Uniqueness of the complex diffraction amplitude in x-ray Bragg diffraction

Inversion of the diffraction pattern from an inhomogeneously strained crystal using an iterative algorithm

“…The experimental determination of the local strain remains, however, an open issue: electron microscopy has the required spatial resolution but suffers from the need for sample thinning down to electron transparency, which modifies the strain field 2 ; High Resolution X-Ray Diffraction (HRXRD) is both strain sensitive and non-destructive but, as the phase of the scattered field is not experimentally accessible, the strain profile at the nanometer scale is only achieved through a model dependent approach 3,4 . In this context, direct inversion based on x-ray diffraction is a rapidly progressing technique 5,6,7 . The possibility to directly determine the structure from a diffraction pattern alone was first mentioned by Sayre 8 .…”

Application of the Phase-Retrieval X-Ray Diffractometry to an Ultra-High Spatial Resolution Mapping of SiGe Films near the Absorption Edge of Ge

“…In the kinematical approximation we can write the relationship between the complex diffraction amplitude R(Q(E)), diffracted by a single crystal under Bragg diffraction conditions, and the crystal complex structure function, wðE; zÞ ¼ c h ðE; zÞ e 2pihuðzÞ , as [2,4,7]:…”

“…It is important to reiterate here that the true, physical zeros of the CDA, Q m , are presumed to not depend on the radiation energy, E [4]. Furthermore, it is important to note that, since the parasitic scattering effects have an incoherent nature, they do not produce the physical zeros of the CDA that should be included in the complete phase profile (3) [7].…”

“…Furthermore, it is important to note that, since the parasitic scattering effects have an incoherent nature, they do not produce the physical zeros of the CDA that should be included in the complete phase profile (3) [7]. Thus, the CDA profile with the minimal (Hilbert) phase [2,4], retrieved from the experimentally measured intensity and analytically continued into the whole complex plane as [4,7] ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi…”

“…The method exploits the X-ray phase-retrieval formalism [2], that is based on the theorem that all causal physical functions are analytical [3] and, therefore, a logarithmic dispersion relation can be used to retrieve the X-ray phase of the complex diffraction amplitude (CDA) [4] scattered by a single crystal under Bragg conditions. The method has been applied successfully to a high-resolution determination of one-and two-dimensional lattice-strain distributions in silicon crystals with SiGe/Si superlattices and with implanted high-energy ions [1].…”