Neutron incoherent scattering law for diffusion in a cosine potential in one dimension: Application to self-diffusion in smectic phases

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Abstract
The properties of a particle undergoing translational diffusion in one dimension, in the presence of a cosine potential, are investigated. Translational order parameters are defined and the neutron incoherent scattering law is calculated. The simple diffusion model and the jump model between equidistant sites are found as limiting cases of zero and infinite potential heights, respectively. This formalism is then used to re-analyse neutron quasi-elastic scattering data in smectic A terephthal-bis-4-n-butyl aniline. Combining these results and N.M.R. spin echo measurements of the long-range translational diffusion coefficient across the layers, it is found that the height of the barrier which maintains the molecules in the smectic planes, at 184°C, is about 1·2 k b T an , where T an is the smectic A-nematic transition temperature. This corresponds to a value of the translational order parameter of about 0·29.