PH YSICA L RE VIE% B VO LUME 16, N UMBER 2 15 J U LY 1977 Elastic continuum theory of interface-atom mean-square displacements B. Djafari-Rouhani Service de Physique Atornique, Section d'Etude des Interactions Gaz-Solides, Centre d'Etudes Nucleaires de Saclay, Boite Postale No. 2, 91190 Gif-sur-Yvette, France L. Dobrzynski Equipe de Physique des Solides du Laboratoire des Surfaces et Interfaces, ~ Institut Superieur d'Electronique du Nord, 3 rue Francis Baes, 59046 Lille Cedex, France R. F. Wallis~ Laboratoire de Physique des Solides, Universite Pierre et Marie Curie, Paris and University of California, Irvine, California (Received 7 September 1976) The mean-square displacements of particles near an interface between two different isotropic elastic continua are calculated for the first time. A Green's-function method was used in the high-temperature limit. The dependence of the mean-square displacements on distance from the interface is exhibited explicitly. I. INTRODUCTION II. INTERFACE GREEN'S FUNCTIONS Extensive experimental' and theoretical2 investi- gations of surface-atom mean-square displace- ments (MSD) have been carried out during the last few years. The experimental values can be ob- tained by low-energy electron diffraction, scat- tering of atoms, ' or Mossbauer effect. These techniques provide the atom MSD for clean sur- faces as well as for surfaces covered with an ad- sorbed monolayer. One can expect that the atom MSD of surfaces covered with several monolayers will also be measured. The atom MSD near an in- terface between two solids can be measured quali- tatively in this way, and may be measured quanti- tatively by more sophisticated techniques. In the present paper, we study for the first time the MSD of particles near an interface between two different solids. We use an approach based on the Green's functions for two semi-infinite sotropic elastic media bounded by a planar interface. This approach' was used previously to calculate the sur- face-atom MSD as function of distance from the surface in the high-temperature limit. The values at the interface are obtained here in a way similar to that used for the surface. ' We find in the pre- sent case that the difference between the interface and the bulk MSD becomes inversely proportional to the distance to the interface. This behavior was also obtained for the difference between the free surface and the bulk MSD. ' ' The necessary interface Green's functions have already been used for the calculation of the inter- face specific heat at low temperatures. ' We will derive them in Sec. II. Then, (Sec. III) we will de- rive the atom MSD for atoms near an interface. We consider now two different elastic isotropic media 1 and 2 occupying, respectively, the half- spaces x, &0 and x, &0. We need to know the Green's function U for the two crystals connected by this planar interface. The procedure is similar to the one used' for the surface. Let us first in- troduce the Green's function for crystal 1 which, inside the crystal, satisfies the equation 1~ Q2 =6 „5(x- x'), (2.1) gv xti exv 5(x- x') . (2. 2) In order to obtain the Qreen's function U for the two crystals connected by a planar surface, we have to solve Eqs. (2.1) and (2. 2) subject to the boundary conditions at x, = 0: where C»„are the position-independent elastic moduli of the material, p is the mass density, ~ is the frequency of the time-dependent elastic dis- placement field in the medium, and n, P, p. , v are the Cartesian indices x, y, or z. For an isotropic crystal, the C»„are functions of the more usual elastic constants C„, C, and C„=C„-2C„. The equation satisfied by the Green's function for crystal 2 is obtained from the one above by chang- ing p and C ~„„, respectively, to p' and C ~», . 16 741