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Dynamical treatment for elastic constants and phonon spectra of alkali metals
Volume 135, number 6,7
PHYSICSLETTERSA
6 March 1989
DYNAMICAL TREATMENT FOR ELASTIC CONSTANTS A N D P H O N O N SPECTRA OF ALKALI METALS N. SINGH, N.S. BANGER and S.P. SINGH Department of Physics, M.D. University, Rohtak 124001, India
Received 25 February 1988; revised manuscript received 24 October 1988;accepted for publication 21 December 1988 Communicatedby A.A. Maradudin
The exponentially damped two-bodyinteraction obtained in second-order perturbation theory using the rational dielectric function and the Heine-Abarenkov model potential is used to calculate the elastic constants of simple cubic metals. An approximate temperature dependence has been included in the effective potential through an asymptotic factor. The interaction is also used to predict the phonon spectra of cesium. The phonon spectra of Cs have also been calculated in wave-number space using the Lindhard-Taylordielectric function. The results obtained using the two screeningfunctions are almost the same. The use of the rational dielectricfunctionfor selectionseliminates the complexityand slowconvergenceof the two-bodyinteraction obtained using the Hartree dielectric function.
The Hartree dielectric function for the screening due to electrons with or without modifications combined with a pseudopotential is used to write down the ion-ion interaction in the second-order perturbation theory [ 1 ]. Because of its poor convergence in real space, the calculations of phonon spectra are done in reciprocal space. Using this ion-ion interaction Soma et al. [ 2 ] obtained the elastic constants of alkali metals. A damping factor suggested by Takanaka and Yamamoto [ 3 ] was multiplied by them in the ion-ion interaction to obtain a temperature dependence. The logarithmic singularity present in the Lindhard [ 4 ] function gives rises to long range Friedel oscillations in the pair potential. The weak logarithmic singularity in the slope of it at q=2kF cannot be detected as most elemental metals do not have reciprocal lattice vectors spanning the Fermi surface [5 ]. These very long range oscillations interfere destructively in the lattice sum; nevertheless they are responsible for structural stability of the crystal. The long range Friedal oscillations are responsible for the poor convergence of the force constants obtained with the two-body interaction. The calculations in wave-number space indicated that this singularity has very limited impact on the properties of interest. While it does give Kohn anomalies in the vibration spectrum, the general form of the spectrum does not depend upon the singularity. This suggests that we may simplify the screening to obtain a suitable and simple interionic interaction which could give a good account of most properties of the metals [ 6 ]. The replacement of the Hartree dielectric function with its logarithmic singularity by a rational dielectric function [ 7 ] is a good choice for describing s electron screening. The exchange correlation corrections due to Taylor [ 8 ] are inherited [ 9 ] in the rational dielectric function through its parameters as they are obtained by matching it with the Lindhard-Taylor [4,8 ] dielectric function. The rational function reproduces the Lindhard [ 4 ] function exactly except for its logarithmic singularity. It has the correct low and high q behaviour and is 0.5 at q = 2kF. The use of the rational dielectric function allows us to evaluate the integral appearing in the ionion interaction obtained in second-order perturbation theory analytically [9 ]. This, in turn, simplifies the numerical calculations drastically, as the exponentially damped pair potential has the advantage of not containing the very long range Friedel oscillations. This effective pair potential is then multiplied by an asymptotic factor obtained by Takanaka and Yamamoto [ 3 ] to obtain a temperature dependence. The two-body interaction so obtained in conjunction with the Heine-Abarenkov model potential is defined as 368
0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 135, number 6,7
V(r) = - 7
PHYSICS LETTERS A
.=l A. cos(k.r+a.) e x p ( - 2 j )
exp(-kaTr/hVF) ,
6 March 1989
(l)
where the amplitude A. is given by A. =2d. Iz3(q.)12 ,
(2)
and the phase or. is given by a . =d. +2 arg b(q.).
(3)
The modulus of the normalized Heine-Abarenkov model potential
b(q.) =D sin(qnr~) /q.r~ + ( 1-D)cos(q.r~) , where D and r~ are potential parameters, is defined as
2(k~ +2~)r~ (cosh 22~rc-cos 2k.r~) + ½(1 -D)2(cosh 22.r~ +cos 2k.r~)
Ib(q~) [ =
D(I-D)
+ (k2+22)r ¢ (2. sinh22.r¢+k, sin2k.rc)
11/2
.
(4)
and arg
1)(q.) =tan
_,( [D/(k 2 +22)r¢] (k. tanh 2 . r c - 2 . t a n k.r~)- ( 1 - D ) tanh 2.r¢ tan k.r~) [O/(k 2 +2~)r~] (2. tanh 2.r~ +k. tan/qr~) + ( 1- D ) _"
(5)
The parameters 2., k., d. and d. are functions of the electron radius rs [ 9 ]. q. = k. + i2. is the pole of the inverse dielectric function. The exponential factor in eq. ( 1 ) outside the square brackets is the temperature dependent damping factor due to Takanaka and Yamamoto [3]. In the limits k.--.0 and d.~0, eq. (1) reduces to the analytic expression for the effective potential
VTF(r)=~(Dsinh2r 2rc
c
+ ( l - D ) cosh 2re
)2
exp(-.~r),
(6)
with ~(=2+ nkaT
(7)
hvv ' in the Thomas-Fermi approximation, provided one sets 2n equal to the inverse screening length 2 and 2 ~ = l dn=l. Using the interionic potential V(r), we obtain the radial and tangential force constants Kr and Kt. From Kr and Kt at any interionic separation we obtain the interionic force constant K=a defined as the tensor d2V(r)
(
r~ra\
r,~ra
K.p= dr,~dra =_tS,~a- --~-)Kt + - ~ Kr.
(8)
From the interionic force constants K~a at nth-neighbour separation, we obtain the elastic constants for cubic metals using the dynamical long-wave-phonon method [ 10 ]. Finally, the dynamical matrix
D~a(q ) = ~ K~a(rt) [1 - e x p ( - i q . r t ) ] /
(9)
was used to generate the phonon frequencies. 369
Volume 135, number 6,7
PHYSICS LETTERS A
6 March 1989
In the numerical calculations we used the same e x p e r i m e n t a l d a t a for the t e m p e r a t u r e variation o f the atomic v o l u m e o f alkali metals as has been used by S o m a et al. [ 2 ]. The potential p a r a m e t e r s reported by P o p o v i c et al. [ 1 1 ] were used for numerical calculations. The elastic m o d u l i o b t a i n e d at r o o m t e m p e r a t u r e are found to be in good agreement with their e x p e r i m e n t a l results for all alkali metals a n d are c o m p a r e d in table 1. The exponentially d a m p e d pair potential leads to rapidly converging calculations. Meaningful but not very accurate results can be o b t a i n e d in some cases including only a few nearest neighbours, here we carry each calculation to convergence. On the other hand, one has to include the c o n t r i b u t i o n b e y o n d the tenth shell to get meaningful results using the two-body interaction o b t a i n e d using the L i n d h a r d - T a y l o r [4,8] dielectric function, here we have included c o n t r i b u t i o n s up to the sixteenth shell. The difference of the elastic stiffness constants between absolute zero t e m p e r a t u r e and r o o m t e m p e r a t u r e o f alkali metals are s u m m a r i z e d in table 2 along with the results o b t a i n e d by S o m a et al. [ 2 ] in the static a p p r o x i m a t i o n . The p h o n o n frequencies o f Cs metal along three s y m m e t r y directions are shown in fig. 1 as o b t a i n e d with eq. (9). It is found that the simple interionic pair potential defined by eq. ( 1 ) reproduces the e x p e r i m e n t a l results [ 12 ] fairly well. The c o n t r i b u t i o n s up to the seventh shell have been found sufficient to achieve convergence for these calculations. However, to achieve convergence o f the same orders in p e r f o r m i n g the calculation o f p h o n o n spectra in reciprocal space with the usual practice, i.e. using the effective p a i r potential defined with the L i n d h a r d dielectric function in second-order p e r t u r b a t i o n theory, one has to include contributions up to the 27th shell. Thus a numerical calculation o f p h o n o n spectra in all seven branches o f three s y m m e t r y directions using the present pair potential requires about one tenth o f c o m p u t e r time o f that required by the calculation o f it in w a v e - n u m b e r space at the same n u m b e r o f points using the effective p a i r potential defined with the L i n d h a r d dielectric function. The p h o n o n spectra o f Cs is also o b t a i n e d in q-space using the Table 1 The calculated elastic constants of alkali metals at 300 K. The experimental data is summarized by Shimada [ 13 ] and Huntington [ 14 ]. Ctl (Mbar)
Li Na K Rb Cs
C12 (Mbar)
C44 (Mbar)
a)
b)
expt.
a)
b)
expt.
a)
b)
expt.
0.162 0.094 0.039 0.029 0.021
0.162 0.105 0.042 0.030 0.022
0.157 0.095 0.042 0.032 0.026
0.139 0.083 0.034 0.024 0.017
0.140 0.088 0.035 0.025 0.018
0.133 0.078 0.034 0.026 0.022
0.107 0.058 0.025 0.019 0.013
0.108 0.061 0.026 0.019 0.014
0.116 0.062 0.029 0.021 0.016
Results using the Lindhard-Taylor dielectric function [4,8 ]. b) Results using the Pettifor dielectric function [ 7 ]. al
Table 2 The difference of the elastic stiffness constants AC,j= C,j( T=0 K) - C,j( T= 300 K) (in 10L~dyn cm-2). ACt t
Li Na K Rb Cs
AC, 2
a)
b)
c)
a)
b)
c)
a)
b)
c)
0.203 0.180 0.093 0.068 0.065
0.224 0.204 0.108 0.074 0.070
0.230 0.172 0.105 0.084 0.062
0.181 0.164 0.086 0.060 0.057
0.188 0.176 0.093 0.063 0.059
0.154 0.126 0.063 0.046 0.031
0.083 0.059 0.031 0.024 0.022
0.090 0.066 0.035 0.026 0.023
0.077 0.086 0.048 0.041 0.030
"~ Results using the Lindhard-Taylor dielectric function [4,8 ]. b~ Results using the Pettifor dielectric function [ 7 ]. c~ Values obtained by Soma et al. in the static approximation [2 ]. 370
AC44
Volume 135, number 6,7
PHYSICS LETTERS A
6 March 1989
i
[,I,]
rloo]
|,0
,/
[,o]
[lool
Cs "N. . . . .
_.
1.0
f ~
L t"
L O~Q' i
0.~
0
tlI
/"
o
'
/ // o i z/
1.0
0.0 o'.~ REDUCED WAVE VECTOR"~(2~f/al)
O.S
0.5
Fig. 1. Dispersion curves of Cs along symmetry directions. The dashed and solid curves represent the present calculations for rs=5,624 a.u. and r,=5.735 a.u. at 0 and 300 K, respectively. The points are the experimental data by Mizuki and Stassis [ 12 ] at 280 K. Here a~ is the lattice parameter.
[110]
C$
/
t
0
1-0 REDUCED
0
(
O!S WAVE ¥ACTOR"~(2 X/aLl
Fig. 2. Dispersion curves of Cs at 0 K calculated using the room temperature value of r, = 5.735 a.u. The solid and dashed curves represent the results obtained using the Pettifor [ 7 ] dielectric function in real space and the Lindhard-Taylor [4,8 ] dielectric function in wave number space, respectively.The solid and dashed curves overlap each other except at a few points shown in the [ 110 ] direction. The rest of the description is the same as that of fig. 1.
L i n d h a r d - T a y l o r [4,8 ] dielectric function and is compared with the present theoretical results in fig. 2. Evidently, the two calculations yield the same results for all the wave vectors, it is to be noted that the results shown in fig. 2 have been calculated from the force constants obtained using the effective potential defined at T = 0 K. However, the results so calculated represent the values o f p h o n o n frequencies near room temperature as the room temperature values of r , = 5.735 a.u. for Cs has been used in these calculations. As a result, our calculated p h o n o n frequencies shown in fig. 2 are also in good agreement with the experimental values determined at 280 K. It is to be noted further that in wave-number space one has to calculate the Coulomb and electronic contributions to p h o n o n spectra separately. However, in the present method there is no need to calculate the electrostatic Coulomb part o f the p h o n o n frequencies, as a part of the band structure term exactly cancels the electrostatic Coulomb repulsion in the effective potential. In summary, we have used the Pettifor [ 7 ] dielectric function for s electron screening to obtain a theoretical expression for the pair potential. The use o f the rational dielectric function [9 ] with its obvious drawbacks for s electrons has eliminated the complexity and slow convergence o f two-body interaction obtained using the Hartree dielectric function. Its prime virtue is its simplicity and ease o f application. It is found that even properties such as elastic constants and p h o n o n spectra o f alkali metals can be treated well in terms o f simple expressions o f i o n - i o n interaction. This approach can easily be extended to calculate the elastic properties o f the entire range o f simple metals and their alloys. Use o f this pair potential for calculating the elastic and dynamical properties o f a metal reduces the computer time m a n y times. We wish to thank Professor D,G. Pettifor for helping us in calculating the parameters o f the rational dielectric function. The authors are also grateful to Professor K.N. Pathak, Professor S. Prakash and Professor H.H. Lal for their encouragements throughout this work. Financial assistance from CSIR New Delhi and M D U Rohtak is gratefully acknowledged.
371
Volume 135, number 6,7
PHYSICS LETTERS A
References [ 1 ] W.A. Harrison, Pseudopotentials in the theory of metals (Benjamin, New York, 1966 ). [2] T. Soma, S. Kagaya and H. Matsuo, Phys. Stat. Sol. (b) 106 ( 1981 ) 309. [ 3 ] K. Takanaka and R. Yamamoto, Phys. Stat. Sol. (b) 84 ( 1977 ) 813. [4] J. Lindhard, K. Dan. Vindensk. Selsk, Mat. Fys. Medd. 28 (1954) 8. [5 ] N.F. Mott and H. Jones, Properties of metals and alloys (Dover, New York, 1936). [6] W.A. Harrison and J.M. Wills, Phys. Rev. B 25 (1982) 5007. [7] D.G. Pettifor, Phys. Scr. T1 (1982) 26. [8] R. Taylor, J. Phys. F 8 (1978) 1699. [9] D.G. Pettifor and M.A. Ward, Solid State Commun. 49 (1984) 291. [ 10] G.L. Squires, Ark. Fys. (Sweden) 25 (1963) 21. [ 11 ] Z.D. Popovic, J.P. Carbotte and G.R. Pierey, J. Phys. F 4 (1974) 351. [ 12] J. Mizuki and C. Stassis, Phys. Rev. B 34 (1986) 5890. [ 13 ] K. Shimada, Phys. Star. Sol. (b) 61 (1974) 325. [ 14] H.B. Huntington, in: Solid state physics, Vol. 7, eds. F. Seitz and D. Turnbull (Academic Press, New York, 1964).
372
6 March 1989
PHYSICSLETTERSA
6 March 1989
DYNAMICAL TREATMENT FOR ELASTIC CONSTANTS A N D P H O N O N SPECTRA OF ALKALI METALS N. SINGH, N.S. BANGER and S.P. SINGH Department of Physics, M.D. University, Rohtak 124001, India
Received 25 February 1988; revised manuscript received 24 October 1988;accepted for publication 21 December 1988 Communicatedby A.A. Maradudin
The exponentially damped two-bodyinteraction obtained in second-order perturbation theory using the rational dielectric function and the Heine-Abarenkov model potential is used to calculate the elastic constants of simple cubic metals. An approximate temperature dependence has been included in the effective potential through an asymptotic factor. The interaction is also used to predict the phonon spectra of cesium. The phonon spectra of Cs have also been calculated in wave-number space using the Lindhard-Taylordielectric function. The results obtained using the two screeningfunctions are almost the same. The use of the rational dielectricfunctionfor selectionseliminates the complexityand slowconvergenceof the two-bodyinteraction obtained using the Hartree dielectric function.
The Hartree dielectric function for the screening due to electrons with or without modifications combined with a pseudopotential is used to write down the ion-ion interaction in the second-order perturbation theory [ 1 ]. Because of its poor convergence in real space, the calculations of phonon spectra are done in reciprocal space. Using this ion-ion interaction Soma et al. [ 2 ] obtained the elastic constants of alkali metals. A damping factor suggested by Takanaka and Yamamoto [ 3 ] was multiplied by them in the ion-ion interaction to obtain a temperature dependence. The logarithmic singularity present in the Lindhard [ 4 ] function gives rises to long range Friedel oscillations in the pair potential. The weak logarithmic singularity in the slope of it at q=2kF cannot be detected as most elemental metals do not have reciprocal lattice vectors spanning the Fermi surface [5 ]. These very long range oscillations interfere destructively in the lattice sum; nevertheless they are responsible for structural stability of the crystal. The long range Friedal oscillations are responsible for the poor convergence of the force constants obtained with the two-body interaction. The calculations in wave-number space indicated that this singularity has very limited impact on the properties of interest. While it does give Kohn anomalies in the vibration spectrum, the general form of the spectrum does not depend upon the singularity. This suggests that we may simplify the screening to obtain a suitable and simple interionic interaction which could give a good account of most properties of the metals [ 6 ]. The replacement of the Hartree dielectric function with its logarithmic singularity by a rational dielectric function [ 7 ] is a good choice for describing s electron screening. The exchange correlation corrections due to Taylor [ 8 ] are inherited [ 9 ] in the rational dielectric function through its parameters as they are obtained by matching it with the Lindhard-Taylor [4,8 ] dielectric function. The rational function reproduces the Lindhard [ 4 ] function exactly except for its logarithmic singularity. It has the correct low and high q behaviour and is 0.5 at q = 2kF. The use of the rational dielectric function allows us to evaluate the integral appearing in the ionion interaction obtained in second-order perturbation theory analytically [9 ]. This, in turn, simplifies the numerical calculations drastically, as the exponentially damped pair potential has the advantage of not containing the very long range Friedel oscillations. This effective pair potential is then multiplied by an asymptotic factor obtained by Takanaka and Yamamoto [ 3 ] to obtain a temperature dependence. The two-body interaction so obtained in conjunction with the Heine-Abarenkov model potential is defined as 368
0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 135, number 6,7
V(r) = - 7
PHYSICS LETTERS A
.=l A. cos(k.r+a.) e x p ( - 2 j )
exp(-kaTr/hVF) ,
6 March 1989
(l)
where the amplitude A. is given by A. =2d. Iz3(q.)12 ,
(2)
and the phase or. is given by a . =d. +2 arg b(q.).
(3)
The modulus of the normalized Heine-Abarenkov model potential
b(q.) =D sin(qnr~) /q.r~ + ( 1-D)cos(q.r~) , where D and r~ are potential parameters, is defined as
2(k~ +2~)r~ (cosh 22~rc-cos 2k.r~) + ½(1 -D)2(cosh 22.r~ +cos 2k.r~)
Ib(q~) [ =
D(I-D)
+ (k2+22)r ¢ (2. sinh22.r¢+k, sin2k.rc)
11/2
.
(4)
and arg
1)(q.) =tan
_,( [D/(k 2 +22)r¢] (k. tanh 2 . r c - 2 . t a n k.r~)- ( 1 - D ) tanh 2.r¢ tan k.r~) [O/(k 2 +2~)r~] (2. tanh 2.r~ +k. tan/qr~) + ( 1- D ) _"
(5)
The parameters 2., k., d. and d. are functions of the electron radius rs [ 9 ]. q. = k. + i2. is the pole of the inverse dielectric function. The exponential factor in eq. ( 1 ) outside the square brackets is the temperature dependent damping factor due to Takanaka and Yamamoto [3]. In the limits k.--.0 and d.~0, eq. (1) reduces to the analytic expression for the effective potential
VTF(r)=~(Dsinh2r 2rc
c
+ ( l - D ) cosh 2re
)2
exp(-.~r),
(6)
with ~(=2+ nkaT
(7)
hvv ' in the Thomas-Fermi approximation, provided one sets 2n equal to the inverse screening length 2 and 2 ~ = l dn=l. Using the interionic potential V(r), we obtain the radial and tangential force constants Kr and Kt. From Kr and Kt at any interionic separation we obtain the interionic force constant K=a defined as the tensor d2V(r)
(
r~ra\
r,~ra
K.p= dr,~dra =_tS,~a- --~-)Kt + - ~ Kr.
(8)
From the interionic force constants K~a at nth-neighbour separation, we obtain the elastic constants for cubic metals using the dynamical long-wave-phonon method [ 10 ]. Finally, the dynamical matrix
D~a(q ) = ~ K~a(rt) [1 - e x p ( - i q . r t ) ] /
(9)
was used to generate the phonon frequencies. 369
Volume 135, number 6,7
PHYSICS LETTERS A
6 March 1989
In the numerical calculations we used the same e x p e r i m e n t a l d a t a for the t e m p e r a t u r e variation o f the atomic v o l u m e o f alkali metals as has been used by S o m a et al. [ 2 ]. The potential p a r a m e t e r s reported by P o p o v i c et al. [ 1 1 ] were used for numerical calculations. The elastic m o d u l i o b t a i n e d at r o o m t e m p e r a t u r e are found to be in good agreement with their e x p e r i m e n t a l results for all alkali metals a n d are c o m p a r e d in table 1. The exponentially d a m p e d pair potential leads to rapidly converging calculations. Meaningful but not very accurate results can be o b t a i n e d in some cases including only a few nearest neighbours, here we carry each calculation to convergence. On the other hand, one has to include the c o n t r i b u t i o n b e y o n d the tenth shell to get meaningful results using the two-body interaction o b t a i n e d using the L i n d h a r d - T a y l o r [4,8] dielectric function, here we have included c o n t r i b u t i o n s up to the sixteenth shell. The difference of the elastic stiffness constants between absolute zero t e m p e r a t u r e and r o o m t e m p e r a t u r e o f alkali metals are s u m m a r i z e d in table 2 along with the results o b t a i n e d by S o m a et al. [ 2 ] in the static a p p r o x i m a t i o n . The p h o n o n frequencies o f Cs metal along three s y m m e t r y directions are shown in fig. 1 as o b t a i n e d with eq. (9). It is found that the simple interionic pair potential defined by eq. ( 1 ) reproduces the e x p e r i m e n t a l results [ 12 ] fairly well. The c o n t r i b u t i o n s up to the seventh shell have been found sufficient to achieve convergence for these calculations. However, to achieve convergence o f the same orders in p e r f o r m i n g the calculation o f p h o n o n spectra in reciprocal space with the usual practice, i.e. using the effective p a i r potential defined with the L i n d h a r d dielectric function in second-order p e r t u r b a t i o n theory, one has to include contributions up to the 27th shell. Thus a numerical calculation o f p h o n o n spectra in all seven branches o f three s y m m e t r y directions using the present pair potential requires about one tenth o f c o m p u t e r time o f that required by the calculation o f it in w a v e - n u m b e r space at the same n u m b e r o f points using the effective p a i r potential defined with the L i n d h a r d dielectric function. The p h o n o n spectra o f Cs is also o b t a i n e d in q-space using the Table 1 The calculated elastic constants of alkali metals at 300 K. The experimental data is summarized by Shimada [ 13 ] and Huntington [ 14 ]. Ctl (Mbar)
Li Na K Rb Cs
C12 (Mbar)
C44 (Mbar)
a)
b)
expt.
a)
b)
expt.
a)
b)
expt.
0.162 0.094 0.039 0.029 0.021
0.162 0.105 0.042 0.030 0.022
0.157 0.095 0.042 0.032 0.026
0.139 0.083 0.034 0.024 0.017
0.140 0.088 0.035 0.025 0.018
0.133 0.078 0.034 0.026 0.022
0.107 0.058 0.025 0.019 0.013
0.108 0.061 0.026 0.019 0.014
0.116 0.062 0.029 0.021 0.016
Results using the Lindhard-Taylor dielectric function [4,8 ]. b) Results using the Pettifor dielectric function [ 7 ]. al
Table 2 The difference of the elastic stiffness constants AC,j= C,j( T=0 K) - C,j( T= 300 K) (in 10L~dyn cm-2). ACt t
Li Na K Rb Cs
AC, 2
a)
b)
c)
a)
b)
c)
a)
b)
c)
0.203 0.180 0.093 0.068 0.065
0.224 0.204 0.108 0.074 0.070
0.230 0.172 0.105 0.084 0.062
0.181 0.164 0.086 0.060 0.057
0.188 0.176 0.093 0.063 0.059
0.154 0.126 0.063 0.046 0.031
0.083 0.059 0.031 0.024 0.022
0.090 0.066 0.035 0.026 0.023
0.077 0.086 0.048 0.041 0.030
"~ Results using the Lindhard-Taylor dielectric function [4,8 ]. b~ Results using the Pettifor dielectric function [ 7 ]. c~ Values obtained by Soma et al. in the static approximation [2 ]. 370
AC44
Volume 135, number 6,7
PHYSICS LETTERS A
6 March 1989
i
[,I,]
rloo]
|,0
,/
[,o]
[lool
Cs "N. . . . .
_.
1.0
f ~
L t"
L O~Q' i
0.~
0
tlI
/"
o
'
/ // o i z/
1.0
0.0 o'.~ REDUCED WAVE VECTOR"~(2~f/al)
O.S
0.5
Fig. 1. Dispersion curves of Cs along symmetry directions. The dashed and solid curves represent the present calculations for rs=5,624 a.u. and r,=5.735 a.u. at 0 and 300 K, respectively. The points are the experimental data by Mizuki and Stassis [ 12 ] at 280 K. Here a~ is the lattice parameter.
[110]
C$
/
t
0
1-0 REDUCED
0
(
O!S WAVE ¥ACTOR"~(2 X/aLl
Fig. 2. Dispersion curves of Cs at 0 K calculated using the room temperature value of r, = 5.735 a.u. The solid and dashed curves represent the results obtained using the Pettifor [ 7 ] dielectric function in real space and the Lindhard-Taylor [4,8 ] dielectric function in wave number space, respectively.The solid and dashed curves overlap each other except at a few points shown in the [ 110 ] direction. The rest of the description is the same as that of fig. 1.
L i n d h a r d - T a y l o r [4,8 ] dielectric function and is compared with the present theoretical results in fig. 2. Evidently, the two calculations yield the same results for all the wave vectors, it is to be noted that the results shown in fig. 2 have been calculated from the force constants obtained using the effective potential defined at T = 0 K. However, the results so calculated represent the values o f p h o n o n frequencies near room temperature as the room temperature values of r , = 5.735 a.u. for Cs has been used in these calculations. As a result, our calculated p h o n o n frequencies shown in fig. 2 are also in good agreement with the experimental values determined at 280 K. It is to be noted further that in wave-number space one has to calculate the Coulomb and electronic contributions to p h o n o n spectra separately. However, in the present method there is no need to calculate the electrostatic Coulomb part o f the p h o n o n frequencies, as a part of the band structure term exactly cancels the electrostatic Coulomb repulsion in the effective potential. In summary, we have used the Pettifor [ 7 ] dielectric function for s electron screening to obtain a theoretical expression for the pair potential. The use o f the rational dielectric function [9 ] with its obvious drawbacks for s electrons has eliminated the complexity and slow convergence o f two-body interaction obtained using the Hartree dielectric function. Its prime virtue is its simplicity and ease o f application. It is found that even properties such as elastic constants and p h o n o n spectra o f alkali metals can be treated well in terms o f simple expressions o f i o n - i o n interaction. This approach can easily be extended to calculate the elastic properties o f the entire range o f simple metals and their alloys. Use o f this pair potential for calculating the elastic and dynamical properties o f a metal reduces the computer time m a n y times. We wish to thank Professor D,G. Pettifor for helping us in calculating the parameters o f the rational dielectric function. The authors are also grateful to Professor K.N. Pathak, Professor S. Prakash and Professor H.H. Lal for their encouragements throughout this work. Financial assistance from CSIR New Delhi and M D U Rohtak is gratefully acknowledged.
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PHYSICS LETTERS A
References [ 1 ] W.A. Harrison, Pseudopotentials in the theory of metals (Benjamin, New York, 1966 ). [2] T. Soma, S. Kagaya and H. Matsuo, Phys. Stat. Sol. (b) 106 ( 1981 ) 309. [ 3 ] K. Takanaka and R. Yamamoto, Phys. Stat. Sol. (b) 84 ( 1977 ) 813. [4] J. Lindhard, K. Dan. Vindensk. Selsk, Mat. Fys. Medd. 28 (1954) 8. [5 ] N.F. Mott and H. Jones, Properties of metals and alloys (Dover, New York, 1936). [6] W.A. Harrison and J.M. Wills, Phys. Rev. B 25 (1982) 5007. [7] D.G. Pettifor, Phys. Scr. T1 (1982) 26. [8] R. Taylor, J. Phys. F 8 (1978) 1699. [9] D.G. Pettifor and M.A. Ward, Solid State Commun. 49 (1984) 291. [ 10] G.L. Squires, Ark. Fys. (Sweden) 25 (1963) 21. [ 11 ] Z.D. Popovic, J.P. Carbotte and G.R. Pierey, J. Phys. F 4 (1974) 351. [ 12] J. Mizuki and C. Stassis, Phys. Rev. B 34 (1986) 5890. [ 13 ] K. Shimada, Phys. Star. Sol. (b) 61 (1974) 325. [ 14] H.B. Huntington, in: Solid state physics, Vol. 7, eds. F. Seitz and D. Turnbull (Academic Press, New York, 1964).
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