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Many-body effects in molecular solids
Solid State Communications, Vol. 18, pp 799—801, 1976.
Pergamon Press.
Printed in Great Britain
MANY-BODY EFFECTS IN MOLECULAR SOLIDS N.P. Gupta and P.K. Garg Physics Department, University of Jodhpur, Jodhpur, India (Received 14 October 1975 by S. Amelinckx) The effects of many-body interactions in the molecular (rare-gas) solids have been investigated, on the basis of Axilrod—Teller approximation, by a rigid-atom model. It is found that the 3-body interaction is the most dominant of all and the rest may be safely ignored. The discrepancy seen in the phonon dispersion curves is expected to be removed by the inclusion of appropriate anharmonic effects. 1. INTRODUCTION BY THE STUDIES of stability of lattices’3 and various solid state properties,4~’2this is more or less established that n-body interactions are present in the molecular (rare-gas) solids. Arguments have been given both for’315 and against’6”7 neglecting the terms beyond the triplet. However, Copeland and Kestner’s suggestion~in favour of neglecting the quadruplet and higher contributions is sound enough, as they explicitly show that these terms are negligibly insignificant in comparison to the triplet one. As such the n-body effects in the solids may almost be represented by the 3-body contribution to the cohesive energy. This conclusion is based upon the calculations made by several workers~7’17~using one-piece power! exponential 2-body potential functions, the validity of which has been questioned by many including Guggenhelm and McGlashan.~They are of the view that any interaction potential cannot be adequately represented over the whole range of the intermolecular separation by only a two/three parameters and one-piece potential function. The analytical potentials used by Barker and coworkers,~’~ Munn and Smith ~ and Dymond and Alder’6 contain more than three parameters, but all are single-piece functions. As such these also may not fully describe the interaction potential on the two-sides of the potential minimum, Recently, the authors have developed a rigid-atom model2426 for these solids, which derives the intermolecular forces through the 4-parameter and 2-piece Buckingham—Corner potential function.27 It is seen that the potential is not only superior in form to the other conventional potential functions, but it also successfully describes the phonon dispersion,24’25 and the anharmonic effects26 in this class of solids. It is, therefore, thought desirable that the effect of the n-body interaction in these solids be studied by the rigid-atom model, through “argon” the representative of the class, 799
2. THE RIGID-ATOM MODEL AND THE MANY-BODY INTERACTION The lattice dynamical mode12426 assumes the closed-shell atoms of the rare-gas solids to be rigid-hard spheres and to be vibrating about the f.c.c. lattice sites. The potential energy of the solid may be written in the form i~.
=
~
)(j/) +
1< i
+
Ø~”~(i/k. . .) +
~
i < .i <
k <...
(1) where the n-body interaction potential function ~t2k~/k .) vanishes when one of the atoms i, j, k, . is remote from the others. The function Ø(2)(i/) is a 2-body potential and depends only on the distance betwe~two atoms, and as such has been represented by the buckingham—Corner “pair-potential” function27 [~c(Ru)]. Following Copeland and Kestner13 and others cb~”ki/k. .) is replaced by Axilrod—Teller’s2~30triple—dipole interaction term Ø~)~(j/k) = fl(1 + 3 cos 0. cos 0~cos 0k)/(R~RfkR~i), . .
. .
.
(2) where 0~0,~and °k are the interior angles of the dipolar triangle with sides R~,Rh, and Rkg, and ~l [ (9/1 6)1a2] is a constant depending upon the ionization energy (I) and polarizability (a) of the interacting atoms. The AT-term merely represents the long-range part of the three-body potential. The short-range part31 of this potential, which is almost identical in nature (repulsive) and form to that of the long-range one, has rather doubtful validity32 and as such is not taken into account. The model estimates the zero-point quantum contribution of the pair-interaction by the Debye theory of specific heats and includes it in the calculation through the potential parameters.33 The zero-point contribution of the 3-body forces, which may be evaluated by the
800
MANY-BODY EFFECTS IN MOLECULAR
Einstein modeI,18*34 is expected to be quite small, and as such has been neglected. For obtaining the phonon frequencies, the usual secular equation det lM(K, kk’) - Imo’l
= 0,
k $,&l
*
&&~ij>
t
1.6
g
1
G%(@).
i’
.z
I-
0 .B
0.4
0
(4) of kth and k’th atoms in
+
(5)
The potential parameters (cu = 13.20343, /3 = 0.15, r,,, = 3.78 x lo-‘cm, and E = 206.0836 x lo-l6 erg) have been evaluated by a method of iteration developed by Gupta and Dayal and using the latest observed data.24 Also Fisher and Watts’s valpe34 of the constant Sl (= 73.2 x 10wwerg cm? in &$(ijk) for argon has been adopted, as it is regarded to be the best of all found out so far. The phonon dispersion [u(q), q] curves in the principal symmetry directions are drawn by picking up frequencies corresponding to the points (Q, 0, 0), (Q, Q, 0), and (Q, Q, Q). The phonon wave vector K and the reduced wave vector are related as q = aK, a being the lattice constant. These curves have been presented (Fig. 1) along with the neutron scattering points of Egger et aZ.36 (““Ar, 4.2 K) and FujiietaL37 (36Ar, 10 K), after making proper allowance of the isotopic masses. It is obvious that the latest and improved data of Fujii et aL3’ shows better agreement than that of Egger et aZ.36 The many-body effect changes the frequencies in the
,’
/’
/’
;:
The phonon frequencies (w) have been computed by solving the secular equation (3), for the solidified 40Ar at 0 K, corre sp onding to the nonequivalent 262 wave vectors (K). The “force constants” && - I’) are derived from the “effective potential” function
9.“. T.BI
L
(3)
3. PHONON DISPERSION
+ =
.o
-
- 1’) exp [- 27riK*r(Z - I’)],
where r is the vector separation the Zth and I’th cells.
Vol. 18. No.
2.4
has been derived following the Born-von Karman formalism. The dynamical matrix M is defined as M(K, kk’) = 7
--r
SOLIDS
0
000xl10) 9-
6.5.5)
0 I-
%-
Fig. 1. Dispersion curves for 40Ar our theoretical at 0 K and zero atm. pressure: (-) Buckingham-Corner pair potential with Axilrod-Teller 3-body term, (- - -) Buckingham-Corner potential. Experimental points: Egger et al. (reference 36) at 4.2 K and 1 atm. pressure: (x) longitudinal; (A) transverse. Fujii et al. (reference 37) at 10 K and 1 atm. pressure: (A) longitudinal; (0) transverse Ti; (0) transverse T,. right direction, but carries them a little ahead almost in every branch of the curves. Some reduction of the discrepancy might be achieved‘by including the anharmonic effects appropriate to the temperature. However, in the present context, it may be said that the Axilrod-Teller term [equation (2)] overestimates the three-body interaction to the extent of exceeding the interaction of four and all higher terms. 4. CONCLUSION In view of the results and their analysis it may be concluded that the presence of many-body effects in the molecular (rare-gas) solids is significant, and may well be represented only by the 3-body interaction, i.e. all higher-body interactions are negligible in comparison to the 3-body one. Moreover, the present rigid-atom model is quite appropriate for this class of solids. Acknowledgements - The senior author (N.P.G.) is grateful to the Indian National Science Academy (New Delhi) for a project-grant under its Basic Sciences Research Programme and P.K.G. expresses thanks to the University of Jodhpur for providing a scholarship after the termination of the INSA Project.
REFERENCES 1.
JANSEN L., Phys. Rev. A135, 1292 (1964).
2.
WALLACE D.C., Phys. Rev. A133, 153 (1964).
3.
HULLER A., 2. Phys. 241,340
4.
ZUCKER I.J.,Nuovo
5.
ZUCKER I.J. & CHELL G.G.,J. Phys. Cl, 1505 (1968).
(197 1).
Pimento B54, 177 (1968).
Vol. 18, No. 7
MANY-BODY EFFECTS IN MOLECULAR SOLIDS
6.
BARKER J.A., KLEIN M.L. & BOBETIC M.V.,Phys. Rev. B2, 4176 (1970).
7. 8.
BARKER J.A., BOBETIC M.V. & KLEIN M.L.,Phys. Lett. A34, 415 (1971). JANSEN L.,Phil. Mag. 8, 1305 (1963).
9.
LOSEE D.L. & SIMMONS R.O., Phys. Rev. Lett. 18, 457 (1967).
10. 11.
COULSON C.A.,Disc. Faraday Soc. 43, 285 (1965). LURIE N.A. & SKALYO J., Jr.,Phys. Lett. A46, 357 (1974).
12.
ZUBOV V.1., Ann. Phys. (Leipzig) 32,93 (1975).
13. 14.
COPELAND D.A. & KESTNER N.R.,J. Chem. Phys. 49, 5214 (1968). BARKER J.A., HENDERSON D. & SMITH W.R.,Mol. Phys. 17,79(1969).
15.
BARKER J.A. & POMPE A., Aust. J. Phys. 21, 1683 (1968).
16.
DYMONDJ.H.&ALDERB.J.,J. Chem. Phys. 51, 309 (1969).
17. 18.
KLEIN M.L., HORTON G.K. & FELDMAN J.L.,Phys. Rev. 184,968(1969). GOETZE W. & SCHMIDT H., Z. Phys. 192,409 (1966).
19.
HULLER A., GOETZE W. & SCHMIDT H., Z. Phys. 231, 173 (1970).
20.
BOBETIC M.V. & BARKER J.A., Phys. Rev. B2, 4169 (1970).
21.
GUGGENHEIM E.A. & McGLASHAN, Proc. R. Soc. A255, 456 (1960).
22.
BOBETIC M.V., BARKER J.A. & KLEIN M.L.,Phys. Rev. B5, 3185 (1972).
23.
MUNN R.J. & SMITH F.J.,J. Chem. Phys. 43, 3998 (1965).
24. 25.
GARG P.K. & GUPTA N.P.,Ann. Phys. (U.S.A.) 91, 375 (1975). GUPTA N.P. & GARG P.K.,Ann. Phys. (U.S.A.) (in press).
26.
GARG P.K. & GUPTA N.P., Solid State Commun. 17, 89 (1975).
27.
BUCKINGHAM R.A. & CORNER J., Proc. R. Soc. 189, 118(1947).
28.
AXILROD B.M. & TELLER E.,J. Chem. Phys. 11,299(1943).
29.
AXILRODB.M.&TELLERE.,J. Chem.Phys. 17,1349(1949).
30.
AXILROD B.M. & TELLER E.,J. Chem. Phys. 19, 724 (1951).
31.
JANSEN L.,Phys. Rev. A135, 1292 (1964).
32.
SWENBERG C.E.,Phys. Lett. A24, 163 (1967).
33.
GUPTA N.P. & DYAL B.,Proc. NatL Acad. Sci. (India) A37, 79 (1969).
34. 35.
CHELL G.G. & ZUCKER LI., J. Phys. Cl, 35 (1968). FISHERR.A.&WATTSR.O.,MoLPhys. 23, 1051 (1972).
36. 37.
EGGER H., GSANGER M., LUSCHER E. & DORNER B.,Phys. Rev. Lett. A28, 433 (1968). FUJI! Y., LURIE N.A., PYNN R. & SHIRANE G.,Phys. Rev. BlO, 3647 (1974).
801
Pergamon Press.
Printed in Great Britain
MANY-BODY EFFECTS IN MOLECULAR SOLIDS N.P. Gupta and P.K. Garg Physics Department, University of Jodhpur, Jodhpur, India (Received 14 October 1975 by S. Amelinckx) The effects of many-body interactions in the molecular (rare-gas) solids have been investigated, on the basis of Axilrod—Teller approximation, by a rigid-atom model. It is found that the 3-body interaction is the most dominant of all and the rest may be safely ignored. The discrepancy seen in the phonon dispersion curves is expected to be removed by the inclusion of appropriate anharmonic effects. 1. INTRODUCTION BY THE STUDIES of stability of lattices’3 and various solid state properties,4~’2this is more or less established that n-body interactions are present in the molecular (rare-gas) solids. Arguments have been given both for’315 and against’6”7 neglecting the terms beyond the triplet. However, Copeland and Kestner’s suggestion~in favour of neglecting the quadruplet and higher contributions is sound enough, as they explicitly show that these terms are negligibly insignificant in comparison to the triplet one. As such the n-body effects in the solids may almost be represented by the 3-body contribution to the cohesive energy. This conclusion is based upon the calculations made by several workers~7’17~using one-piece power! exponential 2-body potential functions, the validity of which has been questioned by many including Guggenhelm and McGlashan.~They are of the view that any interaction potential cannot be adequately represented over the whole range of the intermolecular separation by only a two/three parameters and one-piece potential function. The analytical potentials used by Barker and coworkers,~’~ Munn and Smith ~ and Dymond and Alder’6 contain more than three parameters, but all are single-piece functions. As such these also may not fully describe the interaction potential on the two-sides of the potential minimum, Recently, the authors have developed a rigid-atom model2426 for these solids, which derives the intermolecular forces through the 4-parameter and 2-piece Buckingham—Corner potential function.27 It is seen that the potential is not only superior in form to the other conventional potential functions, but it also successfully describes the phonon dispersion,24’25 and the anharmonic effects26 in this class of solids. It is, therefore, thought desirable that the effect of the n-body interaction in these solids be studied by the rigid-atom model, through “argon” the representative of the class, 799
2. THE RIGID-ATOM MODEL AND THE MANY-BODY INTERACTION The lattice dynamical mode12426 assumes the closed-shell atoms of the rare-gas solids to be rigid-hard spheres and to be vibrating about the f.c.c. lattice sites. The potential energy of the solid may be written in the form i~.
=
~
)(j/) +
1< i
+
Ø~”~(i/k. . .) +
~
i < .i <
k <...
(1) where the n-body interaction potential function ~t2k~/k .) vanishes when one of the atoms i, j, k, . is remote from the others. The function Ø(2)(i/) is a 2-body potential and depends only on the distance betwe~two atoms, and as such has been represented by the buckingham—Corner “pair-potential” function27 [~c(Ru)]. Following Copeland and Kestner13 and others cb~”ki/k. .) is replaced by Axilrod—Teller’s2~30triple—dipole interaction term Ø~)~(j/k) = fl(1 + 3 cos 0. cos 0~cos 0k)/(R~RfkR~i), . .
. .
.
(2) where 0~0,~and °k are the interior angles of the dipolar triangle with sides R~,Rh, and Rkg, and ~l [ (9/1 6)1a2] is a constant depending upon the ionization energy (I) and polarizability (a) of the interacting atoms. The AT-term merely represents the long-range part of the three-body potential. The short-range part31 of this potential, which is almost identical in nature (repulsive) and form to that of the long-range one, has rather doubtful validity32 and as such is not taken into account. The model estimates the zero-point quantum contribution of the pair-interaction by the Debye theory of specific heats and includes it in the calculation through the potential parameters.33 The zero-point contribution of the 3-body forces, which may be evaluated by the
800
MANY-BODY EFFECTS IN MOLECULAR
Einstein modeI,18*34 is expected to be quite small, and as such has been neglected. For obtaining the phonon frequencies, the usual secular equation det lM(K, kk’) - Imo’l
= 0,
k $,&l
*
&&~ij>
t
1.6
g
1
G%(@).
i’
.z
I-
0 .B
0.4
0
(4) of kth and k’th atoms in
+
(5)
The potential parameters (cu = 13.20343, /3 = 0.15, r,,, = 3.78 x lo-‘cm, and E = 206.0836 x lo-l6 erg) have been evaluated by a method of iteration developed by Gupta and Dayal and using the latest observed data.24 Also Fisher and Watts’s valpe34 of the constant Sl (= 73.2 x 10wwerg cm? in &$(ijk) for argon has been adopted, as it is regarded to be the best of all found out so far. The phonon dispersion [u(q), q] curves in the principal symmetry directions are drawn by picking up frequencies corresponding to the points (Q, 0, 0), (Q, Q, 0), and (Q, Q, Q). The phonon wave vector K and the reduced wave vector are related as q = aK, a being the lattice constant. These curves have been presented (Fig. 1) along with the neutron scattering points of Egger et aZ.36 (““Ar, 4.2 K) and FujiietaL37 (36Ar, 10 K), after making proper allowance of the isotopic masses. It is obvious that the latest and improved data of Fujii et aL3’ shows better agreement than that of Egger et aZ.36 The many-body effect changes the frequencies in the
,’
/’
/’
;:
The phonon frequencies (w) have been computed by solving the secular equation (3), for the solidified 40Ar at 0 K, corre sp onding to the nonequivalent 262 wave vectors (K). The “force constants” && - I’) are derived from the “effective potential” function
9.“. T.BI
L
(3)
3. PHONON DISPERSION
+ =
.o
-
- 1’) exp [- 27riK*r(Z - I’)],
where r is the vector separation the Zth and I’th cells.
Vol. 18. No.
2.4
has been derived following the Born-von Karman formalism. The dynamical matrix M is defined as M(K, kk’) = 7
--r
SOLIDS
0
000xl10) 9-
6.5.5)
0 I-
%-
Fig. 1. Dispersion curves for 40Ar our theoretical at 0 K and zero atm. pressure: (-) Buckingham-Corner pair potential with Axilrod-Teller 3-body term, (- - -) Buckingham-Corner potential. Experimental points: Egger et al. (reference 36) at 4.2 K and 1 atm. pressure: (x) longitudinal; (A) transverse. Fujii et al. (reference 37) at 10 K and 1 atm. pressure: (A) longitudinal; (0) transverse Ti; (0) transverse T,. right direction, but carries them a little ahead almost in every branch of the curves. Some reduction of the discrepancy might be achieved‘by including the anharmonic effects appropriate to the temperature. However, in the present context, it may be said that the Axilrod-Teller term [equation (2)] overestimates the three-body interaction to the extent of exceeding the interaction of four and all higher terms. 4. CONCLUSION In view of the results and their analysis it may be concluded that the presence of many-body effects in the molecular (rare-gas) solids is significant, and may well be represented only by the 3-body interaction, i.e. all higher-body interactions are negligible in comparison to the 3-body one. Moreover, the present rigid-atom model is quite appropriate for this class of solids. Acknowledgements - The senior author (N.P.G.) is grateful to the Indian National Science Academy (New Delhi) for a project-grant under its Basic Sciences Research Programme and P.K.G. expresses thanks to the University of Jodhpur for providing a scholarship after the termination of the INSA Project.
REFERENCES 1.
JANSEN L., Phys. Rev. A135, 1292 (1964).
2.
WALLACE D.C., Phys. Rev. A133, 153 (1964).
3.
HULLER A., 2. Phys. 241,340
4.
ZUCKER I.J.,Nuovo
5.
ZUCKER I.J. & CHELL G.G.,J. Phys. Cl, 1505 (1968).
(197 1).
Pimento B54, 177 (1968).
Vol. 18, No. 7
MANY-BODY EFFECTS IN MOLECULAR SOLIDS
6.
BARKER J.A., KLEIN M.L. & BOBETIC M.V.,Phys. Rev. B2, 4176 (1970).
7. 8.
BARKER J.A., BOBETIC M.V. & KLEIN M.L.,Phys. Lett. A34, 415 (1971). JANSEN L.,Phil. Mag. 8, 1305 (1963).
9.
LOSEE D.L. & SIMMONS R.O., Phys. Rev. Lett. 18, 457 (1967).
10. 11.
COULSON C.A.,Disc. Faraday Soc. 43, 285 (1965). LURIE N.A. & SKALYO J., Jr.,Phys. Lett. A46, 357 (1974).
12.
ZUBOV V.1., Ann. Phys. (Leipzig) 32,93 (1975).
13. 14.
COPELAND D.A. & KESTNER N.R.,J. Chem. Phys. 49, 5214 (1968). BARKER J.A., HENDERSON D. & SMITH W.R.,Mol. Phys. 17,79(1969).
15.
BARKER J.A. & POMPE A., Aust. J. Phys. 21, 1683 (1968).
16.
DYMONDJ.H.&ALDERB.J.,J. Chem. Phys. 51, 309 (1969).
17. 18.
KLEIN M.L., HORTON G.K. & FELDMAN J.L.,Phys. Rev. 184,968(1969). GOETZE W. & SCHMIDT H., Z. Phys. 192,409 (1966).
19.
HULLER A., GOETZE W. & SCHMIDT H., Z. Phys. 231, 173 (1970).
20.
BOBETIC M.V. & BARKER J.A., Phys. Rev. B2, 4169 (1970).
21.
GUGGENHEIM E.A. & McGLASHAN, Proc. R. Soc. A255, 456 (1960).
22.
BOBETIC M.V., BARKER J.A. & KLEIN M.L.,Phys. Rev. B5, 3185 (1972).
23.
MUNN R.J. & SMITH F.J.,J. Chem. Phys. 43, 3998 (1965).
24. 25.
GARG P.K. & GUPTA N.P.,Ann. Phys. (U.S.A.) 91, 375 (1975). GUPTA N.P. & GARG P.K.,Ann. Phys. (U.S.A.) (in press).
26.
GARG P.K. & GUPTA N.P., Solid State Commun. 17, 89 (1975).
27.
BUCKINGHAM R.A. & CORNER J., Proc. R. Soc. 189, 118(1947).
28.
AXILROD B.M. & TELLER E.,J. Chem. Phys. 11,299(1943).
29.
AXILRODB.M.&TELLERE.,J. Chem.Phys. 17,1349(1949).
30.
AXILROD B.M. & TELLER E.,J. Chem. Phys. 19, 724 (1951).
31.
JANSEN L.,Phys. Rev. A135, 1292 (1964).
32.
SWENBERG C.E.,Phys. Lett. A24, 163 (1967).
33.
GUPTA N.P. & DYAL B.,Proc. NatL Acad. Sci. (India) A37, 79 (1969).
34. 35.
CHELL G.G. & ZUCKER LI., J. Phys. Cl, 35 (1968). FISHERR.A.&WATTSR.O.,MoLPhys. 23, 1051 (1972).
36. 37.
EGGER H., GSANGER M., LUSCHER E. & DORNER B.,Phys. Rev. Lett. A28, 433 (1968). FUJI! Y., LURIE N.A., PYNN R. & SHIRANE G.,Phys. Rev. BlO, 3647 (1974).
801