Model for the lattice vibrations of a crystal of diatomic molecules—II. Quasiharmonic equation of state

J. Phys. Chem. Solids, 1972,Vol. 33, pp. 129l- 1299. PergamonPress. Printed in Great Britain

MODEL FOR THE LATTICE VIBRATIONS OF A CRYSTAL OF DIATOMIC M O L E C U L E S * - I I . Q U A S I H A R M O N I C EQUATION OF STATE G. E. JELINEK Sandia Laboratories, Albuquerque, N e w Mexico 87115, U.S.A. and A. M. KARO Lawrence Livermore Laboratory, Livermore, Calif. 94550, U.S.A. and

L. J. SLUTSKY Department of Chemistry, University of Washington Seattle, Wash. 98105, U.S.A. ( R e c e i v e d I S e p t e m b e r 1971 ) A b s t r a c t - T h e equation of state of the diatomic solids oxygen, nitrogen, fluorine, and carbon monoxide

is presented for a calculation based upon a homogeneous deformation description of the crystal Hamiltonian and a Lennard-Jones[6-12] type phenomenological description of the interaction between non-bonded atoms in the solid. The zero-pressure specific heats, thermal expansivities and compressibilities of our model calculation describe most satisfactorily the low-temperature-phase thermal data of 02, N~, F2, and CO. The predictions of our model are in good agreement with the existing (albeit limited)experimental non-zero pressure data. 1. INTRODUCTION

the thermodynamic properties. The heat capacities of all of these solids have been carefully investigated. The intermolecular forces are of relatively short range. They are, therefore, a reasonable set of systems for the testing of models and methods for the estimation of the effects of volume anharmonicity on thermal properties. In this paper we shall be concerned with volume effects calculated in the quasiharmonic approximation. Purely anharmonic 3 and 4 phonon interaction terms shall not be considered. In the first paper in this series (hereafter referred to as I) a model structure (approximating the/3-oxygen structure) and potential (Lennard-Jones [6-12] centered on the individual atoms) for 02, F2, N~, CO, Clz, and Br2 was described. The parameters of the Lennard-Jones potential were evaluated *Work supported by the U.S. Atomic Energy from heats of vaporization and crystal density. Commission. The .normal modes corresponding to these THE DISCUSSION of the thermal properties of crystals in terms of a harmonic model is useful qualitatively and often successful quantitatively. There are, however, many crystals with substantial anharmonic contributions to their thermodynamic properties over the larger part of their solid range. There are significant properties of all crystals (i.e., thermal expansion, lattice thermal resistance, zero-point dilation) which are either predicted to be zero or grossly underestimated in a purely harmonic model. There is evidence for relatively largeamplitude torsional or librational motion in the solidified 'diatomic gases': O~, F2, N2, and COl1]. One might expect a correspondingly large volume dependent contribution to

1291

1292

G . E . J E L I N E K , A. M. K A R O and L. J. S L U T S K Y

parameters were computed. The thermal amplitudes and the i.r. spectra were discussed on the basis of the dispersion curves and frequency distribution spectra of the harmonic model. It was, however, found that this model implies a large volume dependent contribution to the thermal properties. Anharmonic contributions due to thermal expansion have been taken into account via a quasiharmonic calculation based upon the Born and Huang method of homogeneous deformations[2]. The derivation of the relevant equations constitutes the main argument of Sections 4 and 6 of Ref. [2] and will only be outlined here. 2. CALCULATIONS

A lattice is said to be homogeneously deformed if the a m Cartesian component of the displacement of the k th atom in the Ith unit cell, ua(lk), can be expressed

u.(lk) = E u.brb(lk) b

(1)

where rb(lk) is the b th Cartesian-coordinate of the atom (lk). Such a deformation preserves the homogeneity of a lattice and the methods of the Born-von Kfirmfin theory may be appropriately applied to the deformed structure provided that the force constants are evaluated in the deformed configuration. Elastic strain and thermal expansion are examples of homogeneous deformations. Born and Huang seek an expansion of the Helmholz free energy, A, in the form A =A~

ab

A(ab)a~b+89 ~, A(ab,cd)fiaoaca. ab

ca

(2)

From such an expression all the thermodynamic properties may be derived by appropriate differentiations[3]. Their program is in outline: (i) to express the potential energy as a power series in the internal coordinates (those describing the displacement of the atoms from their positions in the homogeneously deformed s t r u c t u r e ) a n d in the macroscopic

parameters Uab which characterize the deformation. The resulting Hamiltonian is partitioned into a part (HI) describing an assembly of harmonic oscillators with frequencies obtained by routine solution of the purely harmonic problem and a part (Hti) containing terms linear and quadratic in the uab. To the desired degree of accuracy H = HI + Hn. The remaining terms in H, of third or higher order in the internal coordinates or the macroscopic parameters, are ignored. (ii) Using HII as the perturbing Hamiltonian the energies are evaluated by means of second-order perturbation theory. (iii) The partition function and free energy are evaluated assuming that, for all levels, the first and second order corrections to the energy are small compared to kT. In terminating the expansion of the free energy with terms quadratic in the strain parameters it is implied that the elastic constants are given simply by the termsA (ab, cd) and are thus explicitly functions of the temperature, but not of the strain. This is contrary to the existing body of experimental information on the temperature and pressure variation of elastic constants which suggest that any realistic equation of state must provide for an explicit volume-dependence. We have included terms third and fourth order in the strain parameter, Go, where it is now assumed that the third and higher order elastic constants, like the second order elastic constants, are largely accounted for by the corresponding term in the expansion of the static lattice energy. * Once the coefficients in equation (2) are computed the calculation of the thermodynamic properties is relatively straightforward. The equilibrium values of the strain parameters Gb are calculated at each temperature from

( aA p.ob

(3)

*For oxygen at 0~ the static lattice contribution to EA (ab, cd) is 81 per cent. ab cd

A CRYSTAL OF DIATOMIC MOLECULES-II The volume in terms of the equilibrium volume, V0, is V

V0(det [1 + 2a,~l) 'n.

=

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(4)

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of thermal

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Fig. l(a).

Cv=-- W-i%o. The coefficient obtained from

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The zero point dilatation and theoretical volumes used in the quasiharmonic frequency calculation of paper I are given by the P = 0 solution of equations (3) and (4). From the above Cp is easily obtained from C p - C v = TVod//3. For the diatomic rhombohedral model the tTa~have the form ~11 = ~22 ~ tT~ ~ 0 and tTo~= 0 for a ~ b, where a refers to the Xa m Cartesian coordinate of Fig. I-1.

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Fig. l(b). I

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3. RESULTS N2

In Fig. 1 the computed heat capacities are compared with the experimental results for oxygen*[4], fluorine[8], nitrogen[9, 10] and carbon monoxide[l l, 12]. The agreement is very good for the low-temperature forms of 02, F2, and CO. Our calculated Cp for or-N2 are generally 15 per cent lower than the experimental data. It will be seen that this

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*The smoothed heat capacities given by Orlova were derived utilizing also the experimental results of Kostryukova and Strelkov[5], Clusius[6] and Giauque and Johnston[7].

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Fig. 1(c).

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1294

G.E. JEL1NEK, A. M. KARO and L. J. SLUTSKY

mined coefficients of thermal expansion of oxygen[13] and nitrogen[13, 14] with our calculation is made in Fig. 2 where the 2.0 -CpFJR. THEORY f 9 - - - Cv/3R, THEORY predicted volume expansivity for F2 is also plotted. Agreement with the experimental 1.5 data of Manzhelii, Tolkachev and Voitovich (MTV)[13] for N2 is exceptionally good, l.O although the expansion curve of Heberlein, Adams and Scott (HAS)[14] differs substanrip tially from that of MTV. Our integrated volume change for the a-phase of N2 is virtually 0 1 I I I I lO 20 " 30 40 SO 60 70 the same as the H A S curve, since we obtain TEMPERATURE PKI [ V ( 3 5 . 6 ) - - V ( 0 ) ] V ( 0 ) = 2 . 6 per cent which Fig. l(d). compares to their 2.2 per cent. The temperaFig. 1. Constant volume (Cv)and constant pressure (Cp) ture-dependence of the experimental expansivheat capacities vs. temperature. The experimental data ity of O3 at the lowest temperatures is some(all are Cp) are from the following: 02, Ref. [4]; F2, Ref. [8], N2, Ref. [10] (closed circles) and Ref. [7] (open what unusual and there is a possibility of circles); CO, Ref. [11] (closed circles) and Ref. [12] substantial error. (closed triangles). Our results for the isothermal compressibility /3 are compared with experiment in discrepancy in Cp cannot be due to a poor Fig. 3. The experimental results [ 15-18] have description of the thermal expansion or been obtained by estimating the zero-pressure compressibility. The effect of the strain- slope of published PV isotherms and may be induced contribution to Cv is large, compar- in error by about 20 per cent. The data for able with the harmonic contribution at the y-oxygen[19] (stable above 43-8~ and/3-N2 although they are higher temperatures and, owing to the con- (stable above 35-6~ siderable zero-point dilation, is not negligible plotted in Fig. 3 are not really comparable with these calculations. The agreement of the at low temperatures. A comparison of the experimentally deter- theoretical and experimental low-temperature ~5

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A CRYSTAL OF DIATOMIC MOLECULES-II I

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TEMPERA'lURE (~

Fig. 2(c). Fig. 2. Volume expansivity vs. temperature. For F2 shown also are the anisotropic linear coefficients parallel (lower broken curve) and perpendicular (upper broken curve) to the 3-fold axis. The experimental data are from the following: 02, Ref. [13]; N2, Ref. [13] (closed circles) and Ref. [14] (broken curve).

compressibilities of 02 and N2 is quite satisfactory since the 6-12 model potential has a fixed repulsive parameter.

1295

In a previous calculation[20] it was observed that while the theory does not predict the correct temperature derivatives of the Heimholz free energy for temperatures greater than one third the melting temperature for argon, krypton, and xenon it does obtain nearly correct volume derivatives throughout the temperature range of the solid. Figure 4 contains the zero-pressure temperature dependence of the molar volumes of 02, N2, and F2. The low temperature volumes must of course agree with the experimental volumes since the recursive refinement procedure for the potential parameters utilized the density of the solid at its lowest observed temperature. These temperatures are 28~ for oxygen [21], 4-2~ for nitrogen [22], and 23~ for fluorine [23]. Here, as observed in the case of the expansion coefficient (Fig. 2), the theoretical and experimental curves for 02 are not in good agreement even for the low temperature a-form. The theoretical curve for N2 is in good agreement with the measurement of Tolkachev and Manzhelii[24] for both the a- and /3-phases of Nz. Our calculated temperature-dependence of the molar volume of F2 is in virtual agreement with the X-ray densities as measured by Meyer, Barrett and Greer[25]. The molar volume of O2 and CO as a function of the external pressure is plotted in Fig. 5. The theoretical calculation for O2 agrees well with the experimental result of Stewart [16] at the higher temperature and pressures. Our theoretical result for CO agrees with the data of Stevenson [ 15]. The pressure dependence of the compressibility of N2 is compared with the data of Stewart[17] in Fig. 6. It is seen that the agreement, in fact, increases with increasing pressure. 4. CONCLUSIONS

The possibility of further elaboration in the calculations exists within the basic framework of our model. In establishing a connection between the equilibrium internuclear distance

1296

G . E . J E L I N E K , A. M. KARO and L. J. S L U T S K Y

and the position of the minimum in the 6-12 potential the pair-potential was summed over all shells of neighbors for which the interaction is significant. The lattice-dynamical treatment considers interactions between an atom and its first- and second-nearest non-bonded neighbors. The inclusion of interaction with more distant neighbors would complicate the calculation, whereas the use of more elaborate potential functions only alters the numerical values of the force constants and thus does not significantly add to the difficulty of the calculation. I

It does appear that the simple model and method discussed here, without adjustable parameters, give a good account of the equation of state of a number of solids where there is reason to believe that the anharmonicity due to thermal expansion is large. In the first paper of this series, it was argued that the combination spectrum of solid oxygen could be explained and the far-i.r, spectrum of CO and N~ rationalized on the basis of this model. Thus for simple structures the method of homogeneous deformations seems to offer a means of evaluating the anharmonic volume

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A CRYSTAL OF DIATOMIC MOLECULES-II 36

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Fig. 3 (c).

Fig. 4(b). Fig. 3. Temperature dependence of the zero pressure compressibility. The experimental data are from the following: O~, Ref. [15] (closed circles), Ref. [16] (open triangles), and Ref. [19] (open circles); N2, Ref. [14] (broken curve), Ref. [ 17] (open triangle), Ref. [ 18] (closed square), and Ref. [ 19] (closed circles).

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Fig. 4(c). Fig. 4. Temperature dependence of the zero-pressure molar volume. The experimental data are from the following: 02, Ref. [26]; N2, Ref. [24]; F2, Ref. [25].

60

TEMPERATURE (~

Fig. 4(a).

contribution to the equation of state of anisotropic materials without unreasonable computational effort. The extension of these methods

to molecular crystals with complex structures would be very tedious, but the results obtained here seem to offer some hope that simplified tractable models, structures of artificially high symmetry, or 'rigid body approximations' might give adequate representations of the

1298

G . E . J E L I N E K , A. M. K A R O and L. J. S L U T S K Y O.o o , I ~

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T 9 5I~ 9 S'~WA RT

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density of energy levels for structures with rather large numbers of atoms in the translational unit.

REFERENCES l?

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16

5

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10

15

20

P Iltbar)

Fig. 5(b). 0.0

I

I

I

I

T 9 ~K

----

0.3 0

STEVENSON

I

I

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2

4

6

's

1o

P (kbar)

Fig. 5(c). Fig. 5. Isotherms for O2 (32 and 51~ and CO (60~ The experimental data are from the following: O2, Ref. [16]; CO, Ref. [15].

1. C A H I L L J. E. and L E R O I G. E., J. chem. Phys. 51, 97 (1969) ibid. 51, 1324 (1969). 2. B O R N M. and H U A N G K. Dynamical Theory of Crystal Lattices, Oxford University Press, New York (1954). 3. GIBBS J. W., Collected Works, Yale University Press, New Haven (1952). 4. O R L O V A M. P., Zh. Fiz. Khim. 40, (12) 2986 (1966). 5. K O S T R Y U K O V A M. O. and S T R E L K O V P. G., Dokl. akad. nauk SSSR 90, 525 (1953). 6. C L U S 1 U S K.,Z. Phys. Chem. 133,41 (1929). 7. G I A U Q U E W. F. and J O H N S T O N H., J. Am. Chem. Soe. 51, 2300 (1929). 8. H U J. H., W H I T E D. and J O H N S T O N H. L., J. Am. Chem. Soc. 75, 5642 (1953). 9. G I A U Q U E W. F. and C L A Y T O N J. 0., J. Am. Chem. Soc. 55, 4875 (1933). 10. B A G A T S K I I M. I., K U C H E R Y A V Y V. A., M A N Z H E L I I V. G. and POPOV V. A., Phys. Status Solidi 26, 453 (1968). 11. G I L L E. K. and M O R R I S O N J. A., J. chem. Phys. 45, 1585 (1966). 12. C L A Y T O N J. O. and G I A U Q U E W. F., J. Am. Chem. Soc. 54, 2610 (1932). 13. M A N Z H E L l l V. G., T O L K A C H E V A. M. and V O I T O V I C H E. I. Phys. Status Solidi 13, 351 (1966). 14. H E B E R L E I N D. C., A D A M S E. D. and S C O T T T . A., J. low temp. Phys. 2,449 (1970). 15. S T E V E N S O N R.,J. chem. Phys. 27,673 (1957).

A CRYSTAL OF DIATOMIC M O L E C U L E S - I I 16. STEWART J. W., J. Phys. Chem. Solids 12, 122 (i 960). 17. STEWART J. W. Phys. Rev. 97, 578 (1955). 18. SWENSON C. A.,J. chem. Phys. 23, 1963 (1955). 19. BEZUGLYI P. A., TARASENKO L. M. and IVANOV Yu S., Soviet Phys.-solid State 10, 1660 (1969). 20. JELINEK G. E.,Phys. Rev. B3,2"~16 (1971). 21. BARRETT C. S. and MEYER f_.., Phys. Rev. 160, 694 (1967).

1299

22. BOLZ L. H., BOYD M. E., MAUER F. A. and PEISER H. S. ,4 cta Crystallogr 12, 247 (1959). 23. MEYER L., BARRETT C. S. and GREER S. C., J. chem. Phys. 49, 1902 0968). 24. TOLKACHEV A. M. and MANZHELII V. G., Soviet Phys.-solidState 7, 1711 (1966). 25. MEYER L., BARRETT C. S. and GREER S. C., J. chem. Phys. 49, 1902 (1968). 26. BARRETT C. S., MEYER L. and WASSERMAN J.,Phys. Rev. 163, 851 (1967).