Divergent Beam Method Research Articles

From Borrmann to Super-Borrmann Effect: from 2-Beam ton-Beam Diffraction

Several ways in which the Borrmann effect manifests itself in X-ray topography (the 2-beam case) are described. Some of these manifestations add significantly to the information derivable from topograph images. Examples are the sign of dislocation Burgers vectors and the sign of fault vectors of stacking faults. Phenomena associated with coherent n-beam diffraction (n ≥ 3) can occur in conventional X-ray topography and are common in divergent-beam photography. In the latter case, the enhanced Borrmann effect (the ‘Super-Borrmann’ effect) gives rise to fine-structures within images of Kossel line intersections. These have been recorded using high-resolution photographic techniques and have been usefully applied in lattice-parameter measurement by the divergent-beam method.

Characterization of Lattice Defects and Concomitant Strain Distribution

AbstractA number of X-ray methods characterizing lattice defects are described. They were developed in response to a variety of challenging problems in materials science, A method based on a computer-aided rocking curve analysis, CARCA, was developed which offers a rapid mapping of the dislocation structure in an epitaxial film, as well as a tensor analysis of nonuniform elastic strains. Characterization methods were developed in an attempt at bridging, systematically, the gap between micro and macromechanics when the problem arose to clarify the distribution of elastic strains emanating from stress concentrators such as notches, cracks, holes and to elucidate strain interactions. Gradients of elastic strains were characterized by a method of local intensity measurements. For crystal with homogeneous elastic strain distribution a tensor analysis is described, based on precision measurements obtained by the backreflection divergent beam method. A direct linkage between the imaging of the micro structure by TEM and the macro-response of deformation and recovery of commercial alloys was achieved by a version of the CARCA method, designed to characterize the lattice defects in polycrystalline materials. Example applications of the methods are presented with the hope that their usefulness may find adaptations in other areas of investigation.

Premartensitic strain tensor analysis of Ni58Al42by the X-ray divergent beam method

A strain tensor analysis of Al42Ni58g crystals was performed characterizing the strain distribution of this alloy in the premartensitic stage. The experimental basis of this analysis was provided by the precision input data obtained from a modified X-ray back-reflection divergent beam method. The strain components ej(j = 1,2,... ,6) and strain tensor ejjwere determined by a least square method, after collecting strain data in excess of the minimum number required. The analysis disclosed abnormal strain values for the (2macr11) and (2macr11) planes, which were attributed to strain-field distortions induced by omega-like displacements. After exclusion of these strain values, the eigenvectors, defining the principal strains, were determined. Their magnitude and directions referred to the crystallographic coordinate system are λ,= 4.74 × 10-3 [111], λ2= 5.8 × 10-4 [121], and λ3 = -1.93 × 10-3 [101]. In the light of recent experimental observations by TEM and of studies on the temperature dependence of elastic constants, the results of the X-ray strain tensor analysis were interpreted as manifestations of induced lattice anharmonicity, the shear strains (101) [101] being intimately associated with strains along [macr111] and [121] directions. Regarding the strain distribution as a precursor of the martensitic transformation, it is predicted that martensite is formed by (101) [101] shear, accompanied simultaneously by faulting of the (101) planes.

Application of x-ray topographic methods to thin film electronic materials

X‐ray methods are presented which, besides offering a topographic survey of the defect structure in electronic materials, give quantitative information on the induced strain distribution. Complete strain tensor analyses are obtained by the back‐reflection divergent beam method and by the method of computer‐aided rocking curve analysis (CARCA). The latter is based on the double‐crystal diffractometer principle. An example of the strain analysis by CARCA is given showing the strain dependence on the thickness variation of an InGaAsP film grown on a substrate of InP. Valuable information on the density of interface dislocation is obtained by combining these methods with lattice curvature measurements using scanning topography with automatic Bragg angle control. The elucidation of the strain distribution in depth becomes enhanced by application of x radiation of various wavelengths.

The kinematical theory of X-ray spherical-wave diffraction

It is shown that the Fraunhofer approximation, used in the kinematical theory of X-ray diffraction, may fail for a wide variety of crystals with different perfection. The kinematical theory describing the diffraction pattern in the general case is developed. The case of spherical-wave diffraction by a plane parallel crystal is considered in detail. The intensity distribution and the diffraction line width are ascertained to be essentially dependent on the region of diffraction in which the observation plane is located. On the other hand, the diffraction pattern geometry is independent of the diffraction region and is determined only by the crystal structure and the optics of diffraction. The geometry of the diffraction pattern recorded by the divergent-beam method is analysed in detail.

Heteroepitaxial Growth of GaP on Silicon by Molecular Beam Epitaxy

Heteroepitaxial GaP layers were grown on Si substrates, misoriented by 5° from the (111) face toward the [100] axis, by molecular beam epitaxy (MBE). The RHEED pattern of GaP is of elongated streaks, indicating that the quality of the grown layers is good. From observation by the X-ray divergent beam method, the [111] axis of the grown layers is parallel to that of the substrate. Autodoping of Si into the epitaxial layers from the substrate is very small compared with the layers grown by CVD. Photoluminescence was measured for GaP on Si which was irradiated with an ionized nitrogen beam during MBE.

Stress-strain analysis of Fe and Cu single crystals induced by absorbed hydrogen

Iron and copper single crystals were studied with the aid of the X-ray divergent beam method after various stages of cathodic polarization. The stress-strain analysis of the various stages of the hydrogenation disclosed an anisotropy of strain distribution in the matrix. The maximum strain direction of the principal strain was found to coincide with one of the (111) directions for iron crystal and one of the (100) directions for copper respectively.

水素吸蔵したFeおよびCu単結晶のひずみ解析

Iron and copper single crystals were studied with the aid of the X-ray divergent beam method after various stages of cathodic polarization. The stress-strain analysis of the various stages of hydrogenation disclosed an anisotropy of strain distribution in the matrix. The maximum strain direction of the principal strain was found to coincide with one of the 〈111〉 directions for the iron crystal and one of the 〈100〉 directions for the copper crystal, respectively. The elastic stored energy occurring during the hydrogen absorption in the matrix increased with charging time. The strain distribution became highly assymmetric with time, especially in iron single crystals. The anisotropy was explained in terms of the formation of platelet segregation of hydrogen atoms in the matrix.

Measurement of Misalignments between the [111] Axes of GaP Deposits and Si Substrates by the X-Ray Divergent Beam Method

Single-crystal GaP layers, about 3 µm thick, were deposited by the vapor growth technique on Si substrate wafers which were intentionally misorientated up to 18° from the (111) face. The divergent beam technique of X-ray diffraction was used for analyzing the lattice misorientation on the GaP on Si. The [111] axis of the GaP deposits and that of Si substrates were not exactly parallel. The dependence of the degree of angular misalignment between the two [111] axes on the substrate orientations is described.

Qualitative Analysis of the Kossel Back Reflection Pattern from Selected Semiconductors

AbstractThe value of the divergent beam method as a powerful tool for investigating crystal perfection over rather wide limits has been known since the analytical work of K, Lonsdale. This method has been treated experimentally and theoretically by numerous authors and was popularized by Sigmund Weissmann. Although not nearly as useful as the reflection and transmission topographic techniques of Berg-Barrett and Lang for crystal characterization, it is potentially very useful for three dimensional quantitative strain analysis. The “technique phase” as Harvey Yakowitz remarked, is essentially over; however, refinements and optical improvements are certain to evolve if quantitative measurements are to become useful. Very short exposures and the non-destructive nature of the reflection divergent beara - Kossei method is a useful survey tool that has gained limited acceptance.Presently, the reflection mode is useful in the qualitative survey of numerous semiconductor materials and processes, some of which have been considered from time to time by various authors. A range of crystal perfection, as well as some surface modifications, are surveyed. Kossel reflection pattern artifacts and irregularities are discussed.

Synergy of line profile analysis and selected area topography by the X-ray divergent beam method

Synergy of line profile analysis and selected area topography by the X-ray divergent beam method

Strain inhomogeneities in lightly compressed tungsten crystals

A study of strain inhomogeneities in a zone-refined tungsten crystal oriented for (123) slip was carried out before and after a slight compression. The study was based on changes of interplanar spacings measured along different crystallographic directions and samples from different areas of the crystal by means of the X-ray back-reflection divergent beam method. Upon compression a redistribution of the locked-in elastic strains took place and the induced strains assumed a more homogeneous distribution for one of the sets of (123) planes considered. This homogeneity in the strain distribution along definite crystallographic directions is believed to result from the interaction of the primary slip dislocations with the pre-existing dislocation configuration and appears to be characteristic of the initial stage of work-hardening. The dislocation interaction induced by a small strain found practical application in the elimination of a troublesome substructure in highly zone-refined tungsten crystals and appears to form the scientific basis of certain stress-relief treatments long practised in metallurgical technology.

Stress-Strain Analysis of Metal Crystals by the Divergent X-Ray Beam Method

A stress-strain analysis of single cubic crystal is developed which utilizes the strain data supplied by the X-ray back reflection divergent beam method. The principal strains and their direction are determined, and the three dimentional stress-strain configuration is obtained from these data. The stored energy of the crystal is also deduced from a knowledge of the stress-strain configuration.This analysis is applied to the study of the characteristics of various Cu-alloy crystals after various deformation and annealing process.

Stress-Strain Analysis of Single Cubic Crystals and Its Application to the Ordering of CuAu I. Paper II

A stress-strain analysis of single cubic crystals is developed which utilizes the strain data supplied by the x-ray back-reflection divergent beam method. The principal strains and their directions are determined and from the principal strains and the known elastic constants the complete stress-strain configuration is obtained. Thus the maximum magnitude and the direction of the shearing strain on a given set of crystallographic planes are obtained and the set of planes on which the maximum value of the shearing maxima occurs is also determined. From a knowledge of the stress-strain configuration, the stored elastic energy of the crystal is deduced; it can be partitioned into two components, that due to shearing strains and that due to a mixture of normal and shearing strains. The conditions under which the principal stress system coincides with the principal strain system are also investigated. Furthermore, a number is constructed that measures the distortion of the crystal in terms of the energy increments associated with the elastic constants. The stress-strain analysis applied to the ordering of a CuAu crystal at 125°C corroborates quantitatively the qualitative results previously obtained by transmission electron microscopy. The dependence of stored elastic energy on annealing time is determined and it is shown that the first maximum and decline are associated with the maximum and decline of coherency strains set up between the ordered CuAu I nuclei and the disordered matrix. Upon increasing the annealing time, twinning occurs to relieve the tetragonality strains introduced by the ordered CuAu I domains. The second maximum is compounded by twinning on certain (110) planes and delayed ordering on other (110) planes of the matrix. The subsequent decline of the stored elastic energy is associated with twinning on all (110) planes. The shearing stress necessary to initiate microtwinning does not exceed 7×108 dyn/cm2.

Strain and Precision Lattice Parameter Measurements by the X-Ray Divergent Beam Method. I

An X-ray method is described which permits a precise determination of the interplanar spacings in a crystal. The method is capable of disclosing small differences in interplanar spacings even between various sets of (hkl) planes of the same form and, therefore, serves as a basis for a stress-strain analysis. An error analysis of this method shows that the minimum measurable strains in the lattice, expressed as Δd/d, are a function of the diffraction angle θ. For θ varying from 35° to 65°, the minimum measurable strains vary from 0.03% to 0.009%. A precision determination of the lattice parameter of a zone-refined tungsten crystal yielded a0=3.16566±0.00002 Å. Limitations and possible applications of the method to problems in physical metallurgy and solid state physics are presented.

Divergent-beam X-ray photography of crystals

Divergent-beam X-ray photography of single crystals by transmission can be used to study the ‘extinction’, that is, the diminution of the transmitted radiation that takes place at the Bragg reflexion angles. The intensity and geometry of the absorption lines observed give useful information about the texture of the crystal. Divergent beam photographs have shown that many crystals of organic compounds are unexpectedly perfect, and that sudden cooling to liquid-air temperatures will increase the mosaic character of their structure by an important factor and make them more suitable for structural analysis by the usual methods. Type I diamonds, and natural ice even near to its melting-point, are also found to possess a high degree of perfection, which cannot be removed by liquid-air treatment. The divergent beam method may be used for the determination of orientation, but it is important that the wave-length of X-rays employed should be correctly related to the size and nature of the crystal. In certain favourable cases it is possible to make precision measurements of lattice constant or of wave-length from divergent beam photographs, without the use of any kind of precision apparatus. By such means it has been shown that the C—C distance in individual diamonds varies from 1541.53(± 0-02) to 1541.27X, (1.54465-1-54440A), a difference presumably due to varying impurity content. Using diamond and a brass anticathode, the Zn Ka 1 wave-length, relative to Cu K Ka 1 as 1537.40X, is found to be 1432.21 ( ± 0-04) X. Temperature control would improve the accuracy of this measurement, which is, however, in good agreement with the latest value obtained by orthodox precision methods.