Rotational motion of molecules and molecular groups in the solid state studied by neutron scattering

Rotational motion of molecules and

molecular groups in the solid state

studied by neutron scattering

Chr. Steenbergen

Rotational motion of molecules and

molecular groups in the solid state

studied by neutron scattering

iMiiiiiiiiiiiiiitipiii

iliilHl.ill llli:llttllllllPillllnliilllM:iii:illl III!

C1002

7

9830

2

PI

15

8

730

9

BIBLIOTHEEK TU Delft

P 1158 7309

C

₂₇₉₈₃₀

Rotational motion of molecules and

molecular groups in the solid state

studied by neutron scattering

Proefschrift ter verkrijging van

de graad van doctor in de

technische wetenschappen

aan de Technische Hogeschool Delft,

op gezag van de rector magnificus

prof. dr. ir. F.J. Kievits,

voor een commissie aangewezen

door het college van dekanen

te verdedigen op

woensdag 21 februari 1979

te 16.00 uur door

Chrjstjaan Steenbergen

natuurkundig ingenieur

geboren te Rotterdam

y^^sx?

79 'Q

^^OQ

fé> DC£L£NSIR,101

^.

: L f ^

Dit proefschrift is goedgekeurd door

de promotor: DR. IR. L.A. DE GRAAF

CONTENTS

I PREFACE AND GUIDE 1 I I INELASTIC NEUTRON-SCATTERING STUDY OF THE METHYL GROUP MOTIONS IN

DIMETHYLTIN DIFLUORIDE 5 1. I n t r o d u c t i o n 6 2 . M e a s u r e m e n t s 6 3 . T h e o r y 7 4 . R e s u l t s 8 R e f e r e n c e s 10 I I I NEUTRON SCATTERING STUDY OF THE METHYL-GROUP REORIENTATIONS AND

THE LOW TEMPERATURE PHASE TRANSITION IN (CH ) 2 S n F 2 "

I n t r o d u c t i o n 12 T h e o r y 12 E x p e r i m e n t s 14 A n a l y s i s 18 D i s c u s s i o n 23 R e f e r e n c e s 25 IV ROTATIONAL MOTIONS OF NH^ GROUPS IN NH.ZnF, STUDIED BY

QUASI-4 QUASI-4 3 '

ELASTIC NEUTRON SCATTERING 27

1. I n t r o d u c t i o n 28 2. S t r u c t u r a l d a t a and t h e o r e t i c a l models 29 2 . 1 . C r y s t a l l o g r a p h i c d a t a on NH.ZnF, 29 2 . 2 . Theory f o r t h e i n c o h e r e n t q u a s i - e l a s t i c s c a t t e r i n g 30 2 . 3 . J u m p - r e o r i e n t a t i o n m o d e l s 32 3 . E x p e r i m e n t a l d a t a 34 3 . 1 . S p e c t r o m e t e r 34 3 . 2 . E x p e r i m e n t a l r e s u l t s and d a t a h a n d l i n g 34 4 . C o n c l u s i o n 40 R e f e r e n c e s 42 VI

43 V ROTATIONAL CORRELATION FUNCTIONS OF PLASTIC-CRYSTALLINE

NEOPENTANE FROM NEUTRON SCATTERING EXPERIMENTS

1. I n t r o d u c t i o n 44 2. Theory 45 2.1. Translational motion 46

2.2. Models for the rotational motion 47

2.3. Vibrational motion 51 2.4. Relation to infrared absorption and Raman scattering 51

3. Experimental procedure 52

4. Data analysis 53 4.1. Correction for the instrumental resolution 55

4.2. Renormalisation of I(K,t) 56 4.3. Determination of the translational and vibrational

parts of I(K,t) 57 4.4. Determination of the orientational correlation functions 59

5. Discussion 61 5.1. Comparison with infrared and Raman results 61

5.2. Models for the reorientational motion 62 5.3. The reorientational motions in the disordered solid

phase of neopentane: Conclusions 64

Appendix 65 References 68

VI NEUTRON SCATTERING STUDIES OF THE SOLID TETRAMETHYL COMPOUNDS

OF SILICON, GERMANIUM AND TIN 71

1. Introduction 72 2. Theory 73 3. Experiments and data handling 75

4. Analysis 78 4 . 1 . Q u a s i - e l a s t i c s c a t t e r i n g 78 4 . 2 . I n e l a s t i c s c a t t e r i n g 80 4 . 2 . 1 . R e s u l t s w i t h 5 meV n e u t r o n s 80 4 . 2 . 2 . R e s u l t s w i t h 32 meV n e u t r o n s 82 4 . 2 . 3 . Review of t h e d i f f e r e n t r e s u l t s 84 5 . M e t h y l g r o u p r e o r i e n t a t i o n i n t h e M(CH ) , s e r i e s 85 A p p e n d i x 88 R e f e r e n c e s 89 V I I

VII INVESTIGATION OF METHYL REORIENTATIONS IN 1,4-DIMETHOXY-2,6-DIMETHYLBENZENE BY NMR AND INELASTIC NEUTRON SCATTERING.

1. GENERAL INTRODUCTION AND INELASTIC SCATTERING 91

1. General introduction 92 2. Introduction to the INS measurements 94

3. Theory 95 4. Experiments and data handling 97

5. Analysis 101 6. Concluding remarks 104

• References 104

VIII INVESTIGATION OF METHYL REORIENTATIONS IN 1,4-DIMETHOXY-2,6-DIMETHYLBENZENE BY NMR AND INELASTIC NEUTRON SCATTERING.

3. COMPARISON OF THE DIFFERENT RESULTS AND DISCUSSION. 105

1. Introduction 106 2. Discussion 106 2.1. Discrepancy between the liquid NMR and the other results 107

2.2. Discussion of the solid state NMR and INS results for

methyl group 6 110 2.3. Information about the molecular geometry 112

References 113 SUMMARY 115 SAMENVATTING 117 NAWOORD ' 119 CURRICULUM VITAE 119 VIII

PREFACE AND GUIDE

Inelastic neutron scattering is a spectroscopic technique which is particular-ly suited for the study of motions in hydrogenous substances. Hydrogen nuclei have a cross section for incoherent neutron scattering which is an order of magnitude larger than for any other nucleus. This means that in an inelastic neutron scattering study essentially the self motion of the hydrogen atoms is probed. The reorientational motions of molecular groups like NH,, CH_ and NH,, and of simple organic molecules are of special interest, since they are often not observable by conventional spectroscopic techniques because of optical selection rules, whereas neutrons are not plagued by this restriction, the ^ neutron scattering studies reported hereafter are intended to contribute to the knowledge about these reorientations in the solid state.

In these studies conventional neutron spectrometers have been used, which means that energy transfers less than about 30 meV have been studied with a resolution varying between 0.O6 and 2 meV. Usually two energy transfer ranges are distinguished: the quasi-elastic scattering range below about 2 meV, and the inelastic scattering range above that value. These regions should not be taken too absolutely, as a considerable overlap is possible. In the inelastic scattering range the transitions between discrete levels in the rotational potentials of groups and/or molecules may be present, yielding information about the shape and height of that potential. The quasi-elastic scattering often is determined by the reorientational motion of a molecule or molecular group, whereas the observable reorientation times are determined by the energy resolution of the spectrometer. A resolution of 0.06 meV corresponds with about 50 ps observation time, one of 2 meV with 1.5 ps. The temperature dependence of the reorientational motion yields additional information on the height of the torsional potential. Furthermore, from the wave-vector-transfer dependence of the quasi-elastic scattering information about the number of equilibrium orientations of a molecule or group, and of their configuration, can be abstracted which allows for a determination of (part of) the symmetry of the rotational potential.

The subjects of the neutron scattering studies to be reported have been chosen for the following reasons. The compound (CH,)„SnF. can be regarded as a simple model system in which, the methyl groups are known to be identical and are expected to have a very low (one-dimensional) barrier to reorientation,

In the course of the present experiments a phase transition was found in this substance, which allowed for a study of the change in methyl group motion at this transition.

A calculation of the three dimensional reorientation potential of the tetra-hedrat NH, ions in NH,ZnF_, assuming only electrostatic interactions with the ions in the crystal, already existed including a calculation of the energy levels therein. A quasi-elastic neutron study makes a comparison with those predictions and with earlier inelastic neutron data possible.

The M(CH,), series, with M=C, Si, Ge and Sn, provides more complex systems with reorientational freedom. With increasing central atom-carbon bond length the molecules become less globular and the mutual hindering of the peripheral methyl groups within a molecule diminishes. Hence, C(CH,), exhibits an orien-tationally disordered solid phase over a large temperature range in which the entire (tetrahedral) molecules rotate on the picosecond time-scale. On this time-scale the methyl groups in C(CH,), do not rotate, whereas the reorien-tations of those in Sn(CH-), can be readily observed above about 80 K. A quasi-elastic neutron scattering study on C(CH,), was undertaken to resolve existing disagreement about motional details, as well as to make a direct comparison with infrared and Raman data possible. The other compounds in the M(CH,), series have not been studied with quasi-elastic neutron scattering before. Furthermore, it was expected that accurate values of the methyl torsional transitions in Ge(CH,), and Sn(CH_), could be obtained in an-inelastic scattering experiment, improving existing data.

In the molecule l,4-dimethoxy-2,6-dimethylbenzene four methyl groups can be distinguished, for which different reorientation behaviour may be expected. It is of importance to investigate whether the contributions of the individual methyl groups of such a complex molecule to neutron spectra can be unraveled. A comparison of the neutron results for this molecule with those of NMR investigations is possible, and may even lead to more detailed information regarding the methyl reorientations.

The investigations indicated above are reported in several papers published in or submitted to various journals. Hence, it could not be avoided that parts of the present publication show different notations for references or different styling of the figures. The sequence of the different papers is according to

their introduction in the paragraphs written above. Part of the study on (CH,),SnF„ has been carried out at the National Bureau of Standards (Gaithers-bifrg Md, USA) in cooperation with dr. J.J, Rush. The experiment on NH.ZnF, has been initiated by and carried out in cooperation with dr. L.J. de Jongh, dr. J. Bartolomé and dr. L. Bevaart. It was also reported (in a slightly different version) in the thesis of the latter. The investigation on 1,4-dimethoxy-2,6-dimethylbenzene has been suggested by dr. R.A. Wind of the Magnetic Resonance group of the Delft University of Technology and it has been carried out in cooperation with members of that group. It is originally reported in three separate papers of which the second one presenting the solid state NMR study is not given here. Section 2.1 in the last paper of the series has to be attributed to dr, W.M.M.J. Bovée.

The most detailed treatment of neutron scattering theory and handling of time-of-flight data can be found in the paper on C(CH,),. The theory of jump-re-orientational motion is treated most comprehensively in the paper on NH.ZnF , while examples for specific cases are mentioned in most papers. The neutron data are compared extensively with data from various spectroscopic techniques in the papers on C(CH,), and the other M(CH,), compounds, and the last paper on 1,2-dimethoxy-2,6-dimethylbenzene.

Conversion factors for the various energy units used are given in the following table. 1 meV 1 kJmol"' 1 K meV

1

10.4 0.086 kJmol"' 0.096

1

8.3*10~^

K

11.6 120

1

3

L e t t e r to the e d i t o r

I n e l a s t i c n e u t r o n - s c a t t e r i n g study of the methyl group motions in

dimethyltin d i f l u o r i d e

Chr. Steenbergen and L.A. de Graaf

I n t e r u n i v e r s i t a i r Reactor I n s t i t u u t

Mekelweg 15, 2600 GA Delft

The Netherlands

SUMMARY

An i n e l a s t i c neutron s c a t t e r i n g study of (CH ) SnF. in the temperature range

9-150 K has been performed. From the e l a s t i c and q u a s i - e l a s t i c i n t e n s i t i e s at

the higher temperatures the symmetry of the b a r r i e r to methyl group r e o r i e n

-t a -t i o n has been de-termined -to be -three fold. The -tempera-ture dependence of -the

residence time of the methyl group in a p o t e n t i a l well appeared to be

non-exponential .

1. INTRODUCTION The s t r u c t u r e of d i m e t h y l t i n d i f l u o r i d e , (CH ) SnF_, h a s b e e n d e t e r m i n e d by X-ray d i f f r a c t i o n [ 1 ] t o b e t e t r a g o n a l w i t h a = 4 . 2 4 A a n d c = 14.16 A, and two f o r m u l a u n i t s p e r u n i t c e l l . I t c o n s i s t s of an i n f i n i t e t w o - d i m e n s i o n a l n e t w o r k of t i n a t o m s , and b r i d g i n g f l u o r i n e a t o m s , w i t h t h e methyl g r o u p s above and b e l o w t h i s p l a n e , c o m p l e t i n g o c t a h e d r a l c o o r d i n a t i o n of t h e t i n a t o m s . From t o t a l c r o s s - s e c t i o n m e a s u r e m e n t s w i t h s u b t h e r m a l n e u t r o n s i t was d e d u c e d [ 2 ] t h a t t h e m e t h y l g r o u p s i n t h i s compound h a v e a low b a r r i e r t o r e o r i e n t a t i o n (< 12 meV). T h i s means t h a t t h e r e i s o n l y a weak i n t e r a c t i o n w i t h t h e s u r r o u n d i n g atoms i n t h e c r y s t a l l a t t i c e . When i n t e r a c t i o n w i t h t h e t i n and f l u o r i n e atoms would b e d o m i n a n t , t h e symmetry of t h e b a r r i e r t o r e o r i e n t a t i o n i s t w e l v e f o l d . A s t r o n g e r m e t h y l - m e t h y l i n t e r a c t i o n c a u s e s a l o w e r i n g of t h e symmetry t o s i x o r t h r e e f o l d . I n t h e f o l l o w i n g we r e p o r t an a t t e m p t t o d e t e r m i n e t h e symmetry and h e i g h t of t h e b a r r i e r from an i n e l a s t i c n e u t r o n - s c a t t e r i n g e x p e r i m e n t .

2 . MEASUREMENTS

The e x p e r i m e n t was p e r f o r m e d u s i n g t h e r o t a t i n g - c r y s t a l s p e c t r o m e t e r f o r t h e r m a l n e u t r o n s RKS 2 a t t h e D e l f t r e a c t o r , which i s d e s c r i b e d i n d e t a i l e l s e w h e r e [ 3 J . I n t h i s s p e c t r o m e t e r a r o t a t i n g p y r o l y t i c g r a p h i t e c r y s t a l

( 1 7 , 0 0 0 rpm) p r o d u c e d a p u l s e d n e u t r o n beam w i t h an e n e r g y of 14.5 meV. The e n e r g y r e s o l u t i o n of t h e s p e c t r o m e t e r was a b o u t 7% (FWHM). A f t e r s c a t t e r i n g 3 from t h e s a m p l e , t h e n e u t r o n s were d e t e c t e d i n s e v e n d e t e c t o r - b a n k s of 4 He d e t e c t o r s e a c h , p l a c e d a t t h e end of a 1.45 m H e - f i l l e d f l i g h t p a t h . The s c a t t e r i n g a n g l e s w e r e b e t w e e n 15 and 85 , c o v e r i n g a m o m e n t u m - t r a n s f e r r a n g e a t z e r o e n e r g y t r a n s f e r of 0 . 7 - 3 . 6 A ( r e s o l u t i o n 0 . 0 5 A FWHM), and c h o s e n s u c h as t o a v o i d a n g l e s a t w h i c h B r a g g ( e l a s t i c c o h e r e n t ) s c a t t e r i n g from s a m p l e a n d / o r c o n t a i n e r o c c u r s . D i m e t h y l t i n d i f l u o r i d e powder was p l a c e d i n . a f l a t s a m p l e c o n t a i n e r i n a h e l i u m - f l o w c r y o s t a t . The s c a t t e r e d f r a c t i o n was 0 . 1 5 . M e a s u r e m e n t s w e r e done a t 9 K, 45 K, 77 K and 150 K. F u r t h e r m o r e a t 10 K m e a s u r e m e n t s of t h e empty c o n t a i n e r and a v a n a d i u m sample w e r e p e r f o r m e d . The i n t e g r a t e d s c a t t e r e d i n t e n s i t i e s p e r d e t e c t o r for t h e v a n a d i u m and t h e 9 K, 45 K, 77 K and 150 K s a m p l e m e a s u r e m e n t s w e r e about 30 x 10 , 30 X l o ' ' , 2 . 5 X lo'*, 10 X 10^ and 4 x 10^, r e s p e c t i v e l y .

The o b t a i n e d t i m e - o f - f l i g h t s p e c t r a were c o r r e c t e d f o r b a c k g r o u n d and d e t e c t o r e f f i c i e n c y . An e x a m p l e i s shown i n f i g . 1. I t was c o n c l u d e d from t h e l o w -t e m p e r a -t u r e v a n a d i u m and (CH ) SnF s p e c -t r a , -t h a -t -t h e 9 K d i m e -t h y l -t i n

energy trarvtv/mé/ 6 4 3 O - 2

Fig. 1. .

Corrected time-of-flight spectra of (CH ).SnF„ at 65° scattering angle, for 45 K (full circles), 77 K (cros-ses), 150 K (open circles). The full line indicates the resolution.

80 KX)

time-of-ftight channel number

difluoride measurement represented the resolution of the apparatus. So the latter was chosen as the resolution run, which facilitated a direct normalisa-tion of the measurements. Further data handling was done using the fast Fourier method of Bregman and de Mul [4], in which the spectra are directly Fourier transformed into the time domain. In the time domain the measurements are easily corrected for resolution. A full account of this procedure is given elsewhere [5].

3. THEORY

In a neutron s c a t t e r i n g experiment on a hydrogenous p o l y c r y s t a l l i n e compound

e s s e n t i a l l y the s c a t t e r i n g function S (K,(ij) i s measured, which i s r e l a t e d t o

the intermediate s c a t t e r i n g function I ( < , t ) through a Fourier transform. Here

K i s the momentum t r a n s f e r , hoj the energy t r a n s f e r and t the time. The

sub-s c r i p t sub-s sub-standsub-s f o r sub-s e l f , t h a t i sub-s : only the i n d i v i d u a l motionsub-s of the hydrogen

atoms are included. The intermediate s c a t t e r i n g function i s defined as

N

Ï

i=l

1

I ^ ( < , t ) = jj I < e x p ( i K . r ^ ( 0 ) ) e x p ( - i K . r . ( t ) ) ) > ,

( 1 ) where r. is the position vector of a hydrogen atom and the summation is over

all N protons.

At high temperatures the classical jump reorientation model can be applied to our problem [6]. In this model a proton, situated in a potential well, jumps"^ after a mean residence time T to a next well. For a methyl group with 3, res-pectively 6, wells the intermediate scattering functions become:

l3(<,t) = { A ^ ( K ) + Cl - Ag(K)]exp(-3t/2T)}

x exp(-K^u^) , • (2)

I^(K,t) = {A|(<) + A2(K)exp(-t/2T)

+ A,(<)exp(-3t/2T) + A (K)exp(-2t/T)}

(3)

<ith

A (K) = [1 + 2 sin(<r/3)/(Kr/3)]/3 , (4)

A (<) = [1 + 2 sin(Kr)/(Kr) + 2 sin (tcr/3) / (icr/3)

+ sin(2Kr)/(2K:r)]/6 , (5)

2 .

where r i s the r o t a t i o n a l axis-H d i s t a n c e and u i s the mean squared amplitude

of the proton in the w e l l .

4. RESULTS

In f i g . 2 the e l a s t i c i n t e n s i t y , t h a t i s the long time l i m i t of I ( K , t ) , i s

given as a function of K for the 77 K and 150 K measurements. Due to the

l i m i t e d spectrometer r e s o l u t i o n , which determines the maximum o b s e r v a t i o n

time, the e l a s t i c i n t e n s i t y at 45 K could not be determined. These d a t a can be

f i t t e d b e s t with a 3-fold jump r e o r i e n t a t i o n model with u = 0.02 A and

0.05 A for the 77 K and 150 K measurements, r e s p e c t i v e l y . At 150 K a 6 - or

12-fold r e o r i e n t a t i o n cannot be excluded e n t i r e l y , but i t is improbable i n

view of the 77 K r e s u l t s . Apparently methyl-methyl i n t e r a c t i o n s dominate the

hindering of the group. The accuracy of the experiment was not good enough to

be able to decide whether there are one or more exponentials p r e s e n t in the

F i g . 2 .

E l a s t i c i n t e n s i t y as function of

momentum t r a n s f e r for the

measure-ments a t 77 K (crosses) and 150 K

( c i r c l e s ) . The v e r t i c a l bars i n d i c a t e

the u n c e r t a i n t y in the 77 K

measure-ment. Also drawn are the r e s u l t s of

some junp r e o r i e n t a t i o n models (d =

1.03 X). The f u l l and d o t t e d l i n e s

give the long time l i m i t of l 3 ( K , t ) ,

for u = 0 A and u = 0.04 S , r e s

p e c t i v e l y . The dotdash l i n e i n d i c a

-t e s -the long -tijne l i m i -t of I , ( K , -t )

for u = 0 A . The arrows i n d i c a t e

the p o s i t i o n s of coherent (Bragg)

s c a t t e r i n g from the sample, t h e i r .

length being a rough c l a s s i f i i r a t i o n

in s t r o n g , medium and weak r e f l e c

-t i o n s .

time dependence of I ( K , t ) . A f i t with one e x p o n e n t i a l - ( 3 fold jumps) y i e l d e d

T = 0 . 8 + 0.1 ps at 77 K and T = 0.4 + 0.1 ps at 150 K.

The r e s i d e n c e time a t 45 K, provided that the model s t i l l a p p l i e s , i s 30 ps

or l o n g e r . The temperature behaviour of T between 45,K and 150 K i s d i f f e r e n t

from t h a t expected for a c l a s s i c a l t e m p e r a t u r e - a c t i v a t e d p r o c e s s . A more

d e t a i l e d investigation~of t h i s behaviour between 30 K and 300 K has been

s t a r t e d . The spectrum a t 9 K appeared to be only p a r t l y r e s o l v e d , making a

d i r e c t accurate f i t t o the i n e l a s t i c p a r t impossible.

2 3 wave-vector tronsfer/A'

ACKNOWLEDGEMENT

We are indebted to dr. R.B. Helmholdt for the calculation of the Brag scattering intensities.

REFERENCES

[ 1 ] E.O. S c h l e m p e r and W.C. H a m i l t o n , I n o r g . C h e m . 5 ( 1965)995. [ 2 ] J . J . Rush and W.C. H a m i l t o n , I n o r g . Chem. 5 ( 1 9 6 6 ) 2 2 3 8 . [ 3 ] L.A. de G r a a f and P . V e r k e r k , t o b e p u b l i s h e d .

[ 4 ] J . D . Bregman and F . F . M , de Mul, N u c l . I n s t r . M e t h . 9 3 ( 1971)109. [ 5 ] C h r . S t e e n b e r g e n and L.A. d e G r a a f , t o b e p u b l i s h e d .

[ 6 ] C . T . Chudley and R . J . E l l i o t t , P r o c . P h y s . S o c . L o n d . 77(1961) 3 5 3 , K. S k ö l d , J . C h e m . P h y s . 4 9 ( 1 9 6 8 ) 2 4 4 3 .

Neutron s c a t t e r i n g study of the methyl-group r e o r i e n t a t i o n s and the low

temperature phase t r a n s i t i o n i n (CH ) SnF

Chr. Steenbergen

I n t e r u n i v e r s i t a i r Reactor I n s t i t u u t

Mekelweg 15, 2600 GA Delft

the Netherlands

and

J . J . Rush

National Measurement Laboratory

National Bureau of Standards

Washington, D.C. 20234, USA.

SUMMARY

The r e o r i e n t a t i o n of the methyl groups in s o l i d , (CH)„SnF„ has been i n v e s t i g a t

-ed by neutron q u a s i - e l a s t i c s c a t t e r i n g . I t i s found t h a t a phase t r a n s i t i o n

occurs in the s o l i d a t about 70K, which might be of second o r d e r . The shape of

the measured q u a s i - e l a s t i c l i n e s can be described by i n s t a n t a n e o u s t h r e e - f o l d

jumps of the methyl groups. From the derived residence times as a function of

temperature below the phase t r a n s i t i o n , an a c t i v a t i o n energy E /k = 250 K and

a residence time at i n f i n i t e temperatures T„ = 0.10 ps have been determined.

Above the phase t r a n s i t i o n the a c t i v a t i o n energy i s at l e a s t a f a c t o r 2 smaller.

INTRODUCTION R e c e n t l y a number of q u a s i - e l a s t i c and i n e l a s t i c n e u t r o n - s c a t t e r i n g s t u d i e s h a v e b e e n p e r f o r m e d on s m a l l m o l e c u l e s and m o l e c u l a r g r o u p s l i k e m e t h a n e , am-m o n i u am-m - i o n s and am-m e t h y l g r o u p s . At t e am-m p e r a t u r e s c o am-m p a r a b l e w i t h t h e b a r r i e r t o r e o r i e n t a t i o n of s u c h an e n t i t y t h e m e a s u r e m e n t s can be d e s c r i b e d [ 1 , 2 ] v e r y w e l l by a jump r e o r i e n t a t i o n m o d e l , w h i c h assumes l i b r a t i o n a l m o t i o n s a l t e r -n a t e d by i -n s t a -n t a -n e o u s jumps b e t w e e -n d i f f e r e -n t e q u i l i b r i u m p o s i t i o -n s . At low t e m p e r a t u r e s h o w e v e r , t h e r o t a t i o n a l e n e r g y l e v e l s become w e l l d e f i n e d and t r a n s i t i o n s b e t w e e n t h e l e v e l s i n t h e s p l i t g r o u n d s t a t e i n p r i n c i p l e can b e m e a s u r e d [ 3 , 4 , 5 ] . As t h e s e s p l i t t i n g s r e d u c e s t r o n g l y w i t h i n c r e a s i n g r e o r i e n t a t i o n a l b a r r i e r , g r o u p s w i t h a low p o t e n t i a l b a r r i e r a r e of p a r t i c u l a r i n t e -r e s t . I t i s known [ 6 , 7 ] t h a t t h e m e t h y l g r o u p s i n d i m e t h y l t i n d i f l u o r i d e h a v e a v e r y low b a r r i e r t o r e o r i e n t a t i o n i n t h e t e m p e r a t u r e r e g i o n above 77K. T h i s i s due t o t h e c r y s t a l s t r u c t u r e ( d e t e r m i n e d by X - r a y d i f f r a c t i o n a t 300 K) [ 8 ] , w h i c h i s t e t r a g o n a l w i t h a = 4 . 2 4 A and c = 14.16 A and two f o r m u l a u n i t s p e r u n i t c e l l . I t c o n s i s t s of an i n f i n i t e t w o - d i m e n s i o n a l n e t w o r k of t i n a t o m s and b r i d g i n g f l u o r i n e a t o m s , w i t h t h e m e t h y l g r o u p s above and b e l o w t h i s p l a n e , c o m p l e t i n g o c t a h e d r a l c o o r d i n a t i o n of t h e t i n a t o m s . The h i g h l y s y m m e t r i c e n -v i r o n m e n t of f l u o r i n e and t i n atoms and t h e l a r g e s p a c i n g b e t w e e n n e i g h b o u r i n g m e t h y l g r o u p s r e s u l t i n a low b a r r i e r t o r e o r i e n t a t i o n .

I n view of t h e e a r l i e r r e s u l t s [ 7 ] , a d e t a i l e d s t u d y of t h e r e o r i e n t a t i o n of t h e m e t h y l g r o u p s i n ( C H , ) . S n F . a t t e m p e r a t u r e s b e l o w 77K seemed i n t e r e s t i n g b e c a u s e of t h e p o s s i b l e o b s e r v a t i o n of t h e g r a d u a l t r a n s i t i o n from c l a s s i c a l jump r e o r i e n t a t i o n t o t h e quantum m e c h a n i c a l r e g i m e . The m e a s u r e m e n t s p r e s e n t

-ed h e r e , h o w e v e r , show-ed t h a t i n t h i s compound a p h a s e t r a n s i t i o n o c c u r s b e t w e e n 78K and 61K. Below t h i s t r a n s i t i o n t h e b a r r i e r t o r e o r i e n t a t i o n of t h e m e t h y l g r o u p s becomes l a r g e r and t h e r e s o l u t i o n of t h e s p e c t r o m e t e r a p p e a r e d i n s u f f i c i e n t t o r e s o l v e any t u n n e l s p l i t t i n g . T h e r e f o r e , we were f o r c e d t o r e s t r i c t o u r s e l v e s m a i n l y t o t h e o b s e r v a t i o n of t h e c l a s s i c a l , t h e r m a l l y a c t i v a t e d , r e o r i e n t a t i o n b e h a v i o u r of t h e m e t h y l g r o u p s . THEORY I n a n e u t r o n s c a t t e r i n g e x p e r i m e n t on a h y d r o g e n o u s compound t h e d o m i n a n t c o n t r i b u t i o n i s from t h e h y d r o g e n atoms b e c a u s e of t h e i r l a r g e i n c o h e r e n t s c a t t e r -i n g l e n g t h b . The t o t a l c o h e r e n t s c a t t e r -i n g c o n t r -i b u t -i o n does n o t e x c e e d f -i v e p e r c e n t o f t h e t o t a l s c a t t e r i n g and i s m a i n l y l o c a t e d a t p a r t i c u l a r v a l u e s o f t h e w a v e - v e c t o r t r a n s f e r e e (Bragg s c a t t e r i n g ) . The d o u b l e d i f f e r e n t i a l c r o s s

s e c t i o n for incoherent s c a t t e r i n g i s given by

d4^ = '^'''^ï'^s(^'") ' • . (')

where k. and k a r e the magnitudes of the i n c i d e n t and the s c a t t e r e d neutron

wave vectors r e s p e c t i v e l y - , and S (K,tü) is the s e l f p a r t of the s c a t t e r i n g

function at energy t r a n s f e r tico and wave-vector t r a n s f e r K_.

At high temperatures the methyl group motion can be described by the jump

r e o r i e n t a t i o n model [ 1 , 9 ] . In t h i s model a methyl group, having a c e r t a i n

o r i e n t a t i o n at t = 0, jumps a f t e r a mean residence time T to an equivalent

o r i e n t a t i o n . A f t e r averaging over a l l random o r i e n t a t i o n s ( p o l y c r y s t a l l i n e

sample) of the groups, the s c a t t e r i n g function for a t h r e e - f o l d b a r r i e r

becomes [10]

S^(K:,tü) = exp(-K2u^) I j (1+2 JQ(Kd)) S (w)

H- ^ ( l - j „ ( K d ) ) ^^^'^ I , (2)

^"^ ° (3/2T)^+a.^ >

where d is the jump distance and u^ is the mean squared amplitude of the methyl

group in the h i n d e r i n g p o t e n t i a l . A s i x f o l d b a r r i e r y i e l d s a s i m i l a r e x p r e s

-s i o n with a d e l t a function and t h r e e Lorentzian-s. The i n t e n -s i t y f a c t o r -s of the

d e l t a function ( e l a s t i c l i n e ) for these two models are e f f e c t i v e l y i d e n t i c a l

for <•< 1.8 A , and only for K > 2.5 A can they be d i s t i n g u i s h e d c l e a r l y .

Usually the temperature dependence of the residence time i s described by the

Arrhenius-law for a thermally a c t i v a t e d process

T = TQ exp (E^/kg T) . (3)

where E is the a c t i v a t i o n energy and k„ Boltzmann's constant. E i s r e l a t e d to

_{a '=•' B a}

the height of t h e h i n d e r i n g p o t e n t i a l and can be defined as the average energy

r e q u i r e d to cross the b a r r i e r . The residence time a t i n f i n i t e temperature T„

can be seen as t h e inverse of the ground-state frequency in the t o r s i o n a l

p o t e n t i a l ( i . e . the frequency with which a methyl group t r i e s to cross the

b a r r i e r ) .

The s c a t t e r i n g function of a methyl group at low temperatures (kT«E ) c o n s i s t s

of sharp peaks, o r i g i n a t i n g from t r a n s i t i o n s between the t o r s i o n a l l e v e l s of

the group in i t s h i n d e r i n g p o t e n t i a l . The l e v e l scheme can be c a l c u l a t e d from

t h e Schrodinger equation in one dimension

21 da^

V(a)iJ; = Eijj

( 4 )

where I i s the moment of i n e r t i a of the methyl group, V(a) the angle dependent

p o t e n t i a l with r o t a t i o n angle a, ijj the eigenfunction and E the (energy)

eigen-value of the r o t o r . Eigeneigen-values in t h r e e - and s i x - f o l d cosine p o t e n t i a l s with

V(a) = 4- V (1-cos na) are given in the l i t e r a t u r e [ 1 1 , 1 2 ] , and are p l o t t e d in

Fig. 1 for a l i m i t e d range of V and E . The s p l i t t i n g s of the ground s t a t e

l e v e l s (the tunnel s p l i t t i n g s ) for these two p o t e n t i a l shapes d i f f e r

dramatical-ly for a given b a r r i e r h e i g h t , and hence provide a s e n s i t i v e probe for the

sym-metry of the p o t e n t i a l .

V./meV ZOO S

Fig. 1.

E n e r g y l e v e l s of a m e t h y l group i n t h e p o t e n t i a l J V ( 1 - c o s na) f o r n = 3 ( f u l l l i n e s ) and n = 6 ( d a s h e d l i n e s ) . The d o t - d a s h l i n e i n d i c a t e s t h e top of t h e p o t e n t -i a l . E n e r g y l e v e l s a r e c a l c u l a t e d w i t h I = 5 . 4 * lO"'^^ k g m^. EXPERIMENTS

The m e a s u r e m e n t s w e r e done u s i n g c o m m e r c i a l l y a v a i l a b l e (CH ) „ S n F , powder i n an a l u m i n i u m s a m p l e c o n t a i n e r ( f l a t p l a t e g e o m e t r y ) , w h i c h was mounted i n a h e l i u m flow c r y o s t a t . N e u t r o n s p e c t r a w e r e t a k e n on t h e BT-4 t r i p l e a x i s s p e c t r o m e t e r a t t h e NBS r e a c t o r . The a n g l e b e t w e e n sample and i n c o m i n g n e u t r o n beam

was about 45 , r e s u l t i n g in a transmission of about 0.84 for 4 meV neutrons

and 0.89 for 15 meV n e u t r o n s . Multiple s c a t t e r i n g i s not expected to have a

s i g n i f i c a n t e f f e c t on the r e s u l t s for these transmission v a l u e s .

The f i r s t p a r t of the measurements was c a r r i e d out using a fixed incident

energy of 14.74 meV, obtained from the 004 planes of a p y r o l y t i c graphite

mono-chromator. A p y r o l y t i c graphite f i l t e r reduced the orde.r contamination, which

mainly c o n s i s t e d of 33.17 meV n e u t r o n s , to 18%. After s c a t t e r i n g , the energy

d i s t r i b u t i o n of .the neutrons was determined using the 002 planes of another

p y r o l y t i c graphite c r y s t a l .

800

600

c E ^ 4 0 0 c i 200

I 0

S 400 2 0 0 -0 200 0 200 • T • ^i • •

-•

I = 78K = 14.74 meV . . . - • ^ ^öe-ao"^

. .-....••••

,^'

. • • o^""

••

'\

, ^

'

. » -.

VA*

K--\.z6ir' •

-' ^ • • • • — . «r=I.88A"' • " " " • • ' " - . . . K = Z.56ir' • "'''--"I - , K=3.80&"' • 1 * 1 * 1

Fig. 2.

Q u a s i - e l a s t i c s c a t t e r i n g s p e c t r a

of (CH ) SnF^ a t 78K for various

wave-vector t r a n s f e r s . The

energy t r a n s f e r i s p o s i t i v e for

neutron energy l o s s . The f u l l

l i n e i n d i c a t e s the r e s o l u t i o n .

- 6 - 4 - 2 0 2 4 6 energy transfer/ meV

In order to be able to compare the present measurements with previous ones [ 7 ] ,

q u a s i e l a s t i c s c a t t e r i n g measurements were f i r s t done a t 78K. From a d i f f r a c t

-ion p a t t e r n , taken a t t h i s temperature, four values of the wave-vector t r a n s f e r K,

where no Bragg s c a t t e r i n g was observed, were s e l e c t e d . The measured q u a s i

-e l a s t i c s p -e c t r a ar-e display-ed in Fig. 2. Each sp-ectrum shows a v-ery broad

Lórentzian underneath a r e s o l u t i o n broadened d e l t a function, i n d i c a t i n g f a s t

r e o r i e n t a t i o n of the methyl groups. A s i m i l a r experiment a t 45 K, however,

revealed no broadened Lorentzian a t a l l within the applied r e s o l u t i o n (fwhm =

0.65 meV). Because t h i s r e l a t i v e l y sudden change in the s p e c t r a might i n d i c a t e

a phase t r a n s i t i o n , a d i f f r a c t i o n p a t t e r n was measured a t 45K t o compare with

the 78K data. The two d i f f r a c t i o n measurements showed s i g n i f i c a n t differences

only in a narrow region around 3.35 A , whereas wave-vector t r a n s f e r s up to

3.8 X were covered. This region was measured for varying temperatures with

zero energy t r a n s f e r and the r e s u l t s are displayed in Fig. 3.

T'=78K1 . T=59Kt ,T = 54Kt 3.4 3.3 34 33 3.4 wove-vector transfer/i"

F i g ; 3 .

Diffraction p a t t e r n ,measurai

with zero energy t r a n s f e r

and in a small wave-vector

t r a n s f e r range, for d i f f e

-rent temperatures. The

arrows i n d i c a t e whether the

temperature was reached by

cooling or h e a t i n g of the

s amp 1 e.

From these r e s u l t s i t i s c l e a r , considering the growth of a peak a t ' 3 . 3 2 A

and the gradual s p l i t t i n g of the peak a t 3.38 A when the temperature i s

lowered, t h a t a phase t r a n s i t i o n occurs between 61 and 78K.

I t was then decided to concentrate the experiments mainly on the observation

of the q u a s i - e l a s t i c s c a t t e r i n g below t h i s t r a n s i t i o n . Q u a s i - e l a s t i c

measure-ments were done with 14.74 meV neutrons and a r e s o l u t i o n of 0.65 meV (fwhm) at

78, 59, 55 and 45 K. In the s p e c t r a , obtained below 60K a c l e a r s e p a r a t i o n

between d e l t a function and Lorentzian cannot be observed, but a broadening i s

s t i l l p r e s e n t at 59K and 55K.

For the second p a r t of the measurements the 002 planes of the monochromator

wereused to obtain a fixed i n c i d e n t energy of 3.69 meV. The wavelength of the

i n c i d e n t neutrons i s beyond the Bragg cut-off for aluminium, and t h e r e f o r e the

background s c a t t e r i n g from container and c r y o s t a t i s v i r t u a l l y eliminated. A

cooled p o l y c r y s t a l l i n e beryllium f i l t e r (-15 cm long) e f f e c t i v e l y removed

neutrons with energies above 5.2 meV from the primary beam. The energy d i s t r i

-bution of the s c a t t e r e d neutrons was measured, using the 002 planes of a

pyro-l y t i c g r a p h i t e c r y s t a pyro-l .

Since a lower i n c i d e n t energy l i m i t s the observable wave-vector t r a n s f e r range,

another s e t of wave-vector t r a n s f e r s had t o be chosen for the q u a s i - e l a s t i c

s c a t t e r i n g measurements. Scans at 45.2 and 38.2 K were made, using a r e s o l u t i o n

of 79 and 56 peV r e s p e c t i v e l y . From Fig. 4, where the measurement at 38.2 K is

displayed, i t can be concluded t h a t even at such a low temperature a d e t e c t

-able broadening of the l i n e s i s s t i l l p r e s e n t .

800 600 400 200 O 400 200 0 400 200 0 200 0 T = 38.2K • E| = 3.69meV

..••"

...--y

' '*

'

•'f\

' \

K-'OTOX"' f l 3 6 A " ' ' if = l.88A"' " -c = 2,l4«"'

-<^--, . .,. .

Fig. 4.

Q u a s i - e l a s t i c s c a t t e r i n g

s p e c t r a of (CH,),SnF. a t

38.2 K for d i f f e r e n t

wave-vector t r a n s f e r s .

0.2 -0.1 0 0.1 0,2 energy transfer/meV T = 5.2 K E| = 3 6 9 meV = 1884"' 0 02 04 energy tronsfer/meV

F i g . 5 .

Spectrum of (CH„)2SnF2 at

5.2 K and 1 .88 ft~^

wavevector t r a n s f e r . The r e s o

l u t i o n function i s i n d i c a t

-ed by the f u l l l i n e .

The attempt t o measure a t r a n s i t i o n between the l e v e l s in the s p l i t ground

s t a t e of the methyl groups f a i l e d . A scan with 79 peV r e s o l u t i o n at 5.2 K and

1.88 A wave-vector t r a n s f e r (Fig. 5) revealed no i n e l a s t i c l i n e s up to 0.5

meV energy t r a n s f e r . Also the e l a s t i c peak, measured with 56 yeV r e s o l u t i o n ,

showed no significant deviation from the resolution function, which was deter-mined from a measurement on vanadium.

ANALYSIS

The d i f f r a c t i o n measurement at 45 K was not performed with s u f f i c i e n t accuracy

over a wide enough K range to allow a determination of t h e c r y s t a l s t r u c t u r e

below the phase t r a n s i t i o n . S t i l l some information about t h i s t r a n s i t i o n can be

obtained from the d i f f r a c t i o n measurements at zero energy t r a n s f e r a s a function

of temperature ( F i g . 3 ) . The incoherent background, taken as the i n t e n s i t y at

the base of the peaks, r i s e s with decreasing t e m p e r a t u r e , as displayed in

Fig. 6, due to the concentration, of the q u a s i - e l a s t i c s c a t t e r i n g around zero

energy t r a n s f e r . Within the small energy window, determined by the r e s o l u t i o n

of the spectrometer, the amount of s c a t t e r e d neutrons w i l l increase with the

narrowing of the q u a s i - e l a s t i c l i n e at lower t e m p e r a t u r e s [ 1 3 ] . From the r e s u l t s

in Fig. 6, we can only s e t an upper l i m i t of 72 K and a lower limit of 61 K

for the t r a n s i t i o n temperature, because the behaviour of t h e incoherent

back-ground above the phase t r a n s i t i o n cannot be estimated a c c u r a t e l y and a jump in

the q u a s i - e l a s t i c l i n e - w i d t h (and hence in the incoherent background) a t t h i s

t r a n s i t i o n cannot be excluded. The gradual broadening and f i n a l l y the s p l i t

-t i n g of -the d i f f r a c -t i o n peak a-t 3.38 A as -the -tempera-ture i s lowered, and

the absence of a s i g n i f i c a n t h y s t e r e s i s in t h i s p r o c e s s , i n d i c a t e a second

order or nearly second order phase t r a n s i t i o n . The r i s i n g i n t e n s i t y around

3.32 A could provide a s t r a i g h t f o r w a r d check of t h i s s u p p o s i t i o n .

Unfortun-Fig. 6.

Temperature dependence of

the incoherent background,

determined from the

diff r a c t i o n measurements d i s

-played in Fig. 3. The dashed

l i n e i s merely a guide to

the e y e .

6 300 2 0 0 -50 60 70 temperoture/K

a t e l y the accuracy and the l i m i t e d number of the measurements do not allow a

d e f i n i t e conclusion about the teitperature dependence of the i n t e n s i t y of t h i s

peak.

I t has been shown previously [7] t h a t t h r e e - f o l d r e o r i e n t a t i o n of the methyl

groups i s the dominant c o n t r i b u t i o n to the q u a s i - e l a s t i c s c a t t e r i n g above the

phase t r a n s i t i o n . Since the s i t e symmetry of the methyl groups below the

. t r a n s i t i o n temperature must be equal or lower than the symmetry above t h i s

t e m p e r a t u r e , something close to t h r e e - f o l d jump r e o r i e n t a t i o n i s the only

p o s s i b l e mechanism t h a t can account for the broadening of the e l a s t i c l i n e s at

low temperatures. Therefore, the q u a s i - e l a s t i c s c a t t e r i n g r e s u l t s at a l l

tempe-r a t u tempe-r e s and wave-vectotempe-r t tempe-r a n s f e tempe-r s wetempe-re f i t t e d to a sum of a d e l t a function and

a s i n g l e Lorentzian (both r e s o l u t i o n broadened), which have an i n t e n s i t y r a t i o

as given by E q . ( 2 ) .

3

In t h i s procedure the c o r r e c t i o n s for analyser e f f i c i e n c y k cot 0 , where 0 i s

t h e Bragg angle of the analysing c r y s t a l , and for d e t a i l e d balance were taken

i n t o account. A c o r r e c t i o n for the s c a t t e r i n g c o n t r i b u t i o n from the aluminium

sample-container and cryostat' was omitted, because i t i s expected to be smaller

T=55K

J

« «-'258^

.L

T=45K K-.3B0i:'

Fig. 7.

Examples of f i t s with the

t h r e e - f o l d jump model to

q u a s i - e l a s t i c s c a t t e r i n g

d a t a , obtained at d i f f e r e n t

temperatures with an i n c i d e n t

energy of 14.74 meV. The

f u l l l i n e i n d i c a t e s the t o

-t a l f i -t , -the dashed l i n e -the

Lorentzian p a r t and the d o t

-dash l i n e the background.

2 4 - 2 0 2 energy transfer/meV

F i g . 8. Examples of f i t s w i t h the t h r e e - f o l d jump model to q u a s i - e l a s t i c s c a t t e r i n g d a t a , obtained w i t h an i n -c i d e n t energy of 3.69 meV. The t o t a l f i t i s i n d i c a t e d by the f u l l l i n e . The dashed and dot-dash l i n e s represent r e s p e c t i v e l y t h e Lorentzian p a r t and the background.

' 0 4 -0.2 0 Q2 04 -02 0 02 energy transfer/m^

than the statistical accuracy of the measurements. Some of the resulting fits are given in Fig. 7 and 8. The free parameters, applied in the fitting proced-ure, were the intensity of the neutron background, the residence time T(Eq.(2)) and the total intensity of the peak. The offset of the peak and the width of the resolution function (Gaussian shape) were checked to be constant, and equal to a measurement on vanadium, for all measurements with a given resolut-ion. In addition the intensity of the fast neutron background was measured to be constant for comparable measurements.

T/K 78 59 55 4 5 . 2 3 8 . 2 0 . 7 0 40-HO 1.36 0 . 7 8 + 0 . 0 6 19 + 1 63 + 6 wave-1.88 0 . 8 1 + 0 . 0 4 6 . 0 + 0 . 2 1 0 . 6 + 0 . 6 23 + 1 6 0 + 4 - v e c t o r 2 . 1 4 18+1 70+5 c r a n s f e r / A 2 . 5 8 0 . 7 4 + 0 . 0 4 6 . 4 + 0 . 3 1 1 . 4 + 0 . 7 3 . 8 0 0 . 6 1 + 0 . 0 7 6 . 0 + 0 . 4 9 . 4 + 0 . 8 T a b l e 1. R e s i d e n c e t i m e s T / p s o b t a i n e d from t h e q u a s i - e l a s t i c s c a t t e r i n g m e a s u r e m e n t s ( s e e t e x t ) . A T'45 2K 1 I d 36*" T-38.2K L^TOW T=45.2K 1C.2J4A-' T=382K<V488jri

uiJ

In Table 1 the residence times obtained are displayed along with t h e i r standard

d e v i a t i o n s . For each temperature these x values are approximately constant with

only two obvious d i s c r e p a n c i e s . The low value at T = 38.2 K and K = 0.70 8 i s

caused by the i n e v i t a b l e inaccuracy of a f i t to a r e l a t i v e l y sharp Lorentzian

with only about 6% of the h e i g h t of the d e l t a function. The low value at

T = 78 K and K = 3.80 A must be due to the inadequacy of the t h r e e - f o l d jump

model at high jump frequencies and for a low b a r r i e r to r e o r i e n t a t i o n . In t h a t

case an appreciable f r a c t i o n of the methyl groups occupies the (semi) f r e e

-r o t o -r l e v e l s (see Fig. 1), and thus the instantaneous jumps a-re mixed with

free r o t a t i o n . I t can be shown t h a t t h i s would give r i s e to a d i f f e r e n t s c a t

-t e r i n g law for large K. An i n -t e r p r e -t a -t i o n based upon s i x - f o l d jumps or jumps of

even higher symmetry can be excluded since t h i s r e o r i e n t a t i o n behaviour would

have given much more dramatic changes in the e x t r a c t e d x values with r i s i n g K.

From the data in Table 1, omitting the two u n r e l i a b l e v a l u e s , an average i for

each temperature has been determined. A p l o t of t h e s e residence times as a

function of the inverse temperature i s given in Fig. 9. The values obtained

T/K 150 100 70 60 50 40 50.0 20.0 10.0 5 0 2 0 10 0 5

' .

o

'

B

' ' ' ' D'

/

/

/

/

/

A

/

V

/

F i g . 9 . S e m i - l o g a r i t h m i c p l o t of t h e r e s i d e n c e t i m e x a s a f u n c t i o n of t h e i n v e r s e t e m p e r a t u r e . The r e c t a n g l e s a r e t h e v a l u e s o b t a i n e d i n t h i s i n v e s t i g a t i o n and t h e open c i r c l e s a r e from Ref. 7 . The d a s h e d l i n e r e -p r e s e n t s E q . ( 3 ) w i t h E / k = 250 K and ! „ = 0 . 1 0 p s . 15 20 25 1000. T7K"' 21

from R e f . [ 7 ] a r e included a l s o . From t h i s figure i t i s concluded t h a t below the

t r a n s i t i o n temperature the r e o r i e n t a t i o n of the methyl groups can be described

by thermally a c t i v a t e d jumps with E /k = 250 + 30K and TQ = 0.10 + 0.02 ps

( E q . ( 3 ) ) . If the a c t i v a t i o n energy i s roughly i n t e r p r e t e d as the d i s t a n c e of

the t o r s i o n a l ground l e v e l to the top of a t h r e e - f o l d b a r r i e r (see F i g . 1 ) ,

t h i s b a r r i e r w i l l be about 320 K. Above the t r a n s i t i o n temperature the a c t i v

-a t i o n energy is -approxim-ately h-alved.

From t h e dependence of the t o t a l i n t e n s i t y of the q u a s i - e l a s t i c s p e c t r a from

2

the wave-vector t r a n s f e r , the mean squared amplitude u of the protons about

t h e i r e q u i l i b r i u m p o s i t i o n s can be e x t r a c t e d (Fig. 10). I t should be noted

2

t h a t the determined u does not s o l e l y c o n s i s t of a c o n t r i b u t i o n from t h e

l i b r a t i o n s of the methyl group ( E q . ( 2 ) ) , but t h a t c o n t r i b u t i o n s from l a t t i c e

2

v i b r a t i o n s and v i b r a t i o n s of the CH bond are included as w e l l . At 78K u =

0.059+0X)03 &^ i s found, while below 60K i t appears t h a t u = 0.O37+ODO3X .

From t h e s e values i t i s obvious t h a t at the phase t r a n s i t i o n a rapid change in

2

u o c c u r s , but t h i s change i s not n e c e s s a r i l y discontinuous. A l a r g e p a r t of

2

the difference in u below and above the t r a n s i t i o n must be a t t r i b u t e d t o the

D

>>

D

15

3

-^ - a -

4L

_

__ A ~~ —+-^ - —+-^ - - —+-^ , _ —+-^ —+-^ — - —+-^ - j —+-^

10

C7A-^

F i g . 10.

Semi-logarithmic p l o t of the i n t e n s i t y of the q u a s i - e l a s t i c peaks as a function

of K^. The symbols r e p r e s e n t data at d i f f e r e n t temperatures; dots - 78 K,

crosses - 59 K, open c i r c l e s - 55 K, plusses - 45 K, t r i a n g l e s - 45.2 K and

2

a s t e r i s k s - 38.2 K. From the slope of the dashed l i n e s u i s determined to be

0.059+0.0038^ at 78K and 0.037+0.003 8^ below 60K.

change i n r o t a t i o n a l b a r r i e r , and h e n c e i n t h e l i b r a t i o n a l a m p l i t u d e of t h e methyl g r o u p s . The n o n - e x i s t e n c e of a w e l l r e s o l v e d t r a n s i t i o n b e t w e e n t h e t o r s i o n a l g r o u n d s t a t e l e v e l s o f t h e m e t h y l g r o u p s a t 5.2 K ( F i g . 5 ) , and t h e a b s e n c e of s i g n i -f i c a n t d e v i a t i o n s -from t h e i n s t r u m e n t a l r e s o l u t i o n i n t h e s c a n w i t h 56 peV r e s o l u t i o n , l e a d t o t h e c o n c l u s i o n t h a t t h e t u n n e l s p l i t t i n g ( s ) must be l e s s t h a n 20 yeV. I n t h i s e s t i m a t e i t i s t a k e n i n t o a c c o u n t , t h a t a t a w a v e - v e c t o r t r a n s f e r of a b o u t 1.9 A t h e t o r s i o n a l - g r o u n d - s t a t e t r a n s i t i o n s u s u a l l y h a v e an i n t e n s i t y of 10-20% of t h e e l a s t i c i n t e n s i t y [ 3 , 1 3 , 1 4 ] . The 20 peV maximum f o r t h e t u n n e l s p l i t t i n g l e a d s t o a n - e s t i m a t e o f 290 K f o r t h e l o w e s t p o s s i b l e b a r r i e r of a t h r e e - f o l d c o s i n e p o t e n t i a l (Eq. ( 4 ) , F i g . 1 ) . A s i x - f o l d b a r r i e r would h a v e t o be much h i g h e r t h a n 1000 K i n o r d e r t o c o r r e s p o n d w i t h s u c h a s m a l l t u n n e l s p l i t t i n g , a n d , t h e r e f o r e , i t i s r u l e d o u t c o n s i d e r i n g t h e o b s e r v e d E , a DISCUSSION In t h e p r e c e e d i n g s e c t i o n s we h a v e p r e s e n t e d e v i d e n c e f o r t h e e x i s t e n c e of a phase t r a n s i t i o n i n d i m e t h y l t i n d i f l u o r i d e b e t w e e n 61 and 72K. The o b s e r v e d changes i n t h e powder d i f f r a c t i o n p a t t e r n and i t s t e m p e r a t u r e b e h a v i o u r ( F i g . 3 ) i n p a r t i c u l a r i n d i c a t e t h a t t h i s p h a s e t r a n s i t i o n p r o b a b l y i s of s e c o n d o r d e r . I n a r e c e n t s p e c i f i c - h e a t m e a s u r e m e n t [ 1 5 ] , p e r f o r m e d on o u r r e q u e s t , t h e second o r d e r b e h a v i o u r h a s b e e n c o n f i r m e d and a t r a n s i t i o n t e m p e r a t u r e of 70.5K was found. T h i s t r a n s i t i o n t e m p e r a t u r e i s a l s o c o n s i s t e n t w i t h a r e c e n t nmr e x p e r i m e n t [ 1 6 ] , i n w h i c h a c h a n g e i n t h e m e t h y l g r o u p b a r r i e r h e i g h t was d e t e c t e d a s w e l l . MÖssbauer e f f e c t s p e c t r a of t h i s compound r e v e a l no d i s c o n t i n u i t y i n t h e t e m -p e r a t u r e d e -p e n d e n c e of t h e v i b r a t i o n a n i s o t r o -p y of t h e t i n n u c l e u s [ 1 7 . 1 8 ] , which i m p l i e s t h a t t h e t i n v i b r a t i o n s a r e n o t a f f e c t e d by t h e p h a s e t r a n s i t i o n . T h e r e f o r e , t h e f o r c e s on t h e t i n a t o m s must r e m a i n c o n s t a n t , w h i c h i s c o n s i s t e n t w i t h t h e s e c o n d o r d e r c o n c e p t and i n d i c a t e s t h a t t h e p l a n a r t i n f l u o r i n e c o n -f i g u r a t i o n i s l i k e l y t o r e m a i n u n c h a n g e d a t t h e p h a s e t r a n s i t i o n . I t i s o b v i o u s t h a t a d e t e r m i n a t i o n of t h e l o w - t e m p e r a t u r e c r y s t a l s t r u c t u r e i s n e c e s s a r y t o gain more i n f o r m a t i o n on t h e s t r u c t u r a l a s p e c t s of t h e p h a s e t r a n s i t i o n . To t h i s e f f e c t a n e u t r o n powder d i f f r a c t i o n s t u d y i s p l a n n e d , i n c l u d i n g a more a c c u r a t e d e t e r m i n a t i o n of t h e t e m p e r a t u r e d e p e n d e n c e of t h e d i f f r a c t i o n p a t t e r n between 3 . 2 a n d 3 . 5 A

The d e c r e a s e of t h e mean s q u a r e d a m p l i t u d e of t h e h y d r o g e n atoms a t t h e p h a s e t r a n s i t i o n ( F i g . 1 0 ) must be p r e d o m i n a n t l y due t o a c h a n g e i n l i b r a t i o n (and

v i b r a t i o n ) amplitudes of the methyl groups, since the MÖssbauer effect

measure-ments did not reveal a deviation in the v i b r a t i o n s of the t i n atoms. A l a r g e

2

p a r t of the reduction of u must be caused by the change in r e o r i e n t a t i o n a l

2 . .

b a r r i e r . A c a l c u l a t i o n of u (one-dimensional) from the wave function ijJ i n

the ground s t a t e of a t h r e e - f o l d cosine p o t e n t i a l with V /k = 340 K ( E q . ( 4 ) )

•} J J J D

y i e l d s u = 0.030 A , while u in a s i m i l a r p o t e n t i a l with V /k < 150 K i s

• .. 3 B

estimated to be at l e a s t 0.045 A above 70 K (excited l e v e l s are s i g n i f i c a n t l y

? . . .

occupied here and t h e r e f o r e u~ i s i l l - d e f i n e d ) . Also the v i b r a t i o n of the

methyl group as a whole might a l t e r s u b s t a n t i a l l y at the phase t r a n s i t i o n .

The a v a i l a b l e data on the temperature dependence of the r e s i d e n c e time T

n e i t h e r confirm nor c o n t r a d i c t a second order phase t r a n s i t i o n (Fig. 9 ) .

Clearly more data are needed in the 6080 K temperature range. Neutron s c a t t e r

-ing measurements are i n progress at these and higher temperatures in o r d e r

to verify the second order concept.

The good f i t of the measurements to a t h r e e - f o l d jump model indicates t h a t the

main c o n t r i b u t i o n t o the r e o r i e n t a t i o n a l b a r r i e r has t h r e e - f o l d symmetry. This

i s supported by the absence of a s p l i t t i n g of the ground t o r s i o n a l l e v e l i n

excess of 20 yeV, which i s c o n s i s t e n t with the p o t e n t i a l h e i g h t V,/k^ a 320 K

e x t r a c t e d from E . ^ e ground t o r s i o n a l energy in t h i s p o t e n t i a l is about

^ -1

5.9 meV (implying a t o r s i o n a l frequency ii) = 9 rad ps ) , which yields an

estimate for T„ of 0.11 ps in very good agreement with the value determined

from the measurements T = 0. 10 + 0.02 p s . The d e t a i l e d shape of the p o t e n t i a l ,

however, cannot be derived from our measurements.

A f u r t h e r v e r i f i c a t i o n of the shape and height of the h i n d e r i n g p o t e n t i a l i s

p o s s i b l e from a d i r e c t measurement of the t r a n s i t i o n from t h e ground l e v e l to

the f i r s t e x c i t e d l e v e l . For a p o t e n t i a l with V-/k ' = 320 K an i n e l a s t i c peak

should be found a t 11 meV neutron energy t r a n s f e r . This energy t r a n s f e r range

was not covered in the present experiment, nor in previous experiments [ 7 ] ,

but i t w i l l be included in the new measurements mentioned above. Also a d i r e c t

measurement of the tunnel s p l i t t i n g , u t i l i z i n g a v e r y h i g h r e s o l u t i o n s p e c t r o

-meter [ 3 , 1 4 ] , could f u r t h e r e s t a b l i s h the p o t e n t i a l shape.

I ( K , t ) = S. (K,to) exp(iu)t) du) (2)

— ' m e —

which can be expressed as

1 ^

I(l£>t) = ^ I <exp{iK.r^(0)}exp{-iK.r. ( t ) } > (3)

i=l

where N is the number of the protons and r,(t) is the position operator of the

proton at time t.

This function describes the motion of the hydrogen atoms due to rotations and

librations of the ammonium ion, internal vibrations and vibrations of the

lattice. The development in time of the rotational part of I(K,t) for the

motion of a molecule or molecular group in a crystal usually is slow compared

to the time dependence of the librational and vibrational parts. Therefore, the

explicit time development of the thermal cloud due to librations and vibrations

need not be calculated. The effect of these motions is taken into account in

the Debye-Waller factor.

A simple and effective way of describing the rotational motion of the ammonium

ion is the application of jump reorientation models ' . In these models the

molecular group is assumed to librate about an equilibrium orientation during

an average residence time T,_and to jump between equilibrium sites in an

infi-nitely short time. Now the probability P.(t) for an H atom to be at a site i

can be expressed as:

P.(t)

= I a n

'exp(-b t/T) (4)

1 k=l "-^ ^

which i s the general s o l u t i o n of the well known r a t e equation for the jump

models:

d P . ( t ) _ n '

-—-' = (n'T) I p ( t ) - P . ( t ) (5)

j=l

Here n' is the number of sites from which a jump to site i is possible and n is

the total number of sites. The coefficients a., and b. can be calculated

straightforwardly and solutions for different symmetries have appeared in the

literature*"'. Eq. 3 can be written simply as

2 2 "

I(K,t) = exp(-u K )

I

P. (t)exp(ii<._r.) (6)

i=l "^

^

2 2

where exp(-u K ) i s the Debye-Waller f a c t o r and £ . the jump d i s t a n c e . Since i n

our experiments a p o l y c r y s t a l l i n e sample was used an averageing procedure over

a l l d i r e c t i o n s of < has t o be applied, which y i e l d s :

I ( K , t ) = exp(-u^<^) [ P . ( t ) j ^ ( < r . ) (7)

i=l

where j ( < r . ) = ( s i n K r . ) / K r . i s a s p h e r i c a l Bessel function.

•"o 1 1 1

2 . 3 . Jump-reorientation models.

As mentioned above the experiments are r e s t r i c t e d to the cubic phase of NH.ZnF .

For the symmetry of the NH, i o n , we now have t o consider the p o s s i b i l i t i e s of a

r e g u l a r or a d i s t o r t e d tetrahedron . This leads t o the following

jump-reorien-t a jump-reorien-t i o n models jump-reorien-t h a jump-reorien-t were u l jump-reorien-t i m a jump-reorien-t e l y used in jump-reorien-the d a jump-reorien-t a a n a l y s i s .

( i ) Model I : The NH group i s a r e g u l a r o r s l i g h t l y d i s t o r t e d tetrahedron and

each H atom can occupy 24 p o s s i b l e e q u i l i b r i u m p o s i t i o n s . These p o s i t i o n s may

be arranged in 12 p a i r s , where the s i t e s of a p a i r are at equal distance

(5 0.18 A) from the l i n e connecting the c e n t r a l n i t r o g e n with a fluorine atom.

This d i s t a n c e i s comparable t o the v i b r a t i o n a l amplitude of the p r o t o n s .

More-over, models with 12 o r more equivalent e q u i l i b r i u m p o s i t i o n s tend t o have

s o l u t i o n s t h a t a r e almost i d e n t i c a l and equal t o t h e r o t a t i o n a l diffusion l i m i t .

Therefore, we take as an approximation 12 e q u i l i b r i u m p o s i t i o n s s i t u a t e d at t h e

N-F l i n e s . Then t h i s model i s equivalent t o t h e s o - c a l l e d [110] model in R e f . 6 .

The intermediate s c a t t e r i n g function can now be w r i t t e n as

I ( i c , t ) = exp(-u^K^) (AQ + Aj e"'^^'^+A2 e^'^'^^+A^ e"^'^^'^) (8)

where:

% = -iT " Ï JO^^") ^ I JQ^-^^Kd) + i Jo(/3Kd) + ^ Jo(2<d)

^ = i ^ho^^^^ - iJ0(^3<d) - | j o ( 2 K d ) _

*2 = I -1 JO^*''^'') * i JO^'^"^^

S'Tl'I^O^^^^ -\i0^^''''^ 4 J0(^3<d) - ^ j,(2Kd)

Here d i s the N-H d i s t a n c e . C l e a r l y , the time-independent p a r t of I(K,t) i s

given by A , which thus gives the e l a s t i c p a r t Idc,"") of I(K:,t) as a function

of K. Because the experimental data w i l l not be accurate enough t o make a f i t

to the sum of exponentials as in Eq. 8 u s e f u l , we approximate the

time-depen-dent p a r t of I ( < , t ) for small ,t as follows:

I ( K , t ) - !(<,»>) = (Aj + A^ + Aj) - (t/2T)(A| + 2k^ + 3Aj)

= (A| + Aj + Aj) e x p ( - t / 2 T ^ j j ) (9)

with a s i n g l e K-dependent decay time

, - 1

•^eff ° '^^^1 •" ^^2 "" AjX^i -^ 2A2 + 3Aj) (10)

This s h o r t - t i m e approximation can be shown to be v a l i d for t <<8T .

( i i ) Model I I : In t h i s model 90 - r e o r i e n t a t i o n s around only one two-fold

sym-metry axis of the NH, ion are taken i n t o accouijit, so t h a t each proton can jump

to four d i f f e r e n t p o s i t i o n s . Reorientations around o t h e r axes may occur too

but on a time s c a l e , which i s long compared to t h a t of the experiment. The

preference for r e o r i e n t a t i o n around one axis could be caused by a s l i g h t

d i s t o r t i o n of the NH, ion, reducing the moment of i n e r t i a for t h a t a x i s .

Because the degree of d i s t o r t i o n i s not known', two extreme cases are given:

Model I I . The NH group is u n d i s t o r t e d and I ( < , t ) i s determined to be:

I ( K , t ) = exp(-u^K^) (Bg + Bj <r^''^ + B^ e'^''''^) (11)

with BQ = I (1 + 2 jg(|/3icd) + JQ(|/6<d)}

B, - ^ {1 - JQ(|/6Kd)}

B^ = I {1 - 2 JQ(|/3Kd) + j p ( | / 6 K d ) }

Analogous to the former model we may c h a r a c t e r i z e the q u a s i - e l a s t i c p a r t by a

s i n g l e exponential with a < dependent residence time:

"^eff " ^^^1 •" ^2^ "*1 * ^ " 2 ^ " ' - ^'^^

tfcdel I I . The NH, group i s t e t r a g o n a l l y elongated with two 90 H-N-H angles and

two 120 ones. Here we find formulae, t h a t are analogous to (11) and (12) with

2

-^3d replaced by d.

(iii) Model III. As other multi-axis reorientation models one could consider models having 180 reorientations around the diad axes or 120 reorientations • around the triad axes of a regular tetrahedron. They result in*:

I ( K . - ) = { (1 + 3 J Q ( / 3 K d ) ) • (13) and a t i m e d e p e n d e n t p a r t of I ( K , t ) w i t h o n l y one e x p o n e n t i a l ( h e n c e T ^ ^ ^ = '^) • 3. EXPERIMENTAL DATA 3 . 1. S p e c t r o m e t e r The e x p e r i m e n t was p e r f o r m e d on t h e r o t a t i n g c r y s t a l t i m e o f f l i g h t s p e c t r o m e -t e r RKS-2 a -t -t h e r e s e a r c h r e a c -t o r (HOR) i n D e l f -t . T h i s s p e c -t r o m e -t e r h a s b e e n d e s c r i b e d i n d e t a i l e l s e w h e r e . To s e l e c t a p u l s e d m o n o c h r o m a t i c n e u t r o n beam we c h o s e a p y r o l y t i c g r a p h i t e c r y s t a l , w h i c h r o t a t e d a t a s p e e d of 19000 rpm. An i n c o m i n g w a v e l e n g t h of X = 1.60 A (E = 32 meV) was o b t a i n e d from t h e 006 p l a n e s a t a s c a t t e r i n g a n g l e of 90 by p h a s i n g two F e r m i c h o p p e r s and t h e r o t a t i n g p y r o l y t i c g r a p h i t e c r y s t a l . The two Fermi c h o p p e r s s e r v e t o s u p p r e s s l o w e r and h i g h e r - o r d e r r e f l e c t i o n s , and t o r e d u c e t h e b a c k g r o u n d o r i g i n a t i n g from t h e r m a l and f a s t n e u t r o n s c a t t e r i n g by t h e c r y s t a l . Less t h a n 0 . 8 % c o n t a m i n a t i o n by 1.2 A n e u t r o n s was p r e s e n t . The l e n g t h of t h e s a m p l e d e t e c t o r f l i g h t -p a t h was 1,44 m. The f l i g h t - -p a t h was f i l l e d w i t h h e l i u m gas i n o r d e r t o s u -p -p r e s s a i r s c a t t e r i n g p r o c e s s e s . 2 8 He d e t e c t o r s h a v e b e e n p l a c e d i n g r o u p s o f f o u r a t s e v e n d i f f e r e n t a n g l e s b e t w e e n 10 and 90 , c o r r e s p o n d i n g t o w a v e v e c t o r t r a n s -f e r s i n t h e r a n g e 0 . 6 8 - 5 . 5 2 A . C a r e was t a k e n t o a v o i d a n g l e s a t w h i c h B r a g g

s c a t t e r i n g m i g h t o c c u r . The d e t e c t o r s a t c|) = 90 t u r n e d o u t t o b e l a r g e l y c o v e r e d by s p e c t r o m e t e r s h i e l d i n g and t h u s gave v e r y p o o r s t a t i s t i c s . The d a t a w e r e a c c u m u l a t e d i n a T r i d a c m u l t i c h a n n e l a n a l y z e r , u s i n g 512 t i m e c h a n n e l s of 3 ps e a c h .

The s a m p l e was k e p t i n a f l a t a l u m i n i u m c o n t a i n e r and had a t r a n s m i s s i o n of a b o u t 88%. We h a v e c a r r i e d o u t e x p e r i m e n t s i n a t e m p e r a t u r e i n t e r v a l

80 K<T<300 K by u s i n g a n i t r o g e n f l o w c r y o s t a t . The b a c k g r o u n d was d e t e r m i n e d from a m e a s u r e m e n t w i t h t h e empty san5)le h o l d e r a t T = 120 K. A s p e c t r u m of NH.ZnF a t 8 5 . 5 K showed no b r o a d e n i n g a s compared w i t h a v a n a d i u m s a m p l e and t h e r e f o r e was t a k e n a s a r e s o l u t i o n m e a s u r e m e n t , f a c i l i t a t i n g n o r m a l i z a t i o n and c o r r e c t i o n p r o c e d u r e s .

The e n e r g y r e s o l u t i o n amounted t o 2 meV f u l l w i d t h a t h a l f maximum (FWHM), and t h e momentum t r a n s f e r r e s o l u t i o n t o 0 . 1 5 A FWHM.

3 . 2 . E x p e r i m e n t a l r e s u l t s and d a t a h a n d l i n g

The d a t a of e a c h g r o u p of f o u r d e t e c t o r s w e r e c o r r e c t e d f o r b a c k g r o u n d and d e t e c t o r e f f i c i e n c y . An e x a m p l e of a s o - o b t a i n e d t i m e - o f - f l i g h t (TOF) s p e c t r u m i s g i v e n i n F i g . 2. I n t h i s f i g u r e t h e q u a s i - e l a s t i c i n c o h e r e n t s c a t t e r i n g peak