- Email: [email protected]
Comparative study of electrical resistivity of d and f-shell liquid metals
ARTICLE IN PRESS
Physica B 337 (2003) 245–248
Comparative study of electrical resistivity of d and f-shell liquid metals J.K. Baria V. P. & R. P. T. P. Science College, Vallabh Vidyanagar–388 120, Gujarat, India Received 9 February 2003; received in revised form 6 April 2003; accepted 17 May 2003
Abstract A model potential depending on an effective core radius but otherwise parameter free is used for the comparative study of electrical resistivity of d and f-shell liquid metals. In the present paper electrical resistivity of d and f-shell liquid metals have been calculated using Ziman’s formula, Ziman’s formula modified and used by Khajil and Tomak and tmatrix formulation given by Evans and Evans et al. Previously no one have reported such comparative study using pseudopotentials. In the electrical resistivity calculations we have used structure factor derived through charge hard sphere approximation. The beauty of this approximation is that it needs pseudopotential form factor for the calculation of structure factor. So this gives the better explanation of structure factor than any other approximations. From present investigations it is found that t-matrix formulation results are excellently agrees with the experimental findings. A successful application is an evidence our potential to predict wide class of physical properties of d and f-shell metals. r 2003 Elsevier B.V. All rights reserved. PACS: 71.15.Dx; 72; 71.45.Gm Keywords: Electrical resistivity; Pseudopotential; Exchange and correlation
1. Introduction The Ziman’s nearly free electron (NFE) theory has been fairly successful in describing the quantitative behavior of the electrical resistivity in simple liquid metals. This is because in these metals the mean free path is about one hundred times the interatomic distance and the weak scattering picture should be valid. Even for the heavy polyvalent metals (e.g. mercury, thallium and lead) where the mean free path is only about two interatomic distances, the NFE model can E-mail address: jay [email protected] (J.K. Baria).
yield results, which are in reasonable in agreement with experiments. Calculations of electrical resistivity using structure factor from various experiments or different versions of bare ion potential and dielectric function, gives correct order of magnitude but differ among themselves. In fact sometimes the theory is trusted sufficiently to use measured value of resistivity to determine the parameter of the potential. The Ziman’s formula is not expected to apply to d-state transition metals because the unfilled d-state cause strong resonant scattering which seems inappropriate for description by pseudopotential. Nevertheless, Evans et al. [1] put forward a version of equation of resistivity
0921-4526/03/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0921-4526(03)00410-1
ARTICLE IN PRESS J.K. Baria / Physica B 337 (2003) 245–248
246
in which jW ðqÞj2 was replaced by exact value of the squared matrix element for the scattering of a plane wave by a transition metal ion. The potential of the latter was taken to be Muffin-Tin potential as derived in solid state physics for band structure calculations and the exact scattering can be calculated by the phase shift method.
2. Electrical resistivity of liquid metals For both liquid and amorphous metals the electrical resistivity is given by [1,2] rZiman ¼
3pm2 4Ze2 _3 nkF6 Z N q3 SðqÞjW ðqÞj2 Yð2kF qÞ dq:
ð1Þ
0
Here, SðqÞ is the structure factor. In the present work we have used the charged hard sphere (CHS) [3] structure factor. The structure factor derived through CHS has an advantage that it involves the pseudopotential form factor. In this sense it differs from the Percus–Yevick (PY) [4] theory. So the structure factor derived through CHS gives the better explanation than PY theory. Recently Baria and Jani [5,6] have proposed model potential which is quite successful in describing lattice mechanical properties of d and f-shell metals is used in the present calculation of electrical resistivity. The unit step function Y which cuts off the integration at 2kF corresponding to a preferably sharp Fermi surface, is defined as Yð2kF qÞ ¼ 0 for q > 2kF ; ¼ 1 for qp2kF :
ð2Þ
But the finite mean free path corresponds to a finite uncertainty in the electron position. This corresponds to a finite uncertainty in the electron momentum. Thus the Fermi surface is not perfectly sharp as implied by Eq. (1) but it is blurred. The attempts to take this blurring in to account in the formulation of resistivity is reviewed by Mc Caskill and March [7]. Ferraz
and March [8] approach yields in place of Eq. (1). rsc ¼
3pm2 4Ze2 _3 nkF6 Z N q4 SðqÞjW ðqÞj2 Gðq; kF ; lÞ dq:
ð3Þ
0
This equation must be self-consistently solved. Very few explicit approximations are proposed for the function Gðq; kF ; lÞ [7]. In this work we have used the form for Gðq; kF ; lÞ as used by Laakkonen and Nieminen [9] and Khajil and Tomak [10] 2 Gðq; kF ; lÞ ¼ 3 tan1 ðqlÞ pq 1 2ql tan1 2 1 þ 4ðkF lÞ2 ðqlÞ2 p 1 2 Y q 2 þ 4kF : ð4Þ 2 l The mean free path is determined self-consistently as follows. The first step of the selfconsistency loop is to calculate resistivity using Eq. (1) i.e. with mean free path ‘‘l’’ is infinity. A new mean free path ‘‘l’’ is then obtained from the Drude relation as rL ¼
_kF : ne2 l
ð5Þ
The iterations are continued till rL converges.
3. Resistivity (t-matrix) approach We are concerned in this section with the scattering occurring in liquid metals. For example, a flux of electrons is incident on a scattering center and it is important to know the number of electrons that are scattered in a definite direction. This problem is most conveniently discussed using phase analysis. This analysis will throw much additional light on the use of pseudopotential see for more detail [11,12]. For many purposes a detailed knowledge of how the electronic wave functions behave in the neighborhood of the perturbing potential is not required, it is only their asymptotic form that matters. This, of course, is completely determined by phase shifts. Since,
ARTICLE IN PRESS J.K. Baria / Physica B 337 (2003) 245–248 Table 1 Presently calculated values of phase shift of some liquid d and f-shell metals Metal
Z0
Z1
Z2
Cu Ag Au Ni Pd Pt Rh Ir La Yb Ce Th
0.7374 0.6649 0.6141 0.9458 0.7780 0.7630 0.8184 0.8013 0.6618 0.6466 0.6488 0.6381
0.2157 0.2444 0.2540 0.3344 0.3762 0.3796 0.3663 0.3706 0.1908 0.1852 0.1789 0.2264
0.0319 0.0323 0.0360 0.0637 0.0732 0.0741 0.0708 0.0718 0.0138 0.0791 0.1214 0.1672
when the potential is due to something as small as metallic ions, the phase shifts are normally insignificant beyond say Z3 therefore Z0 ; Z1 and Z2 are to be calculated. The phase-shift Z1 of the lth partial wave related to the pseudopotential is given by Z mkF O0 2 Zl ¼ W ðyÞyPl ðcos yÞ dy; ð6Þ 4p_2 0 where y ¼ q=2kF and Pl ðcos yÞ is a Legendre polynomial and 1 q 2 cos y ¼ 1 : 2 kF Using the above equation phase-shift Z0 ; Z1 ; Z2 are calculated for s, p and d components and are shown in Table 1. Note that the low-energy phaseshift obeys the rule Z0 > Z1 > Z2 : The larger the phase-shift, the stronger the scattering. Evidently it is liable to be especially strong if the energy of the incident electron happens to coincide with that of a virtual bound state. In the case of noble and transition metals, the NFE approach is not appropriate due to presence the presence of a d band in the conduction band, which complicates the picture. We use the t-matrix of the pseudopotential to calculate the scattering cross-section rather than the usual screened ion model potentials. The single scatter t-matrix form factor for energy conserving transition is given by
247
Evans [13] and Evans et al. [14] 2p_3 1 X ð2l þ 1Þsin Z1 ðEF Þ tðk~; k~0 Þ ¼ 1=2 O 0 1 mð2mEÞ exp ½iZ1 ðEF Þ P1 ðcos yÞ:
ð7Þ
The t-matrix has the dimensions of energy and is normalized to the atomic volume. The resistivity takes the form with y ¼ q=2kF Z 1 3pO0 ~ ~0 2 3 rt ¼ 4y SðqÞ tð k ; k Þ ð8Þ dy: e2 _2 v2F 0
4. Results and discussion The resistivity calculated using Ziman’s formula, self-consistent approach and t-matrix formulations are tabulated in Table 2 with the experimental and other such theoretical findings. Daver et al. [15] have calculated resistivity and the mean free path using self-consistent method of about twenty liquid metals inclusive noble metals. They have used the Ashcroft’s empty-core pseudopotential, the parameter of the potential were chosen to yield the best agreement with the experimental structure factor. As they have fitted the potential parameter their results are very close to the experimental findings for simple metals while for Cu and Ag it deviates little. Ononiwu [16] has calculated the resistivity of all 3d, 4d and 5d transition metals using Ziman and t-matrix formula. Their t-matrix results are little close to the experimental findings for Ag, Au, Ni and Pd while Ziman’s formula gives very high values for Au, Pt and Rh. Gorecki and Popielawski [19] have calculated resistivity using the self-consistent quasi-crystalline approximation. Their results are found very poor. In our present investigations, the resistivity calculations using Ziman’s formula are very inferior for all d and f-shell metals, selfconsistent results for Cu and Au are little close to the experimental findings while for t-matrix formulation result shows an excellent agreement with the experimental findings for all d and f-shell liquid metals moreover, it shows significant improvements in the previously reported [16–19] theoretical results. The experimental data for Rh,
ARTICLE IN PRESS J.K. Baria / Physica B 337 (2003) 245–248
248
Table 2 Electrical resistivity of some liquid d and f-shell metals Metal
Cu Ag Au Ni Pd Pt Rh Ir La Yb Ce Th
Present work
rExpt
Others
l in (au)
rsc
rTmat
rZiman
[15,16]
[15,16]
[17]
[17]
[18]
67.96 63.99 49.39 48.33 148.85 199.65 110.50 178.12 133.92 139.02 176.00 172.18
19.85 26.97 34.95 55.36 65.99 60.62 94.74 70.17 74.37 90.17 89.13 108.23
20.05 17.55 32.24 81.78 84.30 99.75 72.47 105.85 147.58 118.56 127.17 137.21
30.78 9.96 41.26 41.02 41.07 44.83 50.10 96.64 332.07 177.02 267.63 186.47
20 19 31 85 83 90 — — 140 110 125 —
26.7 28.5 36.2 54.9 58.1 94.6 — — 165 137 134 —
10 20 27 87 79 22 96 90 — — — —
19 40 108 27 16 295 196 95 — — — —
21.0 21.7 21.4 41.4 36.0 62.0 — — — — — —
Ir and Th are not available so our results are predictive in nature. Here we mention clearly that we have not fitted the parameter of the potential either with the resistivity or with the structure facture. It is strongly emphasis previously by Esposito et al. [20] that the Ziman’s formula describes electrical transport correctly for simple liquid metals while self-consistent results of resistivity are over estimates if mean free paths are comparable or smaller than an interatomic distances, which happens in the d and f-shell metals. To the best of our knowledge after 1993 no one has reported the resistivity calculations of liquid d and f-shell metals using any method viz, Ziman’s formula, self-consistent approach and t-matrix formula. The present investigations confirm that the t-matrix formulations are the better chose for the resistivity calculations of d and f-shell liquid metals. References [1] R. Evans, D.A. Greenwood, P. Lloyd, Phys. Lett. A 35 (1971) 57.
[2] O. Dreierach, R. Evans, H.J. Guntherodt, H.U. Kunzi, J. Phys. F 2 (1972) 709. [3] H.B. Singh, A. Holz, Phys. Rev. A 24 (1983) 1108. [4] J.K. Percus, G.J. Yevick, Phys. Rev. 110 (1958) 1. [5] J.K. Baria, A.R. Jani, Physica B 328 (2003) 317. [6] J.K. Baria, A.R. Jani, Pramana J. Phys. 60 (2003) 1. [7] J.S. Mc Caskili, N.H. March, Phys. Chem. Liquids 12 (1982) 1. [8] A. Ferraz, N.H. March, Phys. Chem. Liquids 8 (1979) 271. [9] J. Laakkonen, R.M. Nieminen, J. Phys. F 13 (1983) 2265. [10] T.M.K. Kahjil, M. Tomak, Phys. Stat. Sol. B 134 (1986) 321. [11] T.E. Faber, Theory of Liquid Metals, Cambridge University Press, Cambridge, 1972, p. 30. [12] L.I. Yastrebov, A.A. Katsnelson, Foundation of One Electron Theory of Solids, Mir Publishers, Moscow, 1987, p. 42. [13] R. Evans, J. Phys. C (Met. Phys. Suppl.) 2 (1970) 137. [14] R. Evans, B.L. Gyorffy, N. Szabo, J.M. Ziman, in: S. Takeuchi (Ed.), Properties of Liquid Metals, Taylor and Francis, London, 1973. [15] F. Daver, T.M.A. Khajil, M. Tomak, Phys. Stat. Sol. B 138 (1986) 373. [16] J.S. Ononiwu, Phys. Stat. Sol. B 177 (1993) 413. [17] Y. Waseda, A. Jain, S. Tamaki, J. Phys. F. 8 (1978) 125. [18] K. Hirata, Y. Waseda, A. Jain, R. Srivastava, J. Phys. F. 7 (1977) 419. [19] J. Gorecki, J. Popielawski, J. Phys. F. 13 (1983) 2107. [20] E. Esposito, H. Ehrenreich, C.D. Gelatt Jr., Phys. Rev. B 18 (1978) 3913.
Physica B 337 (2003) 245–248
Comparative study of electrical resistivity of d and f-shell liquid metals J.K. Baria V. P. & R. P. T. P. Science College, Vallabh Vidyanagar–388 120, Gujarat, India Received 9 February 2003; received in revised form 6 April 2003; accepted 17 May 2003
Abstract A model potential depending on an effective core radius but otherwise parameter free is used for the comparative study of electrical resistivity of d and f-shell liquid metals. In the present paper electrical resistivity of d and f-shell liquid metals have been calculated using Ziman’s formula, Ziman’s formula modified and used by Khajil and Tomak and tmatrix formulation given by Evans and Evans et al. Previously no one have reported such comparative study using pseudopotentials. In the electrical resistivity calculations we have used structure factor derived through charge hard sphere approximation. The beauty of this approximation is that it needs pseudopotential form factor for the calculation of structure factor. So this gives the better explanation of structure factor than any other approximations. From present investigations it is found that t-matrix formulation results are excellently agrees with the experimental findings. A successful application is an evidence our potential to predict wide class of physical properties of d and f-shell metals. r 2003 Elsevier B.V. All rights reserved. PACS: 71.15.Dx; 72; 71.45.Gm Keywords: Electrical resistivity; Pseudopotential; Exchange and correlation
1. Introduction The Ziman’s nearly free electron (NFE) theory has been fairly successful in describing the quantitative behavior of the electrical resistivity in simple liquid metals. This is because in these metals the mean free path is about one hundred times the interatomic distance and the weak scattering picture should be valid. Even for the heavy polyvalent metals (e.g. mercury, thallium and lead) where the mean free path is only about two interatomic distances, the NFE model can E-mail address: jay [email protected] (J.K. Baria).
yield results, which are in reasonable in agreement with experiments. Calculations of electrical resistivity using structure factor from various experiments or different versions of bare ion potential and dielectric function, gives correct order of magnitude but differ among themselves. In fact sometimes the theory is trusted sufficiently to use measured value of resistivity to determine the parameter of the potential. The Ziman’s formula is not expected to apply to d-state transition metals because the unfilled d-state cause strong resonant scattering which seems inappropriate for description by pseudopotential. Nevertheless, Evans et al. [1] put forward a version of equation of resistivity
0921-4526/03/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0921-4526(03)00410-1
ARTICLE IN PRESS J.K. Baria / Physica B 337 (2003) 245–248
246
in which jW ðqÞj2 was replaced by exact value of the squared matrix element for the scattering of a plane wave by a transition metal ion. The potential of the latter was taken to be Muffin-Tin potential as derived in solid state physics for band structure calculations and the exact scattering can be calculated by the phase shift method.
2. Electrical resistivity of liquid metals For both liquid and amorphous metals the electrical resistivity is given by [1,2] rZiman ¼
3pm2 4Ze2 _3 nkF6 Z N q3 SðqÞjW ðqÞj2 Yð2kF qÞ dq:
ð1Þ
0
Here, SðqÞ is the structure factor. In the present work we have used the charged hard sphere (CHS) [3] structure factor. The structure factor derived through CHS has an advantage that it involves the pseudopotential form factor. In this sense it differs from the Percus–Yevick (PY) [4] theory. So the structure factor derived through CHS gives the better explanation than PY theory. Recently Baria and Jani [5,6] have proposed model potential which is quite successful in describing lattice mechanical properties of d and f-shell metals is used in the present calculation of electrical resistivity. The unit step function Y which cuts off the integration at 2kF corresponding to a preferably sharp Fermi surface, is defined as Yð2kF qÞ ¼ 0 for q > 2kF ; ¼ 1 for qp2kF :
ð2Þ
But the finite mean free path corresponds to a finite uncertainty in the electron position. This corresponds to a finite uncertainty in the electron momentum. Thus the Fermi surface is not perfectly sharp as implied by Eq. (1) but it is blurred. The attempts to take this blurring in to account in the formulation of resistivity is reviewed by Mc Caskill and March [7]. Ferraz
and March [8] approach yields in place of Eq. (1). rsc ¼
3pm2 4Ze2 _3 nkF6 Z N q4 SðqÞjW ðqÞj2 Gðq; kF ; lÞ dq:
ð3Þ
0
This equation must be self-consistently solved. Very few explicit approximations are proposed for the function Gðq; kF ; lÞ [7]. In this work we have used the form for Gðq; kF ; lÞ as used by Laakkonen and Nieminen [9] and Khajil and Tomak [10] 2 Gðq; kF ; lÞ ¼ 3 tan1 ðqlÞ pq 1 2ql tan1 2 1 þ 4ðkF lÞ2 ðqlÞ2 p 1 2 Y q 2 þ 4kF : ð4Þ 2 l The mean free path is determined self-consistently as follows. The first step of the selfconsistency loop is to calculate resistivity using Eq. (1) i.e. with mean free path ‘‘l’’ is infinity. A new mean free path ‘‘l’’ is then obtained from the Drude relation as rL ¼
_kF : ne2 l
ð5Þ
The iterations are continued till rL converges.
3. Resistivity (t-matrix) approach We are concerned in this section with the scattering occurring in liquid metals. For example, a flux of electrons is incident on a scattering center and it is important to know the number of electrons that are scattered in a definite direction. This problem is most conveniently discussed using phase analysis. This analysis will throw much additional light on the use of pseudopotential see for more detail [11,12]. For many purposes a detailed knowledge of how the electronic wave functions behave in the neighborhood of the perturbing potential is not required, it is only their asymptotic form that matters. This, of course, is completely determined by phase shifts. Since,
ARTICLE IN PRESS J.K. Baria / Physica B 337 (2003) 245–248 Table 1 Presently calculated values of phase shift of some liquid d and f-shell metals Metal
Z0
Z1
Z2
Cu Ag Au Ni Pd Pt Rh Ir La Yb Ce Th
0.7374 0.6649 0.6141 0.9458 0.7780 0.7630 0.8184 0.8013 0.6618 0.6466 0.6488 0.6381
0.2157 0.2444 0.2540 0.3344 0.3762 0.3796 0.3663 0.3706 0.1908 0.1852 0.1789 0.2264
0.0319 0.0323 0.0360 0.0637 0.0732 0.0741 0.0708 0.0718 0.0138 0.0791 0.1214 0.1672
when the potential is due to something as small as metallic ions, the phase shifts are normally insignificant beyond say Z3 therefore Z0 ; Z1 and Z2 are to be calculated. The phase-shift Z1 of the lth partial wave related to the pseudopotential is given by Z mkF O0 2 Zl ¼ W ðyÞyPl ðcos yÞ dy; ð6Þ 4p_2 0 where y ¼ q=2kF and Pl ðcos yÞ is a Legendre polynomial and 1 q 2 cos y ¼ 1 : 2 kF Using the above equation phase-shift Z0 ; Z1 ; Z2 are calculated for s, p and d components and are shown in Table 1. Note that the low-energy phaseshift obeys the rule Z0 > Z1 > Z2 : The larger the phase-shift, the stronger the scattering. Evidently it is liable to be especially strong if the energy of the incident electron happens to coincide with that of a virtual bound state. In the case of noble and transition metals, the NFE approach is not appropriate due to presence the presence of a d band in the conduction band, which complicates the picture. We use the t-matrix of the pseudopotential to calculate the scattering cross-section rather than the usual screened ion model potentials. The single scatter t-matrix form factor for energy conserving transition is given by
247
Evans [13] and Evans et al. [14] 2p_3 1 X ð2l þ 1Þsin Z1 ðEF Þ tðk~; k~0 Þ ¼ 1=2 O 0 1 mð2mEÞ exp ½iZ1 ðEF Þ P1 ðcos yÞ:
ð7Þ
The t-matrix has the dimensions of energy and is normalized to the atomic volume. The resistivity takes the form with y ¼ q=2kF Z 1 3pO0 ~ ~0 2 3 rt ¼ 4y SðqÞ tð k ; k Þ ð8Þ dy: e2 _2 v2F 0
4. Results and discussion The resistivity calculated using Ziman’s formula, self-consistent approach and t-matrix formulations are tabulated in Table 2 with the experimental and other such theoretical findings. Daver et al. [15] have calculated resistivity and the mean free path using self-consistent method of about twenty liquid metals inclusive noble metals. They have used the Ashcroft’s empty-core pseudopotential, the parameter of the potential were chosen to yield the best agreement with the experimental structure factor. As they have fitted the potential parameter their results are very close to the experimental findings for simple metals while for Cu and Ag it deviates little. Ononiwu [16] has calculated the resistivity of all 3d, 4d and 5d transition metals using Ziman and t-matrix formula. Their t-matrix results are little close to the experimental findings for Ag, Au, Ni and Pd while Ziman’s formula gives very high values for Au, Pt and Rh. Gorecki and Popielawski [19] have calculated resistivity using the self-consistent quasi-crystalline approximation. Their results are found very poor. In our present investigations, the resistivity calculations using Ziman’s formula are very inferior for all d and f-shell metals, selfconsistent results for Cu and Au are little close to the experimental findings while for t-matrix formulation result shows an excellent agreement with the experimental findings for all d and f-shell liquid metals moreover, it shows significant improvements in the previously reported [16–19] theoretical results. The experimental data for Rh,
ARTICLE IN PRESS J.K. Baria / Physica B 337 (2003) 245–248
248
Table 2 Electrical resistivity of some liquid d and f-shell metals Metal
Cu Ag Au Ni Pd Pt Rh Ir La Yb Ce Th
Present work
rExpt
Others
l in (au)
rsc
rTmat
rZiman
[15,16]
[15,16]
[17]
[17]
[18]
67.96 63.99 49.39 48.33 148.85 199.65 110.50 178.12 133.92 139.02 176.00 172.18
19.85 26.97 34.95 55.36 65.99 60.62 94.74 70.17 74.37 90.17 89.13 108.23
20.05 17.55 32.24 81.78 84.30 99.75 72.47 105.85 147.58 118.56 127.17 137.21
30.78 9.96 41.26 41.02 41.07 44.83 50.10 96.64 332.07 177.02 267.63 186.47
20 19 31 85 83 90 — — 140 110 125 —
26.7 28.5 36.2 54.9 58.1 94.6 — — 165 137 134 —
10 20 27 87 79 22 96 90 — — — —
19 40 108 27 16 295 196 95 — — — —
21.0 21.7 21.4 41.4 36.0 62.0 — — — — — —
Ir and Th are not available so our results are predictive in nature. Here we mention clearly that we have not fitted the parameter of the potential either with the resistivity or with the structure facture. It is strongly emphasis previously by Esposito et al. [20] that the Ziman’s formula describes electrical transport correctly for simple liquid metals while self-consistent results of resistivity are over estimates if mean free paths are comparable or smaller than an interatomic distances, which happens in the d and f-shell metals. To the best of our knowledge after 1993 no one has reported the resistivity calculations of liquid d and f-shell metals using any method viz, Ziman’s formula, self-consistent approach and t-matrix formula. The present investigations confirm that the t-matrix formulations are the better chose for the resistivity calculations of d and f-shell liquid metals. References [1] R. Evans, D.A. Greenwood, P. Lloyd, Phys. Lett. A 35 (1971) 57.
[2] O. Dreierach, R. Evans, H.J. Guntherodt, H.U. Kunzi, J. Phys. F 2 (1972) 709. [3] H.B. Singh, A. Holz, Phys. Rev. A 24 (1983) 1108. [4] J.K. Percus, G.J. Yevick, Phys. Rev. 110 (1958) 1. [5] J.K. Baria, A.R. Jani, Physica B 328 (2003) 317. [6] J.K. Baria, A.R. Jani, Pramana J. Phys. 60 (2003) 1. [7] J.S. Mc Caskili, N.H. March, Phys. Chem. Liquids 12 (1982) 1. [8] A. Ferraz, N.H. March, Phys. Chem. Liquids 8 (1979) 271. [9] J. Laakkonen, R.M. Nieminen, J. Phys. F 13 (1983) 2265. [10] T.M.K. Kahjil, M. Tomak, Phys. Stat. Sol. B 134 (1986) 321. [11] T.E. Faber, Theory of Liquid Metals, Cambridge University Press, Cambridge, 1972, p. 30. [12] L.I. Yastrebov, A.A. Katsnelson, Foundation of One Electron Theory of Solids, Mir Publishers, Moscow, 1987, p. 42. [13] R. Evans, J. Phys. C (Met. Phys. Suppl.) 2 (1970) 137. [14] R. Evans, B.L. Gyorffy, N. Szabo, J.M. Ziman, in: S. Takeuchi (Ed.), Properties of Liquid Metals, Taylor and Francis, London, 1973. [15] F. Daver, T.M.A. Khajil, M. Tomak, Phys. Stat. Sol. B 138 (1986) 373. [16] J.S. Ononiwu, Phys. Stat. Sol. B 177 (1993) 413. [17] Y. Waseda, A. Jain, S. Tamaki, J. Phys. F. 8 (1978) 125. [18] K. Hirata, Y. Waseda, A. Jain, R. Srivastava, J. Phys. F. 7 (1977) 419. [19] J. Gorecki, J. Popielawski, J. Phys. F. 13 (1983) 2107. [20] E. Esposito, H. Ehrenreich, C.D. Gelatt Jr., Phys. Rev. B 18 (1978) 3913.