A role of second-order derivatives of amplitudes in X-ray beam dynamical diffraction equations

Dynamical diffraction equations for X-ray beams are considered with retaining the second-order derivatives of amplitudes in the direction perpendicular to the diffraction plane. For the dynamical diffraction problem, the retarded Green function is determined in case of a perfect crystal. Amplitudes of transmitted and diffracted waves in the crystal are represented as convolution over the crystal surface with use of determined Green function. Such representation can be used for solving diffraction problems in Laue and Bragg geometries in perfect crystals with both plane and not plane entrance and exit surfaces.     

Precipitation in silicon wafers after high temperature preanneal studied by X-ray diffraction methods

We have investigated oxygen precipitation in Czochralski silicon wafers focusing on influence of nucleation temperature and high temperature pre-anneal during common three step treatment. Thick Si wafers were studied mainly by x-ray diffraction in Laue transmission geometry using Mo x-ray tube, but were also compared to reciprocal space maps obtained in Bragg reflection geometry. The analysis of measured diffraction scans in Laue geometry was performed by means of Takagi equations and statistical dynamical theory of diffraction. From the simulated Laue diffraction curves we find the size of the individual defect area and the fraction of strain area volume in the wafer. The results obtained from x-ray diffraction were compared to loss of interstitial oxygen according to infrared absorption spectroscopy and the size of SiO₂ precipitate core was estimated. These techniques are in agreement with transmission electron microscopy images.     

Direct measurement of the delay time for a neutron in a crystal in the case of the Laue diffraction

The dynamical Laue diffraction has been studied for a direct beam diffracted from a thin (～3.5 cm) α-quartz crystal at the Bragg angles close to 90°. It is shown that diffraction occurs at Bragg angles up to 87°. The time-of-flight method is used to measure the time of the neutron presence in the crystal under the diffraction conditions. The time delay for a scattered neutron inside the crystal predicted earlier for the Bragg angles close to 90° confirmed experimentally. The effective velocity of the neutron propagation in a crystal measured at the incident-neutron velocity of 810 m/s and the diffraction angle of 87° equals (43±1) m/s. The result obtained confirms the possibility of increasing, by an order of magnitude, the sensitivity of the diffraction method of determining the dipole moment of a neutron at Bragg angles close to 90° predicted earlier theoretically.     

Diffraction of X‐ray free‐electron laser femtosecond pulses on single crystals in the Bragg and Laue geometry

A solution of the problem of dynamical diffraction for X‐ray pulses with arbitrary dimensions in the Bragg and Laue cases in a crystal of any thickness and asymmetry coefficient of reflection is presented. Analysis of pulse form and duration transformation in the process of diffraction and propagation in a vacuum is conducted. It is shown that only the symmetrical Bragg case can be used to avoid smearing of reflected pulses.     

On the role of secondary extinction in the measurement of the integrated intensity of X-ray diffraction peaks and in the determination of the thickness of damaged epitaxial layers

The integrated intensity of X-ray diffraction reflections has been measured for a series of epitaxial layers of AIII nitrides (GaN, AlN, AlGaN) grown on different substrates (sapphire, SiC) and characterized by different degrees of structural perfection. It has been shown that, despite a high density of dislocations and a significant broadening of the diffraction peaks, the obtained values are not described by the kinematic theory of X-ray diffraction and suggest the existence of extinction. The results have been analyzed on the basis of the Darwin and Zachariasen extinction models. The secondary extinction coefficients and the thicknesses of epitaxial layers have been determined using two orders of reflection both in the Bragg geometry (0002 and 0004) and in the Laue geometry (\(10\bar 10\)) and \(10\bar 20\)). It has been demonstrated that the secondary extinction coefficient is the greater, the smaller is the broadening of the diffraction peaks and, consequently, the dislocation density. It has been found that, for epitaxial layers with a regular system of threading dislocations, the secondary extinction coefficient for the Laue reflections is substantially greater than that for the Bragg reflections.     

Anisotropy of elastic strains and specific features of the defect structure of <Emphasis Type="Italic">a</Emphasis>-plane GaN epitaxial films grown on <Emphasis Type="Italic">r</Emphasis>-plane sapphire

The structural state of GaN epitaxial layers grown on r-plane sapphire through metal-organic vapor phase epitaxy has been investigated using X-ray diffraction. The interplanar spacings in two directions in the (11$ \bar 2 $ \bar 2 0) plane of the interface and in the direction perpendicular to it, as well as the diffraction peaks in the ϑ and ϑ-2ϑ scan modes in the Bragg and Laue geometries, are measured on double- and triple-crystal diffractometers. The intensity distribution maps for asymmetric Bragg reflections are constructed for two azimuthal positions of the sample. An analysis of the data obtained has demonstrated that the elastic strain is anisotropic and that the X-ray diffraction pattern parallel to the interface plane is broadened. The layers are contracted in the [1$ \bar 1 $ \bar 1 00] direction and unstrained in the [0001] direction. The broadening of the Bragg reflections in the [1$ \bar 1 $ \bar 1 00] direction is considerably larger than that in the [0001] direction. It is shown using the Williamson-Hall plots for the Bragg and Laue reflections that these broadenings are not related to different degrees of mosaicity but are determined by the local dilatations and misorientations around defects. The data obtained are analyzed, and the conclusions regarding the dislocation structure of the samples are drawn.     

Aperture (slit) effects on the dynamical diffraction of X-rays and thermal neutrons

With increasing accuracy of perfect crystal methods, the effects of apertures (slits) on angular and spacial intensity distributions in dynamical perfect crystal diffraction have become quite important. We present an analytical treatment of such effects. The results are first applied to the diffraction of rays delivered by a source of given lateral extension, i.e. where the critical size of the source as such is essentially influencing the dynamical diffraction phenomena. Both the Laue and the Bragg geometry are considered. Next we calculate the change which a rocking curve measured in either geometry suffers from a slit with varying width and position in front of or between the crystals. Finally we report a pertinent experiment which we have performed in the Laue geometry. We find good agreement with the calculations. It follows that whenever dynamical diffraction is used in quantitative measurement the possible influence of diffraction from diaphragms requires special attention and analysis. For the latter the theory presented here is very well suited.     

Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: Diffraction-induced splitting, selective compression, and focusing of pulses

A theory for the dynamical Bragg diffraction of a spatially confined laser pulse in a linear photonic crystal with a significant modulation of the refractive index in the Laue geometry has been developed. The diffraction-induced splitting of a spatially confined pulse into the Borrmann and anti-Borrmann pulses localized in different regions of the photonic crystal and characterized by different dispersion laws is predicted. The selective compression or focusing of one of these pulses with the simultaneous broadening or defocusing of the other pulse is shown to be possible.     

Magnetic properties of Y2Fe15Cr2 single crystal

A Y₂Fe₁₅Cr₂ single crystal with the Th₂Ni₁₇-type structure has been prepared by the Czochralski method and investigated by means of Laue back-reflection, metallographic observation, X-ray diffraction, the singular point detection technique and magnetic measurements. A magnetohistory effect has been observed at a low temperature. Magnetization curves have been measured along the easy and hard directions in fields up to 6.5 T. The saturation magnetization and magnetocrystalline anisotropy field decrease with increasing temperature. The experimental magnetocrystalline anisotropy constant is in good agreement with the calculation results on first approximation.     

Dynamical scattering of X-rays by dislocations in the Bragg and Laue case

The interbranch scattering of X-rays at dislocations is studied experimentally and with the aid of numerical simulations. Narrow beam of a small divergence is used for the study which is made both in the Laue and Bragg case of diffraction. The role of interbranch scattering in the formation of defect contrast in X-ray topography is discussed.     

Diffraction reflection of x-ray radiation with a two-dimensionally bounded wavefront from perfect crystals

The problem of dynamical diffraction of x-ray radiation with a two-dimensionally bounded wavefront is solved in the Bragg and Laue geometries in a crystal with an arbitrary thickness and an arbitrary reflection asymmetry parameter. An analysis of the wavefront deformation during the diffraction and subsequent propagation is carried out. It is shown that the most favorable conditions for the reflected beam to retain its shape are accomplished in a crystal whose thickness is less than the extinction depth.     

X-ray Bragg diffraction on periodic surface gratings

We report on the X-ray Bragg diffraction analysis of periodic surface gratings on (100) GaAs crystals. We show that the X-ray Bragg diffraction is a very powerful method to investigate ultrafine surface gratings and allow us to determine not only the grating period but, in particular, the width of the grooves and to characterize the quality of the sidewalls, which is not easily possible with optical methods. The X-ray diffraction on surface crystal gratings is analogue to the Fraunhofer diffraction of multiple slits or reflection gratings.     

X-ray Laue diffraction with allowance for two-dimensional curvature of the wavefront: The concept a locally plane wave

Symmetrical Laue diffraction in a perfect crystal with a plane entance surface is considered. The two dimensional curvature of the wave front of the incident beam is taken into account. Using the corresponding Green function, a general expression for the amplitude of diffracted wave in the crystal is presented. Based on this expression, the concept of a locally plane wave is analyzed taking into account two-dimensional curvature of the wave front. This concept is used for obtaining the rocking curves depending on the angles of deviation from the chosen exact Bragg direction in both the diffraction plane and in perpendicular direction. The obtained result is compared with the result of the standard dynamical diffraction theory.     

由透射波的摇摆曲线求GaAs中Ga的反常散射因数

利用同步辐射光，在对称Ｌａｕｅ情况下，于Ｇａ的Ｋ吸收限附近，测定了ＧａＡｓ（２００）的衍射波和透射波的摇摆曲线．通过测定的透射波的摇摆曲线和Ｘ射线衍射动力学理论的相应计算结果的定量比较，求得ＧａＡｓ中的Ｇａ原子在其Ｋ吸收限附近的反常散射因数 关键词：     

Study of the influence of higher—order Laue zones on the exit wavefunction

Based on an analytical expression of electron dynamical diffraction derived recently,the exit wavefunctions of nonperiodic samples have been calculated.The results have shown that the amplitude and phase obtained by the analytical expression were in good agreement with those obtained using the multi-slice method.Besides,the influence of the higher-order Laue zone on the wavefunction for different sample thickness was studied in detail and the results showed that the exit wavefunction is dominated by the zero-order Laue zone.When the sample is very thin,the influence of the higher-order Laue zone should be considered.     

A study of the contrast on growth striations in silicon by X-ray double crystal topography in the Laue case

The image of growth striations in Si formed by the double crystal X-ray topography in the Laue case (DCTL) is investigated. The results of the dynamical diffraction theory for crystals with small microdefects have been compared with the contrast behaviour determined experimentally. It has been found that the theory explains qualitatively the contrast on the striations and it has been demonstrated that the DCTL method is suitable for detection of the presence of small microdefects in growth striations. Since the paper completes a series of papers on X-ray topography of growth striations, some general conclusions are formulated concerning the applicability of the X-ray topographical methods to the investigation of the structure of the growth striations     

Rocking curves in the Laue geometry depending on the departure of incident plane X-ray wave from the Bragg condition in the diffraction plane and in the direction perpendicular to the diffraction plane

Using the method of Green functions in the symmetric Laue geometry the rocking curves of plane X-ray wave were theoretically investigated depending on the departure of a beam from exact Bragg condition in the diffraction plane and in the direction perpendicular to the diffraction plane. The Si(220) reflection of MoKα radiation was considered by way of illustration.     

Diffracted transition radiation of relativistic electrons in an artificial periodic structure

A theory concerning the coherent X-ray radiation of relativistic electrons crossing an artificial periodic medium in Laue scattering geometry has been constructed. Expressions describing the spectralangular characteristics of radiation in the Bragg scattering direction have been obtained and studied. Radiation is considered in analogy with that in the crystal medium as a result of the coherent summation of the contributions of two radiation mechanisms, in particular, parametric X-ray radiation (PXR) and diffracted transition radiation (DTR). It is shown that the DTR yield from a layered target can exceed the particle radiation yield in a single-crystal radiator by one order or more under similar conditions.     

Scanning LLL x-ray interferometry

A new formalism is presented for the theory of scanning LLL x-ray interferometry, which takes account also of the amount of x-ray absorption and of crystal yawing. It is based on the Takagi approach to the dynamical theory of x-ray diffraction and uses, to a great extent, the formalism of quantum mechanics, in order to reduce the algebraic complexity of the Ewald-von Laue approach. The formalism presented here is an easy-to-handle tool for the study of x-ray propagation in multicrystal systems and for the investigation into deviations of the travelling-fringe period from the spacing of diffracting planes, as it explains how interference features change when moving in paramater space.     

A nonstandard dynamical theory of x-ray scattering in crystals

A new version of the X-ray diffraction dynamical th eory in perfect crystals is presented. The theory is based on direct analysis of Maxwell equations taking into account the model approximations of the character of X-ray-crystal interaction corresponding to the Evald—Laue theory. The method modification of many scales is used as a mathematical method for approximate solution of the diffraction equation. The obtained results correspond to the known conclusions of the diffraction dynamical theory beyond the area of full external reflection. The expressions for the reflection amplitude factors of diffracted and specular waves for arbitrary angles including the area of full external reflection have been obtained.