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The effect of small changes in the effective ionic radius on the properties of liquid lithium at the melting point
Volume
109A, number
7
PHYSICS
lOJune
LETTERS
THE EFFECT OF SMALL CHANGES IN THE EFFECTIVE IONIC RADIUS ON THE PROPERTIES OF LIQUID LITHIUM AT THE MELTING POINT J.M. GONZALEZ Deportamento Received
MIRANDA
de Termologio,
14 February
Facultad de Fisico, Universidod de Barcelona, Diagonal 645, 08028 Barcelona, Spain
1985; revised manuscript
received
15 April 1985; accepted
The sensitivity of the radial distribution function and the self-diffusion effective ionic radius of liquid lithium has been investigated by the method self-diffusion constant on this parameter has been observed.
In a series of papers Gonzalez Miranda and Torra [ I] and Gonzalez Miranda [2,3] have shown that the pseudopotential of Ashcroft [4] and the dielectric function of Geldart and Vosko [S] allow us to obtain reliable pair potentials for Li and Na when a BornMayer core [6] is used to approximate the core overlap interaction. This is achieved by means of a careful selection of the parameter R which represents an effective ionic radius for the various interactions involved. This parameter enters as input in the pseudopotential model and in the Born-Mayer term. In this paper we investigate the sensitivity of the properties of the liquid on these parameter of the potential. We consider here three pair potentials for Li near its melting point computed from values of R which differ only by a small quantity. In their calculation we have used the Born-Mayer term and the above pseudopotential and dielectric function. For details on the potential model and its calculation the reader is refered to ref. [3 1. The parameters involved are: the valence of the ions, 2 = 1; the number of electron states in the most external shell of the ions, n = 2; the specific volume of the conduction electron gas, u = 151 .I au, and the effective ionic radius R. Three values have been considered for this parameter R = 1.27 au, R = 1.29au and R = 1.30 au, giving rise to three different pair potentials which will be labeled respectively I_iA, LiB and LiC throughout the paper. They have been drawn in fig. 1. There we observe an identical long range behavior for all three potentials 0.375-9601/8.5/$ 03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
for publication
18 April 1985
constant on changes of the order of 1% in the of molecular dynamics. A strong sensitivity of the
and only small differences on the position and hardness of the repulsion. A molecular dynamics run has been performed for each one of the above pair potentials in a state near to the melting point. That is at a density of 0.515 g cmp3 and a temperature of 455 K. We have solved the classical equations of motion for a system of 686 particles subject to periodic boundary conditions. The integration algorithm used was the predictor-corrector of Rahman [7] with a time step 1 = 0.24 X IO- l4 s. A series of 1500 iterations, with the system in equilibrium, was computed for each potential. The statistics obtained with the above parameters is similar to that reported by other authors in anlogous computer simulation studies of liquid metals [8-lo], and the statistical errors in the quantities computed are of the same order of magnitude as the errors of the same quantities when obtained from experiment. The influence of R on the structure of the liquid has been investigated by means of the radial distribution function, g(r). It is easily computable from a molecular dynamics run and contains the same information as the static structure factor which is commonly obtained by means of X-ray or neutron diffraction experiments. Our results for g(r) are drawn in fig. 2. The three curves for g(r) are almost identical except for the heights of the first peak, whose values are displayed in table 1. We have also included there the relative differences between the heights of the maxima taken with respect to that of LiA together with the 333
Volume 109A, number 7
PHYSICS LETTERS
3.2
3.6
4.0
4n4 r
3
4
5
6
I
lOJune
In6
6
r
Snk
(adhl~
(ah.';1
II
5.6
li
6.0
13
6.4
I4
15
Fig. 1. Effective pair potentials for Li: (- - -) LiA, (0 * . ) LiB and (-) LE. In the inset the details of the potential walI are shown. (The unit of length equals 0.529 X lo-lo m, and that of energy 0.436 x 10-l’ J.)
corresponding values and relative differences for R. An estimate of the error on the values of the heights is the standard deviation of the g(r) points which is about 0.2 near the main peak. From table 1 we see that the relative differences between the heights of the peaks of g(r) are of the same order of magnitude 334
as the corresponding relative changes in the parameter R of the pair potential function. It seems then, that the effect of changing the ionic radius has an influence on the structure of the same order of the change performed. The influence of R on the diffusive behavior has
lOJune
PHYSICS LETTERS
Volume 109A, number 7
rkhud Fig. 2. Radial distribution functions for liquid Li at the melting point: (- - -) LiA, (* . 0) LiB and (-)
LiC.
Table 1 Ionic radii, R, and molecular dynamics results for the radial distribution function at the main peak, g, and for the self-diffusion constant, D, for the three pair potential functions considered in this paper. For these three quantities we have also included the relative differences with respect to the corresponding values for LiA, defined as AX/XA = 1X - XAl/XA for each of the quantities R,gandD.
Potential
R (au)
AMA
LiA LiB LiC
1.27 1.29 1.30
0.0 1.6 2.4
(%)
g(r)
&/&?A (%)
D (1O-5 cm2 s-l)
ADIDA (W)
2.52 2.56 2.60
0.0 1.7 3.2
8.01 * 0.21 7.20 i 0.18 6.32 * 0.21
0.0 10.1 21.1
335
Volume 109A, number 7
PHYSICS LETTERS
lOJune 1985
a
Fig.3. Mean square displacements for Li at the melting point: (- - -) LiA, ( . . .) LiB and (-) 0.103 x 10-14 s.) been investigated by means of the calculation of the self-diffusion constant D. It has been computed from the slope of the long time behavior of the mean square displacement r2(t). The three functions r2(t) computed from our molecular dynamics series are drawn in fig. 3. They spread widely showing a great influence of the effective ionic radius on the diffusive phenomena. The fitting of the calculated points for r2(t) to a straight line and the calculation of the error in the slope have been performed in the way described by Bevington [ 1 l] where the points are wheighted inversely to its statistical error. The values obtained for D together with their errors and relative differences with respect to the value for LiA are also displayed in table 1. The differences between each pair of values of D are much greater than the respective errors. We observe here that the increase in the ionic radii gives rise to a decrease in the values for the self-diffusion constant. From the comparison of the relative differences of D and those of R we see that the former 336
LiC. (The unit of time equals
are an order of magnitude greater than the latter. Then, it follows that small changes in R give rise to important changes in D. Finnally, two points deserve to be noted: (1) The relative changes for D observed here are of the same order of magnitude as those reported by Gonzalez Miranda and Torra [l] in a similar study performed for Na, where the sensitivity of the properties on the change of the theoretical model for the dielectric function was studied. Then, a carefull fit of the value of the effective ionic radius has the same importance as the selection of an accurate model for the dielectric function in order to obtain a reliable pair potential. (2) More frequently the values for effective ionic radii in pseudopotential theory are derived by fitting experimental values for data related to the structure of the ions in the liquid [ 121. From our results it follows that it will be better to select this value by means of a fit of atomic transport properties.
PHYSICS LETTERS
Volume 109A, number I
The calculations
reported in this paper have been
performed with the computer UMVAC 1100 of the “Centro de Proceso de Datos de1 Ministerio de Educacitjn
y Ciencia”
of the “Centro Politdcnica
through the DCT2000 terminal
de Calculo
de la Universidad
de Cataluiia”.
References [ 1 ] J.M. Gonzalez Miranda and V. Torra, J. Phys. F13 (1983) 281. [2] J.M. Gonzalez Miranda and V. Terra, Phys. Lett. 103A (1984) 126. [3] J.M. Gonzaez Miranda, Phys. Lett. 108A (1985) 35.
lOJune
[4] N.W. Asheroft, Phys. Lett. 23 (1966) 48. [5] D.J.W. Geldart and S.H. Vosko, Can. J. Phys. 44 (1966) 2137. [6] F. Seitz, Modern theory of solids (McGraw-Hill, New York, 1940). [7] A. Rahman, Phys. Rev. 159 (1964) 98. [8] T. Lee, J. B&chop, W. van der Lugt and W. van Gnnsteren, Physlca 93B (1978) 591. (91 R.D. Mountain and S.H. Haan, J. Res. N.B.S. 84 (1979) 439. [lo] I. Ebbsjo, T. Klnell and J. Wailer, J. Phys. Cl3 (1980) 1865. [ 111 P.R. Bevinton, Data reduction and error analysis for the physical sciences (McGraw-Hill, New York, 1969). 1121 M.L. Cohen and V. Helne, Solid State Phys. 24 (1970) 37.
337
109A, number
7
PHYSICS
lOJune
LETTERS
THE EFFECT OF SMALL CHANGES IN THE EFFECTIVE IONIC RADIUS ON THE PROPERTIES OF LIQUID LITHIUM AT THE MELTING POINT J.M. GONZALEZ Deportamento Received
MIRANDA
de Termologio,
14 February
Facultad de Fisico, Universidod de Barcelona, Diagonal 645, 08028 Barcelona, Spain
1985; revised manuscript
received
15 April 1985; accepted
The sensitivity of the radial distribution function and the self-diffusion effective ionic radius of liquid lithium has been investigated by the method self-diffusion constant on this parameter has been observed.
In a series of papers Gonzalez Miranda and Torra [ I] and Gonzalez Miranda [2,3] have shown that the pseudopotential of Ashcroft [4] and the dielectric function of Geldart and Vosko [S] allow us to obtain reliable pair potentials for Li and Na when a BornMayer core [6] is used to approximate the core overlap interaction. This is achieved by means of a careful selection of the parameter R which represents an effective ionic radius for the various interactions involved. This parameter enters as input in the pseudopotential model and in the Born-Mayer term. In this paper we investigate the sensitivity of the properties of the liquid on these parameter of the potential. We consider here three pair potentials for Li near its melting point computed from values of R which differ only by a small quantity. In their calculation we have used the Born-Mayer term and the above pseudopotential and dielectric function. For details on the potential model and its calculation the reader is refered to ref. [3 1. The parameters involved are: the valence of the ions, 2 = 1; the number of electron states in the most external shell of the ions, n = 2; the specific volume of the conduction electron gas, u = 151 .I au, and the effective ionic radius R. Three values have been considered for this parameter R = 1.27 au, R = 1.29au and R = 1.30 au, giving rise to three different pair potentials which will be labeled respectively I_iA, LiB and LiC throughout the paper. They have been drawn in fig. 1. There we observe an identical long range behavior for all three potentials 0.375-9601/8.5/$ 03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
for publication
18 April 1985
constant on changes of the order of 1% in the of molecular dynamics. A strong sensitivity of the
and only small differences on the position and hardness of the repulsion. A molecular dynamics run has been performed for each one of the above pair potentials in a state near to the melting point. That is at a density of 0.515 g cmp3 and a temperature of 455 K. We have solved the classical equations of motion for a system of 686 particles subject to periodic boundary conditions. The integration algorithm used was the predictor-corrector of Rahman [7] with a time step 1 = 0.24 X IO- l4 s. A series of 1500 iterations, with the system in equilibrium, was computed for each potential. The statistics obtained with the above parameters is similar to that reported by other authors in anlogous computer simulation studies of liquid metals [8-lo], and the statistical errors in the quantities computed are of the same order of magnitude as the errors of the same quantities when obtained from experiment. The influence of R on the structure of the liquid has been investigated by means of the radial distribution function, g(r). It is easily computable from a molecular dynamics run and contains the same information as the static structure factor which is commonly obtained by means of X-ray or neutron diffraction experiments. Our results for g(r) are drawn in fig. 2. The three curves for g(r) are almost identical except for the heights of the first peak, whose values are displayed in table 1. We have also included there the relative differences between the heights of the maxima taken with respect to that of LiA together with the 333
Volume 109A, number 7
PHYSICS LETTERS
3.2
3.6
4.0
4n4 r
3
4
5
6
I
lOJune
In6
6
r
Snk
(adhl~
(ah.';1
II
5.6
li
6.0
13
6.4
I4
15
Fig. 1. Effective pair potentials for Li: (- - -) LiA, (0 * . ) LiB and (-) LE. In the inset the details of the potential walI are shown. (The unit of length equals 0.529 X lo-lo m, and that of energy 0.436 x 10-l’ J.)
corresponding values and relative differences for R. An estimate of the error on the values of the heights is the standard deviation of the g(r) points which is about 0.2 near the main peak. From table 1 we see that the relative differences between the heights of the peaks of g(r) are of the same order of magnitude 334
as the corresponding relative changes in the parameter R of the pair potential function. It seems then, that the effect of changing the ionic radius has an influence on the structure of the same order of the change performed. The influence of R on the diffusive behavior has
lOJune
PHYSICS LETTERS
Volume 109A, number 7
rkhud Fig. 2. Radial distribution functions for liquid Li at the melting point: (- - -) LiA, (* . 0) LiB and (-)
LiC.
Table 1 Ionic radii, R, and molecular dynamics results for the radial distribution function at the main peak, g, and for the self-diffusion constant, D, for the three pair potential functions considered in this paper. For these three quantities we have also included the relative differences with respect to the corresponding values for LiA, defined as AX/XA = 1X - XAl/XA for each of the quantities R,gandD.
Potential
R (au)
AMA
LiA LiB LiC
1.27 1.29 1.30
0.0 1.6 2.4
(%)
g(r)
&/&?A (%)
D (1O-5 cm2 s-l)
ADIDA (W)
2.52 2.56 2.60
0.0 1.7 3.2
8.01 * 0.21 7.20 i 0.18 6.32 * 0.21
0.0 10.1 21.1
335
Volume 109A, number 7
PHYSICS LETTERS
lOJune 1985
a
Fig.3. Mean square displacements for Li at the melting point: (- - -) LiA, ( . . .) LiB and (-) 0.103 x 10-14 s.) been investigated by means of the calculation of the self-diffusion constant D. It has been computed from the slope of the long time behavior of the mean square displacement r2(t). The three functions r2(t) computed from our molecular dynamics series are drawn in fig. 3. They spread widely showing a great influence of the effective ionic radius on the diffusive phenomena. The fitting of the calculated points for r2(t) to a straight line and the calculation of the error in the slope have been performed in the way described by Bevington [ 1 l] where the points are wheighted inversely to its statistical error. The values obtained for D together with their errors and relative differences with respect to the value for LiA are also displayed in table 1. The differences between each pair of values of D are much greater than the respective errors. We observe here that the increase in the ionic radii gives rise to a decrease in the values for the self-diffusion constant. From the comparison of the relative differences of D and those of R we see that the former 336
LiC. (The unit of time equals
are an order of magnitude greater than the latter. Then, it follows that small changes in R give rise to important changes in D. Finnally, two points deserve to be noted: (1) The relative changes for D observed here are of the same order of magnitude as those reported by Gonzalez Miranda and Torra [l] in a similar study performed for Na, where the sensitivity of the properties on the change of the theoretical model for the dielectric function was studied. Then, a carefull fit of the value of the effective ionic radius has the same importance as the selection of an accurate model for the dielectric function in order to obtain a reliable pair potential. (2) More frequently the values for effective ionic radii in pseudopotential theory are derived by fitting experimental values for data related to the structure of the ions in the liquid [ 121. From our results it follows that it will be better to select this value by means of a fit of atomic transport properties.
PHYSICS LETTERS
Volume 109A, number I
The calculations
reported in this paper have been
performed with the computer UMVAC 1100 of the “Centro de Proceso de Datos de1 Ministerio de Educacitjn
y Ciencia”
of the “Centro Politdcnica
through the DCT2000 terminal
de Calculo
de la Universidad
de Cataluiia”.
References [ 1 ] J.M. Gonzalez Miranda and V. Torra, J. Phys. F13 (1983) 281. [2] J.M. Gonzalez Miranda and V. Terra, Phys. Lett. 103A (1984) 126. [3] J.M. Gonzaez Miranda, Phys. Lett. 108A (1985) 35.
lOJune
[4] N.W. Asheroft, Phys. Lett. 23 (1966) 48. [5] D.J.W. Geldart and S.H. Vosko, Can. J. Phys. 44 (1966) 2137. [6] F. Seitz, Modern theory of solids (McGraw-Hill, New York, 1940). [7] A. Rahman, Phys. Rev. 159 (1964) 98. [8] T. Lee, J. B&chop, W. van der Lugt and W. van Gnnsteren, Physlca 93B (1978) 591. (91 R.D. Mountain and S.H. Haan, J. Res. N.B.S. 84 (1979) 439. [lo] I. Ebbsjo, T. Klnell and J. Wailer, J. Phys. Cl3 (1980) 1865. [ 111 P.R. Bevinton, Data reduction and error analysis for the physical sciences (McGraw-Hill, New York, 1969). 1121 M.L. Cohen and V. Helne, Solid State Phys. 24 (1970) 37.
337