Influence of anisotropic vibrational motion on diffraction from grain boundaries

Abstract

*Theoretical Division, Los Alamos National Laboratory Lxra Alamos, NM 87545 USA (Received October 23, 1989) (Revised November 28, 1989) INTRODUCTION X-ray diffraction has been used extensively, in recent years, to analyze the structure of high angle grain boundaries (1-4). A major difficulty in the interpretation of the X-ray results involves correcting for the vibrational motion of the atoms in the boundary (the Debye-Waller factor). Common practice is to assume that all of the atoms in the grain boundary have the same vibrational amplitude, independent of their local environment (1-4). Recent simulations (5) show that the magnitude and the direction of the atomic vibrations vary from atom to atom in the grain boundary region, suggesting that the assumption of isotropic vibrations is not valid. The experimental scattering factor is generally deconvoluted assuming isogopic vibrations in order to determine the atomic structure of the grain boundary. Fitzsimmons and Sass (3) have attempted to take the assymetric nature of the vibrations into account by fitting their scattering data with two vibrational frequencies (parallel and perpendicular to the grain boundary). In the present paper, we demonstrate that the assumptions of uniform and isotropic vibrations (one or two frequencies) does not sufficiently take the highly inhomogeneous environment of a grain boundary into account and could lead to marked changes in the experimentally derived grain boundary structure. We have recently introduced ~tn approximate method for calculating the free energy of defects in solids, which we called the local-harmonic fl.,H) model (6). In this method, the vibrations of each atom are determined by calculating the local-dynamical matrix (ignoring the coupling of vibrations of different atoms) and are employed in a determination of the free energy within the fi'amework of an Einstein model. The su'ucture of a defect is calculated by minimizing the free enertw with respect to the atomic coordinates. The vibrational frequencies of each atom thus reflect the local atomic distribution around the atom at finite temperature. In this paper, we calculate the scattering intensities from Y,=5 and Y,=13 (001) twist boundaries in gold based upon atomic structure determined from finite temperature atomic relaxations that include the vibrational distributions found within the LH model (6). We show that the assumption of isotropic, homogeneous atomic vibrations leads to predicted scattering factors that differ greatly from those obtained with the true anisotropic, inhomogeneous vibrations. THEORY The integrated X-ray scattering intensity is given by Co , where (hkl) is the reflecting plane and Co is a constant that includes information about the l.xnentz polarization, beam absorption, illuminated area, resolution, atomic scattering factor, etc. and will be ignored hereafter. The effects of thermal vibration are included in the finite-temperature structure factor, Fhkl(T), which is defined as