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On the importance of many-body forces in solid xenon
Volume 46A, number 5
PHYSICS LETTERS
14 January 1974
ON THE IMPORTANCE OF MANY-BODY FORCES IN SOLID XENON* N.A. LURIE** Brookhaven National Laboratory, Upton, New York 11973, USA and Brandeis University, Waltham, Massachusetts 02154, USA
and J. SKALYO, Jr. Brookhaven National Laboratory, Upton, New York 11973, USA
Received 2 November 1973 Measurements of the elastic constants of a xenon single crystal at 10°K are used to evaluate the importance of three-body and many-body forces. The evidence suggests that inadequacies exist in the theories as applied to xenon. One of the more intringuing aspects of rare gas crystals is the possibility for studying the importance of three-body and many-body forces. Although manybody studies have been vigorously pursued in rare gas crystals, contradictory calculations have left the situation less than clear. Part of the problem has been the lack of definitive experimental verification of many-body effects. This is a particularly difficult question since cases where the contribution of manybody forces can be compared with experiment often involve a contribution from two-body forces which themselves are not well known. We discuss here some new experimental evidence based on a measurement o f the elastic constants using neutron scattering [1 ]. In this experiment the elastic constants were determined by measuring the small wave vector portions o f the phonon dispersion relations at 10°K, with careful consideration given to the effects of resolution; the method is similar to that o f Peter et al. [2] used on krypton. The corrected experimental data are shown in fig. 1. The solid lines in the figure are a nearest neighbor harmonic force constant fit. Details o f these experiments will be reported elsewhere [I ]. The resulting elastic constants are Cll = 527 + 9, c12 = 282 + 8, and c44 = 295 + 4 X 10~ dynes cm - 2 . A commonly used indicator of many-body effects * Work performed under the auspices of the U.S. Atomic Energy Commission. ** Supported by the National Science Foundation.
ENERGY,meV 0 i
i
.0
z 0
0
;K
Fig. 1. Small wave vector portions of the phonon dispersion relations for xenon at looK. The points are the corrected experimental data and the solid tines represent a nearneighbor harmonic force constant fit. ~"is the reduced wave vector, ~"= qa/2n, where a is the lattice parameter (a = 6 . 1 2 9 A). The uncertainties in the data a~e of the order of the size of the points. in fee rare gas crystals is the Cauchy condition, c12 = c44. This relation is satisfied for central pair-wise interaction with no zero-point motion. The parameter
= (C44--C12)/C12 is defined as the deviation from the Cauchy condition.
357
Volume 46A, number 5
PHYSICS LETTERS
Table 1 Deviation from the Cauchy relation, 6 = ( c 4 4 - c 1 2 ) / c 1 2 xenon at 0°K. Ref. Experiment Models: Two-body L-J, near neighbor Two-body L-J, all neighbors Three-body, with L-J two-body Three-body, with exp-6 two-body All order dipolar, with L-J two-body All order dipolar, with exp-6 two-body All order dipolar and three-body multipolar with L-J two-body
for 8
[1]
0.05±0.04
[21 [21 [51 [51 [9] [9]
0.015 0.014 -0.092 -0.075 -0.068 -0.059
Ill]
-0.114
We shall confine our discussions here to the low temperature limit. Barron and Klein [3] showed that at T = 0°K, with only pair interactions, the effect of zeropoint motion was to make (5 slightly positive. Subsequently, several attempts were made [ 4 - 7 ] to include static three-body forces of the Axiltod-Teller type [8], i.e., triple-dipolar forces. These tended to give a negative contribution to (5, opposite to the zero-point motion. The restriction to just tripledipolar forces was questioned by Lucas [9], and all orders of dipolar interactions were explicitly studied by Huller [ I 0 ] . The conclusion is that the fourthorder dipolar term is not negligible, but that it seems to be canceled by still higher-order terms [ 11 ]. Threebody higher multipolar interactions have been studied by Zucker and Doran [12], and again the contributions are not negligible. Whether there is a similar cancellation among higher-order multipolar terms is not yet known. A serious problem in evaluating the correctness of these various calculations is that to compare with an experimental (5 one must include the zero-point motion part which depends on the two-body potential; an effect which the heavy mass of xenon minimized. For xenon, in particular, there is no reliable pair interaction [13], and most of the reported calculations use a Lennard-Jones 6 : 12 potential. Numerical values of (5 for xenon at 0°K using the various models mentioned are summarized in the table. The experimental value deduced from our measurements of the elastic constants is given with its estimated uncertainty, (5 = 0.05 -+ 0.04. We note that the experiments were performed at 10°K whereas the calculations are for 0°K; however, no essential difference in (5 is expected. It seems from this comparison that all 358
14 January 1974
the theories are inappropriate in their description of many-body forces in xenon. Even allowing for the slight uncertainty due to the two-body interaction, there appears to be a definite discrepancy. Hfiller's results [6, 10] allow for comparison of two different two-body potentials; it is seen that there is only a small difference in the values of (5. Thus, this is probably not the reason for the discrepancy. Multipolar forces that have not been included [12] and short-range exchange interactions [14] are not expected to be large enough to account tbr the observed differences between experiment and theory, but these need to be more quantitatively evaluated. One additional possibility is the uncertainty in the interaction strenghts which govern the higher order effects. Moreover, the difficulty here may be most prominent for xenon because it is expected that the interatomic forces in xenon are the largest of the rare gases since it has the largest number of electrons. Accurate data on the other rare gas crystals, especially argon where a better two-body potential exists and where the many-body multipolar forces are weaker, would be helpful in verifying these conclusions. Our thanks to J.D. Axe and G. Shurane for their generous advice and encouragement.
References [1] N.A. Lurie, G. Shirane and J. Skalyo, Jr., to be published. [2] H. Peter et al., J. Phys. Chem. Solids 34 (1973) 255. [3] T.H.K. Barron and M.L. Klein, Proc. Phys. Soc.(London) 85 (1965) 533. [4] W. G~tz and H. Schmidt, Z. Phys. 192 (1966) 409. [5] I.J. Zucker and G.G. Chell, J. Phys. C1 (1968) 1505. [6] A. Hiiller, W. G6tze and H. Schmidt, Z. Phys. 231
(1970) 173. [7] J.A. Barker, M.L. Klein and M.V. Bobetic, Phys. Rev. B2 (1970) 4176. [8] B.M. Axilrod and E. Teller, J. Chem. Phys. 11 (1943) 299. [9] A.A. Lucas, Physica 35 (1967) 353;Phys. Rev. 176 (1968) 1093. [10] A. Huller, Z. Phys. 245 (1971) 324. [ 11 ] J.A. Barker, C.H.J. Johnson and T.H. Spurling, Australian J. Chem. 25 (1972) 1811. [12] I.J. Zucker and M.B. Doran, J. Phys. C5 (1972) 2302; M.B. Doran and I.J. Zucker, J. Phys.C4 (1971) 307. [13] M.L. Klein, G.K. Jorton and J.L. Feldman, Phys. Rev. 184 (1969) 968. [14] C.E. Swenberg, Phys. Lett. 24A (1967) 163.
PHYSICS LETTERS
14 January 1974
ON THE IMPORTANCE OF MANY-BODY FORCES IN SOLID XENON* N.A. LURIE** Brookhaven National Laboratory, Upton, New York 11973, USA and Brandeis University, Waltham, Massachusetts 02154, USA
and J. SKALYO, Jr. Brookhaven National Laboratory, Upton, New York 11973, USA
Received 2 November 1973 Measurements of the elastic constants of a xenon single crystal at 10°K are used to evaluate the importance of three-body and many-body forces. The evidence suggests that inadequacies exist in the theories as applied to xenon. One of the more intringuing aspects of rare gas crystals is the possibility for studying the importance of three-body and many-body forces. Although manybody studies have been vigorously pursued in rare gas crystals, contradictory calculations have left the situation less than clear. Part of the problem has been the lack of definitive experimental verification of many-body effects. This is a particularly difficult question since cases where the contribution of manybody forces can be compared with experiment often involve a contribution from two-body forces which themselves are not well known. We discuss here some new experimental evidence based on a measurement o f the elastic constants using neutron scattering [1 ]. In this experiment the elastic constants were determined by measuring the small wave vector portions o f the phonon dispersion relations at 10°K, with careful consideration given to the effects of resolution; the method is similar to that o f Peter et al. [2] used on krypton. The corrected experimental data are shown in fig. 1. The solid lines in the figure are a nearest neighbor harmonic force constant fit. Details o f these experiments will be reported elsewhere [I ]. The resulting elastic constants are Cll = 527 + 9, c12 = 282 + 8, and c44 = 295 + 4 X 10~ dynes cm - 2 . A commonly used indicator of many-body effects * Work performed under the auspices of the U.S. Atomic Energy Commission. ** Supported by the National Science Foundation.
ENERGY,meV 0 i
i
.0
z 0
0
;K
Fig. 1. Small wave vector portions of the phonon dispersion relations for xenon at looK. The points are the corrected experimental data and the solid tines represent a nearneighbor harmonic force constant fit. ~"is the reduced wave vector, ~"= qa/2n, where a is the lattice parameter (a = 6 . 1 2 9 A). The uncertainties in the data a~e of the order of the size of the points. in fee rare gas crystals is the Cauchy condition, c12 = c44. This relation is satisfied for central pair-wise interaction with no zero-point motion. The parameter
= (C44--C12)/C12 is defined as the deviation from the Cauchy condition.
357
Volume 46A, number 5
PHYSICS LETTERS
Table 1 Deviation from the Cauchy relation, 6 = ( c 4 4 - c 1 2 ) / c 1 2 xenon at 0°K. Ref. Experiment Models: Two-body L-J, near neighbor Two-body L-J, all neighbors Three-body, with L-J two-body Three-body, with exp-6 two-body All order dipolar, with L-J two-body All order dipolar, with exp-6 two-body All order dipolar and three-body multipolar with L-J two-body
for 8
[1]
0.05±0.04
[21 [21 [51 [51 [9] [9]
0.015 0.014 -0.092 -0.075 -0.068 -0.059
Ill]
-0.114
We shall confine our discussions here to the low temperature limit. Barron and Klein [3] showed that at T = 0°K, with only pair interactions, the effect of zeropoint motion was to make (5 slightly positive. Subsequently, several attempts were made [ 4 - 7 ] to include static three-body forces of the Axiltod-Teller type [8], i.e., triple-dipolar forces. These tended to give a negative contribution to (5, opposite to the zero-point motion. The restriction to just tripledipolar forces was questioned by Lucas [9], and all orders of dipolar interactions were explicitly studied by Huller [ I 0 ] . The conclusion is that the fourthorder dipolar term is not negligible, but that it seems to be canceled by still higher-order terms [ 11 ]. Threebody higher multipolar interactions have been studied by Zucker and Doran [12], and again the contributions are not negligible. Whether there is a similar cancellation among higher-order multipolar terms is not yet known. A serious problem in evaluating the correctness of these various calculations is that to compare with an experimental (5 one must include the zero-point motion part which depends on the two-body potential; an effect which the heavy mass of xenon minimized. For xenon, in particular, there is no reliable pair interaction [13], and most of the reported calculations use a Lennard-Jones 6 : 12 potential. Numerical values of (5 for xenon at 0°K using the various models mentioned are summarized in the table. The experimental value deduced from our measurements of the elastic constants is given with its estimated uncertainty, (5 = 0.05 -+ 0.04. We note that the experiments were performed at 10°K whereas the calculations are for 0°K; however, no essential difference in (5 is expected. It seems from this comparison that all 358
14 January 1974
the theories are inappropriate in their description of many-body forces in xenon. Even allowing for the slight uncertainty due to the two-body interaction, there appears to be a definite discrepancy. Hfiller's results [6, 10] allow for comparison of two different two-body potentials; it is seen that there is only a small difference in the values of (5. Thus, this is probably not the reason for the discrepancy. Multipolar forces that have not been included [12] and short-range exchange interactions [14] are not expected to be large enough to account tbr the observed differences between experiment and theory, but these need to be more quantitatively evaluated. One additional possibility is the uncertainty in the interaction strenghts which govern the higher order effects. Moreover, the difficulty here may be most prominent for xenon because it is expected that the interatomic forces in xenon are the largest of the rare gases since it has the largest number of electrons. Accurate data on the other rare gas crystals, especially argon where a better two-body potential exists and where the many-body multipolar forces are weaker, would be helpful in verifying these conclusions. Our thanks to J.D. Axe and G. Shurane for their generous advice and encouragement.
References [1] N.A. Lurie, G. Shirane and J. Skalyo, Jr., to be published. [2] H. Peter et al., J. Phys. Chem. Solids 34 (1973) 255. [3] T.H.K. Barron and M.L. Klein, Proc. Phys. Soc.(London) 85 (1965) 533. [4] W. G~tz and H. Schmidt, Z. Phys. 192 (1966) 409. [5] I.J. Zucker and G.G. Chell, J. Phys. C1 (1968) 1505. [6] A. Hiiller, W. G6tze and H. Schmidt, Z. Phys. 231
(1970) 173. [7] J.A. Barker, M.L. Klein and M.V. Bobetic, Phys. Rev. B2 (1970) 4176. [8] B.M. Axilrod and E. Teller, J. Chem. Phys. 11 (1943) 299. [9] A.A. Lucas, Physica 35 (1967) 353;Phys. Rev. 176 (1968) 1093. [10] A. Huller, Z. Phys. 245 (1971) 324. [ 11 ] J.A. Barker, C.H.J. Johnson and T.H. Spurling, Australian J. Chem. 25 (1972) 1811. [12] I.J. Zucker and M.B. Doran, J. Phys. C5 (1972) 2302; M.B. Doran and I.J. Zucker, J. Phys.C4 (1971) 307. [13] M.L. Klein, G.K. Jorton and J.L. Feldman, Phys. Rev. 184 (1969) 968. [14] C.E. Swenberg, Phys. Lett. 24A (1967) 163.