Analysis of crossover of dispersion curves of alkali metals in the [ζ00] direction

Physica 125B (1984) 63-74 North-Holland, Amsterdam

ANALYSIS OF CROSSOVER OF DISPERSION CURVES OF ALKALI METALS IN THE [~00] D I R E C T I O N V. R A M A M U R T H Y a n d M. S A T I S H K U M A R * Department of Physics, Indian Institute of Technology, New Delhi 110016, India Received 11 August 1983 Revised in final form 16 December 1983 The dispersion curves of alkali metals, deduced on the basis of DAF, CGW, AS and GTF models, are analysed to determine the circumstances under which they crossover along the [st00] direction. All these models incorporate the same electron-ion interactions, satisfy the symmetry requirements of the lattice and reproduce the observed crossover in lithium with a positive value for the electron gas constant, C'. However, none of them produces a crossover in any other alkali metal. It is shown that these dispersion curves are hardly influenced by the nature or range of the ion-ion interactions in a model, but the stronger electron-ion interactions in lithium produces the crossover while the weaker electron-ion interactions in other alkali metals do not. The role played by the umklapp contributions in producing a crossover, apparent differences in the values of C' from one model to another and the consequent variation in the wave vector at which the crossover occurs in lithium are discussed.

I. Introduction E v e r since t h e e x p e r i m e n t s using t h e inelastic s c a t t e r i n g of n e u t r o n s [1--4] r e v e a l e d t h e a n o m a l o u s c r o s s o v e r of d i s p e r s i o n curves a l o n g t h e [~'00] d i r e c t i o n in l i t h i u m [1], a t t e m p t s h a v e b e e n m a d e to r e p r o d u c e this c r o s s o v e r on t h e basis of t h e p h e n o m e n o l o g i c a l m o d e l s [5-21] as well as t h e p s e u d o p o t e n t i a l m o d e l s [22-40]. O n t h e c o n t r a r y , t h e d i s p e r s i o n curves of s o d i u m [2], p o t a s s i u m [3], r u b i d i u m [4] a n d c e s i u m [41] d o n o t e x h i b i t any c r o s s o v e r a l o n g t h e [~'00] d i r e c tion. N e v e r t h e l e s s , n e i t h e r the c i r c u m s t a n c e s u n d e r which t h e d i s p e r s i o n curves c r o s s o v e r in alkali m e t a l s h a v e b e e n i n v e s t i g a t e d n o r t h e app a r e n t c o n t r a d i c t i o n b e t w e e n this a n o m a l y a n d t h e e x i s t e n c e of h o m o l o g y a m o n g t h e p h o n o n f r e q u e n c i e s of alkali m e t a l s [4,5] has b e e n r e s o l v e d so far. In spite of all t h e e x p e r i m e n t a l a n d t h e o r e t i c a l a t t e n t i o n r e c e i v e d b y t h e i r lattice d y n a m i c s , we d o n o t y e t k n o w w h e t h e r t h e n a t u r e a n d r a n g e of a t o m i c i n t e r a c t i o n s v a r y f r o m o n e alkali m e t a l to a n o t h e r . In this c o n t e x t , it is of u t m o s t i m p o r t a n c e to u n d e r s t a n d t h e * Now at the Department of Applied Physics, Regional Engineering College, Kurukshetra 132 119, India.

factors r e s p o n s i b l e for t h e o b s e r v e d c r o s s o v e r in l i t h i u m which has i n v a r i a b l y o b s t r u c t e d t h e d e v e l o p m e n t of a unified t h e o r y for the lattice d y n a m i c s of alkali m e t a l s , b y a n a l y s i n g t h e i r d i s p e r s i o n curves a l o n g t h e [~'00] d i r e c t i o n . H o w e v e r , m o s t of t h e lattice d y n a m i c a l m o d e l s [5-15], which are b a s e d on a v a r i e t y of a s s u m p tions r e g a r d i n g t h e n a t u r e a n d r a n g e of a t o m i c i n t e r a c t i o n s , h a v e failed to r e p r o d u c e this crosso v e r in lithium. O n t h e o t h e r h a n d , the few m o d e l s [16--18] which h a v e s u c c e e d e d in p r o d u c ing a c r o s s o v e r in l i t h i u m with a n e g a t i v e v a l u e for the e l e c t r o n gas c o n s t a n t , C', w o u l d p r o d u c e s i m i l a r c r o s s o v e r s in o t h e r alkali m e t a l s as well! B e s i d e s R a m a m u r t h y a n d N e e l a k a n d a n [19] h a v e e s t a b l i s h e d that t h e n e g a t i v e v a l u e s of C ' a r e a c o n s e q u e n c e of artificially m a t c h i n g t h e results of a lattice d y n a m i c a l m o d e l h a v i n g no t r a n s l a t i o n a l s y m m e t r y with t h o s e of e x p e r i m e n t s which c o n f o r m with t h e s y m m e t r y of t h e lattice. H e n c e t h e c r o s s o v e r p r o d u c e d b y t h e s e ina d e q u a t e m o d e l s is a s p u r i o u s result a n d t h e analysis of t h e s e m o d e l s d o e s n o t s e r v e any p u r p o s e as far as i s o l a t i n g t h e b a s i c f a c t o r s res p o n s i b l e for this a n o m a l y in l i t h i u m is c o n c e r n e d . O n t h e c o n t r a r y , a v a r i e t y of p s e u d o p o t e n t i a l m o d e l s [22-32] h a v e s u c c e e d e d in p r o d u c i n g a

0378-4363/84/$03.00 O E l s e v i e r Science P u b l i s h e r s B.V. ( N o r t h - H o l l a n d Physics P u b l i s h i n g D i v i s i o n )

64

v. Ramamurthy and M. Satishkumar / Crossover of dispersion curves of alkali metals

crossover of dispersion curves in lithium at different wave vectors while a few of them [3440] did not. However, it is not at all clear from these investigations whether this crossover is an anomalous property or is linked with a characteristic feature of a pseudopotential. For instance, Ono [28] has attributed this crossover to the exchange interaction of conduction electrons with the ion core whereas Liu and Moore [33] have associated it with the asymmetry of the electron-ion matrix elements. In any case, detailed analysis of the dependence of the phonon dispersion relations of alkali metals on these pseudopotential models is precluded by their mathematical complexities. Besides, more or less the same short range ion-ion interactions are incorporated in all these models. Consequently, it is rather difficult to ascertain any of the basic factors responsible for the observed crossover in lithium from these pseudopotential calculations. On the other hand, Ramamurthy and Singh [20] and Satishkumar [21] have also reproduced the crossover of dispersion curves in lithium with a large positive value of C' by making use of five-constant and six-constant axially symmetric force models, respectively. These models which satisfy the symmetry requirements of the lattice did not produce a crossover of dispersion curves in any other alkali metal. The success of these models seems to suggest that the analysis of phonon dispersion relations of alkali metals deduced on the basis of a variety of force constant models with different assumptions regarding the nature and range of atomic interactions is likely to yield the relevant information, provided these models are free from any basic deficiencies. The present paper describes such an analysis which reveals the dependence of dispersion curves of alkali metals on various aspects of atomic interactions and the factors responsible for their crossover in lithium along the [,~00] direction.

form I O ( q ) - o~211= 0,

(1)

where 0% is the phonon frequency with wave vector q and I is a 3 × 3 unit matrix. To facilitate the analysis of dispersion relations along the [(00] direction, the dynamical matrix, D ( q ) is split into an ionic part, D ~, representing the contributions from the short-range ion-ion interactions and an electronic part, D e, representing the contributions from the long range electron-ion interactions. Further, the ion-ion interactions are expressed in terms of central and angular forces, axially symmetric forces or general tensor forces while their range is varied from one lattice dynamical model to another. The diagonal and off-diagonal elements of the matrix D i of the corresponding models: the De Launay angular force (DAF) model, the Clark, Gazis and Wallis angular force (CGW) model, the axially symmetric force (AS) model as well as the general tensor force (GTF) model, are given, respectively, in [14], [42], [21], [9], and these expressions are not reproduced here. As a consequence of their assumptions regarding the nature and range of these interactions, the respective models have five, six, six and seven force constants. On the contrary, the electron-ion interactions are expressed in terms of the same electron gas model [43] which is based on the deformation potential approximation. The translational symmetry is restored to all these models by incorporating the umklapp contributions summed over several reciprocal lattice vectors. The elements of the matrix D e so obtained contribute to the longitudinal as well as transverse modes of vibration and are given by [20] C' [ (g - q)x(g - q)ySOg - q ) F ( g - q)] MDexy = ~ ~ -~---q)2~ q)/a2 ,

(2) 2. Theory The secular determinant for the phonon frequencies of alkali metals may be written in the

where C' is the electron gas constant, g is a reciprocal lattice vector, a is the lattice parameter, S(r/) is the interference factor which has been

11.65 38.18 64.91 141.95 220.71

Metal

Li Na K Rb Cs

3.484 4.240 5.226 5.628 6.050

a (10 -t° m)

*Homologous phonon frequencies.

M (10-27 kg) 1.435 0.816 0.415 0.312 0.247

1.208 0.679 0.340 0.262 0.206

1.075 0.570 0.284 0.186 0.148

[44] [45] [46] [47] [48]

Cll C12 C,u (10 l° N m -2) (10 l° N m -E) (10 -1° N m -2) Ref.

Table I Relevant experimental data for the alkali metals

8.80 3.58 2.21 1.32 0.94

rE'r(100) (THz) 9.15 3.89 2.38 1.45 1.05

zllox VLt~] ) (THz)

1.90 0.93 0.53 0.34 0.26

11 V'rl(i~0) (THz)

7.00 2.88 1.78 1.11 0.79

111 VL,X(i~) (THz)

98 90 9 120 100

T (K)

[1] [2] [3] [4] [41]*

Ref.

Lit

g~

g~

g,

t~

g~

66

v. Ramamurthy and M. Satishkumar / Crossoverof dispersion curves of alkali metals

required number of zone boundary frequencies as well as the relevant experimental data for alkali metals from table I. The numerical values of C' which differ from model to model are listed in table II while those of other force constants are given elsewhere [42].

evaluated exactly over the atomic polyhedron for the b.c.c, structure and F(r/) is the realistic screening function which takes into account the exchange and correlation effects. The force constants are evaluated in all cases by making use of their relations with the three elastic constants and the

3. Analysis of dispersion relations In order to determine the factors which are responsible for the crossover of longitudinal and transverse branches of alkali metals along the [st00] direction, it is necessary to investigate the variation of different contributions to the respective phonon frequencies with the reduced wave vector, ~', as well as with the nature and range of atomic interactions. For this purpose the dispersion relations for the DAF, CGW, AS and GTF models are deduced by solving the corresponding secular determinants (1). These dispersion relations along the [~r00] direction could be written in a general form: S

C 47r2Mv2(sr00) = r l sin 2 (rrsr/2) +/'2 sin 2 (zr() + -~ SL(,~O0)

(3)

Ct 4rr2MVT2(~'00) = F, sin 2 (~-~'/2)+ F3 sin: (~r~')+ -~- ST(~'00),

(4)

and

where F~,/'2 and F3 are combinations of the force constants which depend on the nature and range of the ion-ion interactions in a lattice dynamical model, SL(~O0) and ST(~O0) correspond to umklapp contributions at a wave vector ~r represented by [ ] in eq. (2) for the longitudinal and transverse modes of vibration, respectively. It is obvious from these expressions that /'1 contributes equally and umklapp processes with C' contribute unequally to these modes at any wave vector. On the contrary,/'2 and the normal processes included in SL(~O0) contribute to the former while F3 contributes to the latter exclusively. Nevertheless, the crossover of these branches of an alkali metal occurs at ~'c only if its transverse phonon frequencies become equal to and greater than the corresponding longitudinal phonon frequencies at and above ~'¢, respectively. Consequently, ~r should satisfy the condition, obtained by equating (3) and (4), viz. Z -= F~CF3 = Tu - Lu -= Z¢,

(5)

where Tu = ST(~eO0)/4 sin 2 (Trsr~)

(6)

Lu = SL(~cO0)/4 sin 2 (~-~'c).

(7)

and

The left-hand side of eq. (5), denoted by a characteristic constant, Z, is independent of sr whereas the right-hand side of eq. (5) corresponding to its critical value, Z~, is independent of a lattice dynamical

V. Ramamurthyand M. Satishkumar/ Crossoverof dispersioncurvesof alkali metals

67

model as long as the electron-ion interactions are incorporated in the same manner. It should therefore be obvious that the dispersion curves of a model crossover only when its Z becomes equal to Zc at some value of ~'¢ in the range 0.0 to 1.0. Hence the dependence of Z on the nature and range of ion-ion interactions as well as that of Zc on the wave vector are analysed in sections 3.1 to 3.5 to ascertain the circumstances under which eq. (5) is satisfied in the case of an alkali metal.

3.1. The De Launay angular force model [14] The dispersion relations along the [sr00] direction are given by eqs. (3) and (4) with F1 = ~-~ (/31 + 2($1),

F2 = 4fl2

and

F3 = 432,

(8)

where/3, and din are the nth neighbour central and angular force constants, respectively. Besides, Z for this model could be expressed, by eliminating these force constants, as

ZOAF = a(CuS C44)_ 1,

(9)

where C' = [8rr2M{v~(100) + 4~'L(~.~.)} 2111 -- 16a(Cu + 2C44)]

(10)

[SL(100) + 4SL(~½½)-16]

On the other hand, a crossover of the dispersion curves of the D A F model is produced at ~¢ by a critical value of C', which is given by

C'=

a ( C n - C44)

(11)

(1 + Z¢)

3.2. Clark, Gazis and Wallis angular force model [42] The dispersion relations along the [~'00] direction for this model are given by eqs. (3) and (4) with

F1 = - ~ (J~l -~- 4 y l + 6y2),

8 F2 = 4(fl2+ ~ yl + 3y3)

and

(2

)

F3 = 4 - 3 Yl -I- Y2 -I- 3y3 ,

(12)

where Yn are the angular force constants associated with the angle On. Nevertheless, the expression for Z reduces to the form Zcow = a(C' b, C44)_ 1,

(13)

where 2 111 2 11 8"n"2M{~'L2 100 + 4~'L(~.~.)-8b'L(~0)}-b 16a(Cll + 2C12+ 2C44)]

C'= [

( [)L(lOO)+ 4SL(~½)- 8SL(~O)+ 48]

(14)

V. Ramamurthy and M. Satishkuraar/ Crossoverof dispersion curves of alkali metals

68

Besides, C' which produces a crossover of dispersion curves of the CGW model at ~'c is given by eq. (11).

3.3. The axially symmetric force model [21] The dispersion relations of the AS model along the [,;'00] direction are expressed by eqs. (3) and (4) with F1 = 1 6 ( ¢ + B1),

F2=4(AE+BE+2A3)

and

F3=4(BE+A3),

(15)

where A, a n d / 3 , are the nth neighbour radial and tangential force constants, repectively. When these force constants are eliminated, Z could be written in the form ZAS = a(C'~C,

C4,)

- 1,

(16)

where C' = [8rrEM{vZL(100) + 4~'L(~) 2111 -- 4~'L(~O)} 211 + 16aC12] [SL(100) + 4 S L ( ~ ) - 4SL(~0) + 16]

(17)

Nevertheless, C'c given by eq. (11) produces a crossover of the dispersion curves of the AS model at sr¢.

3.4. General tensor force model [9] When the electron-ion interactions are incorporated into this model, the dispersion relations along the [~'00] direction are given by eqs. (3) and (4) with F1 = 16trb

/'2 = 4(0"2+ 40"3)

and

/'3 = 4(A2 + 20.3 + 2A3),

(18)

where 0., and A. are the nth neighbour tensor force constants. However, Z for this model could be expressed in terms of elastic constants as Zcav =

a ( C n - C44) C' - 1,

(19)

where 2

2 11

2

11

C ' = [8~ M{VL(~.~.0)- VT,~.~0)}- 8 a { C n + C44}] [SL(~O) - STI(½~O) - 8]

(20)

Besides, the corresponding expression for C~ at ~rc is given by eq. (11).

3.5. Contributions from umklapp processes [20] In spite of the differences in the numerical values for C' of an alkali metal, its critical value does not vary with the lattice dynamical model. On the contrary it follows from eq. (11) that C~ varies with the

V. Ramamurthy and M. Satishkumar / Crossover of dispersion curves of alkali metals

69

wave vector for any alkali metal as a consequence of its dependence on Zc, while at any wave vector, it differs from one alkali metal to another. In order to understand the former, the variation of Zc is shown in fig. 1 by plotting the numerical values of Tu and Lu obtained from eq. (2) as a function of sr~ for various alkali metals. It is obvious from this figure that the umklapp contributions to the tranverse modes increase faster than the normal and the u m k l a p p contributions to the longitudinal modes of vibration, with ~'c. H e n c e the difference, Z~, increases from a negative value at ~'~= 0.3 to a positive value at sr~>/0.8, passing through zero at the point of intersection of these curves. Besides, Zc increases from lithium to cesium at any wave vector and as a consequence the point of intersection shifts towards smaller wave vectors. Nevertheless, eq. (5) is satisfied and the longitudinal and transverse branches crossover only if Z for an alkali metal becomes equal to its Z~ or the corresponding C' becomes equal to its critical value at some wave vector ~'~. Numerical values of C' obtained from eqs. (10), (14), (17) and (20) as well as those of C'~ at sr¢ = 0.5 obtained from eq. (11) in the case of different alkali metals are listed in table II. Comparison of the f o r m e r with the latter, facilitates the speculation regarding the wave vectors at which the dispersion curves of different alkali metals crossover. Since the differences in the numerical values of C' invariably lead to corresponding differences in Z, the wave vector, src, at which the dispersion curves crossover varies with the lattice dynamical model. To demonstrate this effect which is of some consequence in the case of lithium, numerical values of Z for each alkali metal calculated using different values of C' as well as those of Zc at src = 0.5, taken from fig. 1, are included in table II. In addition, phonon dispersion curves along the [~r00] direction deduced on the basis of D A F ,

tsoo]

2.5

t~

I

--~O- . . . . .

-O-

- - O "' . . . .

-O- ....

~__

---0

d3

. . . .

-0--

_0~- ~

--

_-0--

0'.5

-

I

0.7

01.8

Fig. 1. Variation of Zc with src in the case of alkali metals. T . ~ ,

L.--O--.

V. Ramamurthy and M. Satishkumar / Oossover of dispersion curves of alkali metals

70

L~oo]

//

3.0

Na

// /! // // /

2.0 N "I1--

T

;//

/

.

K

///

N T

1.0

0

0.2

0.4

0.6

0.8

1.0

0

Fig. 2. Dispersion curves of lithium along the [~'00] direction. D A F model: L Iq, T I ; C G W model: L O. T Q; AS model: L O, T O; G T F model: L A, T &; Experimental: L . . . . . , T • The arrows indicate the wave vector g'c at which the crossover occurs in different models.

~

0.2

I

I

0.4

0.6

I

0.8

I

1,0

Fig. 3. Dispersion curves of sodium, potassium, rubidium and cesium along the [~'00] direction. D A F model: L D, T I ; C G W model: L O, T Q; AS model: L O T O; G T F model: L A, T &; Experimental: L . . . . , T - -

CGW, AS and G T F models in the case of lithium and other alkali metals are shown, respectively, in figs. 2 and 3. T o avoid overlapping and the consequent confusion, each pair of lines is referred to a different origin in fig. 2 while the number of points corresponding to each model is reduced to a minimum in fig. 3. Experimental phonon frequencies obtained using the inelastic scattering of neutrons [1--4] and the homology [41] are also plotted as a function of sr in these figures to facilitate their comparison.

Table II Numerical values of C', Z and their critical values at sr~= 0.5 for alkali metals C'(N m -1)

Z

c~ Metal

DAF

CGW

AS

GTF

(N m -~)

DAF

CGW

AS

GTF

Li Na K Rb Cs

2.947 0.293 0.339 0.511 0.260

3.485 0.143 0.294 0.430 0.347

2.437 0.444 0.379 0.579 0.193

5.035 0.269 0.571 0.711 0.401

4.007 2.842 1.538 1.484 1.160

-0.574 2.559 1.019 0.388 1.292

-0.640 6.295 1.329 0.649 0.718

-0.485 1.349 0.807 0.225 2.087

-0.751 2.877 0.199 0.003 0.486

Zc -0.687 -0.633 -0.555 -0.522 -0.486

V. Ramamurthy and M. Satishkumar / Crossover of dispersion curves of alkali metals

4. Discussion It should be obvious from the above analysis that the characteristic constants, Z, of the DAF, CGW, AS and G T F models denoted, respectively, by eqs. (9), (13), (16) and (19) are identical. In view of the fact that these expressions involve only the force constant, C', it is reasonable to conclude that Z is independent of the nature of the ion-ion interactions assumed or the number of force constants associated with a lattice dynamical model. However, the numerical values of Z which depend on the lattice parameter, the elastic constant difference, as well as on C' vary from one alkali metal to another. In spite of the fact that a ( C 1 1 - C 4 4 ) decreases rather slowly from lithium to cesium, the values of Z for lithium are negative whereas those for all other alkali metals are positive in the case of all models. On the contrary, Zc is negative for all alkali metals at ~'c = 0.5 and becomes positive at ~ c - 0.8. Nevertheless, it is obvious from table II that the actual values of Z for all alkali metals except lithium are very much larger than the corresponding values of Z~ at ~r¢= 0.5 and the former become equal to the latter only at wave vectors in the immediate vicinity of the zone boundary. As a consequence, the phonon dispersion curves of lithium along the [~r00] direction, (shown in fig. 2) crossover around ~r ~ 0.5 in the case of all models while those of other alkali metals (shown in fig. 3) do not crossover in any model. Besides establishing the invariance of Z with respect to the lattice dynamical models considered, the present analysis reveals that the electron-ion interactions, which were incorporated into these models in a manner described in section 2, are mainly responsible for the crossover of dispersion curves of lithium along the [~'00] direction. The numerical values of C' which represent the strength of these interactions, associated with lithium, are an order of magnitude larger than the corresponding values associated with other alkali metals. This implies that the binding between the electrons and the lattice is very weak in all alkali metals except lithium. Smaller values of C', in turn, give rise to

71

smaller excess umklapp contributions to the transverse modes of vibration which can never match the excess ion-ion contributions to the longitudinal modes of vibration at any wave vector in the range 0.0 to 1.0. As a consequence, the numerical values of Z are very much larger than the corresponding values of Zc in all models and the dispersion curves of these metals do not crossover at any wave vector along the [~r00] direction. Further, the weak electron-ion interactions have hardly any influence on the lattice dynamics of these alkali metals. It is precisely because of this reason that their phonon dispersion curves are insensitive to the way in which the volume forces were incorporated in a force constant model. On the contrary, the strong electron-ion interactions which manifest themselves with large values of C' or the corresponding negative values of Z in all models, produce the crossover of dispersion curves in lithium. It should therefore be clear that the neglect of umklapp contributions which invariably destroys the translational symmetry [5, 10, 11, 13, 16, 18], their incorrect inclusion [7, 9, 14, 17] or ignoring the volume forces altogether [6, 8, 9, 13] resulted in the failure of earlier models to reproduce the crossover in lithium without producing a crossover in any other alkali metal. In addition, lattice dynamical study of sodium and potassium cannot be used to judge the validity of a force constant model or to ascertain the nature and range of atomic interactions. It appears from eqs. (3) and (4) that any lattice dynamical model which takes into consideration ion-ion interactions extending up to next-nearest neighbours and beyond, should reproduce the observed crossover in lithium so long as it incorporates the electron-ion interactions correctly and satisfies the symmetry requirements of the lattice. Further at any wave vector along the [~00] direction, the ion-ion contributions to the longitudinal and transverse modes of vibration vary with the assumptions regarding the nature and range of these interactions. Nevertheless, it is obvious from this analysis that the difference between these contributions is independent of the model and consequently its characteristic constant, Z, is determined by the relative

72

V. Ramamurthy and M. Satishkumar / O'ossover of dispersion curves of alkali metals

strength of electron-ion interactions. In spite of using the same volume forces to incorporate these interactions, the numerical values of C', Z and the wave vector, ~'¢, at which the crossover occurs in lithium, differ considerably from one model to another. For instance, Z for the G T F and AS models are respectively lower and higher than Zc at ~'~= 0.5. This variation is mainly responsible for the crossover of dispersion curves, deduced on the basis of GTF, C G W , D A F and AS models, respectively, at st, = 0.46, 0.50 0.60 and 0.60 and these are indicated by arrows in fig. 2. It should be obvious from table II that the numerical value of Z in the case of other alkali metals vary with the lattice dynamical model. However, these values of Z which are larger than the corresponding values of Z¢ at all wave vectors along the [st00] direction, do not produce a crossover in any case. This is confirmed by the dispersion curves of sodium, potassium, rubidium and cesium shown in fig. 3. Nevertheless, the apparent variation in the numerical values of Z invariably contradicts the earlier assertion that the occurrence of crossover in the dispersion curves of lithium is independent of the nature and range of the ion-ion interactions assumed in a lattice dynamical model. In order to resolve this apparent contradiction regarding the variation of Z, it is necessary to recall that the numerical values of C' are determined by using eqs. (10), (14), (17) and (20) respectively in the D A F , C G W , AS and G T F models and the differences in these values of C' lead to corresponding differences in the values of Z for any alkali metal. Since the total atomic interactions are split into ion-ion and electronion interactions, the strength of the latter represented by C' in an alkali metal is highly sensitive to the differences in the assumptions regarding the nature and range of the former. The relations between the force constants and the elastic constants or zone boundary frequencies differ from one model to another. Besides the experimental data required to evaluate these force constants depends on their n u m b e r in any model. As a consequence, C' cannot be evaluated using the same set of elastic constants and zone boundary frequencies in all lattice dynamical models. In addition, there is no compatibility

between the elastic constants determined from the ultrasonic velocity m e a s u r e m e n t s and the phonon frequencies at the zones boundary points measured using the inelastic scattering of neutrons [49]. A b o v e all, the present models, which express the atomic interactions in terms of a variety of interatomic forces, suffer from a basic deficiency, viz. they do not satisfy the crystal equilibrium condition [50]. The cumulative effect of all these factors manifests itself in the form of different values for C', leading to corresponding variation in the numerical values of Z. Nevertheless, the dispersion curves of C G W and G T F models crossover at the correct wave vector while those of D A F and AS models crossover at a higher wave vector, (c = 0.6, in the case of lithium. However, this variation in src becomes insignificant especially when several pseudopotential models [34--40] have failed to reproduce this crossover and others [22-32] are incapable of reproducing it at the correct wave vector. Irrespective of the differences in the numerical values of C' and Z, it is obvious from figs. 2 and 3 that all these models reproduce the crossover of dispersion curves along the [~00] direction in lithium whereas none of them produces a crossover in any other alkali metal. Hence it is reasonable to conclude that the dispersion curves of alkali metals along the [~r00] direction are hardly affected by the variations in the ion-ion interactions of a lattice dynamical model. Besides, the present analysis clearly establishes that the nature and range of the ion-ion interactions do not vary from one alkali metal to another. On the contrary, it is the strong electron-ion interactions which are responsible for the observed crossover in lithium whereas the weaker electron-ion interactions forbid this crossover in other alkali metals. Since there are no p electrons in lithium, the 'size' of its ion core is reduced by spherical symmetry of the s electron wave functions. The consequent decrease in the 'effective' distance between the conduction electrons and the ion core invariably strengthens the binding between them. Besides the numerical values of C', the anisotropy of the Fermi surface and the values of effective mass associated with electrons in lithium seem to

V. Ramamurthy and M. Satishkumar / Crossover of dispersion curves of alkali metals

support this view. Nevertheless, the observed crossover in lithium is consistent with the existence of homology among the phonon frequencies of alkali metals [51] and therefore it is not an anomalous property. Unfortunately, even the present force constant models which are free from any basic deficiency, are incapable of analysing the electron-ion interactions beyond this point. In order to understand the basic features of these many body interactions which are responsible for the observed crossover in lithium, it is necessary to analyse a variety of pseudopotentials which have been used in the lattice dynamical studies of alkali metals. However, this is beyond the scope of the present investigations.

5. Conclusions It is obvious that all lattice dynamical models reproduce the crossover of dispersion curves in lithium along the [st00] direction, without producing a crossover in any other alkali metal, provided they incorporate the electron-ion interactions correctly and conform to the translational symmetry of the lattice. When the excess ion-ion contributions to the longitudinal phonon frequencies are compensated by the excess umklapp contributions to the transverse phonon frequencies at some wave vector, the dispersion curves crossover. The former are independent of the model while the latter increase with the strength of electron-ion interactions. Consequently, the larger values of C' reproduce the observed crossover in lithium and the smaller values of C' in other alkali metals suppress it. Variations in the nature and range of ion-ion interactions from one model to another give rise to difference in the values of C' in all alkali metals as well as in ~'c at which the crossover occurs in lithium, but neither of these differences is significant.

Acknowledgements The authors are grateful to Dr. S.B. Rajendraprasad for many valuable and stimulating discussions. One of the authors (M.S.) is greatly

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indebted to Prof. L.V. Sud and Prof. B.L. Sharma of Regional Engineering College, Kurukshetra for granting him leave.

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