Some Nuclear Resonance Properties of Nearly Free Methyl Rotors in the Solid State BY PETERS. ALLEN Department of Physics University of Nottingham Nottingham NG7 2RD Received 31st July 1978 This paper is primarily concerned to show how nuclear resonance can be used to investigate the rotational energy states and transition rates of weakly hindered symmetric rotors in the solid state while at the same time developing an explanation for the nuclear resonance data from the methyl protons in CHJLiCOO * 2D20. The discourse introduces the ideas involved in the context of a simple threefold free rotor before extending them first to a weakly hindered rotor and then to a pair of coaxial rotors. The kernel is that by exploiting the angular variation of temperature and frequency dependent studies experimental data of sufficient precision are available to enable meaningful molecular information to be obtained.Though space restricts this presentation to a limited explanation of the spectrum quantitative derivations from transient signals and relaxation data will be presented for discussion. 1. OUTLINE OF THE FREE METHYL ROTOR A free two dimensional rigid rotor whose orientation relative to some arbitrary datum can be described in terms of a single angular coordinate x,has energy states which are readily derived from a simple wave equation. In fact the energy eigen- functionsU (x)are given by U&) = (2n)-+exp (imx) (1.1) and the eigenvalues Emby h2 m2 Em = -21 where rn = 0 &I &2 . . . is the rotational quantum number h is Planck’s constant divided by 271 and I is the moment of inertia about the rotor axis.Substituting into eqn (1.2) the moment of inertia of a methyl group and converting to temperature units gives Em 2 7.5m2K. In this free rotor limit the energy eigenvalues are not affected by the detailed symmetry of the rotor but only by its moment of inertia. However if we are to proceed into a more detailed discussion involving hindering potential barriers nuclear spin func- tions nuclear dipole-dipole interactions and transition rates it is expedient to classify these free rotor states under a symmetry group to which the rotor corresponds. Now the group of permutations to which the feasible transformations of a methyl triangle do correspond is isomorphous with the rotational symmetry group C3and it follows that the free rotor representation can be reduced under that rotation group.To be precise the angle x only enters the rotational energy eigenfunctions by way of the factor exp(imX). Thus the specific transformation C [-e(1 23) for a triangle labelled clockwise looking along the positive rotation axis] has the effect of multiplying the METHYL ROTORS IN THE SOLID STATE energy eigenfunctions by exp(-iim2~/3). We can see therefore that the three pos- sible effects on the eigenfunctions of the transformation C3 will be to transform an eigenfunction into itself into E times itself or into E* times itself where E = exp(i2~/3). These three possibilities are identical to the behaviour of the three irreducible repre- sentations (respectively A E and E as shown in the character table of table 1) of the free hindered free hindered 0 0.5 1.0 1.5 0 0.5 1.0 1.5 V3/ kJ mol-' Vs/ kJ mol-' FIG.1.-Classification of the free methyl rotor energy states under the symmetry groups C3and C,.Also shown is how these states are perturbed in both cases by the growth of a hindering potential barrier. V3 is the amplitude of a three-fold sinusoidal barrier whereas VSrepresents the amplitude of a six-fold sinusoid. symmetry group C3,under the specific transformation C,. Thus each of free rotor functions transforms as one or other of the irreducible representations of the C symmetry group. The classification of the lower free rotor states under the group C is indicated in fig.1 together with their classification under the Cbsymmetry group to which reference will be made later. Turning to the nuclear spins the most simple methyl group spin representation R,,is the set of eight functions Iaaa) [cap> l&> Ipacc) IpPa) Ipap) I&?> and IPpD) where ct represents a proton spin parallel to the applied field B and p an anti- parallel proton spin. Under the symmetry group C3,R,has the characters given in table 1 which may be decomposed to show that R,= 4A + 2E + 2E,. TABLEI .-CHARACTERS UNDER SYMMETRY GROUP C, OF THE IRREDUCIBLE REPRESENTA-TIONS A EaAND Eb AND OF THE SPIN REPRESENTATION R, OF THREE SPINS 1 1*. c3 E c3 c5 A 1 1 1 Ea 1 &* & E = exp (i2ni3) Eb 1 E &* R 8 2 2 P.S. ALLEN In other words an alternative but symmetrically more acceptable spin representation would consist of four linear combinations of the original functions which transform as the A irreducible representation and two pairs each of which has either € or Eb symmetry respectively. The form of these linear combinations may be obtained by applying the theorem stated by Tinkham’ and the resulting methyl spin functions VR,mIare given in table 2. TABLE 2.-METHYL GROUP NUCLEAR SPIN FUNCTIONS FOR POSITIVE ml (WHERE mi IS THE METHYL SPIN COMPONENTS PARALLEL TO &). THE NEGATIVE 1711 COMPONENTS ARE OBTAINED BY REPLACING c1 BY etc. The coupling of these spin functions to the previously described rotor functions is by no means arbitrary; it is the result of this coupling which forms the basis for the interesting nuclear resonance behaviour of methyl groups and which in turn enables nuclear resonance to probe their rotational states.Because each symmetry element of the group C3is equivalent to an even number of interchanges of a pair of identical particles Pauli’s exclusion principle demands that the overall wavefunction must be totally symmetric to the operation of any symmetry element. The overall wave-function must therefore transform according to the A symmetry species. As a result the acceptable rotor-spin product wavefunctions are restricted by the C3multiplication table to the three types A x A € x & and Eb x €,. Before sketching the nuclear spin energy states of a free methyl rotor it is neces- sary to take account of the nuclear dipole-dipole interaction which operates on both the methyl spin and rotor functions.Each of the six standard parts (A to F) of the dipolar hamiltonian XD,can be rewritten in terms of the orientation x and of the angle ly between B and the methyl C axis. Moreoever the terms can be subse- quently collected together in a symmetry adapted fashion (see for example Haupt)2 to formcomponents for each of the six parts which are either A x A € x Eb or Eb x E space x spin operator products. The dipolar energy shifts depending on the first two parts of XD(A and B) are straightforward in the limit when the dipolar energy is very much less than the separation of the A and € rotor states. They are zero for the E states and d for the A states where y’h’ -.r3 (3 cos2 ly -&t,l .A>. (1.3) In eqn (1.3)’ 1’ is the proton gyromagnetic ratio Y is the intramethyl proton separation and the Kronecker delta is zero unless the subscripts are identical in which case it is unity. As a result the nuclear spin states of the lower rotational levels of a free methyl rotor are as illustrated in fig. 2. 2. WEAKLY HINDERED METHYL ROTORS In real systems in their solid states either the crystalline or the intramolecular environment of the methyl group invariably introduces a hindering potential which in turn modifies the energy states of the methyl rotor. The nature of these modifica- tions depends both upon the symmetry of the hindering potential and upon its magni- METHYL ROTORS IN THE SOLID STATE m m I FIG.2.-Effects on the methyl nuclear energy states of the application of the rotational the Zeeman and the dipolar haniiltonians (XR, Z and XD)shown schematically.E is Zeeman energy split- ting and dis the dipolar shift of eqn ( 1.3). Also illustrated is the r.f. absorption spectrum to which these energy states would give rise. The contribution of each r.f. transition is indicated by the side of the corresponding component. tude. For example changes in symmetry modify the number and classification of states which comprise a complete basis set. This is illustrated in fig. 1 where the free rotor states are classified under the symmetry group C6,which is isomorphous with the permutation group of a triangle in a six-fold potential as well as under C3.Increases in the magnitude on the other hand modify the energy eigenvalues bringing together all those states which comprise a complete set while at the same time separat- ing one set from another. This is also illustrated in fig. 1 where the effect of increas- ing a six-fold potential is shown to be different from that of increasing a three-fold barrier. This difference which effectively changes the number of lower energy states has a significant effect on the nuclear resonance properties. To digress for a moment we can see from fig. 1 that if the magnitude of the barrier is increased indefinitely then (in the case of C3 symmetry for example) the first pair of degenerate E states converges on the ground A state and their small separation is then referred to as a tunnelling splitting.Together they will form the torsional ground state of a highly hindered methyl group. Returning to the weakly hindered methyl groups we are now in a position to appreciate how their nuclear resonance properties in addition to being governed by the ordinary nuclear transition processes are compounded by restrictions due to the symmetry species of the various nuclear energy states. For example the phonon reservoir by itself is unable to maintain thermal equilibrium between all the weakly hindered rotor levels because alone it cannot promote symmetry conversion transi- tions. As a result the situation following a sudden temperature change is expected to be the rapid re-establishment of internal equilibrium within each symmetry species but a departure of the absolute populations from their overall thermal equilibrium values because of the inequality of the partition functions of the symmetry species sub-systems.On the basis of this expectation if Z,(L) and Z,(L) represent the partition functions of the A and E sub-systems respectively at a lattice temperatureL and p(L) = Z,(L)/Z,(L) then the post-jump deviation Ansl,from thermal equilibrium population n,, for the state i of symmetry s is given by = n.4 -P(LJl/[l + P(U1) (2.1) and At? = AnE,, }?El {[p(L.2) -P(L1)1/dL2)[1 + dLI)11* (2.2 P. S. ALLEN 137 Thus the A and the E deviations will be opposite in sign of a relative magnitude which depends on the ratio of their partition functions and will increase as the magnitude of the temperature jump increases.In consequence such observables as the spectrum which depends on the populations through the intensities of its components and the spin-lattice relaxation rates which are dominated by the symmetry conversion transi- tions and are therefore driven by the A to E population difference both vary with time during the approach to overall thermal equilibrium. Moreover when such equili- brium is established they continue to depend on these population differences and therefore provide a means by way of the Boltzmann factor to evaluate the rotor energies. The approach to equilibrium of the Zeeman spin-lattice relaxation of methyl protons in lithium acetate is illustrated in fig.3 from which it is clear that I1 II d .. a a 4 a I1 II .a II time Ih FIG.3.-Experimental data on the efficiency of the spin-lattice relaxation process for the methyl protons in lithium acetate following a sudden temperature change from 77 to 4.2 K when ty = 0. TIrepresents a “ global ” spin-lattice relaxation time which turns a blind eye to minor deviations from a perfect exponential recovery of the magnetization. Vertical dotted lines represent alternately either a lowering or raising of the temperature between 77 and 4.2K. The zero of time is defined as the time when the cryostat equilibrated at 4.2 K following the first temperature change. the relaxation efficiency increases markedly immediately following the temperature jump before slowly falling back to its equilibrium value at 4.2 K as the A to E popu-lation imbalance is destroyed.An interesting transient phenomenon which occurs for weakly hindered symmetric groups and which owes its very existence to this population imbalance between symmetry species is the dipolar polarization generated by a sudden temperature jump. This phenomenon was first observed by Haupt3 in 7-picolene and is illustrated for lithium acetate in fig. 4. This dipolar signal whose intensity can be lo4 times greater than its equilibrium value appears (following a short r.f. pulse) in quadrature with the Zeeman signal. It reflects an asymmetry about the central Larmor frequency uL,in the frequency distribution of the rotating frame magnetization.For a collec- tion of isolated but aligned methyl groups whose spectrum would be simply the stick pattern generated from fig. 2 this quadrature signal could only arise from an asym- metry of the intensities of the r.f. transitions within the A symmetry species since only they produce satellites away from uL. The E states in such a system can only generate magnetization precessing at uLitself. The growth and decay of the intensity of the quadrature signal could then reflect a transient departure of the A species from a unique spin temperature in accompaniment with the equilibration of relative popula- tions of the symmetry species. The identification of which relaxational transitions are dominant in the generation and decay of the dipolar polarization and in the transient and equilibrium spin-lattice relaxation process is important to the understanding of symmetric molecular groups.METHYL ROTORS IN THE SOLID STATE In the past isotropic sums over all possible transitions have been used to compare with experimental data from polycrystalline samples. The different angular depend- ence of these transitions has not been exploited. However issues such as inter- versus intramethyl transitions the relative roles of E type perturbations (caus- ing A to E transitions) and A type perturbations (causing symmetry retention transitions) and competition between A,,, = 1 and A,,, = 2 transitions are all amen- able to angular investigation. For example under the C3 symmetry classifications .-Y c 3 x a .--I I 1-1 I 1 1 1 I 1 I 0 200 LOO 600 800 time Is FIG.4.-Time evolution of the intensity of the dipolar polarization signal from the methyl protons in lithium acetate following a temperature change from 4.2 to 10 K when ty = 0.The zero of time is defined by the equilibration of the cryostat at 10 K 20 s after initiation of the temperature jump. when t,v = 0 all AmI = 1 intramethyl transition rates go to zero but an intramethyl perturbation of E symmetry causing A,,, = 2 transitions remains. The reason for the neglect of angular studies has been the complete absence of single crystal samples. Lithium acetate has removed this restriction and provided the first example of a weakly hindered methyl rotor in single crystal form. In the following section we shall out- line the exploitation of angular effects in lithium acetate and because of limitations of space we shall concentrate on the resonance spectrum.3. METHYL ROTORS IN LITHIUM ACETATE In the solid state of the deuterated analogue of lithium acetate dihydrate (CH,COOLi 2D,O) the methyl groups are essentially arranged in coaxial pairs with a methyl plane separation of 0.25 nm.4 The next nearest magnetic nuclei ('Li) are at a distance of 0.41 nm from the methyl centres and for the purposes of this discussion we shall neglect their presence in comparison with the mutual effects of all the methyl protons. The proton-proton dipolar interaction can be sub-divided into the intra- methyl and the intermethyl parts; the fact that the inter part is significant when com- pared to its intra counterpart makes it expedient to treat the whole of the proton- proton interaction in terms of a single symmetry group.If the two methyl triangles are labelled clockwise looking along their respective positive C3 axes with group I comprising protons 1 2 and 3 and group I1 containing protons 4 5 and 6 then their feasible transformations can be represented by the nine permutation elements formed from (123) (132) (456) (465) and their products. These nine elements form a group which is isomorphous with the direct-product P. S. ALLEN 139 group 'C x "C,. Since all of these nine elements are conjugate there will be nine one-dimensional irreducible representations which we shall label rl to T9,where rl is the totally symmetric representation and the remaining identities are given in table 3.If the rotor functions are simply taken to be the products of the two individual free TABLE 3.-IDENTIFICATION OF THE IRREDUCIBLE REPRESENTATIONS OF THE DIRECT PRODUCT GROUP 'c,X "c3WITH THE ROTATIONAL AND SPIN STATES OF A PAIR OF FREE COAXIAL METHYL ROTORS. ~~~ energy/K rotor states spin states rotor functions the energy states for the methyl pair together with their symmetry classifications will also be as illustrated in table 3. The spin representation R,,for the six spins Z = + is of order 64 but can be reduced under the direct-product group according to R,= i6r1+ ~(r, + r3+ r4+ r,)+ 4(r + r7+ rS+ r9). The sixteen-fold degeneracy of the rlspin state is made up from a septet a quintet a triplet and a singlet.The eight-fold r2to Tsspin states each comprise a quintet and a triplet whereas a triplet and a singlet constitute the four-fold r6to T9 spin states. The coupling of these spin functions to the free rotor states are as shown in table 3. The nuclear resonance properties of this six spin symmetry conscious system are undoubtedly time consuming to describe and so we shall focus attention only on the spectrum and the truncated part of XD. The intramethyl components of XD(one from each group) each have a rlsymmetry spin operator whose angular dependence is (3 cos2 ry -1). In addition r2to T5spin operator terms exist for the intra interac- tion which depend on sin2 ry. The intermethyl interaction on the other hand con- tains spin operator terms of all symmetry species.Those which transform as rl follow (3 cos2 ry -l) the same angular dependence holds for T6and T9terms where- as terms of all other symmetry species are proportional to sin ry. The multiplication table of the direct product group will establish which terms in XDmight have finite matrix elements between the various spin states. However it is clear that all diagonal elements if they are finite will vary as (3 cos2 ry -1). The predominance of these elements is apparent from the magic-angle spectrum of fig. 5(a) (ie. ry = 54"44') which has a second moment of (1.8 & 0.2)G' at 4.2 K no discernible structure and which broadens only very slightly with increasing temperature as it picks up small amounts of intermethyl dipolar shifted intensity as the E x E rotor states increase in population.Changing to a parallel orientation of the C3axes relative to B will re- METHYL ROTORS IN THE SOLID STATE move all the finite off-diagonal elements depending on sin ty and at the same time maximise the dipolar shifts of the diagonal elements. Unfortunately the angular dependence of both the inter and intra diagonal interactions is the same and it is necessary to evaluate the matrix elements to see their effects. The intra dipolar interaction by itself produces a spectrum identical to that generated by a collection of isolated groups whose energy states are illustrated in fig. 2. It is the inter interaction which gives the ty = 0 spectrum its characteristic 0 20 400 20 40 60 0 20 40 60 kHz k Hz kHz Fig.5.-Equilibrium spectra (normalized to the same peak height) of the methyl protons in lithium acetate. The conditions relating to each spectrum are (a) tp = 54" 44' and T = 4.2 K (b) ty = 0 and T = 4.2 K and (c) v/ = 0 and T = 77 K. shape [see fig. 5(b) and (c)]. For the ground state alone the inter interaction lifts all but one of the sixteen-fold spin degeneracies produces twelve symmetrically shifted pairs of transitions and most importantly leaves no ground state transition at the Zeeman energy. The central intensity shown in the experimental data of fig. 5(b) arises from unshifted components from the higher E x E rotor states and from overlap due to extra-methyl broadening.Fig. 5(c) shows how the ty = 0 spectrum changes in going to 77 K. The relative increase in peak heights is due to the increased contribu- tions from the next higher rlrotational states which require temperature increases of this magnitude to populate them appreciably. If as is likely the barrier to rotation is not zero then the states of table 3 will converge with increasing barrier height in a manner analogous to that portrayed in fig. 1. If moreoever the hindering barrier is six-fold the problem must be re- analysed in terms of a larger direct-product group. Thus a detailed analysis of the spectral components of each rotational state enables the temperature dependence of the angular variations in the spectrum to be employed to obtain an estimate of the rotor energies and through them the symmetry and magnitude of the hindering potential.In addition to explaining the equilibrium spectrum this analysis is also needed to account for the shape of the transient dipolar signal depending as it does on the non-equilibrium-population-differencedriving force for transitions between spin states. The growth and decay of the intensity of that signal meanwhile will require the analysis of the matrix elements of the non-secular parts of X,,. The foregoing is intended to provide a meaningful framework within which more extensive examples relating the behaviour of the spectrum the dipolar polarization signal and the relaxation data to the evaluation of methyl rotor states and transition rates can be presented for discussion at the meeting.P. S. ALLEN 141 I am grateful to both Peter Branson and Michael McCall for their painstaking collection of the experimental data presented here and to the S.R.C. for equipment grants which furnished much of the apparatus. M. Tinkham Group Theory and Quantum Mechanics (McGraw Hill New York 1964) pp. 39-43. J. Haupt Z. Naturforsch. 1971 26A 1578. J. Haupt Phys. Letters 1972 38A 389. J. L. Galigne M. Mouret and J. Falgueirettes Acta Cryst. 1970 B26 368.