Surface Effects in the Second-Order Doppler Shift of the Mössbauer Resonance

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Abstract
The second-order Doppler shift in the Mössbauer effect depends upon the mean-square velocities of the emitting and absorbing atoms. On the basis of a theorem discussed by Born in connection with the lattice dynamical theory of the Debye-Waller factor, a general expression has been obtained for the mean-square velocity of an arbitrary atom in a crystal lattice, assuming harmonic forces. The results is valid for any temperature and may be applied to lattices having free surfaces or impurities. Approximate expressions are developed for the high- and low-temperature limits. The general results are applied to specific calculations of the mean-square velocity for atoms at or near a free surface. Ordinarily, the mean-square velocity turns out to be smaller for an atom at the surface than for one in the interior of the crystal. This is a consequence of the surface atom being linked to fewer neighboring atoms than is the case for an interior atom. It is concluded, however, that whether or not a crystal lattice possesses surface modes of vibration has little direct bearing on the mean-square velocity of surface atoms.