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Model for the lattice vibrations of a crystal of diatomic molecules—I. Frequency distributions, Debye-Waller factors, and infrared spectra
J. Phys. Chem. Solids, !972, Vol.33, pp. 1279-1290. PergamonPress. Printedin Great Britain
MODEL FOR THE LATTICE VIBRATIONS OF A CRYSTAL 9 F DIATOMIC M O L E C U L E S * - I . FREQUENCY DISTRIBUTIONS, DEBYE-WALLER FACTORS, AND INFRARED SPECTRA G. E. JELINEK
Sandia Laboratories, Albuquerque, New Mexico 87115, U.S.A. and
L. J. SLUTSKY Department of Chemistry, University of Washington, Seattle, Wash. 98105, U.S.A. and A. M. KARO
Lawrence Livermore Laboratory, Livermore, Calif. 94550, U.S.A. ( R e c e i v e d 1 S e p t e m b e r 197 I)
Abstract-Dispersion curves, frequency distribution spectra, and anisotropic X-ray temperature factors are computed for a model of the low-temperature forms of crystalline 03, N2 and F2 which assumes that the static equilibrium structure involves the rhombohedral close-packing of diatomic molecules. The interaction between nonbonded atoms is described by a Lennard-Jones [6, 12] potential centered on the atoms. An interpretation of the far-i.r, and lattice-combination spectra of these crystals in reasonable accord with the computed dispersion curves is offered. Theoretical root-meansquare amplitudes are in good agreement with predictions derived from X-ray and Raman scattering data.
1. INTRODUCTION BORN AND HUANG[1] have developed a computationally straightforward approach for the evaluation of the thermodynamical properties of a quasiharmonic lattice. Herein we apply the formulae of Born and Huang to calculate the thermal and optical properties of a model of solidified oxygen, fluorine, nitrogen, carbon monoxide, chlorine, and bromine. In this paper we obtain the secular equation and its eigenvalues and eigenvectors which constitute the basis for the evaluation of the volume dependent properties discussed in the following paper. A description of our model is given in Section 2. A comparison of the model with
the low temperature crystal structures of the solidified diatomic gases indicates it might constitute a reasonable model for the description of the lattice vibrations. The assumed interaction between nonbonded and between covalently bonded atoms is presented ir~ Section 3. The parameters of the LennardJones[6, 12] potential are derived from the low temperature solid state data in a recursive refinement procedure. The force constant matrices and elements of the secular determinant are derived in Section 4. In Section 5 the dispersion curves, frequently distribution spectra, and thermal displacement correlations are compared with experimental data. In the following paper (referred to as II) the general equation of state based upon the method of homogeneous deformation is presented.
*Work supported by the U.S. Atomic Energy Commission. 1279
1280
G.E.
J E L I N E K , L. J. S L U T S K Y and A. M. K A R O
2. THE MODEL Interactions with more distant neighbors will The static equilibrium structure is assumed be neglected. to be the rhombohedral close-packing of Between 23.9 and 43-6~ neutron[2] and diatomic molecules with the axes of the electron[3] diffraction data indicate that molecules being parallel to the threefold axis B-oxygen has a primitive monomolecular of the crystal. The assumed crystal structure rhombohedral structure with the axis of the is illustrated schematically in Fig. 1 and in molecule oriented parallel to the threefold projection on a plane normal to the threefold axis of the crystal. In the/3-oxygen structure [111] axis in Fig. 2. In the case of 'ideal' the nine nearest nonbonded neighbors of a packing of diatomic molecules, each atom has given atom in the ideally-packed structure one covalently-bonded nearest neighbor at divide into a set of six atoms lying in the [111] distance do, nine nonbonded near neighbors at plane at a distance of 3.31 ,~ and a set of three a0 and six nonbonded neighbors at (a02 + d02)1/2. atoms at a distance of 3.18 ,~. In the monoclinic structure of a-oxygen[4,5] (stable below 23-9~ the 0 - 0 axes, remain parallel; but the six near-neighbors in the plane normal to the 0 - 0 axes, equivalent in the (2,0 ) /3-oxygen structure, are split into a set of four at 3-20 and two at 3-43 ,~. I) In B-fluorine [6] stable between 45.55~ and the melting point at 53.54~ the unit cell is cubic and contains eight molecules. Alphafluorine [7] stable below 45.55~ is monoclinic 10,01 with four molecules per unit cell. For/3-nitrogen stable above 35.6~ Streib (6,1) (0,1) et a/.[8], prefer a precession model in which the molecular centers form a hexagonal closepacked lattice with a precession angle of 56.0deg. Nitrogen[9], below 35-6~ has a Fig. 1. The interaction of atom (0,0) with a few of its near neighbors for the assumed crystal structure. cubic structure in which a given atom has three nonbonded neighbors at 3.42 ,~ and six at 3.55 A. The low temperature a form of carbon monoxide[10] is isomorphous with a-nitrogen. Bader, Henneker and Cade [11] calculated electron density contours from H a r t r e e - F o c k ; 9 ~, A ,.: ground-state wave functions for oxygen, fluorine, and nitrogen. They concluded that the 0-002 contour, which includes 95 per cent of the charge density, represents the dimensions of the molecules. According to M e y e r f ' 7 ; [12] the packing density in a-oxygen is 71.4 per cent as calculated from the unit cell volume and the 0.002 contour. Since a structure of close packed spheres has a packing density of 74 per cent, taking into account the Fig. 2. The coordinates s y s t e m s - t h e projection is on a plane normal to the molecular axes. oblong shape of the oxygen molecule, a-
(6,0~)
G
G
@
@
A CRYSTAL OF DIATOMIC MOLECULES-I
oxygen must be considered a close-packed structure. In a similar analysis a-fluorine must also be considered a close-packed structure. The main difference between a-oxygen and a-fluorine is a doubling o.f the c-axis of the unit cell of fluorine due to'~ilt (about 11 ~ in alternating directions of the molecules perpendicular to the basal plane. It will be seen in the subsequent discussion that if our ideally packed equilibrium ~tructure is assumed, when anharmonic terms in the potential are considered, an anisotropic zero-point dilation in the observed sense is predicted. Thus, when considering the dynamics of our static model for 02 at 30~ it is observed that the anisotropic dilation of the lattice splits the 9 nearest nonbonded neighbors into a set of 6 at a distance of 3.23 A and 3 at 3.21 ,A. This dynamical configuration of our model conforms to the observed r oxygen structure. Thus, it would seem possible that a primitive rhombohedral equilibrium structure with large vibrational excursions from the equilibrium configuration might constitute a reasonable model for the description of the lattice vibrations of solid oxygen and fluorine and perhaps for nitrogen and carbon monoxide as well. While a lattice dynamical calculation for the true crystal structures (such as monoclinic with 8 molecules per unit cell) may be unjustified, our model simulates reasonably well the mean number and distances of the near neighbors in the true crystal structures. 3. FORCE CONSTANTS AND THE POTENTIAL REFINEMENT
The coordinate systems pertinent to a discussion of this problem are illustrated in projection in Fig. 2 where the 'A's' are the basis vectors of the rhombohedral lattice and the 'X's' are Cartesian unit vectors. The position of the k-th atom in the/-th unit cell with respect to the atom (0,0) chosen to be at the orig!n will be specified by r ( l , k ) = r(/) + r(k) = l~al + 12a2 +/3a3 + r(k) with the '/'s' integers. The origin of the unit cell will
1281
be taken to be at the atom indexed by k = 0; thus, r(0) = 0 while r(1) = -- doX3. Let the displacement of the k-th atom in the /-th unit cell from its equilibrium position be denoted by the vector u(/,k) with Cartesian components ua(l,k), then the quadratic and higher order interatomic force constants are defined as the partial derivatives of the potential energy ~ with respect to the components ofu. a2~p cbab ( ll',kk') =
OUa (l,k)Oub (l',k') '
~abc ( ll' l",kk' k") a3~ = aua(l,k)aub(l . ,k. ). a.u ,. ( l. .,k ) etc.
(1)
The quantities gaab(ll',kk') which specify the harmonic interaction of a given pair of atoms may be written as a 3 x 3 matrix which will be designated ~ ( l l ' , k k ' ) . If the Lennard-Jones[6, 12], potential is written in the form V = - ~[2(ro/r) 6 - (ro[r)12], then ~s, the cohesive energy (per molecule) of the crystal, may be expressed.
r = - - ~ ~,' {2[ro/r(l,k) ]6--[ro/r(l,k) ]12} 9 (2) lk
If Wn is defined in terms of a, the distance between nearest nonbonded neighbors, by
Wn = ~,' [a/r(l,k)]", lk
then ~o, = --e[ 2We(ro/a) 6 -
W12(ro/a)12].
(3)
The equilibrium Value of a ((a~os/Oa)= 0) is thus ao (I/V12/W6)I/6Fo and the energy ~P8= =
EW62/W12 .
We have obtained nonarbitrarily adjustable potential parameters in a recursive refinement procedure. If ~o, plus the zero-point energy of the model* is associated with the experimental *The contribution of the optical branch associated with the intramolecular bond stretching must be omitted of course.
1282
G.E.
JELINEK,
L. J. S L U T S K Y
energy of vaporization (at 0~ then the parameters of the Lennard-Jones potential may readily be evaluated. For the first cycle the zero-point energy is given by means of a Debye temperature estimated from heat capacity data, and a0 is equated to the average crystallographically-determined nearest neighbor distance, a~, (when known) or computed from the density. In succeeding cycles the zero-point energy of the model plus the zeropoint dilation are considered in the refinement procedure. Note that due to the zero-point dilation (see Paper II) a0 is less than a~. The values of the lattice sums, heats of vaporization, Debye temperature, bond distances, and force constants used in this calculation are collected i n Table 1. The quadratic (K,K2), cubic (G,G2), and quartic (D,Dz) force constants are obtained by evaluating the second, third, and fourth derivatives of the Lennard-Jones potential at a0 and (a02 + d02)~12.The force constants (K~) between covalently bonded atoms have been derived from the gas-phase spectroscopic data and procedure referenced in Table 1.
a n d A. M. K A R O
We shall refer to the lattice modes of the hexagonal (D3h) model. For the hexagonal model (with the two molecules per unit cell) molecules 7, 8 and 9 of the rhombohedral model, Fig. 1, now occupy the hexagonal positions directly below molecules 1, 2 and 3. The interactions between nearest and nextnearest nonbonded atoms are assumed to be given by the Lennard-Jones [6, 12] potential derived for the rhombohedral model. The force constants and the 12 by 12 secular determinant are derived from the symmetry operations of the hexagonal lattice.
4. T H E H A R M O N I C A P P R O X I M A T I O N
In the notation developed in the preceding section, the matrices specifying the harmonic part of the interaction between atom (0,0) and its nearest nonbonded neighbors, designated (1,1) and (6,0) in Fig. 1, are --! 9 (06,00) =
0 0 0 0 0 0
Table 1. The force constants and quantities entering into the calculation o f the force constants Constant
do(/~)[al Vs(cma/mole) -- A H [~o(cal/mole) O o ( D e b y e temp.) ao(/~) ra(A) -- e(cal/mole) We W12 K(dyn/cm) K2(dyn/cm) K1(105 d y n / c m ) Ea~ G ( 1 0 n d y n / c m 2) G2(I 0 ix d y n / c m z) D(1020 d y n / c m a) D2 ( 102~ d y n / c m 3) t"lSee Ref. [13]. tblSee Ref. [5]. t~lSee Ref. [4]. tdlSee Ref. [14].
Oxygen 1"2074 20.91 tbl 2071.0 m 104.5 m 3.134 3"136 114.0 16"695 11 "904 1525-0 393.0 11 "77 -- 8-63 -- 2"61 4-60 1.40
Fluorine 1"4 i 77 19.3 to) 2194.0 tgl 110.0 m 2-979 3-153 120-0 15"367 10"943
1915.0 218.0 4.45 Eu -- 11.39 -- 1-88 6.38 1 "09
Nitrogen 1" 104 27.076 Edj ~1780-0 thl 83.5 3-486 3"685 93-0 17"894 12"833 989"0 401-0 22.97 -- 5"04 -- 2"23 2"42 1 "07
t~qSee Ref. [I0]. mSee Ref. [ 15]. t~JSee Ref. [ 16]. thlSee Ref. [I7].
Carbon monoxide 1" 128 27-019 t~) 2083.0 tu 79"5 tu 3.477 3-676 108-1 17"750 12"720 1159"0 449.0 19-04 -- 5.92 -- 2"52 2.84 1 "21
Chlorine
Bromine
2" 02 34.775 [e~ 7168-0 ig~ 115.0 3.527 3-727 408-4 14"214 10"218 4218"0 88-4 3"29 -- 21.27 -- 1 "56 10"07 0.83
2" 27 39" 241 [el 11066-0 tgl 80.0 3.838 4-052 628"2 14"018 10"109 5436.0 48" 7 2"21 -- 25.23 -- 1 "55 10.99 0.77 mSee Ref. [18]. mSee Ref. [ 19]. tklSee Ref. [20]. tIISee Ref.[21].
A CRYSTAL OF DIATOMIC M O L E C U L E S - I
9 (01,01) =
i - - Ko/ 3
o [
--X/2K/3; 2K/3 [
-- V ~ K / 3
Substituting equation (6) into equation (5) (4a) 4~av2mku,(k[y) = ~, un(k'lY)dPo,(ll',kk ') l'ktb
and for its second nearest n o , b o n d e d neighbor (6,1) the matrix is
o
o6o,
=
0
(._o).
0
\ao/ 0
1283
• exp {--2r
9 [r(/,k) - - r ( l ' , k ' ) ]}.
(7)
F o r the equations to have nontrivial solutions, it is required that the determinant of the coefficients vanish, or, defining (4b) C . b ( y , k k ' ) = ~, dPab(ll',kk') exp {--27riy l
o (do?. \ao/
• [r(/,k) - - r ( / ' , k ' ) ]}, where H = KJ[1 + (do/ao)2]. The force constant matrix of atom (0,0) with its covalently bonded neighbor (0,1) is
9 (00,01) =
i!0 o} 0 0
.
(4c)
--KI
The remaining matrices ~ ( O l ' , k k ' ) may readily be generated by the procedure suggested by Born and Begbie [22, 23] the relevant operations being (TO reflection in the X2X~ plane and (T2) reflection in the X2 = -- ~/3-X, plane.
T1 = TI -I =
T~ =
T2 -1 =
89
Jl~ 0 0
1 0
--
[Cab(y,kk') = 4r
= 0
(9)
where I is the 6 • 6 unit matrix. F o r the primitive diatomic rhombohedral lattice, the 6 • 6 matrix C(y,kk') is of the form A(00) B* (01)
B(01) A(00)
(10)
The elements of the symmetric 3 x 3 submatrices A(00) and B(01) in terms of the force constants and components of the wave vector are
.
mkii,(l,k) = -- t~b dP"b(ll"kk')ub(l"k')"
+3H (7K/2) -- (3K/2) (C~ + C~) + 3H 2 K + K ~ +6(do/ao)2H -- (M3-K/2) (Ca2 - Can) A2a (00) = 0 - (K/4) (E2 + E3) -- (H/2) (C12 + Cla + aC2a) (1 1) B22 (01) = -- (K/12) ( E 2 + E a + 4 E x ) -- (3H/2) (C,2 + C,3) = -(2K/3) (E~ + E2 + Ez) B33 (01) - - 2 (dolao) 2 H (C12 -1- Cla -4- C23 ) -- K1 B12 ( 0 l ) = -- (V~K/12) (E2--Ea)
A22 (00) A33 (00) A12 (00) A13 (00) Bn(01)
If only the harmonic terms in the potential are considered, the equations of motion are (5)
Travelling wave solutions of the form ua(l,k,t) = u,(kly) exp {2zri[y. r(/,k) + vt]} (6) are sought.
we obtain
A~ (00) = (7K/2) -- (K/2) (C~2 + Can + (723)
,
1 0
(8)
= = = = =
( v r 3 " n / 2 ) (C12 -- C13)
B13 (01) = (X/-6K/6) (E2--ga) + i [ P~_Sx2-- Sla -- 2S2a) ] B23 (01) = -- (X/2K/6) (2E1 -- E~ -- E3)
+ i [ x / 3 e ( s ~ + S,3) ]
1284
G. E. J E L I N E K , L. J. S L U T S K Y and A. M. K A R O
with C~ = [cos 2~r(y,--yj) ], So = sin [2,r (y~ --y~) ], Ej = exp (2~'iy~),
P = (dolao)H,
of the lattice point group. These correspond to 512 points in the first Brillouin zone. This, then, is the set of normal modes used as a basis in the calculation of the volume dependence of the thermal properties in Paper II.
i = X/'Z-(
The approach of Born and von KLrmfin is to require that, for crystal L-unit cells on a side, the u's b e periodic with period L. Thus, if y is expressed in terms of the reciprocal lattice basis vectors as y = ylbl+y,b2+y3ba, it is required that Yl = milL, Y2 = milL, and Ys = ma/L with the m's integers running from 0 to L - - 1. Or, since the addition or subtraction of a reciprocal lattice basis vector (or any combination or integral multiples of reciprocal lattice basis vectors) from y does not alter equation (6), the 'y's' may be considered to run from-- 89to + 89in L equal steps. Thus, to obtain the normal modes of a crystal L atoms on a side it is necessary to solve equation (9) for each of the values of y consistent with the cyclic boundary conditions. In the case of our model with two atoms in the primitive unit cell six roots, v(y,j) are obtained at each value of y. If the Cartesian components of the eigenvectors, E(k[y,j) associated with j th normal mode with wavevector y are e~(kly,j) the displacement of the k th a t o m in the l th u n i t cell is 1
N
u~(l,k) - ( N m k ) m ~, Q ( y , j ) yJ
•
ea(kl y , j )
exp 27riy 9r(l)
(12)
where Q(y,j) is the amplitude of the j t h normal mode for wavevector y. We have solved the secular equation,* equation (9), using the harmonic force constants given in Table 1 on a mesh of 85 points evenly distributed throughout the portion of the first Brillouin zone which cannot be further reduced by the symmetry operations *We have assumed a homonuclear molecule of C O with a mass equal to the geometric mean of the masses of the carbon and oxygen atoms.
5. RESULTS IN THE HARMONIC APPROXIMATION
(a) Thermal displacement correlations The thermal average of the square of the displacement of the k th atom along the a m Cartesian direction can be calculated from the eigenvalues and eigenvectors obtained in solution to equation (9) as 1
N
(Ua2(k) ) = N m ~ ( q 2 ( y , j ) ) e~(k[y,j) yJ
•
(13)
N is the number of unit cells in the crystal; m is the mass of an atom; and ( q 2 ( y , j ) ) = [ h/(8~.2v(y,j) ] x coth [hv(y,j)12kT]
(14)
is the thermal-average squared amplitude of a harmonic oscillator with frequencies v(y,j). The quantities ea(k[y,j) are the components along the a th Cartesian axis of the motion of the k th atom in t h e j th normal mode with wave vector y. The asterisk denotes complex conjugation. Temperature factors enter explicitly into all structure analyses made by diffraction methods in the form e x p { - - B ( s i n O / k ) 2} where B = 16~2(u82) and (us 2) is the mean square amplitude perpendicular to the scattering planes. We have calculated anisotropic temperature factors, 167rz (ua z), for our model of a diatomic solid. F o r oxygen at 23~ we obtain B = 3.30 and 2.54,~ z respectively perpendicular and parallel to the threefold axis. Barrett, M e y e r and Wasserman [4] find values of B ranging between 1-75-4.89 ,~z for a-oxygen at 23~ the temperature factor
A CRYSTAL OF DIATOMIC MOLECULES-I
being quite sensitive to the weighting of the observed intensities. Our values are in very reasonable agreement with the results of Barrett et al. Meyer et al.[7] utilized an isotropic B of 6___0.5/~2 in the crystal structure determination of t~-fluorine at 23~ We obtain substantially smaller anisotropic values of 3.04 and 2.22 ,~z (perpendicular and parallel, respectively). Jordan, Smith, Streib and Lipscomb[9] have obtained an isotropic temperature factor of 9-93 A2 for the/37phase of N2 at 50~ assuming a hexagonal-closepacked fixed-molecule equilibrium structure. Our anisotropic values of 9.50 and 9.13 ,~2 give an excellent account of the thermal amplitudes in/3-nitrogen. The packing in the orthorhombic structure of solid chlorine and bromine is sufficiently far removed from closest packing of cylinders as to render the significance of any agreement between computed results and the observed properties doubtful. As can be seen from Table 1, the force constant representing resistance to shear perpendicular to the plane of the molecular axes (Ks) is v~ry small for chlorine and bromine. The model calculation, in fact, is completely unstable (negative K2) for the molecular input data appropriate for solid iodine. However, the isotropic temperature factor 3.70___0.24/~2 cited by Donahue and Goodman[24] or C!2 (at l13~ is in reasonably good agreement with o u r calculated anisotropic value of 4.12,~ ~ parallel to the threefold axis. Our 7.04/~2 value for perpendicular displacement is anomalously large due to the small K2 force constant. These data are summarized in Table 2. As originally noted by Cruickshank[25], the amplitudes of torsional motions m~iy be estimated from lattice mode frequencies observed by Raman scattering. The average mean-square torsional amplitude for linear molecules on centrosymmetric sites for harmonic ground state librations is given by [26]:
(|
= (h/87r2nl) ~ (gj/vj) coth (hvj/2kT) J (15)
1285
Table 2. Temperature factors, B = 16w2(ua2) Crystal Oxygen Fluorine Nitrogen Chlorine
Temp. (*K)
BLI[a] (/~2)
B L(b] (,~z)
23 23 50 113
2.54 2.22 9.13 4-12
3.30 3.04 9-50 7.04
Bobserved (/~2) 1.75 --4.89 tc~ 6 --b0.5[dl 9-93 tel 3.70+--0.24 m
taJTheoretical B parallel to 3-fold axis. tbrrheoretical B perpendicular to 3-fold axis. to'See Ref. [4]. tOJSee Ref. [7]. teISee Ref. [9]. mSee Ref. [24].
where O is the angular displacement from equilibrium, j is summed over all zone-center librational frequencies, vj, of degeneracy, gj; I is the moment of inertia of the molecule and n is the sum of the g~'s. For the assumed intramolecular bond lengths (do) of Table 1 and the r.m.s, amplitude given by equation (13), we can calculate the mean librational angle measured from the static equilibrium configuration by sin O = 2(up*)In/do
(16)
(up2) l~a is the r.m.s, amplitude perpendicular to the 3-fold axis. Our calculation, equation (16), as opposed to equation (15) considers all the quasiharmonic spectrum frequencies. The results of our calculation are shown in Fig. 3 and compared with the data of Cahill and Leroi [27, 28] in Table 3. Equation (13), which considers all frequencies in the mean thermal displacement, yields a mean librational angle generally 40 per cent higher than the zone-center only librational frequencies calculation. The theoretical calculation does, however, agree qualitatively with the amplitudes derived from experimental data for o~,/3, and y-02, a -- N2, and a -- CO. As indicated the effective dimensions of the molecules may be taken as the 0.002 electron density contour of Bader et al. We find excellent agreement between their molecular dimensions and those predicted by
1286
(3. E. J E L I N E K , L. J. S L U T S K Y and A. M. K A R O
I
s
10
I
I
I
,
30
40
20
N2
1
TEMPERATURE PKI
Fig. 3. Mean librational angle measured from Xs (the molecular axes orientation in the static configuration).
Table 3. R.M.S. amplitudes for librational motion Temp. Crystal
a-O2 /3-O~ 3~-O2 a-N2 a-CO
(~
10 22 28 48 48 16 24 36 12 28 44 60
( 02)89 . . . . . d
9-1 tal 9-6 10.4 tal 12.6 17la] 14tb] 16 18 11r~ 12 14 17
(0) 289
12.8 13-8 14.5 18.9 18.9 16.3 17-5 20.6 14-9 16-8 20-4 26-4
t~lSee Ref. [27]. r~JSee Ref. [28].
our model. For oxygen the 0.002 contour gives the width of the 02 molecule as 3-18/~. As seen in Table 1 our calculated width of the 02 molecule (equivalent to the nearest neighbor nonbonded atom-atom separation in the ideally packed model) is 3-13 ,A for the static lattice. Due to the zero-point dilation this is not the mean separation. When considering the zero-point dilatation (the procedure is derived in Paper II), the mean separation of two parallel 02 molecules in the ideally packed structure at T = 0 ~ is 3.22/~.
According to Bader et al. the maximum dimension of the F2 molecule perpendicular to the internuclear axis is 3.0/~. We obtain 2.98 A for the molecular width in the static lattice and 3-06,~ for the mean width at T=0~ The 0-002 contour of nitrogen gives 3.39 ~ as the width of the N2 molecule. Our model yields 3-49/~ for the static equilibrium width and 3.59 ,~ for the mean width. Hence, in the case of 02 and F2, at least, our rhombohedral ideal packing model predicts correctly the molecular dimensions as given by the 0.002 electron density contours. There has been much interest in the characterization of the structural transformation in oxygen, fluorine, and nitrogen[12]. We find it informative to look for a correlation between the observed a-/3 phase transformation temperature and a critical root-mean-square amplitude in our model, comparable to the Lindemann type criterion for melting. In Fig. 4 the r.m.s, displacements parallel (a) and perpendicular (b) to the three-fold axis have been plotted as a percentage of the mean nearest neighbor distance. The observed a-/3 phase transformation temperatures have been plotted as the solid circles. Accordingly, we observe that the a-fl phase transformation in oxygen, fluorine, and nitrogen occurs when the r.m.s, amplitude parallel to the threefold
A CRYSTAL OF DIATOMIC M O L E C U L E S - I
axis reaches a critical value of 5.2___0-1 per cent of the nearest neighbor distance. Note, however, that this is a correlation; the model itself, in fact, provides no a priori criterion or mechanism for predicting the phase transformation temperature. (b) l.r. spectra .1 The quasiharmonic 30~ frequency distribution spectrum of the rhombohedral model of/3-oxygen is shown in Fig. 5(a). Dispersion curves calculated for /3-oxygen for' waves propagating along the threefold [111] and twofold [110] axes are sketched in Fig. 5(b). The optical branch associated with 0-0 stretching (not shown) is essentially fiat with frequency differing negligibly from the gasphase value ve = 1580-4 cm-1[13] from which the force constant (K1) was derived. Neither the stretching mode nor the twofold degenerate (E,) zero-wavevector optical mode (TO) associated with the torsional oscillation of the 02 molecule occurring at 44 cm -~ for I
I
I
02
=.'N
I
I
10
20
I
I
30 40 TEMPERATURE{~
Fig. 4(a).
I
I
I
'~
i
F2
02
0
I
I
I
I
I
I
10
20
30
40
50
60
TEMPERATURE(~K|
Fig. 4(b). Fig. 4. R.m.s. displacements divided by the mean nearestneighbor spacing for directions: (a) parallel and (b) perpendicular to the 3-fold axis.
I
~hF2
I
I
1287
I
I
50
60
30~ is expected to be i.r. active. Thus, no fundamental absorption in the far i.r. is predicted. The conservation of energy and crystal momentum requires that, for normal processes the sum of the frequencies of the phonons produced be equal to the frequency of the incident radiation and that the sum of the wavevectors of the phonons be equal to the wavevector (essentially zero) of the incident photon. Thus, one might expect to find allowed combinations with the 0-0 stretching mode displaced from the fundamental by all frequencies between zero and the maximum lattice frequency. However, since the dispersion curves are fiat at the zero boundaries and there is therefore an accumulation of modes with frequencies close to the zone boundary yalues, and because of the larger volume in phase-space associated with
1288
G . E . J E L I N E K , L.
J.
S L U T S K Y and A. M. KARO
larger absolute values of the wavevector one would expect the frequencies of maximum absorption to represent the frequencies at the zone boundaries. Calms and Pimentel[29] observe a relatively sharp but extremely weak absorption in B-oxygen at 1549 cm -~ which is attributed to the 0 - 0 stretching mode (gas-phase v0 = 1556-2 cm -~) weakly active in the neighborI
I
I
i ~!ii
o2
8O
~o--
".%
20_
~
J
/~
,00 ~ 02 \::::::o/~ Z.O,
~ Y
0
Y~
Z.B.
Fig. 5(c).
, -- 10
30
20
\ lii I
i
I
I~
i
40 50 60 FREQUENCY(cm"11
Fig. 5. (a). Frequency spectrum G (~o) and quasiharmonic dispersion curves for the (b) rhombohedral and the (c) hexagonal model of 02 for the theoretical 30~ volume.
70
80
Fig. 5(a).
I[,| If ~ I [.o]I I
_
I ira ]
I__
/-::::" .?"/.~.~: :-, :: ]:
TA
* -
/
-
i i t "!"~L
I I-
3O
:] 2O 10 0
/
.~
"l
"2
t I
y~
ZB.
~ y
Fig. 5(b).
0
Y --~
Z.8.
hood of a vacancy or impurity. Since this portion of our calculation presumes no quasiharmonic intramolecular frequency shift, we naturally obtain a frequency for the 0 - 0 stretching mode close to the gas-phase Ve rather than v0, thus the comparison with the observed spectrum will be made in terms of displacements from the calculated value of the 0 - 0 fundamental frequency. The saEent feature of the combination spectrum is a broad relatively intense absorp,tion centered at 69 cm -1 above the fundamental which is attributed by Cairns and Pimentel to combinations with modes representing translation in the plane (xy) perpendicular to the threefold (z) axis. This band is superimposed on a broad general absorption extending from a b o u t 40 to 100 cm -1 above the 0 - 0 frequency. The frequencies in Fig. 5(b) extend from 28.9 to 80.9cm -1. F o r waves propagating along the twofold axis [110] the frequencies, at the zone boundary, of the longitudinal acoustic mode (LA) and the transverse acous-
A CRYSTAL OF DIATOMIC
tic mode (TA) polarized in the xy plane, both of which might be described qualitatively as translation in the xy plane, are 80.9 and 53-6 cm -1 in agreement with the assignment of Cairns and Pimentel. H o w e v e r the longitudinal acoustic mode for waves propagating along the threefold axis, qualitjatively T=, has a frequency of 57.8 cm -1 at the zone boundary and the torsional mode (TO) for waves propagating along the twofold axis a frequency of 49.1 cm-L Thus the absorption centered at 69 cm -~ would, in terms of this model, be due to the superposition of a variety of modes. In general, for a model of this sort, in which there is considerable curvature to the optical branches and no significant separation between optical and acoustic branches, a combination spectrum consisting of broad general absorption rather than well-defined features is expected. Indeed the combination spectrum might be as readily discussed in terms of the frequency distribution spectrum as in terms of the dispersion calves.
The lattice modes of the hexagonal D3h model for 02 are shown in Fig. 5(c) for wavevectors perpendicular (ii) and parallel (i) to the molecular axis. The frequency distribution spectra were obtained from a modification of a frequency interpolation routine of Gilat and Raubenheimer[30]. The frequency spectra and dispersion curves of N2, F2, CO, C12, and Brz are generally similar to those shown for 02. H o w e v e r due to the relative differences between the K,K2 force constants splitting in the optical branches can result in various peaks in the spectrum separating or converging. In the hexagonal structure (for zero-v~avevector) only T=(A2,) and R(E,) are i.r. active whereas there are i.r. active modes (Txu) corresponding to translation of the molecule in the plane normal to its axis in the true P2a3 structure. It is, of course, not to be expected that a model structure of incorrect symmetry will reproduce the selection rules of the correct structure. N o r do the nine optical
MOLECULES-I
1289
modes of the hexagonal model fully enumerate the 21 zero-wave optical modes of a structure which contains four molecules per primitive unit cell. The analogs, if any exist, to the remaining zone-origin modes of the more complex structure must be sought in the zone boundary modes of the hexagonal structure. The zero-wavevector mode of the hexagonal crystal polarized in the xy plane is simply a shear normal to the 3-fold axis with alternate layers of molecules moving in opposite directions. A more reasonable analogy to the optically active Tx~ modes of the P213 structure might be the zone boundary limit of a longitudinal optical mode propagating along the two-fold axis. These occur at 80 cm -1 for N2 and 84 cm -~ for CO. The zero-wavevector T=(A2,) and R(E,) modes are 55 and 41 cm -1 respectively for N2 and 58 and 42 cm -1 for CO. The far-i.r, spectrum of crystalline nitrogen and carbon monoxide has been investigated by Anderson and Leroi[31] and by Ron and Schnepp[32]. In the case of nitrogen two bands are found centered at 48 and 69 cm -1. Thus the values of 41, 55 and 80 cm -1 estimated here are in reasonable agreement with experiment. In CO the observed bands are at 49 and 85 cm -1 again in rough agreement with the calculated 42, 58 and 84 cm -~. While we would contend that the simple rhombohedral and hexagonal structures give a picture of the density of vibrational states in carbon monoxide and nitrogen which is adequate for the discussion of the thermodynamic properties away from phase boundaries, we would hesitate to suggest on the basis of this calculation that the 48-49 cm -a band be assigned to torsion rather than translation as suggested by Anderson and Leroi. However, if the values" of the torsional frequency calculated here are qualitatively correct then the rather broad 48-49 cm -a band might be interpreted as a superposition of translational and torsional components.
1290
G . E . JELINEK, L. J. SLUTSKY and A. M. KARO 6. SUMMARY
We have developed a model of the low temperature forms of crystalline 0~, N2, F2, CO, C12, and Br~ which assumes the static equilibrium structure to be the rhombohedral or hexagonal close packing of diatomic molecules. The nonarbitrarily adjustable parameters of a Lennard-Jones [6-12] potential between nonbonded atoms have been obtained in a recursive refinement procedure from observed solid-state data. The dynamical properties of the model are in good agreement with both the temperature factors derived from structural analyses and molecular dimensions of calculated electron density contours. For N2, 02 and F2 a correlation was found between the observed o~-fl solid-solid phase transformation temperature and a critical r.m.s, amplitude (5.2__+0-1 per cent of the nearest nonbonded atom-atom distance). Frequency distribution spectra and dispersion curves for the rhombohedral and hexagonal models were presented. It was shown that the combination spectrum of solid oxygen could be explained and the far-i.r. spectrum of CO and N2 rationalized on the basis of this model. REFERENCES
1. BORN M. and H U A N G K., The Dynamical Theory of Crystal Lattices, Oxford University Press, New York (1954). 2. ALIKHANOV R. A., Soviet Phys.--JETP 45, 812 (1963). 3. HORL E. M.,Acta crystallogr. 15, 845 (1962). 4. BARRETT C. S., MEYER L. and WASSERMAN J.,J. chem. Phys. 47,592 (1967). 5. BARRETT C. S. and MEYER L., Phys. Rev. 160, 694 (1967). 6. JORDAN T. H., STREIB W. E. and LIPSCOMB W. N.,J. chem. Phys. 41,760 (1964).
7. MEYER L., BARRETT C. S. and GREER S. C., J. chem. Phys. 49, 1902 (1968). 8. STREIB W. E., JORDAN T. H. and LIPSCOMB W. N.,J. chem. Phys. 37, 2962 (1962). 9. JORDAN T. H., SMITH H. W., STREIB W. E. and LIPSCOMB W. N., J. chem. Phys. 41, 756 (1964). 10. WYCKOFF R. W. G., Crystal Structures, Vol. I, John Wiley, New York (I 958). 11. BADER R. F., HENNEKER W. H. and CADE P. E.,J. chem. Phys. 46, 3341 (1967). 12. MEYER L., Phase Transitions in van der Waals Lattices, Advances in Chemical Physics XVI, 343 (1969). 13. HERZBERG G., Molecular Theory and Molecular Structure, I. Spectra of Diatomic Molecules, van Nostrand, New York (1950). 14. BOLZ L. H., BOYD M. E., MAUER F. A. and PEISER H. S.,Acta crystallogr. 12,247 (1959). 15. ORLOVA M. P.,zh. Fiz. Khim. 40, 2986 (1966). 16. KELLY K. K., U.S. Department of the Interior Bureau of Mines Bulletin No. 383 (1935). 17. BONDI A., Physical Properties of Molecular Crystals, Liquids, and Glasses, John Wiley, New York (1968). 18. MASON E. A. and RICE W. E.,J. chem. Phys. 22, 843 (1954). 19. BURFORD J. C. and GRAHAM G. M., Can. J. Phys. 47, 23 (1969). 20. HU J. H., WHITE D. and JOHNSON H. C.,J. Am. Chem. Soc. 75, 5642 (1953). 21. ANDRYCHUK D., Can.J. Phys. 29, 151 (1951). 22. BEGBIE G. H. and BORN M., Proc. R. Soc. (London) A I ~ , 179 (1946). 23. BEGBIE G. H.,ibid. A185, 189 (1946). 24. DONOHUE J. and GOODMAN S. H., Acta crystallogr. 18,568 (1965). 25. CRUICKSHANK D. W. J., Acta crystallogr. 9, 1005 (1956). 26. CAHILL J. E., Ph.D. thesis, Princeton University (1968). 27. CAHILL J. E. and LEROI G. E., J. chem. Phys. 51, 97 (1969). 28. CAHILL J. E. and LEROI G. E., J. chem. Phys. 51, 1324 (1969). 29. CAIRNS B. R. and PIMENTEL G. C., J. chem. Phys. 43, 3432 (1965). 30. GILAT G. and RAUBENHEIMER L. J., Phys. Rev. 144, 390 (1966). 31. ANDERSON A. and LEROI G. E., J. chem. Phys. 45, 4359 (1966). 32. RON A. and SCHNEPP O.,J. chem. Phys. 46, 3991 (1967).
MODEL FOR THE LATTICE VIBRATIONS OF A CRYSTAL 9 F DIATOMIC M O L E C U L E S * - I . FREQUENCY DISTRIBUTIONS, DEBYE-WALLER FACTORS, AND INFRARED SPECTRA G. E. JELINEK
Sandia Laboratories, Albuquerque, New Mexico 87115, U.S.A. and
L. J. SLUTSKY Department of Chemistry, University of Washington, Seattle, Wash. 98105, U.S.A. and A. M. KARO
Lawrence Livermore Laboratory, Livermore, Calif. 94550, U.S.A. ( R e c e i v e d 1 S e p t e m b e r 197 I)
Abstract-Dispersion curves, frequency distribution spectra, and anisotropic X-ray temperature factors are computed for a model of the low-temperature forms of crystalline 03, N2 and F2 which assumes that the static equilibrium structure involves the rhombohedral close-packing of diatomic molecules. The interaction between nonbonded atoms is described by a Lennard-Jones [6, 12] potential centered on the atoms. An interpretation of the far-i.r, and lattice-combination spectra of these crystals in reasonable accord with the computed dispersion curves is offered. Theoretical root-meansquare amplitudes are in good agreement with predictions derived from X-ray and Raman scattering data.
1. INTRODUCTION BORN AND HUANG[1] have developed a computationally straightforward approach for the evaluation of the thermodynamical properties of a quasiharmonic lattice. Herein we apply the formulae of Born and Huang to calculate the thermal and optical properties of a model of solidified oxygen, fluorine, nitrogen, carbon monoxide, chlorine, and bromine. In this paper we obtain the secular equation and its eigenvalues and eigenvectors which constitute the basis for the evaluation of the volume dependent properties discussed in the following paper. A description of our model is given in Section 2. A comparison of the model with
the low temperature crystal structures of the solidified diatomic gases indicates it might constitute a reasonable model for the description of the lattice vibrations. The assumed interaction between nonbonded and between covalently bonded atoms is presented ir~ Section 3. The parameters of the LennardJones[6, 12] potential are derived from the low temperature solid state data in a recursive refinement procedure. The force constant matrices and elements of the secular determinant are derived in Section 4. In Section 5 the dispersion curves, frequently distribution spectra, and thermal displacement correlations are compared with experimental data. In the following paper (referred to as II) the general equation of state based upon the method of homogeneous deformation is presented.
*Work supported by the U.S. Atomic Energy Commission. 1279
1280
G.E.
J E L I N E K , L. J. S L U T S K Y and A. M. K A R O
2. THE MODEL Interactions with more distant neighbors will The static equilibrium structure is assumed be neglected. to be the rhombohedral close-packing of Between 23.9 and 43-6~ neutron[2] and diatomic molecules with the axes of the electron[3] diffraction data indicate that molecules being parallel to the threefold axis B-oxygen has a primitive monomolecular of the crystal. The assumed crystal structure rhombohedral structure with the axis of the is illustrated schematically in Fig. 1 and in molecule oriented parallel to the threefold projection on a plane normal to the threefold axis of the crystal. In the/3-oxygen structure [111] axis in Fig. 2. In the case of 'ideal' the nine nearest nonbonded neighbors of a packing of diatomic molecules, each atom has given atom in the ideally-packed structure one covalently-bonded nearest neighbor at divide into a set of six atoms lying in the [111] distance do, nine nonbonded near neighbors at plane at a distance of 3.31 ,~ and a set of three a0 and six nonbonded neighbors at (a02 + d02)1/2. atoms at a distance of 3.18 ,~. In the monoclinic structure of a-oxygen[4,5] (stable below 23-9~ the 0 - 0 axes, remain parallel; but the six near-neighbors in the plane normal to the 0 - 0 axes, equivalent in the (2,0 ) /3-oxygen structure, are split into a set of four at 3-20 and two at 3-43 ,~. I) In B-fluorine [6] stable between 45.55~ and the melting point at 53.54~ the unit cell is cubic and contains eight molecules. Alphafluorine [7] stable below 45.55~ is monoclinic 10,01 with four molecules per unit cell. For/3-nitrogen stable above 35.6~ Streib (6,1) (0,1) et a/.[8], prefer a precession model in which the molecular centers form a hexagonal closepacked lattice with a precession angle of 56.0deg. Nitrogen[9], below 35-6~ has a Fig. 1. The interaction of atom (0,0) with a few of its near neighbors for the assumed crystal structure. cubic structure in which a given atom has three nonbonded neighbors at 3.42 ,~ and six at 3.55 A. The low temperature a form of carbon monoxide[10] is isomorphous with a-nitrogen. Bader, Henneker and Cade [11] calculated electron density contours from H a r t r e e - F o c k ; 9 ~, A ,.: ground-state wave functions for oxygen, fluorine, and nitrogen. They concluded that the 0-002 contour, which includes 95 per cent of the charge density, represents the dimensions of the molecules. According to M e y e r f ' 7 ; [12] the packing density in a-oxygen is 71.4 per cent as calculated from the unit cell volume and the 0.002 contour. Since a structure of close packed spheres has a packing density of 74 per cent, taking into account the Fig. 2. The coordinates s y s t e m s - t h e projection is on a plane normal to the molecular axes. oblong shape of the oxygen molecule, a-
(6,0~)
G
G
@
@
A CRYSTAL OF DIATOMIC MOLECULES-I
oxygen must be considered a close-packed structure. In a similar analysis a-fluorine must also be considered a close-packed structure. The main difference between a-oxygen and a-fluorine is a doubling o.f the c-axis of the unit cell of fluorine due to'~ilt (about 11 ~ in alternating directions of the molecules perpendicular to the basal plane. It will be seen in the subsequent discussion that if our ideally packed equilibrium ~tructure is assumed, when anharmonic terms in the potential are considered, an anisotropic zero-point dilation in the observed sense is predicted. Thus, when considering the dynamics of our static model for 02 at 30~ it is observed that the anisotropic dilation of the lattice splits the 9 nearest nonbonded neighbors into a set of 6 at a distance of 3.23 A and 3 at 3.21 ,A. This dynamical configuration of our model conforms to the observed r oxygen structure. Thus, it would seem possible that a primitive rhombohedral equilibrium structure with large vibrational excursions from the equilibrium configuration might constitute a reasonable model for the description of the lattice vibrations of solid oxygen and fluorine and perhaps for nitrogen and carbon monoxide as well. While a lattice dynamical calculation for the true crystal structures (such as monoclinic with 8 molecules per unit cell) may be unjustified, our model simulates reasonably well the mean number and distances of the near neighbors in the true crystal structures. 3. FORCE CONSTANTS AND THE POTENTIAL REFINEMENT
The coordinate systems pertinent to a discussion of this problem are illustrated in projection in Fig. 2 where the 'A's' are the basis vectors of the rhombohedral lattice and the 'X's' are Cartesian unit vectors. The position of the k-th atom in the/-th unit cell with respect to the atom (0,0) chosen to be at the orig!n will be specified by r ( l , k ) = r(/) + r(k) = l~al + 12a2 +/3a3 + r(k) with the '/'s' integers. The origin of the unit cell will
1281
be taken to be at the atom indexed by k = 0; thus, r(0) = 0 while r(1) = -- doX3. Let the displacement of the k-th atom in the /-th unit cell from its equilibrium position be denoted by the vector u(/,k) with Cartesian components ua(l,k), then the quadratic and higher order interatomic force constants are defined as the partial derivatives of the potential energy ~ with respect to the components ofu. a2~p cbab ( ll',kk') =
OUa (l,k)Oub (l',k') '
~abc ( ll' l",kk' k") a3~ = aua(l,k)aub(l . ,k. ). a.u ,. ( l. .,k ) etc.
(1)
The quantities gaab(ll',kk') which specify the harmonic interaction of a given pair of atoms may be written as a 3 x 3 matrix which will be designated ~ ( l l ' , k k ' ) . If the Lennard-Jones[6, 12], potential is written in the form V = - ~[2(ro/r) 6 - (ro[r)12], then ~s, the cohesive energy (per molecule) of the crystal, may be expressed.
r = - - ~ ~,' {2[ro/r(l,k) ]6--[ro/r(l,k) ]12} 9 (2) lk
If Wn is defined in terms of a, the distance between nearest nonbonded neighbors, by
Wn = ~,' [a/r(l,k)]", lk
then ~o, = --e[ 2We(ro/a) 6 -
W12(ro/a)12].
(3)
The equilibrium Value of a ((a~os/Oa)= 0) is thus ao (I/V12/W6)I/6Fo and the energy ~P8= =
EW62/W12 .
We have obtained nonarbitrarily adjustable potential parameters in a recursive refinement procedure. If ~o, plus the zero-point energy of the model* is associated with the experimental *The contribution of the optical branch associated with the intramolecular bond stretching must be omitted of course.
1282
G.E.
JELINEK,
L. J. S L U T S K Y
energy of vaporization (at 0~ then the parameters of the Lennard-Jones potential may readily be evaluated. For the first cycle the zero-point energy is given by means of a Debye temperature estimated from heat capacity data, and a0 is equated to the average crystallographically-determined nearest neighbor distance, a~, (when known) or computed from the density. In succeeding cycles the zero-point energy of the model plus the zeropoint dilation are considered in the refinement procedure. Note that due to the zero-point dilation (see Paper II) a0 is less than a~. The values of the lattice sums, heats of vaporization, Debye temperature, bond distances, and force constants used in this calculation are collected i n Table 1. The quadratic (K,K2), cubic (G,G2), and quartic (D,Dz) force constants are obtained by evaluating the second, third, and fourth derivatives of the Lennard-Jones potential at a0 and (a02 + d02)~12.The force constants (K~) between covalently bonded atoms have been derived from the gas-phase spectroscopic data and procedure referenced in Table 1.
a n d A. M. K A R O
We shall refer to the lattice modes of the hexagonal (D3h) model. For the hexagonal model (with the two molecules per unit cell) molecules 7, 8 and 9 of the rhombohedral model, Fig. 1, now occupy the hexagonal positions directly below molecules 1, 2 and 3. The interactions between nearest and nextnearest nonbonded atoms are assumed to be given by the Lennard-Jones [6, 12] potential derived for the rhombohedral model. The force constants and the 12 by 12 secular determinant are derived from the symmetry operations of the hexagonal lattice.
4. T H E H A R M O N I C A P P R O X I M A T I O N
In the notation developed in the preceding section, the matrices specifying the harmonic part of the interaction between atom (0,0) and its nearest nonbonded neighbors, designated (1,1) and (6,0) in Fig. 1, are --! 9 (06,00) =
0 0 0 0 0 0
Table 1. The force constants and quantities entering into the calculation o f the force constants Constant
do(/~)[al Vs(cma/mole) -- A H [~o(cal/mole) O o ( D e b y e temp.) ao(/~) ra(A) -- e(cal/mole) We W12 K(dyn/cm) K2(dyn/cm) K1(105 d y n / c m ) Ea~ G ( 1 0 n d y n / c m 2) G2(I 0 ix d y n / c m z) D(1020 d y n / c m a) D2 ( 102~ d y n / c m 3) t"lSee Ref. [13]. tblSee Ref. [5]. t~lSee Ref. [4]. tdlSee Ref. [14].
Oxygen 1"2074 20.91 tbl 2071.0 m 104.5 m 3.134 3"136 114.0 16"695 11 "904 1525-0 393.0 11 "77 -- 8-63 -- 2"61 4-60 1.40
Fluorine 1"4 i 77 19.3 to) 2194.0 tgl 110.0 m 2-979 3-153 120-0 15"367 10"943
1915.0 218.0 4.45 Eu -- 11.39 -- 1-88 6.38 1 "09
Nitrogen 1" 104 27.076 Edj ~1780-0 thl 83.5 3-486 3"685 93-0 17"894 12"833 989"0 401-0 22.97 -- 5"04 -- 2"23 2"42 1 "07
t~qSee Ref. [I0]. mSee Ref. [ 15]. t~JSee Ref. [ 16]. thlSee Ref. [I7].
Carbon monoxide 1" 128 27-019 t~) 2083.0 tu 79"5 tu 3.477 3-676 108-1 17"750 12"720 1159"0 449.0 19-04 -- 5.92 -- 2"52 2.84 1 "21
Chlorine
Bromine
2" 02 34.775 [e~ 7168-0 ig~ 115.0 3.527 3-727 408-4 14"214 10"218 4218"0 88-4 3"29 -- 21.27 -- 1 "56 10"07 0.83
2" 27 39" 241 [el 11066-0 tgl 80.0 3.838 4-052 628"2 14"018 10"109 5436.0 48" 7 2"21 -- 25.23 -- 1 "55 10.99 0.77 mSee Ref. [18]. mSee Ref. [ 19]. tklSee Ref. [20]. tIISee Ref.[21].
A CRYSTAL OF DIATOMIC M O L E C U L E S - I
9 (01,01) =
i - - Ko/ 3
o [
--X/2K/3; 2K/3 [
-- V ~ K / 3
Substituting equation (6) into equation (5) (4a) 4~av2mku,(k[y) = ~, un(k'lY)dPo,(ll',kk ') l'ktb
and for its second nearest n o , b o n d e d neighbor (6,1) the matrix is
o
o6o,
=
0
(._o).
0
\ao/ 0
1283
• exp {--2r
9 [r(/,k) - - r ( l ' , k ' ) ]}.
(7)
F o r the equations to have nontrivial solutions, it is required that the determinant of the coefficients vanish, or, defining (4b) C . b ( y , k k ' ) = ~, dPab(ll',kk') exp {--27riy l
o (do?. \ao/
• [r(/,k) - - r ( / ' , k ' ) ]}, where H = KJ[1 + (do/ao)2]. The force constant matrix of atom (0,0) with its covalently bonded neighbor (0,1) is
9 (00,01) =
i!0 o} 0 0
.
(4c)
--KI
The remaining matrices ~ ( O l ' , k k ' ) may readily be generated by the procedure suggested by Born and Begbie [22, 23] the relevant operations being (TO reflection in the X2X~ plane and (T2) reflection in the X2 = -- ~/3-X, plane.
T1 = TI -I =
T~ =
T2 -1 =
89
Jl~ 0 0
1 0
--
[Cab(y,kk') = 4r
= 0
(9)
where I is the 6 • 6 unit matrix. F o r the primitive diatomic rhombohedral lattice, the 6 • 6 matrix C(y,kk') is of the form A(00) B* (01)
B(01) A(00)
(10)
The elements of the symmetric 3 x 3 submatrices A(00) and B(01) in terms of the force constants and components of the wave vector are
.
mkii,(l,k) = -- t~b dP"b(ll"kk')ub(l"k')"
+3H (7K/2) -- (3K/2) (C~ + C~) + 3H 2 K + K ~ +6(do/ao)2H -- (M3-K/2) (Ca2 - Can) A2a (00) = 0 - (K/4) (E2 + E3) -- (H/2) (C12 + Cla + aC2a) (1 1) B22 (01) = -- (K/12) ( E 2 + E a + 4 E x ) -- (3H/2) (C,2 + C,3) = -(2K/3) (E~ + E2 + Ez) B33 (01) - - 2 (dolao) 2 H (C12 -1- Cla -4- C23 ) -- K1 B12 ( 0 l ) = -- (V~K/12) (E2--Ea)
A22 (00) A33 (00) A12 (00) A13 (00) Bn(01)
If only the harmonic terms in the potential are considered, the equations of motion are (5)
Travelling wave solutions of the form ua(l,k,t) = u,(kly) exp {2zri[y. r(/,k) + vt]} (6) are sought.
we obtain
A~ (00) = (7K/2) -- (K/2) (C~2 + Can + (723)
,
1 0
(8)
= = = = =
( v r 3 " n / 2 ) (C12 -- C13)
B13 (01) = (X/-6K/6) (E2--ga) + i [ P~_Sx2-- Sla -- 2S2a) ] B23 (01) = -- (X/2K/6) (2E1 -- E~ -- E3)
+ i [ x / 3 e ( s ~ + S,3) ]
1284
G. E. J E L I N E K , L. J. S L U T S K Y and A. M. K A R O
with C~ = [cos 2~r(y,--yj) ], So = sin [2,r (y~ --y~) ], Ej = exp (2~'iy~),
P = (dolao)H,
of the lattice point group. These correspond to 512 points in the first Brillouin zone. This, then, is the set of normal modes used as a basis in the calculation of the volume dependence of the thermal properties in Paper II.
i = X/'Z-(
The approach of Born and von KLrmfin is to require that, for crystal L-unit cells on a side, the u's b e periodic with period L. Thus, if y is expressed in terms of the reciprocal lattice basis vectors as y = ylbl+y,b2+y3ba, it is required that Yl = milL, Y2 = milL, and Ys = ma/L with the m's integers running from 0 to L - - 1. Or, since the addition or subtraction of a reciprocal lattice basis vector (or any combination or integral multiples of reciprocal lattice basis vectors) from y does not alter equation (6), the 'y's' may be considered to run from-- 89to + 89in L equal steps. Thus, to obtain the normal modes of a crystal L atoms on a side it is necessary to solve equation (9) for each of the values of y consistent with the cyclic boundary conditions. In the case of our model with two atoms in the primitive unit cell six roots, v(y,j) are obtained at each value of y. If the Cartesian components of the eigenvectors, E(k[y,j) associated with j th normal mode with wavevector y are e~(kly,j) the displacement of the k th a t o m in the l th u n i t cell is 1
N
u~(l,k) - ( N m k ) m ~, Q ( y , j ) yJ
•
ea(kl y , j )
exp 27riy 9r(l)
(12)
where Q(y,j) is the amplitude of the j t h normal mode for wavevector y. We have solved the secular equation,* equation (9), using the harmonic force constants given in Table 1 on a mesh of 85 points evenly distributed throughout the portion of the first Brillouin zone which cannot be further reduced by the symmetry operations *We have assumed a homonuclear molecule of C O with a mass equal to the geometric mean of the masses of the carbon and oxygen atoms.
5. RESULTS IN THE HARMONIC APPROXIMATION
(a) Thermal displacement correlations The thermal average of the square of the displacement of the k th atom along the a m Cartesian direction can be calculated from the eigenvalues and eigenvectors obtained in solution to equation (9) as 1
N
(Ua2(k) ) = N m ~ ( q 2 ( y , j ) ) e~(k[y,j) yJ
•
(13)
N is the number of unit cells in the crystal; m is the mass of an atom; and ( q 2 ( y , j ) ) = [ h/(8~.2v(y,j) ] x coth [hv(y,j)12kT]
(14)
is the thermal-average squared amplitude of a harmonic oscillator with frequencies v(y,j). The quantities ea(k[y,j) are the components along the a th Cartesian axis of the motion of the k th atom in t h e j th normal mode with wave vector y. The asterisk denotes complex conjugation. Temperature factors enter explicitly into all structure analyses made by diffraction methods in the form e x p { - - B ( s i n O / k ) 2} where B = 16~2(u82) and (us 2) is the mean square amplitude perpendicular to the scattering planes. We have calculated anisotropic temperature factors, 167rz (ua z), for our model of a diatomic solid. F o r oxygen at 23~ we obtain B = 3.30 and 2.54,~ z respectively perpendicular and parallel to the threefold axis. Barrett, M e y e r and Wasserman [4] find values of B ranging between 1-75-4.89 ,~z for a-oxygen at 23~ the temperature factor
A CRYSTAL OF DIATOMIC MOLECULES-I
being quite sensitive to the weighting of the observed intensities. Our values are in very reasonable agreement with the results of Barrett et al. Meyer et al.[7] utilized an isotropic B of 6___0.5/~2 in the crystal structure determination of t~-fluorine at 23~ We obtain substantially smaller anisotropic values of 3.04 and 2.22 ,~z (perpendicular and parallel, respectively). Jordan, Smith, Streib and Lipscomb[9] have obtained an isotropic temperature factor of 9-93 A2 for the/37phase of N2 at 50~ assuming a hexagonal-closepacked fixed-molecule equilibrium structure. Our anisotropic values of 9.50 and 9.13 ,~2 give an excellent account of the thermal amplitudes in/3-nitrogen. The packing in the orthorhombic structure of solid chlorine and bromine is sufficiently far removed from closest packing of cylinders as to render the significance of any agreement between computed results and the observed properties doubtful. As can be seen from Table 1, the force constant representing resistance to shear perpendicular to the plane of the molecular axes (Ks) is v~ry small for chlorine and bromine. The model calculation, in fact, is completely unstable (negative K2) for the molecular input data appropriate for solid iodine. However, the isotropic temperature factor 3.70___0.24/~2 cited by Donahue and Goodman[24] or C!2 (at l13~ is in reasonably good agreement with o u r calculated anisotropic value of 4.12,~ ~ parallel to the threefold axis. Our 7.04/~2 value for perpendicular displacement is anomalously large due to the small K2 force constant. These data are summarized in Table 2. As originally noted by Cruickshank[25], the amplitudes of torsional motions m~iy be estimated from lattice mode frequencies observed by Raman scattering. The average mean-square torsional amplitude for linear molecules on centrosymmetric sites for harmonic ground state librations is given by [26]:
(|
= (h/87r2nl) ~ (gj/vj) coth (hvj/2kT) J (15)
1285
Table 2. Temperature factors, B = 16w2(ua2) Crystal Oxygen Fluorine Nitrogen Chlorine
Temp. (*K)
BLI[a] (/~2)
B L(b] (,~z)
23 23 50 113
2.54 2.22 9.13 4-12
3.30 3.04 9-50 7.04
Bobserved (/~2) 1.75 --4.89 tc~ 6 --b0.5[dl 9-93 tel 3.70+--0.24 m
taJTheoretical B parallel to 3-fold axis. tbrrheoretical B perpendicular to 3-fold axis. to'See Ref. [4]. tOJSee Ref. [7]. teISee Ref. [9]. mSee Ref. [24].
where O is the angular displacement from equilibrium, j is summed over all zone-center librational frequencies, vj, of degeneracy, gj; I is the moment of inertia of the molecule and n is the sum of the g~'s. For the assumed intramolecular bond lengths (do) of Table 1 and the r.m.s, amplitude given by equation (13), we can calculate the mean librational angle measured from the static equilibrium configuration by sin O = 2(up*)In/do
(16)
(up2) l~a is the r.m.s, amplitude perpendicular to the 3-fold axis. Our calculation, equation (16), as opposed to equation (15) considers all the quasiharmonic spectrum frequencies. The results of our calculation are shown in Fig. 3 and compared with the data of Cahill and Leroi [27, 28] in Table 3. Equation (13), which considers all frequencies in the mean thermal displacement, yields a mean librational angle generally 40 per cent higher than the zone-center only librational frequencies calculation. The theoretical calculation does, however, agree qualitatively with the amplitudes derived from experimental data for o~,/3, and y-02, a -- N2, and a -- CO. As indicated the effective dimensions of the molecules may be taken as the 0.002 electron density contour of Bader et al. We find excellent agreement between their molecular dimensions and those predicted by
1286
(3. E. J E L I N E K , L. J. S L U T S K Y and A. M. K A R O
I
s
10
I
I
I
,
30
40
20
N2
1
TEMPERATURE PKI
Fig. 3. Mean librational angle measured from Xs (the molecular axes orientation in the static configuration).
Table 3. R.M.S. amplitudes for librational motion Temp. Crystal
a-O2 /3-O~ 3~-O2 a-N2 a-CO
(~
10 22 28 48 48 16 24 36 12 28 44 60
( 02)89 . . . . . d
9-1 tal 9-6 10.4 tal 12.6 17la] 14tb] 16 18 11r~ 12 14 17
(0) 289
12.8 13-8 14.5 18.9 18.9 16.3 17-5 20.6 14-9 16-8 20-4 26-4
t~lSee Ref. [27]. r~JSee Ref. [28].
our model. For oxygen the 0.002 contour gives the width of the 02 molecule as 3-18/~. As seen in Table 1 our calculated width of the 02 molecule (equivalent to the nearest neighbor nonbonded atom-atom separation in the ideally packed model) is 3-13 ,A for the static lattice. Due to the zero-point dilation this is not the mean separation. When considering the zero-point dilatation (the procedure is derived in Paper II), the mean separation of two parallel 02 molecules in the ideally packed structure at T = 0 ~ is 3.22/~.
According to Bader et al. the maximum dimension of the F2 molecule perpendicular to the internuclear axis is 3.0/~. We obtain 2.98 A for the molecular width in the static lattice and 3-06,~ for the mean width at T=0~ The 0-002 contour of nitrogen gives 3.39 ~ as the width of the N2 molecule. Our model yields 3-49/~ for the static equilibrium width and 3.59 ,~ for the mean width. Hence, in the case of 02 and F2, at least, our rhombohedral ideal packing model predicts correctly the molecular dimensions as given by the 0.002 electron density contours. There has been much interest in the characterization of the structural transformation in oxygen, fluorine, and nitrogen[12]. We find it informative to look for a correlation between the observed a-/3 phase transformation temperature and a critical root-mean-square amplitude in our model, comparable to the Lindemann type criterion for melting. In Fig. 4 the r.m.s, displacements parallel (a) and perpendicular (b) to the three-fold axis have been plotted as a percentage of the mean nearest neighbor distance. The observed a-/3 phase transformation temperatures have been plotted as the solid circles. Accordingly, we observe that the a-fl phase transformation in oxygen, fluorine, and nitrogen occurs when the r.m.s, amplitude parallel to the threefold
A CRYSTAL OF DIATOMIC M O L E C U L E S - I
axis reaches a critical value of 5.2___0-1 per cent of the nearest neighbor distance. Note, however, that this is a correlation; the model itself, in fact, provides no a priori criterion or mechanism for predicting the phase transformation temperature. (b) l.r. spectra .1 The quasiharmonic 30~ frequency distribution spectrum of the rhombohedral model of/3-oxygen is shown in Fig. 5(a). Dispersion curves calculated for /3-oxygen for' waves propagating along the threefold [111] and twofold [110] axes are sketched in Fig. 5(b). The optical branch associated with 0-0 stretching (not shown) is essentially fiat with frequency differing negligibly from the gasphase value ve = 1580-4 cm-1[13] from which the force constant (K1) was derived. Neither the stretching mode nor the twofold degenerate (E,) zero-wavevector optical mode (TO) associated with the torsional oscillation of the 02 molecule occurring at 44 cm -~ for I
I
I
02
=.'N
I
I
10
20
I
I
30 40 TEMPERATURE{~
Fig. 4(a).
I
I
I
'~
i
F2
02
0
I
I
I
I
I
I
10
20
30
40
50
60
TEMPERATURE(~K|
Fig. 4(b). Fig. 4. R.m.s. displacements divided by the mean nearestneighbor spacing for directions: (a) parallel and (b) perpendicular to the 3-fold axis.
I
~hF2
I
I
1287
I
I
50
60
30~ is expected to be i.r. active. Thus, no fundamental absorption in the far i.r. is predicted. The conservation of energy and crystal momentum requires that, for normal processes the sum of the frequencies of the phonons produced be equal to the frequency of the incident radiation and that the sum of the wavevectors of the phonons be equal to the wavevector (essentially zero) of the incident photon. Thus, one might expect to find allowed combinations with the 0-0 stretching mode displaced from the fundamental by all frequencies between zero and the maximum lattice frequency. However, since the dispersion curves are fiat at the zero boundaries and there is therefore an accumulation of modes with frequencies close to the zone boundary yalues, and because of the larger volume in phase-space associated with
1288
G . E . J E L I N E K , L.
J.
S L U T S K Y and A. M. KARO
larger absolute values of the wavevector one would expect the frequencies of maximum absorption to represent the frequencies at the zone boundaries. Calms and Pimentel[29] observe a relatively sharp but extremely weak absorption in B-oxygen at 1549 cm -~ which is attributed to the 0 - 0 stretching mode (gas-phase v0 = 1556-2 cm -~) weakly active in the neighborI
I
I
i ~!ii
o2
8O
~o--
".%
20_
~
J
/~
,00 ~ 02 \::::::o/~ Z.O,
~ Y
0
Y~
Z.B.
Fig. 5(c).
, -- 10
30
20
\ lii I
i
I
I~
i
40 50 60 FREQUENCY(cm"11
Fig. 5. (a). Frequency spectrum G (~o) and quasiharmonic dispersion curves for the (b) rhombohedral and the (c) hexagonal model of 02 for the theoretical 30~ volume.
70
80
Fig. 5(a).
I[,| If ~ I [.o]I I
_
I ira ]
I__
/-::::" .?"/.~.~: :-, :: ]:
TA
* -
/
-
i i t "!"~L
I I-
3O
:] 2O 10 0
/
.~
"l
"2
t I
y~
ZB.
~ y
Fig. 5(b).
0
Y --~
Z.8.
hood of a vacancy or impurity. Since this portion of our calculation presumes no quasiharmonic intramolecular frequency shift, we naturally obtain a frequency for the 0 - 0 stretching mode close to the gas-phase Ve rather than v0, thus the comparison with the observed spectrum will be made in terms of displacements from the calculated value of the 0 - 0 fundamental frequency. The saEent feature of the combination spectrum is a broad relatively intense absorp,tion centered at 69 cm -1 above the fundamental which is attributed by Cairns and Pimentel to combinations with modes representing translation in the plane (xy) perpendicular to the threefold (z) axis. This band is superimposed on a broad general absorption extending from a b o u t 40 to 100 cm -1 above the 0 - 0 frequency. The frequencies in Fig. 5(b) extend from 28.9 to 80.9cm -1. F o r waves propagating along the twofold axis [110] the frequencies, at the zone boundary, of the longitudinal acoustic mode (LA) and the transverse acous-
A CRYSTAL OF DIATOMIC
tic mode (TA) polarized in the xy plane, both of which might be described qualitatively as translation in the xy plane, are 80.9 and 53-6 cm -1 in agreement with the assignment of Cairns and Pimentel. H o w e v e r the longitudinal acoustic mode for waves propagating along the threefold axis, qualitjatively T=, has a frequency of 57.8 cm -1 at the zone boundary and the torsional mode (TO) for waves propagating along the twofold axis a frequency of 49.1 cm-L Thus the absorption centered at 69 cm -~ would, in terms of this model, be due to the superposition of a variety of modes. In general, for a model of this sort, in which there is considerable curvature to the optical branches and no significant separation between optical and acoustic branches, a combination spectrum consisting of broad general absorption rather than well-defined features is expected. Indeed the combination spectrum might be as readily discussed in terms of the frequency distribution spectrum as in terms of the dispersion calves.
The lattice modes of the hexagonal D3h model for 02 are shown in Fig. 5(c) for wavevectors perpendicular (ii) and parallel (i) to the molecular axis. The frequency distribution spectra were obtained from a modification of a frequency interpolation routine of Gilat and Raubenheimer[30]. The frequency spectra and dispersion curves of N2, F2, CO, C12, and Brz are generally similar to those shown for 02. H o w e v e r due to the relative differences between the K,K2 force constants splitting in the optical branches can result in various peaks in the spectrum separating or converging. In the hexagonal structure (for zero-v~avevector) only T=(A2,) and R(E,) are i.r. active whereas there are i.r. active modes (Txu) corresponding to translation of the molecule in the plane normal to its axis in the true P2a3 structure. It is, of course, not to be expected that a model structure of incorrect symmetry will reproduce the selection rules of the correct structure. N o r do the nine optical
MOLECULES-I
1289
modes of the hexagonal model fully enumerate the 21 zero-wave optical modes of a structure which contains four molecules per primitive unit cell. The analogs, if any exist, to the remaining zone-origin modes of the more complex structure must be sought in the zone boundary modes of the hexagonal structure. The zero-wavevector mode of the hexagonal crystal polarized in the xy plane is simply a shear normal to the 3-fold axis with alternate layers of molecules moving in opposite directions. A more reasonable analogy to the optically active Tx~ modes of the P213 structure might be the zone boundary limit of a longitudinal optical mode propagating along the two-fold axis. These occur at 80 cm -1 for N2 and 84 cm -~ for CO. The zero-wavevector T=(A2,) and R(E,) modes are 55 and 41 cm -1 respectively for N2 and 58 and 42 cm -1 for CO. The far-i.r, spectrum of crystalline nitrogen and carbon monoxide has been investigated by Anderson and Leroi[31] and by Ron and Schnepp[32]. In the case of nitrogen two bands are found centered at 48 and 69 cm -1. Thus the values of 41, 55 and 80 cm -1 estimated here are in reasonable agreement with experiment. In CO the observed bands are at 49 and 85 cm -1 again in rough agreement with the calculated 42, 58 and 84 cm -~. While we would contend that the simple rhombohedral and hexagonal structures give a picture of the density of vibrational states in carbon monoxide and nitrogen which is adequate for the discussion of the thermodynamic properties away from phase boundaries, we would hesitate to suggest on the basis of this calculation that the 48-49 cm -a band be assigned to torsion rather than translation as suggested by Anderson and Leroi. However, if the values" of the torsional frequency calculated here are qualitatively correct then the rather broad 48-49 cm -a band might be interpreted as a superposition of translational and torsional components.
1290
G . E . JELINEK, L. J. SLUTSKY and A. M. KARO 6. SUMMARY
We have developed a model of the low temperature forms of crystalline 0~, N2, F2, CO, C12, and Br~ which assumes the static equilibrium structure to be the rhombohedral or hexagonal close packing of diatomic molecules. The nonarbitrarily adjustable parameters of a Lennard-Jones [6-12] potential between nonbonded atoms have been obtained in a recursive refinement procedure from observed solid-state data. The dynamical properties of the model are in good agreement with both the temperature factors derived from structural analyses and molecular dimensions of calculated electron density contours. For N2, 02 and F2 a correlation was found between the observed o~-fl solid-solid phase transformation temperature and a critical r.m.s, amplitude (5.2__+0-1 per cent of the nearest nonbonded atom-atom distance). Frequency distribution spectra and dispersion curves for the rhombohedral and hexagonal models were presented. It was shown that the combination spectrum of solid oxygen could be explained and the far-i.r. spectrum of CO and N2 rationalized on the basis of this model. REFERENCES
1. BORN M. and H U A N G K., The Dynamical Theory of Crystal Lattices, Oxford University Press, New York (1954). 2. ALIKHANOV R. A., Soviet Phys.--JETP 45, 812 (1963). 3. HORL E. M.,Acta crystallogr. 15, 845 (1962). 4. BARRETT C. S., MEYER L. and WASSERMAN J.,J. chem. Phys. 47,592 (1967). 5. BARRETT C. S. and MEYER L., Phys. Rev. 160, 694 (1967). 6. JORDAN T. H., STREIB W. E. and LIPSCOMB W. N.,J. chem. Phys. 41,760 (1964).
7. MEYER L., BARRETT C. S. and GREER S. C., J. chem. Phys. 49, 1902 (1968). 8. STREIB W. E., JORDAN T. H. and LIPSCOMB W. N.,J. chem. Phys. 37, 2962 (1962). 9. JORDAN T. H., SMITH H. W., STREIB W. E. and LIPSCOMB W. N., J. chem. Phys. 41, 756 (1964). 10. WYCKOFF R. W. G., Crystal Structures, Vol. I, John Wiley, New York (I 958). 11. BADER R. F., HENNEKER W. H. and CADE P. E.,J. chem. Phys. 46, 3341 (1967). 12. MEYER L., Phase Transitions in van der Waals Lattices, Advances in Chemical Physics XVI, 343 (1969). 13. HERZBERG G., Molecular Theory and Molecular Structure, I. Spectra of Diatomic Molecules, van Nostrand, New York (1950). 14. BOLZ L. H., BOYD M. E., MAUER F. A. and PEISER H. S.,Acta crystallogr. 12,247 (1959). 15. ORLOVA M. P.,zh. Fiz. Khim. 40, 2986 (1966). 16. KELLY K. K., U.S. Department of the Interior Bureau of Mines Bulletin No. 383 (1935). 17. BONDI A., Physical Properties of Molecular Crystals, Liquids, and Glasses, John Wiley, New York (1968). 18. MASON E. A. and RICE W. E.,J. chem. Phys. 22, 843 (1954). 19. BURFORD J. C. and GRAHAM G. M., Can. J. Phys. 47, 23 (1969). 20. HU J. H., WHITE D. and JOHNSON H. C.,J. Am. Chem. Soc. 75, 5642 (1953). 21. ANDRYCHUK D., Can.J. Phys. 29, 151 (1951). 22. BEGBIE G. H. and BORN M., Proc. R. Soc. (London) A I ~ , 179 (1946). 23. BEGBIE G. H.,ibid. A185, 189 (1946). 24. DONOHUE J. and GOODMAN S. H., Acta crystallogr. 18,568 (1965). 25. CRUICKSHANK D. W. J., Acta crystallogr. 9, 1005 (1956). 26. CAHILL J. E., Ph.D. thesis, Princeton University (1968). 27. CAHILL J. E. and LEROI G. E., J. chem. Phys. 51, 97 (1969). 28. CAHILL J. E. and LEROI G. E., J. chem. Phys. 51, 1324 (1969). 29. CAIRNS B. R. and PIMENTEL G. C., J. chem. Phys. 43, 3432 (1965). 30. GILAT G. and RAUBENHEIMER L. J., Phys. Rev. 144, 390 (1966). 31. ANDERSON A. and LEROI G. E., J. chem. Phys. 45, 4359 (1966). 32. RON A. and SCHNEPP O.,J. chem. Phys. 46, 3991 (1967).