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Lattice heat capacities of solid argon
Solid State Communications Vol. 17, Pp. 89—92, 1975
Pergamon Press.
Printed in Great Britain
LA~FICEHEAT CAPACITIES OF SOLID ARGON P.K. Garg and N.P. Gupta Physics Department, University of Jodhpur, Jodhpur, India. (Received 7 March 1975 by S. Amelinckx)
An anharmonic rigid-atom model has been used to study the heat capacities of solid argon. The model derives the interatomic forces by Buckinghani—Comer potential and takes account of all neighbour interaction. The contribution of the cubic and quartic terms of the potential energy expansion to the heat capacities have been accounted for through Helmholtz-free energy by perturbation theory. It is concluded that the model is most suitable among the ones designed for the class of solids to which argon belongs.
1. INTRODUCTION
certain model to all members of the series, one may
IN RECENT YEARS keen interest has been shown in the study of solid state properties of the inert-gases.’8 It has been due to a number of reasons. Firstly, the atoms constituting the inert gas solids have closed electronu shells, and as such there are neither free. electrons nor magnetic moments to contribute to the thermal properties. Therefore, these properties may be determined by lattice dynamics only. Secondly, the interparticle interaction in the solids to a good approximation are of spherically symmetric central type. Hence the interatomic potential may be written in simple analytic form. However, there are reasons to believe that the interactions are not completely pairwise additive and many-body forces9 may be present. But it is generally believed4’10 that they do not play significant role in the thermal properties of solids. Moreover, the solids form a series of substances (He, Ne, Ar, Kr and Xe), the behaviour of which ranges from classical to quantum.5 The heavier ones are ahnost classical solids, whereas the lighter Ne and He are predominated by quantum effects.’1 Also the vacancy contribution to the thermal properties at the significant.’~3As such the study of the thermal low temperatures of these solids is not much
also be able to study the change in character (from classical to quantum) of the solids. In view of the situation lattice heat capacities of solid argon, a substance centrally situated between the well known quantum solid “helium” and classical “xenon”, have been studied by an “Anharmonic Rigid-Atom Model (ARM)”,’4”5 only to acertain its improvement on replacement of the original Buckingham pair.potential16 by the Buckingharn—Corner17”8 potential. Comparison of the results shows that the modified ARM is better than the original one. 2. ANHARMONIC RIGID-ATOM MODEL The model14’15 assumes inert gas atoms to be rigid-hard spheres and executing quasiharmonic motion at f.c.c. lattice sites. It derives the weak long. range attractive van der Waals and strong short.range exchange interatomic forces by the two-piece Buckingham—Corner (B—C) potential function; r
[
Fm
c1(r)
=
e g
(~)
1 (a, j3) exp a 1 +~ 2~~] r r —
properties of these solids is not only easy, but it also provides ideal testing ground to various proposed lattice dynamical models. In addition, by applying a
(em) 6
and 89
,
—g2 (a, j3) for r > Fm,
90
LAUICE HEAT CAPACITIES OF SOUD ARGON
=
e
Ig’ (a, 13)expa 6
(i +
(1__) r —g
(~)j
~
m
2’~exp
I,
31010 dyn C~2 Bulk modulusenergy = 2.67 Sublimation = x1846 cal mol~ z.p.E.= 187 calmoV’. The anharmonic potential parameters so obtained are: ~
1,
2(a,~3)
4
—
L~
.~
\
forr~
a
where
gi(a,j3)
=
Vol. 17, No. 1
[6+8131[a(1+$)—(6+813)]’,
=
13.02836, e
=
=
0.15,
3.85 x
rm
=
169.5329 x 108
1016
ergs
cm.
3. VIBRATION SPECTRUM AND HEAT CAPACITIES
g 2(a,13)
=
a[a(1+13)—(6+813)1’,
defined for a pair of atoms separated by a distance r. Here e is the depth of the potentiai well for the two atoms at equilibrium separation Tm, a determines the steepness of repulsion at small interatomic distance, and j3 is the dimensionless ratiodipole—dipole of the coefficients 8) and (r6) of dipole—quadrupole (r terms. The inverse six-power term in the potential represents the van der Waals attraction, and the exponential term accounts for the overlap repulsion. l’his B—C potential is a variation of the Buckingham potential}6 The basic improvement in this potential over the original Buckingham potential is the inclusion of a parametrically variable and additive dipole—quadrupole correction to the attractive dipole—dipole term in conformity with quantum mechanical calculations. The presence of the former is considered necessary as the multipole series does not converge rapidly. Secondly it being a two-piece potential function describes the potential curves on both sides of the potential minimum. The quastharmonic values of the parameters e, a and Tm are determined~in terms of the dimensionless parameter 13 by using equiffibrium conditions of the lattice, and the compressibility-free energy relation. The parameter (3 is computed by an expression given by Margenau.~The effect of zero-point vibrations, so appreciably present in the solid,’ has been estimated by Debye theory of specific heats, and is included in the calculation through the potential parameters by a method of iteration.22 The contribution of cubic and quartic terms of the potential energy expansion has been included in the potential parameters through the Helniholtz free energy by perturbation theory.23 The computation takes care of all neighbour interactions,~and uses the latest crystal data at absolute zero of temperature: Lattice parameter = 5.3111 x 10-8 cm.25
In order to compute the lattice vibration spectrum and heat capacities, the Brillouin zone has been divided into 8000 cells of equal volumes. By the cyclic boundary conditions and the symmetry properties of the octahedral zone, the 8000 wave vectors corresponding to eachthat cellcorresponding reduce to 262tonon-equivalent ones including the origin. The computed phonon spectra ranges up to about 2 THZ. Histogram is, therefore, prepared with the interval length I~v= 0.1 THZ. The function g(v) giving the total number of frequencies in the interval (v ~v) to (v + ~~v) is calculated with the histogram. The frequency distribution function g(v) vi v curve (Fig. 1) has been drawn in such a manner, that the area under —
~
2
~
04
00
10
lZ
*4
IS
1$
FIG. 1. Frequency spectrum curve for solid argon at OK obtained with Buckinghani—Corner (exp, 6, 8) potential.
the curve is equal to the area under the histogram. Since small fluctuations in the [g(v), v] curve are of no indication of any real effect in the simple solid under study, all these have been smoothed out. The contribution of each step of frequency spectrum to heat capacity is obtained by multiplying
20
Vol. 17, No.1
LA11ICE HEAT CAPACITIES OF SOUD ARGON
the Einstein function corresponding to mid-point of each of the intervals by the statistical weight of the point. Just to study the variation of specific.heat with temperature, C,, values have been obtained at various temperatures ranging from the lowest to the melting point. The resulting (C,,, T) curve has been presented (Fig. 2) along with recent observed data.25
________________________________
/ /
/
/
‘~
91
2
• /
As usual all the heat capacities have also been converted to equivalent Debye O’~S,and (0, T) curve has been drawn (Fig. 3) for comparison with the latest experimental25 as well as theoretical’4 results.
/
*
___________________________
0
20
80
40 T
(
0*1
40
00
)—~
FIG. 2. Variation of specific heat at constant volume with temperature for solid argon: (—) our theoretical obtained with Buckingham—Corner (exp, 6, 8) potential. (- - -) theoretical obtained with Buckingham type potential (reference 14). (0) Experimental of Peterson et aL (reference 25).
\ \
appreciable
4. DISCUSSION OF RESULTS The frequency distribution function g(v) vs v curve for argon (Fig. 1) refers to OK. It possesses all the characteristics of a monatomic solid. It has two maxima,27 one near about ~ and the other close to the upper frequency limit VL. It is seen that our values of specific heats (Fig. 2) are not only better than the theoretical values of Gupta and Gupta14 by ARM using Buckingham potential, but they show excellent agreement with the latest experimental results of Peterson et al.25 Our (0, T) curve (Fig. 3) is not only similar in shape to the theoretical curve of Gupta and Gupta,14 but it also shows much better agreement with experiment.25 At about 17 K, our curve crosses the experimental curve and as such our 0-values below this point are lower and above it are higher than the observed values. With an increase of temperature beyond 17 K our curve rises and shows excellant agreement with the experiment. The little discrepancy might have crept in during conversion of C,, into °D~ The maximum discrepancy at temperatures lower than 17 K is of the order of 7 per cent. Really it is in comparison to the discrepancy at higher
—-—
10
________________________________ 0
10
30 T
40
temperatures. Perhaps it may be due to the coarseness of the mesh of reciprocal points that we have chosen for the calculation. The main contribution to specific heat in this range of temperature arises from the central part of the Brillouin zone,28 and we have chosen only a few points there for calculation.
(°K)
FiG. 3. Equivalent Debye temperature for the specific heat for solid argon: (—) our theoretical obtained with Buckingham—Corner (exp, 6, 8) potential. (....) theoretical obtained with Buckingham type potential (reference 14). (— —) Experimental of Peterson et al. (reference 25).
5. CONCLUSIONS In view of the above analysis of the results, it may be concluded that the present anharmonic rigid-atom model using the Buckingham—Corner potential function is not only superior to the one using Buckingham
92
LATFICE HEAT CAPACITIES OF SOLID ARGON
potential function, but it is also the most suitable of all such models designed for the study of argon. The model being rigid and consistent may be applied to other members of the °rou~
Vol. 17, No. 1
Acknowledgements — The authors express their gratitude to the Indian National Science Academy (New Delhi) for financial assistance under a Project Grant at earlier stages of the and for to the Computer Umversity ofwork; Jodhpur, allowing us to Center, use their TDC.l2 machine.
REFERENCES 1.
DOBBS E.R. and JONES G.O.,Rep. Frog. Phys. 20, 516 (1957).
2.
BOATO G., Cryogenics. 4,65 (1964).
3.
POLLACK G.L, Rev. Mod. Phys. 36, 748 (1964).
4.
HORTON G.K., Amer. J. Phys. 36, 93 (1968).
5.
WERTHAMER N.R., Amer. J. Phys. 37, 763 (1969).
6.
SMITHB.L, Contemp. Phy& 11, 125 (1970).
7.
GLYDE H.R. and KLEIN M.L., CriticalRev. Solid State Sci. 2(2), 181 (1971).
8.
SIMMONS R.O., Solid State Phys. (to be published).
9.
BOBETIC M.V. and BARKER J.A,Fhys. Rev. B2(10), 4169 (1970).
10.
KLEIN M.L, HORTON G.K. and FELDMAN J.L.,Phys. Rev. 184,968(1969).
11.
GUPTA NP. and DAYAL B, Phys. Status Solidi. 18, 731 (1966).
12.
PETERSON O.G., BATCHELDER D.N. and SIMMONS RO, Phil. Mag. 120, 1193 (1965).
13.
LOSEE D.L. and SIMMONS RO.,Fhys. Rev. 172, 934 (1968).
14.
GUPTA R.K. and GUPTA N.P., Nuovo Cimento B66, 1 (1970).
15.
GUPTA R.K. and GUPTAN.P., ibid. B6, 29(1971).
16.
GUPTA N.P. and GUPTA R.K., Can. J. Phys. 47, 617 (1969).
17.
CORNER J., Trans. Faraday Soc. 44, 914 (1948).
18.
BARUAA.K.,J. Chem.Phys. 31, 957 (1959).
19.
GUGGENHEIM EA. and McGLASHAN M.L., Proc. R. Soc. A255, 456 (1960).
20.
GUPTA N.P, Aust. J. Phys. 22, 471 (1969).
21.
MARGENAUH.,ReV. Mod. Phys. 11, 1(1939).
22.
GUPTA N.P. and DAYAL B., Proc. Natl. Acad. Sci. (India) A37, 79 (1969).
23.
FLINN P.A. and MARADUDIN A.A., Ann. Phys. 22, 223 (1963).
24.
SUD K.K., DAVE A.K. and GUPTA N.P.,J. Chem. Phys. 54,4518 (1971).
25.
PETERSON O.G., BATCHELDER D.N. and SIMMONS R.O., Phys. Rev. 150, 703 (1966).
26. 27.
URVAS A.O., LOSEE D.L. and SIMMONS R.O., J. Phys. Chem. Solids. 28,2269 (1967). BORN M. and HUANG K., in Dynamical Theory of Crystal Lattices, p. 38. Oxford University Press (1954).
28.
DAYAL B. and SHARAN B.,froc. R. Soc. (London). A262, 136 (1961).
Pergamon Press.
Printed in Great Britain
LA~FICEHEAT CAPACITIES OF SOLID ARGON P.K. Garg and N.P. Gupta Physics Department, University of Jodhpur, Jodhpur, India. (Received 7 March 1975 by S. Amelinckx)
An anharmonic rigid-atom model has been used to study the heat capacities of solid argon. The model derives the interatomic forces by Buckinghani—Comer potential and takes account of all neighbour interaction. The contribution of the cubic and quartic terms of the potential energy expansion to the heat capacities have been accounted for through Helmholtz-free energy by perturbation theory. It is concluded that the model is most suitable among the ones designed for the class of solids to which argon belongs.
1. INTRODUCTION
certain model to all members of the series, one may
IN RECENT YEARS keen interest has been shown in the study of solid state properties of the inert-gases.’8 It has been due to a number of reasons. Firstly, the atoms constituting the inert gas solids have closed electronu shells, and as such there are neither free. electrons nor magnetic moments to contribute to the thermal properties. Therefore, these properties may be determined by lattice dynamics only. Secondly, the interparticle interaction in the solids to a good approximation are of spherically symmetric central type. Hence the interatomic potential may be written in simple analytic form. However, there are reasons to believe that the interactions are not completely pairwise additive and many-body forces9 may be present. But it is generally believed4’10 that they do not play significant role in the thermal properties of solids. Moreover, the solids form a series of substances (He, Ne, Ar, Kr and Xe), the behaviour of which ranges from classical to quantum.5 The heavier ones are ahnost classical solids, whereas the lighter Ne and He are predominated by quantum effects.’1 Also the vacancy contribution to the thermal properties at the significant.’~3As such the study of the thermal low temperatures of these solids is not much
also be able to study the change in character (from classical to quantum) of the solids. In view of the situation lattice heat capacities of solid argon, a substance centrally situated between the well known quantum solid “helium” and classical “xenon”, have been studied by an “Anharmonic Rigid-Atom Model (ARM)”,’4”5 only to acertain its improvement on replacement of the original Buckingham pair.potential16 by the Buckingharn—Corner17”8 potential. Comparison of the results shows that the modified ARM is better than the original one. 2. ANHARMONIC RIGID-ATOM MODEL The model14’15 assumes inert gas atoms to be rigid-hard spheres and executing quasiharmonic motion at f.c.c. lattice sites. It derives the weak long. range attractive van der Waals and strong short.range exchange interatomic forces by the two-piece Buckingham—Corner (B—C) potential function; r
[
Fm
c1(r)
=
e g
(~)
1 (a, j3) exp a 1 +~ 2~~] r r —
properties of these solids is not only easy, but it also provides ideal testing ground to various proposed lattice dynamical models. In addition, by applying a
(em) 6
and 89
,
—g2 (a, j3) for r > Fm,
90
LAUICE HEAT CAPACITIES OF SOUD ARGON
=
e
Ig’ (a, 13)expa 6
(i +
(1__) r —g
(~)j
~
m
2’~exp
I,
31010 dyn C~2 Bulk modulusenergy = 2.67 Sublimation = x1846 cal mol~ z.p.E.= 187 calmoV’. The anharmonic potential parameters so obtained are: ~
1,
2(a,~3)
4
—
L~
.~
\
forr~
a
where
gi(a,j3)
=
Vol. 17, No. 1
[6+8131[a(1+$)—(6+813)]’,
=
13.02836, e
=
=
0.15,
3.85 x
rm
=
169.5329 x 108
1016
ergs
cm.
3. VIBRATION SPECTRUM AND HEAT CAPACITIES
g 2(a,13)
=
a[a(1+13)—(6+813)1’,
defined for a pair of atoms separated by a distance r. Here e is the depth of the potentiai well for the two atoms at equilibrium separation Tm, a determines the steepness of repulsion at small interatomic distance, and j3 is the dimensionless ratiodipole—dipole of the coefficients 8) and (r6) of dipole—quadrupole (r terms. The inverse six-power term in the potential represents the van der Waals attraction, and the exponential term accounts for the overlap repulsion. l’his B—C potential is a variation of the Buckingham potential}6 The basic improvement in this potential over the original Buckingham potential is the inclusion of a parametrically variable and additive dipole—quadrupole correction to the attractive dipole—dipole term in conformity with quantum mechanical calculations. The presence of the former is considered necessary as the multipole series does not converge rapidly. Secondly it being a two-piece potential function describes the potential curves on both sides of the potential minimum. The quastharmonic values of the parameters e, a and Tm are determined~in terms of the dimensionless parameter 13 by using equiffibrium conditions of the lattice, and the compressibility-free energy relation. The parameter (3 is computed by an expression given by Margenau.~The effect of zero-point vibrations, so appreciably present in the solid,’ has been estimated by Debye theory of specific heats, and is included in the calculation through the potential parameters by a method of iteration.22 The contribution of cubic and quartic terms of the potential energy expansion has been included in the potential parameters through the Helniholtz free energy by perturbation theory.23 The computation takes care of all neighbour interactions,~and uses the latest crystal data at absolute zero of temperature: Lattice parameter = 5.3111 x 10-8 cm.25
In order to compute the lattice vibration spectrum and heat capacities, the Brillouin zone has been divided into 8000 cells of equal volumes. By the cyclic boundary conditions and the symmetry properties of the octahedral zone, the 8000 wave vectors corresponding to eachthat cellcorresponding reduce to 262tonon-equivalent ones including the origin. The computed phonon spectra ranges up to about 2 THZ. Histogram is, therefore, prepared with the interval length I~v= 0.1 THZ. The function g(v) giving the total number of frequencies in the interval (v ~v) to (v + ~~v) is calculated with the histogram. The frequency distribution function g(v) vi v curve (Fig. 1) has been drawn in such a manner, that the area under —
~
2
~
04
00
10
lZ
*4
IS
1$
FIG. 1. Frequency spectrum curve for solid argon at OK obtained with Buckinghani—Corner (exp, 6, 8) potential.
the curve is equal to the area under the histogram. Since small fluctuations in the [g(v), v] curve are of no indication of any real effect in the simple solid under study, all these have been smoothed out. The contribution of each step of frequency spectrum to heat capacity is obtained by multiplying
20
Vol. 17, No.1
LA11ICE HEAT CAPACITIES OF SOUD ARGON
the Einstein function corresponding to mid-point of each of the intervals by the statistical weight of the point. Just to study the variation of specific.heat with temperature, C,, values have been obtained at various temperatures ranging from the lowest to the melting point. The resulting (C,,, T) curve has been presented (Fig. 2) along with recent observed data.25
________________________________
/ /
/
/
‘~
91
2
• /
As usual all the heat capacities have also been converted to equivalent Debye O’~S,and (0, T) curve has been drawn (Fig. 3) for comparison with the latest experimental25 as well as theoretical’4 results.
/
*
___________________________
0
20
80
40 T
(
0*1
40
00
)—~
FIG. 2. Variation of specific heat at constant volume with temperature for solid argon: (—) our theoretical obtained with Buckingham—Corner (exp, 6, 8) potential. (- - -) theoretical obtained with Buckingham type potential (reference 14). (0) Experimental of Peterson et aL (reference 25).
\ \
appreciable
4. DISCUSSION OF RESULTS The frequency distribution function g(v) vs v curve for argon (Fig. 1) refers to OK. It possesses all the characteristics of a monatomic solid. It has two maxima,27 one near about ~ and the other close to the upper frequency limit VL. It is seen that our values of specific heats (Fig. 2) are not only better than the theoretical values of Gupta and Gupta14 by ARM using Buckingham potential, but they show excellent agreement with the latest experimental results of Peterson et al.25 Our (0, T) curve (Fig. 3) is not only similar in shape to the theoretical curve of Gupta and Gupta,14 but it also shows much better agreement with experiment.25 At about 17 K, our curve crosses the experimental curve and as such our 0-values below this point are lower and above it are higher than the observed values. With an increase of temperature beyond 17 K our curve rises and shows excellant agreement with the experiment. The little discrepancy might have crept in during conversion of C,, into °D~ The maximum discrepancy at temperatures lower than 17 K is of the order of 7 per cent. Really it is in comparison to the discrepancy at higher
—-—
10
________________________________ 0
10
30 T
40
temperatures. Perhaps it may be due to the coarseness of the mesh of reciprocal points that we have chosen for the calculation. The main contribution to specific heat in this range of temperature arises from the central part of the Brillouin zone,28 and we have chosen only a few points there for calculation.
(°K)
FiG. 3. Equivalent Debye temperature for the specific heat for solid argon: (—) our theoretical obtained with Buckingham—Corner (exp, 6, 8) potential. (....) theoretical obtained with Buckingham type potential (reference 14). (— —) Experimental of Peterson et al. (reference 25).
5. CONCLUSIONS In view of the above analysis of the results, it may be concluded that the present anharmonic rigid-atom model using the Buckingham—Corner potential function is not only superior to the one using Buckingham
92
LATFICE HEAT CAPACITIES OF SOLID ARGON
potential function, but it is also the most suitable of all such models designed for the study of argon. The model being rigid and consistent may be applied to other members of the °rou~
Vol. 17, No. 1
Acknowledgements — The authors express their gratitude to the Indian National Science Academy (New Delhi) for financial assistance under a Project Grant at earlier stages of the and for to the Computer Umversity ofwork; Jodhpur, allowing us to Center, use their TDC.l2 machine.
REFERENCES 1.
DOBBS E.R. and JONES G.O.,Rep. Frog. Phys. 20, 516 (1957).
2.
BOATO G., Cryogenics. 4,65 (1964).
3.
POLLACK G.L, Rev. Mod. Phys. 36, 748 (1964).
4.
HORTON G.K., Amer. J. Phys. 36, 93 (1968).
5.
WERTHAMER N.R., Amer. J. Phys. 37, 763 (1969).
6.
SMITHB.L, Contemp. Phy& 11, 125 (1970).
7.
GLYDE H.R. and KLEIN M.L., CriticalRev. Solid State Sci. 2(2), 181 (1971).
8.
SIMMONS R.O., Solid State Phys. (to be published).
9.
BOBETIC M.V. and BARKER J.A,Fhys. Rev. B2(10), 4169 (1970).
10.
KLEIN M.L, HORTON G.K. and FELDMAN J.L.,Phys. Rev. 184,968(1969).
11.
GUPTA NP. and DAYAL B, Phys. Status Solidi. 18, 731 (1966).
12.
PETERSON O.G., BATCHELDER D.N. and SIMMONS RO, Phil. Mag. 120, 1193 (1965).
13.
LOSEE D.L. and SIMMONS RO.,Fhys. Rev. 172, 934 (1968).
14.
GUPTA R.K. and GUPTA N.P., Nuovo Cimento B66, 1 (1970).
15.
GUPTA R.K. and GUPTAN.P., ibid. B6, 29(1971).
16.
GUPTA N.P. and GUPTA R.K., Can. J. Phys. 47, 617 (1969).
17.
CORNER J., Trans. Faraday Soc. 44, 914 (1948).
18.
BARUAA.K.,J. Chem.Phys. 31, 957 (1959).
19.
GUGGENHEIM EA. and McGLASHAN M.L., Proc. R. Soc. A255, 456 (1960).
20.
GUPTA N.P, Aust. J. Phys. 22, 471 (1969).
21.
MARGENAUH.,ReV. Mod. Phys. 11, 1(1939).
22.
GUPTA N.P. and DAYAL B., Proc. Natl. Acad. Sci. (India) A37, 79 (1969).
23.
FLINN P.A. and MARADUDIN A.A., Ann. Phys. 22, 223 (1963).
24.
SUD K.K., DAVE A.K. and GUPTA N.P.,J. Chem. Phys. 54,4518 (1971).
25.
PETERSON O.G., BATCHELDER D.N. and SIMMONS R.O., Phys. Rev. 150, 703 (1966).
26. 27.
URVAS A.O., LOSEE D.L. and SIMMONS R.O., J. Phys. Chem. Solids. 28,2269 (1967). BORN M. and HUANG K., in Dynamical Theory of Crystal Lattices, p. 38. Oxford University Press (1954).
28.
DAYAL B. and SHARAN B.,froc. R. Soc. (London). A262, 136 (1961).