Angular parameters in X-ray diffraction theory

In this article, we provide a detailed mathematical analysis of the Helmholtz equation, in the context of dynamical wide-angle X-ray scattering by deformed crystals. We show that the dynamical theory of diffraction may be written without introducing a sophisticated dispersion theory, by using the following simple condition for wavevectors: |K_h|=|K₀|=k. This choice gives simple way to describe the reciprocal space mapping phenomena in a dynamical approach and simulate the diffraction curves for strained crystals over many orders of reflection. Several theoretical approaches to obtaining the angular distribution of scattered intensity are discussed.     

Fractal analysis of powder X-ray diffraction patterns

X-ray diffraction (XRD) patterns with broad background are commonly found in the characterization of materials with a certain degree of amorphicity, so the sharp intensity peaks associated with material phases are not well defined. This work used rescaled range (denoted by R/S) analysis, a method intended for fractal analysis of noisy signals, to characterize XRD patterns with broad background. It is found that XRD patterns with broad background are not random at all, but contain information on regularities expressed as autocorrelations of the intensity signal. Sol-gel alumina fired at different temperatures was used as an example to illustrate the applicability of the method. It is shown that fractal R/S analysis is able to locate angular regions that can be associated to ideal International Centre for Diffraction Data Powder Diffraction File (ICDD PDF) lines of diverse alumina phases.     

Kinematic simulation of convergent beam low-energy electron diffraction patterns

Kinematic simulations are used to investigate the geometrical properties of convergent beam low-energy electron diffraction patterns from an Si(0 0 1) surface. Compression of a low-energy electron microscope immersion lens is included explicitly and the sensitivity of patterns to surface reconstruction and dimer buckling is investigated. Key pattern features and whole pattern symmetries are identified from the simulations and interpreted geometrically. Advantages over conventional low-energy electron diffraction techniques are identified.     

Magnification at diffraction of widely divergent beam of X-ray radiation

A new technique for realization of diffraction of a widely divergent beam of characteristic X-ray radiation in crystals is developed. It is shown that the X-ray diffraction image is a set of hyperbolas, which are formed by radiation diffracted from certain points of the crystal surface (active points). The active points are also located along hyperbolas and to every point of the diffraction image corresponds an active point on the crystal surface. The diffraction pattern is a magnified image of the distribution of active points. The diffraction magnification has been determined experimentally and it was shown to coincide to high accuracy with the calculated value. A method is proposed for detection of structurally distorted local domains and determination of their sizes.     

Digital processing of X-ray diffraction patterns from oriented fibers

Accurate measurement of intensity data from x-ray diffraction patterns of oriented fibers is one of the major difficulties in the determination of polymer structures. Accurate diffraction data are particularly important when discrimination between different, but similar, molecular models is required. Although digital processing of diffraction patterns has provided more accurate data than was obtainable by traditional measurements, many of the methods used have been partly subjective and restricted to well-resolved diffraction patterns. Background is a major source of error in intensity measurement. A general method of two-dimensional background removal has been developed. Objective methods of measuring the intensities of Bragg reflections from x-ray diffraction patterns of polycrystal-line fibers are described. A profile fitting method (which is an extension of the previously reported angular deconvolution procedure) for measuring diffraction data from noncrystalline oriented polymers is also presented. These methods have been used to collect data from diffraction patterns of a variety of nucleic acid polymers.     

Determination of the crystallographic texture of sintered PrFeB magnets based on X-ray diffraction patterns

An alternative method for determining the degree of crystallographic alignment (cos Θ) of the magnetic Pr₂Fe₁₄B phase (Φ) is proposed. The method is based on the relative X-ray diffracted intensities of the Φ planes of sintered magnets with and without texture. The degree of crystallographic alignment is also determined by X-ray pole figures using the (0 0 4) reflection and considered as a standard reference for comparison. The method is applied to Pr₁₆Fe₇₆B₈ magnets with 0.51cos Θ0.97. The difference between the crystallographic alignments determined by these two methods is 3% within the experimental error. The advantages and limitations of using X-ray diffraction patterns to quantitatively evaluate the texture of sintered magnets are also discussed.     

Thermally induced adjustment of the parameters of X-ray diffraction peaks from crystals

The effect of nonuniform thermal action of a light beam on the parameters of a diffraction peak for a calcite crystal is studied. It is shown that the spatial intensity distribution in an X-ray beam can be corrected by controlling the thermally induced mismatch between Bragg reflections from local areas in a crystal.     

Thin-film characterization by grazing incidence X-ray diffraction and multiple beam interference

An overview is given of the application of two-beam and multiple-beam approach to grazing incidence X-ray diffraction (GIXD) for single-crystal thin film characterization. Crystallographic orientation, lattice mismatch, order parameter, and phase of in-plane reflection for various thin film systems are determined by using conventional two-beam GIXD and resonance three-beam GIXD. The diffraction technique, analysis procedure, and theoretical consideration are also presented and discussed.     

Computer simulations of the X-ray diffraction patterns of imperfect Al/Nb superlattices

In order to obtain more structural details from X-ray diffraction (XRD) patterns of metallic multilayers we developed a simulation program for XRD patterns of Al/Nb multilayers. We followed the theory of an imperfect one-dimensional superlattice described by Z. Mitura and P. Mikolajczak. Computer simulated patterns are compared with experimentally obtained XRD spectra.     

X-ray diffraction analysis of double-layer systems irradiated by a high-power ion beam

Systems of thin Ni films on a copper substrate, thin Ni films on an aluminum substrate, and St20 steel covered with graphite irradiated by a high-power ion beam of nanosecond duration have been investigated by x-ray diffractometry. The mixing of nickel with aluminum and St20 steel with graphite is revealed. It is shown that the mixing predominantly occurs in the liquid phase and depends on the thermal properties of a film and substrate.     

Simulation of X-ray diffraction patterns for silk fibres using paracrystalline statistics

Using paracrystalline statistics, we have simulated one-dimensional and three-dimensional X-ray diffraction patterns from natural silk fibres for various values of crystal size and lattice distortion parameters. This is in agreement with the experimental observation of X-ray pattern reported earlier.     

衍射光学元件设计参数对束匀滑衍射效率的影响

经过理论推导,分析了320 mm口径衍射光学元件在束匀滑实验中光强分布出现高级次衍射斑和元件实际衍射效率变低的原因,模拟计算得到了接近实验的光强分布。模拟分析发现:通过调整设计参数,如适当增加抽样点数,使设计时焦斑主瓣占输出计算窗口的比例减小至0.2以下,可以大大降低由于高级次衍射斑造成的衍射效率损失,控制在2%以内,使台阶分布式相位片实际衍射效率得到提高,在对口径为70及320 mm的台阶分布式相位片样品测试中得到了验证。     

X-ray diffraction study of sub-lattice order parameters in ferrielectric ammonium sulphate

Single-crystal X-ray diffraction has been used to study the sublattice order parameters corresponding to the ordering of the two kinds of symmetrically independent ammonium ions in (NH₄)₂SO₄. The diffraction data were collected on an automatic computer-controlled diffractometer using the 2θ-ω scan technique in the temperature range 22 to 130°C. The temperature dependence of the integrated intensities of Bragg reflections is well explained not by a pseudo-proper ferroelectric model, but by a ferrielectric one. It is concluded that (NH₄)₂SO₄ undergoes a ferrielectric phase transition at about ?50°C.     

衍射光学元件设计参数对束匀滑衍射效率的影响

经过理论推导,分析了320 mm口径衍射光学元件在束匀滑实验中光强分布出现高级次衍射斑和元件实际衍射效率变低的原因,模拟计算得到了接近实验的光强分布。模拟分析发现:通过调整设计参数,如适当增加抽样点数,使设计时焦斑主瓣占输出计算窗口的比例减小至0.2以下,可以大大降低由于高级次衍射斑造成的衍射效率损失,控制在2%以内,使台阶分布式相位片实际衍射效率得到提高,在对口径为70及320 mm的台阶分布式相位片样品测试中得到了验证。     

高斯光束通过非线性介质后的远场衍射图样的研究

 利用菲涅尔－基尔霍夫衍射积分公式，对发散和会聚高斯光束通过薄非线性介质时形成的远场衍射图样进行了研究。模拟计算结果表明：当发散高斯光束通过自散焦介质或会聚高斯光束通过自聚焦介质时，远场均会出现中央较暗、向外围逐渐增强、分布尺度较大的粗衍射环；当发散高斯光束通过自聚焦介质或会聚高斯光束通过自散焦介质时，远场均会出现中心强度最大、向外围逐渐减弱、分布尺度较小的细衍射环。不同远场衍射图样归根结底是入射高斯光束因介质折射率变化造成的空间自相位调制及其波前曲率共同作用的结果。     

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.     

The dynamic symmetry and its application for interpretation of X-ray diffraction patterns

It is shown here that real crystals besides their crystallographic symmetry (230 space groups) display another kind of symmetry called in this paper — the dynamic symmetry (1651 Shubnikov groups). The dynamic symmetry manifests itself during transformation by the crystal structure of quantities whose symmetry has two-color operations.An example is given of application of dynamic symmetry for interpretation of phenomenon of forbidden reflexions in an X-ray pattern of ethylidene-N, N′-diacetamide crystals.     

X-ray diffraction on denatured ferritin

Denatured ferritin, in contrast to other denatured corpuscular proteins, gives several diffractions, even in the regiond>15 Å. These diffractions correspond to the mutual arrangement of the micelles and proteinic component of ferritin. Diffractions in the regiond<15 Å correspond both to the internal structure of the denatured proteinic component — apoferritin — and to the internal structure of ferric micelles of ferritin. Regions containing only the denatured proteinic component with polypeptide chains in-configuration, occur in the structure of denatured ferritin in addition to the crystallites or crystalline regions with regular arrangement of micelles and the proteinic component.

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, , d>15 Å. . d<15 Å — , — . , gb-.

     

Dependence of hydrogen released on the charge state of incident ions

The behavior of polymers under heavy ion bombardment is of great interest. In the present study, hydrogen released from polypropylene (PP) and polyethylene terephthalate (PET) was investigated as a function of charge state (11⁺, 14⁺, and 25⁺) for 130 MeV ¹⁰⁷Ag ions. It was found that hydrogen released from the polymers varies as α q ⁿ , where n was found to be 2.98 and 1.94 for PP and PET, respectively, when compared with the value of ～3.0 reported in the literature for different polymers and ion combinations. Radii of the damaged zones or ion track formed were deduced from the slope of the hydrogen released versus ion fluence curves. This radius was also found to depend upon the charge state of the incident ion. It varies as β q ^m , where m is 1.25 and 0.741 for PP and PET, respectively.     

Dependence of the electron-cloud instability on the beam energy

The electron cloud (EC) can be formed in the beam pipe of a circular accelerator if the secondary emission yield (SEY) of the inner surface is larger than 1, and it can detrimentally affect the circulating beam. Understanding the underlying physics and defining the scaling laws of this effect is indispensable to steer the upgrade plans of the existing machines and the design of new ones. The single bunch EC instability (ECI) is shown to be strongly affected by the transverse beam size. Transversely, smaller beams going through an electron cloud generate higher electron peak densities and lower the intensity threshold to make the beam unstable. In particular, since higher energy beams have smaller transverse sizes (for equal normalized transverse emittances), the scaling of the ECI threshold with the beam energy turns out to be surprisingly unfavorable.