S Clough

Nuclear magnetic resonance line shapes of methyl groups undergoing tunnelling rotation

Nuclear magnetic resonance spectra are calculated for an equilateral triangle of protons tunnelling between equivalent orientations. It is shown that the effect of the motion may be described by introducing an interproton exchange interaction into the nuclear spin Hamiltonian, the coefficient J being related to the height of the potential barrier to rotation. This term has matrix elements equivalent to those of the rotational energy operator. Spectra are obtained for arbitrary J, giving a quantum-mechanical solution of a motional narrowing problem and demonstrating the formal identity of the phenomenon, in this case with exchange narrowing. The limit of large J is in agreement with the classical result of Andrew and Bersohn. Isotropic averages of the spectra with Gaussian broadening to take account of inter-methyl-group interactions are computed for comparison with measurements on powders at low temperatures.

The rate of thermally activated methyl group rotation in solids

Evidence is presented which suggests that the thermally activated reorientation rate tau -1 of methyl groups in a very wide range of solids closely follows a universal law which relates tau -1 to the hindering barrier height V and the temperature theta . Thus tau -1 simply measures theta with a calibration curve dependent on V. A simple theory of the calibration curves is described, using a thermal average over all methyl group torsion-rotation states of the modulus of the expectation value of angular momentum. No adjustable parameters are involved if a simple cosine hindering potential is assumed. The relationship is tested on a large number of methyl-containing solids for which V has been deduced from measurements of the tunnel splitting at low temperatures. The predicted temperature-dependent reorientation rate turns out to be very similar to that given by a modified version of the theory of Stejskal and Gutowsky (1958). The relationship between the theories is discussed. Comparison is made with experimental data derived from magnetic resonance relaxation time measurements, and from inelastic neutron scattering studies.

Study of Tunneling Rotation of Methyl Groups by Electron Spin Resonance and Electron Nuclear Double Resonance

Free radicals formed by γ irradiation of single crystals of 4‐methyl‐2,6‐di‐tert‐iarybutylphenol are studied by resonance methods at low temperatures. The important part of the radical is the fragment Ċ–CH3, where the free‐electron spin occupies a 2p orbital centered on the carbon atom attached to the methyl group and experiences a hyperfine interaction with each of the three protons dependent on the proton position, giving rise to structure in the electron spin resonance spectrum which is sensitive to motion of the protons. The hindered methyl group performs torsional oscillations and also undergoes tunneling rotation between its three equivalent orientations, the nuclear spin states being correlated with the motion through the exclusion principle, at least at temperatures sufficiently low for the fragment to be regarded as isolated. An almost symmetrical seven line ESR spectrum is then expected and is observed at 4.2°K. In first order the spectrum is independent of the tunneling frequency when this is l...

Coupled tunnelling motion of a pair of methyl groups in lithium acetate studied by inelastic neutron scattering

The tunnelling and torsional motions of methyl groups in lithium acetate dihydrate (CH3COOLi·2H2O) have been studied in detail by incoherent inelastic neutron scattering. The results are interpreted by a model of pairs of methyl groups performing a coupled tunnelling motion. The strength of the coupling term is estimated to be about twice as strong as the threefold hindering barrier for the single particle motion.

The correlation of methyl tunnelling and thermally activated reorientation

A strong correlation is demonstrated between two parameters connected with methyl group motion, and measured in a wide variety of solids. The parameters are h nu t, the tunnel splitting of the ground torsional state, and theta min, the temperature at which the rate constant tau -1 associated with thermally activated reorientation has the value 2*108s-1. The results suggest that there exists a universal law of the form f( nu , tau -1, theta )=0 governing the temperature ( theta ) dependence of tau -1. This is independent of the phonon spectrum of the host lattice, in conflict with much recent theoretical work but in harmony with the model of Das (1957).

Proton spin-lattice relaxation due to tunnelling motions

A theory of proton spin-lattice relaxation due to the rotation of hindered methyl groups is described. The theory bridges the gap between low temperatures when the motion may be described as a tunnelling rotation, and high temperatures when it is better visualized as a random jumping between three equivalent orientations. In this higher temperature range the theory reduces to that of Bloembergen, Purcell and Pound (1948). Two kinds of thermally activated transitions are identified between eigenstates of the tunnelling methyl group. Together these give rise to a frequency spectrum for the motion which differs very considerably from the Lorentzian spectrum assumed by the BPP theory. As a consequence, the temperature dependence of the spin-lattice relaxation time should be much more complex at low temperatures than has previously been recognized. The theory offers an explanation for some hitherto unexplained published results and indicates that T1 studies are a sensitive way of studying the details of low temperature rotational motion.

The approach to classical dynamics of reorienting methyl groups

The authors have studied the approach towards classical behaviour of a dynamical system which exhibits strong quantum effects at low temperature. By combining nuclear magnetic relaxation and inelastic neutron scattering data, the rate of thermally activated methyl-group reorientation is measured from 10 to 65K for the 4-methyl group of MDBP (4-methyl-2,6-ditertiarybutylphenol). At low temperatures reorientation occurs by thermally assisted quantum tunnelling processes. The reorientation rate converges towards the predictions of classical theory with increasing temperature, but still deviates significantly at 65K. It appears that the classical theory may be used without important errors only at temperatures for which kT exceeds the methyl torsion splitting. In this high-temperature range a simple classical calculation of the reorientation rate is within a factor of 2 or 3 of the measured value.

The transition from free quantum tunnelling to thermally driven motion of methyl groups

Methyl groups in solids are representative of systems which exhibit tunnelling at low temperatures and thermally activated hopping motion at high temperatures. They exhibit very clearly all the features of the transition between free and thermally driven quantum tunnelling, the latter being describable in terms of classical hopping concepts. A satisfactory theoretical description has been lacking because the usual approaches, which treat the dynamic part of the scalar hindering potential by time-dependent perturbation theory are, the author suggest, incomplete. It is shown that a better description is possible if a vector potential arising from the motion of the lattice is introduced in addition to the scalar potential. This leads to a new Hamiltonian and a representation in which the important terms, which lead to the experimentally observed spectroscopic changes with increasing temperature, are diagonal. In this new representation a simple model emerges which exhibits most, if not all, of the observed experimental features. It also resolves an old problem: how to reconcile the quantum mechanical requirement that the rotation angles phi =0 and phi =2 pi are identical, with the classical assumption that they are distinguishable, and thus provides a bridge between the quantum and classical descriptions of rotation.

The Haupt effect: coupled rotational and dipolar relaxation of methyl groups

A theory is described for the dynamic proton dipolar polarization observed by Haupt (1972) in 4-methylpyridine following a sudden temperature change. The theory differs from that of Haupt in assuming that transitions which change the rotational quantum number of the 4-methyl group by +or-3 occur very rapidly, maintaining thermal equilibrium within each of the three subsets of rotational levels corresponding to the three methyl group proton spin symmetry species A, Ea and Eb. The difference of A and E species populations approaches the new equilibrium value slowly and exponentially, following the temperature jump, and generates dipolar polarization in the process. Transitions between Ea and Eb species lead to destruction of the polarization, whose evolution from zero due to these competing processes has the simple form C(exp(-at)-exp(-bt)). This is checked by a modified version of Haupts experiment in which the initial temperature jump is followed by a later burst of RF pulses which reduces the dipolar polarization to zero.

METHYL TUNNELLING SPECTROSCOPY AND LEVEL-CROSSING PHENOMENA IN SOLID ACETONE

Reports the observation in acetone at low temperature of a number of new phenomena connected with methyl tunnelling motion and field-cycling nuclear magnetic resonance spectroscopy. The authors observe three strong anomalies in the field-cycling nuclear magnetic resonance spectroscopy. The authors observe three strong anomalies in the field dependence of proton spin-lattice relaxation at 4K for fields B0, B0/3 and B0/3 and B0/3 and B0/3 with B0=2.25T. They are due to internal resonances (or level crossings) occurring for n nu 0= nu t, (n=1, 2, 3) where nu 0 is the field-dependent nuclear Larmor frequency and nu t is the methyl tunnel frequency. This structure is a consequence of the field-cycling technique which involves rapid switches of the magnetic field to enable all measurements of proton magnetisation to be made at a single field 0.6 T. A simple thermodynamic model is described in which there is very rapid thermal equilibration between the nuclear Zeeman and methyl tunnel reservoirs at the fields corresponding to the n=1 and 2 anomalies and the nuclear magnetisation is otherwise unaffected by the rapid field changes. This model successfully simulates the observed spectrum and a variety of other field-cycling experiments.