Methyl group tunnelling studies in calixarenes

ELSEVIER

Physica B 202 (t994) 279-286

Methyl group tunnelling studies in calixarenes R. Caciuffo a'*, G. Amoretti b, C.J. Carlile c, F. Fillaux d, O. Francescangeli a, M. Prager e, F. Ugozzoli f aDipartimento di Scienze dei Materiali e della Terra, Sezione Fisica, Universitit di Ancona, Via Brecce Bianche, 1-60131 Ancona, ltaly bDipartimento di Fisica, Universitlt di Parma, 1-43100 Parma, ltaly ~Rutherford Appleton Laboratory, Chilton, Oxon 0 X l l OQX, UK d LASIR, CNRS, 2 rue Henry Dunant, 94320 Thiais, France elnstitut f ~ r Festk~rperforschung, KFA J~lich D-5170 J~lich, Germany f lstituto di Chimica Generale ed Inorganica, Unioersith di Parma, and Centro di Strutturistica Diffrattometrica CNR, 1-43100 Parma, Italy

Abstract Inelastic neutron scattering has been used to study the tunnelling of methyl groups belonging to several guest molecules (toluene, p-xylene, 7-picoline) incarcerated in a host calixarene matrix. In all the cases investigated, the low temperature neutron spectra show a number of bands which may be interpreted as being due to transitions between tunnel-split librational states of the guest methyl groups. The main line occurs near 0.63 meV, very close to the C H 3 quantum free rotor limit. In the toluence complex, the quantum regime persists at least up to 60 K. Effects of coupling between rotational and vibrational modes are discussed. Very subtle structural changes not revealed by diffraction measurements are suggested by the neutron spectroscopy results.

1. Introduction Molecular inclusion phenomena of organic guest molecules inside ordered matrices of natural origin such as cyclodextrins, and of synthetic origin such as crown ethers have been extensively studied in biomimetic chemistry, enzyme catalysis and analytical chemistry. The design and synthesis of molecular systems able to give selective inclusion of ions or neutral molecules is one of the major aims of h o s t - g u e s t or s u p r a m o l e c u l a r chemistry [1, 2]. Among the classes of synthetic matrices which are able to form inclusion complexes with neutral * Corresponding author.

organic molecules, a group of cup-shaped macrocyclic phenol-methylene oligomers have attracted much attention in the past few years and deserve particular mention. They are formed by the basecatalysed condensation of phenols and formaldehyde and are known as the calix I-n] arenes I-3, 4] where n refers to the number of molecules which comprise the calix. Until recently, the calixarenes have provided the only example of intramolecular cavity complexes formed by neutral molecules and, in perspective, they m a y become another milestone in receptor chemistry. Their most interesting property is their ability to incarcerate neutral and ionic guests in supramolecular three-dimensional arrays. Because of their potential for selective

0921-4526/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 0 2 0 5 - A

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R. Caciuffo et al./ Physica B 202 (1994) 279-286

complexation of alkali metal cations a n d / o r organic neutral molecules, the calixarenes may be used in various applications, from sensors with high ion selectivity to carriers of ions through different liquid media. In this paper, we report the results of several inelastic neutron scattering (INS) experiments performed to study the tunnelling of methyl groups belonging to several guest molecules (toluene, pxylene, y-picoline) incarcerated in a host p-tert butylcalix [4] arene matrix. It is known that the intramolecular potential barrier for internal rotations in the chosen guest molecules is extremely low and, therefore, if the reorientational barrier provided by the host lattice is also small, they may exhibit a low temperature quantum rotational spectrum. In fact, in all the cases investigated the C H 3 torsional dynamics approaches the limit of a quantum free rotor, and the neutron spectroscopy results presented here show that the calixarene complexes are among the best examples discovered up to now of systems where methyl groups freely rotating in a solid matrix are present.

2. Structural and experimental details The p-tert-butylcalix [4] arene complexes crystallise in the tetragonal P4/n space group and are characterised, at room temperature (RT), by a fourfold symmetry axis running through the centre of the hydrophobic cage formed by the host molecule [5]; a cyclic network of intramolecular hydrogen bonds stabilises the conformation of the tetramer which is called, following the nomenclature introduced by Gutsche [3], a cone. Such a conformation leaves a large intramolecular cavity suitable for the complexation of aromatic molecules such as benzene, toluene, p-xylene or y-picoline. Previous work has shown that the tert-butyl group of the host plays a crucial role in the complexation of aromatic guests which seems favoured by specific attractive CH3 ... n interactions between the methyl groups of the host and the aromatic rt orbital of the guest

[6]. The guest molecules are aligned with their C - C H 3 bond along the fourfold symmetry by the presence of two-fold dynamic disorder about the

long molecular axis [5]. A symmetry lowering of the entire host-guest unit can therefore occur at low temperature when the motion slows down to some critical rate. Fig. l(a) shows a view of the cage of the (1 : 1) toluene or y-picoline complex perpendicular to the four-fold axis. The RT lattice parameters for the toluene complex are a = 12.756(5)/~ and c = 13.792(6)~. In the case of the p-xylene complex, the guest molecule is held in a closed cavity formed by two calixarene units facing their tert-butyl groups (Fig. l(b)); the RT lattice parameters are a = 12.823(5)/~ and c = 25.618(6)/~. The inelastic neutron scattering experiments were carried out on the direct-geometry time-offlight (TOF) chopper spectrometers M I B E M O L at the Laboratoire L6on Brillouin (Saclay, France), IN5 at the Institut Laue-Langevin (Grenoble, France), and on the inverse-geometry T O F spectrometer IRIS at the Rutherford Appleton Laboratory (Chilton UK). INS spectra were recorded at different temperatures between 1.8 and 96 K, operating the spectrometers so as to provide different values for the energy resolution and explore different energy-transfer (he)) ranges. The highest resolution at the elastic line of 4 geV was achieved on IRIS using (0 0 4) mica crystal analysers. Measurements were made both on complexes with deuterated guest molecules and on different specimens of fully protonated compounds. All samples were prepared at the University of Parma and consisted of about 3 g of powder contained in flat AI cans, mounted into standard liquid-helium

(a)

(b)

Fig. 1. The intramolecular cavity of (a) the (1 : 1) complex of p-tert-butylcalix [4] arene with toluene or y-picoline and (b) the complex with p-xylene. The disordered p-xylene molecule is indicated by the van der Waals radii.

281

R. Caciuffo et al./ Physica B 202 (1994) 279-286

cryostats. T h e samples were shown to be single phase by X-ray diffraction.

energy resolution at the elastic position Aho9 of (a) 97 IxeV, (b) 37 peV). Several n a r r o w bands are visible between 0.16 and 0.7meV; a peak at ho~ = 1.92 m e V is also visible in the I N S spectrum obtained with the higher incident energy (21 = 5.2 A, Ahto = 97 peV). A b r o a d feature at 2.6 m e V is further identified, at 30 K, on the neutron-energy-gain side of the T O F spectrum. O n the other hand, there are no visible peaks in the corresponding crosssection recorded for the complex with toluene-ds. This confirms that the excitations observed in the p r o t o n a t e d sample correspond to torsional levels of

3. R e s u l t s a n d d i s c u s s i o n

3.1. p-tert-butylcalix [4] arene(1 ." 1) toluene

The t e m p e r a t u r e dependence of the neutron scattering spectrum m e a s u r e d for the fully p r o t o n a t e d c o m p o u n d on the I N 5 spectrometer is shown in Fig. 2 (incident wavelength (a) 2i = 5.2 A, (b) 7.2/~;

0.5

.

,

o

•

.

0.8

0.4 20K

0.6

0.3

2K

0.2

0.4

0.1

0.2

0

0

o.21=

I

o.i

"~ •

0.2

'8'

0

.

.

.

.

.

.

~'~

0.4

0.2

~

0

""

0.4

. . . . .

50 0

-

o.,

O.l

0.2

0

l

-3

-2

-1

0

Energy Transfer

(a)

0

I

2 (meV)

"

-0.8

-0.4

0

Energy Transfer

0.4

0.8

(meV)

Co)

Fig. 2. The temperature dependence of the INS function measured on the IN5 spectrometer for the (1 : 1) complex of p-tertbutylcalix[4]arene and toluene: (a) incident wavelength 2~ = 5.2/~, energy resolution at the elastic position of 97 peV; (b) incident wavelength 2i = 7.2 ,~, energy resolution at the elastic position of 37 peV.

282

R. Caciuffo et al./ Physica B 202 (1994) 279-286

the toluene C H 3 group. As the temperature increases, there is a continuous broadening of the inelastic lines up to about 60 K, where the quantum spectrum merges into the classical quasielastic one. The positions of the peaks are almost temperature independent, suggesting that the coupling between the CH3 rotational modes and the phonon bath is very weak. To analyse these results within the single-particle model, consisting of one-dimensional isolated internal rotors in a periodic hindrance potential, it is necessary to assume the presence of at least two inequivalent crystallographic sites, occupied by the toluene guest in different proportions [7]. In fact, numerical diagonalization of the corresponding Schr6dinger equation h2~2!Iv 21 ~(~2 t- I15 , ~N V3,(I -- cos(3n~b)) - E 1 ~ = 0,

(1) where I is the reduced moment of inertia of the methyl group and the rotational constant B = h2/2! --- 0.655 meV, shows that the dominant lines in the INS spectra can be accounted for by considering a site characterised by a six-fold potential with a small threefold contribution (V6 = 5.24 meV, V3 = 0.92 meV), and a second one having a pure threefold symmetry (II3 = 5.04 meV). The methyl groups in the latter site would then be responsible for the line at 0.40 meV, corresponding to the tunnel splitting of their ground torsional state, whilst the lines at 0.63, 1.92 and 2.6 may be interpreted as being due to torsional transitions of the methyl groups experiencing the six-fold barrier. The comparison with the experimental results is shown in Table 1. The weak bands visible below 0.4 meV are not accounted for and could correspond to additional sites. The reorientational barrier provided by the host lattice to the rotation of the methyl group is very small. It is therefore more appropriate to talk of free rotor levels rather than tunnel states. The energies of the lowest free rotational transitions for the methyl rotator are in fact 0.655, 1.965 and 2.62 meV, which are very close to the observed values. It is noticeable that the most intense band at 0.63 meV is, to our knowledge, the highest frequency ever reported for methyl tunnelling.

Table 1 Comparison between observed and calculated transition energies for the p-tert-butylcalix[4]arene (1:1) toluene complex. The calculations are performed in the framework of the single particle model assuming two inequivalent sites. Site (a) is characterised by a six-fold potential with parameters V6 = 5.24 meV and V3 = 0.92 meV; the potential on site (b) has threefold symmetry (V3 = 5.04 meV). Energy values are in meV units

Observed

Calculated

Site

0.16(1) 0.198(2) 0.36(2) 0.402(9) 0.630(2) 1.92(5) 2.60(1)

0.401 0.628 1.906 2.534

(b) (a) (a) (a)

According to the room temperature crystallographic structure, only one 12-fold site should be present. The tunnelling spectroscopy results therefore suggests the occurrence of a structural transition leading to guest-host units where the symmetry of the calix matches that of the toluene through the appearance of either a two-fold axis or a mirror plane. This symmetry reduction could be the result of a freezing of the toluene dynamical disorder about the long molecular axis. High resolution neutron diffraction measurements, performed on the H R P D diffractometer at ISIS, failed to detect any structural distortion down to 4.2 K but a clear evidence of symmetry lowering at 248 K was obtained by solid state 13C N M R and Differential Scanning Calorimetry measurements [8]. Alternative interpretations of the measured neutron spectra could be given assuming that no structural distortions occur but that instead the CH3 rotors are coupled [9]. The crystal structure provides no evidence of close pairs of methyl groups but rather suggests the presence of linear chains along the c-axis and two-dimensional square lattices parallel to the (a,b) plane. The C H a - C H 3 distances are almost equivalent along the three crystallographic directions and the strength of the coupling in the plain and along the chain should in principle be comparable. However, if the coupling along the chain is supposed to be significatively larger, the quantum sine-Gordon breather model

R. Caciuffo et al./ Physica B 202 (1994) 279-286

[10] can be applied. In this case, spatially localised incoherent excitations having a soliton character may occur in the chain and some of the observed transitions should correspond to excitations of quantized travelling states of breather modes [11, 12]. A breather, in this case, may be viewed as a superposition of roton states corresponding to spatially localised nonlinear oscillations of the methyl groups around their equilibrium orientation. As shown in Ref. [9], the results of the quantum sine-Gordon model are in excellent agreement with the observations, provided that no renormalization of the soliton rest mass is assumed, a fact which could be attributed to inter-chains interactions. However, the C H 3 - C H 3 distance being of the order of 13/~, it is difficult to justify the strength needed for the coupling term.

3.2. p-tert-butylcalix [4] arene(2." 1)p-xylene The INS spectrum obtained for the p-xylene complex at 1.8 K using the M I B E M O L spectrometer is shown in Fig. 3. The inset shows the results given by higher resolution measurements performed with IRIS in a restricted energy transfer range. Also in this case, the main line at 0.626(1)meV has an energy which is very close to the first rotational level of a freely rotating CH3

2.5

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2

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Energy

1.5 Transfer

2

2.5

3

(meV)

Fig. 3. INS spectrum obtained at 1.8 K on M I B E M O L for the (2:1) complex with p-xylene. The inset shows higher resolution measurements of IRIS at the same temperature.

283

group. The number of lines observed in the investigated range is larger than that expected for isolated internal tops but low energy bands are absent, at least down to 50 ~teV. With a c-axis larger than 25/~, coupling between the methyl groups belonging to different guest molecules are not expected. On the other hand, the width of the peak at 626 laeV is quite narrow (smaller than 20 laeV), and this seems to exclude the occurrence of direct coupling between the two CH3 groups belonging to the same p-xylene molecule. We therefore postulated that a coupling exists between the rotational motion of the CH 3 groups and librations of the whole p-xylene molecule [13]. As in the case of the calixarene (1: 1) toluene complex, a symmetry lowering distortion related to an ordering of the guest is assumed to occur at low temperature, despite no change in the neutron diffraction pattern having been observed down to 4.2 K. The multiplicity of the potential is then reduced by a factor 2, and the Hamiltonian to be used is -

82

+ V2~p2 + H12,

82

+

1 V6 {1 - cos(6t~l)} (2)

where the first two terms are the rotational kinetic energies of the methyl group and the whole molecule, respectively (B1 = 0.655 meV; Bz = 0.022 meV); the third term is the hindering potential for a onedimensional threefold rotator in an environment of twofold symmetry, and the fourth term is the harmonic approximation for the potential energy associated to the librational motion of the whole molecule in a cage of twofold symmetry, providing a hindering potential of peak-to-peak height 1/2. Finally, H~z describes the coupling between rotational and vibrational modes which would allow cross transitions involving simultaneous excitations or de-excitations of both rotational and librational levels. The Schr6dinger equation corresponding to the Hamiltonian (2) has been solved numerically under the assumption that H12 = 0; as shown in Table 2, the experimental observations can be satisfactorily accounted for with potential barriers V6 = 6.5 meV and V2 = 55 meV. The free rotor torsional quantum numbers are used to label the energy levels,

284

R. Caciuffo et al./ Physica B 202 (1994) 279-286

Table 2 Comparison between transition energies observed at 1.8 K for p-tert-butylcalix[4]arene (2: 1) p-xylene and values calculated using the Hamiltonian (2) with I/6 = 6.5 meV, V2 = 55 meV and H~2 = 0. Energy values are in meV Transition

Observed

Calculated

0A,0~ 1E,0 1~,0--' 0A,1 1E,0~ 2E,0 0A,0-o 0A,1 0A,0"-*2E,0

0.626(1) 1.62(1) 1.85(1) 2.20(1) 2.48(1)

0.626 1.62 1.82 2.25 2.45

12

A

0

,I 0

,I ,I , I , 4

8

Energy

Transfer

I,

, 12

, 16

, 20

(meV)

Fig. 4. The high energy transfer neutron spectrum obtained with the TFXA spectrometer suggests the presence of narrow peaks superimposed upon the phonon density of states at the positions expected for the higher harmonics of the p-xylene librational modes.

with the addition of an index, A or E, giving the symmetry of the corresponding rotational wave functions; the second q u a n t u m number refers to the librational states of the whole guest molecule. The 1E,0--* 0A,1 transition is allowed if the coupling term H12 is different from zero. The 3A,0 level is expected at 4.4 meV and, in fact, a strong peak at this energy is visible in the INS cross-section measured on the TFXA high resolution broad-band spectrometer of the ISIS spallation source (Fig. 4). At almost the same energy, our model predicts the n = 2 librational level of the p-xylene molecule; peaks in the neutron spectrum shown in Fig. 4 also appear at the positions corresponding to the higher librational harmonics, up to n = 5. Calculations of the transition matrix

elements for a free rotor are in agreement with the relative intensities of the 0A,0-o 2E,0 transition at 2.45 meV to the 1E,0 ~ 2E,0 transition at 1.82 meV [14].

As the temperature increases, the peak position of the tunnelling line shifts towards even higher energies, reaching a maximum of 0.660 meV at 30 K (Fig. 5). This value is larger than the rotational constant assumed here and used in most studies. The unusual shift to higher energies can be explained as a consequence of the coupling to the p-xylene librational modes. In fact, when the molecule is not in the librational ground state, the charge density of the protons in the benzene ring is smeared out and the repulsive interaction with the methyl protons is reduced. Therefore, the libration-mediated modulation of the potential amplitude is equivalent to a reduction of the internal hindrance barrier. Because phonons are not coupled to the spin system of the methyl protons, the temperature broadening of the A-level ground state 0A,0 is due to phonon-induced transitions to the first excited A-state (3A,0), which in our case appears at 4.4 meV. Similarly, the E-level ground state 1E,0 at 0.626 meV broadens due to transitions to the level 2E,0 at 2.45meV. The observed temperature broadening of the tunnelling transition shows an Arrhenius behaviour with an activation energy of 4.6meV,~ 7B1 and a prefactor F0 = 1.36meV. This means that the temperature effect is

0.67

E

0.66

,-

0.65

I

I

I

I

O

"~ O..

fit.

0.64

0.63 0.62

, 0

I 10

,

t

,

20 Temperature

I

,

30

, 4O

5

(K)

Fig. 5. Temperature dependence on the main tunnelling line in the p-xylene spectrum.

285

R. Caciuffo et al./ Physica B 202 (1994) 279-286

dominated by the broadening of the A states and that the prefactor for the E states is significantly smaller due to a weaker coupling to phonons [15, 16]. This is justified by the fact that the 3A rotational state of the methyl group has almost the same energy as the 2nd librational state of the p-xylene molecule while no phonon band exists as counterpart of the methyl 2~ state.

Table 3 Transition energies hto and full width at half maximum (FWHM) measured at 5 K for the p-tert-butylcalix[4]arene (1:1)y-picolinecomplex h~o(meV)

FWHM (~teV)

0.174(2) 0.227(3) 0.377(4) 0.629(1)

60(1) 101(1) 110(1) 73.7(5)

3.3. p-tert-butylcalix [ 4 ] arene (1 : 1) ?-picoline

In this case, the crystal packing deduced from room temperature single crystal X-ray diffraction patterns is the same as for the toluene complex and, therefore, similar tunnelling spectra were expected. However, difference in the attractive CH3 ... ~ interaction between the methyl groups of the host and the aromatic n orbital of the guest could result in a different position inside the intramolecular cavity of the guest methyl group and, therefore, in a different rotational barrier. In fact, as shown in Fig. 6 where the INS cross-section measured at 5 K on M I B E M O L is reported, excitation bands are visible at almost the same energies as were observed for the toluene case (see Table 3), but with important differences. Firstly, the most intense line is now the one at 0.174 meV; secondly, the bandwidths are much larger than those observed for the toluence

'

"14

I

'

I

'

I

'

I

'

I

'

t

'

I

I

'

I

'

12

and the p-xylene complex. Higher resolution spectra taken on IRIS by using mica (004) analysers (energy resolution at the elastic peak of 4 ~teV) confirm that the width of the excitation lines is due to intrinsic broadening. The third difference is in the temperature dependence of the relative intensity of the observed peaks. In the toluene complex the most stable line is the one at 0.63 meV which is still visible above 60 K; in the y-picoline spectrum this line is already washed out at 15 K, whilst the peak at 0.17meV disappears above T,~ 40 K. The number of peaks observed in the investigated energy range suggest the occurrence of a symmetry lowering structural distortion, leading to inequivalent crystallographic sites for the y-picoline guest. Static disorder or a modulation of the the distortion could be responsible for the width of the tunnelling lines. Work is in progress to determine a suitable rotational barrier; N M K and DSC measurements are planned to have further evidence and information on the structural distortion.

..,="10 4. Conclusions

m

2 0

,

-0.8

-0.6

-0.4

-0.2

Energy Transfer

I

,

I

0

,

I

,

0.2

(meV)

Fig. 6. The INS intensity distribution obtained at 5 K for the (1 : 1) complexof p-tert-butylcalix[4] arene with 7-picoline.Substantial differenceswith the spectrum of the toluence complex may be appreciated, in spite of the very similar room temperature crystal packing.

Only a few solids are known at present in which the rotational dynamics of small molecular groups approach the limit of quantum mechanical free rotations. Solid hydrogen, which is a quantum crystal [17], and methane, in its low temperature partially ordered phase [18], are among the best examples. Methyl groups show free rotation rather rarely. The closest approach to the free rotor so far is found in y-picoline [19]. The next weakly hindered system, lithium acetate [20], shows already a tunnel splitting smaller than B/2. The results reported

286

R. Caciuffo et al./ Physica B 202 (1994) 279-286

in this p a p e r s h o w t h a t the calixarene c o m p l e x e s are a new class o f systems where the t o r s i o n a l d y n a m i c s in the q u a n t u m regime can be s t u d i e d in a n e x t e n d e d t e m p e r a t u r e range. This is of p a r t i c u lar interest b e c a u s e it p r o v i d e s new possibilities of testing the p r e d i c t i o n s o f theoretical m o d e l s of m o lecular t u n n e l l i n g processes and, in p a r t i c u l a r , the effects of c o u p l i n g between r o t a t i o n a l a n d vibrational m o d e s a n d the t r a n s i t i o n from q u a n t u m to classical r o t a t i o n a l behaviour. O n the o t h e r hand, the tunnelling s p e c t r o s c o p y results give useful i n f o r m a t i o n o n the h o s t - g u e s t i n t e r a c t i o n m e c h a n i s m s which lead to the highly ion-selective p r o p e r t i e s of this class of c o m p l e x e s which are the subject of extensive investigations in the field of s u p r a m o l e c u l a r chemistry.

References [1] J.M. Lehn, Angrew, Chem. Int. Ed. Engl. 27 (1988) 89. [2] J.L. Atwood, J.E.D. Davies and D.D. MacNicol (eds.), Inclusion Compounds, Key Organic Host Systems, Vol. IV (Oxford Science Publications, Oxford, 1991). [3] C.D. Gutsche, in: Calixarenes, Monographs in Supramolecular Chemistry, ed. J.F. Stoddard (RSC, Cambridge, 1989). [4] C.D. Gutsche, in: Calixarenes, a Versatile Class of Macrocyclic Compounds, eds. J. Vicens and V. B6hmer (Kluwer Academic Publishers, Dordrecht, 1991).

[5] G.D. Andreetti and F. Ugozzoli, in: Calixarenes, eds. J. Vicens and V. B6hmer (Kluwer Academic Publishers, Dordrecht, 1991). [6] G.D. Andreetti, O. Ori, F. Ugozzoli, C. Alfieri, A. Pochini and R. Ungaro, J. Incl. Phenom. 6 (1988) 523. [7] R. Caciuffo, O. Francescangeti, S. Melone, M. Prager, F. Ugozzoli, G.D. Andreetti, G. Amoretti, G. Coddens and H. Blank, Physica B 180&181 (1992) 691. I-8] G.A. Facey, R.H. Dubois, M. Zakrzewski, C.I. Ratcliffe, J.L. Atwood and J.A. Ripmeester, preprint. [9] R. Caciuffo, G. Amoretti, F. Fillaux, O. Francescangeli, S. Melone, M. Prager and F. Ugozzoli, Chem. Phys. Lett. 201 (1993) 427. [10] F. FiUaux and C.J. Carlile, Phys. Rev. B 42 (1990) 5990. [11] J.F. Currie, J.A. Krumhansl, A.R. Bishop and S.E. Trullinger, Phys. Rev. B 22 (1980) 477. [12] R. Rajaraman, Solitons and Instantons (Elsevier, Amsterdam, 1989). [13] M. Prager, R. Caciuffo, G. Amoretti, C.J. Carlile, G. Coddens, F. Fillaux, O. Francescangeli and F. Ugozzoli, to be published. [14] B. Asmussen, private communication, to be published. [15] A.C. Hewson, J. Phys. C 15 (1982) 3841, 3855. [16] M. Prager, N. Wakabayashi and M. Monkenbush, Z. Phys. B, submitted. [17] I.F. Silveira, Rev. Mod. Phys. 52 (1980) 382. [18] W. Press and A. Kollmar, Solid State Commun. 17 (1975) 405. [19] B. Alefeld, A. Kollmar and B.A. Dasannacharya, J. Chem. Phys. 63 (1975) 4415. [20] D. Cavagnat, J. Lascombe, J.C. Lassegues, A.J. Horsewill, A. Heidemann and J.B. Suck, J. Physique 45 (1984) 97.