Thick Perfect Crystals Research Articles

Two-dimensional recurrence relations and Takagi–Taupin equations. I. Dynamical X-ray diffraction by a perfect crystal

The dynamical diffraction of spatially restricted X-ray beams in a thick perfect crystal is studied using two-dimensional recurrence relations and the Takagi–Taupin (T-T) equations. It is shown that the two-dimensional recurrence relations are transformed into T-T equations when passing from a crystal with an array of discrete lattice planes to a model of continuous periodic electron density. The results of calculations of the X-ray diffraction field inside the crystal and the angular distribution of the scattering intensity in reciprocal space based on these two approaches are presented. It is shown that, when using the two-dimensional recurrence relations and T-T equations, the calculated contours of reciprocal-space maps and their qx sections are similar to each other, and the qz sections completely coincide.

Quantum-information approach to dynamical diffraction theory

We present a simplified model for dynamical diffraction of particles through a periodic thick perfect crystal based on repeated application of a coherent beam splitting unitary at coarse-grained lattice sites. By demanding translational invariance and a computationally tractable number of sites in the coarse-graining we show how this approach reproduces many results typical of dynamical diffraction theory and experiments. This approach has the benefit of being applicable in the thick, thin, and intermediate crystal regimes. The method is applied to a three-blade neutron interferometer to predict the output beam profiles, interference patterns, and contrast variation.

Features of X-ray diffraction in thick perfect crystals with deformed subsurface layers

Diffraction of X-rays in thick perfect crystals with deformed subsurface layers has been investigated. It is shown that in deformed layers kinematic scattering occurs, while in the bulk of perfect part dynamic scattering takes place. It is revealed that in the beams diffracted on surface defects, an essential role plays the second harmonic of the employed characteristic MoKα1 radiation.

Deterministic Retrieval of Complex Green’s Functions Using Hard X Rays

A massively parallel deterministic method is described for reconstructing shift-invariant complex Green's functions. As a first experimental implementation, we use a single phase contrast x-ray image to reconstruct the complex Green's function associated with Bragg reflection from a thick perfect crystal. The reconstruction is in excellent agreement with a classic prediction of dynamical diffraction theory.

Unambiguous reconstruction of the complex amplitude reflection coefficient of a laterally homogeneous crystal using analyser-based phase-contrast imaging

A theoretical approach is developed to invert analytically a hard X-ray analyser-based phase-contrast image of a known weak object to recover the complex amplitude reflection coefficient (ARC) of a laterally homogeneous crystal. Numerical simulations test the method to recover the ARC from two systems of interest: a thick perfect crystal and a linearly strained thin film. For the latter model, a kinematical diffraction approximation was used to recover the one-dimensional deformation profile from the ARC.

Investigation of distorted subsurface layers of thick perfect crystals

The effect of crystal surface defects on the intensity of X-ray scattering has been investigated. It was found that in the presence of distorted subsurface layers, the occurrence of reflection structures (fine lines) depended on the divergence and monochromaticity of the incident beam, as well as on the absorption and character of X-ray scattering on distorted layers and in the main bulk (perfect part) of the crystal.

Diffractometer alignment and polarization determination of X-ray beams by means of the Borrmann effect

The Borrmann effect, that is, the anomalous transmission of X-ray beams through sufficiently thick perfect crystals in Laue geometry, is a sensitive diffraction tool which has many fundamental and practical applications: the accurate alignment of any full four-circle instrument (equipped with a cradle in asymmetric design) with respect to the incident beam, measurement of the divergence of any X-ray beam, and the precise determination of the beam polarization, incident or diffracted. For these applications, it is shown that no modifications to the diffractometer are required. The measurements simply consist of measuring φ and χ rocking curves of the anomalous transmitted beam where the ω angle is used as a parameter. The analyses require the findings of the peak centroid positions, the full widths at half maximum and the rocking-curve integrated intensities. Some results are presented as illustrations of the method. These Borrmann-effect applications promise to be more helpful for diffractometers installed at synchrotrons than for in-house laboratories.

X‐ray diffraction in thick perfect crystals with deformed subsurface layers

AbstractThe diffraction of X‐rays in thick perfect crystals with deformed subsurface layers has been investigated. It was shown that the kinematic scattering of X‐rays took place in deformed layers, while the dynamical one occurred in the perfect part of the crystal, ‐its bulk; also, that in beams of X‐rays diffracted on surface defects, the role played by the second harmonics of the applied characteristic radiation MoKα1 was essential.

ENHANCED NUCLEAR RESONANCE SCATTERING IN DYNAMICAL DIFFRACTION OF GAMMA-RAYS

ibstract. The angular and energy dependence of the dynamical diffraction of gamma rays was studied experimentally in the symmetric Brag case. Mossbauer diffraction spectra were measured with high angular resolution in a pure nuclear Bragg reflection of a nearly perfect 57FeB0, single crystal. In the spectrum taken at the exact Bragg position, very broad hyperfine resonance peaks were observed. In the spectra just above and below the Bragg angle the reflection peaks were much smaller and were shifted to energy positions above and below resonance respectively. The results were interpreted by means of the dynamical theory and compared with computer calculations. The measurements represent the first direct observation of the enhancement of nuclear resonance scattering in the case of a thick crystal reflection. Furthermore the angular dependence of the resonance scattering by a perfect single crystal has been measured for the first time. Soon after the discovery of the Mossbauer effect the dynamical diffraction of gamma radiation by a regular ensemble of resonant nuclei was considered theoretically. First the diffraction of gamma rays incident on a thin perfect single-crystal film near the Bragg angle was studied (Trammell 1961). It was found that in this case the halfwidth of the resonance scattering is larger than for an isolated nucleus and increases with the thickness of the crystal film. The energy broadening of the resonance scattering was called the enhancement effect. A general treatment of the dynamical diffraction of resonant gamma rays in thick perfect single crystals was given for the hue case (Afanas’ev and Kagan 1965) and also for the Bragg case (Kagan et al1968). In the latter paper it was shown that in the case of a Bragg reflection by an infinitely thick perfect crystal, under certain conditions, an angular position can be found where the reflection becomes total in spite of the presence of a strong inelastic channel and independent of the energy distance from resonance. In the region around this angular position the reflectivity falls off rapidly. Thereby the reflectivity remains unchanged when the energy distance from resonance and the angular deviii:ion are both changed in sign. Also the Bragg reflection by thin perfect crystals was treated in a general way (Hannon and Trammell 1969). The contributions of other authors may be found summarised in review articles (Belyakov 1975, Champeney 1979). Recently the problem of filtering of

偏光Ｘ線源

Bragg reflections with≅45° Bragg angle, Laue-case diffraction from thick perfect crystals and synchrotron X-radiation are now utilized as plane-polarized X-ray sources. The characteristics of these sources are surveyed putting stress on the monochromator systems for the synchrotron radiation.

Simultaneous Diffraction of X Rays and the Borrmann Effect

The Borrmann effect (the anomalous transmission of x rays through thick perfect crystals) has been used as the primary diffraction in a study of simultaneous diffraction. The major part of the work is an analysis of multiple diffraction using the dynamical theory of x-ray diffraction. In the analysis we have utilized the concept of normal modes in describing the propagation of the x-ray wave field. We have determined these modes when the Bragg condition is exactly and simultaneously satisfied for all the diffracting planes. The particular configuration studied was one in which only three noncoplanar waves have appreciable amplitudes. We have shown that at least one of the proper modes of any three-field case of the type considered is a nonabsorbing mode of propagation (i.e., having nodes of the electric field at the atomic planes). This indicates that the primary anomalous transmission will persist with little or no change in intensity when a third reciprocal lattice point enters the Ewald sphere. In particular, when the third reciprocal lattice point is of the same family as the primary, simultaneous anomalous transmission is predicted. For this case, a relatively small change in the primary reflected beam and a twofold enhancement of the forward beam is calculated which are seen to be due to a superposition of two degenerate Borrmann effects having a common forward diffraction direction. Experimental results using polarized and unpolarized incident radiation are presented and discussed.