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Methyl group hopping reorientation and the non-existence of spin symmetry species at room temperature
CHEMICAL PHYSICS LETTERS
Volume 80, number 2
1 June 1981
COMMENT METHYL GROUP HOPPING REORIENTATION AND THE NON-EXISTENCE
OF SPiN SYMMETRY
SPECIES AT ROOM TEMPERATURE
S. CLOUGH Deparrment
of Physics, ihiversrty
of Nottmgham,
Nottingham
NC7 ZRD, UK
Received 10 January 1981
A series of experiments is discussed which, it has been claimed, demonstrate that hmdered methyl groilps exist at room temperature as relatively long-lived proton spin symmetry species. It is concluded that this concept lacks a theoretical foundation, conflicts with the model of random hopping reonentation between localised states, and leads to physically implausible predictions.
It is ROWwell known that hindered methyl groups in solids at low temperatures may be described in terms of long-lived A and E proton spin symmetry species. Over a number of years experiments have been carried out [l-6] which it is claimed demonstrate the existence of similar long-lived species at room temperature. It has not been gene&y realised though how radically this concept conflicts with the commonly accepted view of methyl group reorientation at room temperature, by random hopping around the symmetry axis through angles off 2n/3, nor how physically implausible from the standpoint of conventional thinking, are the predicted consequences of room-temperature spin symmetry species (RTSSS). ‘Ihis is perhaps because these consequences typically appear in the rather unspectacular form of departures from exponentiality of spin-lattice relaxation, and because the physical meaning of the mathematical formalism is not always transparent. An impression may have been gained that the RTSSS model is in some way a refmement of the traditional one. The first objective of this letter is to correct this impression and to show that the two are completeIy_incompatible and cannot both be correct. The second objective is to examine the theoretical foundations of both models. A third objective is to relate methyl group reorientation to anaIog0u.s transport phenomena of the solid state, to fit it into a wider framework of existing theory. Finally we discuss the implications of some of 0 009-2614/81/0000-0000/S
02.50
0 No&-Holland
the experiments which have been used to support RTSSS. (Emid and co-workers use the name SRSD symmetry restricted spin diffusion.) The traditional view, roughly stated, is that at room temperature, a hindered methyl group hops randomly through +- 2~13, between the three orrentations favoured by the hindering barrier, at a rate described by a correlation time T=_A more careful statement would discriminate between two cases: (a) when all three protons have the same spin state, i.e. Ima+ or IPflfi>,and (b) when one proton spin state is different from the other two, e.g. Idi>. In case (a) reorienof the group through -C21r/3 is not a useful concept si?ce it changes nothing, the protons being indistinguishable, so we may restrict ourselves to case (b). Then the observable consequence of reorientation is that the odd spin state (i.e. /3 in the example above) hops from one proton site to another. All the wellknown consequences of CH3 reorientation like nuclear spin-lattice relaxation, inelastic neutron scattering, etc. follow from the random walk performed by the odd proton spin state between the three sites. Between hops it is localized on one site and the hopping process results in a modulation of spin- and space-dependent interactions at a rate of order r;‘. We shall refer to this well-known model as the hopping model. The contrasting claims of the RTSSS model have been most explicitly stated by Ernid in a pape; [7] which is the most recent of a series [7-141. The odd Publishing Company
389
Volun~c
80.
number
2
CHEMICAL PHYSICS LETTLRS
spin state is always delocalized, I.e. always equally drstrlbuted among the three sites and the lifetime of a rotatronal state is not rc but perhaps lOlo times longer “m distinct contrast to the common view of the socalled reonentation process in which rt is assumed that the hfetrmes of clock\\qse and anticlockwise motions are of order T=“_ Between hops the group is in a state of coherent reorientation which is unaffected by the hops: “This means that for example rotors in clockwise rotation will remam rotating in the same direction“. Long-lived coherent rnorron and rapid incoherent motion therefore coexist Ciearly the two models are incompatrble. Emrd [7] bases the case for RTSSS on symmetry (Le. on the indistingurshabrhty of the protons) so it is important first to make the elementary point that the localized states of the traditional model are consistent with indistinguishability. If we attach notional ldbe!s, say p,q_r to the three protons. ensunng that indistinguishability is satisfied by appropriate symmetrization of the total wavefunction. then the probabihty of finding proton p in site 1 (or 2 or 3) is $ in each case. since otherwrse we could identify which was proton p by its location. Simrlarly we cannot say whether it 1s proton p, 4 or r which has the odd spur state fl_ Thus in the symmetrized state, proton p (like 4 and r) is equally distributed between the three sites. and is two thirds in spin state o and one third in spin state /3. This is of course entirely consistent with the proton at site 1 being entirely in state 0, whrle those at sates 2 and 3 have spin state (Y. Such localized states have been given in detail 1151 many times. Like many others before [ 161, Emid [7] uses as a basis, a collective description of methyl group proton space and spin states, which consists of products AA, EaEb, EbEa, of a spatial function and a spin function, each chosen to belong to an irreducible representation of C, _ Indistinguishability is satisfied by restricting the products to those above, which belong as a whole to the A representation. Each of the products describes a state in which the odd spin state is delocalized. A localized state though can be constructed as a linear combination $Joc = 3-l/‘(AA
+ hE”Eb + x*EbEa) ,
(1)
h being one of the cube roots of unity, 1 or exp(+ 211i/3), the choice determining whether the odd spin state is localized on site 1,2 or 3. Thus both models satisfy undistinguishability_ 390
1 June 1981
Now the RTSSS model assumes in addition that the state of the hopping methyl group is best described by a single product, i.e. AA, EaEb or EbEa. These are the spin symmetry species. There appears to be no justification for this assumption and clearly it is not dictated either by symmetry considerations, or by the magnitude of the frequency of the coherent motion (i.e. the tunnel frequency) which is acknowledged to be very small or zero [7] at room temperature. The assumption has the effect of conferring on the odd spin state fl the property of delocalization and of introducing the coexistence of coherent and incoherent reorientation. It 1s from tlus that all the differences from conventional theory follow. It will be recognised that methyl reorientation (the transport of a P spin state among three sites) is closely analogous to many other transport phenomena of the solid state, like exciton diffusion, low mobility conductivity, etc. where hopping models are usual. At very low temperatures when phonons are not excited, these phenomena are described by wave-like propagation, the delocalized states being then appropriate because the lifetime of localized states is limited by tunnelling between neighbour localized states. With increasing temperature, the mean free path of the wave-like states becomes shorter due to phonon scattering, and one goes over to localized states and hopping. This is the case with methyl group reorientatron. At high temperature the methyl group can be regarded as isolated from the lattrce for only a very short time (less than rc). On this very short time scale tunnelling can be ignored, the three protons can be treated as fixed and are identified by their locations; they are therefore assigned individual spin states (Yor 0 as they would be if there were no reorientation at all. This is the basis for the traditional approach. There are some difficulties in describing the transition from low-temperature wave-like tunnelling reorientation to high-temperature hopping, which are well recognised in other areas. Writing in 1971 about exciton transport, Grover and Silbey [ 171 describe the position as follows: “.._ time-dependent perturbation theory is used to obtain a formal expression for the probability of an exciton moving between two different lattice sites. It is found that the terms in this expression corresponding to coherent motion in which an exciton moves without disturbing the thermal equilibrium of the lattice, decrease in importance with in-
Volume 80, number 2
CHEMICAL PHYSICS LETTERS
creasing temperature while the terms corresponding to incoherent motion, in which exciton migration is always accompanied by a change in the thermal phonon distribution, increase in Importance as the temperature is raised. Unfortunately, the perturbation theory diverges in practice and one is ultimately forced to treat these two forms of motion separateIy by adopting a model of purely coherent migration in the low-temperature Iimit and a hopping model in the limit of high temperatures”. This IS exactly the positson with methyl group reorientation (though approaches not based on perturbation theory can describe the whole range [IS]). The above quotation Illustrates clearly -why the recent advances in our knowledge and understanding of methy group tunnelling at Iow temperatures should not affect the status of the traditional hopping model at high temperatures. An exact treatment would of course give the same results whether a localized or delocalized basis is used, but as the above quotation makes clear. when the concept of transition probabilIties is used, the wrong choice of basis may lead to error. A detailed discussion of the limits of the validity of perturbation theory has been given by Holstein [ 191 in the context of polaron transport, and provides a justification For the model of hoppmg between localized states in the high-temperature regime_ The source of the difference in the experimental predictions of the RTSSS and hopping models can be summarized in two words: rotational memory. In the hopping model the rotational memory lasts only about 7, as the name correlation time implies. In the RTSSS model it is of the order of the spin-lattice relaxation time T, , perhaps I O’O Q-~or even longer. This is because a state function consisting of a single product like EaEb can only be changed mto a different product (EbEa say) by the weak magnetic interactions, whereas a Iocalized state Iike (1) can be changed into a different localized state by a change of h caused by the strong methyl group-phonon interactions. All of the experiments which have been claimed to support RTSSS depend on this long memory. Thus it is possible to induce a small rotation of the methyl groups through a change in the nucIear magnetism. This has been called rotational polarization [9] and is a consequence of a transfer of angular momentum from nuclear magnetism to methyl rotation. According to the hopping model this rotational polarization is destroyed by the hopping as fast as it is created, the ro-
1 June 198:
tational polarization relaxation time bemg of order rcc, and the steady-state magnitude is extremely small. According to RTSSS, the rotational polarization is not destroyed but builds up during spin-lattice relaxation, and then the rotational polarization tends to restore the nuclear magnetism. The consequence is that the nuclear Zeeman relaxation is relatively rapid to begin with, but (according to RTSSS) slows down as the rotational polarization builds up. This non-exponential relaxation has been a key feature in the work supporting RTSSS. Another type of experiment mvolves reversing the sample orientation [2] which reverses the sign of the rotatIonal polarization, and so (again according to RTSSS) speeds up nuclear relaxation instead of slowing it down. From the standpoint of the hopping model, there is of course no possibility that information can be stored in the rotation of the group for times of the order lOlo TV. Nonetheless an Impressive amount of experimental evidence has been presented which appears to demonstrate these memory effects. However there are also many reports [20-221 of exponential relaxation due to CH, groups, which therefore conflict Lvlth the RTSSS model. Furthermore the. hopping model can itself lead to non-exponential relaxation [23,24] as mdeed may spin temperature gradients and powder averaging of anisotropic relaxation rates. Thus the experimental evidence may be less strongly in favour of RTSSS than has been supposed. The conclusion of this survey must be that the hopping model has a clear and plausible physical basis and is m harmony with what is known of similar processes, while neither of these statements applies to RTSSS. The experimental evidence for the latter may therefore need to be re-assessed.
References [l]
R.A. Wind and S. Cmid, Phys. Rev. Letters 33 (1974) 1422. [ 2 J R.A. Wind, S. Emid and J.F JN. Pourquie, Phys Letters 53A (1975) 310. [ 31 S. Emid and R.A. Wmd, Chem. Phys. Letters 33 (1975) [4] ?&.hI. Pourquid, S. Emid, J. Smidt and R.A. Wmd, Phys. Letters 61A (1977) 198. [5] R.A. Wind, S. Emid, J.F.J.M. Pourquid and J Smidt, J. Chem. Phys. 67 (1977) 2436. 161 R.A. Wind, S. Emid, J.T.J.M. Pourquib and J. Smidt, J. Phys. C9 (1976) 139. 391
Volume
SO, number 2
CHEhfICAL
[7 ] S. Emid, Chcm. Phys. Letters 72 (1980) 189. [S] S. Emid and R-A_ Wind, Chem_ Phys. Letters 27 (1974) 312. [9] S. Emid, R.A. Wind and S. Clough, Phys. Rev. Letters
33 (1974) 769. [lo]
S_ Emid and R-A. Wnd, in: Proceedings of the 18th Amp&e Congress, Nottingham, eds_ P.S. AlIen. C-R. Andrew and C.4. Bates (1974) p_ 373. [l l] S. tmid, D. Lankhurst, J. Smidt and R.A. Wind, inProceeding of the 19th Ampere Congress, Heidelberg, eds. H. Brummer, R.H. Hausser and D. Schweitzer (1976) p_ 320. [I?] R A. Wind, S. Emid, J.F.J.M. PourquiB and J. Smidt, in. Proceedmgs of the 19th Amp&e Congress, f Heidelberg, eds. H. Brummer, K.H. Hausser and D. Schweitzer (1976) p_ 56.5. [I 3j S. Hmld, R J. Baarda, J. Schmidt and R.A. Wind, Physkx NC93 (1978) 327.
392
PHYSICS
LETTERS
1141 fls] [i6] [ 171
1 June 1981
S. Emid and R.A. Wind, Physica BiC93 (1978) 344. F. Apaydin and S. Ciough, J. Phys. Cl (1968) 932. J.H. Treed, J. Chcm. Phys 43 (1965) 1710. M. Grover and R. Silbey, J. Chem. Phys. 54 (1971) 4843. [ 18 ] S. CIough, J. Phys. C, to be published. f19] T. Holstein, Ann. Phys. 8 (1959) 343. 1201 E-R. Andrew, WS. Hinshaw, ,M.G. Hutchins and R.O.I. Sjdblom, &fol. Phys. 31 (1976) 1479. [Zl ] E-R. Andrew, D-J. Bryant and E.M. CasheU, Chem. Phys Letters 69 (1980) 551. [22] W. LMdlIer-Warmuth, R. Schiiler, M. Prayer and A. KoIlmar, J. Chem. Phys 69 (1978) 2382. 1231 R-L. Hilt and P.S. Hubbard, Phys. Rev_ 134A (1964) 392. 1241 Xf. hfehxing and H. Rnber, J. Chem. Phys. 59 (1973) 1116.
Volume 80, number 2
1 June 1981
COMMENT METHYL GROUP HOPPING REORIENTATION AND THE NON-EXISTENCE
OF SPiN SYMMETRY
SPECIES AT ROOM TEMPERATURE
S. CLOUGH Deparrment
of Physics, ihiversrty
of Nottmgham,
Nottingham
NC7 ZRD, UK
Received 10 January 1981
A series of experiments is discussed which, it has been claimed, demonstrate that hmdered methyl groilps exist at room temperature as relatively long-lived proton spin symmetry species. It is concluded that this concept lacks a theoretical foundation, conflicts with the model of random hopping reonentation between localised states, and leads to physically implausible predictions.
It is ROWwell known that hindered methyl groups in solids at low temperatures may be described in terms of long-lived A and E proton spin symmetry species. Over a number of years experiments have been carried out [l-6] which it is claimed demonstrate the existence of similar long-lived species at room temperature. It has not been gene&y realised though how radically this concept conflicts with the commonly accepted view of methyl group reorientation at room temperature, by random hopping around the symmetry axis through angles off 2n/3, nor how physically implausible from the standpoint of conventional thinking, are the predicted consequences of room-temperature spin symmetry species (RTSSS). ‘Ihis is perhaps because these consequences typically appear in the rather unspectacular form of departures from exponentiality of spin-lattice relaxation, and because the physical meaning of the mathematical formalism is not always transparent. An impression may have been gained that the RTSSS model is in some way a refmement of the traditional one. The first objective of this letter is to correct this impression and to show that the two are completeIy_incompatible and cannot both be correct. The second objective is to examine the theoretical foundations of both models. A third objective is to relate methyl group reorientation to anaIog0u.s transport phenomena of the solid state, to fit it into a wider framework of existing theory. Finally we discuss the implications of some of 0 009-2614/81/0000-0000/S
02.50
0 No&-Holland
the experiments which have been used to support RTSSS. (Emid and co-workers use the name SRSD symmetry restricted spin diffusion.) The traditional view, roughly stated, is that at room temperature, a hindered methyl group hops randomly through +- 2~13, between the three orrentations favoured by the hindering barrier, at a rate described by a correlation time T=_A more careful statement would discriminate between two cases: (a) when all three protons have the same spin state, i.e. Ima+ or IPflfi>,and (b) when one proton spin state is different from the other two, e.g. Idi>. In case (a) reorienof the group through -C21r/3 is not a useful concept si?ce it changes nothing, the protons being indistinguishable, so we may restrict ourselves to case (b). Then the observable consequence of reorientation is that the odd spin state (i.e. /3 in the example above) hops from one proton site to another. All the wellknown consequences of CH3 reorientation like nuclear spin-lattice relaxation, inelastic neutron scattering, etc. follow from the random walk performed by the odd proton spin state between the three sites. Between hops it is localized on one site and the hopping process results in a modulation of spin- and space-dependent interactions at a rate of order r;‘. We shall refer to this well-known model as the hopping model. The contrasting claims of the RTSSS model have been most explicitly stated by Ernid in a pape; [7] which is the most recent of a series [7-141. The odd Publishing Company
389
Volun~c
80.
number
2
CHEMICAL PHYSICS LETTLRS
spin state is always delocalized, I.e. always equally drstrlbuted among the three sites and the lifetime of a rotatronal state is not rc but perhaps lOlo times longer “m distinct contrast to the common view of the socalled reonentation process in which rt is assumed that the hfetrmes of clock\\qse and anticlockwise motions are of order T=“_ Between hops the group is in a state of coherent reorientation which is unaffected by the hops: “This means that for example rotors in clockwise rotation will remam rotating in the same direction“. Long-lived coherent rnorron and rapid incoherent motion therefore coexist Ciearly the two models are incompatrble. Emrd [7] bases the case for RTSSS on symmetry (Le. on the indistingurshabrhty of the protons) so it is important first to make the elementary point that the localized states of the traditional model are consistent with indistinguishability. If we attach notional ldbe!s, say p,q_r to the three protons. ensunng that indistinguishability is satisfied by appropriate symmetrization of the total wavefunction. then the probabihty of finding proton p in site 1 (or 2 or 3) is $ in each case. since otherwrse we could identify which was proton p by its location. Simrlarly we cannot say whether it 1s proton p, 4 or r which has the odd spur state fl_ Thus in the symmetrized state, proton p (like 4 and r) is equally distributed between the three sites. and is two thirds in spin state o and one third in spin state /3. This is of course entirely consistent with the proton at site 1 being entirely in state 0, whrle those at sates 2 and 3 have spin state (Y. Such localized states have been given in detail 1151 many times. Like many others before [ 161, Emid [7] uses as a basis, a collective description of methyl group proton space and spin states, which consists of products AA, EaEb, EbEa, of a spatial function and a spin function, each chosen to belong to an irreducible representation of C, _ Indistinguishability is satisfied by restricting the products to those above, which belong as a whole to the A representation. Each of the products describes a state in which the odd spin state is delocalized. A localized state though can be constructed as a linear combination $Joc = 3-l/‘(AA
+ hE”Eb + x*EbEa) ,
(1)
h being one of the cube roots of unity, 1 or exp(+ 211i/3), the choice determining whether the odd spin state is localized on site 1,2 or 3. Thus both models satisfy undistinguishability_ 390
1 June 1981
Now the RTSSS model assumes in addition that the state of the hopping methyl group is best described by a single product, i.e. AA, EaEb or EbEa. These are the spin symmetry species. There appears to be no justification for this assumption and clearly it is not dictated either by symmetry considerations, or by the magnitude of the frequency of the coherent motion (i.e. the tunnel frequency) which is acknowledged to be very small or zero [7] at room temperature. The assumption has the effect of conferring on the odd spin state fl the property of delocalization and of introducing the coexistence of coherent and incoherent reorientation. It 1s from tlus that all the differences from conventional theory follow. It will be recognised that methyl reorientation (the transport of a P spin state among three sites) is closely analogous to many other transport phenomena of the solid state, like exciton diffusion, low mobility conductivity, etc. where hopping models are usual. At very low temperatures when phonons are not excited, these phenomena are described by wave-like propagation, the delocalized states being then appropriate because the lifetime of localized states is limited by tunnelling between neighbour localized states. With increasing temperature, the mean free path of the wave-like states becomes shorter due to phonon scattering, and one goes over to localized states and hopping. This is the case with methyl group reorientatron. At high temperature the methyl group can be regarded as isolated from the lattrce for only a very short time (less than rc). On this very short time scale tunnelling can be ignored, the three protons can be treated as fixed and are identified by their locations; they are therefore assigned individual spin states (Yor 0 as they would be if there were no reorientation at all. This is the basis for the traditional approach. There are some difficulties in describing the transition from low-temperature wave-like tunnelling reorientation to high-temperature hopping, which are well recognised in other areas. Writing in 1971 about exciton transport, Grover and Silbey [ 171 describe the position as follows: “.._ time-dependent perturbation theory is used to obtain a formal expression for the probability of an exciton moving between two different lattice sites. It is found that the terms in this expression corresponding to coherent motion in which an exciton moves without disturbing the thermal equilibrium of the lattice, decrease in importance with in-
Volume 80, number 2
CHEMICAL PHYSICS LETTERS
creasing temperature while the terms corresponding to incoherent motion, in which exciton migration is always accompanied by a change in the thermal phonon distribution, increase in Importance as the temperature is raised. Unfortunately, the perturbation theory diverges in practice and one is ultimately forced to treat these two forms of motion separateIy by adopting a model of purely coherent migration in the low-temperature Iimit and a hopping model in the limit of high temperatures”. This IS exactly the positson with methyl group reorientation (though approaches not based on perturbation theory can describe the whole range [IS]). The above quotation Illustrates clearly -why the recent advances in our knowledge and understanding of methy group tunnelling at Iow temperatures should not affect the status of the traditional hopping model at high temperatures. An exact treatment would of course give the same results whether a localized or delocalized basis is used, but as the above quotation makes clear. when the concept of transition probabilIties is used, the wrong choice of basis may lead to error. A detailed discussion of the limits of the validity of perturbation theory has been given by Holstein [ 191 in the context of polaron transport, and provides a justification For the model of hoppmg between localized states in the high-temperature regime_ The source of the difference in the experimental predictions of the RTSSS and hopping models can be summarized in two words: rotational memory. In the hopping model the rotational memory lasts only about 7, as the name correlation time implies. In the RTSSS model it is of the order of the spin-lattice relaxation time T, , perhaps I O’O Q-~or even longer. This is because a state function consisting of a single product like EaEb can only be changed mto a different product (EbEa say) by the weak magnetic interactions, whereas a Iocalized state Iike (1) can be changed into a different localized state by a change of h caused by the strong methyl group-phonon interactions. All of the experiments which have been claimed to support RTSSS depend on this long memory. Thus it is possible to induce a small rotation of the methyl groups through a change in the nucIear magnetism. This has been called rotational polarization [9] and is a consequence of a transfer of angular momentum from nuclear magnetism to methyl rotation. According to the hopping model this rotational polarization is destroyed by the hopping as fast as it is created, the ro-
1 June 198:
tational polarization relaxation time bemg of order rcc, and the steady-state magnitude is extremely small. According to RTSSS, the rotational polarization is not destroyed but builds up during spin-lattice relaxation, and then the rotational polarization tends to restore the nuclear magnetism. The consequence is that the nuclear Zeeman relaxation is relatively rapid to begin with, but (according to RTSSS) slows down as the rotational polarization builds up. This non-exponential relaxation has been a key feature in the work supporting RTSSS. Another type of experiment mvolves reversing the sample orientation [2] which reverses the sign of the rotatIonal polarization, and so (again according to RTSSS) speeds up nuclear relaxation instead of slowing it down. From the standpoint of the hopping model, there is of course no possibility that information can be stored in the rotation of the group for times of the order lOlo TV. Nonetheless an Impressive amount of experimental evidence has been presented which appears to demonstrate these memory effects. However there are also many reports [20-221 of exponential relaxation due to CH, groups, which therefore conflict Lvlth the RTSSS model. Furthermore the. hopping model can itself lead to non-exponential relaxation [23,24] as mdeed may spin temperature gradients and powder averaging of anisotropic relaxation rates. Thus the experimental evidence may be less strongly in favour of RTSSS than has been supposed. The conclusion of this survey must be that the hopping model has a clear and plausible physical basis and is m harmony with what is known of similar processes, while neither of these statements applies to RTSSS. The experimental evidence for the latter may therefore need to be re-assessed.
References [l]
R.A. Wind and S. Cmid, Phys. Rev. Letters 33 (1974) 1422. [ 2 J R.A. Wind, S. Emid and J.F JN. Pourquie, Phys Letters 53A (1975) 310. [ 31 S. Emid and R.A. Wmd, Chem. Phys. Letters 33 (1975) [4] ?&.hI. Pourquid, S. Emid, J. Smidt and R.A. Wmd, Phys. Letters 61A (1977) 198. [5] R.A. Wind, S. Emid, J.F.J.M. Pourquid and J Smidt, J. Chem. Phys. 67 (1977) 2436. 161 R.A. Wind, S. Emid, J.T.J.M. Pourquib and J. Smidt, J. Phys. C9 (1976) 139. 391
Volume
SO, number 2
CHEhfICAL
[7 ] S. Emid, Chcm. Phys. Letters 72 (1980) 189. [S] S. Emid and R-A_ Wind, Chem_ Phys. Letters 27 (1974) 312. [9] S. Emid, R.A. Wind and S. Clough, Phys. Rev. Letters
33 (1974) 769. [lo]
S_ Emid and R-A. Wnd, in: Proceedings of the 18th Amp&e Congress, Nottingham, eds_ P.S. AlIen. C-R. Andrew and C.4. Bates (1974) p_ 373. [l l] S. tmid, D. Lankhurst, J. Smidt and R.A. Wind, inProceeding of the 19th Ampere Congress, Heidelberg, eds. H. Brummer, R.H. Hausser and D. Schweitzer (1976) p_ 320. [I?] R A. Wind, S. Emid, J.F.J.M. PourquiB and J. Smidt, in. Proceedmgs of the 19th Amp&e Congress, f Heidelberg, eds. H. Brummer, K.H. Hausser and D. Schweitzer (1976) p_ 56.5. [I 3j S. Hmld, R J. Baarda, J. Schmidt and R.A. Wind, Physkx NC93 (1978) 327.
392
PHYSICS
LETTERS
1141 fls] [i6] [ 171
1 June 1981
S. Emid and R.A. Wind, Physica BiC93 (1978) 344. F. Apaydin and S. Ciough, J. Phys. Cl (1968) 932. J.H. Treed, J. Chcm. Phys 43 (1965) 1710. M. Grover and R. Silbey, J. Chem. Phys. 54 (1971) 4843. [ 18 ] S. CIough, J. Phys. C, to be published. f19] T. Holstein, Ann. Phys. 8 (1959) 343. 1201 E-R. Andrew, WS. Hinshaw, ,M.G. Hutchins and R.O.I. Sjdblom, &fol. Phys. 31 (1976) 1479. [Zl ] E-R. Andrew, D-J. Bryant and E.M. CasheU, Chem. Phys Letters 69 (1980) 551. [22] W. LMdlIer-Warmuth, R. Schiiler, M. Prayer and A. KoIlmar, J. Chem. Phys 69 (1978) 2382. 1231 R-L. Hilt and P.S. Hubbard, Phys. Rev_ 134A (1964) 392. 1241 Xf. hfehxing and H. Rnber, J. Chem. Phys. 59 (1973) 1116.