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Structure factors with shoulder in liquid metals
Volume 74A, number 3,4
PHYSICS LETTERS
12 November 1979
STRUCTURE FACTORS WITH SHOULDER IN LIQUID METALS C. REGNAUT, J.P. BADIALI and M. DUPONT Gmupe de Recherche No. 4 du CNRS “Physique des Liquides et Electrochimie”, Associk ir Wniversith Pierre et Marie Curie, 75 230 Paris Cidex 05, France Received 21 May 1979
The nonlocal Shaw pseudopotential and the optimized random phase approximation are used to calculate five liquid polyvalent metal structure factors. Some particular features are predicted. Effective mass contributions are found to be important for some metals.
In a very recent publication Day et al. [l] have shown that the liquid structure factor S(q) of simple alkali metals as Li, Na, K, could be predicted from a first-principles pseudopotential calculation. S(q) is deduced from the effective pair potential via a Monte Carlo calculation (MC). These metals have hard-spherelike structure factors. Some polyvalent metals exhibit a low lying hump in S(q) on the high angle side of the principal peak [2]. Silber and Young (SY) [3] have recently proposed an explanation for such a feature on the basis of a shouldered hard-sphere potential. The calculation of S(q) has been done for Bi with the random phase approximation @PA). Levesque and Weis [4] using both the more refined optimized random phase approximation (ORPA) and the MC technique have validated the SY model. However, a shoulder is also obtained in the second and subsequent peaks in S(q), which has never been established experimentally. This schematic model may be supported by second-order pseudopotential theory and other contributions such as Van der Waals components in the interatomic forces, as suggested in the work of SY. In this paper we .focus on the exact form of the effective pair potential calculated from second-order pseudopotential theory and we consider five polyvalent liquid metals: Al, Mg, Ga, Zn, Sn. The first two have normal structure factors, while the last three exhibit deformations of the main peak in S(q): asymmetry on the left side for Zn, shoulder on the right side
for Sn and Ga. Gallium is an interesting metal to test because it has a low melting point and it has been shown that in supercooled and amorphous states the shoulder becomes a subsidiary peak [5,6]. The structure factors are calculated by ORPA using potentials computed from the fully nonlocal Shaw pseudopotential [9]. Two effective pair potentials are used, the first one with effective mass corrections, and the second one without these corrections. The potentials are cut at a node near 20 au (1 atomic unit = 0.529 A); exchange and correlation in the electron gas are calculated with the Vashista and Singwi dielectric function [7]. With these conditions seven optimization parameters are used in the ORPA procedure. Moreover we have checked, with a long-range oscillating-type potential, that the ORPA and MC techniques obtain very close results for Ga. This metal having a lower melting point than the others, we thus verify the reliability of the ORPA scheme near normal temperatures. Our calculated S(q) are compared with the experimental data of Waseda near the melting temperature [2]. A good description is obtained for Mg (fig. la). The effective mass corrections give a better overall agreement with experiment. For Al the best results are obtained without these corrections. Including them, the amplitude of the main peak of S(q) is too much reduced, and its position is shifted to high 4 values out of the range of experimental errors (fig. 1b). In fig. lc to le are given the results for the three anomalous metals. 245
Volume 74A, number 3,4
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PHYSICS LETTERS
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Calculated structure factor in ORPA with the nonlocal model of Shaw and the dielectric function of Vashista and Singwi for five metals: (a) Mg, 953 K; (b) Al, 943 K, (c) Zn, 723 K, (d) Sn, 523 K, (e) Ga, 353 K. Results are given both with effective mass corrections (full line) or without these corrections (broken line). The dash—dotted curve in (c) refers to Fig. 1.
the soft sphere calculation. Points represent experimental data.
In fig. lc we point out how the liquid structure is sensitive to the long-range part of the potential. We compare the ORPA S(q) with the soft model of Weeks et al. [8], which only takes into account the repulsive short-range part. It is clear that the long-range contribution of the pair potential may induce the asymmetry of the left side in S(q) for Zn, which is a particular feature of this metal. Fig. Ic exhibits a major discrepancy between the model and the experimental data of Waseda in the range 0.5—1.3 au. We note that other measurements have been done for Zn [9] ;these more recent works give better agreement with our results ineluding effective mass corrections in this low q range. For Sn (fig. ld) the whole structure factor is better described without effective mass corrections. For Ga (fig. le) the calculated S(q) predicts an important deformation of the main peak. Without the effective mass corrections in the potential, the calculated and the experimental S(q) of liquid gallium near the melting point have qualitatively the same features. On the other hand, these corrections lead to a double peak andS(q) presents some analogy
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12 November 1979
with the structure factor of the supercooled liquid or the amorphous state. In a subsequent paper we shall discuss the temperature dependence of liquid structure via ORPA. It is interesting to notice that the pair potentials used here give reasonably good agreement for the entropy of the five liquid metals we consider [10]. Moreover Shaw’s development gives a phonon spectrum for Al which compares well with experimental data [111. Depending on the considered metal, we show that effective mass corrections may have an important influence on the calculated liquid structure factor. Finally we conclude that the pair potentials used contain the principal interatomic force components involving the liquid structure. For “anomalous” metals the expected features of S(q) can be described at least qualitatively, from a first-principles model. Our work indicates that further improvements in the analysis of the interionic forces via pseudopotential theory are needed in order to obtain better agreement with experiment. The authors wish to thank Dr. R. Evans for providing them with detailed numerical values of pair potentials and helpful correspondence. Thanks are also expressed to Dr. D. Levesque and J.J. Weis for their Monte Carlo checking of our results. References Sun and P.H. Cutler, Phys Rev. A19 (1979) 328. [2] Y. Waseda, in: Liquid metals 1976, Eds. R. Evans and D. A. Greenwood, Inst. Phys. Conf. Ser. No. 30 (1977) p. 230; and private communication (1975). [3] M. Silbert andW.H. Young, Phys. Lett. 58A (1976) 469. [1] R.S. Day, F.
[4] D. Levesque and J.J. Weis, Phys. Lett. 60A (1977) 473. [5] A. Bizid, L Bosio, H. Curien, A. Defrain and M. Dupont, Phys. Stat. Sol. (a) 23 (1974) 135. [6] A. Bererhi, L. Bosio and R. Cortes, J. Non. Cryst. Solids 30 (1979) 253. [7] P. Vashista and K.S. Singwi, Phys. Rev. B6 (1972) 875. [8] J.D. D. Chandler Phys.Weeks, 54 (1971) 5237. and H.C. Andersen, J. Chem. [9] W. Knoll, in: Liquid metals 1976, eds. R. Evans and D.A. Greenwood, Inst. Phys. Conf. Ser. No. 30(1977) p. 117; C.N.J. Wagner, in: Liquid metals 1976, eds R. Evans and D.A. Greenwood, Inst. Phys. Conf. Ser. no. 30 (1977) p. 110. [10) R. Kumaravadivel and R. Evans, J. Phys. C9
(1976) 3877. [11] P.V.S. Rao, J. Phys. Chem. Solids 35 (1974) 669.
PHYSICS LETTERS
12 November 1979
STRUCTURE FACTORS WITH SHOULDER IN LIQUID METALS C. REGNAUT, J.P. BADIALI and M. DUPONT Gmupe de Recherche No. 4 du CNRS “Physique des Liquides et Electrochimie”, Associk ir Wniversith Pierre et Marie Curie, 75 230 Paris Cidex 05, France Received 21 May 1979
The nonlocal Shaw pseudopotential and the optimized random phase approximation are used to calculate five liquid polyvalent metal structure factors. Some particular features are predicted. Effective mass contributions are found to be important for some metals.
In a very recent publication Day et al. [l] have shown that the liquid structure factor S(q) of simple alkali metals as Li, Na, K, could be predicted from a first-principles pseudopotential calculation. S(q) is deduced from the effective pair potential via a Monte Carlo calculation (MC). These metals have hard-spherelike structure factors. Some polyvalent metals exhibit a low lying hump in S(q) on the high angle side of the principal peak [2]. Silber and Young (SY) [3] have recently proposed an explanation for such a feature on the basis of a shouldered hard-sphere potential. The calculation of S(q) has been done for Bi with the random phase approximation @PA). Levesque and Weis [4] using both the more refined optimized random phase approximation (ORPA) and the MC technique have validated the SY model. However, a shoulder is also obtained in the second and subsequent peaks in S(q), which has never been established experimentally. This schematic model may be supported by second-order pseudopotential theory and other contributions such as Van der Waals components in the interatomic forces, as suggested in the work of SY. In this paper we .focus on the exact form of the effective pair potential calculated from second-order pseudopotential theory and we consider five polyvalent liquid metals: Al, Mg, Ga, Zn, Sn. The first two have normal structure factors, while the last three exhibit deformations of the main peak in S(q): asymmetry on the left side for Zn, shoulder on the right side
for Sn and Ga. Gallium is an interesting metal to test because it has a low melting point and it has been shown that in supercooled and amorphous states the shoulder becomes a subsidiary peak [5,6]. The structure factors are calculated by ORPA using potentials computed from the fully nonlocal Shaw pseudopotential [9]. Two effective pair potentials are used, the first one with effective mass corrections, and the second one without these corrections. The potentials are cut at a node near 20 au (1 atomic unit = 0.529 A); exchange and correlation in the electron gas are calculated with the Vashista and Singwi dielectric function [7]. With these conditions seven optimization parameters are used in the ORPA procedure. Moreover we have checked, with a long-range oscillating-type potential, that the ORPA and MC techniques obtain very close results for Ga. This metal having a lower melting point than the others, we thus verify the reliability of the ORPA scheme near normal temperatures. Our calculated S(q) are compared with the experimental data of Waseda near the melting temperature [2]. A good description is obtained for Mg (fig. la). The effective mass corrections give a better overall agreement with experiment. For Al the best results are obtained without these corrections. Including them, the amplitude of the main peak of S(q) is too much reduced, and its position is shifted to high 4 values out of the range of experimental errors (fig. 1b). In fig. lc to le are given the results for the three anomalous metals. 245
Volume 74A, number 3,4
__________________________________________
________________________
PHYSICS LETTERS
—
_________________________
Calculated structure factor in ORPA with the nonlocal model of Shaw and the dielectric function of Vashista and Singwi for five metals: (a) Mg, 953 K; (b) Al, 943 K, (c) Zn, 723 K, (d) Sn, 523 K, (e) Ga, 353 K. Results are given both with effective mass corrections (full line) or without these corrections (broken line). The dash—dotted curve in (c) refers to Fig. 1.
the soft sphere calculation. Points represent experimental data.
In fig. lc we point out how the liquid structure is sensitive to the long-range part of the potential. We compare the ORPA S(q) with the soft model of Weeks et al. [8], which only takes into account the repulsive short-range part. It is clear that the long-range contribution of the pair potential may induce the asymmetry of the left side in S(q) for Zn, which is a particular feature of this metal. Fig. Ic exhibits a major discrepancy between the model and the experimental data of Waseda in the range 0.5—1.3 au. We note that other measurements have been done for Zn [9] ;these more recent works give better agreement with our results ineluding effective mass corrections in this low q range. For Sn (fig. ld) the whole structure factor is better described without effective mass corrections. For Ga (fig. le) the calculated S(q) predicts an important deformation of the main peak. Without the effective mass corrections in the potential, the calculated and the experimental S(q) of liquid gallium near the melting point have qualitatively the same features. On the other hand, these corrections lead to a double peak andS(q) presents some analogy
246
12 November 1979
with the structure factor of the supercooled liquid or the amorphous state. In a subsequent paper we shall discuss the temperature dependence of liquid structure via ORPA. It is interesting to notice that the pair potentials used here give reasonably good agreement for the entropy of the five liquid metals we consider [10]. Moreover Shaw’s development gives a phonon spectrum for Al which compares well with experimental data [111. Depending on the considered metal, we show that effective mass corrections may have an important influence on the calculated liquid structure factor. Finally we conclude that the pair potentials used contain the principal interatomic force components involving the liquid structure. For “anomalous” metals the expected features of S(q) can be described at least qualitatively, from a first-principles model. Our work indicates that further improvements in the analysis of the interionic forces via pseudopotential theory are needed in order to obtain better agreement with experiment. The authors wish to thank Dr. R. Evans for providing them with detailed numerical values of pair potentials and helpful correspondence. Thanks are also expressed to Dr. D. Levesque and J.J. Weis for their Monte Carlo checking of our results. References Sun and P.H. Cutler, Phys Rev. A19 (1979) 328. [2] Y. Waseda, in: Liquid metals 1976, Eds. R. Evans and D. A. Greenwood, Inst. Phys. Conf. Ser. No. 30 (1977) p. 230; and private communication (1975). [3] M. Silbert andW.H. Young, Phys. Lett. 58A (1976) 469. [1] R.S. Day, F.
[4] D. Levesque and J.J. Weis, Phys. Lett. 60A (1977) 473. [5] A. Bizid, L Bosio, H. Curien, A. Defrain and M. Dupont, Phys. Stat. Sol. (a) 23 (1974) 135. [6] A. Bererhi, L. Bosio and R. Cortes, J. Non. Cryst. Solids 30 (1979) 253. [7] P. Vashista and K.S. Singwi, Phys. Rev. B6 (1972) 875. [8] J.D. D. Chandler Phys.Weeks, 54 (1971) 5237. and H.C. Andersen, J. Chem. [9] W. Knoll, in: Liquid metals 1976, eds. R. Evans and D.A. Greenwood, Inst. Phys. Conf. Ser. No. 30(1977) p. 117; C.N.J. Wagner, in: Liquid metals 1976, eds R. Evans and D.A. Greenwood, Inst. Phys. Conf. Ser. no. 30 (1977) p. 110. [10) R. Kumaravadivel and R. Evans, J. Phys. C9
(1976) 3877. [11] P.V.S. Rao, J. Phys. Chem. Solids 35 (1974) 669.