Low frequency excitations of aromatic molecules in the solid state

Chemical Physics 166 (1992) 425-439

North-Holland

Low frequency excitations of aromatic molecules in the solid state

Jeffrey Williams and Bernhard Frick Instrtut Max van Laue-Paul Langevln. 156 X, 38042 Grenoble, France

Recetved 23 April 1992

We have used inelastic and near-elastic neutron scattering to investigate the motion of aromatic molecules in the solid state. In

particular we have studied benzene, toluene, ethylbenzene, 1,3,5-trtfluorobenzene and the van der Waals adduct formed between

benzene and hexafluorobenzene, C6H6: C6F6. From the angle dependence of the inelastic scattering we have determined the mean

squared amplitude of oscillation of the protons responsible for the scattering at a number of sample temperatures. From experi-

ments involving near-elastic or quasi-elastic neutron scattering we have measured the change of linewtdth induced in the neutron

beam during the scattering process relative to a fixed energy window, thts can also be used, in some cases, as a measure of the

mean square amphtude of scatterer oscillation.

1. Introduction

The theory of the vibrational spectroscopy of molecular crystals has been given general treatment by several authors [ 1,2]. In all cases one of the out- standing problems is the separation of true vibrations from large amplitude angular oscillations or librations. Such considerations are of great importance in the understanding of the vibrational spectra of or- ganic crystals as most solid state spectra are measured at temperatures well in excess of absolute zero and as a consequence molecules as a whole are certainly undergoing oscillations in addition to zero- point vibrations. Such vibrational motions are not the only type of dynamics it is possible to observe in molecular solids. It was suggested long ago by Pauling [ 31 that for small molecules with small moments of inertia the rotational motion at elevated temperatures may well be free.

We present here an investigation of proton dynamics within aromatic solids. In the case of non-substituted aromatic rings we observe the protonaceous molecules begin to reorient at a particular temperature, that is, there is a substantial activation barrier to be overcome. Then this rotation-like reorientation

Correspondence to: J. Williams, Institut Max von Laue-Paul

Langevin, 156 X, 38042 Grenoble, France.

continues at an ever increasing frequency as the temperature mounts right up to the melting temperature of the solid. In some materials, phase transitions in- tervene to complicate the interpretation. We observe that in substituted benzene there is no rotational reorientation of the aromatic molecule as a whole and that the solid melts at the temperatures where one would expect it to commence. We have measured the zero point oscillations of the benzene molecule in C6H,: C6F6 and show that it is of similar magnitude in the other molecules investigated. Finally some ideas concerning the melting process in these types of solid are presented.

2. Neutron scattering

As is well known, in inelastic neutron scattering the characteristics of the scattered neutron do not de- pend solely upon the magnitude of the nuclear scattering cross section. We may write for the observable in a neutron scattering experiment, the double differential scattering cross section [ 41, which describes the probability that a neutron will be a scattered into a particular solid angle and into a particular energy interval,

0301-0104/92/$05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.

426 J. Willlams and B. Frrck / Chemcal Physrcs 166 (I 992) 425-439

d2a - = L [al”&“c(&, a) +GP.L,(&, w)l . dadw k.

(1)

Here we represent the scattering of a single particle or of a system where the scattering is dominated by a particular type of particle. In eq. ( 1) k. and k are the incoming and scattered neutron wavevectors, respectively. The nuclear scattering cross sections are crcO,,

and a,,,, the coherent and incoherent cross sections, S,,,( $, w) is the incoherent scattering law which defines single particle behaviour and Scoh (&, w) is the coherent scattering law which describes collective scattering, for example Bragg scattering. For the molecules we will be considering here, hydrogenated ar- omatics, the incoherent scattering cross section of the proton will dominate eq. ( 1).

The incoherent scattering law for Nscatterers is re- lated to the experimental observable, that is, the double differential cross section by

(2)

It may be shown [ 41 that the scattering into w space, for w f 0, is governed by the vibrational density of states, g(w), which defines the number of oscillations per unit range of angular frequency close to the frequency IX= E/A. We may write [ 41 the incoherent scattering law for energy gain of the neutron as

Xexp( -2W,)n(w) gON, w

where lk^-$ 1 =Q is the momentum transfer, M is the mass of the scattering nucleus and N is the number of scatterers. The exponential term is the Debye- Waller factor, which weakens the observed intensity due to the uncertainty in the scatterers position caused by thermal motions. The term, n(o), is the Boltz- mann factor for the target molecule, it represents the probability of the scattering molecule being capable of giving energy to the neutron, that is, up scattering an anti-Stokes process.

For a multi-component scatterer, the observed scattering will be weighted by the individual nuclear scattering cross sections, o,, and the scattering from

each vibration is weighted by the square of the atomic displacement, ( u f ) , of the oscillating atom of mass

M [41. A number of simplifications to eq. ( 3 ) can be read-

ily made. For example, the spectrometer used for our inelastic measurements is supplied with highly energetic neutrons which directly excite molecular transitions in a Stokes-like process. For inelastic measurements the sample is maintained at low temperatures, x2 K, and we therefore observe only

transitions which originate from the ground state. The observed scattering intensity at a particular w for a scatterer of mass M then becomes

S(Q,W)=(U~)Q2exp(-2,), (4)

where we see clearly how the scattering intensity will be a function of the momentum transfer and amplitude of oscillation. The Debye-Waller factor is the expectation value of the scattering amplitude of motion of the scattering particle, the proton, and the scattering vector.

Consider a particle whose position is nominally fixed but also contains a term, u(t), fluctuating in time. If each particle or molecule undergoes independent fluctuations the thermal average of the intensity scattered from a lattice will contain terms,

exp( ( - itime) ), where & is the change of wavevec- tor. Expanding the exponential and using the average, (tie&) = 0, for & and (ri) uncorrelated, we find that the powder average of the Debye-Waller factor is given by

exp(-2W,)=exp(-(uf)Q2/3). (5)

The motion of the scatterer will arise from a number of sources. At low temperatures there will be the zero-point vibrational oscillations. As the temperature rises there may be sufficient thermal energy to excite low frequency molecular vibrations, together with intermolecular phonon modes and large amplitude librational angular oscillations or rotations. All will contribute to the measured ( u2) , the mean squared amplitude of the proton oscillations. How- ever, when we consider in detail the origin of the measured ( u2) it can be seen that all vibrations excited in the sample will not contribute equally. The theory of the mean square atomic vibrations in solids was developed by James [ 5 ] after whom we may write

J. Wdlrams and B. Fruzk / Chemrcal Physrcs 166 (1992) 425-439 421

(u’> =3kT/47c2mv’, (6)

where we relate the mean square amplitude, (u*), of an oscillator of mass m and vibrational frequency v, assuming that the available thermal energy goes into molecular vibrations only. As the inelastic measurement Gill be a sum of the contributions from all the excited vibrations in the solid, the contribution of each vibration will be weighted by l/v*, that is, strongest for the lowest frequency oscillations.

From eq. (5) we see that there is a Q2 dependence upon the Debye-Waller factor dampening of the scattered intensity. Such a picture of the effect of the Debye-Waller factor on the measured scattering intensity implies that as the sample temperature increases so the amplitude of oscillation, ( u*) , also increases and consequently the scattered elastic or downscattered intensity will fall in a continuous smooth manner. This is provided the scatterers remain localised undergoing vibrations around their rest positions. As soon as there is local motion or large scale, presumably anharmonic displacements, for example near phase transitions, then this will cause the measured scattered intensity to fall more rapidly than is implied by eqs. (4) and ( 5 ) .

3. Measurement and interpretation of S(&, w z 0)

The near-elastic neutron scattering experiments described here were performed at the Institut Laue- Langevin, Grenoble, on the high energy resolution, high-Q backscattering spectrometer IN 13 [ 6 1. This spectrometer has a CaFz monochromator which se- lects the energy of the incident neutrons, 2.23 A. The sample is placed in a thin walled aluminium container. The thickness of the sample depending upon the amount of hydrogen present, the ideal experimental situation being a thickness corresponding to zz 10% scattering of the incident neutrons. The data are corrected for the container scattering and detec- tor characteristics.

The resolution of this instrument, x 8 ueV z 2000 MHz, is determined by its design being as close as possible to 180” scattering angle for monochromator and analyser and by the properties of the optical ele- ments used in its construction. When the analysing part of a spectrometer such as IN 13 is set to the same

energy as the incident neutron beam we define an energy window, the instrument resolution function. This window may serve as a means of determining linewidth changes during scattering as an increase in linewidth will correspond to a decrease in analysed scattering.

In such a “fixed window experiment” (FWS) we determine the spectral intensity at fiw z 0. At low temperature the purely elastically scattered line and any quasi-elastic spectrum, with linewidth < 8 PeV, are contained within our energy window. The phonon spectrum will be at a much higher frequency, of order a few meV, and because of the high energy resolution of the spectrometer will not contribute to our measurements. Then as the temperature increases the measured elastic line intensity falls, for example see fig. la, below 150 K, due to the Debye-Waller factor. Here the molecules are fixed in the solid and are

undergoing small amplitude oscillations and from eq. (5) we may estimate the magnitude of these motions by plotting In S( Q, w = 0) versus Q2.

As the temperature increases the quasi-elastic spectrum will continue to broaden and will fall outside the energy window of the spectrometer. In the case of benzene, fig. la, this is clearly seen to commence at Tz 150 K. As thequasi-elastic spectrum broadens with increasing temperature the observed scattered intensity continues to fall, Eventually, the quasi-elastic spectrum is entirely outside the energy window and we are only left with the remainder of the elastic line. Thus giving the observed step in the temperature dependence of the In S( Q, w z 0) curve. In this temperature region the characteristic fre-

quency, v. of any motion, the origin of the quasi- elastic scattering, is of order fi/AE, where AE is the width of the energy window.

To understand these observations we make use of an analogy with NMR spectroscopy. In solid state NMR the narrowing of an observed resonance line with temperature is analysed in terms of the Debye correlation time, rc. If we write T=Z 1 /vcr which is the average rate of oscillation of the molecule containing the resonating or scattering atom we may write after the theory developed by Bloembergen et al. [ 7 ]

(Av)‘=(Av,)‘ftan-‘(aAv/v,) , (7)

where the linewidth (a frequency) is Au, Au0 is the

428 J Williams and B. Fnck / Chemcal Phvsm 166 (1992) 425-439

(a) 0 (b) 0

s i

0 -0.5 g -0.1

ul cn

5 6

-0.2

-10 50 150 250 50 150 250

TEMPERATURE/K TEMPERATURE/K

(4 o

-5 ti -0.5

G UJ

UC

-1.0

50 150 250 TEMPERATURE / K TEMPERATURE/K

Fig. 1. (a) A plot of In S( Q, w z 0) versus temperature for benzene. the onset of stxfold reortentatlon is clearly seen at TZ 150 K. the solid

melts at 278 K. The scattering was recorded at Q= 1.19 A- I. The linear dashed line is an extrapolation of the Debye-Wailer attenuatton

and 1s intended as a guide for the eye only. (b) An expanded part of the plot of In S( Q, w-0) versus Tfor (x) benzene together with

data, measured at the same Q ( 1.19 A- ’ ), for (0 ) 1,3,5-tnfluorobenzene. (c) A plot of In S( Q. o zz 0) versus temperature for polycrys-

talline samples of ( A ) benzene and ( + ) toluene, both samples measured at Q= 1.19 A-‘. The toluene melts at the temperature where

the benzene begins to rotationally reorient. (d) A plot of In S( Q. w z 0) versus temperature for polycrystalline samples of ( + ) benzene

and ( A ) ethylbenzene. both samples measured at Q= 1.19 A-‘. See text for details.

second moment of the rigid lattice which contains the scattering/resonating atom, it represents the unnar- rowed linewidth. The constant, cy, is of order unity

[71. Neutron scattering observes phenomena in CO space

whilst NMR probes directly in time. Thus the rigid lattice frequency limit in eq. ( 7 ) corresponds to slow dynamical processes which on a neutron scattering time scale would be indistinguishable from the instrument resolution function, R (co).

If we think of the rigid lattice linewidth as our energy window, AE, we may consider eq. (7) as repre- senting the loss of scattered intensity as the quasi- elastic component of the scattering broadens due to the thermally activated reorientational motion de-

fined by v,. In this thermally activated motion we as- sume that the rate of molecular transitions from one equivalent site to another, characterised by the correlation frequency, v. is low and the lattice is still practically rigid. It may then be assumed that v, is of the Arrhenius form

v, = v. exp ( - EJRT) , (8)

where E, is an activation energy corresponding to the height of the potential barrier between two equivalent molecular positions and v. is the attempt frequency which may be of order the frequency of the lowest energy librational mode. In our near-elastic scattering experiment we may interpret our measurement as

J. Wdllams and B. Frock / Chemical Physics 166 (1992) 425-439 429

cu

z I S(&, w, T)R(w’) do for WZO , -cc

where S( &, w) can, for protons (purely incoherent scatterers) which are undergoing rotational motion confined within a given volume, be decomposed into an inelastic component with a residual elastic line [ 8 1. The spatial information is contained in the structure factors, A,( &) the elastic structure factor and A,( &) the inelastic structure factors. Thus

SC&, W)=exp(-2W,) L&((i) s(m)

+ c A,(&)L,(w) 1 > (9) I

where C,A,( &) = 1 and L(o,) is a Lorentzian function describing the frequency of proton motion.

Following the NMR model, we consider the increasing frequency of the molecular motion leading to a broadening of the Lorentzian functions in eq. (9 ). Thus for w z 0 and assuming a rectangular resolution function

+ CA(Q) tan-‘(~/~,) , > I (10)

both terms in eq. ( 10) being subject to a Debye- Waller attenuation. Eq. (9) may be used to directly analyse our data, for example, in fig. 1 a below 150 K the scattering falls off with temperature according to the first term on the rhs of eq. ( lo), the second term being too slow to be observable. Above 150 K the second term on the rhs of eq. ( 10) starts to become important.

4. Measurement of S’($ co)

Above we showed how quasi-elastic scattering, that is scattering about the incident elastic line, can be used to derive values for the amplitude of oscillations about fixed lattice positions in solids. Similar information may be obtained from truly inelastic features. That is, scattering from excitations well separated from the

broadened elastic line (energy changes of the order z lo- 100 meV). Such measurements, if made at a variety of scattering angles, will through eq. (4) allow a determination of the average ( u’) .

Our determination of S( 0, w ) for C6H6 : C6F6 was made on the high-flux high resolution spectrometer IN4 at the Institut Laue-Langevin, Grenoble [ 6 1. The sample was contained in an aluminium sachet and incident neutrons of wavelength 2.2 8, or 17 meV were used for the experiment. The sample temperature was controlled by a helium cryostat and the Stokes spectrum was recorded. The scattered neutron signal was corrected for sample container scattering and detec- tor efficiency and using a vanadium sample studied at the same wavelength and orientation the scattering law was derived.

The inelastic spectra displayed in fig. 4 were obtained, in part on IN6 at the ILL, Grenoble [ 61 and also at the Rutherford Appleton Laboratory, UK. At the latter institute the measurements were made on the crystal spectrometer TFXA, using energetic neutrons to obtain the Stokes spectrum at low sample temperatures. The hexafluorobenzene spectrum was recorded on IN6, Grenoble, in neutron energy gain, thus the higher sample temperature for this measurement.

5. General discussion

In many solids, even at temperatures well below their melting point the temperature dependence of the quasi-elastically scattered neutrons, the Debye-Waller factor, falls more rapidly with temperature than would be expected from consideration of a rigid lattice. We have investigated the following systems.

5.1. Benzene

Consider fig. la, this is plot of In S(Q, 0~0) versus temperature for a polycrystalline sample of benzene measured at Q= 1.19 A-‘. The measurement begins at 40 K and as the temperature mounts to about 150 K the intensity is seen to fall slowly. This is as one would expect from the Debye-Waller factor. However, above 150 K the scattered intensity is seen to fall much more rapidly and falls continuously to the melting point at 278 K whereafter it plummets as

430 J. Williams and B. Fnck / Chemrcal Physics 166 (1992) 425-439

the material melts. The dashed line, an extrapolation of the Debye-Waller behaviour, is a guide for the eye.

The more abrupt decrease in the scattered flux above 150 K arises because the molecules at these temperatures have sufficient thermal excitation energy to undergo sixfold reorientations in the solid.

This onset of reorientational motion in solid benzene was first observed by Andrew and Eades [ 91 in NMR studies of solid benzene (between 90 and 278 K) and the temperature dependence (between 200 and 273 K) of the neutron quasi-elastic linewidths was reported by Fujara et al. [ lo]. It was found that an Arrhenius model fitted the observed temperature dependence of the NMR and quasi-elastic linewidths giving r,=l/~,=9.2~10-‘~ s and E,=-17.6 kJ/ mol for the latter technique and TV = 3.2 x 1 O- I4 s and E,= - 15.5. kJ/mol for the NMR measurement.

Although this activation energy, E,, corresponds to a temperature much higher than 150 K we are able to see the onset of motion because there are NA exp( - EJRT) molecules with sufficient thermal excitation energy to overcome the sixfold reorientational barrier and undergo a rotation at the observed temperatures. When these few, thermally activated molecules execute a large amplitude motion, a rotation of 60” will move a proton in benzene by about 2.5 A, they disproportionally weight the loss of scattering intensity, see eq. ( 5 ) . As the temperature increases more and more molecules are able to perform this reorientation and hence In S( Q, o z 0 ) falls rapidly. In fig. 2a we give our analysis of the data displayed in fig. 1 a using eq. ( 9 ). The necessary Debye- Waller factor is obtained from the low temperature, 50 KG T< 110 K, data by plotting In S( Q, or0) versus Q2 (to derive ( u2) ) and then plotting ( u2) versus temperature; our experimental values of ( u2( T) ) are given in fig. 3a and are seen to agree with available literature values. From data displayed in fig. 3a we see how the relative mean square amplitude of motion of the benzene ring changes as the temperature of the sample increases. In the low temperature region it is linear with a slope 6.48x 1O-4 AZ/K (60 KG T< 110 K) which we use to calculate our Debye-Waller correction. The onset of a large amplitude rotational motion at temperatures above 150 K is clearly seen. At low temperatures, 2.5 =G T< 50 K, in fig. 3a we display absolute values of ( u2) determined from the variation of S( &, w)

with Q2 via eq. (4), for C6H6 : C6F6, these are a measure of the zero point oscillations for the protonaceous partner in this system. The two sets of data, quasi-elastic and inelastic are seen to the complementary.

In fig. 2a we analyse our FWS data for benzene, the best lit values for the Arrhenius parameters of the sixfold reorientational jumps are vo=2.3X lOI s-’ ~9.5 meV and E,= - 12.5 kJ/mol. Also given in fig. 2a is the theoretical curve obtained with the Arrhenius parameters derived from the other quasi-elastic neutron scattering study of the benzene solid state reorientation [ lo]. These latter workers do not report the inclusion of a Debye-Waller function in their analysis and the given parameters cannot account therefore for the low temperature behaviour. We note that their preexponential factor, 449 meV, does not represent any reasonable solid state intermolecular excitation. Whereas in our case the prefactor of 9.5 meV will be shown later to be very close to the energy of the excitation about the sixfold axis of a benzene molecule in the solid state. The value for v. from the NMR work, 129 meV, was derived from measurements in an entirely different frequency regime than ours. We note that our parameters agree within the error bars with the reported neutron linewidths at high temperatures and with the NMR data.

The fact that the benzene rings are rotating in the solid at temperatures near the melting point of the material does not, provided the molecules remain planar, affect the intermolecular forces which hold the solid together. When it rotates the molecules appear disc-like as opposed to hexagonal and this will not affect the mean value of the quadrupole-quadrupole interaction energy, the main electrostatic contribution to the lattice energy, U(R) =0.75~@,@~,/4rre~~, where x is dependent upon the relative orientation of the two benzene molecules, 8 is the molecular quadrupole moment and R is the distance between the centres-of-mass of the molecular pairs.

In fig. 5a we display In S( Q, w= 0) versus Q2 and versus sample temperature for benzene and we clearly observe the onset of rotational motion and see how this motion changes with Q* and temperature. We also note that the changes are continuous. There is most probably only one thermally activated dynamical process going on in solid benzene below its melting point.

J. Williams and B. Frrck / Chemrcal Phystcs 166 (1992) 425-439

(all 0 r

08-

0.21

((3 0

z -0.2

A d ‘si -0.4

E

-0.6

‘t

1 OO

Benzene Q = 1.19 A-’ \

1 1 100 200 300

TEMPERATURE (K)

50 150 250 TEMPERATURE / K

The neutron diffraction experiments of Cox et al.

[ 111 and Bacon et al. [ 121 on solid benzene have given a detailed picture of the different types of motion going on in this solid at a number of temperatures. They were able to determine the anisotropic thermal parameters for both the carbon and the hydrogen atoms of the benzene ring. At the lowest tern- perature studied, 138 K, Bacon et al. [ 121 found that for both types of atoms the largest contribution to the overall mean squared amplitude, ( u2 ) overall

((U2)overal, = <U2)0UWplane + (U2)rad,al -t

<u2> tangent,a,, this sum is what the quasi-elastic experiment measures), comes from the out-of-plane motions. This motion being twice the sum of the radial and tangential contributions. The amplitude of all three types of motion increases with temperature, however, the tangential amplitude increases most

(b)l.o

0.8 c

0.2

I 01 I L_ I

0 100 200 31 TEMPERATURE ( K )

431

Fig. 2. (a) Plot of normalised elastic intensity versus temperature

for polycrystalline benzene at Q = 1.19 A- ‘, together with a best

tit line derived from eq. (9). See text for details. (b) Plot of nor-

malised elastic intensity versus temperature for polycrystalline

toluene at Q= 3.76 A-’ together with best tit lines derived from

eq. (9). See text for details. (c) Plot of lnS(Q, WOO) versus

temperature for (x ) polycrystalline C6H6 : C,F, at Q= 1.19 A- ‘,

together with that of ( A ) benzene at the same Q.

rapidly being due to reorientations. At 2 18 K it is, from the diffraction data [ 11,12 1, larger than either the radial or the out-of-plane contribution. At 270 K, for the carbon atom, ( u2) tangentla, is almost equal to the sum of the other two contributions to (~*)~~~,~i,. The largest part of the data displayed in fig. 3a and all the data in fig. 3b were derived from the quasi- elastic scattering results by plotting lnS(Q, wx0) versus Q* at each temperature. In fig. 3a the lowest temperature values of ( u2), which we take as a measure of the zero-point oscillation, (ui), were obtained by fitting S( Q, w), measured between 2 and 12 meV, to eq. (4). As can be seen from fig. 3a the diffraction results agree quite well with our quasi- elastic value of ( u 2 ) overall.

The mean square angular amplitude of oscillation about the hexad axis is determined by Bacon et al. to

432 J. Williams and B. Frick / Chemical Physics 166 (1992) 425-439

TEMPERATURE/K

. (b)l5OE-2

!*: 0

ethylbenzene. I ,

12 OE-2 - ioluene 0 /I /’ o

I 200

TEMPERATURE (K)

Fig. 3.(a) Plot of ( u2), derived from quasi-elastic measure-

ments, versus temperature for ( + ) polycrystalline benzene. The

data represented by (+ ) are values derived from the Q-depen-

dence of S( 0, o) in C6H6 : C6F6 as measured on the inelastic neu-

tron spectrometer, IN4 of the ILL, Grenoble (see text for de-

tails) and the data represented by (0) are values derived from

the neutron diffraction studies of Bacon et al. [ 121 on crystalline

samples of benzene. The value taken from Bacon et al. corre-

sponds to the largest contribution to the thermal parameters of

the H atoms at a particular temperature. The sum of all of the

thermal parameters (out-of-plane+ tangential + radial) give

(uZ> ,ssK=0.147AZand (~*)~,sk=0.247A*. (b) Plotof (u*)

derived from quasi-elastic measurements versus temperature for

polycrystalline toluene and ethylbenzene.

be 7.9” at 270 K, quite a considerable motion when compared to the 60” sixfold symmetry. But consis- tent with the picture given above of sufficiently energetic molecules being free to undergo rotation at temperatures > 150 K.

As mentioned above it is possible to relate the magnitude of the amplitude of a vibrational motion to the frequency of the oscillation (see eq. (6) ). Cruickshank [ 13,14 ] has developed a model in which it is possible to describe solid state rotational motions. For a tangential oscillation or a rotation, the

mean amplitude of oscillation in radians, (@) is re- lated to the frequency, V, through the moment of inertia, I,

(11)

This model has been successful in comparing Raman vibrational frequencies with the anisotropic thermal

parameters derived from crystallography [ 13,14 1. Bacon et al. [ 12 ] give ( @) for the hexad axis of benzene at a number of temperatures from which we obtain a value 1.5 ? 0.2”, by extrapolation to 25 K, the temperature at which we have measured the neutron inelastic spectrum of benzene displayed in fig. 4. Then from eq. ( 1 1 ), using 1,, =29.6 x 1O-46 kg m2 we obtain vibrational frequencies lying between 80 and 6 1 cm-‘. Harada and Shimanouchi [ 15 ] have assigned, from laser Raman measurements, the low frequency lattice modes of solid benzene and they identify two vibrations as involving the rotation of the aromatic ring about its hexad axis. For 270 K and at 35 cm-’ there is an in-phase rotation of the four benzene rings (B,,) which occupy a unit cell and at 63 cm- ’ there is the out-of-phase ( B2,J partner of a pair of benzene rings. The frequencies are a function of temperature, decreasing with increasing temperature. The Raman measurements were made at a number of temperatures and extrapolating their data to 25 K we identify the intense peak seen in the benzene inelastic neutron spectrum in fig. 4 at about 77 cm-’ as arising from excitations about the hexad axis. From our analysis of the quasi-elastic data, figs. la and 5a, the derived value of the pre-exponential factor, vo= 76.7 cm- ‘, is in agreement with the mean frequency of the phonon spectrum of benzene displayed in fig. 4.

For small harmonic torsional oscillations, within a potential of the form 4 U(R) ( 1 - cos n/3) where U(R) is the barrier height which may be equated with the activation barrier determined from quasi-elastic neutron scattering, IZ is the symmetry of the reorientation (n = 6 for benzene and C6H6 : CsF6 ) and p is the coordinate; these torsional frequencies may be represented as [ 16 ]

n

u=2n J

U(R) 21 . (l-2)

For benzene, from the available values of the temperature dependence of the reorientational frequency we

J. Williams and B. Frrck / Chemrcal Physics 166 (1992) 425-439 433

01 0

* 00 160

9

Cs&

I OO 40 80

‘) 120 160

ENERGY TRANSFER/cm-l

Fig. 4. Neutron inelastic spectra for C6H6:C6F6, benzene and

hexafluorobenzene. The intensities are in arbitrary units. The

C6H6: C,F6 and the benzene spectra were measured at the RAL,

UK on TFXA and the C6F6 spectrum on IN6 at the ILL, Greno-

ble. The sample temperatures were, benzene 25 K, ben-

zene’hexafluorobenzene 25 K and hexafluorobenzene 95 K.

may determine U(R) at each temperature. We may also, from the literature lattice constants [ 11,121, find the intermolecular distances at the same temperatures. This allows us to attempt to rationalize the possible origin of the intermolecular potential, as deter-

mined from the fundamental torsional frequencies, in terms of the interaction of the molecular quadrupole moments.

From our measurements of the inelastic spectrum of benzene at temperatures less than 200 K together with the laser Raman measurements [ 15 ] we have determined the temperature dependence of the frequency of the sixfold torsions, 67.1 cm-’ at 138 K, 59.6 cm-’ at 218 K and 54.7 cm-’ at 270 K. Assum- ing eq. ( 12) to be valid at these temperatures, we calculate barrier potentials of 15.8 kJ/mol at 138 K, 12.5 kJ/mol at 218 K and 10.5 kJ/mol at 270 K. As the experimental temperatures are much lower than the barrier heights the harmonic approximation in- volved in eq. ( 12) is valid. The experimental values for the barrier height to rotation are 15.5 kJ/mol determined from NMR measurements [ 8 1, 17.6 kJ/mol determined from quasi-elastic neutron scattering [ 91 and 12.5 kJ/mol from our present work, all these experiments were made at temperatures of order and above 150 K. This comparison shows the limitations of a temperature independent representation of the solid dynamics. As the c direction is the smallest intermolecular distance and for two interacting quadrupole moments, U( R ) varies as 1 /R 5 [ 17 1, it is in- teractions along this direction which will dominate.

5.2. Toluene and ethylbenzene

With toluene and ethylbenzene, there is no rotational channel for molecular relaxation in the solid state. Figs. 1 c and 1 d show the In S( Q, o =: 0) versus temperature plots for polycrystalline samples of toluene and ethylbenzene, respectively, together with that for benzene at the same Q( 1.19 A- ’ ). It is read- ily seen that near the temperature where the benzene molecule started to reorient in the solid the toluene and ethylbenzene solids melt, 178 K. We may think of this as the inability of the lattice of these substituted benzene molecules to allow for the rotational degree of freedom available in solid benzene.

From figs. 3a and 3b we see that the relative mean

amplitude of oscillation, ( u’) , for toluene between 50 and 150 K increases linearly as in benzene. We note that at these temperatures, the methyl group in toluene is already undergoing rapid rotational reorientation. Both molecules are oscillating about fixed positions to, nearly, the same extent.

434 J. Williams and B. Fnck / Chemcal Physm 166 (1992) 425-439

Fig. 2b represents an analysis of our FWS data for toluene at high Q, again the Debye-Waller factor is derived from the lowest temperature data, giving d(u2(T))/dT=9.3x10-4 AZ/K (50 K<T<170

K). The temperature dependence of the data between 50 K and the melting point 178 K can be modelled almost completely by use of a Debye-Waller factor only. The free rotation of the -CH3 group has already become established by the start of our mea-

surements. The literature [ 27 ] values for the Arrhe- nius parameters necessary to describe the rotation of the -CH3 group, i.e. vo= 5.8 meV and E,= - 1.2 kJ/ mol, give the dashed line.

The situation becomes particularly complex when we consider ethylbenzene, fig. 3b. At the lowest temperatures the behaviour of ( u2) is similar to benzene and toluene, a Debye-Waller attenuation caused by localised motions. However, at 100 K ( u2 ( T) ) has started to increase more rapidly than

Fig. 5. (a) Plot of In S( Q, w =Z 0) versus sample temperature ver-

sus Q2 for benzene. (b) Plot of lnS( Q, 0x0) versus sample

temperature versus Q2 for ethylbenzene. (c) Plot of In S( Q, wx 0)

versus sample temperature versus Q* for C6H6 : C6F6.

either benzene or toluene. Here we are seeing the side chain beginning to execute motions which increase the relative mean molecular ( u2) more rapidly than is possible in either benzene or toluene. The spin relaxation measurements of Beckmann et al. [ 18 ] show how low these C2Hs side group reorientational barriers can be, 9.3 kJ/mol, and how they change with the presence of other similar side groups. From our ethylbenzene data we obtain a Debye-Waller factor of 10x 1O-4 AZ/K (50 K< T< 100 K).

In fig. Id the In S( Q, 0% 0) curve for ethylbenzene reaches a minimum at E 140 K, then rises in stages to reach the same value as benzene and toluene at GZ 160 K before melting at 178 K. This behaviour may be seen more clearly in the In S( Q, w x 0) versus T

versus Q2 plot in fig. 5b. A similar observation of the change of molecular dynamics in the polycrystalline and glassy forms of ethylbenzene has been made by Beckmann et al. [ 18 ] in their NMR spin relaxation

J. Williams and B. Fnck / Chemrcal Physrcs 166 (1992) 425-439 435

study. A detailed investigation of the Q dependence of the quasi-elastic scattering [ 191 shows that the step-like changes in the temperature dependence of the scattering may well be due to dynamic effects.

Such fluctuations in the temperature dependence of In S( Q, oz 0) are not seen in toluene, see fig. lc, one may therefore conclude that it is the length of the side chain which gives rise to this anomalous behaviour; the pendant -CH,-CH, groups interacting amongst themselves.

A further point of interest in the ethylbenzene and toluene data, seen most clearly in figs. lc, Id and 5b is the presence of a substantial amount of quasi-elastic scattering above the melting point. This is particularly the case in ethylbenzene. In both samples the liquid state quasi-elastic scattering is seen to fall with both increasing temperature and with increasing Q2 at a particular temperature.

To see if this effect is significant a measurement of S(Q, w) was made in ethylbenzene well above its melting point at 188 K. The energy transfer range investigated by this measurement, made on IN 13, was from - 80 to 120 l.teV. Fig. 6 displays the spectra recorded at different Q, we see significant quasi-elastic broadening whose width increases rapidly with increasing Q2. At Q= 1.29 A-’ the quasi-elastic width is 20 peV, whilst at Q= 2.1 A-’ the quasi-elastic width is found to be 60 peV, we are able therefore to derive a liquid state diffusion constant of 1.5~ lo-’ cm2/s

Fig. 6. The scattering law, S( Q, o) for ethylbenzene determined just above its melting point ( 188 K) at three different values of Q.

at 188 K for ethylbenzene. We may compare the magnitude of this diffusion constant with that of benzene measured by quasi-elastic neutron scattering, Winfield and Ross [ 201 found a macroscopic diffusion constant of 2.5x lo-’ cm2/s for benzene at 280 K, that is, just above its melting point. This difference in diffusion constant we ascribe to sterical hindrance which not only affects the reorientational motions in the solid but also the translational motion in the liquid.

5.3. 1,3,5triJluorobenzene

With the molecule 1,3,Strifluorobenzene we have a different behaviour. Fig. lb shows how In S( Q, wz 0) for a polycrystalline sample of 1,3,5-trifluorobenzene varies with temperature at Q= 1.19 A-’ and allows us to make a comparison with the data for benzene. Below 150 K the 1,3,Mifluorobenzene data is reminiscent of that of benzene, a Debye-Waller factor attenuation of the intensity due to localised measurements. However, the fluorinated aromatic cannot undergo the sixfold reorientational motion seen in benzene, consequently, its scattering above 150 K is very different. FOJ benzene a sixfold reorientation moves the ring by 60” between two equivalent sites. In 1,3,Mifluorobenzene, 60” rotations between equivalent sites are no longer possible, the molecule can only reorient by making 120” rota-

0=2.09/k’ 3.4 ..;

436 J. Willlams and B. Frock /Chemical Physics 166 (1992) 425-439

tions which would give a very large value for ( u’) . Presumably, the barrier to such a reorientation is too large and the plot of lnS( Q, WZO) appears to fall smoothly over the entire range of temperature to its melting point at 268 K. However, an estimate exists for the barrier to reorientation in this molecule from spin-lattice relaxation measurements, - 46.0 kJ/mol [ 2 11. It is therefore not unexpected than we see no noticeable onset of rotational reorientation. The large difference in the heights of the barriers to thermal reorientation in benzene and trifluorobenzene can be clearly seen in the different rates of decrease in the plots of lnS( Q, wz.0) versus Tin fig. lb. The benzene with an activation barrier of between - 12.5 and - 17.6 kJ/mol starts to fall at a lower temperature than the 1,3,%trifluorobenzene with an activation barrier of - 46 kJ/mol [ 2 11.

From our measurements at low Q we see a slight change of slope of the In S( Q, oz 0) versus temperature plot at temperatures greater than 200 K, however, it is not very clear. With the much larger activation barrier there will be fewer molecules at solid state temperatures with sufficient thermal energy to pass over the barrier and we see only the slightest in- dication of the onset of reorientation. We note that in our analysis of the FWS scans for 1,3,5-trifluorobenzene we have needed to consider a Debye-Waller factor only; our derived value is 7.9 x 10e4 AZ/K (60 K< T< 110 K).

5.4. Benzene-hexajluorobenzene

The other aromatic system investigated was the solid formed by mixing equimolar quantities of benzene and hexafluorobenzene C6Hs : C6F6 [ 22,23 1.

Values for the mean square amplitude of oscillation as a function of temperature have been given previously for this material [ 22 1. What is displayed in fig. 3a are our measurements of the amplitudes of the low temperature oscillation of the protons obtained for the different excitations within the low frequency phonon band determined from inelastic experiments on solid C6H, : CsF6. These measurements were made at 2.5, 10, 20 and 50 K. The scattering law, S(&, w), derived from the corrected data was fitted to eq. (4) to obtain thamean square amplitude of oscillation at each temperature. It can be seen from fig. 3a that the two sets of data, quasi-elastic and in-

elastic, give complementary results. The excitations monitored in this temperature dependence study of S(Q, o) are at 29.2, 42.2, 66.9, 90.2 and 110 cm-’ at 2.5 K. In all cases it can be seen that with increasing availability of thermal excitation energy the average amplitude of oscillation increases. This being particularly so for the higher frequency transitions, which are likely to be librational in character.

Fig. 2c shows the In S( Q, ox 0) versus temperature plot for C6H6: C6F6 and for benzene at Q= 1.19 A-‘. We see clearly how at temperatures less than 150 K the Debye-Waller factor for benzene is larger than that for the C6H6:CbF6. However, at temperatures above 150 K it is the scattering from the binary mix- ture which falls first, that is, the barrier to sixfold reorientation is lower in C6H, : C6F6 than in pure benzene; -9.4 kJ/mol [22] as opposed to - 17.6 kJ/ mol [ 9 1. The phase transitions seen in the binary complex complicate comparisons above 200 K. From figs. 5a and 5c we may compare In S( Q, w= 0) versus T versus Q2 for benzene and C6H6: C6F6, in the for- mer over the entire temperature-Q’ range there is only one dynamical change, that is, at about 150 K the aromatic rings start to rotate. In C6H6:C6F6, above Q’z 9 A-’ this is also what is seen, a smooth change from localised vibrations to sixfold reorientations. However, below Q2 = 9 A-’ the behaviour is very different. It is here at low Q, that is, large distances in the solid that the three phase transitions seen in this material below its melting point are manifest [ 22-241. Indeed, above Q’ z 9 Ap2, that is at a molecular level in the solid, C6H6 : C6Fs is the same as the benzene as far as neutron scattering is concerned.

Our FWS experiments analysed via eq. (9) gave Arrhenius parameters of v,=2.2~ lOI* s-’ (9 meV) and E, = - 12 kJ/mol with an increase of mean square displacement of 2.46x lop4 A2/K (60 KG T< 100 K). This activation energy is in reasonable agreement with the previous value [ 221 and the preexponential factor is close to the mean of the phonon spectrum of C6H6 : C6F6 displayed in fig. 4.

In an earlier section we discussed how the intense line seen in the inelastic neutron spectrum of benzene, given in fig. 4, at 77 cm-’ is due to the rotation- like excitation of the benzene ring in the solid. These excitations take place in a moderately deep potential z 183 meV. In the case of C6H6 : C6Fb there will be excitations of the same type at similar frequencies,

J. Williams and B. Frrck / Chemical Physics 166 (1992) 425-439 431

the neutrons probe mainly the dynamics of the pro- tonated components and the intermolecular distances in the two solids are very much the same as are the lattice forces. The main difference being the depth of the potential over which the benzene molecule has to move to execute a sixfold reorientation. In the case of C6H,: C6F, this potential is x 98 meV deep [22] almost half the depth of the potential in pure solid benzene, a difference explained in terms of the component molecular quadrupole moments

[221. What was also noticeable in this earlier study of

CbH6: C6F6 was the pronounced red shift of the line at 66.9 cm-’ up to the point of the first phase transition seen in this material at 205 K. At 2.5 K it occurs at 66.9 cm-’ whilst at 200 K it has moved to 50.5 cm-‘. Using eq. ( 12) we may then calculate the height of the potential within which the benzene ring

is undergoing sixfold librations. Taking 1,, = 29.6 x 1O-46 kg m2 for the benzene molecule, we find with n=6 that at 2.5 K, U(R)=15.7 kJ/mol whilst at 200 K, U(R) has become 9.0 kJ/mol. The presence of a structural phase transition at 205 K is perhaps not unexpected when we consider this substantial decrease in the barrier height for molecular reorientation, having fallen from 80kT at 25 K to 5kT

at 205 K. The origin of this barrier may be seen qual- itatively by considering chains of hexagons, closely interpenetrating, attempting to rotate against one another.

Earlier we mentioned that the sixfold libration of benzene is red shifted on heating. Indeed, from 25 to 200 K it is red shifted by 16 cm-‘. Benzene in C6H6 : C6F6 over the same temperature range has a red shift of 16.5 cm-‘. There will therefore be a similar large decrease in U(R) in both solids. In the case of benzene the red shifting continuous right up to its melting point where the same line appears at 54 cm-‘, a total softening of the mode by 26.5 cm-‘, 33% of its low temperature value over the temperature range of the solid. We note that at the melting point U(R) has fallen to 4kT, whereas for C6H6: C6H6 at its first phase transition it had fallen to 5kT.

The question naturally arises as to why does C6H6: C6F6 undergo a phase transition at 205 K and indeed two others at 25 1 and 275 K, whilst solid benzene has no phase transitions below its melting point? In response we point out that it is the nature of the

quadrupole moment-quadrupole moment interac- tions between a benzene molecule and its neighbours in the two solids which produces the two different solid structures. In pure benzene, quadrupole moments of like polarity interact to give a solid com- posed of slipped parallel and T ordered pairs of molecules, i.e. (1 and E. Whilst in C6H6: C6F6 the opposite polarity of the quadrupole-quadrupole interaction produces stacked chains of alternating, parallel benzene and hexafluorobenzene molecules. The one structure being more stable to thermal excitations particularly excitations transverse to the main chain direction than the other.

6. Melting of aromatic solid lattices

Although the solid-liquid transition is one of the most familiar physical processes, the detailed micro- scopic understanding of the phenomenon is still rather

vague [25]. Earlier we were interested in the change of quasi-

elastic linewidth with sample temperature, that is, as the scatterer underwent thermally activated oscillations. This analysis has a connection with melting and lattice dynamics (instability of the solid phase). One of the earliest theories of solid state instability leading to melting was due to Lindemann [ 26 ] who con- jectured that melting should occur at a temperature, TM, when the average displacement, ( u) , of the atom or molecule which constitute the solid due to thermal fluctuations reach a given fraction, r* z 0.1, of the intermolecular or interatomic spacing, R. That is

r*=(u2)&c/Rx0.1. (13)

If we consider a Debye solid and neglect the frequency difference between longitudinal and transverse phonons we may write [ 25 ]

(u~>~, =9A’T,/mkT&, (14)

where T,, is the Debye temperature and m the molecular mass. This equation suggests the existence of a liaison between relative amplitudes of molecular oscillation in the solid and the melting point of the solid. Alternatively, we may consider the amplitude of angular oscillation, ( g2). Cruickshank has given a means of relating (@) to the vibrational frequency of an appropriate molecular excitation, see eq. ( 11).

438 J. Wdhams and B. Frxk / Chemrcal Physics 166 (V992) 425-439

Bacon et al. have given the large magnitude, 7.9’) near the melting point of benzene. We propose that melting occurs in these samples when this mean amplitude of oscillations reaches some fraction of the free rotation amplitude, for example, some fixed fraction of 60” for benzene. An analogy with the Lindemann theory of melting.

Eq. ( 11) gives (@) in radians, if melting occurs when @ is some fraction of the sixfold rotation angle, 60’ for benzene, hexafluorobenzene and the adduct benzene : hexafluorobenzene, we may write &,,, o (7.9”/60”)‘. Eq. (11) may then be rearranged for the maximum value of T,

TM z@&,,4n21u’/k, (15)

the melting point. Above we pointed out how low frequency vibrations would dominate in any consideration of the origin of (@) . Again it is the lowest frequency vibrations which need to be considered when looking at the melting point. This weighting in favour of low frequency and hence large mean vibrational amplitudes reflects the coupling to the motions of the neighbouring molecules.

Consider benzene, for @f,, x (7.9”/60” )2, with a parallel moment of inertia of 29.6 x 1O-46 kg m2 and given T,=278.8 K we find v=46 cm-‘. Similarly for hexafluorobenzene with I,, = 164.2 x 1O-46 kg m2, T,=278.5 K and &,,, z (7.9”/60”)2 we find v= 19.5 cm-‘. For C6H6: C6F6, with a reasonable estimate for the moment of inertia taken as I= (reduced mass)x(3.77A)2=130~10-46kgm2,@~,,~(7.9”/ 60”)2andTM=299Kwetindv=23cm-‘.Aconsid- eration of fig. 4 shows that these calculated frequencies agree rather well with the lowest frequency excitations measured in these systems by inelastic neutron scattering. A similar analysis of 1,3,5-trifluorobenzene, with I,, =95x 1O-46 kg m2, T,=267.5 K and assuming a similar magnitude for the amplitude of oscillation yields v = 25 cm- I.

We may extend this simple model to toluene and ethylbenzene. Taking the moment of inertia for the methyl rotor to be 53.45 x 1O-48 kg m2 and the lowest torsional excitation of the methyl rotor in solid toluene as 46.8 cm- ’ [ 271, we calculate (@) via eq. ( 11) to be 44” at 178 K, which is certainly a large fraction of the free rotation angle 120”. A similar ar- gument would apply for ethylbenzene. Cavagnat et al. [ 271 point out how there is a strong coupling be-

tween methyl rotations and the phonon density of states in solid toluene at temperatures well below the solids melting point. It seems reasonable then to apply our model and say that fusion occurs when the side group is no longer able to store the available thermal energy, which in consequence has to be stored in other degrees of freedom which are more important in destabilizing the lattice.

In passing we note how the addition of a methyl

group strongly depresses the melting point of a series of aromatic molecules. Benzene, naphthalene and anthracene melt at 278.5, 353.5 and 489 K, respectively. The addition of a CHs group changes the melting points to 178, 251 and 358 K, respectively. The possibility of low frequency excitations in the side group strongly effects the ability of the molecules to relax in the solid, they cannot now undergo rotational relaxation. The effect persists in the homolo- gous series, toluene, ethylbenzene, n-propylbenzene and n-butylbenzene with melting points of 178, 178, 173.8 and 184.5, respectively [ 281. Whereas in benzene the molecule is able to store a great deal of thermal energy and yet remain a solid, once a side group has been introduced, strong coupling to the rotation of other molecules is produced. Then if this side group can no longer store the available thermal energy it must lead to whole molecule motion and thus to cou- pled lattice instabilities and finally to melting.

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Low frequency excitations of aromatic molecules in the solid state