X-Ray Tomographic Imaging of Crystal Structure at the Atomic Level

Abstract

A direct nondiffractive tomographic algorithm is proposed for the determination of the crystal structure from real-space projections obtained by illuminating the sample with white x rays. This approach was applied to the pattern of the directional fine structure in absorption of white x rays recorded for a GaP crystal and allowed for a determination of the electron density distribution within the unit cell. DOI: 10.1103/PhysRevLett.96.035502 PACS numbers: 61.10.ÿi, 42.30.Rx, 81.70.Tx The majority of x-ray methods for crystal structure investigations are based on diffraction phenomena and sample the information in the reciprocal space by measuring the intensities at discrete Bragg peak positions [1]. Usually, a Fourier transformation is used to recover the electron density distribution of the object. However, a direct back transform is hindered by the lack of phase information in the recorded intensity data and the inversion algorithms are ambiguous. The phase information can become directly accessible, e.g., by using anomalous [2] or multibeam x-ray diffraction [3]. Another approach involves measuring the x-ray wave field intensity at atomic sites inside a crystal as it is performed in the x-ray standing wave technique [4] or in x-ray holography (XH) [5,6]. The absorption cross section in these methods is modulated by x-ray diffraction, which results in an angular dependent absorption fine structure measured through the secondary yield coming from specific sorts of atoms. This directional fine structure can be inverted to real space by using Fourier [7] or holographic reconstruction [8]. All these methods utilize monochromatic x-ray radiation.In this Letter we propose and demonstrate experimentally that crystal structure can be recovered from real-space projections obtained with polychromatic x rays (hereafter called white x rays), by using a nondiffractive tomographic algorithm. This approach is based on recent work [9], which showed that the directional absorption fine structure is related to the spherical projection of crystal structure. For a white x-ray beam, the wave field, formed by interference of the incident beam with the waves scattered on single atoms, cancels out by energy integration for all directions, except for the forward scattering component, coinciding with the incident beam. In a practical case, the white x-ray radiation is not uniform over the whole energy range, which leads to remnant high order diffraction effects. However, in the presence of long-range order, the distortion of the real-space projection is small for directions coinciding with the lattice planes and, as it will be shown, does not significantly influence tomographic reconstruction.Scattering of spherical waves from atoms of the sample onto an absorbing atom results in directional absorption fine structure [10]. For a white x-ray beam this fine structure can be written as [9] k ÿ2r 0 Rewherek is the beam direction, r 0 is the Thomson scattering length, Nk is the effective x-ray spectrum, r is the electron density, r is...

Directional fine structure in absorption of white x rays: A tomographic interpretation

“…The calculation is verified by comparison with the data recorded in experiment, preliminarily described in Ref. 21, performed for a GaP crystal.…”

“…⌫ is a small positive number corresponding to the imaginary part of the wave vector having real part k. The directional fine structure in absorption of polychromatic x rays can be calculated as 20,21 …”

“…In addition, by virtue of the reciprocity theorem, they are connected to the Kossel diffraction lines. 19 In recent work, 20,21 it was demonstrated that the patterns of the directional fine structure in x-ray absorption recorded using white x rays have a simple real-space interpretation. For a white x-ray beam, the wave field variations, due to interference of the incident beam with the waves scattered from single atoms, cancel out by energy integration for all directions, except for the forward scattering component, coinciding with the incident beam.…”

Projections of local atomic structure revealed by wavelet analysis of x-ray absorption anisotropy

“…2,3 This approach analyzes absorption anisotropy of polychromatic x rays, which arises due to the interaction between the incident plane wave with spherical waves scattered inside the sample ͓see Figs. 1͑a͒ and 1͑b͔͒.…”

“…3 It analyzes the intensity of bands, corresponding to projections of atomic planes, that are formed due to the superposition of signals from a large number of periodically arranged scatterers as shown in Fig. 1͑d͒.…”