Lattice-dynamical calculations on solid nitrogen

CHEMICAL.PHYSIC~ LETTER< --‘., _’

Volume 40;dumber . .2

1 Iune Li76

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LATTICE-DYNAMICAL Giuseppe lkIPPIh!I,

CALCL’LATIONS

Carlo M. GRAMACCIOLI,

di Chimim F&m e Centro CN.R.,

Istituto

ON SOLiD

NiTROGEN

M;issimo SIMONETI’A and Giuseppe 8. SUFFRiTTI

Universittri diM&rz,

20133 &ii&n. Itaiy

Received2 February 1976 Latticedynamical calctdations for solid nitrogen in the neighborhood of 0 K have been performed. using the Born-von Kjrtin method and Luty-kwiey potential function for non-bonded atom-atom interactions. For CY-N2there js a fairly good agreement between calculated and observed values of lattice vibration freauencies, unit-cell parameters and heat capacities. In proximity of the tiansition from a to P or 7 phase, some discontinukies in the c&xdated values of cell parameters are observed.

1. Introduction

tributions, each of them due to an interaction between two non-bonded atoms, according to a semi-empirical

In these years, there has been considerable interest in lattice dynamics of solid nitrogen; calculations of vibration frequencies, unit-cell parameters, Grtineisen coefficients,

specific

heats, etc. have been performed

using both simpIe and sophisticated models [l--6], with essentially good agreement with experiment. Solid nitrogen in the neighborhood of 0 K is observed in three crystalline phases (see fig. 1). Of these, one (ar-Nz)is cubic, with space group PI1 3 or Pa3, according to different authors [4,5, ‘7-131, the latter group being consistent with a minimum of packing energy; another @-N2) is hexagonal, disordered [7,9,14], and the thud (y_Nz) is tetragonal, with space grogp’P42/mnm [I 51.

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nl

In line with this general interest, and encouraged by the essential success of our calculations, as applied

to other substances [16-181, we tried to consider solid nitrogen, with an emphasis upon free-energy calculations’in the neighbqrhood of the transition region between different phases.

2. Method of cakukxtion OO

In our calculations we refer lo a perfect crystal nxxde of %gid”

molecules, of which the oscillations are h&no& and independent of each other. The potential energy E is assumed to be a sum of various con- ‘.

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Fig,. 1_ Phase diagram for nitrogen, as reported by Stewart 130). The dashed lines AB and EB represent the zone where -we found discontinuities in calculated unit-ceil parameters _ as a function of temperature and przsuse..

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potential which is function only of histance. To this purpose, 211packing‘distances below 12 A have been considered: Since the experimental molecular position do& not always correspond to a minimum of energy according to the semi-empirical functibns used, a preliminary treatment is usually necessary to modify the rotational and translational coordinates, in order to bring the molecule into the desired minimum; for this purpose, a Raphson-Newton procedure is used [ 16-181. Since unit-cell parameters, being obviously temperature-dependent, do not correspond to the minimum of energy, as evaluated from lattice&ms. our preliminary minimization is not performed with respect to these parameters: they are determined at the end of latticedynamical calculations, considering the minimum of Gibbs free energy G. For the particular cases of CCand 7-Nz , if the molecule is assumed

to be rigid, no preliminary minimization of packing energy is necessary, because of symmetry constraints.

After the minimum of energy is reached, from force constants we start to build up dynamical matrices iWl(q)for various values of the wave vector 4 in the frrst Brillouin zone. As it is well-known,

vibration-

al frequencies are obtained from diagonalization of these dynamical matrices, and from them a density of . states can be obtained. For sampling the Brillouin zone, we adopted the same procedure as in our previous studies [16-181.

The sampling is thicker around the origin, and the grid over the Brillouin zone contains 256 points: in order to spare computing time, symmetry equivalence has been accounted for [19]; density of states has been evaluated with a channel width of 2 cm-‘. This “fast” procedure agrees very well with results obtained with 2 considerably ‘%hicker” sampling. A check of

this kind was necessary in order to be sure that the irregularities observed in free energy G (see below) did not derive from too drastic an approximation. From density of states g(Ui>, thermodynamic functions can be easily derived, according to statistical thermodynamics. In particular, the Gibbs free energy is given by: G=-RThZ+PV+EO,

0)

where In Z, the logarithm of the vibraticnal partition function in the crystal is given by:

t June’1976

In Z = !n Zint i- X

g(Vj)

i

ItI

[

1-exp(-hv,/kT)1

AVj ..(2)

where Z.,,,,.the partition function relative to the internal degrees of freedom, can be considered to be a constant with respect to variation of cell parameters,. as required by our -assumption of molecular rigidity. The zero-point energy E. is evaluated as: Eo = R. g

(hvi,,/2k) f 5R C i

g(Ui)(hvi/2k)hj

-- (3)

As is we11 known, the mdst probable configuration of a system at equilibrium for a given temperature and pressure corresponds to a minimum of Gibbs free energy G. In order to compare values of G relative to different crystals of the same compound (the crystals may differ in unit-cell parameters only, or may be of quite different structures) it is necessary to use a common reference system. For this purpose, the energy levels can be considered relative to a reference zero of “separated” (i.e. non-interacting) “stationary molecules”_ All this results in writing: G=-RTlnZ,,,

+Epck +PVfEo~e,t~.

(4)

where EpaA is the packing energy in the crystal, as obtained from lattice sums, Eon,,,) is the zero-point energy relative to external (lattice) modes and In Z,,, is the second term on the right side of (2). At a given temperature and pressure, the terms on the right side _

of (4) clearly depend upon cell parameters (and crystal structure), and the values of G relative to different crystals can be compared with each other. In practice, a calculation as described above is performed for a series of points corresponding to different cell parameters and structures in the neighborhood of a plausrble minimum of G (for instance, an experimental result); the theoretically most stable structure will correspond to the minimum of G. This is the same prpcedure we used for some hydrocarbons [18j .

3. Results and discussion For N-N interaction between non-bonded atoms, semiempirical potential functions, as given in the literature, are mainly of two kinds: a “6-exp” and a “12-6” form. Among the former, the function proposed by

Luty and Pawley [4] seems to be especially promising. 211

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This function has recently been derived just for having best fit to data relative to LY-and 7-Q; consequently, we chose it for our calculations. Internal mode contributions have been derived using the experimental frequency of 2327 cm-l for stretching vibrations in the N, molecule 1201. In table 1 experimental data for a-N2 are reported: the agreement with our calculations is fair. As already shown by some authors (e.g. ref. [18]), thermal expansion comes out reasonabIy well from quasiharmonic approximation. The thermal expansion coefficient 0: = (1 /v) (a Y/a7)p in the neighborhood of 0 K is not, however, in excellent agreement with experimental data (cr,bs = I .95 X 10d4; QI,~, = 3.3 X lo-” K-l): in a previous study on some hydrocarbons our results were more satisfactory [18]_ It seems that, at least for this purpose, Huler-Zunger’s function gives better results (frorn their calculations [6] (Y= 2.8 X 1O-4 K-l). The compressibility coefficient fl= -(l/v) X (a V/W), in the region here considered is in very gocd agreement with experimental data [2 l-231 QQlc = 0.0355; &bs = 0.0312 kbar-I); also the entropy and the heat capacity cP (assumed to be equal to cV) seem to be reasonable, taking account of the perturbations occurring in the real crystal in proximity of -phase transitions. Whereas the thermal expansion coefficient is - on the whole - constant from 0 K to about 30 K, at higher temperature our calculations indicate some discontinuities. In fact, as shown in fig. 2, if the free energy G iaplotted at various temperatures (in this figure, pressure is zero) as a function of cell parameter a, each temperature shows more than one minimum. The absolute minimum should, of course, correspond to the calculated cell parameter at that temperature: we see tha? increasing temperature brings about an exchange in absolute minima, so that at about 33.5 K the unit-cell parameter a shows a r;udden variation. Increasing temperature again, a similar situation happens at 44 and 55 K; all these variitions should correspond to a sudden expansion of the ideal crystal. Owing to this remarkable situation, we repeated our calculations at intervals of 0.005 A in the cell parameter (z, in order to have the lines in fig. 2 relatively free from interpolation errors: our results, however, remained unchanged: A similar situation is ob-

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Fe. 2. The Gibbs free energy asa function of unit-cell prameter a for (E-N* at various temperatures (here at zero pressure). Each isotherm is vertically shifted in crrder not to have all the lines crowded together_

Fig. 3. The Gibbs frez energy of cr-Nz at 20 K. as it results from our calculations for various values of pressure. as il

served with respect to variation in pressure: as shown in fig. 3, for P> 3 kbar the cell undergoes a sudden contraction. It may be particularly interesting to report the zones where these discontinuities are observed on the phase diagram: these zones correspond to the dashed lines in fig. 1. They are generally close to the limits of stability of OLphase with respect to either /3 or y phases: this supports our point of view that such discontinuities are physically meaningful, and do not derive from inadequacies of our model. It might have. been interesting to compare the calculated free energy of the Q!phase with the corresponding values for p and r-Q. However, for the p phase our calculation scheme cannot be applied at present. For 7-N2, unfortunately we found negative eigenvalues of dynamical matrices in a certain region of the Brillouin zone, situated approxiz&tely along the [l IO] direction; these negative eigenvalues peisist also when other potential functions are used, like for

instance Huler-Zunger’s [6]. These inconveniences clearly reflect some inadequacies of the present treatment. As is well known, imaginary frequencies often appear if large anharmonicities have to be dealt with. !n such cases, a harmonic treatment fails and further quantum corrections become important. Since it is difficult, however, to ascertain whether most of this trouble is really due to anharmonicity, as these effects are probably not so different for LYand 7 phases, before declaring our approximation to 5e inadequate, we are considering a reinvestigation of this subject using anisotropic atom-atom potential functions.

function

of unit-cell p2rametera.

References [l] 0. Schnepp and A. Ron, Discuss&s

Faraday Sot. 48

(1969) 26. [2] T.S. Kuan, A. Warshel and 0. Schnepp, J. Chem. Phys. 52 (1970) 3012.

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1’,

-I- [3], ~k.~~cob~~nd -3647.

” [4].T.

O.%hnepp,~J.

Chem. Phys; 58 (1973)

Luty and G-S: F‘awley. Chem. Phys. Letters 28 (1974) 593; J.C. Fiaich. N.S. Gillis and-A&. Anderson. J. Chcm. Phys. 61 (1974) 1399. A. &urger and E. Huler. J. Chem. Phys. 62 (1975) 3010; private communic=ation_ T.H. Jordan. H-W. Smith. W.E. Streib and W.N. Lipscomb, J. Chem. Phys. 41 (1964) 756. LH. Bolz, M.E. Boid. F-A. Mauer and H.S. Peimr, Acta Cryst. 12 (1959) 247. A:F. Schuch and R.L hlills, J. Chem. Phys. 52 (1970) 6000. S.J. La Plaw and W.C. Hamilton, Acta Ckyst. B28 (1972) 984. E.J. Wachtel. J. Chem. Phys. 57 (1972) 5620. 1.R. Brookeman and T.A. Scott, Acta Cryst. B28 (1972) 983. J.A. Vettables and CA. English, Acta Gyst. B30 (1974) 929. W.E. Streib, T.H. Jordan and W.N. Lipscomb, J. Chem. f’hys. 37 (1962) 2962. R.L. Mills and A.F. Schuch, Phys. Rev. Letters 23 (1969) 1154.

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(161 G. Filippini; C&. G&wcioIi, M. Simonetta and G-B. Suffritti, J. Chem. Phys. 59’(1973) 5088. [ 17’3 G. Filippini, CM. Gramaceioli. M. Simonetta and G.B. Suffritti. Chem. Phys. 8 (1975) 136. [iSI G. Fiiippini, C.M. GramkioIi, M. Simonetta and G.B. Suffritti, Chem. Phys. Letters 35 (1975) 17. [19] G. Filippim, CM. Grarnacdioli. M. Simcnetta and G.B. Suffritti, Acta Cryst. A32 (1976), to be published. [ZO] A. Anderson,T.S. Sun and M.C.A. Donkersloot. Can. J. Phys. 48 (1970) 2265. 1211C.A. Swenson. J. Chem. Phys. 23 (1955) 1963. 1221R.L. Millsand E.R. Grilly, Phys. Rev. 99 (1955) 480. 1231 E.R. Grilly and R.L. hfills, Phys. Rev. 105 (1957) 1140. 1241 W-F. Giauqce and J-0. Clayton. J. Am. Chem. Sot. 55 (1933) 487.5. I251 hf. Brith, A. Ron and 0. Schnepp. J. Chem. Phys. 51 (1969) 1318. WI K.K. Kelly, Bureau of Mints Report 389, Washington D.C. (1962). [271 MM. Thiery, D. Fabre, M.J. Louis and H. Vu, J. Chem. Phys. 61 (1974) 4559. I281 A. Anderson and G.E. Leroi, J. Chem. Phys. 45 (1966) 4359. I291 A. Ron and 0. Schnepp, J. Chem. Phys. 46 (1967) 3991. 1391 J-W. Stewart. J. Phys. Chcm. Solids 1 (1956) 146.