Theory of Anharmonic Lattice Vibration in Metallic Fine Particles. I

\Ve apply the self-consistent Einstein moclel to the theory of anharmonic lattice vibration in metallic fine particles, and discuss the size dependence of the melting and superconducting transition temperatures in the connection to the softening of the surface lattice vibrations.Assuming a simple interatomic potential and certain distribution of the particle size, we express both transition temperatures as functions of the average radius of fine particles and show that the numerical results calculatecl from the theoretical expressions arc in fairly good accord with the experiments. § I. IntroductionIn a previous paper, one of the present authors has discussed, on the basis of a self-consistent Einstein model, that the atoms on the surface of metal crystal perform oscillations with much larger amplitudes compared with the interior atoms.])The large surface lattice vibration is related to the softening of the frequencies of the surface atoms and also closely related to the so-called surface relaxation of the lattice constant.Since the surface to volume ratio of metallic fine particles is increased with decreasing size, the surface softening should give appreciable effects in the various thermal properties of metallic fine particles vvith sufficiently small radii.Up to the present time, there are many accumulated experimental facts to sho1v that this is indeed the case.It has long been known for instance that the metallic fine particles have melting points much lower than those of the corresponding bulk metals.The first observation of this depression of the melting point was made by Takagi 2 l on Pb, Sn and I3i thin films by the use of the electron diffraction method.Since then similar observations have been reported, for example, on Sn by vVronski, 31 on Pb and In by Coombes, 41 on Au by Buffat and BorePJ and so on.This phenomena are undoubtedly due to the surface effects, and phenomenological theories based on the thermodynamic arguments have been presented.61 A simple microscopic theory 1vas first proposed by one of the present authorsn with a pretty success in explaining the main experimental facts.This paper contains a sophistication of the previous theory.Another example of the phenomena 1vhich seem to be related with the surface