Temperature diffuse scattering for cubic powder patterns

Abstract

In measuring the short range order diffuse scattering from powder patterns of binary alloys, the correction for temperature diffuse scattering becomes very important, particularly if the sample is held above the critical temperature. The correction is sometimes made by assuming that for powder patterns the approximation of independent vibrations is adequate: IT.D = Nlef ~" (1-e-~'M). (1) The rigorous theory of temperature diffuse scattering of X-rays is available (James, 1948), and by a few approximations it is possible to compute a temperature diffuse scattering curve for cubic powder patterns which is a much better approximation than equation (1). From James, equation (5"45), ITD = Nlepe-2MkT [R]2-c°s~ aR; where, for an element: N is the number of atoms, I e the Thomson scattering per electron, f the atomic seat-tering factor, 2M the usual Debye factor, ]c the Boltzman constant, T the absolute temperature, m the mass of the atom, IRI (= 2sin0/2) the diffraction vector drawn from the origin of the reciprocal lattice to the position of measurement, Igl (= l/A) the vector from the nearest relevant reciprocal-lattice point to the position of measurement and equal to the reciprocal of the wavelength of the elastic wave, V~i the velocity of the elastic wave ~j, aRj the angle between R and the vibration direction of wave ~j, and ~ the summation over the three inde-i pendent waves. By assuming that all elastic waves have the same average velocity IF, that 2M is small, and that each Brillouin zone can be replaced by a sphere of equal volume, (2) becomes I rD = N le f2 (1-e-2M) g2max./3g2. (3) Within any Brillouin zone, g is expressed by g~-= R~÷R~kl-2RRhktCOS% where Rhkl is the vector from the origin to the center of zone hkl. The contribution to the powder pattern intensity, at fixed ]R I-2 sin 0/2, from one Brillouin zone is then given by gg. i~ o sin ~d~ Irjv = NIef ~ (1-e-2M) ~ o Rg" + R~kl-2RRh~l cos (4) where ~v 0 is determined by gmax.9" = R e +R]kz-2RRhktCOS q%. Multiplying by the appropriate multiplicity factor 3"hkt, and adding values from all Brillouin zones which contribute at a fixed value of IRI, gives the powder pattern intensity. As an example consider a face-centered cubic element. In this case gmax. = (3/n)x/a/a, where a is the edge of the cubic cell. In terms of the general variable x = 2a sin 0/2 and the values xhkt = 2a sin Ohkt/~ ~ for each unmixed * Research sponsored by the U.S. Office of Naval Research. hkl, the ratio of Ir~ to that for independent vibrations is given by ITD