Rotational motion in plastic quinuclidine : Results from quasielastic neutron scattering

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Rotational motion in plastic quinuclidine : Results from quasielastic neutron scattering

C. Brot, B. Lassier-Govers, R.E. Lechner, F. Volino

To cite this version:

C. Brot, B. Lassier-Govers, R.E. Lechner, F. Volino. Rotational motion in plastic quinuclidine : Results from quasielastic neutron scattering. Journal de Physique, 1979, 40 (6), pp.563-571.

�10.1051/jphys:01979004006056300�. �jpa-00209140�

Rotational motion in plastic quinuclidine :

Results from quasielastic ^neutron scattering

C. Brot, ^B. Lassier-Govers, ^R. E. Lechner (*) and F. Volino (*)

Laboratoire de Physique ^{de la Matière} Condensée, Faculté des Sciences, 06034 Nice Cedex, ^France

(*) ^Institut Laue-Langevin, 156X, ^{Centre de} Tri, ^Grenoble Cedex, ^France

(~) Equipe ^de Physico-Chimie Mol., S.P.S., D.R.F., CEN-G, 85X, 38041 Grenoble Cedex, France

(Reçu ^{le 27 octobre} 1978, ^{révisé le 7} février 1979, accepté ^{le 20} février 1979)

Résumé.

Dans cette publication ^sont présentés des résultats de diffusion incohérente quasiélastique ^{de neutrons}

sur

la quinuclidine, C7H13N, ^dans

phase plastique (température ambiante). ^Les spectres quasiélastiques ^obser-

vés sont comparés ^à plusieurs modèles pour le ^mouvement rotationnel des molécules. On trouve que les résultats

expérimentaux ^sont

^accord

modèle où les molécules tournent par ^sauts de 90° autour des

cris-

tallographiques C4

temps de résidence de (24,8 ± 2,5).10-12 ^{s, et par sauts de 120° autour des}

moléculaires C3

temps de résidence de (9,6 ± 2,0).10-12

Abstract.

Results of incoherent quasielastic ^neutron scattering experiments

quinuclidine, C7Hl3N, ⁱⁿ

its plastic phase (room temperature)

presented. ^{The observed} quasielastic spectra

compared ^{to several}

models for the rotational motion of the molecules. It is found that the data support

picture, where the mole- cules perform 90°-reorientations about crystallographic C4-axes ^with

^{residence time of} (24.8 ^± 2.5) ^{.10-12 s,}

and 120°-reorientations about the molecular C3-axis ^with

residence time of (9.6 ± 2.0). ^10-12

Classification

Physics ^Abstracts

61.50

1. Introduction and theoretical background.

Molecular crystals ^of roughly globular ^molecules

such as admantane, cyclohexane, methane, ^{etc. form}

an interesting class of materials which have différent solid phases [1, 2, 3] : in the lower temperature phase,

the molecules either do ^not reorient or perform ^rare

orientational jumps corresponding ^{to their own} sym- metry operations. ^A change ^of phase ^{occurs when a}

reorientational motion of the molecules introduces new, distinguishable, orientations, ^thus increasing ^the

apparent (statistical) symmetry ^{of the} crystal. ^There

can be a certain number of phases corresponding ^to

the occurrence of different reorientational motions ; the highest temperature phase ^{1 is most} often cubic.

One must remember that this only ^{means that}

^on

an average ^over time or over space

the probability

distribution of molecular orientations has cubic symmetry.

Our aim was to study ^{the nature} of the molecular motion in such a (plastic) phase, namely ^{that of} qui- nuclidine, C7H13N. ^{For this kind of} study ^{a certain}

number of techniques ^are available ; ^{we chose NMR} and incoherent quasielastic scattering ^{of neutrons. In}

the present paper ^we wish to report ^{on the neutron} results.

The quinuclidine ^molecule (Fig. 1) ^has symmetry

C3v ^{but is} nearly perfectly globular : ^the proton which is on the molecular axis is only ^{a few} per ^cent

Fig. 1.

^Schematic drawing ^{of the} quinuclidine molecule, C7Hi3N.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004006056300

564

nearer to the centre of the molecule than the twelve off-axis protons. ^A transition occurs at Tt

¹⁹⁸ K, the melting point ^{is at} T.

^{413 K.}

If any precise interpretation ^{of the} dynamical ^data

is desired a knowledge ^{of the} average atomic positions

is necessary. The cubic structure of the high tempe-

rature phase has been determined on a powder by X-rays [4] ^and recently ^the orientations of the mole- cule in the cubic cell have been determined on a

single crystal [5]. ^{All the atoms of the molecule lie}

in a general position ^{of the} group 0, ^{since the mole-} cular and cubic C3 ^{axes are not} parallel ^to each-other.

This means that there is no reduction due to symme- try of the number of the positions ^{allowed to each}

atom. By reorientation around the crystallographic

axes, 24 distinguishable orientations for one molecule

are obtained in the cell. Due to the C3 symmetry ^of the molecule there exists a strong probability ^that

there is also a reorientational motion about the mole- cular C3 axis. This has been confirmed by ^{the NMR}

results [6].

In the following ^{we will use the} method described

by Rigny [7] and Thibaudier and Volino [8, 13] ^to

calculate the correlation functions and the interme- diate scattering ^functions pertinent ^{for neutrons.}

1.1 CUBIC MODELS. -1.1.1 General.

_{In pre-} vious publications [9-12] ^we studied the neutron inco- herent intermediate scattering functions, Is(Q, t), ^for

the different numbers of allowed orientations (poten-

tial wells) compatible with the cubic symmetry.

Is(Q, t) ^{is the} average of the correlation function F(t)

defined in ref. [8], ^{over all initial atom} positions r. -

where f(rj)

^{e+iQrj and} where it is assumed that all

(Nv) ^initial positions ^{have the same} probability (Nv)-1.

Rr j is the atomic position after the rotation R.

P(R, t) ^{is the} probability ^{that the} molécule ^{has been} brought from its orientation at t

0 to its orientation at time t by the rotation R.

For our present, ^more complicated, problem, ^where

reorientations around crystallographic (fixed) ^{as well}

as molecular (mobile) ^axes occur, ^we need the more

elaborate _group theoretical treatment of ref. [13], applied ^{to the} special ^{case of a finite} group.

The _group G to be used for the combined motion considered here (i.e. rotations with cubic symmetry plus rotations around the molecular C3 axis which is not aligned along ^the C3 axis of the lattice) ^{is the}

direct product ^{of the} crystallographic ^{cubic 0 and}

molecular C3 subgroups : ^{0 Q} C3. It is of order _g.

In the following ^we will label by prime ^{and double} prime ^the quantities which refer respectively ^{to the} crystallographic and molecular subgroups. ^{Thus the}

latter will be called G’ and G" ; g’ and g" ^{are the orders}

of these subgroups ^{and we} have g

g’ . g".

To evaluate the probability P(R, t) ^{we will use}

eq. (10) ^{of ref.} [13] ^{but to be coherent with our nota-}

tions we will replace Qrx, Q’ ^{of that} equation by Q’

and Q ". Since these matrices are scalar matrices

(Qrx = q« I ) they ^commute [8] ^{and we obtain the} following expression ^{for the} probability ^P

where

a refers to the irreducible representation Tx ^{of the}

-

product group,

da is the dimension of r (x,

xa ^are the characters of ra.

The other quantities ^{are defined as} follows : if

p’(R) ^{is the} probability per unit time of the occurrence

of a rotation R

and similarly ^for p" ^and qÏ.

The second equalities ⁱⁿ eq. (3) ^and (4) ^{come from}

the fact that p’(R) ^and p"(R) ^are different from zero

only ^{in each} subgroup G’ and G" respectively.

Let us call rl,. and T Jl" the ^average times between two consecutive operations (i.e. specific ^reorienta- tions) of each of the individual classes 1À’ and li" ^of

the subgroups G’ and G".

If R is an operation ^of the class _/1

/1’

/1" ^of

the product G, ^{and if} Xf7.(/1’ /1") is the character of this class, ^{we can write}

-1

with

and

where E’ and E" are the identity ^classes of the sub- groups G’ and G" and nll’ ^and nll" ^are the numbers of elements of the classes y’ ^and li", respectively.

Combining eqs. (1) ^and (5) ^{the final} expression ^for

the intermediate scattering function is :

where the structure factors a ^are given by

Note that the first sum in eq. (10) goes ^over all the classes of the product group G ; ^the .p CI ^are purely geometrical quantities calculated from the atomic

positions.

The reciprocal characteristic times for each irre- ducible representations ^are obtained from _eqs. (6), (7), (8) ^{as follows :}

1.1.2 Case of ^the group 0 (D C3.

^The group 0 has five real irreducible representations, ^the C3 group has three irreducible representations, ^{two of them} being complex conjugated. Consequently ^the product

Table 1.

Characteristic decay ^rates for ^the

irreducible representations of ^the product group of crystallographic (cubic) and molecular (C3) ^rotations,

group U Q9 C3 nas 5 x 3 =15 irreaucioie represen- tations a, ten of them being complex.

More precisely they ^{are two} by ^two complex conju- gated. Physically ^it is necessary that the characteristic times of two complex conjugated representations ^are

the _same, consequently ^{we are left with} only ^{ten diffe-}

rent modes which we will call : A1, A1, A2, A2, E, E’, F1, F’1,

F2, F2. ^{In this set the} prime ^{refers to the} degenerated ^modes.

Let us now specify ^the times r,. ^{as T4, T2, T3 and} T2’ where the labels refer to the classes C4, C2, C3

and C2, ^{of the} crystallographic group. The correlation time r ,, for ^a reorientation around the molecular axis will be denoted as r.. Using ^this notation the characteristic times of the ten modes are listed in table I.

To compute ^the quantities g/, ^we remark that :

1) for the five modes without a prime ^{index the}

calculation can be done directly ^from eq. (10) ^since

the characters are non imaginary ;

2) ^{for the five} degenerated ^{modes we use} also eq. (10)

but for the _purpose of calculation the representation ^a

is replaced by ^{the sum of the two} complex imaginary representations. This is summarized in table II which

Table II.

Character table for ^the product group, 0 Q C3. The characters for ^the representations ^with

a prime superscript ^{are sums} of pairs of complex conjugated characters.

represents ^a simple ^{real table} of characters used in

practice for the numerical computation.

The preceding ^{formulae are} valid for a single

566

crystal ^with f (rj)

exp(iQrj). ^{For a powder, one}

has to average ^over ail the orientations of Q :

where jo(x) ^{is the} spherical ^{Bessel function of order}

zero

1.1.3 Application ^to quinuclidine.

^{In order to}

evaluate expression (10) for each of the twelve off- axis protons of the molecules, ^{we have to consider} the 24 x 3

72 operations ^{of the} product group, which bring ^the proton ^{into the 72} positions (Rr).

This has been done with a computer using ^{the atomic} positions known from X-ray ^{data. For the axial} pro- ton only ^{24 different} positions ^are generated. ^The

effective structure factors to be used are obtained by averaging ^{over the 13} protons of the molecule. For

simplicity ^{we will use in the} following ^{the same nota-}

tion t/J a for these averaged ^{structure factors.}

The theoretical results for the structure factors are

shown on figure ^{2a as a} function of Qd, d being ^the

common gyration radius of the protons (d

^2.35 Â).

It is found that for the _range of Q ^values of interest here t/J AÍ’ t/J A2" t/J Al ^{and IPE are} negligible. ^{As is well}

known and obvious from table I, t/J Al

is the weight

of the elastic peak (« Elastic Incoherent Structure Factor » EISF) ^{in the most} general ^{case when all the}

L;l (except ^for TÀ,’) ^are different from zero.

For comparison ^{we also} performed ^these compu- tations for the simpler ^{24 wells} model, ^i.e. assuming

that the molecular axis reorientations do not exist

(Fig. 2b).

The theoretical incoherent rotational scattering

function is the time Fourier transform of IS(Q, t) : r

with :

This function has to be multiplied by

to take into account the effects of translational and torsional vibrations, where U2 > ^{is the mean} square

displacement (along ^{the vector} Q) corresponding ^to

these small amplitude motions. This has been taken from the X-ray ^data.

Also, ^{since our} reorientational model is classical,

we have to multiply the above theoretical scattering

function by exp( - hco/2 kT), ^a quantity ^{which is}

Fig. ^2.

Structure factors of the incoherent rotational scattering

function for quinuclidine : a) corresponding ^{to the} general

eq. (10) ^i.e. including ^both crystallographic (cubic) and molecular

(C3) ^rotations (model c). ^The

theoretical, ^the empty circles

the experimental ^EISF t/J Al ; b) crystallographic ^rotations only, theoretical.

close to unity ^{in the} interesting ^Q) region. ^The appli-

cation of these two corrections will be described in

more detail in the next section.

2. Experimental.

^The experiments ^were per- formed at the high ^{flux reactor} of the Institute Laue-

Langevin ^{at Grenoble} using the time of flight ^multi- chopper spectrometer ^IN5.

Since quinuclidine ^is hygroscopic ^{and sublimes} readily ^{even at room} temperature, ^our samples ^have

been dried under vacuum and loaded and sealed in

an air tight container in ^a dry ^box. During ^{the mea-}

surements taken ^{at room} temperature ^the powdered sample ^{was held in a slab} shaped ^aluminium container, with a diameter of 50 _mm, and a thickness of 0.3 mm ; the sample ^{had a} transmission of 83 %.

We first did two different experiments ^{at = 6.25 A}

(resolution ^{FWHM = 0.11} meV) ^{and = 8.22 A}

(resolution ^0.05 meV). The result for the EISF was

not consistent with a cubic model. This was due to

the resolution which was not sufficient to resolve all the quasielastic components ^{from the} purely ^elastic scattering. ^{In later} experiments ^{we used Â}

^{10.95 À} (resolution ^0.014 meV) ^and

^{8.84 A} (resolution

0.025 meV). ^The corresponding Q ^{values were bet-}

ween 0.1 and 1.05 A -1 and 0.13 and 1.3 A -1, ^res- pectively. ^{In each of these runs} spectra scattered into twelve angles ^{were recorded} simultaneously. ^The scattering angles ^{were chosen so as to} exclude the

Bragg peaks ^{which are situated at} Q

^{1.215 A-1} (111)

and Q

^{1.402 A-1} (200).

The maximum of the Q values available is only ^of

the order of 1.2 Â - 1 due to the long ^{incident wave-} length ^{that we were} obliged ^to adopt ^{in order to} get the best possible energy resolution. This is unfortu- nate because the EISFs of the différent models for a

cubic system ^{do not} differ much for Qd ^{less than n} (i.e. for Q ^{1.34 Â for} quinuclidine).

In a first analysis ^{the raw data were treated : |: the}

time of flight spectra ^{are obtained at constant} angle ;

these spectra ^are corrected for detector efficiency, sample ^holder scattering, absorption ^and self-shielding

but not for multiple scattering, this last correction

being only possible ^{when a} theoretical S(Q, m) ^func-

tion is available. One must remember that the spectra

are taken at constant angle ^so that there is a Q ^varia-

tion in each spectrum; ^the largest variation in one

spectrum ^{for À}

^8.84 Â and 0

1330 is from

Q

^{1.27 A -1 for m = - 0.01 meV to} Q

^{1.50 for}

eu

0.7 meV. This Q variation is taken into account in the comparison of theoretical to experimental spectra, ^{which was made} by fitting différent models

including those described in the preceding ^{section to}

the data.

The following fitting ^{function was used :}

where :

a is a trivial scaling parameter, determined experi- mentally ;

b takes care of a constant background (which ⁱⁿ practice ^{turns out to be} negligible) ;

c accounts for the weak scattering by ^low frequency phonons;

s(Q, m) is the convolution of a theoretical classical

scattering function, ^{such as} S(Q, m) ^of eq. (9) ^with

the instrument resolution function.

It is important ^{to note that the} parameter ^{c is} only weakly coupled ^{to the width} parameters characterizing S(Q-, w), ^{because the term} cQ2 ^varies very slowly ⁱⁿ

the Q-range ^{of each} spectrum.

The resolution function, ^measured through ^the scattering ^{of a vanadium} sample, ^is nearly triangular.

All reasonable simple theoretical S(Q, m) ^{for mole-}

cular rotations being composed ^{of a} b function plus

a certain number of Lorentzians, the convolution of

these components by ^a triangle ^{could be} performed analytically [14].

Writing ^{for the} general rotational scattering ^func-

tion :

» .

where the factors Ci and the number of Lorentzians N depend ^{on the model} (e.g. ^N

^oo for rotational diffusion [15]), ^{it is seen that} Co ^{is the EISF.}

For rotational diffusion ^as well as for a cubic model

Co(Q) goes nearly ^{to zero at or in the} neighbourhood

of Qd

^1t, whereas for jump-wise reorientation in a

plane ^{it will not. In our} first series of experiments ^the

resolution was relatively ^{modest so that} part ^{of the}

quasielastic intensity ^was inside the resolution func- tion : apparent ^{values of} Co ^{were too} high ^{for a}

cubic model. In the second series of (higher resolution) experiments ^the separation ^{of the} purely ^elastic peak

was achieved to a much higher degree. Examples ^of

spectra ^{taken at} higher resolution are given ^on figure ^3.

3. Fit of various dynamical ^models.

^{Let us now}

consider in some detail different dynamical ^models.

The first three are variants of the jump reorientation model described in section 1, the fourth is the isotropic

rotational diffusion model. Consider the jump ^reorien-

tation model, which includes crystallographic ope- rations of cubic symmetry combined with reorienta- tions about molecular axes. As it seems a priori ^diffi- cult to get ^the five correlation times (see ^table I)

necessary ^to fully describe the model we tried different

limiting ^cases.

a) ^{FIRST MODEL : 24 WELLS « ANY WELL JUMP}

HYPOTHESIS ».

In this model one supposes :

i) That the molecule reorients only ^{around the} crystallographic ^{axes and not} around the molecular

axes.

ii) ^{That the} probabilities of reorientation corres-

ponding ^to single operations of all the different classes

are the same :

which we denote -r-1

From table I it is seen that the spectrum ^{will be}

composed ^{of an elastic} peak plus ^{one Lorentzian}

with half width :

and weight :

b) ^{SECOND MODEL : 72 WELLS « ANY WELL JUMP}

HYPOTHESIS ».

This is similar to the first model

except that the molecule can also reorient around its

568

own axis. The quasielastic part ^{of the} spectrum ^is

composed ^{of two} Lorentzians with half widths :

and weights :

Fig. ^3.

Examples ^of quasielastic spectra ^of quinuclidine ^taken

at

temperature ^with

^{incident neutron} wavelength ^of

10.95 Á. The scattering angles 0

10.250, 82.450 and 119.40 for

figures 3a, 3b, ^{and 3c} respectively. ^Points

measured, ^{the full} line is

least _squares fit of model

(crystallographic C4 ^{and mole-}

cular C3 reorientations) ^{to the} experimental ^{data. The}

^vector

transfers at the elastic peak

^0.101 Á -1, ^{0.75 A -1 and 0.99 A -1}

respectively, with little variation along ^the energy axis.

The residence time for the molecular C3 motion,

is obviously Lm/2 while that for the cubic motion is

TAw/23, ^{because among the 24} operations only ^{23 are}

different from identity ^{in the} subgroup ^0.

c) ^{THIRD MODEL : :} C4 JUMPS PLUS MOLECULAR

AXIS REORIENTATIONS.

We consider that among

the crystallographic reorientations only ^{one class of}

operations ^{is allowed,} namely C4. ^For this model the

characteristic times are listed in table III. The resi-

Table III.

Characteristic decay ^rates (L(X)-l for ^the simplified ^model including crystallographic C4 ^and

molecular C3 reorientations.

dence times are now respectively Tm 2 again ^{for the}

molecular motion and L4/6 for the cubic motion.

d) ^FOURTH MODEL : ROTATIONAL DIFFUSION.

We consider also a model which is not a jump ^{model :}

the isotropic rotational diffusion ^{model. Here the}

molecules perform ^{more or} less continuous small

angle random rotations with rapidly changing angular

momentum.

On a time _average this allows _any orientation to be reached. For quinuclidine ^{this is in} contradiction with the recent X-ray results, but nevertheless it seems

worthwhile to introduce it for the sake of completeness.

Then following [15] ^{one has}

where j, (Qd) ^{is the} spherical Bessel function of order 1 and Tl

[DR 1(l ⁺ 1)] -1 ^{is the} decay time for the

spherical ^harmonic of order 1. Here DR ^{is the rotational}

diffusion constant. It is known that for small values of Qd it is sufficient to let 1 run between 0 and 4 or 5.

We now give briefly the results of the fit with the different models above. The residual factor _expresses

the quality of the fit and is defined as :

where we have chosen the weight ^W

1 / Yexp ^{so that}

the wings, which contain information about the wider

Lorentzians, ^{be not} underweighted. ^The quantities Rw given below have been obtained by ^a simultaneous fit of all spectra ^{taken at different} angles.

a) ^{24 WELLS MODEL, ANY} WELL JUMP.

We find ri

6.2 ps ; R

1.82, ^{which is a} very bad residual factor.

b) COMBINED 72 ^WELLS MODEL, ^{ANY WELL JUMP.}

As mentioned above, ^the spectra ^contain only ^two

Lorentzians of half widths 24 r - 1 ^and (24 Aw ^{+ 3} -r - 1) respectively.

For the incident wavelength Â

^{8.84 A we find :}

with 1 For

with R

0.121.

It is seen that we do not get ^a good agreement between the two wavelengths. ^{On the} average ^we obtain

The corresponding residence times are :

c) COMBINED 72 WELLS MODEL, C4 ^JUMPS ONLY, ^FOR

THE CRYSTALLOGRAPHIC OPERATIONS.

The spectra then contain 5 Lorentzians of non negligible weight (Fig. 2a) ^{but these cannot be} distinguished visually.

The fit yields :

For A

8.84 A

with R

0.106.

The agreement between the two wavelengths ^is

much better here.

Examples ^{of fits are shown for} Ào

^{10.95 A in} figure 3, ^{and for} Âo

^{8.84 A in} figure ^4.

Averaging ^{the two sets of data we} get :

570

Fig. ^4.

^Same

figure ^{3 at incident neutron} wavelength ^{8.84 Â.}

In figures ^{4a and} 4b, ^the scattering angles

96° and 119.40°

with

vector transfers at elastic peak 1.06 and 1.23 Â -1, respectively.

which yields :

Moreover then LF1 = T4/4

33.7 ps.

d) ROTATIONAL DIFFUSION.

We obtain

DR- 1

^22.6 ps, which yields ^a correlation time Ti for the first spherical ^harmonic equal ^{to 11.3} ps and ’r2

for the second spherical ^harmonic equal ^{to 3.8 ps.}

The residual factor is 0.15 but the fit is visually ^much

worse than for model 3 (see Fig. 5).

Fig. ^5.

Fit of the isotropic rotational diffusion model to the

experimental spectrum of quinuclidine ^for Â.o

^{8.84 A and scatter-} ing angle ^119.4°.

It should be mentioned that our results were not

corrected for multiple scattering. ^From previous experience [12] ^we know that under the conditions of the present experiment the correlation times obtained from the fit to the data (not corrected for multiple scattering) ^are systematically ^{too small} by ^{about 10} % [16]; since however the error has the same sign ^for

the different wavelengths this does not change ^our

conclusion. Thus our best estimate of the true corre-

lation times of model c (obtained after this 10 % correction) ^{is :}

4. Conclusion and discussion.

The isotropic

rotational diffusion model (model d) is excluded both because of the X-ray data and of the bad agreement between the present ^neutron study ^{and the NMR data} (see following paper, ref. [6]).

Considering ^now the discrete site models, ^the pure

crystallographic reorientation (model a) is excluded because of its much too high residual factor. The two versions of the model with combined motion yield,

at each incident wavelength, ^{low but} comparable

residual factors, ^{but the «} any well jump » (model b)

version does not produce parameters ⁱⁿ agreement ^for the two wavelength ; therefore, ^we consider that the situation described by ^the C4 ^model including ^reorien-

tation about the molecular axis (model c) ^{is much}

more likely ^{to be valid.} Moreover, ^this will be confirm- ed by ^a later paper [6] ^as this is the only model which

yields LF1

² LF2. It will be ^seen indeed that the NMR

of 14N which is essentially ^{sensitive to} LF2 indicates

that iF2

16.6 _ps, i.e. half the value found here for

iF, in model c.

It is worth noting ^that C4 operations being gene- rators of the crystallographic groups C4 jumps ^suffice

to ensure that all 24 orientations are reached and are

equally populated ^{on a time} average. Also _{the array} of the molecular centres of mass on a c.f.c. lattice maintain the existence of the other crystallographic

axes although ^{these are not active with} respect ^to the orientational jumps.

Finally, C4 jumps imply ^smaller jumping ^distances

than C2 ^or C2, jumps. ^With respect ^to crystallogra-

phic C3 jumps, ^{one must remember that at a} given

instant of time the molecular axis does not lie far from a 111 direction ; among the 8 C3 virtually possi-

ble C3 jumps, six would also carry the molecule

across large angular distances, while the remaining

two (about ^the neighbouring ¹¹¹ axis) ^{do not differ}

much from a self reorientation. All the above larger angle jumps presumably ^involve higher barriers. This would explain why C4 jumps ^are privileged ⁱⁿ quinu-

clidine.

References [1] Proceedings ^{of the} symposium

^Plastic Crystals ^{and Rota-}

tion in the Solid State » ed. J. Timmermans, ^J. Phys.

Chem. Solids 18 n° 1 (1961).

[2] Mouvements et Changements de Phase dans les Solides Molé- culaires (XVe Réunion annuelle de la Société de Chimie

Physique), ^{J. Chim.} Phys. ^{63 n° 1} (1966).

[3] The Plastic Crystalline State, ed. J. N. Sherwood (John Wiley

and Sons) ^to appear in 1978.

[4] BRÜESCH, P., Spect. ^{Chem. Acta 22} (1966) ^861.

[5] FOURME, R., ^{To be} published.

[6] BROT, C., VIRLET, J., ^{To be} published.

[7] RIGNY, P., Physica ⁵⁹ (1972) ^707.

[8] THIBAUDIER, C., VOLINO, F., ^Mol. Phys. ²⁶ (1973) ^1281.

[9] BROT, C., LASSIER-GOVERS, B., ^Ber. Bunsenges. Phys. ^{Chem. 80} (1976) 3.

[10] LECHNER, R. E., HEIDEMANN, A., ^Commun. Phys. 1 (1976) ^213.

[11] BROT, C., LASSIER-GOVERS, B., ^Ber. Bunsenges. Phys. ^Chem.

81 (1977) ^444.

[12] LECHNER, R. C., in Proc. Conf. Neutron Scattering, ^Gat- linburg, June 1976. Vol. I, p. 310 Natl. Techn. Information Service, ^U.S. dept. of Commerce Springfield, Virginia

22161 (1976).

[13] THIBAUDIER, C., VOLINO, F., ^Mol. Phys. ³⁰ (1975) ^1159.

[14] VOLINO, F., DIANOUX, ^A. J., LECHNER, ^R. E., HERVET, H., J. Physique Colloq. ³⁶ (1975) ^C1-84.

[15] SEARS, ^V. F., ^{Can. J.} Phys. ⁴⁵ (1967) ^237.

[16] Although the relative statistical

of the shorter correlation time is larger than the relative

of the larger ^{time due}

to weak coupling with the inelastic background para- meter c1 the relative

due to multiple scattering ^is

about the

This is different from another

studied recently, ^{where due to} the small concentration of fast rotating molecules the effect of multiple scattering

the shortest correlation time

much

important [17, 18].

[17] LECHNER, ^R. E., AMOUREUX, ^J. P., BÉE, ^{M. and} FOURET, R., Commun. Phys. ² (1977) ^207.

[18] AMOUREUX, ^J. P., BÉE, M., FOURET, ^{R. and} LECHNER, R. E.,

Proc. Conf. Neutron Inelastic Scattering 1977, ^Vol. I,

p. 397, IAEA Vienna (1978).