Dynamical Diffraction Theory Research Articles

Comparison of EBSD patterns simulated by two multislice methods.

The extraction of crystallography information from electron backscatter diffraction (EBSD) patterns can be facilitated by diffraction simulations based on the dynamical electron diffraction theory. In this work, the EBSD patterns are successfully simulated by two multislice methods, that is, the real space (RS) method and the revised real space (RRS) method. The calculation results by the two multislice methods are compared and analyzed in detail with respect to different accelerating voltages, Debye-Waller factors and aperture radii. It is found that the RRS method provides a larger view field of the EBSD patterns than that by the RS method under the same calculation conditions. Moreover, the Kikuchi bands of the EBSD patterns obtained by the RRS method have a better match with the experimental patterns than those by the RS method. Especially, the lattice parameters obtained by the RRS method are more accurate than those by the RS method. These results demonstrate that the RRS method is more accurate for simulating the EBSD patterns than the RS method within the accepted computation time.

Depth profiles of the interfacial strains of Si0.7Ge0.3/Si using three-beam Bragg-surface diffraction.

Interfacial strains are important factors affecting the structural and physical properties of crystalline multilayers and heterojunctions, and the performance of the devices made of multilayers used, for example, in nanowires, optoelectronic components, and many other applications. Currently existing strain measurement methods, such as grazing incidence X-ray diffraction (GIXD), cross-section transmission electron microscope, TEM, and coherent diffractive imaging, CDI, are limited by either the nanometer spatial resolution, penetration depth, or a destructive nature. Here we report a new non-destructive method of direct mapping the interfacial strain of [001] Si0.7Ge0.3/Si along the depth up to ~287 nm below the interface using three-beam Bragg-surface X-ray diffraction (BSD), where one wide-angle symmetric Bragg reflection and a surface reflection are simultaneously involved. Our method combining with the dynamical diffraction theory simulation can uniquely provide unit cell dimensions layer by layer, and is applicable to thicker samples.

The Scherrer equation and the dynamical theory of X-ray diffraction.

The Scherrer equation is a widely used tool to determine the crystallite size of polycrystalline samples. However, it is not clear if one can apply it to large crystallite sizes because its derivation is based on the kinematical theory of X-ray diffraction. For large and perfect crystals, it is more appropriate to use the dynamical theory of X-ray diffraction. Because of the appearance of polycrystalline materials with a high degree of crystalline perfection and large sizes, it is the authors' belief that it is important to establish the crystallite size limit for which the Scherrer equation can be applied. In this work, the diffraction peak profiles are calculated using the dynamical theory of X-ray diffraction for several Bragg reflections and crystallite sizes for Si, LaB6 and CeO2. The full width at half-maximum is then extracted and the crystallite size is computed using the Scherrer equation. It is shown that for crystals with linear absorption coefficients below 2117.3 cm(-1) the Scherrer equation is valid for crystallites with sizes up to 600 nm. It is also shown that as the size increases only the peaks at higher 2θ angles give good results, and if one uses peaks with 2θ > 60° the limit for use of the Scherrer equation would go up to 1 µm.

Formulation of dynamical theory of X-ray diffraction for perfect crystals in the Laue case using the Riemann surface.

The dynamical theory for perfect crystals in the Laue case was reformulated using the Riemann surface, as used in complex analysis. In the two-beam approximation, each branch of the dispersion surface is specified by one sheet of the Riemann surface. The characteristic features of the dispersion surface are analytically revealed using four parameters, which are the real and imaginary parts of two quantities specifying the degree of departure from the exact Bragg condition and the reflection strength. By representing these parameters on complex planes, these characteristics can be graphically depicted on the Riemann surface. In the conventional case, the absorption is small and the real part of the reflection strength is large, so the formulation is the same as the traditional analysis. However, when the real part of the reflection strength is small or zero, the two branches of the dispersion surface cross, and the dispersion relationship becomes similar to that of the Bragg case. This is because the geometrical relationships among the parameters are similar in both cases. The present analytical method is generally applicable, irrespective of the magnitudes of the parameters. Furthermore, the present method analytically revealed many characteristic features of the dispersion surface and will be quite instructive for further numerical calculations of rocking curves.

Spacing between graphene and metal substrates studied with total-reflection high-energy positron diffraction

The spacing between graphene and metal substrates has been investigated using total-reflection high-energy positron diffraction (TRHEPD). The rocking curve analysis performed, based on the dynamical diffraction theory, shows that the spacing between the graphene layer and a substrate of Cu(111) and that between graphene and a Co(0001) substrate are 3.34 Å and 2.06 Å, respectively. The former is close to the interlayer spacing in graphite, suggesting a weak interaction with the substrate material, whereas the latter value experimentally verifies that, in the case of graphene on Co(0001), there is a strong interaction with the substrate.

Spectroscopy of ultracold neutrons diffracted by a moving grating

Spectra of ultracold neutrons that appeared in experiments on neutron diffraction by a moving grating were measured using the time-of-flight Fourier spectrometer. Diffraction lines of five orders were observed simultaneously. The obtained data are in good agreement with the theoretical predictions based on the multiwave dynamical theory of neutron diffraction by a moving grating.

Quantitative x-ray diffraction analysis of bimodal damage distributions in Tm implanted Al0.15Ga0.85N

In this work radial symmetric x-ray diffraction scans of Al0.15Ga0.85N thin films implanted with Tm ions were measured to determine the lattice deformation and crystal quality as functions of depth. The alloys were implanted with 300 keV Tm with 10° off-set to the sample normal to avoid channelling, with fluences varying between 1013 Tm cm−2 and 5 × 1015 Tm cm−2. Simulations of the radial 2θ–ω scans were performed under the frame of the dynamical theory of x-ray diffraction assuming Gaussian distributions of the lattice strain induced by implantation defects. The structure factor of the individual layers is multiplied by a static Debye–Waller factor in order to take into account the effect of lattice disorder due to implantation. For higher fluences two asymmetric Gaussians are required to describe well the experimental diffractograms, although a single asymmetric Gaussian profile for the deformation is found in the sample implanted with 1013 Tm cm−2. After thermal treatment at 1200 °C, the crystal quality partially recovers as seen in a reduction of the amplitude of the deformation maximum as well as the total thickness of the deformed layer. Furthermore, no evidence of changes with respect to the virgin crystal mosaicity is found after implantation and annealing.

Dynamic theory of neutron diffraction from a moving grating

In the framework of the approximation of slowly varying amplitudes a multiwave dynamical theory of neutron diffraction on a moving phase grating was developed. The influence of the velocity of the grating, its period and height of the slits on the discrete energy spectrum and intensity of various diffraction orders was analyzed.

Algorithms for determining the phase of RHEED oscillations

Oscillations of reflection high-energy electron diffraction (RHEED) intensities are computed using dynamical diffraction theory. The phase of the oscillations is determined using two different approaches. In the first, direct, approach, the phase is determined by identifying the time needed to reach the second oscillation minimum. In the second approach, the phase is found using harmonic analysis. The two approaches are tested by applying them to oscillations simulated using dynamical diffraction theory. The phase of RHEED oscillations observed experimentally is also analysed. Experimental data on the variation of the phase as a function of the glancing angle of incidence, derived using the direct method, are compared with the values computed using both the direct and harmonic methods. For incident-beam azimuths corresponding to low-symmetry directions, both approaches produce similar results.

Time-dependent Bragg diffraction by multilayer gratings

Time-dependent Bragg diffraction by multilayer gratings working by reflection or by transmission is investigated. The study is performed by generalizing the time-dependent coupled-wave theory previously developed for one-dimensional photonic crystals (André J-M and Jonnard P 2015 J. Opt. 17 085609) and also by extending the Takagi–Taupin approach of the dynamical theory of diffraction. The indicial response is calculated. It presents a time delay with a transient time that is a function of the extinction length for reflection geometry and of the extinction length combined with the thickness of the grating for transmission geometry.

Structure refinement using precession electron diffraction tomography and dynamical diffraction: tests on experimental data.

The recently published method for the structure refinement from three-dimensional precession electron diffraction data using dynamical diffraction theory [Palatinus et al. (2015). Acta Cryst. A71, 235-244] has been applied to a set of experimental data sets from five different samples - Ni2Si, PrVO3, kaolinite, orthopyroxene and mayenite. The data were measured on different instruments and with variable precession angles. For each sample a reliable reference structure was available. A large series of tests revealed that the method provides structure models with an average error in atomic positions typically between 0.01 and 0.02 Å. The obtained structure models are significantly more accurate than models obtained by refinement using kinematical approximation for the calculation of model intensities. The method also allows a reliable determination of site occupancies and determination of absolute structure. Based on the extensive tests, an optimal set of the parameters for the method is proposed.

An In Situ Study of Sintering Behavior and Phase Transformation Kinetics in NiTi Using Neutron Diffraction

The powder sintering behavior of NiTi from an elemental powder mixture of Ni/Ti has been investigated, using an in situ neutron diffraction technique. In the sintered alloys, the overall porosity ranges from 9.2 to 15.6 pct, while the open-to-overall porosity ratio is between 8.3 and 63.7 pct and largely depends on the sintering temperature. In comparison to powder compacts sintered at 1223 K and 1373 K (950 °C and 1100 °C), the powder compact sintered at 1153 K (880 °C) shows a much smaller pore size, a higher open-to-overall porosity ratio but smaller shrinkage and a lower density. Direct evidence of eutectoid transformation in the binary Ni-Ti system during furnace cooling to ca. 890 K (617 °C) is provided by in situ neutron diffraction. The intensities of the B2-NiTi reflections decrease during the holding stage at 1373 K (1100 °C), which has been elaborated as an extinction effect according to the dynamical theory of neutron diffraction, when distorted crystallites gradually recover to perfect crystals. The analysis on the first five reflections clarifies the non-existence of any order–disorder transition in the NiTi phase from B2-to-BCC structure.

Monochromatic X-ray-induced thermal effect on four-reflection “nested” meV-monochromators: dynamical diffraction theory and finite-element analysis

The high energy resolution monochromator (HRM) is widely used in inelastic scattering programs to detect phonons with energy resolution, down to the meV level. Although the large amount of heat from insertion devices can be reduced by a high heat-load monochromator, the unbalanced heat load on the inner pair of crystals in a nested HRM can affect its overall performance. Here, a theoretical analysis of the unbalanced heat load using dynamical diffraction theory and finite element analysis is presented. By utilizing the ray-tracing method, the performance of different HRM nesting configurations is simulated. It is suggested that the heat balance ratio, energy resolution, and overall spectral transmission efficiency are the figures of merit for evaluating the performance of nested HRMs. Although the present study is mainly focused on nested HRMs working at 57Fe nuclear resonant energy at 14.4 keV, it is feasible to extend this to other nested HRMs working at different energies.

Crystal thickness and extinction distance measurements by convergent beam electron diffraction fitting and application in quantitative TEM holography analysis on p-n junctions

A modified model function is proposed to determine Si crystal thickness and extinction distance by fitting of the convergent-beam electron diffraction (CBED) intensity based on the dynamical diffraction and absorption theories. The previous fitting method employs two separate functions to simulate the elastic and inelastic parts of CBED intensity, while the present work uses a single model for the simulation. At an electron beam energy of 300 kV, the model demonstrates very fast and accurate thickness measurements ranging from 58 to 334 nm for the Si crystal. The minimum measurable thickness is around three times smaller compared with the conventional K–A plot method. From the fitting, the extinction distance of the Si (004) diffraction was determined to be 183.03 ± 2 nm, which is useful for the determination of the Si crystal structure factor. In conjunction with the transmission electron microscope holography technique, the authors study the electrostatic built-in potential across the p-n junction of the source and drain of a metal–oxide–semiconductor field effect transistor device.

Effective increase in beam emittance by phase-space expansion using asymmetric Bragg diffraction.

We propose an innovative method to extend the utilization of the phase space downstream of a synchrotron light source for X-ray transmission microscopy. Based on the dynamical theory of X-ray diffraction, asymmetrically cut perfect crystals are applied to reshape the position-angle-wavelength space of the light source, by which the usable phase space of the source can be magnified by over one hundred times, thereby "phase-space-matching" the source with the objective lens of the microscope. The method's validity is confirmed using SHADOW code simulations, and aberration through an optical lens such as a Fresnel zone plate is examined via matrix optics for nano-resolution X-ray images.

Dynamical X-ray diffraction in a crystal with the spatial modulation of incident and reflected waves

A new approach has been proposed to the problem of dynamical scattering of spatially limited X-ray beams. The dynamical diffraction theory in a crystal with the lateral modulation of incident and reflected X-ray waves has been developed within this approach. The unusual angular distribution of the diffracted intensity from the crystal with an absorbing surface grating has been physically explained and numerically simulated. The effect of the instrument function on the formed diffraction pattern has been demonstrated.

Development of variable-magnification X-ray Bragg optics.

A novel X-ray Bragg optics is proposed for variable-magnification of an X-ray beam. This X-ray Bragg optics is composed of two magnifiers in a crossed arrangement, and the magnification factor, M, is controlled through the azimuth angle of each magnifier. The basic properties of the X-ray optics such as the magnification factor, image transformation matrix and intrinsic acceptance angle are described based on the dynamical theory of X-ray diffraction. The feasibility of the variable-magnification X-ray Bragg optics was verified at the vertical-wiggler beamline BL-14B of the Photon Factory. For X-ray Bragg magnifiers, Si(220) crystals with an asymmetric angle of 14° were used. The magnification factor was calculated to be tunable between 0.1 and 10.0 at a wavelength of 0.112 nm. At various magnification factors (M ≥ 1.0), X-ray images of a nylon mesh were observed with an air-cooled X-ray CCD camera. Image deformation caused by the optics could be corrected by using a 2 × 2 transformation matrix and bilinear interpolation method. Not only absorption-contrast but also edge-contrast due to Fresnel diffraction was observed in the magnified images.

Focusing performance of a multilayer Laue lens with layer placement error described by dynamical diffraction theory.

The multilayer Laue lens (MLL) is essentially a linear zone plate with large aspect ratio, which can theoretically focus hard X-rays to well below 1 nm with high efficiency when ideal structures are used. However, the focusing performance of a MLL depends heavily on the quality of the layers, especially the layer placement error which always exists in real MLLs. Here, a dynamical modeling approach, based on the coupled wave theory, is proposed to study the focusing performance of a MLL with layer placement error. The result of simulation shows that this method can be applied to various forms of layer placement error.

Diffracted transient radiation of a relativistic electron in a bilayer target

A theory of coherent X-ray radiation from a relativistic electron in a combined target consisting of an amorphous and a crystalline layer is constructed using the dynamic theory of diffraction. The expressions describing the spectral-angular density of coherent X-rays in such a target are derived and analyzed.

Theoretical study of the properties of X-ray diffraction moiré fringes. I.

A detailed and comprehensive theoretical description of X-ray diffraction moiré fringes for a bicrystal specimen is given on the basis of a calculation by plane-wave dynamical diffraction theory. Firstly, prior to discussing the main subject of the paper, a previous article [Yoshimura (1997). Acta Cryst. A53, 810-812] on the two-dimensionality of diffraction moiré patterns is restated on a thorough calculation of the moiré interference phase. Then, the properties of moiré fringes derived from the above theory are explained for the case of a plane-wave diffraction image, where the significant effect of Pendellösung intensity oscillation on the moiré pattern when the crystal is strained is described in detail with theoretically simulated moiré images. Although such plane-wave moiré images are not widely observed in a nearly pure form, knowledge of their properties is essential for the understanding of diffraction moiré fringes in general.