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One-component plasma structure factor of liquid alkali metals
Volume 110A, number 7,8
PHYSICS LETTERS
12 August 1985
O N E - C O M P O N E N T PLASMA S T R U C T U R E F A C T O R OF L I Q U I D ALKALI M E T A L S J.L. B R E T O N N E T and K.N. K H A N N A 1 Laboratoire de Physique des Liquides Mbtalliques, Universitb de Metz, 57000 Metz, France Received 1 April 1985; revised manuscript received 29 May 1985; accepted for publication 31 May 1985 -
An analytical form of the direct correlation function is proposed to determine the structure factor of liquid alkali metals in the OCP reference system. Assuming that the interionic interaction can be modeled by the effective pair potential in the model potential approach, the structure factor is calculated using random phase approximation. It works well up to 2k v, which we need to calculate the electronic transport properties, showing thus the significant role played by the electron gas response function.
The classical one-component plasma is an idealized system of ions immersed in a uniform sea of electrons so that the whole system should be electrically neutral. The plasma parameter P, determining the role of the Coulomb interaction in this system, is defined by P = (jZ2e2/a, where the ion sphere radius a = (3/4zrp) 1/3 is expressed in terms of the number density p. This system is supposed to be a better reference system than a hard-sphere fluid in the case of alkali metals because of their softer interionic potential at short range. Though recent Monte Carlo simulations [1-3] provide a basis to calculate the structure factor, in practice these calculations require excessive computer time. Thus we need an analytical form of the direct correlation function to describe the OCP structure factor, convenient for calculation as for the hardsphere system. Some attempts [4-6] have been made to develop such an analytical function for the structure factor. Chaturvedi et al. [5] obtained an approximate agreement between the calculated and the measured values of the first peak height in S(q). Evans and Sluckin [6] have calculated S ( q ) in the longwavelength limit by using the correlation function suggested by Baus and Hansen [4]. In the present communication we examine a simple analytical expression to determine the structure factor of liquid alkalis in the Baus and Hansen scheme combined with the random phase approximation (RPA). We
also demonstrate the usefulness of the OCP as a reference system for liquid alkalis by presenting the numerical examples of lithium and sodium. To our knowledge, RPA is a simple but effective approach to describe the structure factor of alkali metals in the OCP reference system. We now consider the analytical function for the direct correlation function suggested by Baus and Hansen [4] P
COCp(r)=/=~ 0.= air2i, _
[3Z2e 2 r ,
r
r>r 0 ,
(1)
where the coefficients a i are determined by requiring C o c p ( r ) and its first p derivatives to be continuous at the radius r 0 which is a scaling parameter. The authors pointed out that p = 3 yields the best results for the internal energies compared with HNC or MC data. With their choice the direct correlation function becomes ~Z2e 2
Cocp(r ) = 16r-----7- [ - 35 + 35(r/ro)2 - 21(r/ro)4
-
+5(r/r0)6],
r<~r 0 ,
[3Z2e2
r >1 r 0 .
(2)
r On leave from Kanpur University, Kanpur, India.
420
Having now an explicit form for C o c p ( q ) obtained 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 110A, number 7,8
PHYSICS LETTERS
by using the Fourier transform of Cocp(r), the simplest approximation to take the electron gas into account in calculating the static structure factor is the random phase approximation which can be written as
S(q) = {1 - p [ C o c p ( q ) - flUsc(q)] }-1 ,
(3)
where Usc(q ) is the indirect interaction between the ions which can be considered as a perturbation to the direct Coulomb repulsion. It comes from the electronic screening and its expression in terms of the normalized energy-wavenumber characteristic FN(q) is Usc(q) -
4nZ2e 2 FN(q) • q2
(4)
Combining eq. (4) with the Fourier transform of eq. (2), it follows that O [ C o c e ( q ) -/3Use(q)] = - ( 3 P x 2 / X 2) {(105/X 4) × [(15/X 2 - 6) sin(X)/X - (15/X 2 - 1) cos(X)] -
FN(q)),
(5)
where X = qr 0 and x 0 = ro/a are two dimensionless parameters. We have to point out that the choice of FN(q ) plays a significant role in the determination of the structure factor in the OCP model. Thus we need a convenient model potential and a suitable dielectric function. We therefore construct FN(q) by using a non-local model potential of Shaw following the same way as Bretonnet and Regnaut [7]. We also mention that a comparative examination of the electron-gas response function of Vashishta and Singwi (VS) and Ichimaru and Utsumi (IU) is carried out by constructing FN(q) with both VS and IU dielectric functions. The zero Fourier transform of PCocp(q) [eq. (5)] is the same as the one obtained by Evans and Sluckin
[6],
12 August 19 85
ordinary fluids where C(q) tends to a finite value at low q. However it has been found that FN(q) presents a similar divergence to C o c p ( q ) , for q = 0, exactly compensating it to give a regular behaviour of longwavelength limit. By using the Ashcroft model potential, S(0) is
x 2 + (3r/a2)(r 2+ 7r/4k v - Yolk 2) s - l ( 0 ) = I - gl~x0
(7) where r e is Ashcroft's parameter and 3,0 is a dimensionless parameter connected to the local-field correction G(q), for instance with the VS dielectric function 3,0 = AB where A and B are defined by eq. (41) o f Vashishta and Singwi [8], whereas with the IU dielectric function 70 corresponds to eq. (4) from Ichimaru and Utsumi [9]. Owing to a novel HNC compressibility equation, Baus and Hansen [4] have determined the x 0 parameter as a function of F. The relation between x 0 and P is complicated but it happens that, for the p = 3 model and for large P, x 0 saturates around x 0 = 1.8, indicating that x 0 and P are weakly coupled; thus the choice of P is to fit the height and position o f the first peak o f the structure factor. Numerical calculations carried out by using the non-local model potential of Shaw [10] combined with the IU dielectric function are displayed in fig. 1 for lithium and sodium (P = 117 for Li and P = 184 for Na). We f'md that both the position and height o f the first peak can be
!
|
~(q~
2o
!
I~
-
oI
;)/°
1.12
oCocp(q o)
~-
.:
,Z /
= - P x 0 2 [3/X 2 - 1/6 + X2/264 + O(X4)] .
(6)
It shows the singular feature of the behaviour o f the direct correlation function for small q, which reflects the long range of the Coulomb potential. This is a typical OCP feature which is not encountered in
I,o
[ 2.0[ 2kdNo) 2k~-CLi)
I
3.0
n¢~
Fig. 1. The structure factor S(q) of lithium (full curve) and sodium (dotted curve); compared with the X-ray diffraction data of Waseda [ 13 ], dots Li, open circles Na. 421
Volume 110A, number 7,8
PHYSICS LETTERS
fitted with good accuracy to the experimental value, contradicting the statement of Chaturvedi et al. [5]. It is very clear that F is mainly responsible for the height of the peak, whereas the role of FN(q) is more dominant in the low,q region. The effect of P, almost insensitive in the low-q region, seems to be localized in a narrow domain around the first peak. In addition we can see that the IU dielectric function gives the values of In well below or close to the value at which the bare OCP freezes [11], i.e. P = 168. Careful observations show that the IU dielectric function is more appropriate than the VS one in the OCP system as already observed in the hard-sphere reference system, particularly in the low,q region [12]. For instance, if we fit exactly the same height of the first peak in both the dielectric functions, poor agreement of S(0) with experiment is found for the VS dielectric function in comparison to the IU one. On the other hand fitting S(0) by means of eq. (4), higher values of F are obtained with the VS dielectric function than with the IU dielectric function. With the IU exchange and correlation correction our results for the low-q region are also very close to the experimental values, predicting thereby that the OCP model presents a more realistic description of liquid alkalis than the hard-sphere system. The case of Li is particularly interesting. It may be due to the fact that the OCP model is better suited to lithium because o f the small ionic radius. The semi-analytic procedure is an efficient tool for deriving the structure factor of alkali metals, especially up to the first peak situated beyond the 2k F value which we need in the calculation of several properties like electronic transport properties. However, our calculations do not yield very statisfactory results beyond the First peak. It is a matter of fact that
422
12 August 1985
the amplitude of the oscillations in S(q) beyond the first peak are rapidly damped indicating that this model corresponds to softer repulsive core in the pair potential. It is unlikely that the discrepancies are due either to the model potential or to the electron-gas response function because FN(q) which contains all this information decays quickly beyond 2k F and is without influence upon this range of S(q). Further investigations are in progress to find the reason for this failure which might be due to the choice of various parameters or to the defect in Baus and Hansen scheme. One of us (K.N.K.) acknowledges partial support from the French Government in high level scientific fellowship.
References [1] S. Gatam and J.P. Hansen, Phys. Rev. A14 (1976) 816. [2] F.J. Rogers, D.A. Young, H.E. Dewitt and M. Ross, Phys. Rev. A28 (1983) 2990. [3] F.J. Rogers and D.A. Young, Phys. Rev. A30 (1984) 999. [4] M. Baus and J.P. Hansen, J. Phys. C12 (1979) L55. [5] D.K. Chaturvedi, M. Rovere, G. Senatore and M.P. Tosi, Physica l l l B (1981) 11. [6] R. Evans and T.J. Sluckin, J. Phys. C14 (1981) 3137. [7] J.L. Bretonnet and C. Regnaut, Phys. Rev. B (1985), to be published. [8] P. Vashishta and K.S. Singwi, Phys. Rev. B6 (1972) 875. [9] S. Ichimaru and K. Utsumi, Phys. Rev. B24 (1981) 7385. [10] R.W. Shaw and W.A. Harrison, Phys. Rev. 163 (1967) 604. [11] W.L. Slattery and G.D. Doolen, Phys. Rev. A21 (1980) 2087. [12] J.L. Bretonnet and C. Regnant, J. Phys. F14 (1984) L59. [13] Y. Waseda, The structure of non-crystalline materials (McGraw-Hill, New York, 1980).
PHYSICS LETTERS
12 August 1985
O N E - C O M P O N E N T PLASMA S T R U C T U R E F A C T O R OF L I Q U I D ALKALI M E T A L S J.L. B R E T O N N E T and K.N. K H A N N A 1 Laboratoire de Physique des Liquides Mbtalliques, Universitb de Metz, 57000 Metz, France Received 1 April 1985; revised manuscript received 29 May 1985; accepted for publication 31 May 1985 -
An analytical form of the direct correlation function is proposed to determine the structure factor of liquid alkali metals in the OCP reference system. Assuming that the interionic interaction can be modeled by the effective pair potential in the model potential approach, the structure factor is calculated using random phase approximation. It works well up to 2k v, which we need to calculate the electronic transport properties, showing thus the significant role played by the electron gas response function.
The classical one-component plasma is an idealized system of ions immersed in a uniform sea of electrons so that the whole system should be electrically neutral. The plasma parameter P, determining the role of the Coulomb interaction in this system, is defined by P = (jZ2e2/a, where the ion sphere radius a = (3/4zrp) 1/3 is expressed in terms of the number density p. This system is supposed to be a better reference system than a hard-sphere fluid in the case of alkali metals because of their softer interionic potential at short range. Though recent Monte Carlo simulations [1-3] provide a basis to calculate the structure factor, in practice these calculations require excessive computer time. Thus we need an analytical form of the direct correlation function to describe the OCP structure factor, convenient for calculation as for the hardsphere system. Some attempts [4-6] have been made to develop such an analytical function for the structure factor. Chaturvedi et al. [5] obtained an approximate agreement between the calculated and the measured values of the first peak height in S(q). Evans and Sluckin [6] have calculated S ( q ) in the longwavelength limit by using the correlation function suggested by Baus and Hansen [4]. In the present communication we examine a simple analytical expression to determine the structure factor of liquid alkalis in the Baus and Hansen scheme combined with the random phase approximation (RPA). We
also demonstrate the usefulness of the OCP as a reference system for liquid alkalis by presenting the numerical examples of lithium and sodium. To our knowledge, RPA is a simple but effective approach to describe the structure factor of alkali metals in the OCP reference system. We now consider the analytical function for the direct correlation function suggested by Baus and Hansen [4] P
COCp(r)=/=~ 0.= air2i, _
[3Z2e 2 r ,
r
r>r 0 ,
(1)
where the coefficients a i are determined by requiring C o c p ( r ) and its first p derivatives to be continuous at the radius r 0 which is a scaling parameter. The authors pointed out that p = 3 yields the best results for the internal energies compared with HNC or MC data. With their choice the direct correlation function becomes ~Z2e 2
Cocp(r ) = 16r-----7- [ - 35 + 35(r/ro)2 - 21(r/ro)4
-
+5(r/r0)6],
r<~r 0 ,
[3Z2e2
r >1 r 0 .
(2)
r On leave from Kanpur University, Kanpur, India.
420
Having now an explicit form for C o c p ( q ) obtained 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 110A, number 7,8
PHYSICS LETTERS
by using the Fourier transform of Cocp(r), the simplest approximation to take the electron gas into account in calculating the static structure factor is the random phase approximation which can be written as
S(q) = {1 - p [ C o c p ( q ) - flUsc(q)] }-1 ,
(3)
where Usc(q ) is the indirect interaction between the ions which can be considered as a perturbation to the direct Coulomb repulsion. It comes from the electronic screening and its expression in terms of the normalized energy-wavenumber characteristic FN(q) is Usc(q) -
4nZ2e 2 FN(q) • q2
(4)
Combining eq. (4) with the Fourier transform of eq. (2), it follows that O [ C o c e ( q ) -/3Use(q)] = - ( 3 P x 2 / X 2) {(105/X 4) × [(15/X 2 - 6) sin(X)/X - (15/X 2 - 1) cos(X)] -
FN(q)),
(5)
where X = qr 0 and x 0 = ro/a are two dimensionless parameters. We have to point out that the choice of FN(q ) plays a significant role in the determination of the structure factor in the OCP model. Thus we need a convenient model potential and a suitable dielectric function. We therefore construct FN(q) by using a non-local model potential of Shaw following the same way as Bretonnet and Regnaut [7]. We also mention that a comparative examination of the electron-gas response function of Vashishta and Singwi (VS) and Ichimaru and Utsumi (IU) is carried out by constructing FN(q) with both VS and IU dielectric functions. The zero Fourier transform of PCocp(q) [eq. (5)] is the same as the one obtained by Evans and Sluckin
[6],
12 August 19 85
ordinary fluids where C(q) tends to a finite value at low q. However it has been found that FN(q) presents a similar divergence to C o c p ( q ) , for q = 0, exactly compensating it to give a regular behaviour of longwavelength limit. By using the Ashcroft model potential, S(0) is
x 2 + (3r/a2)(r 2+ 7r/4k v - Yolk 2) s - l ( 0 ) = I - gl~x0
(7) where r e is Ashcroft's parameter and 3,0 is a dimensionless parameter connected to the local-field correction G(q), for instance with the VS dielectric function 3,0 = AB where A and B are defined by eq. (41) o f Vashishta and Singwi [8], whereas with the IU dielectric function 70 corresponds to eq. (4) from Ichimaru and Utsumi [9]. Owing to a novel HNC compressibility equation, Baus and Hansen [4] have determined the x 0 parameter as a function of F. The relation between x 0 and P is complicated but it happens that, for the p = 3 model and for large P, x 0 saturates around x 0 = 1.8, indicating that x 0 and P are weakly coupled; thus the choice of P is to fit the height and position o f the first peak o f the structure factor. Numerical calculations carried out by using the non-local model potential of Shaw [10] combined with the IU dielectric function are displayed in fig. 1 for lithium and sodium (P = 117 for Li and P = 184 for Na). We f'md that both the position and height o f the first peak can be
!
|
~(q~
2o
!
I~
-
oI
;)/°
1.12
oCocp(q o)
~-
.:
,Z /
= - P x 0 2 [3/X 2 - 1/6 + X2/264 + O(X4)] .
(6)
It shows the singular feature of the behaviour o f the direct correlation function for small q, which reflects the long range of the Coulomb potential. This is a typical OCP feature which is not encountered in
I,o
[ 2.0[ 2kdNo) 2k~-CLi)
I
3.0
n¢~
Fig. 1. The structure factor S(q) of lithium (full curve) and sodium (dotted curve); compared with the X-ray diffraction data of Waseda [ 13 ], dots Li, open circles Na. 421
Volume 110A, number 7,8
PHYSICS LETTERS
fitted with good accuracy to the experimental value, contradicting the statement of Chaturvedi et al. [5]. It is very clear that F is mainly responsible for the height of the peak, whereas the role of FN(q) is more dominant in the low,q region. The effect of P, almost insensitive in the low-q region, seems to be localized in a narrow domain around the first peak. In addition we can see that the IU dielectric function gives the values of In well below or close to the value at which the bare OCP freezes [11], i.e. P = 168. Careful observations show that the IU dielectric function is more appropriate than the VS one in the OCP system as already observed in the hard-sphere reference system, particularly in the low,q region [12]. For instance, if we fit exactly the same height of the first peak in both the dielectric functions, poor agreement of S(0) with experiment is found for the VS dielectric function in comparison to the IU one. On the other hand fitting S(0) by means of eq. (4), higher values of F are obtained with the VS dielectric function than with the IU dielectric function. With the IU exchange and correlation correction our results for the low-q region are also very close to the experimental values, predicting thereby that the OCP model presents a more realistic description of liquid alkalis than the hard-sphere system. The case of Li is particularly interesting. It may be due to the fact that the OCP model is better suited to lithium because o f the small ionic radius. The semi-analytic procedure is an efficient tool for deriving the structure factor of alkali metals, especially up to the first peak situated beyond the 2k F value which we need in the calculation of several properties like electronic transport properties. However, our calculations do not yield very statisfactory results beyond the First peak. It is a matter of fact that
422
12 August 1985
the amplitude of the oscillations in S(q) beyond the first peak are rapidly damped indicating that this model corresponds to softer repulsive core in the pair potential. It is unlikely that the discrepancies are due either to the model potential or to the electron-gas response function because FN(q) which contains all this information decays quickly beyond 2k F and is without influence upon this range of S(q). Further investigations are in progress to find the reason for this failure which might be due to the choice of various parameters or to the defect in Baus and Hansen scheme. One of us (K.N.K.) acknowledges partial support from the French Government in high level scientific fellowship.
References [1] S. Gatam and J.P. Hansen, Phys. Rev. A14 (1976) 816. [2] F.J. Rogers, D.A. Young, H.E. Dewitt and M. Ross, Phys. Rev. A28 (1983) 2990. [3] F.J. Rogers and D.A. Young, Phys. Rev. A30 (1984) 999. [4] M. Baus and J.P. Hansen, J. Phys. C12 (1979) L55. [5] D.K. Chaturvedi, M. Rovere, G. Senatore and M.P. Tosi, Physica l l l B (1981) 11. [6] R. Evans and T.J. Sluckin, J. Phys. C14 (1981) 3137. [7] J.L. Bretonnet and C. Regnaut, Phys. Rev. B (1985), to be published. [8] P. Vashishta and K.S. Singwi, Phys. Rev. B6 (1972) 875. [9] S. Ichimaru and K. Utsumi, Phys. Rev. B24 (1981) 7385. [10] R.W. Shaw and W.A. Harrison, Phys. Rev. 163 (1967) 604. [11] W.L. Slattery and G.D. Doolen, Phys. Rev. A21 (1980) 2087. [12] J.L. Bretonnet and C. Regnant, J. Phys. F14 (1984) L59. [13] Y. Waseda, The structure of non-crystalline materials (McGraw-Hill, New York, 1980).