchanged]

[{2(2v'+t09)exp[i(2v'+t09)/2]

+ i[u 2 - (v' -

2~t09) 2] (cos [v' - 2at09] - cos u)

+ a term with 09 and -09 interchanged}

Electron-ion interactions in close packed metals

769

~._ ~ ~ ? ~ ~ ~-,-,_~/.,:~ (~os i-,,- v, = ~ l r . + , , ~

+ a term with u and - u interchanged + a term with to and - to interchanged

+ a term with u, to a n d - u , - t o i n t e r c h a n g e d ~ / / t ~ (A2) ) J / /

(b)

ccp structure

S , ( q , t) =

(u + v + to) (v' + ~to)

u+ttv

^{- v ' +}

(1 + t l ) s i n ( ~ ) s i n ( u 2 2~,to) (u -- v' + 2otto)

( 1 - t,) sin ( ~ ) sin ( u v ' -

+

(u + v' - 2~tto)

( t , _ l + s ) s i n ( U - V ' + t t o ) s i n ( u + v ' - 2 ° t t o )

_{2 ~} sin (v' + ~tto) + (u + v' - 2~tto) (v' + otto)

+ ( s - t l - 1) (a term with u and - u interchanged)

+ ( 2 + s ) (a term with

(u)- (v') and

2v' i n t e r c h a n g e d ) / / s ) (A3)

I l l

0t

S3(q, t) = (u 2 + v2 + to,)

({sin(u+v'-~oj)-sin(u+v'+tto/2)+sin(u-v'+&o)-sin(u-v'-tco/2)}u ' v ( a c + otto)

+ a term with u and - u interchanged]

+[(u_v,+to~){ sin(u-v'+&°)+sin[(t-tS)to]-sin(u+v'+tt~/2)+sin{2v'-tt"/2) }

(u + v' - 2otto) (v' + otto) + a term with u and - u interchanged

+ a term with ( u ) - (v') and

2v'

interchanged//t ) _ l / /

(A4) where

U=qxa, v=qya, to=q~a, v'=v/tl, ~t=l/3t,

[3 = (t/2 - ct)

and t$ = (fl - at).

770 V Ramamurthy

Appendix B. Expressions for $(q, t): symmetry directions

(a) hcp structure

[~00] direction

[ 1 4 '~ sin (~/2) [OCO] direction

s i n ( x ~ / 2 ) F ( - 3 - ~ ) - - ( 3 n ~ / 2 ) + 3 t 2 (,~)(,~/2) S(~, t) = ~ L 1 2 sin (3n~/2) 2 sin (x~) sin (~/2)

2 sin~ (~/2) ] (B2)

+i9t ~ (,r~/2)2 , [00~] direction

S (~, t) -- 18t 4 sin (~/2) sin2 (~/3t,)/(~03 (B3)

(b) ccp structure

[~00] direction

S ( ~ , t ) = s i n ( ~ / 2 ) [ ( ~ - 9 ~ ) c o s ( ~ / 2 ) + ( ~ + 9 ~ ) s i n ( ~ / 2 ) l _{(~/2) ~ j, (84)}

[0~0] direction

S ( ~ , t ) f s i n ( ~ / 2 ) [ ( 2 ~sin (37r~/2) 2 sin (~r~) sin (~r~/2) ] (BS) 0t~/2) 1 - 3-~] (-3~-~-~ ~ 3t ~ (~tO (~t~/2) '

[00~] direction

S(~, t) ffi 18t ( sin (x~/2) sin z (z~/3t2)/OrO 3,

where the directions,

(96) reduced wavevector, ~ = q~a/~t, qya/Tt v/3 and q,c/~t along x, y and z respectively.

Appendix C. Expression for

G(qr.)

G (qrs) ffi 3 [sin (qrs) - (qr,) cos (qr,) ]/ (qr,) 3,

where

( t ~ '`3

qr.

~ \ ~ / [S(u'+v2+~,2)] 1/2.

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E l e c t r o n - i o n interactions in close p a c k e d metals 771

Bose G, Gupta H C and Tripathi B B 1973 Phys. Lett. A43 365 Bross H and Bohn B 1967 Phys. Status Solidi 20 277

Gupta R P and Dayal B 1965 Phys. Status Solidi 8 115 Ramamurthy V 1978 Pramana 11 233

Ramamurthy V 1979 Pramana 13 373

ltarrmmurthy V and Singh K K 1978 Phys. Status Solidi B85 761 Shatan B and Bajpai R P 1970 J. Phys. Soc. Jpn. 29 46

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Saxena H C and Rathore R P S 1984 Phys. Status Solidi B122 K 119 Upadhyaya J C and Verma M P 1973 Phys. Rev. Bg 593