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Liquid metals with structure factor shoulders
Volume 58A, number 7
PHYSICS LETTERS
18 October 1976
LIQUID METALS WITH STRUCTURE FACTOR SHOULDERS M. SILBERT and W.H. YOUNG School of Mathematics and Physics, University of East Anglia, Norwich, UK Received 29 July 1976 Some liquid metals (e.g. Ga, Sn, Sb, Bi) exhibit a low lying shoulder on the high angle side of the principal peak in the structure factor. This feature is compatible with an interatomic potential consisting essentially of a hard core together with an adjacent ledge. Such a potential is entirely consistent with electron theory.
It is now firmly established (see, for example, the compilation of Waseda and Suzuki [1]) that some liquid metals (Ga, Sn, Sb, Bi and, to a small extent Tl) exhibit a low lying shoulder on the high angle side of the first peak. So far, very little in the way of quantitative explanation has been forthcoming, but in this work we show that the feature is consistent with the interatomic potential 00
(r
c
(ci
o
(Xu
2
I I
/
I
/
1
(1)
for suitable choice of the constants u, X and c. When c <0, the above potential has been used (see, e.g., the review article by Barker and Henderson [2] and references therein) to investigate the effects of an attractive tail on a hard sphere system. What has not been done to our knowledge is to consider > 0. There is no motivation for doing this in classical liquid theory, of course, but as we will see there are good reasons in the theory of metals. We have used the random-phase approximation and the formalism described by Woodhead-Galloway et al. [3] to evaluate the structure factors. Though this approach has its limitations at high momentum transfers, it should be adequate for the present region of interest. In this way, a search was conducted to see if we could find parameters in (1) to satisfy the observed data. With three adjustable variables, it is difficult to be sure that a unique solution has been obtained, but we succeeded in obtaining the result shown in fig. 1, for Bi at T= 573 K, with a = 5.65 au, A = 2.1 and e = 0.44 kB T It will be seen that not only is the shoulder re-
—
0
I
2
3
4
Fig. 1. Structure factors for Bi at 573K. Experiment: crosses; hard sphere fit: dashed line; hard sphere with ledge: full line.
produced but, compared with the hard sphere result a = 5.65 au, X = 0, = 0, there is a shift to lower angles of the principal peak in agreement with experi. ment. For any of order kB TM, where TM is the melting temperature, we can expect two effective diameters to be present. Roughly speaking, for the more energetic collisions a will be appropriate; for the others Xu will be the distance of closest approach. This description appears to provide a natural explanation of the second length that Richter and Breitling [4] and Orton [5] sought for this problem. Furthermore, as the temperature is raised, the inner core diameter a becomes appropriate for a larger fraction of the atoms and in this way the disappearance of the structure factor anomaly at high temperatures is explained. The next question is whether the above model p0tential can be supported by basic electron theory.
469
Volume 58A, number 7
o
4
PHYSICS LETTERS
into interatomic potentials that theory alone, at this time, is unable to provide. It needs to be emphasized that the present explanation is not unique on the evidence so far available. Heine and Weaire [9] have suggested that the anomaly
II
I 1
b >
might result from a conventionally shaped interatomic potential but with a hypothetical hard sphere first coordination shell peaking near the first maximum in
\ \ 2
Zn
‘~...
the latter. This would result in some reshuffling of the atoms in the liquid as they seek to find positions of lower potential energy at essentially constant volume.
,,/____“.-..~_._._.~_._
o
~
Mg -2
8
10
2kFr 12
16
Fig. 2. Interatomic potentials v for Mg and Zn taken from the tables of Appapillai and Heine. The hard core diameter a corresponding to a packing fraction of 0.45 is given by 2kFa = 7.4 in each case.
Fig. 2 shows the results of ab initio pseudopotential calculations of interatomic potentials for Mg and Zn taken from the tables of Appapillai and Heine [6]. That shown for Mg is of conventional form; with a hard core and nearest neighbours around the minimum shown, a structure factor close to hard sphere form is expected and obtained for this metal. That shown for Zn contains a hard core with a ledge-type feature. Shaw and Heine [7] have demonstrated that sometimes an improved description of exchange and correlation in the electron gas will remove this anomaly and we believe this is likely to happen for Zn. But what if it is, in fact, a real feature in other cases? Then the model potential described by eq. (1) should be an appropriate zeroth order approximation. Interatomic forces in polyvalent metals are still very poorly known. For any given polyvalent metal (especially with valency three or greater), a search of the literature often reveals calculated curves of both the types shown in fig. 2. Also, as well as the broad ledge shown here, narrow ledges are also obtained (Shyu et a]. [8]). Consequently it may be that observations on structure factors give us a qualitative insight
470
18 October 1976
The radial distribution function for such liquids (Waseda and Suzuki (bc. corroborate calculations this analysis in a qualitative way butcit.)) no quantitative haveFinally, been performed. Ashcroft, Chester and Mon (private cornmunication) have also arrived at the conclusion that a ledge-type feature is involved but their ledge originates from a van der Waals component in the interatomic force. It might be that a final explanation covering all systems of the present type may require all the features discussed above. We wish to thank Mr. K.S. Brennan for assistance with the computations. One of us (WHY.) is grateful for SRC support.
References [1] Y. Waseda and
K. Suzuki, Sci. Rep. RITU, A-Vol. 24 (1973) 139. [2] J.A. Barker and D. Henderson, Ann. Rev. Phys. Chem. 23 (1972) 439. [3] J. Woodhead-Galloway, T. Gaskell and N.H. March, J. Phys. C: Solid State 1(1968) 271. [4] H. Richter and G. Breitling, Adv. Phys. 16 (1967) 293. [5] B.R. Orton, Z. Naturforsch. 30A (1975) 1500. [6] M. Appapillai andY. Heine, Technical Report No. 5 of Theory of Condensed Matter Group, Cavendish Laboratory, Cambridge (1972). [7] R.W. Shaw and V. Heine, Phys. Rev. B5 (1972) 1646. [8] W.-M.Shyu, J.H. Wehling and M.R. Cordes, Phys. Rev. B4 (1971) 1802. [9] V. Heine and D. Weaire, Solid state physics, Vol. 24, eds. H. Ehrenreich, F. Seitz and D. Turnbull (Academic Press, New York, 1970), p. 249.
PHYSICS LETTERS
18 October 1976
LIQUID METALS WITH STRUCTURE FACTOR SHOULDERS M. SILBERT and W.H. YOUNG School of Mathematics and Physics, University of East Anglia, Norwich, UK Received 29 July 1976 Some liquid metals (e.g. Ga, Sn, Sb, Bi) exhibit a low lying shoulder on the high angle side of the principal peak in the structure factor. This feature is compatible with an interatomic potential consisting essentially of a hard core together with an adjacent ledge. Such a potential is entirely consistent with electron theory.
It is now firmly established (see, for example, the compilation of Waseda and Suzuki [1]) that some liquid metals (Ga, Sn, Sb, Bi and, to a small extent Tl) exhibit a low lying shoulder on the high angle side of the first peak. So far, very little in the way of quantitative explanation has been forthcoming, but in this work we show that the feature is consistent with the interatomic potential 00
(r
c
(ci
o
(Xu
2
I I
/
I
/
1
(1)
for suitable choice of the constants u, X and c. When c <0, the above potential has been used (see, e.g., the review article by Barker and Henderson [2] and references therein) to investigate the effects of an attractive tail on a hard sphere system. What has not been done to our knowledge is to consider > 0. There is no motivation for doing this in classical liquid theory, of course, but as we will see there are good reasons in the theory of metals. We have used the random-phase approximation and the formalism described by Woodhead-Galloway et al. [3] to evaluate the structure factors. Though this approach has its limitations at high momentum transfers, it should be adequate for the present region of interest. In this way, a search was conducted to see if we could find parameters in (1) to satisfy the observed data. With three adjustable variables, it is difficult to be sure that a unique solution has been obtained, but we succeeded in obtaining the result shown in fig. 1, for Bi at T= 573 K, with a = 5.65 au, A = 2.1 and e = 0.44 kB T It will be seen that not only is the shoulder re-
—
0
I
2
3
4
Fig. 1. Structure factors for Bi at 573K. Experiment: crosses; hard sphere fit: dashed line; hard sphere with ledge: full line.
produced but, compared with the hard sphere result a = 5.65 au, X = 0, = 0, there is a shift to lower angles of the principal peak in agreement with experi. ment. For any of order kB TM, where TM is the melting temperature, we can expect two effective diameters to be present. Roughly speaking, for the more energetic collisions a will be appropriate; for the others Xu will be the distance of closest approach. This description appears to provide a natural explanation of the second length that Richter and Breitling [4] and Orton [5] sought for this problem. Furthermore, as the temperature is raised, the inner core diameter a becomes appropriate for a larger fraction of the atoms and in this way the disappearance of the structure factor anomaly at high temperatures is explained. The next question is whether the above model p0tential can be supported by basic electron theory.
469
Volume 58A, number 7
o
4
PHYSICS LETTERS
into interatomic potentials that theory alone, at this time, is unable to provide. It needs to be emphasized that the present explanation is not unique on the evidence so far available. Heine and Weaire [9] have suggested that the anomaly
II
I 1
b >
might result from a conventionally shaped interatomic potential but with a hypothetical hard sphere first coordination shell peaking near the first maximum in
\ \ 2
Zn
‘~...
the latter. This would result in some reshuffling of the atoms in the liquid as they seek to find positions of lower potential energy at essentially constant volume.
,,/____“.-..~_._._.~_._
o
~
Mg -2
8
10
2kFr 12
16
Fig. 2. Interatomic potentials v for Mg and Zn taken from the tables of Appapillai and Heine. The hard core diameter a corresponding to a packing fraction of 0.45 is given by 2kFa = 7.4 in each case.
Fig. 2 shows the results of ab initio pseudopotential calculations of interatomic potentials for Mg and Zn taken from the tables of Appapillai and Heine [6]. That shown for Mg is of conventional form; with a hard core and nearest neighbours around the minimum shown, a structure factor close to hard sphere form is expected and obtained for this metal. That shown for Zn contains a hard core with a ledge-type feature. Shaw and Heine [7] have demonstrated that sometimes an improved description of exchange and correlation in the electron gas will remove this anomaly and we believe this is likely to happen for Zn. But what if it is, in fact, a real feature in other cases? Then the model potential described by eq. (1) should be an appropriate zeroth order approximation. Interatomic forces in polyvalent metals are still very poorly known. For any given polyvalent metal (especially with valency three or greater), a search of the literature often reveals calculated curves of both the types shown in fig. 2. Also, as well as the broad ledge shown here, narrow ledges are also obtained (Shyu et a]. [8]). Consequently it may be that observations on structure factors give us a qualitative insight
470
18 October 1976
The radial distribution function for such liquids (Waseda and Suzuki (bc. corroborate calculations this analysis in a qualitative way butcit.)) no quantitative haveFinally, been performed. Ashcroft, Chester and Mon (private cornmunication) have also arrived at the conclusion that a ledge-type feature is involved but their ledge originates from a van der Waals component in the interatomic force. It might be that a final explanation covering all systems of the present type may require all the features discussed above. We wish to thank Mr. K.S. Brennan for assistance with the computations. One of us (WHY.) is grateful for SRC support.
References [1] Y. Waseda and
K. Suzuki, Sci. Rep. RITU, A-Vol. 24 (1973) 139. [2] J.A. Barker and D. Henderson, Ann. Rev. Phys. Chem. 23 (1972) 439. [3] J. Woodhead-Galloway, T. Gaskell and N.H. March, J. Phys. C: Solid State 1(1968) 271. [4] H. Richter and G. Breitling, Adv. Phys. 16 (1967) 293. [5] B.R. Orton, Z. Naturforsch. 30A (1975) 1500. [6] M. Appapillai andY. Heine, Technical Report No. 5 of Theory of Condensed Matter Group, Cavendish Laboratory, Cambridge (1972). [7] R.W. Shaw and V. Heine, Phys. Rev. B5 (1972) 1646. [8] W.-M.Shyu, J.H. Wehling and M.R. Cordes, Phys. Rev. B4 (1971) 1802. [9] V. Heine and D. Weaire, Solid state physics, Vol. 24, eds. H. Ehrenreich, F. Seitz and D. Turnbull (Academic Press, New York, 1970), p. 249.