Phonon dispersion curves of alkali metals

Physica 104B (1981) 343-349 © North-Holland Publishing Company

PHONON DISPERSION CURVES OF ALKALI METALS M. SATISH KUMAR* Department of Physics, Indian Institute of Technology, Delhi, New Delhi-110016, India Received 18 August 1980

A simple force constant model for alkali metals is proposed. The model takes into account the umklapp processes associated with the electrons and satisfies the symmetry requirements of the lattice. The excellent agreement of the dispersion curves with the experimental data of all alkali metals including the crossoverof branches of lithium at ~"= 0.51 is obtained.

1. Introduction

none of these models satisfy the symmetry requirements of the lattice. Further, Ramamurthy and Singh [22] proposed a model which made use of the umklapp processes and satisfied the symmetry requirements of the lattice. In spite of the excellent interpretation of the crossing in lithium given by this model it leaves certain unanswered questions. 1) A large value of c' produces a cross-over in lithium. Table II [22] shows that the value of c' is very large for lithium, becomes small for sodium and again increases for potassium. The increase in c' for potassium is five times the value o f c ' for sodium. This increasing trend of c' leads to the possibility of further increases in c' for rubidium and cesium and a cross-over in the dispersion curves of these metals also. In fact this model produces a cross-over at ~"= 0.7 for rubidium [23] 2) The model makes use of only two shortest reciprocal lattice vectors in the umklapp processes. This would imply that the higher reciprocal vectors do not contribute to the summation or this cut is purely arbitrary. From these observations it is clearly indicated that none of these models is capable of explaining the phonon dispersion curves of all alkali metals and for different alkali metals different models are required. The additional support to this view point comes from the exact calculation of the bulk modulus of the electron gas [24] which predicts a positive value of the bulk modulus of the electron gas for lithium, sodium and potassium and negative for rubidium and cesium.

Lattice dynamics of alkali metals has been most favourite to the theoretical workers because of the characteristic feature of their dispersion curves. The experimental dispersion curves of lithium [ 1] exhibit a cross-over of [~'00] branches at ~"= 0.46, while no such crossing is observed for the dispersion curves of sodium [2], potassium [3] and rubidium [4]. The phonon frequencies of cesium have not been measured so far, mainly because it is very reactive and sensitive to the compression of the band structure [5]. A number of elastic force models [6-20] have been proposed for these metals. Few models got success in getting the cross-over in lithium [ 6 - 8 ] while several other models failed to do so [ 9 - 1 7 ] . The failure or the success of these models in producing the cross-over however hinged over the predicted value of c', the electron-ion interaction force constant, which is related to the bulk modulus of the electron gas. The models with negative value of c' [ 6 - 8 ] produced the crossing and the models with positive value of c' failed. Ramamurthy and Neelakandan [21 ] however have shown that the production of the cross over using the negative value o f c ' is a spurious result and such models would produce the cross over in other alkali metals also, in contrary to the experimental results. Besides, *Present address: Department of Applied Physics, Regional Engineering College, Kurukshetra 132 119, India. 343

M. Satish Kumar/Phonon dispersion curves of alkali metals

344

On the other hand in contrary to these observations the lattice vibrations of the alkali metals are homologous [25]. This property of the alkali metals suggests that it is possible to have a general lattice dynamical model capable of reproducing the dispersion curves of all alkali metals. The following sections of the paper therefore deal with such a model and the calculation of the phonon frequencies of lithium, sodium, potassium, rubidium and cesium. Since the experimental phonon frequencies of cesium are not reported so far the phonon frequencies obtained by Price et al. [26] have been used in place of experimental data for cesium.

Restricting the range of the interaction of the i o n - i o n interaction to the third neighbour only, the diagonal and off diagonal elements o f D i are given by [27]

MDixy = (8/3)sisjc k + 2A 3s2is2j,

2. Theory

where

The normal modes of vibrations could be obtained solving the secular determinant

Cni =

ID(q) - 0o211 = 0,

Sni = sin (nTr~i),

MDixx = 8(A1/3 + B1)[1 - c i c / c k ] + 2,4211 + 2A 3 [2

]

¢2i(C2j q- C 2 k ) ]

+ 2B 2 [3 - c2i - c 2 / -

c2k ]

and

(1)

where wq is the frequency of the normal mode with wave vector q and [ is the 3 X 3 unit matrix. The elements of the dynamical matrix are split into two parts, ionic and electronic,

D(q) = D i + D e.

-

--c2i

COS

(nrr~i),

(3)

i = X, y , z, n = 1, 2.

A i (i = 1 - 3 ) are the first, second and third neighbour radial force constants and B i (i = 1,2) are tangential force constants fx = qx/qmax = (a/27r)qx, a being the lattice parameter. The elements o f D e are given by [221

(2)

Table I Relevant input data for alkali metals Metal

Lithium Sodium Potassium Rubidium Cesium

M a A (10-27 kg) (10-1 nm)

B

11.52 38.16 64.91 141.95 220.71

.299 .286 .271 .266 .260

3.484 4.240 5.226 5.628 6.050

1.007 1.078 1.167 1.198 1.235

CI1

C12

C44

Ref.

(101° N/m 2) 1.435 0.816 0.415 0.312 0.247

VL(100) VL(~-r0) ii i i i Ref. VL(~-~-r) (1012 rtz)

1.208 0.679 0.340 0.255 0.206

1.075 0.570 0.284 0.186 0.148

[28] [29] [30] [31] [32]

8.820 3.580 2.210 1.320 1.00

A3

c'

9.000 3.820 2.400 1.450 1.094

Table II Force constants of alkali metals in (10 -3 N/m) Metal

A1

B1

Lithium Sodium Potassium Rubidium Cesium

2350 2288 2560 1723 1543

398 292 -87.64 -64.79 -80.85

A2 -1566 -312.0 364.9 231.6 240.3

B2 475 210 27.3 44.64 31.22

217 206 -49.38 -30.89 -17.00

3953 1414 52.06 307.7 149.30

7.000 2.880 1.780 1.100 0.816

[1] [2] [3] [4] [26]

M. Satish Kumar/Phonon dispersion curves of alkali metals c'

(1t - q)x(H - q)yS(H - q)F(H - q)

(H-q)2+K1F(H-q)

H

(4) where c' is the force constant associated with the electron-ion interaction, H is the reciprocal lattice vector and K 1 = a -2. S(~) is the interference factor defined by S(r/) = ~ exp (it/" r) dr ~2 ~2

F(r/) = a2K2Fo(r/) / 1 -I_

metry directions of the crystal. For this purpose the secular determinant was taken to the long-wavelength limit and compared with the elastic determinant. This gives the following relations between the elastic and force constants:

A1 + B1 + B2 + A 3 ~ac44, 3

(9a)

2(-4 2 + A 3) + c'= a(Cll - c44),

(9b)

4 A1 + 4A 3 + c' = a(cl2 + c44). 3

(9c)

- -

=

1

(5)

and the expressions for S(r/) are obtained from [38], F(r/) is the screening function and is determined by the choice of the screening approximation. Although various screening functions [21, 33-37] have been reported, in the present case the screening function of Vashishta et al. [36] is used. The expression of F(r/) according to [36] is given by f-

345

Fo(rl) G(rl)1-1 7?2 J '

(6)

In addition the secular determinant was solved at the (100), (½½0) and (½½½)zone boundary points to obtain the relationship between the force constants and the zone boundary frequencies. These are given by 16I-~

+ ~ 1 1 +C'SLIO0 = 4rr2Mv2(100),

812-~ +Bl+B2+A2+-~I2

+C'SL110

= 4rr2Mv2(½½0) 1 4K2 - r/2 In 2gF + r/ F0 (rl) =---f __ 2 LFT? 2g F -- r/

(7)

and

and

4 I2-~+2Bl+3B2+A21

G(rl) A [1 - exp (-Brl2/K~)]. =

(8)

KF is the Fermi wave vector, gTF is the Thomas Fermi screening parameter. A and B are the constants and their values are given in table I. In order to include the contribution of all the reciprocal lattice vectors to the umklapp processes the summation of eq. (4) was extended up to H < (2n/a)VTO since the summation showed convergence at this limit.

+,C'SLlll

2 ~111 = 4rr2MVL( ~~), where C'SLlO0corresponds to D~x at (100) point, e e c tSL110 corresponds to (Dxx + Dxy) at ( 1: 1: 0) point and C'SLllX corresponds to (De + 2Dxey)at (½½½) point. With the help of these six relations the force constants were obtained. The necessary data for the calculation of the force constants and the numerical values of the force constants are given in tables I and II.

Z 1. Evaluation of the force constants 3. Results and discussion In all six force constants appearing in the dynamical matrix need to be evaluated. They were evaluated using the experimentally known elastic constants and the zone boundary frequencies in the principal sym-

Using the calculated force constants the secular determinant was solved in the principal symmetry directions to obtain the phonon frequencies of lithium,

M. Satish Kumar/Phonon dispersion curves of alkali metals

346

the phonon frequencies of lithium, sodium, potassium

sodium, potassium, rubidium and cesium. These frequencies are plotted as a function of the reduced wave vector ~ and the dispersion curves so obtained are shown in figs. 1-5. Besides the experimental values of

°; Io;, ''

r

and rubidium [1--4] and the phonon frequencies of cesium from [26] are also shown to facilitate the comparison.

. . . .

'

i

~'o1/

F'

!

//)'

",t

~'/ T1///i f

/,

(4 I~ /,

L

0

1.2//

l

I

1

I

0.2

0.4

0.6

0.8

I

1.0 0.8 0.6

I

{

0.4

0.2

i i t 0.2

0.4

Fig. 1. P h o n o n dispersion curves of lithium in [~'00], [~T0] and [~'{T] directions. Calculated values: longitudinal (verse ( . . . . . ). Experimental values: longitudinal (}); transverse (~). 4.0

i

i

i

i

[~'oo1

t~ol

/Y

k

3.0

I

!'\\

L

i///1! ~

); trans-

2.0

f,/ /

1.0

0,0

1

0

0,2

r/

T1

]/ ~oo S /~o //6 I

I

0.4 0.6

I

0.8

I

1.0 0.8

t

0.6

i

i_

0.4 0.2

0

0.2 0.4

Fig. 2. P h o n o n dispersion curves of sodium in [~'00], [ ~ 0 ] and [ ~ ] directions. Calculated values: longitudinal ( verse ( . . . . . ). Experimental values: longitudinal (e); transverse (o).

); trans-

347

M. Satish Kumar/Phonon dispersion curves of alkali metals 2.5

I

i

T~

I

I

[ ~'00]

i - -

/(/'

1.5 L

I

[ ~'~'~']

///

"~

i

T

2.0

1.0

1

J

i

[~'~'0]

L

l

O\\ J[/~

//

L

',T

,,'

TI

1

o.~

,

~//' \

0.0 0

I

I

0.2

0.4

I 0.6

1 0.8

1.0

0.8

I 0.6

I 0.4

I 0.2

T?. - - "_

/// I 0

I

I 0.4

0.2

);

Fig. 3. Phonon dispersion curves of potassium in [tOO], [tt0] and [tttl directions. Calculated values: longitudinal ( transverse ( . . . . . ). Experimental values: longitudinal (e); transverse (o). i

q----i

i

i

I--

1.5

/

1.2

1.0

3

e~

0.8

i

[~%']

{~oo1

I

i

I

i

[~'~-o]

0////

I 0.6 0.4 0.2 0.0

0.2

0.4

0.6

0.8

1.0

]

I

0.8

0.6

~

0.4

~'""~1 t

0.2

~/

I

J

0,2

I

I

0.4

Fig. 4. Phonon dispersion curves of rubidium in [['00], [l'~'0] and [~'~'~'] directions. Calculated values: longitudinal ( transverse ( . . . . . ). Experimental values: longitudinal (e); transverse (o). It is obvious from the figures that the agreement b e t w e e n the theoretical curves and the experimental points is excellent. The m a x i m u m discrepancy

);

between the two is <4%. The dispersion curves of lithium exhibit a cross-over in [~'00] direction at ~" = 0.51 very close to the experimental value of~"

34 8

M. Satish Kurnar/Phonon dispersion curves oj a{kali metals ]

r

I

t

7

I

t

[~[~]

goo]

- W;o] -~I

Ii

1.2

1.0

~

-

-

o,-

v

/,/

T\

J/

0.4--

i \

4

/ 0.0

I

T2,?/~

~ 0.6

0.2

0.4

0.6

0.8

l.O

0.8

0.6

0.4

0.2

\ 0

0.2

0.4

Fig. 5. P h o n o n dispersion curves of cesium in [~'00], [g'g'0] and [g'~'~'] directions. Calculated values: longitudinal ( verse ( . . . . . ). Values f r o m [26] : longitudinal (o); transverse (o).

= 0.46. No crossing is observed in the dispersion curves of other alkali metals. The longitudinal and the transverse frequencies degenerate at the (100)point there by confirming that the present model satisfies the symmetry requirements of the lattice. Besides this model uses the umklapp processes in both the calculation of the force constants as well as in the calculation of the phonon frequencies unlike [22], where the use of normal processes only is made in the calculation of the force constants and the umklapp processes in the calculation of phonon frequencies. The value of c' for all alkali metals is positive in contrary to the earlier predictions of the K e for rubidium and cesium [24]. However, the negative value o f K e would imply that the electron gas provides a force which aids the ionic displacements and it would thus raise the question of the stability of the system. Although the present model takes into consideration the ionic interaction up to the third nearest neighbour, the contribution o f B 3 the third neighbour tangential force constant is not included, in the present calculations, since it was observed that the inclusion or exclusion of this or the higher neighbour force constants do not make any change in the calculated

); trans-

phonon frequencies. However, the inclusion of these force constants would require the additional input data and make the calculations unnecessary tedious. Different expressions for G(r/) make the various screening functions [33-36] differ from each other. The difference between the values of G(r/) in various functions being very small the use of any of these approximations will not affect the phonon frequencies of the alkali metals. However, the screening functions [21, 37] have a limitation that they are valid only for small wave vectors and they do not satisfy the symmetry requirements of the lattice.

4. Conclusions The excellent agreement of the theoretical dispersion curves and the experimental values for all alkali metals indicates that the present force constant model has for the first time reproduced the experimental dispersion curves of all alkali metals including the crossover in the [~'00] branches of lithium. This simple model free from the draw backs of earlier force constant models therefore provides a method to calculate the phonon frequencies of alkali metals.

M. Satish Kumar/Phonon dispersion curves of alkali metals

Acknowledgements The author is grateful to Dr. V. Ramamurthy and Mr. S. B. Rajendra Prasad for many valuable and stimulating discussions. The author is also thankful to the Principal and Prof. L. V. Sud of Regional Engineering College Kurukshetra for granting him a study leave.

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