The temperature dependence of rotational tunnelling, modelled with a harmonic substitute system

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Abstract
In 1990, Hiller modelled the temperature dependence of the inelastic lines of the tunnelling spectrum of methyl groups, measured by neutron scattering, using a harmonic substitution system as a replacement for the methyl group. The results obtained there are based on a cumulant expansion that has been restricted to the first two cumulants for a simplified version of the solution of the equation of motion for the sigma + operator, lacking time ordering. In order to verify the results. we perform the cumulant expansion up to second order including time ordering, and also compute the third and fourth cumulants. but without considering time ordering. Using simple crystal models, we show that the time-ordering symbol may in fact be neglected and that higher cumulants become important only when the density of states is sharply peaked. With the Debye density of states it turns out that for very low temperatures the shift of the inelastic lines is proportional to T4, whereas the width increases proportionally to T4. Finally, we present results for the quasielastic line and discuss the mechanism that causes the shift and broadening of the inelastic lines.

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