Diffuse X-ray scattering from a crystal with spheroidal pores

Abstract

The morphology of porous systems created using electrochemical anodization techniques strongly depends on the structural characteristics and chemical properties of the initial materials, concentration of impurities, current density, process duration, and electrolyte composition [1]. Depending on the tech nological conditions, pores of various shapes and sizes are formed in the crystalline matter. Moreover, the morphology of a porous crystal can change under the action of external factors. For example, heat treatment of porous InP leads to the formation of ellipsolidal (in particular, spherical) pores [2]. This behavior is char acteristic of both current line oriented (vertical) and crystallographic (inclined) pores.Analysis of the diffuse X ray scattering provides information about the pore sizes, their orientation rel ative to the crystal surface, the presence or absence of a spatial order, azimuthal anisotropy, etc. [3][4][5][6][7][8]. An important role in the distribution of diffuse scattering is played by the shape of pores. Previously, the theory of diffuse X ray scattering employed two models, according to which pores had the shape of rectangular parallelepipeds [3] or cylinders [5][6][7][8]. Diffuse scatter ing from a crystal with elongated ellipsoidal pores has not been analyzed so far.This investigation is devoted to diffuse X ray scat tering from a crystal with pores having an elongated spheroidal shape, representing a prolate ellipsoid with equal semiaxes in the lateral plane, which are shorter than the vertical semiaxis.Using the formalism of Kato's statistical theory of diffraction [9], an expression for the intensity of the diffuse X ray scattering as a function of vector q (devi ation from the reciprocal lattice vector h) in the kine matical approximation can be written as follows:(1) Here, V 0 is the x ray irradiated volume of the crystal, f is the static Debye-Waller factor, a h is a parameter that characterizes the scattering ability of the crystal, and τ(q) is the so called correlation volume. It should be noted that relation (1) does not take into account dynamical effects in the diffuse X ray scattering the ory-in particular, the primary extinction of the transmitted X ray intensity and the secondary extinc tion in the scattering of incoherent waves. For addi tional simplification, we will also ignore the spatial correlation of pores [6,10]. Under these conditions, the correlation volume τ(q) is given by the Fourier transform of the Kato intrinsic correlation function g(ρ):(2)The intrinsic correlation function of pores in a crystal can be represented as follows [11,12]:where V p is the pore volume and D(r) is a function that depends on the random deformations field and describes local damage of the crystalline lattice [11,12]. In the case under consideration, these distortions are caused by the presence of voids in the crystalline matrix. Denoting by c p the concentration of pores in the matrix and using the D(r) function, we can express the static Debye-Waller factor as follows [11]:(4)) . e...

High-resolution X-ray diffraction in crystalline structures with quantum dots

“…An analytic expression for the intrinsic correlation function has been derived for certain defect models in Refs [38,39,47,48]. Considering diffuse scattering in the framework of triple-axis diffractometry implies integrating (4) over q y .…”