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Rotational quantization of methylgroups in a rotating frameS. Clough a , A.J. Horsewill a , M.R. Johnson a , J.H. Sutcliffea & I.B.I. Tomsah aa Department of Physics, University of Nottingham, NG72RD, UKPublished online: 26 Oct 2007.

To cite this article: S. Clough , A.J. Horsewill , M.R. Johnson , J.H. Sutcliffe & I.B.I. Tomsah(1994) Rotational quantization of methyl groups in a rotating frame, Molecular Physics: AnInternational Journal at the Interface Between Chemistry and Physics, 81:4, 975-990

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MOLECULAR PHYSICS, 1994, VOL. 81, No. 4, 975-990

Rotational quantization of methyl groups in a rotating frame

By S. CLOUGH, A. J. HORSEWILL, M. R. JOHNSON, J. H. SUTCLIFFE and I. B. I. TOMSAH

Department of Physics, University of Nottingham, NG7 2RD, UK

(Received 29 April 1993; accepted 6 September 1993)

The rotational tunnel spectrum of pairs of coupled methyl groups in dimethylsulphide was studied at low temperatures using low field NMR. Two tunnel frequencies of 750 and 100kHz give rise to sidebands of the NMR spectrum. The sidebands change to a broad asymmetric spectrum near internal resonances where nuclear Larmor and methyl tunnel frequencies coincide. An oscillating field at 100kHz restores the sidebands and removes the asym- metry. It is proposed that the changes are due to methyl rotation induced by the applied oscillating field modulating the dipole-dipole terms, which become secular at the resonances. The mechanism extends to molecular dynamics the concepts of spin thermodynamics, and is also similar to the electromagnetic Aharonov Bohm effect.

1. Combined spin and rotational dynamics

1.1. Collective labelling of fermions

The quantum rotation of hindered symmetrical molecules in solids has presented many puzzling features and engendered much controversy over the last twenty years. Two apparently valid points of departure led to conflicting conclusions (see [1] for a discussion). One of the starting points is the theory of spin symmetry species (SSS). This goes back at least to the early 1930s [2] and has the authority of tradition. The significant feature is that it leads to the conclusion that the potential hindering the motion of the methyl group must always have perfect threefold symmetry in order to satisfy the indistinguishability of the protons. This constraint has been incorporated in almost all methyl theories over the last thirty years [3-16]. The other starting point is equally conventional. It is to define a Hilbert space spanned by Slater determinants and a Hamiltonian which propagates the state vector in this space [17]. The conflict arises because there is no threefold symmetry restriction in this case. The constrained symmetry of the SSS theory eliminates wavepacket coordinates on which the link with the classical hopping model of methyl rotation and the parallels with other topics in quantum dynamics depend. SSS theory is consequently unlike other quantum transport theories, while Slater determinant theory, having the coordi- nates, has parallels with squids, the Aharonov-Bohm effect, electron transport in two-dimensional electron gases, and other topics.

Though the two starting points are both supposed to be expressions of the Pauli principle, they clearly differ fundamentally. The difference is that SSS theory assumes that fermions carry individual labels, and Slater determinant theory implies that they have collective labels [18]. The reason for confusion is the general belief that both imply the existence of individual labels. According to most textbooks, fermions are individually labelled particles which become indistinguishable through

0026-8976/94 $10.00�9 1994 Taylor&Francis Ltd.

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the antisymmetrization of a many-fermion state function (a Slater determinant). Inspection of a Slater determinant, however, shows that each label is equally distributed through the whole set of one-particle space-spin functions Ur. Its mean position in coordinate and spin space is the same as the mean position of the whole set of N particles and the mean position of every other label. A label is thus a global property of the collectivity of particles and not a property of an individual particle. The particles (fermions) are the occupants of the Ur. Each has components from all labels. A Slater determinant does not therefore describe a set of delocalized individually labelled particles but a set of fairly localized multi-labelled particles which are indistinguishable because they are identical. This explains the second way of expressing the Pauli principle, which is to treat fermions as unlabelled but with a rule preventing multiple occupancy in the Ur. The problem is that, while the roles of particles and labels are clearly separated in the mathematics, they are confused in the language used to describe the mathematics. SSS theory is a conse- quence of this confusion. It is supposed to be equivalent to antisymmetrization, but is based on the concept of individually labelled fermions, which is foreign to anti- symmetrized wavefunctions. The symmetry constraint whose role is to hide the individual labels is therefore non-existent. Though its removal appears to be a radical step in methyl dynamics, in view of previous theories, it is a return to more soundly based assumptions, and brings methyl quantum dynamics into harmony with hopping theory and with other quantum and classical transport phenomena.

1.2. Rotational quantization

A second constraint of SSS theory is the cyclic boundary condition ~b(r + 2re, t) = r162 t) where r is a methyl rotation coordinate. This is satisfied by exp (imr where m must be an integer. It therefore describes perfect rotational quantization and perfect methyl angular momentum conservation, implying dynamic isolation from the environment. The more general Slater determinant theory [19] replaces the boundary condition by r162 t + 2~/co) = ~(r t) exp (i2~ka) where r is a wavepacket moving round a ring r with angular velocity co and mean wavenumber k a. In making one circuit of the r circle the phase advances by 2nka, where the value of k a depends on the integrated effect of torques acting in the past on the methyl group and hdka/dt is the torque. Thus k~ is the inertial memory of the system, a role played in classical mechanics by the angular velocity co. In the quantum case, co and k a are related through the energy wavenumber relationship. At high temperatures, the wavepacket experiences a fluctuating torque which maintains it in continual motion. At low temperatures the wavepacket comes to rest with ka equal to zero. The motion of a methyl state vector in the Hilbert space is thus governed by a tendency (friction) to relax towards a stationary state and a tendency due to external influences to be driven away from these special states. The rotational quantization which appears at low temperatures is due to the congre- gation of an ensemble of state vectors near stationary states which are eigen- functions of a time independent Hamiltonian. The disappearance of quantization at high temperatures is due to the spreading of the state vectors through the Hilbert space. The typical trajectory of a state vector in these circumstances is described by the hopping model of random reorientation.

Besides being driven by thermal fluctuations of the hindering potential methyl

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Rotational quantization of Me groups 977

motion may also be driven magnetically through dipole-dipole interaction. At high temperatures, the magnetic influences are very small compared with those due to lattice fluctuations but, at low temperatures, the latter are frozen out and the former become dominant. The deviation from perfect quantization then reflects the dynamic interaction of the methyl group with its magnetic environment. Since deviations from perfect quantization of methyl rotors can be measured with considerable precision [20], and rotational relaxation times are relatively long, it is possible to study the fairly coherent dynamical response of an ensemble of methyl groups to perfectly coherent stimuli in the form of oscillating magnetic fields. This is similar to nuclear magnetic resonance (NMR), in which the almost coherent response of a spin system to oscillating magnetic fields is observed. In methyl rotational spectroscopy, there is a single characteristic frequency, the tunnel frequency, associated with excitations from the stationary states. Experiments change this rotational spectrum through the influence of oscillating fields.

A quantum system confined by the walls of a container has a special reference frame in which the system is permanently at rest. A rotating system has no such boundaries. There is an infinite set of rotating reference frames in which the system may come to rest and display quantization. When the rotor is embedded in a crystal lattice it is reasonable to suppose that the lattice provides a preferred rest frame due to interactions which transfer angular momentum when rotor and environment are in relative motion. Experiments demonstrate that methyl rotors generally do have stationary states which are eigenfunctions of a Hamiltonian which is time indepen- dent in the lattice frame. In the presence of a rotating magnetic field, though, the situation is not so clear, since there are now two reference frames competing to be the rest frame of the group. The parallel in spin dynamics is very well known. The competition between spin-lattice relaxation and spin coupling to a rotating field may be in the latter's favour, and spin quantization occurs in a frame rotating relative to the lattice. This gives rise to the concept of thermal equilibrium in a rotating frame [21,22]. Since nuclear spin and methyl rotational dynamics are coupled by the dipole-dipole interactions, rotational quantization may also occur in a rotating frame. If rotational quantization occurs in a rotating frame, the observed effect is to change the rotational spectrum as viewed from the laboratory frame.

Due to the dipole-dipole interaction the two topics of quantum methyl rotation and NMR are really one. The NMR half of the combined subject is described in terms of the evolution of superpositions of quantum states. We extend this to methyl dynamics. The new feature is inertia. Whereas the precession rate of a spin depends only on the instantaneous magnetic field, the angular velocity of a methyl group depends on the previous dynamical history. This complication means that there is at present no general theory of the combined evolution in the quantum domain of both spin and rotational dynamics. A more limited objective is to describe quasi-steady rotational states which may be set up in the presence of oscillating magnetic fields, thereby extending the concepts of spin thermodynamics. In this paper we consider an experimentally interesting but theoretically complex problem of coupled pairs of groups, and show that experimentally induced modifications of the rotational spec- trum are observable, reproducible and capable of a degree of qualitative explanation in terms of the influence of one kind of group on the rotational state of the other.

We describe a study at 4.2 K of dimethylsulphide (DMS) in which two kinds of methyl group occur, having tunnel frequencies 750 kHz and 100 kHz [23]. In many

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978 S. Clough et al.

materials the observable rotational low temperature spectrum of hindered methyl groups consists of a single well-defined tunnel frequency u t. Through discrete side- bands of low field NMR spectra this tunnel frequency can be measured with con- siderable accuracy [20]. For DMS, two sets of sidebands separated from the main N M R peak by 100 kHz and 750 kHz are observed for many values of the magnetic field. In addition to the sidebands, pure rotational transitions are observed at 100kHz and 750kHz. At certain fields, sidebands and pure rotational transitions cross through the Larmor peak and, near these internal resonances, the rotational spectrum changes quite radically to become strongly asymmetric and broadened, with some evidence of a reduction of the 100kHz frequency. At these fields, terms in the dipole-dipole interactions which are normally non-secular become secular [24] and contribute to the NMR line width. The observed effects are much larger and extend over a wider field range than can be explained merely by taking account of these terms. We interpret this as demonstrating driven rotation. The nature of the broadening strongly supports this. The sidebands broaden on one side only, filling in the region between sideband and main peak. Previous discussions of the effect of coupling between methyl groups have been in the context of incoherent transitions between stationary states, using either SSS theory [25, 26] or excitations of a linear chain [27, 28]. The novel feature of our experiments is the coherent nature of the perturbation and of the rotational response.

2. Rotating wavepackets

2.1. The rotational phase

A full description of the motion of three atoms in a crystal requires many coordinates. For the three hydrogen atoms of a methyl group it is appropriate to choose coordinates so that one, qS, describes the rotation of the triangle of atoms, and the others give the shape of the atoms and the shape and position of the triangle relative to the lattice as a whole. Of the many coordinates only ~b makes large excursions from a mean value, and it is these excursions which result in observable phenomena associated with the spin of the three protons. We may expect therefore that an approximate description of the phenomena can be made without explicit introduction of the other coordinates. Dropping them, however, raises a problem of some subtlety [29]. The rotational trajectory in the space of many coordinates is an open (simply connected) one, reflecting the fact that the methyl group motion is part of the motion of the crystal as a whole with energy and angular momentum passing between group and lattice. The one-dimensional space ~b, however, is a circle on which the same trajectories are represented as closed. It is important not to impose on the trajectories by means of a cyclic boundary condition, a property which merely comes from choosing a special reference frame.

To avoid this, it is essential to consider a wavepacket as a general state. The usual explanation of rotational quantization depends on the cyclic boundary condition. It concerns only the stationary states, and tells us that they are characterized by integer wavenumbers. Wavepackets satisfy the weaker criterion that the difference between their component wavenumbers is integral, leaving the average wavenumber ka able to assume a continuous range of values. By transforming to a suitable rotating frame we change the wavenumbers and may make them integral. Thus a wavepacket is always quantized in some frame, but not necessarily in the lattice frame. The

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Rotational quantization of Me groups 979

emergence of the observable effects of rotational quantization at low temperatures is due to the quantization frames of an ensemble all coming to rest with respect to a single frame, usually the lattice frame. The component wavenumbers having become integers, the wavepacket may then relax towards a single stationary state.

The phase associated with rotation is best known in connection with the Aharonov-Bohm effect [31], in which an electron wavepacket, envisaged to follow a closed path which encloses an energized solenoid, acquires a phase during each circuit which is equal to the integral of the magnetic vector potential along the path. There is a close connection with methyl rotation in which the phase may similarly be described by a vector potential. If the environment of the electron is static (no current in the solenoid) then the phase change after one circuit is zero. The non- zero Aharonov-Bohm phase change is the effect of the dynamic interaction between the electron and other electrons in the solenoid. The corresponding phase change of a methyl wavepacket associated with one rotation of the methyl group is a conse- quence of the dynamic interaction between the group and its environment. The general topic of quantum mechanics on a multiply connected space was first devel- oped during the 1970s using the language of topology and path integral techniques [29, 30]. Similar phase shifts representing the effect of interaction between a quantum system and a driving environment appeared in a more general context with the topological phase [32] in the 1980s. The equivalence of the wavepacket phase and a vector potential [33] led to the discovery of the structure of gauge theory in condensed matter dynamics [34]. Gauge theory is a recognition that there is an infinite number of possible reference frames in which a dynamics theory may be represented. Observable consequences are gauge invariant, i.e. independent of the choice of reference frame. Methyl rotation is a good example of quantum dynamics on a multiply connected space and of gauge theory [19].

The problem of drawing the simply connected methyl trajectory on the multiply connected coordinate space ~b is that the coordinate of the latter does not switch abruptly from 2n to zero. This is resolved by first drawing the trajectory on the simply connected universal covering space ~b c where ~b c extends from - o o to oc and then transferring it to q5 with the mapping

~(~b) = Z ~c(~b + 2nn). (1) n

If a wavepacket ~c(qSc, t) is limited to a length 2re like a wavefunction of a particle in a box, then the corresponding function on q5 is a single valued rotating wavepacket. It has leading and trailing edges, an orientation ~b w an angular velocity co w = dqSw/dt and an average wavenumber k a. To describe hindered methyl rotation we shall assume the existence of a periodic potential Vcos (3~bc), and use a basis of Bloch waves ~(k, qSc) which satisfy ~(k, q5 + 27t/3) = ~(k, q~) exp (ik2n/3). The wave- numbers k lie continuously between 3/2 and -3 /2 . By superposing two Bloch waves whose wavenumbers differ by an integer, a modulated function is obtained which travels on ~b c with a velocity cow determined by the difference between the two eigenvalues. The function has nodes separated by 27t. By isolating a 2n section between two nodes a wavepacket limited to 2n is obtained. When transferred to the q5 circle this forms a single valued rotating wavepacket which is preceeded and followed by itself on q5 in the same way that it is preceeded and followed by similar wavepackets on q5 c. We may assume that it is governed by a similar Hamiltonian with a potential Vcos (30).

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980 S. Clough et al.

2.2. Transformations to rotating reference frames

The coordinate ~b c may be chosen as fixed relative to the crystal. The coordinate system ~b may be chosen to rotate relative to the crystal at an arbitrary frequency p by generalizing (1).

t) = + + p t ) . (2)

In this way an infinite number of reference frames is available in which to describe the relative motion of the wavepacket and lattice. Two values o fp are useful, namely p = 0 which means choosing the lattice as a reference frame and p = co w which means choosing a reference frame in which the wavepacket is stationary. Transforming between two rotating frames converts k to k + p and ((-iO/O~)+cr) to ((-iO/O(o) + a - p ) [35]. It thus changes both the wavenumbers of Bloch waves and also a vector potential term in the angular momentum operator, while leaving the expectation value of this operator unchanged. The operator refers to the relative motion of wavepacket and lattice whose expectation value is the same in any refer- ence frame. Hence, values of observables are independent of the reference frame. A torque due to lattice fluctuations or to a rotating field acting through the dipole- dipole interactions changes a. Depending on the choice of reference frame, it is incorporated into the wavefunction as a change of k a or into the Hamiltonian as a change of vector potential. Both methyl rotation and electromagnetism have vector potentials because both are described by U(1) gauge theories [19]. The differences from electromagnetism are that methyl atoms are electrically neutral, the interaction with the environment is mechanical or through the dipole-dipole interaction, and methyl rotation is represented with only one space coordinate. There is consequently only one equivalent of Maxwell's equations, and this is the familiar equation identifying torque and rate of change of angular momentum.

3. Internal magnetic resonances

The propagation of a methyl wavepacket by the lattice on the space 4~ can be described at low temperatures by the Hamiltonian

H = -(h2/2I)((-iO/OqS) + ~7) 2 - Vcos (3q~) + Wcos (qS, t), (3)

where the wavepacket is a combination of Bloch waves with integer wavenumbers = 4-1 or 0. The effect of previous torques appears as the vector potential or, and the

low symmetry term W describes the effect of the lattice in generating and accelerating wavepackets. We shall suppose that the effect of W on the rotational frequency spectrum can be neglected compared with its integrated effect a. The three methyl orientations correspond to localized states in the wells of the potential Vcos (3q~) and are experimentally distinguishable because of the proton spin states associated with three lattice sites. They may be written l u, v, w) where the three spin states u, v, w are c~ or/3 and the order of the symbols refers to the lattice sites. A wavepacket written in terms of the localized states is

[a,b,c] = a]u, v, w) + blw, u, v ) + ely , w,u). (4)

Its propagation is conveniently described by the spin Hamiltonian

H = Hz + HD -- (2hut/3)(Rexp (2~icr/3) + R -t exp (-2rcia/3)), (5)

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Rotational quantization of Me groups 981

where H z and HDD are the nuclear Zeeman and dipole-dipole terms and the third term is equivalent to equation (3) with W = 0, having the same matrix elements. The operator R converts I u, v, w) to I w, u, v). The parameter 2r~r/3 is the phase associ- ated with wavepacket transport between adjacent wells. The basis functions are the eight spin products like ]c~, c~,/3), etc. The spin operators in Hz and HD pertain to the quasi-particles located at the three sites so that the geometrical factors in HD are time independent. When cr = 0 and ut = 0, equation (5) leads to the rigid methyl group spectrum [36] and, when ut r 0 and cr = 0, it leads to the narrowed spectrum with tunnelling sidebands [17]. With cr r 0 the latter is generalized. The question of interest is whether features indicative of c r r 0 can be observed at low temperatures.

Without HDD the eigenvalues of equation (5) are

E ( h , m ) / h = u z m - (2ut/3) cos (2n(h + @/3), (6)

where h = + 1 or 0, Uz is the Larmor frequency and m is the total magnetic quantum number. The frequencies of the spectrum are given by differences of pairs of eigenvalues

u(Am, ha) = u z A m + (2Pt/31/2) sin (2g(h a -k- or)/3), (7)

where ha = + 1/2 or 0. Frequency anticrossings occur when two of these are equal. I f cr = 0 this occurs when u t = Uz. Because of the dipole-dipole interaction the two frequencies do not cross as the field is increased through the internal resonance, but anticross, the two frequency dependencies reflecting off each other as the properties of the wavepackets are exchanged. Each frequency corresponds to a superposition of two eigenfunctions of equation (5). For example, the frequencies u(1,0) and u(0, 1) correspond, respectively, to a stationary methyl state with precessing transverse magnetization and to a rotating methyl state with only longitudinal magnet- ization. The change from one to the other is equivalent to a 90 ~ pulse so far as the spin state is concerned, and it has a similar character for rotation where a 90 ~ pulse means the creation of a rotating wavepacket from a stationary state. The change may be described as a spin-rotat ion flip-flop in which there is an exchange between the angular momentum of methyl rotation and spin angular momentum mediated by the non-secular terms of the dipole-dipole interaction which become secular at the resonance. This is a coherent version of the process which is respon- sible for spin-lattice relaxation at high temperatures.

For pairs of groups the frequencies u(Aml , hal, Am2, ha2 ) a r e labelled by two sets of parameters and resonances occur when Utl 4- ut2 = Uz or 2Uz. Either or both of the groups may exchange rotational energy with magnetic energy at a resonance. Rotation and spin may be regarded as part of a single system coming to a common equilibrium. It is near these resonances that the largest effects of oscillating magnetic fields are expected. In figure 1 the frequencies of the spectrum of DMS have been calculated by adding the spectra of the separate groups for a particular orientation of the magnetic field. Other orientations give spectra which are similar except in the fine details. The field independent pure tunnelling transitions occur at 100kHz and 750 kHz, and at low fields each has two pairs of sidebands separated by u z and 2Uz. The pure Larmor peak at u = Uz is the main diagonal line and at higher fields it is flanked by 100kHz and 750 kHz sidebands which have their origin as Larmor sidebands of the pure tunnelling peaks. Four anticrossing regions are identified where u t = Uz or 2Uz. We refer to these as Am = 1 and 2 resonances, and they occur at 17.6mT and 8.8roT for the 750kHz methyl group and at 2.4roT and

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982 S. Clough e t a l .

E

LL

20-

15_

10-

5

R/i = - :

"-?-: .".'2-- .:k--" _:.'-" . o

% % . . . . . . . . .

---+. . . :::-- .:-.- . . :

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. -@. . Z : .._'=" :2: :: - .'2-" ::" :: ='. -"" . "2-" . -

. . . . . . . . . . .~~

.= .,=---.:.-:f.-- _-:--.::.:-- _. ..::

0 '200 '400 '600 '800 F r e q u e n c y / k H z

'1000

Figure 1. Frequencies in the low field NMR spectrum of dimethyl sulphide as a function of field. Two tunnel frequencies at 100kHz and 750kHz have Am = 1 and 2 Larmor sidebands at very low field, while at higher field Am = 1 and 2 Larmor peaks have 100kHz and 750kHz tunnel sidebands. Each frequency represents an excitation characterized by methyl rotation and/or nuclear precession.

1.2mT for the 100kHz methyl group. In addition, a Am = 3 resonance can be identified near 5mT. The dipolar interactions which cause anticrossing at reson- ances involving both groups have been omitted, so these frequencies cross in this figure.

4 . F i e l d c y c l i n g N M R s p e c t r a o f D M S

Low field N M R using field cycling [23] is employed. A standard nuclear mag- netization is prepared at high field by first destroying all nuclear magnetization with a train of 90 ~ pulses and then allowing a standard recovery period of 20 s in a field of 0'6 T. The field is then switched in about 2 s to a low magnetic field Br. An oscillating field of f requencyfis switched on for a few (10-30) seconds and then the main field is restored to 0.6 T and the remaining magnetization is measured by a single 90 ~ pulse. The cycle is repeated many times while incrementing f. The plot of magnetization ve r sus f fo rms a spectrum which is a flat plateau in which holes occur where f i s equal to one of the spectrum frequencies of figure 2. The plot is inverted, converting the holes to peaks, for presentation. Spectra obtained with weak oscillating fields have poor signal-to-noise, while stronger fields give spectra which are only weakly depen- dent on the field amplitude. The amplitude of the fields is always small compared with any observable line width. The experimental practice is to maintain the ampli- tude of the field as small as is consistent with good signal-to-noise. The experiment is

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Rotational quantization o f M e groups 983

t - 0

Ir n m

e-

v l Vz 2"v z

u

un

n

[] n

nun n

n

= ='i, 2==

% n [] n mu

n m

nun n n

lip [] u

m m m

[]

m m n =~n

=. == []

in ,

V 2

== &

,== ~, []

in un = = q~

! ! I I ! I I I I

500 750 1000 0 250 1250

Frequency/kHz Figure 2. The dipole-dipole-driven NMR spectrum of DMS at 4.5 mT as indicated by the

horizontal broken line in figure 1. The labelling is explained in the text.

easily elaborated. One variant entails the application of a second oscillating field of fixed frequency fst (stirring frequency) at low field using the same coils as for the scanning frequency f. The two frequencies are applied in an alternating sequence of short bursts of 0'5 s so that there is no direct combined effect. The second frequency has a large effect on the intensity of peaks by changing populations of rotational energy levels. This is quite similar to phenomena in spin thermodynamics [37] and is described using a similar theory. Dimethyl sulphide was used as obtained from the Aldrich Chemical Company. A deuterated sample was prepared by mixing aqueous solutions of CH3S-Na + and CD3I as outlined in [38].

Figure 2 shows a typical spectrum obtained at a field of 4 mT which is indicated by the horizontal broken line in figure 1. The peaks in figure 2 are easily assigned. The Am = 1 and 2 pure Larmor peaks are labelled by Am while the Larmor sidebands of the pure rotational peaks are labelled by Am• Figure 3 shows a set of spectra obtained for a range of magnetic fields while irradiating with an oscillating magnetic field at 100 kHz. This produces the simplest set of spectra. The main effect of the additional frequency is to narrow the sidebands in the vicinity of the main Larmor peak and make them more symmetrical. The main peak of the spectrum flanked by its 100 kHz sidebands can be traced through the whole field range. Figure 4 shows a set of similar spectra, now aligned by their Larmor peaks. These may be compared with figure 5 which shows a similar set obtained like figure 2 without irradiation by the 100kHz field. Figure 6 extends these data to lower fields. The striking differences between figures 4 and 5 are the principal result. Figure 7 shows a set of spectra obtained while irradiating with a second frequency 750 kHz and figure

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984 S. Clough et al.

t l _

, m

r -

20

18

16

14

12

10

~ a

5"

e,-

0 200 400 600 800 1000 Frequency/kHz

Figure 3. A series of NMR spectra of DMS obtained while irradiating with an oscillating field of frequency 100 kHz. By partially decoupling the two kinds of methyl group this results in the simplest spectra which closely follow figure 1.

8 shows data like those of figure 5 but obtained with a partially deuterated sample where each molecule contained one CH 3 and one CD 3 group.

5. Discussion

Figure 5 shows the large change in the 100 kHz sideband spectrum which is observed near 8.8 mT and 17.6 mT, where Uz = 375 kHz and 750 kHz. This consists of (a) an asymmetry of the spectrum, the low frequency side almost vanishing, (b) a broadening which joins the high frequency sideband to the main Larmor peak, and (c) an apparent shift of the broadened tunnel sideband separation to a frequency

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Rotational quantization o f Me groups 985

I '

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" 1 : i . ~ . . ~ 8.5 . . . , , - :

i I *

', I ' ~75 ,,.,,r " , ~ . , ~ " �9

\ ~', t ,

' ~. %- '~ , -16 .5 # ; ,

?

' ' ~ ' ~ . ~ 1 ; , . . , \ 3 .0

,e,

,-; ',,' ', ',...,,,,.,:10.5 I - ,,

e~

.,. , ~,.,~9.5

i i

\ i . .8 .5 J mlt

; i , -

',,: ',j,..7.5

c% c%+I00

F R E Q U E N C Y / k H z

z m --I

0

i

m

B -4

Figure 4. Spectra obtained as in figure 3 with 100 kHz irradiation but showing the region near the main Am = 1 Larmor peak in more detail and with experimental conditions optimized for signal-to-noise. The spectra are aligned on their Larmor frequencies.

below 100kHz. The broadening is not due simply to the presence of overlapping peaks since it is observed on the high frequency side of the Larmor peak at 9 mT and at 18 m T when the pure rotation peak is on the low frequency side of the Larmor peak. The inward shift of the high frequency sideband is particularly marked between 15 mT and 17 mT and perhaps occurs also at 8 roT. In figure 4 the asym- metry and the inward shift have largely been removed. Some broadening is still evident at 8 ' 5mT and 16.5mT and now it appears on both sides of the Larmor peak. The same broadening can be observed in the similar data of figure 3 which was obtained with different field amplitudes and irradiation times from figure 4.

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986 S. Clough et al.

I

I �9 i

�9 I :T' i I i ' I i ' I

I , ; , % 1 , ]:'I'. ,,4,

, ; : r "%,

I ; ' ' !J:|' ~. ='~.o.. 16

i

. ?,

" ' " ;'12.5 ,,,.,

,~,~,.. 10

% ~-~,, 9

8

' ~ 7 J i j i

, , a

-,~,-~ r

m J

I I I c%-100 co z ~+100

FREQUENCY/kHz

:l>

Z m

O -11

m r '- EJ 3

Figure 5. Spectra obtained as in figure 4 but without the 100 kHz decoupling irradiation. The line shape change when the Larmor frequency is near 750 kHz and 375 kHz are the points of principal interest.

Figure 6 also shows an inward shift at 5"5 mT where u z = 250 kHz. This implies a Am = 3 resonance. Since the dipole-dipole interaction contains only Am = 2 matrix elements, an effect at this field is not easily accounted for. However, we have often seen A m = 3 transitions in many materials, so it is not at all unexpected. At 2.5 m T where Uz = 100 kHz figure 6 shows that the spectrum is again broadened.

I r radia t ion with 750 kHz induces a strong asymmetry between 16 mT and 19 mT

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Rotational quantization of Me groups 987

...%. ,

" I " �9149 . r - . . 4 . " . t . 7.0

�9 "... : i " - : . : ' t " ' . " ' . L '. ".. . . . ' . '~. . . j . . . r . ?..

.~ . . . r - . -_. t . . 6.5 . . . , .~ ! ' . ' . ' . I �9 : , - ,

-E " . . " .4., ". �9 I. . I . . .~.... I. : 6.0

" ' . . ' i : I ' . ' " ! . - ; �9 ~ i.-.. & . . E..-, �9 ..4 "-"." I "..'-' j . v 5.5

- - - . , L " I . . . r _ . . . . * . ' l ' . . I '..' I �9 I ' - ' ..--}--;-. t 5.0

..' . I . " 1 " . I" . . "1' . . " I ' . . . " { '.

l . - . . ~ . t . . . . t " 4.5 .'." I .- ," I " ' ' ~

i i [. ';~ .-I ' I 1 . . 3.5 �9 I .---.f .-~. :.L'I..

[

".i"." ~ i "... 2.5 �9 : [ .. ~ L .

I I I c%-100 ~z c%+100

F R E Q U E N C Y / k H z

E

Z rn - t O 7 7

nl r - ~J

3 --t

Figure 6. Spectra extending figure 5 to lower fields.

and destroys the magnetization completely at 17 roT. These features are explained by the usual thermodynamic theory [37] and are a result of the second frequency lying in the same spectral range as the first (scanned) frequency. In spite of the asymmetry a similar pattern of broadening is evident in this figure as in the others. The spectra at 16 mT and 19 mT are strongly broadened compared with those just outside this range and the same is apparent at 9 roT. The Am = 0 pure rotation peak at 100 kHz is enhanced near 20 mT by the second frequency, a dynamic polarization effect which may contain useful information on the mechanism of rotation-lattice relaxation�9 Between 16 mT and 20 mT on this diagram a small peak occurs near Uz/4. A similar peak has been observed for other materials. Considerable care has been taken to exclude harmonics of the irradiation frequencies so we believe this is not the origin of this peak. We have, however, no alternative explanation for it. The experiments with the deuterated sample led to data which were quite similar to those from the ordinary samples, though of lower quality. The assumption that the pairs of methyl groups which we study correspond to the pair on a single molecule is therefore incorrect. We presume that they correspond to pairs on adjacent molecules, and that even in the deuterated sample a CH 3 group is likely to have a CH 3 neighbour.

A major feature is the disappearance of the 100 kHz sideband spectrum on the low frequency side of the Larmor peak near the resonances in figure 5. Because of the broadening it has not proved possible to be sure whether this is a change in the shape of the spectrum or whether it reflects a variation of intensity due to a population dependence across an otherwise symmetrical spectrum. The restitution of both sidebands and symmetry in figure 4 we assume to be a decoupling of the dynamics of the two kinds of group. By responding to the resonant 100 kHz field, the 100 kHz groups are prevented from responding to the modulated dipolar field of the 750 kHz

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988 S. Clough et al.

20

18

16

14

"-- 12

-Q

i2 .o 10

r ,

t13

Figure 7.

0 200 400 600 800 1000 Frequency/kHz

A series of spectra similar to figure 3 but with irradiation at 750 kHz.

group. In maintaining equality of populations of the 100 kHz levels the additional field also ensures the symmetry of the sideband spectra. The residual broadening which is still to be observed in figure 4 shows that the coupling has not been removed entirely. The model which we have outlined leads us to expect that the tunnel frequency 100kHz may be substantially reduced by an oscillating field, but only marginally increased as ka varies. All the spectra agree with this. What is not observed is a Lorentzian (life-time) broadening of unshifted 100kHz sidebands. The broadening occurs over a wide range of fields and is clearly associated with the resonances. The dipole-dipole interaction alone cannot account for it. In spite of the absence of a quantitative theory it seems likely that the variations of the rotational

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Rotational quantization of Me groups 989

/ I

:i~ z-../',.J! ,::--, 21

t ,.' i / t �9 I I ] "

~m-~ .@. ' "-<. 19

+",i/Xl ', : ' ' \ 18

,.I.U ,.'1! / V .I it/ " 16

�9 V' /"::

," " ' \ i 14

, . , , : ' : i t ",

,r/, / i t l j , , ; 10 '-r , ! , " ~ "v'

l.l.t P,, , i t , / I \ / '~ ' kv'~\' 9

y , , I I

m~-lO0 m~ c%+100

F R E Q U E N C Y / k H z

63 Z rrl --I O -n ITI I ' - - ID

3 .-I

Figure 8. Spectra obtained as in figure 4 for a partially deuterated sample having one CD3 group per molecule. The similarity with figure 4 suggests that the two kinds of group cannot be identified with the two groups on each molecule.

spectrum are due to dipolar coupling between the two kinds of groups influencing the effect of oscillating fields in inducing methyl rotation.

The authors are grateful to the BP Venture Research Unit for supporting this work, to the Sudanese government for a fellowship for I.B.I.T. and to Dr D. K.

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990 S. Clough et al.

Knight of the University of Nottingham Chemistry Department for preparing the deuterated sample.

References

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