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Extended Born-Green equation: A new approach for determining pair potentials in disordered systems
Volume 58A, number 6
4 October 1976
PHYSICS LETTERS
EXTENDED BORN-GREEN EQUATiON: A NEW APPROACH FOR DETERMINiNG PAIR POTENTIALS IN DiSORDERED SYSTEMS W. ABEL, R. BLOCK and W. SCHOMMERS Kernforschungszenrrum Karisruhe, Institut fur AngewandreKernphysik, Postfach 3640, D- 7500 Karisruhe, F.R.G Received 9 June 1976 Revised manuscript received 24 July 1976 The Born-Green equation was tested using molecular dynamics data for liquid rubidium at 319 K. The unsatisfactory results of this equation for the pair potential are caused by the insufficient precision of the Kirkwood approximation for describing the triplet correlation function g3(r, s, t). A more realistic ansatz forg 3(r, s, t) was employed in the formulation of an extended Born-Green equation. This new approach gives much better results for the potential than the simple Born-Green equation.
Within the framework of pair potential theory all significant physical properties (e.g. the thermodynamic functions) essentially depend on the shape of the potential used. This means that the knowledge of the pair potential is of fundamental interest in many-particle system analysis. In this paper a new approach of evaluating the pair potential from structure data of disordered systems will be discussed. We restrict ourselves to isotropic and homogeneous systems taking liquid rubidium as a reference. There are several attempts to determine the pair potential from the pair correlation function: (1) The self-consistent method, proposed in ref. [1] ,determines a pair potential by iteration using molecular dynamics. (2) The method of integral equations (Born-Green [2] Percus-Yevick [3] hypernetted chain [4]). Here we want to investigate the Born-Green (BG) equation using the quantities obtained by method (1). By this method a pair potential for liquid rubidium has already been determined, which describes well several experimental data (scattering law, diffusion constant, etc.) [5, 6] This potential will be used as a reference potential and will be designated as uR(r), r being the distance between two particles of the system. ,
,
.
The Born-Green equation. For isotropic and homogeneous systems provided that the potential energy can be expressed as a sum of pair potentials, a general relationship can be given between the pair potential v(r), the pair correlation function g(r) and the triplet correlation function g3(r, s, t) [2]: 2—?—s2)tdt, (1) = —~-lng(r)+-~ ds g3(r, s, 0 (t
7~
where r, s, t are the distance between three particles of the system, T represents the absolute temperature, kB the Boltzmann constant and p the particle number density. By substituting the Kirkwood approximation (KA) [7] g 3(r, s, t) g(r)g(s)g(t)
(2)
into eq. (1) one obtains the BG-equation. The usefulness of this BG.equation depends on the quality of the KA. For liquid rubidium g3(r, s, t) was determined by means of molecular dynamics [8] using the reference potential UR(r). The comparison of this g3 (r, s, t) with the KA shows deviations by up to 35% and the question arises how these deviations are reflected in the BG-potential. To answer this we substitute reference potential UR(r) and the molecular dynamics data for g(r) on the righthand side of the BG-equation. If the BG-equation is a good approach, we expect that the potential VBG(r) obtained from the lefthand side of the equation agrees with UR(r). However, 367
Volume 58A, number 1
PHYSICS LETTERS
4 October 1976
2.0 ‘y~r) vir) 1.5-1
~
1.92
: 1.0
05
:1 11
/I
Li’
:
1.60 1.26
0.32 0.5
0
~ 7~
N a:
25
1O.Or)A)
:\
5~U 7
100
0
-
0.32
;/•id.o~A]
~(‘\7.~ -
‘.
/1
r(Ai~
Fig. 3. Pair potentials for liqu~rubidium. Reference potentiaL ——— Result Fig. 2. The correction function 7(r). from the extended Born-Green equation - 1.5 Molecular dynamics results. This 7(r) using the molecular dynamics data for gives the best result for the pair potential 7(r) (in fig. 2 the broken line). Result Fig. 1. Pair potentials for liquid rubidium. from the extended Born-Green equation from the extended Born-Green equation — Reference potential. ——— Result from (the potential is shown in fig. 3 as a dotted using the correction function 7(r), which the simple Born-Green equation, line), is shown in fig. 2 as a dotted line. -
1.0
fig. 1 shows that these potentials are quite different from each other. The extended Born-Green equation. In order to get a better pair potential it is necessary to improve the BGequation. For this purpose we have taken the following ansatz [91 for g3(r, s, t) in eq. (1) g3(r, s, t) =g(r)g(s)g(t)
—
y(r)-y(s)’,’(t).
(3)
This ansatz was chosen because: (i) The correction function y(r) can be determined experimentally. There exists the following relationship between 7(r) and the measurable function H(r) [9] H(r) = 7(r) f7(r
—
s) y(s) d.c
(4)
H(r) can be expressed in terms of g(r) and [ag(r)/ap]~ H(r) =g(r) f[g(s)
—
1] [g(t)
—
1] d.c
—
k~T[ag(r)/ap]
~
(5)
In order to get H(r) one has to determine g(r) and g(r) as a function of pressure p at constant temperature T. (ii) The shapes of all calculated 7(r) are similar. Using eq. (3) and the molecular dynamic~data for g3(r, s, t) and g(r) one can determine directly 7(r). However, there are as many functions 7(r) as configurations exist for g3(r, s, t). But it has been demonstrated [10] that these functions obtained from different configurations [8] are very much the same over large r-ranges. 368
Volume 58A, number 6
PHYSICS LETTERS
4 October 1976
Fig. 2 shows the molecular dynamics results for a special 7(r) with the best statistics. It has been determined from the configuration g3(r, s = 4.7, t = 4.7) by simple arithmetics. r = 4.7 corresponds to the principal peak in g(r). The magnitude of 7(r = 4.7) is about 52% of the height ofg(r = 4.T~).Already with this 7(r) the extended BGequation eq. (3) inserted in fig. 1 leads to an improvement of the potential as compared with the results following from the simple BG-equation (eq. (2) in eq. (1)). By variation of 7(r) (using an interactive graphical method [12]) within its statistical error limits it is possible to generate a potential, which is in good agreement with the reference potential. Such a potential is shown in fig. 3. The corresponding function 7(r) has been plotted in fig. 2. From fig. 3 it can be seen that the extended BG-equation leads to much better results for the pair potential than the simple BG-equation. Even the details in the reference potential are reproduced very well. The BG.equation has been discussed for liquid rubidium near the melting point. The cause for the unsatisfactory results of this equation was the crudeness of the KA for describing the triplet correlation function. Therefore, the KA has been corrected by the function 7(r) and an extended BG-equation was proposed. An important reason for introducing 7(r) was that this function can be measured. In order to get 7(r) one has to know [ag(r)/ap]T which means that the pair correlation function g(r) has to be studied as a function of pressure p at constant temperature T [9, 11]. In conclusion, if one uses the method of integral equation (BG) to determine a pair potential, the knowledge of g(r) alone will not be sufficient. The expression [ag(r)/ap]~ ought to be added as an additional experimental information. It should be mentioned that the pair potential is very sensitive to variations in g(r) and 7(r). This implies thatg(r) and 7(r) must be determined very accurately in the experiment. Although the present calculations were made for liquid Rb, we believe that an extended BG-equation will give better results for all those systems in which large deviations of g3(r, s, t) from the KA are expected. These deviations will depend on the packing density, which is a function of density and temperature. Finally, it should be mentioned that the present paper is based on molecular dynamics data only and no iteration was needed in our analysis. The numerical problems which may arise in connection with the iterative solution of the extended equation are out of the scope of this paper. —
—
References [1] [2] [3] [4]
W. Schommers, Phys. Lett. A43 (1973) 157. N.H. March, in: Theory of condensed matter (IAEA, Vienna, 1968). J.K. Percus and G.J. Yevick,Phys. Rev. 110(1958)1. J.M.J. van Leeuwen, J. Groeneveld and .1 de Boer, Physica 25 (1959) 792. [5] W. Gl~ser,S. Hagen, U. Löffler, J.-B. Suck and W. Schommers, Properties of liquid metals (Taylor and Fr ancis, London, 1973). [6] W. Schommers, Solid State Commun. lb (1975) 45. [71 J.S. Kirkwood, J. Chem. Phys. 56 (1935) 2034. [8] R. Block and W. Schommers, J. Phys. C8 (1975) 1997. [9] P.A. Egelstaff, D.I. Page and C.R.T. Heard, J. Phys. C4 (1971) 1453. [10] R. Block, Progress Report of the Teilinstitut Nukleare Festkörperphysik (1974) KFK 2054. [11] P. Schofield, Proc. Phys. Soc. 88 (1966) 149. 112] W. Abel, Progress Report of the Teiinstitut Nukleare Festkiirperphysik (1975) KFK 2183.
369
4 October 1976
PHYSICS LETTERS
EXTENDED BORN-GREEN EQUATiON: A NEW APPROACH FOR DETERMINiNG PAIR POTENTIALS IN DiSORDERED SYSTEMS W. ABEL, R. BLOCK and W. SCHOMMERS Kernforschungszenrrum Karisruhe, Institut fur AngewandreKernphysik, Postfach 3640, D- 7500 Karisruhe, F.R.G Received 9 June 1976 Revised manuscript received 24 July 1976 The Born-Green equation was tested using molecular dynamics data for liquid rubidium at 319 K. The unsatisfactory results of this equation for the pair potential are caused by the insufficient precision of the Kirkwood approximation for describing the triplet correlation function g3(r, s, t). A more realistic ansatz forg 3(r, s, t) was employed in the formulation of an extended Born-Green equation. This new approach gives much better results for the potential than the simple Born-Green equation.
Within the framework of pair potential theory all significant physical properties (e.g. the thermodynamic functions) essentially depend on the shape of the potential used. This means that the knowledge of the pair potential is of fundamental interest in many-particle system analysis. In this paper a new approach of evaluating the pair potential from structure data of disordered systems will be discussed. We restrict ourselves to isotropic and homogeneous systems taking liquid rubidium as a reference. There are several attempts to determine the pair potential from the pair correlation function: (1) The self-consistent method, proposed in ref. [1] ,determines a pair potential by iteration using molecular dynamics. (2) The method of integral equations (Born-Green [2] Percus-Yevick [3] hypernetted chain [4]). Here we want to investigate the Born-Green (BG) equation using the quantities obtained by method (1). By this method a pair potential for liquid rubidium has already been determined, which describes well several experimental data (scattering law, diffusion constant, etc.) [5, 6] This potential will be used as a reference potential and will be designated as uR(r), r being the distance between two particles of the system. ,
,
.
The Born-Green equation. For isotropic and homogeneous systems provided that the potential energy can be expressed as a sum of pair potentials, a general relationship can be given between the pair potential v(r), the pair correlation function g(r) and the triplet correlation function g3(r, s, t) [2]: 2—?—s2)tdt, (1) = —~-lng(r)+-~ ds g3(r, s, 0 (t
7~
where r, s, t are the distance between three particles of the system, T represents the absolute temperature, kB the Boltzmann constant and p the particle number density. By substituting the Kirkwood approximation (KA) [7] g 3(r, s, t) g(r)g(s)g(t)
(2)
into eq. (1) one obtains the BG-equation. The usefulness of this BG.equation depends on the quality of the KA. For liquid rubidium g3(r, s, t) was determined by means of molecular dynamics [8] using the reference potential UR(r). The comparison of this g3 (r, s, t) with the KA shows deviations by up to 35% and the question arises how these deviations are reflected in the BG-potential. To answer this we substitute reference potential UR(r) and the molecular dynamics data for g(r) on the righthand side of the BG-equation. If the BG-equation is a good approach, we expect that the potential VBG(r) obtained from the lefthand side of the equation agrees with UR(r). However, 367
Volume 58A, number 1
PHYSICS LETTERS
4 October 1976
2.0 ‘y~r) vir) 1.5-1
~
1.92
: 1.0
05
:1 11
/I
Li’
:
1.60 1.26
0.32 0.5
0
~ 7~
N a:
25
1O.Or)A)
:\
5~U 7
100
0
-
0.32
;/•id.o~A]
~(‘\7.~ -
‘.
/1
r(Ai~
Fig. 3. Pair potentials for liqu~rubidium. Reference potentiaL ——— Result Fig. 2. The correction function 7(r). from the extended Born-Green equation - 1.5 Molecular dynamics results. This 7(r) using the molecular dynamics data for gives the best result for the pair potential 7(r) (in fig. 2 the broken line). Result Fig. 1. Pair potentials for liquid rubidium. from the extended Born-Green equation from the extended Born-Green equation — Reference potential. ——— Result from (the potential is shown in fig. 3 as a dotted using the correction function 7(r), which the simple Born-Green equation, line), is shown in fig. 2 as a dotted line. -
1.0
fig. 1 shows that these potentials are quite different from each other. The extended Born-Green equation. In order to get a better pair potential it is necessary to improve the BGequation. For this purpose we have taken the following ansatz [91 for g3(r, s, t) in eq. (1) g3(r, s, t) =g(r)g(s)g(t)
—
y(r)-y(s)’,’(t).
(3)
This ansatz was chosen because: (i) The correction function y(r) can be determined experimentally. There exists the following relationship between 7(r) and the measurable function H(r) [9] H(r) = 7(r) f7(r
—
s) y(s) d.c
(4)
H(r) can be expressed in terms of g(r) and [ag(r)/ap]~ H(r) =g(r) f[g(s)
—
1] [g(t)
—
1] d.c
—
k~T[ag(r)/ap]
~
(5)
In order to get H(r) one has to determine g(r) and g(r) as a function of pressure p at constant temperature T. (ii) The shapes of all calculated 7(r) are similar. Using eq. (3) and the molecular dynamic~data for g3(r, s, t) and g(r) one can determine directly 7(r). However, there are as many functions 7(r) as configurations exist for g3(r, s, t). But it has been demonstrated [10] that these functions obtained from different configurations [8] are very much the same over large r-ranges. 368
Volume 58A, number 6
PHYSICS LETTERS
4 October 1976
Fig. 2 shows the molecular dynamics results for a special 7(r) with the best statistics. It has been determined from the configuration g3(r, s = 4.7, t = 4.7) by simple arithmetics. r = 4.7 corresponds to the principal peak in g(r). The magnitude of 7(r = 4.7) is about 52% of the height ofg(r = 4.T~).Already with this 7(r) the extended BGequation eq. (3) inserted in fig. 1 leads to an improvement of the potential as compared with the results following from the simple BG-equation (eq. (2) in eq. (1)). By variation of 7(r) (using an interactive graphical method [12]) within its statistical error limits it is possible to generate a potential, which is in good agreement with the reference potential. Such a potential is shown in fig. 3. The corresponding function 7(r) has been plotted in fig. 2. From fig. 3 it can be seen that the extended BG-equation leads to much better results for the pair potential than the simple BG-equation. Even the details in the reference potential are reproduced very well. The BG.equation has been discussed for liquid rubidium near the melting point. The cause for the unsatisfactory results of this equation was the crudeness of the KA for describing the triplet correlation function. Therefore, the KA has been corrected by the function 7(r) and an extended BG-equation was proposed. An important reason for introducing 7(r) was that this function can be measured. In order to get 7(r) one has to know [ag(r)/ap]T which means that the pair correlation function g(r) has to be studied as a function of pressure p at constant temperature T [9, 11]. In conclusion, if one uses the method of integral equation (BG) to determine a pair potential, the knowledge of g(r) alone will not be sufficient. The expression [ag(r)/ap]~ ought to be added as an additional experimental information. It should be mentioned that the pair potential is very sensitive to variations in g(r) and 7(r). This implies thatg(r) and 7(r) must be determined very accurately in the experiment. Although the present calculations were made for liquid Rb, we believe that an extended BG-equation will give better results for all those systems in which large deviations of g3(r, s, t) from the KA are expected. These deviations will depend on the packing density, which is a function of density and temperature. Finally, it should be mentioned that the present paper is based on molecular dynamics data only and no iteration was needed in our analysis. The numerical problems which may arise in connection with the iterative solution of the extended equation are out of the scope of this paper. —
—
References [1] [2] [3] [4]
W. Schommers, Phys. Lett. A43 (1973) 157. N.H. March, in: Theory of condensed matter (IAEA, Vienna, 1968). J.K. Percus and G.J. Yevick,Phys. Rev. 110(1958)1. J.M.J. van Leeuwen, J. Groeneveld and .1 de Boer, Physica 25 (1959) 792. [5] W. Gl~ser,S. Hagen, U. Löffler, J.-B. Suck and W. Schommers, Properties of liquid metals (Taylor and Fr ancis, London, 1973). [6] W. Schommers, Solid State Commun. lb (1975) 45. [71 J.S. Kirkwood, J. Chem. Phys. 56 (1935) 2034. [8] R. Block and W. Schommers, J. Phys. C8 (1975) 1997. [9] P.A. Egelstaff, D.I. Page and C.R.T. Heard, J. Phys. C4 (1971) 1453. [10] R. Block, Progress Report of the Teilinstitut Nukleare Festkörperphysik (1974) KFK 2054. [11] P. Schofield, Proc. Phys. Soc. 88 (1966) 149. 112] W. Abel, Progress Report of the Teiinstitut Nukleare Festkiirperphysik (1975) KFK 2183.
369