Static and Dynamic Aspects of Order and Disorder in CBr, BY MARCEL MORE AND RENE FOUKET Laboratoire de Dynamique des Cristaux Molkculaires (E.R.A. 465), U.E.R. de Physique Fondamentale, Universitk de Lille I, 59655 Villeneuve D'Ascq Cedex, France Received 3rd December, 1979 In the first part of this paper is presented a review of experimental work carried out on CBr4: X-ray and neutron diffraction, neutron diffuse scattering and neutron coherent inelastic scattering. Comparison is made with CD4, and it is deduced that translational and rotational motions are coupled in CBr4. Then in the second part, some consideration is given to the development of a model of translation-rotation coupling using the theoretical work of Yasuda on CD4. 1. INTRODUCTION Below its melting point (365 K), carbon tetrabromide (CRr,) has an orientationally disordered cubic phase (phase I).A first-order transition occurs near 320 K and leads to an ordered monoclinic phase (phase 11). Owing to the similarity of their molecular symmetry (Td), it is interesting to compare CBr, with methane, which has been extensively studied : structure~,~-~ critical ~ c a t t e r i n g , ~ ~ ~ orientational order ~ a r a m e t e r , ~ phonon and librational theoretically etc. The fact that the transition does not appear at the same temperature for CBr, and CD4 is significant. The phenomena are not identical in the two cases mainly because of differences in molecular interaction. The solid-solid transition which occurs at 320 K is of first order and although phase I1 keeps some " memory " of phase I, the orientational order remains unexplained.Pretransitional effects have been seen in methane: critical slowing down for example; they have not been clearly recognized in CBr,. The potential seems to contain components which vary in a different manner with temperature, leading to a few com- peting order parameters. It thus appears interesting to study in more detail the high-temperature phase itself before trying to understand the transition mechanism. Comparison with methane will be made as often as possible. Phases I1 and I, ob- tained by X-ray and neutron diffraction respectively, are briefly schematized in Section 2(a) while in 2(b), the short-range order is analysed in time and space using neutron scattering measurements. The pattern obtained by a neutron diffuse scat- tering experiment is compared with that calculated from a hard-core potential model.Time analysis indicates that correlations have a complicated dynamical origin. Section 2(c) is devoted to neutron coherent inelastic scattering. Examples are given of acoustical phonons measured in the [I 111 direction. They show that acoustic velocities disperse with frequency. The scattering cross-section has a " three-peak- like " soft-mode structure. This probably results from a translation-rotation coup- ling similar to the spin-lattice coupling of the compressible Ising m ~ d e l . ~ ~ . ' ~ In The main experimental results are summarised in Section 2. Section 3 provides a basis for a possible explanation of experimental results.76 ORDER AND DISORDER I N CBr, particular a potential model based on a calculation by Yasudal' on CD, is outlined.This model is adapted to take account of translation-rotation coupling. Although calculations are long and sometimes tedious such a model can account for the com- plexity of the observed phenomena. 2. EXPERIMENTAL 2(a). s T R u c T u R E s The-structures of phases I1 and I have already been described.'6-18 We give only an Low-temperature phase I1 is monoclinic C2,= (a = 21.43, b = 12.12, c = 21.02 A, #? = 110.88", Z = 32). The structure was determined by X-ray diffraction. The 4 molecules in the assym- metric unit are in general positions. Vibrational motions have been analysed in terms of rigid-body tensors T, L and S . Tensor S , which takes account of correlations between translational and rotational motions has been shown to have no significant value. The final R factor has been found equal to R = R, = 0.07.Molecular centres of mass lay approximately on a face-centred cubic lattice, the molecules being tilted a little from the 42m symmetry. The structure of phase I was obtained by neutron diffraction. This phase, extending from 320 to 365 K, is face-centred cubic (a = 8.82 A, Z = 4). The refinement was made with a 6-orientation Frenkel model corresponding to a 42m local symmetry. It gives R = 0.16 with (u$> = 0.2 A2 (isotropic) and an isotropic rotational Debye-Waller factor (ol) = 0.05 Tad2. Introduction of anisotropic coefficients does not improve the fit. A new refinement has been made where the bromine coherent-scattering-length density2 a(r) is expanded as a set of symmetry-adapted functions.This gives a weighted reliability factor R, = 0.07. The probability of orientation of the C-Br bond is a maximum in the [110] direction. Dur- ing reorientation motions, molecules then have a maximum probability of being in a local orientation close to 42m. Nevertheless, Huller and Press l9 have shown that the orientational distribution function f(o) calculated from structure parameters is not always positive. Constraints should be imposed upon the expansion coefficients. The authors prove that f(o) is automatically positive when coefficients are deduced from an appropriate physical potential. To a first approximation it would be an angular potential but, owing to the large value of the trans- lational Debye-Waller factor (a:> = 0.2 A2, translation-rotation coupling must be con- sidered.Such a coupling has been introduced phenomenologically20 and has given new coefficients for the expansion. Nevertheless, we do not know if f(o) is definitely positive, and related coefficients are not connected to a potential. Apart from the fact that molecular centres of mass remain approximately on a f.c.c. lattice, the correspondence between the two phases is not evident. Particularly, orientational order in phase I1 remains unexplained. It could result from lattice distortion leading to a vanishing translation-rotation coupling, as suggested by the zero value of tensor S . outline . The R factor is then reduced to R = 0.037. 2(b).SHORT-RANGE ORDER Measurements made on powder samples by Dolling et aZ." in the cubic phase have re- vealed in the spectrum a diffuse peak for lQl N 2.1 A-'. This peak is not explained by the structure and the authors assign it to correlations. A neutron diffuse-scattering experiment conducted by More et aZ.2' shows that the scattering pattern has cigar-shaped contours near the (220) Bragg peak. The long axes of the cigars point in the [11 13 directions (fig. 1). In this paper,21 we show that scattering cannot be explained by structure-factor fluctuations (AF)2 = <F2> - <F>2. These fluctuations contribute to the scattering intensity in the same region of reciprocal space but the corresponding intensity would be lower and broader.M. MORE AND R .FOURET 77 Coulon and Descamps22 have given an explanation for the existence of this peak. They show that neighbouring molecules are correlated via a potential corresponding to steric hindrance between bromine atoms. This potential has a short-range action and correlates only nearest-neighbour molecules. The authors choose it to be in the form of a static hard- core potential. They use a 6-orientation Frenkel model for the structure. The use of a hard-core potential is justified: among the 6 x 6 possible molecular configurations only 32 give an averaged attractive energy of 2.4 kcal mo1-'. The last 4 give a repulsive energy of 10 kcal mo1-l. The results of their calculation are in good agreement with experiment. Calculated intensity contours are practically identical to those shown in fig.1. 2.E 2 2 .c 1 .E 0 0.5 51 1 FIG. 1 .-Diffuse scattering in the (1, f , 2) plane. The scattering vector is Q = cl(l, 1, 1) + c2( 1 ,I ,O). (Sample 4 cm3, background 400 counts, i.r. = instrumental resolution). However, this model can describe nothing but static disorder with molecules in selected orientations taking account of their surroundings. On the other hand one can imagine that a reorientating molecule translates and rotates in order to find a more stable equilibrium. Then correlations are dynamical. Because of dissipative processes, orientational fluctuations decrease in time as e - r r and give rise to quasi- elastic scattering. Fig. 2 and 3 show energy scans for diffuse scattering at points Q = (2.15, 2.15, 0.) and Q = (2.5, 2.5, 0).We estimate the instrumental resolution width to be 0.054 THz at Ai = 4.05 A. Intensity comprises a purely elastic component arising from the quartz container of the crystal and one (or several) quasielastic component(s). This result is fitted with a Gaussian function for instrumental resolution and a Lorentzian centred at zero energy for the quasi- elastic peak. We then obtain: The corresponding experiments have been made on spectrometer H1 at Saclay. rl = 0.03 0.005 T H ~ (fig. 2) rz = 0.21 i 0.03 THZ (fig. 3). rl is the inverse of the relaxation time of correlations responsible for intensity shown in fig. 1. In making this experiment (fig. 1) our instrumental resolution width was much larger than rl and the result was an integration over energy of correlation functions, that is to say78 ORDER AND DISORDER IN CBr, the static susceptibility.The origin of the (I?,) Lorentzian is not understood at the present. In the case of methane (CD4, phase I), quasielastic diffuse scattering5 additionally reveals anisotropic correlations. As with CBr4, correlations are stronger in [l 111 planes than out of these planes. When the temperature is decreased toward the transition temperature T,, orientational fluctuations slow down,6 the intensity increases, becomes critical and finally we obtain a Bragg peak corresponding to a new order in phase 11. Such critical scattering has not been observed in CBr4. We have only noticed a significant hysteresis of 5 K when 8CO d *z i3 s 600 %+ v) * 1 400 200 A . - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - b -0.1 -0.05 0 0.05 0.1 energy transfer/THz FIG.2.-Constant Q scan (at T = 325 K) at the point Q == (2.1 5,2.15,0). The elastic peak is mainly due to the quartz container. The Lorentzian has half width at half maximum = 0.03 THz. Ai = 5 A. the temperature is lowered toward the transition. At this temperature the crystal breaks. However, when temperature is increased, rl increases and the quasi-elastic intensity decreases (for example: rl = 0.05 THz and intensity is divided by two at T = 358 K). While the transition is first-order, a second-order transition could take place but at To < T , . 2(C). PHONONS Neutron coherent inelastic scattering experiments have been made at Saclay by M. More and B. Hennion. Acoustic phonons were measured in high-symmetry directions and with different polarizations.A few neutron groups are shown in fig. 4 and 5. They corre- spond to the longitudinal branch [ l l l ] . Fig. 5 is a constant Q-scan made at point Q = (1.085, 1.085 and 1.085) with an incoming wave- length Ai = 5 A. It shows a three-peak structure with maxima at fm and 0. At zero energy, an elastic peak due to the quartz container is superimposed. The remaining in- Intensities decrease rapidly and intrinsic widths increase when q increases (fig. 4).M . MORE AND R. FOURET 79 tensity is clearly broader than the instrumental width (0.025 THz in this experiment). In a soft-mode study23 as well as in a theoretical paper worked out for KCN by Michel and N a ~ d t s , ~ ~ we find an expression for the scattering law: where q represents phonon-phonon damping, p= - Cl’, i2 is the phonon frequency with interactions, oo is the phonon frequency without interactions, A-’ is the relaxation time of interactions andf(o) = l/(oz + A2).Using this empirical expression, we have been able to take into account the observed results and then obtain for CBr4 q = 0, 1 21 0.04 THz, n/o0 = 0.6, oo = 0.24 THz. A parametrization of this function for constant-energy scans has allowed us to fit the neutron groups of fig. 4. The full lines in fig. 4 and 5 represent this. A is of the same order as rl previously determined to take account of dynamical correlations. The scattering law [eqn (2. l)] describes soft-mode behaviour. In CBr4, longitudinal acoustic modes behave like this.In fact acoustic velocities are softening with frequency as presented in fig. 6 . A -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 energy transfer/THz FIG. 3,-Constant Q scan (at T = 325 K) at the point Q = (2.5,2.5,0). The Lorentzian has h.w.h.m. = 0.21 THz. 11 = 4.05 A. The anisotropy which has already been seen in diffuse scattering is now characterized by the fact that velocities of longitudinal modes disperse more than transverse ones. The softening of velocities often appears in plastic crystals, for instance in KCN,25 succinonitrileZ6 or ada- man t ane.2 Spectra obtained at several points in reciprocal space (such as zone centres or zone boundaries) are all of the same kind. They show a more or less intense “ background ” centred at o = 0. A de- convolution with the experimental resolution function plus a Lorentzian give an intrinsic half Attempts to measure librational phonons have always failed in CBr4.80 ORDER AND DISORDER I N CBr, width at half maximum of M 1 THz.There seem to exist in CBr4 nothing but diffusive modes characteristic of fluctuations exponentially decreasing with time. In methane,8 acoustic modes broaden out when the frequency increases, but the velocities do not seem to disperse. In the two cases, Librations have not been measured either. 250 200 h Y .-. 1 g - 150 x +J M .4 0) +J .- 100 50 energy transfer/THr f I 1 / I I b 0.1 0.2 0.3 momentum transfer (reduced units) FIG. 4.-Longitudinal acoustic neutron groups in the direction [11 11. molecules are reorientating with too high a frequency and neutrons cannot " see " their oscillations around temporary equilibrium positions.Libration frequencies would be of the order of 1 THz while correlation relaxation frequencies are close to 0.03 THz in CBr, and 0.09 THz in CD4.6 3. THE MODEL In order to explain these experimental features we develop in this Section a Potential energy can be calculated from atom-atom potentials between un- Using a 6-12 Lennard-Jones potential, Yasuda l1 has potential which takes account of translation-rotation coupling. bonded atoms or molecules.M. MORE AND R . FOURET 81 shown that the interaction potential between two tetrahedral molecules (I and 2) such as CBr, or CD, is given by: Vl, = U(R) + V R , 8, q ; w ) + V(R, n - 0, q + ; 7 1 4 + W R , 079; a 1 , m2) (3.0 where (R, 0, q) are the centre-of-mass polar coordinates of molecule 2 with respect to 100 1 O J I I I I I I b -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 energy transfer/THz width: 0.025 THz.FIG. 5.-Constant Q scan (at T = 325 K) at the point Q = (1.085, 1.085, 1.085). Instrumental the centre of mass of molecule 1, w1(co2) are three Euler angles setting the orientation of molecule l(2) in a standard frame. We define our standard orientation for the molecule like Yasuda.ll - U(R) is the isotropic part of the intermolecular potential. To obtain this term, we need an ex- pansion in series of powers of r/R [CA(r/R)p+4] where r = I Y ~ - ~ ~ ~ or In CD,, we have r/R = 0.26 and convergence for the series is correct whenp + q = 24. In CBr,, we have r/R = 0.31 and series has to be taken up to p + q = 36.These expansions have been calculated numerically on a computer. P A v = (R, 0901; m) = 2 u(R, I ) 2 K r n ( 0 , ~ ) Tlrn(w) (3 * 2) 1 2 3 m where the coefficients u(R, 1) given in ref. (1 1) are numerically calculated. are the spherical harmonics, and Tlrn(w) are defined by: YJO, q) 4 (B,, yk) are polar angles for a position of a molecule at a given moment with respect Tetrahedral symmetry of the molecule implies that Al,rn are different from zero to the original position (standard frame). only for I = 0, 3,4, 6, 7, 8, . . . 9(2,,, (co) are the usual Wigner rotation matrices. q, A l , , , are given in ref. (11). W(R, 8, q ; tol, 0.1,) = 2 w(R; 11'1'') 2 C(1I'I";mm') Y&,,,,$ (0, q) (3 -4) 1,1'>3 m,m' l"20 x Tl rn(m 1) Tl rn ( ~ 2 )82 ORDER AND DISORDER IN CBr, 3.0 2 .o 1 .o n 0 .cd - x 3.0 0 .I W w I rn E + 2.0 x 5 4 1.0 s c( w 0 W E c VT A A A A ' A A 500 3.0 7- 0 VT - 500 2 . 0 1 1.04 r1101 1 I I 0 0.1 0.2 0.3 0 . 4 0.5 AEITHz FIG. 6.-Dispersion of the wave propagation velocities.M. MORE AND R . FOURET 83 w(R; ZZ’Z‘’) are numerical coefficients.ll C(ZZ’Z’’; mm’) are the Clebsch-Gordan co- efficients in Rose’s notation.28 This term will not be used here but we can make some remarks about it: (i) The number of relevant coefficients is much greater in CBr4 than in CD, where only a few are preponderant: in particular we notice that James and Keenan” have only used the (3, 3, 6) term leading to the octopole-octopole interaction. (ii) Many of these coefficients have opposite signs and can then cancel each other.(iii) We assume that, in spite of some differences, this term will give results not very different from those obtained in methane. Numerical values (table 1) have been obtained using only interactions between TABLE NUMERICAL VALUES OF u(R, I ) IN EQN (3.2) AND w(R, ll’i”) IN EQN (3.4) IN kcal mol-I v(R, 4) = 1.22 v(R, 6) = 0.688 u(R, 8) = 0.08 I I’ 1” w(R,II‘Z“) 3 3 6 3 3 4 3 3 0 3 4 7 3 4 5 3 4 3 3 4 1 3 6 9 3 6 7 3 6 5 3 6 3 4 4 8 4 4 6 4 4 4 4 4 2 4 4 0 4 6 1 0 4 6 8 4 6 6 4 6 4 4 6 2 - 1.96 - 1.23 1.275 1.13 0.547 -0.649 -0.720 0.762 0.254 -0.362 -0.268 0.665 0.254 0.448 -0.342 -0.253 0.476 0.128 0.124 -0.206 -0.104 bromine atoms (Br-Br). screened by the size of Br atoms. for a Br-Br Buckingham potential Indeed we can assume that C-Br and C-C interactions are We use the coefficients given by Scott and Sheraga2’ VBr-Br = A exp (- Cr) - B/r6 with C = 2.78 A, A = 34 600 and B = 5180 if Yis expressed in kcal mol-’.Yasuda’s calculation needs this potential to be changed into a 6-12 Lennard-Jones potential V = a/r12 - b/r6 where a = 4.374 x lo6 and b = 3924 with V in kcal mol-I.84 ORDER AND DISORDER I N CBr, The crystal potential will now be expanded in series of displacements ui of the centre of mass of the molecule with respect to the averaged equilibrium positions. At zero order, with respect to the displacements, the potential can be written: R i j is the “ equilibrium ” distance between molecules i and j and the second term corresponds to the crystalline field acting on molecule i. Due to disorder, the averaged equilibrium positions are not the true equilibrium positions of the molecules and there are terms of greater order in the expansion. However, owing to cubic symmetry the linear term cancels for the isotropic potential. The second and third terms of eqn (3.1) give: dV = - C ( F [ + Fi) ~ i .(3.7) i If the summation is limited to nearest-neighbour molecules ( r ) at a distance R from molecule i, we obtain : F; = 2 grad V(R, o r , p r ; mi) Fj = 2 grad V(R, n - Or, n + q,; co,). (3.8) (3.9) r r We neglect the linear term arising from the 4th term of eqn (3.1). gradient formula28 to calculate eqn (3.8) and (3.9). Let us use Rose’s (3.10) grad ( u ( R I ) Yirn(or, ~ r ) ) = 2 Elol v(Rt l)TilJrn(or, 9,). I’ Here we have: --I for I’ = I + 1 I + 1 for I‘ = I - 1 0 for I’ # I & 1 and (3.1 1) (3.12) (3.13) where tYl,,rn+p are constants tabulated in ref.(1 1) and differ from zero only if I’ = 4, 6 , 8 . . . and Then we obtain: (3.14) (3.15)M. MORE AND R . FOURET 85 Molecular symmetry gives us I = 0, 3, 4, 6, 7, 8, . . . : crystalline symmetry gives I’ = 4, 6, 8, . . . . Taking eqn (3.10) into account we have I = I’ 1. Then non-zero terms will correspond to : l = 3 , 2 ’ = 4 I = 7, I’ = 6 1 = 7 , 1 ’ = 8 . . . . where the %:’(co,) are the (0,) symmetry-adapted functions belonging to the irre- ducible representation TI,, for I = 3. They have been tabulated by Altmann and Cracknell : 30 This result can be extended to any value of I. bouring molecules. form : The second term Fi implies a summation over all the orientations (co,) of neigh- We can drop out this summation by taking the Fourier trans- (3.18) 12 6 Vlnrn*(q) = 2 eiqR Ylern@r, qr) = 2 [eiqR +.(- 1)”e-’qR] Yl#rnl(Or, qr). (3.19) r = l r = 1 Then : (3.20) where ‘%‘ll.rn(q) is obtained by a generalization of eqn (3.14)-(3.18). For q = 0, we obtain Fi(R, 0) = Fi(R, w). For q # 0, expression (3.19) cannot be reduced like eqn (3.13) and terms of the expansion will be: ((I = 3, I’ = 2 and I’ = 4), ( I = 4, I’ = 3 and I’ = 5), . . .}. Fi(R, q) can be expanded on sets of symmetry-adapted functiQns in the same manner as Fi(R, w). A few of them have large values and for further calculations we will only take these main terms into account. Let us call UB(q) a vector defined with Fourier components %(T)(q) of symmetry adapted angular functions.I varies over all irreducible representations of the rotation group. z numbers all the functions which are classified in irreducible representations of the Oh group. Geometrical factors and coupling constants can be Some coefficients iir.rv(R, 1) have been calculated (table 2).86 ORDER AND DISORDER IN CBr, put into a matrix vorD(q). ments, We denote by ua(q) the Fourier components of displace- Then, the translation-rotation potential can be written in a bilinear form: (3.21) The hamiltonian can be written as where K is the kinetic energy of translation, VT the translational potential energy, Vc the crystalline field and VTR the translation-rotation potential. We have neglected the rotational kinetic energy, assuming that reorientations are slow enough. We have also neglected rotation-rotation coupling W(wl, w,).One can see that V, can easily be included in eqn (3.22). H = C {+P.'(qIPu(q) + +Mufi(q)u,+ (q)%3(q) + NP(!I)US(4)). (3.24) Pu(q) are the conjugate momenta of uu(q) and Mub(q) is the usual dynamical matrix for translational motions. Then we obtain 4 TABLE 2.-NUMERICAL VALUES OF &lie7 U(R, r) IN EQN (3.15) IN kcal m0l-l A-' ~ Z ~ U ( R , 3) = -7.34 &43~(R, 3) = 10.32 634~(R, 4) = -3.64 &54~(R, 4) = 5.36 &56U(R, 6) = 1.84 B76U(R, 6) = 3.04 B ~ ~ v ( R , 7) = 1.83 &67V(R, 7 ) = - 1.06 Let us denote: ua(4) = u;'(q) + U5'"d (3.25) where ui'(q) is an elastic deformation supposed to vary slowly with time and uiib(q) the usual vibration coordinates. From eqn (3.24) we derive the condition for instantaneous equilibrium which is: &'(q) = --(M-')UDNi(q).(3.26) Using (3.27) The first two terms contain translational kinetic and potential energies, the third one has only terms of rotational origin. This result and the fundamental transformation denoted by eqn (3.25) are a generalization of the compressible king model already presented in ref. (14) and (15). This relation connects elastic displacements with angular vector NB(q). (3.24)-(3.26) we get: H = 2 {+PZ ( d P a ( d + 3MuD(q)U2b + ( q ) ~ " % Z ) + 3Nu(q)u,'(d). 4 In this form, the hamiltonian is a sum of independent components.M . MORE AND R. FOURET 87 Although some numerical results have already been obtained, calculations have not yet been completed. However, we think that the theory presented here can explain some experimental features in CBr,.As the translation-rotation potential is bilinear in the rotation and translation coordinates, the dynamical theory written for KCN24 can be applied. In particular, rapid variations with q of matrix ua&) could explain the softening of wave-propagation velocities when q --+ 0. At q = 0, the potential is made of terms of order I = 3 and 7. This could account for additional terms found by Press2* for the structure. The scattering law can be expanded using the hamiltonian derived from the generalized compressible Ising model. So we obtain the translation-translation, translation-rotation and rotation-rotation parts of scattered intensity. Special account of rotation-rotation terms W(R, 8, p; mi, w2) should certainly be taken if we wanted to explain changes occurring at transition point in CBr, as it has been done in CD4.We thank J. Lefebvre (ILL., Grenoble), B. Hennion (L.L.B., Saclay), B. M. Powell (Chalk Rivet, Canada) and M. Bee, J. L. Sauvajol and J. C . Damien (Lille) for their help in experiments and discussions. W. Press, J. Chem. Phys., 1972, 56, 2597. W. Press and A. Huller, Acta Cryst. A , 1973, 29, 252. W. Press, Acta Cryst. A , 1973, 29, 257. W. Press and A. Huller, in Anharmonic Lattices, Structural Transitions and Melting, Nato Advanced Study Institute Series (Noordhoft-Leiden, 1974), p. 185. A. Huller and W. Press, Phys. Rev. Letters, 1972, 29, 266. W. Press, A. Huller, H. Stiller, W. Stirling and R. Currat, Phys. Rev. Letters, 1974, 32, 1354. W. Press and A. Huller, Phys. Rev. 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