The Phase Transition in t-Butyl Cyanide [(CH,),CCN] t BY JONATHON C. FROST AND ALAN J. LEADBETTER Chemistry Department, University of Exeter, Exeter EX4 4QD AND ROBERT M. RICHARDSON 8 Institut Laue-Langevin, Grenoble Received 18th January, 1980 Extensive neutron scattering measurements have been carried out on t-butyl cyanide over a wide temperature range. Crystal structures and the potential for methyl-group rotation have been deter- mined and will be reported elsewhere; this paper is concerned with the whole-molecule motions as determined by incoherent quasielastic and inelastic scattering measurements. One transition only has been observed at 233 K and the whole-molecule reorientational motions determined between 190 and 280 K (m.p. 292 K). At all temperatures the molecules undergo rapid reorientational motion about their C-C-N axes in a 3-fold potential which is strongly temperature dependent.The transition is associated with an order of magnitude decrease in the rotational correlation time and the onset of a slower, strongly cooperative motion involving translational as well as rotational displace- ments of the molecules. These probably involve relaxations between different local structures each having essentially the monoclinic structure of the low-temperature phase and related by rotations of 71/2 resulting in the average tetragonal symmetry of the high-temperature phase. The neutron results lead to a self-consistent reinterpretation of previous n.m.r. data. Tertiary butyl cyanide [(CH,),CCN] is an especially interesting member of the (CH3)3CX series which shows a rich solid-state polymorphism.The dielectric studies of Clemett and Davies' indicated that in the highest temperature crystal phase (I) the molecules may rotate about the CCN axis but the dipolar axis itself does not reorientate until the liquid phase is reached. gave the following phase behaviour : Extensive calorimetric measurements (Crystal 111) T = 213 K; A S = 1.1 J K-I mo1-I (Crystal 11) T = 233 K; A S = 7.8 J K-' mol-I (Crystal I) T = 292 K; A S = 31.8 J K-I mol-I (liquid). The transition III/II was not well-defined but the total entropy increment I11 to I is near R In 3 (9.1 J K-' mo1-I) which was taken to imply that I is a uniaxial rotator phase. Whether or not this interpretation of A S is correct the rotation of the molecules about their CCN axes in Crystal I has been confirmed by n.m.r.3 and very recent incoherent neutron quasielastic (INQES) meas~rements.~ t Experiments carried out at I.L.L., Grenoble and A.E.R.E., Harwell.$ Now at Rutherford Laboratory, Chilton, Didcot, Oxon OX1 1 OQX.J . C . FROST, A. J . LEADBETTER AND R . M . RICHARDSON 33 A number of uncertainties remain, however. First, an earlier n.m.r. study' was interpreted as showing only methyl group rotation below the m.p., while in the later work3 the single step decrease in linewidth near 110 K was interpreted (more reasonably) as due to the onset of both methyl and whole-molecule uniaxial rotations. The additional observation of a single TI minimum near 160 K led to the conclusion that the methyl and molecular rotations are coupled and describable by a single correlation time and energy barrier (16.3 kJ mol-I).The motion is established in phase 111 and almost no changes are observed at the transitions (except a slight 41 0.0 0.5 hw/meV FIG. 1 .-Scattering law, S(Q, o) of t-butyl cyanide at 227 K in the Crystal I1 phase measured on IN5 with lo = 10.0 A. The spectra (which are not to the same vertical scale) have elastic Q values (from bottom to top) of 0.17,0.38,0.76 and 1.15 A-'. The solid curve is a fit of the 3-fold rotational jump model [eqn (6), (7) and (ll)] to the experimental data. The dashed line separates the elastic from the quasielastic scattering. hysteresis around III/II as found calorimetrically). This is a puzzling conclusion because the barriers to methyl rotation in t-butyl compounds have been thoroughly studied '--* and are largely intramolecular in origin whereas the whole-molecule re- orientation must be controlled by intermolcular forces and should have a different temperature dependence. Secondly, the correlation time at the II/I transition from the n.m.r.work is z s in I with a jump at the transition to values too great to be measured in I1 ( 2 2 x s). Questions remain, therefore, about the relation between the different experimental results, the nature and time scale of intra- and inter-molecular rotations in the various phases and the nature of the phase transition(s). s whereas the INQES results gave z 5 x34 PHASE TRANSITION I N (CH3)3CCN t-Butyl cyanide was chosen for an extensive set of measurements on a uniaxial rotator phase with the ultimate objective of understanding not only the single molecule motions but also the collective aspects of the reorientational motions and the nature of the transition.It was selected because both hydrogenous and deuterated versions may be obtained and the melting point being near room temperature should facilitate the preparation and handling of single crystals. Furthermore the strong cyano - 1 0 hw/peV 1 FIG. 2.-Scattering law, S(Q, w) of t-butyl cyanide in the Crystal I1 phase measured on the back scattering spectrometer IN1 0. The solid curves are calculated for the 3-fold jump reorientation model [eqn (6), (7) and ( l l ) ] and the dotted curves separate elastic and quasi-elastic components: (a) Q = 0.57 A-', T = 208 K; (b) Q = 1.51 A-', T = 208 K; (c) Q = 1.51 A-l, T = 190 K.dipole and the uniaxial rotations suggested that the dipoles are probably all aligned (along 0 and n) to give essentially a 2-d system of coupled rotors. So far we have completed low-resolution structural determinations of low- and high-temperature phases and INQES plus incoherent neutron inelastic scattering (INIS) on powder samples.* The structure from < 5 to >227 K is monoclinic (P2Jrn) with two mole- cules per cell in a head to tail arrangement, all dipoles colinear and directed along c. No structural or other changes have been found at the reported III/II transition so we hereafter designate the structure existing below 233 K as Crystal 11. The structure ofJ .C . FROST, A . J . LEADBETTER A N D R . M . RICHARDSON 35 0.0 0.5 ho/meV 0.0 0.5 1.0 U L 1 i .I - .._1-- ho/meV 0 0 0.5 10 1.5 2.0 2.5 FIG. 3.-Scattering law, S(Q, o) of t-butyl cyanide in the Crystal I phase. The solid curves are typical fits to the data using eqn (3), ( 5 ) and (1 1) with arbitrary elastic coefficients, the dashed curves separate elastic and quasielastic components and the horizontal straight lines show the flat inelastic back- ground: (a) 270 K, measured using IN5 at 1, = 10 A, Q values (bottom to top) of 0.17, 0.38, 0.76, 0.92 and 1.15 A-'; (6) 240 K measured using 4H5, Q values (bottom to top) of 1.7, 1.1, 0.75 and 0.53 ii-l.36 PHASE TRANSITION I N (CH3)$CN Crystal I is tetragonal (probably P4/n) with 2 molecules per cell, at the origin and body centre, again in a head to tail configuration with dipoles along the 4-fold axis (parallel to c ) and related to I1 by a straightening of the molecules and a greater spatial requirement in the plane perpendicular to the C-C-N axes.* The methyl- group rotations have been fully characterized and are well described in terms of motion in a 3-fold potential well of depth -18 kJ mol-I and with zo - 0.3 x s where the average residence time in the well is z = zo exp(V3 - EJRT.This paper is concerned with the whole-molecule motions as determined by incoherent neutron scattering measurements in both phases. Structural and methyl rotation data will be used as appropriate in analysing the results but will not be discussed further here.8 Since only incoherent scattering measurements have been made (except for the struc- tural work) only a single-particle description of events is obtained directly but this proves to be very informative and indirectly reveals pronounced co-operative phe- nomena involving coupled rotational and translational displacements in Crystal I, the onset of which appears to be the essential feature of the phase transition.EXPERIMENTAL The sample was obtained from B.D.H. and checked for purity by n.m.r. and gas- liquid chromatography: no impurities were detected suggesting a purity of better than 99.8 %. Differential scanning calorimetry using a Perkin-Elmer DSC-2 gave the following results: (Crystal II)T= 232 k 2 K ; AH = 2.0 i 0.3 kJ mol-' (Crystal I) T = 291 f 2 K; AH -= 9.1 $0.5 kJ mol-' (liquid) in satisfactory agreement with the results of Westrum and Ribner.2 The specimens for the neutron experiments were contained in disc-shaped aluminium alloy cans of diameter ~5 cm and with I mm walls.The sample thickness was ~ 0 . 3 mm giving 900/, transmission. Sample temperatures were controlled to f 1 K and gradients across the sample were probably of similar magnitude. Four spectrometers9 were used in this work and measurements made at several temperatures as follows: (i) The chopper time- of-flight spectrometer 4H5 at AERE Harwell: incident wavelength ,lo = 4.8 A; elastic energy resolution AEo % 100 peV (h.w.h.m.); maximum elastic scattering vector Qh' = 1.85 No other transition was detected despite a careful search. A-1 (Qel == % sin 0). Temperatures T/K = 227, 240, 265, 284.(ii) The multichopper A n time-of-flight spectrometer IN5 at I.L.L., Grenoble: (a) A. == 6.45 A; AEo = 24 peV (h.w.h.m.); Qk' = 1.75 A-l; T/K = 227, 240, 250; (b) ;lo = 10.0 A; AEo = 10 peV (h.w.h.m.); Qh' = 1.15 k'; T/K = 201, 218, 227, 239, 270. (iii) The very high resolution backscattering spectrometer IN10 at T.L.L.: A. = 6.28 A; AEo % 0.5 peV (h.w.h.m.); QK' = 1.8 A-'; T/K = 190, 201, 208. (iv) The rotating crystal spectrometer IN4 at I.L.L. This was used exclusively for inelastic measurements with: Eo = 12.5 meV; AEo = 0.3 meV (h.w.h.m.) QK' = 4.8 A-'; T/K = 5 , 150, 201, 218, 227, 239. Any counters which received Bragg scattering were excluded from the quantitative analyses. The experimental resolution function was determined using both a vanadium sample and the t-butyl cyanide sample itself at low enough temperatures for all rotational motion to be frozen.After subtraction of background and empty-can scattering and correction for absorption and self-shielding the results were normalised by comparison with the scattering from a vanadium sample and were converted, using standard procedures, to the symmetrised scattering law, S(Q, co). This involves only experimentally known quantities and S(Q, w ) is a property only of the system, independent of the experiment; it is also the function most * Crystal I1 (227 K)u = 6.45 A, b = 6.95 A, c =I 6.75 A, p = 92.8"; C-C-N axis parallel to c. Crystal I (239 K) u = b = 6.89 A, c = 6.70 A; C-C-N axis parallel to c.J . C. FROST, A. J . LEADBETTER AND R .M . RICHARDSON 37 1.400 1.050 0.700 1.000 0.500 2.500 2.500 0.99 0 0.540 0.090 4 + + + + + + + + t i + + ' I + + t '+ t i t + + + t + t + + + + t t t t I 1 0.0 5.0 10.0 15.0 20.0 tio/meV FIG. 4.--Inelastic S(Q, w) spectra at various temperatures measured using IN5 in energy gain. Similar results were obtained for energy loss using the IN4 spectrometer. simply related to theoretical calculations. Typical quasielastic spectra are shown in fig. 1- 3 and inelastic results in fig. 4. These results show that some bound stochastic motions are occurring in both crystal phases and these turn out to be whole-molecule rotations about the dipole axis. RESULTS AND DISCUSSION (i) QUASIELASTIC DATA ANALYSIS The incoherent scattering law from a proton in molecule undergoing reorienta- tional motions may be written" S,(Q, m) = e - z w [SXQ, m> -1 SXQ, m)I where e-2w is a Debye-Waller factor {2W = <<Q u>~>>,38 PHASE TRANSITION I N (CH3)3CCN Ss(Q, w ) is the inelastic scattering (broadened by the rotational motion) and SF(Q, a) is the rotational scattering which may be written quite generally (assuming exponential time correlation functions) : where 9 ( z i 1 ) = z i ' / ( z i 2 + a*) and the A,(Q) depend upon the type of motion and 2 A l = 1.l = O Ao(Q) = 1 exp iQ.rP(r) dv l2 (4) where P(r) is the (time independent) distribution function for the proton. Ao(Q) is commonly called the elastic incoherent structure factor (EISF) and gives a direct measure of the trajectory of the proton, averaged over " infinite " time, as the mole- cule rotates. For non-equivalent protons the above results must be averaged over the different protons." Infinite " time is determined by the resolution of the experi- ment, so effectively t , z 5h/AE0. For a molecule such as t-butyl cyanide the number of protons is sufficiently large that except at Bragg reflections the observed scattering is generally completely dominated by the incoherent cross section of the protons. The data were first analysed using eqn (1) and (3) with adjustable coefficients Al to determine the EISF and hence the geometry of the reorientational motions. A model was then devised which specifies the Al (see below) and the data fitted to obtain the correlation times T ~ . In all model calculations the known molecular geometry was used.11*12 Provided that the quasielastic scattering is not too broad S:(Q, w ) may be taken as a flat background in the quasielastic region.The EISF was then determined as the ratio of elastic to total (elastic plus quasielastic) scattering after subtraction of a flat inelastic background. The Debye-Waller factor is given by the Q-dependence of this total intensity. A careful analysis of errors in the EISF was made by taking account of the effects of varying (i) the functional form (number of Lorentzians) in eqn (3) for the computer fitting and also separating elastic and quasi- elastic components by eye for the high temperature phase and (ii) the energy range of the fit. The very small effects of the data being at constant scattering angle and not constant Q were also taken into account.A very useful model for analysing the data is that for instantaneous jump motion among m equidistant points on a circle of radius Y for a powder sample,13 for which the scattering law is of the form of eqn (3) with the quasielastic summation from I = 1 to I = rn - 1, and the coefficients for I 3 0 given by: j , ( X ) = sin X / X and z1 = ( 2 / 2 ) sin2 (nZ/rn) where T is the residence time ( 7 - l is the jump frequency). In determining the EISF, m could be varied and the coefficient of the elastic term allowed to be an arbitrary variable: Do(Q). The model scattering law was fitted to the data after convolution with the experimental resolution function. Reorientational motion between three equidistant sites is important both for methyl and whole-molecule rotations and for this eqn (3) becomes, for a powder sample, S,"(Q, a) = Ao(Q)d(a) + 11 - Ao(Q)12(zi1) A,( Q) = [ 1 + 2j0( Qr.t/3)]/3and z1 = 2 ~ / 3 , with z being the residence time.J .C . FROST, A . J . LEADBETTER A N D R . M . RICHARDSON 39 (ii) EFFECT OF METHYL ROTATIONS For compounds like t-butyl cyanide both methyl and whole molecule rotations may occur. Assuming that these are independent then the total scattering law is a convolution of the two separate components which for a powder sample may be the bar denoting the powder average. general form (3) for the whole-molecule motion (B) gives The methyl rotation (A) has been shown to be described by eqn (6) and taking the where the A and B are functions of Q. Explicit expressions for the coefficients have been given by Schlaak14 for the case where the whole-molecule motion is also a 3-fold uniaxial rotation. For the present purpose, however, we note the fortunate circumstance that the correlation times for methyl rotation in t-butyl cyanide are at least one order of magnitude slower than those for the uniaxial molecular rotation for T 2 190 K, which confirms their inde- pendence in this case.This has been established by investigating the apparent elastic peaks such as those of fig. 3 using an order of magnitude higher resolution. The methyl rotation is thus so slow that it is virtually unobservable using IN5 (and 4H5). For a truly unobservable methyl rotation the EISF would be given by EISF = A,B, + (1 - Ao)Bo = Do. In fact, although it made little difference to the value of the EISF, especially in the high T phase, a significantly better fit to S(Q, co) was obtained by taking account of the very small broadening contribution of the methyl rotation (just observable in the wings of the elastic component).This was simply achieved by writing the scattering law as: SdQ, m) = Bo(Q){4Q>a<a> + [1 - a(Q>ln-lz(~A1)) + 1 B ~ ( Q ) ~ - ' ~ ( z E ~ ) (11) I = 1 where the expression in brackets describes the experimental spectrum observed within the limited energy window of the very high resolution spectrometer IN10 so that a(@ and Z, are experimentally determined quantities. This approach may be used with Bo(Q) as a variable to determine the EISF, or to fit a given model [e.g., eqn (6)] to S(Q, co), and means that a self-consistent treatment is used to analyse all data at all resolutions.(iii) MULTIPLE SCATTERING The correction of experimental data for the effects of multiple scattering is very complicated and expensive of computer time. We have adopted the simpler proce- dure of incorporating the effects of multiple scattering into the model calculations. These calculations were based on the Monte Carlo programme of Johnson15 and parameters such as the EISF or Z~ values from the resultant scattering laws can be used to assess the effects of multiple scattering on the experimental results.40 PHASE TRANSITION I N (CH,),CCN CRYSTAL I1 With the 4H5 spectrometer (A& z 100 peV h.w.h.m.) no quasielastic broadening was observed, in agreement with previous low-resolution work on this phase.4 How- ever, with higher resolution a rotational motion is clearly seen and by also using the back-scattering spectrometer this was observed down to -190 K.Analysis of the data as described above yields the EISF values shown in fig. 5. This shows that t 0.0 0.5 1 .o 1.5 2 .o Q1A-l FIG. 5.-EISF results for the Crystal I1 phase. Experimental values from the IN5 data are shown as filled shapes and from IN10 as open shapes, with the following instrumental resolutions and sample temperatures. AE/peV T/K 0 24 227 4 10 227 218 208 rn 10 0 0.5 n 0.5 20 1 c! 0.5 I90 The error bars (some of which have been omitted for clarity) have been estimated by extracting the EISF for the whole body motion with the value of the time constant for methyl reorientations [rA in eqn (1 I)] changed by i 100% of its most probable value.Line A shows the EISF that would be obtained if only methyl group reorientation were being observed. Line B shows the EISF expected from whole body reorientation about the CCN axis between 3 equivalent sites [eqn (6)] and line C represents the same model with the estimated multiple scattering correction. throughout the low temperature phase the average proton trajectory remains to a good approximation the same. Furthermore the EISF is independent of resolution as well as temperature, which suggests immediately that S(Q, a) is of a rather simple form. Model calculations show clearly that the motion observed is a rotation of the molecule about its CCN axis between three equivalent sites.This model fits the EISF dataJ . C . FROST, A . J . LEADBETTER A N D R . M. RICHARDSON 41 very well although the higher resolution and higher Tresults may be slightly low at the higher Q values indicating some additional motion, probably a precursor of that observed in Crystal I to be discussed below. However, any such effect is difficult to quantify uniquely in Crystal 11 and indeed the full S(Q, w) is fitted very well at all Q values by the sipple 3-fold reorientation model (see fig. 1 and 2). If the residence time is allowed to vary with Q in the fitting computations then except for the lowest Q values, where the EISF tends to unity, z is generally constant to better than 5%. For comparison the EISF expected for rotation of the methyl groups alone is shown in fig.5 which clearly demonstrates that this is not the motion being seen. These results unequivocally demonstrate that the molecules of t-butyl cyanide are undergoing rapid reorientation in a 3-fold potential in the ordered low temperature phase. The correlation times z1 (eqn (7)) are plotted in fig. 6 where the correlation times for the methyl group reorientation are also shown for comparison. z1 varies linearly with T-' and the line in the figure gives the relation z,/s = 1 x exp 23.3 kJ mol-'/RT. The pre-exponential factor is totally unphysical since a value of the order of the free rotator correlation time (Z/kT)1'2 z A more reasonable interpretation is possible using the inelastic data shown in fig. 4. By virtue of its high intensity, which also increases with T, the lowest energy s is expected.I6 TABLE PARAMETERS FOR THE MOLECULAR ROTATION IN CRYSTAL 11.COMPARISON OF INELASTIC AND QUASIELASTIC RESULTS Cia BARRIER HEIGHTS AND RESIDENCE TIMES. 5 150 190 201 208 21 8 227 (239 4.2 f 0.2 3.7 & 0.2 (6.5) & 1.0 7.2 36 34 7.5 3.2 & 0.2 5.8 k 1.5 6.5 23 13 7.2 (5.6) & 1.5 5.8 11.4 10.5 5.2 3.0 f 0.3 5.1 f 1.5 5.1 6.9 6.9 4.5 2.8 & 0.3 4.5 3.1 1.5 4.2 3.6 4.5 3.9 2.8 & 0.4 4.5 i 1.5; Crystal I) 10.0 i 1.0 7.7 i 1.0 z(calc)/s = 4.5 x z(exp>/s = 4.5 x and z(exp) is the experimentally determined residence time. exp ( V3 - Eo)/RT exp ( Vge - Eo)/RT = zo(T) exp(V3 - E,)/RT peak at 3-4 meV may be assigned to the libration of the molecule about its CCN axis. The peak is increasingly swamped by multiphonon scattering with increasing T but remains detectable into the high-temperature phase. Assuming a potential energy of the form V(0) = V3(l - cos 30)/2 then in the harmonic approximation the depth of the potential well is given by V3 = 8 n21(E1 - Eo)/9ir2.The reorientation frequency is then given by the relation r1 = TO' exp - (V, -E,)/RT. Using this very simple model the following calculations were made with the results Residence times were calculated using values of (V, - Eo) derived given in table 1.42 PHASE TRANSITION I N (CH,),CCN from the inelastic results and zo = 4.5 x s. [Note that dipole correlation times (7,) not residence times (7) are shown in fig. 6 for ease of comparison with Crystal I results later.] The agreement between zexp and zCalc is as close as could be expected.The difference is shown in two ways by giving (a) r,(T) values and (b) VSe values required to give the solid line shown in the figure. It is obvious that a slight change of V,, well within the uncertainty of that derived from the librational fre- quencies, or a slight temperature dependence of zo gives perfect agreement with experi- ment for z. Hence inelastic and quasielastic results are self-consistent and the ap- parent Arrhenius behaviour for crystal I1 shown in fig. 6 is an artefact caused by a barrier height to reorientation which is a strong function of T. It is in fact approxi- mately linear in T for 190 < T/K< 230 and in precisely this region the unit cell dimen- sions become strongly temperature dependent.-8 t -9 --. 3.0 4 .O 5.0 1 0 3 ~ 1 T FIG. 6.-Correlation times T~ for t-butyl cyanide. Filled circles are for methyl-group reorientations. Open circles are for whole-molecule motions and show all the results obtained with three different spectrometers at 4 different experimental resolutions (see Experimental section). Vertical line shows the transition temperature. The hatched area shows the estimate of the correlation time for the cooperative translation/rotation relaxation discussed in the text. For Crystal I the line through the points is simply the best fit Arrhenius line. For Crystal 11, see text and table 1 . CRYSTAL I The EISF is determined very accurately from the INS data at both resolutions and also with reasonable accuracy from the lower resolution 4H5 data because of the width of the quasielastic components (200-300 peV, h.w.h.m.).The results are shown in fig. 7 and 8 and for the two higher resolution experiments with the IN5 spectro- meter the EISF is independent both of resolution and temperature showing that on a time-scale of wlO-" s the average proton trajectory is constant throughout theJ . C . FROST, A . J . LEADBETTER AND R . M . RICHARDSON 43 temperature range of Crystal I. However, with the lower-resolution instrument 4H5 the EISF is both considerably higher and also temperature dependent, but approaches the IN5 result with increasing temperature. This shows that the molecular motion is more complex than in Crystal 11 and this is confirmed by detailed model calcula- tions. The 4H5 data at the two lower temperatures (fig.7) are in fair agreement with a model of jump reorientation in a 3-fold potential but in much better agreement with 1 I I I 1 I 0.0 0.5 1 .o 1.5 2.0 QlA - FIG. 7.-The EISF for the Crystal I phase of t-butyl cyanide determined from the 4H5 data. 0,284; V, 265 and 0, 240 K. The lines represent the EISF calculated for: A, 3-fold jump reorientation about the CCN axis [eqn (6)]; B, reorientation about the CCN axis between 6 or more equivalent sites [eqn (3) and (5)J; C, uniaxial rotational diffusion in a 3-fold cosine potential” with barrier height, V = 2kT 21 4 kJ mol-’. a model for diffusive motion in such a potential (effectively a strong collision model) with V3 = 4 kJ mo1-I which is in good agreement with the value deduced from the damped librational peak seen at the lowest temperatures in Crystal I, and with the Arrhenius activation energy deduced from the temperature dependence of the correla- tion times (see below).Furthermore, the simple jump model fits S(Q, co) very well, with a residence time z independent of Q to within 10%. Taken all together this evidence confirms that the faster stochastic motions in Crystal I are the molecular rotations in a 3-fold potential. This motion becomes increasingly diffusive in character (as opposed to librational plus jump reorientation) with increasing T be- cause of the small barrier height: V3 = 2RT. At the highest temperature the 4H5 EISF is close to that for diffusion on a circle among rn sites with rn 3 6, but the high resolution results are still lower at all tempera- tures.Thus it seems clear that there must be at least two components of motion present. The additional motion cannot be methyl-group rotation which is much too44 PHASE TRANSITION IN (CH,),CCN slow, and 3-dimensional molecular rotation is excluded by EISF calculations (fig. 8). In searching for the origin of the additional motion two facts must be noted: first the existence of steric hindrance to rotation even in the high-temperature phase and second the fact that the molecular C3 axis lies parallel to the four-fold axis of the tetra- gonal unit cell of Crystal I. The latter can be taken into account by assuming rota- tion about the CCN axis among 12 sites on a circle but this cannot explain the results because motion additional to diffusion on a circle is required.These considerations lead to the suggestion that the local structure in Crystal I is close to that of Crystal I1 with a coherence length (L) of a few unit cells. Four-fold disorder of this structure in steps of n/2 leads to the average tetragonal symmetry of Crystal I. The molecules then rotate very rapidly in a 3-fold potential about their dipole axes as for Crystal I1 but in addition undergo a slower relaxation of four-fold symmetry among the different 0.0 0.5 1.0 1.5 2.0 Q1A-I FIG. 8.-EISF for the Crystal I phase of t-butyl cyanide. Experimental results obtained using the IN5 spectrometer at the following temperatures and resolutions AElpeV T/K 0 10 270 0 10 239 0 24 240 a 24 250 The error bars represent the probable effect of contamination by coherent " elastic " diffuse scattering at Q > 1.3 A-I.A multiple scattering correction has been applied to all the model EISF calcula- tions shown. The lines A, B and C represent the EISFs calculated for the models described in the text: A (dashed) = 3-jump + 4-fold + z = 1.0 A B C = 3-jump + 4-fold -t (it;>+ = 0.30 8, = 3-jump + 4-fold + 1.2 8, additional displacement parallel to a and b of fig. 9. D is the EISF for a model of reorientation about the CCN axis between 6 or more sites [eqn (3) and ( 5 ) ] and E is the EISF for isotropic tumbling of the moleculeJ . C. FROST, A . J . LEADBETTER AND R. M. RICHARDSON 45 local configurations. These may be generated by 7r/2 rotations of the monoclinic Crystal I1 structure about either c* or c: the former results in molecular rotation plus translational (centre of mass) displacement in theab plane and the latter gives additional translational displacement of the molecules along their dipole axes.The actual magnitudes of the average molecular centre of mass displacements will depend on the coherence length of the local structure, and there must also be additional more- or-less random displacements at the boundaries between local structures of different orientation. The dynamic translational/rotational displacements have therefore been modelled as described below using the known crystal and molecular structures to determine a manifold of N sites for each proton, the molecules also being allowed to undergo 3-fold rotation about their CCN axes in all cases. The EISF was then calculated using the formula for a powder sample : where r l j = Iri - rjl, Y being the position of a proton site of occupation probability P, and in general we assumed P = N-I.Ao(Q) was averaged over non-equivalent protons. The starting point was to rotate a unit cell of Crystal I1 about an axis through its centre of mass parallel to c* in steps of 7c/2 and then displace the resultant structure FIG. 9.-Model for translational/rotational disorder in Crystal I. The bold triangles represent the projections of the 3 methyl groups for one of the molecules in the unit cell onto the ab plane in solid I1 (or the ab plane of solid I). The light triangles are the methyl group positions generated by 72/2 rotations about c* and suitable translations to keep the centre of mass of the 2 molecules in the cell constant.This (arbitrary) procedure generates a pseudo-tetragonal local structure and gives both molecules small lateral displacements. The real displacement could be much larger depending on the coherence length of the Crystal I1 local structure in Crystal I (see text). by appropriate combinations of b/2 and/or 4 2 to give the resultant superposition of structures for each of the two molecules in the unit cell shown in fig. 9. This repre- sents the minimum possible set of translational displacements for this model and it does not describe the EISF. A displacement z parallel to the dipole axis (c in Crystal I) which would result from 4 2 rotations about c in Crystal I1 was therefore added as an adjustable parameter to give a total of 8 molecular positions and the EISF for z = 1.0 8, is shown in fig. 8(A).An as alternative, an isotropic Gaussian displacement46 PHASE TRANSITION IN (CH3)3CCN of the molecular centre of mass was added to the minimum set of four positions from the c* rotation (and the %fold molecule rotation) and the EISF for ( ~ f ) l / ~ = 0.3 A is shown in fig. 8(B). Finally, on the grounds that displacements perpendicular to the direction of the opposed dipoles are perhaps the most likely, additional displace- ments x parallel to the unit cell axes a and b were added to the structure shown in fig. 9 and the resultant EISF for x = 1.2 A is shown in fig. 8 (C). The fits in all cases, but especially for C, are reasonable when taking account of uncertainties (included in the error bars) at the highest Q arising from the possible presence of coherence contri- butions due to the strong coherent diffuse scattering in the disordered phase.An elastic coherence contribution of only a few percent would remove the discrepancies with the models at the highest Q. The parameters for models A, B and C imply a coherence length for the local structure in Crystal 1 of not more than w 5 unit cells. The geometrical nature of the molecular motions in the high temperature phase is thus established as rotation in a weak 3-fold well about the molecule C, axis, plus additional rotational/translational motions which are strongly cooperative, generate the correct high temperature symmetry and involve centre of mass displacements of & l A.The dynamics of these two components of motion have also been deter- mined as follows. The fastest and largest amplitude motion is undoubtedly the 3- fold rotation about the dipole axis which dominates the lower resolution data (see above). For the higher resolution data, using a scattering law of the form of eqn (3) and ( 5 ) then at low Q the term in z1 is the totally dominant quasielastic com- ponent. In fact using rn = 10, the values of z1 were only weakly Q-dependent and for Q < 1.5 A-' fell on a straight line enabling an averaged low-Q value to be determined. These values may then be regarded as the dipole correlation time for the motion in the 3-fold potential and the results are given in fig. 6, where it is seen that self-consistent data are obtained from all experiments with experimental resolutions differing by up to an order of magnitude.The temperature dependence is described by z1 = 0.5 x exp 3.4 kJ inol-l/RT. The zo value is totally reasonable and the activation energy implies V, w 3.5 kJ mol-l which is in agreement both with the EISF analysis and the value derived from the librational frequency. The order of magnitude of the correlation time of the slower co-operative relaxa- tion may be estimated from the fact of its observation with the high resolution but not (at least at lower T ) with the lower resolution experiment and this is also shown in fig. 6. COMPARISON WITH OTHER EXPERIMENTS The previous lower resolution INQES experiments4 on Crystal I were interpreted in terms of a simple 3-fold jump reorientation model and gave z values ~40-50% greater than those reported here and an Arrhenius activation energy of ~2 kJ mol-l.In view of the much lower resolution of these experiments the agreement is as good as could be expected. The n.m.r. measurements (at 16 MHz) showed a single step decrease in linewidth between 95 and 120 K and a single TI minimum (measurements made at 42 MHz) at 162 K from which it was concluded that methyl and whole-molecule uniaxial rotations are coupled and describable by a single correlation time z, and a single energy barrier (16.3 4 0.8 kJ mol-'). The INQES results show clearly that this is not correct but also provide an alternative explanation. Extrapolation to lower T of the neutron correlation times for methyl and whole-molecule reorientation (fig.J .C . FROST, A . J . LEADBETTER AND R . M . RICHARDSON 47 6) show that these become equal near 130 K at r1 NN 10-7-10-8 s so that the corre- lation times indeed become equal at temperatures where their value is just that of the n.m.r. frequency; but this is fortuitous and the motions are not strongly coupled. The n.m.r. activation energy is in excellent agreement with the neutron value for methyl reorientation. The n.m.r. correlation time is about a factor of 10 less than the observed r1 values for methyl rotation but a factor of 2-3 is expected since the spin- lattice relaxation experiment gives r2 [P2 (cosO)]. Hence the n.m.r. Tl experiments are certainly dominated by the methyl reorientations. On the other hand an analysis by El Saffar et aL3 of the linewidth transition gave values for a correlation time, for T < 120 K in agreement with the extrapolated INQES values for whole-molecule rotation and a corresponding activation energy (12 & 1 kJ mo1-l) close to the 5 K value from the INIS data suggesting both that this analysis is sensitive to the whole-molecule motion and that the barrier height is not significantly temperature dependent for T .L( 100 K. Hence the onset of the line width transition is determined by the slower (at this T ) molecular rotations but the spin-lattice relaxation is dominated by the methyl-group rotations.THE PHASE TRANSITION The t-butyl cyanide molecules undergo rotation about their 3-fold axes in a 3-fold potential at all temperatures. In the low temperature phase the residence time 2(>2 x s) is more than an order of magnitude longer than the flight time 248 x s) where this is defined by zf = (kT/Z)-1/2 and also about an order of magnitude or more longer than the librational period zL = v ~ l ( z , w 10-l2 s).Hence the model of librational motion interspersed by relatively rare instantaneous jumps between wells is reasonable. The molecules may be envisaged as jumping to a new orientation when allowed by phonon-type thermal fluctuations. In the high-temperature phase z is only w2 or 3 times longer than zf and zL so that the simple reorientational model is no longer adequate, but more importantly neighbouring molecules must frequently be rotating simultaneously. Hence, strongly co-operative displacements of the centres of mass are required to allow this.Con- sideration of the molecule and site symmetry has led to a picture of relaxation among a set of local configurations comprising different 7rn12 orientations of structure similar to that of the low-temperature phase. The correlation time for this motion is slower than that for the 3-fold molecular rotation. The energy barrier for the 3-fold potential in the low temperature phase is strongly temperature dependent but does not collapse at the transition: in fact it changes rather little. Instead the decrease of the barrier, which is linked to a high thermal expansion, increases the rotation frequency until the concomitant growth of the co-operative translational displacements triggers the transition to the disordered phase. Thus it is the presence of the z 1 A co-operative displacements of the molecu- lar centres of mass and much faster molecular rotation about the C-C-N axis which distinguishes the high temperature phase and not the existence of rotation of the molecules about their C-C-N axes. C. Clemett and M. Davies, Trans. Favaday Soc., 1962, 58, 1705. E. F. Westrum and A. Ribner, J. Phys. Chem., 1967, 71, 1216. Z. M. El Saffar, P. Schultz and E. F. Meyer, J. Chem. Phys., 1972, 56, 1477. S. Urban, J. Mayer, I. Natkaniec, J. Sciensinki and W. Nawrocik, Acta Phys. Polon., 1978, A53, 379. S. L. Segel and A. Mansingh, J. Chem. Phys., 1969,51, 4578. J. R. Durig, S. M. Craven and J. Bragin, J. Chem. Phys., 1970, 53, 38.48 PHASE TRANSITION IN (CH3)jCCN ' C. I. Ratcliffe and T. C. Waddington, J.C.S. Furuduy II, 1976, 72, 1821. J. C. Frost, Ph.D. Thesis (University of Exeter, 1979). A. J. Dianoux, R. E. Ghosh, H. Hervet and R. E. Lechner, I.L.L. Internal Report 75 D16T (1975); W. S. Howells, I.L.L. Internal Report 75 H1 30T (1975); H. A. Baston, A.E.R.E. M5270 Harwell (1972); lo A. J. Leadbetter and R. E. Lechner, in The Plastically Crystalline State, ed. J . N. Sherwood (Wiley, Chichester, 1979, chap. 8). l1 R. L. Livingston and C. N. R. Rao, J. Amer. Chem. SOC., 1959, 81, 3584. l2 L. J. Nugent, D. E. Mann and D. R. Lide Jr, J . Chenz. Phys., 1962, 36, 965. l3 J. D. Barnes, J . Chem. Phys., 1973, 58, 5193. l4 M. Schlaak, Mol. Phys., 1977, 33, 125. lS M. W. Johnson, A.E.R.E. Report 7682 (1974). l6 C. Brot, Chem. Phys. Letters, 1969, 3, 319. A. J. Dianoux and F. Volino, Mol. Phys., 1977, 34, 1263.