Multifunctional Two- and Three-Dimensional Polycrystalline X-Ray Diffractometry

INTRODUCTION

Following the discovery of X-ray by W.C Roentgen in 1895, W.L Bragg (1912) developed a relationship, which is now widely known as Bragg’s law, to relate the reflected X-ray beams incidenting at certain angle to the cleavage faces of crystals, the distance betweem atomic layers in the crystal, and the wavelength of the incident X-ray beam. The works of both Roentgen and Bragg provide us keys to open up mysteries of the Universe and look into the microcosm in atomic level with X-ray “eyes”. Since Bragg’s law is derived according with resomance effects, it plays a critical role in the operation of diffractometry like X-ray diffractometer (XRD) which becomes important in exploring both theoretical and experimental aspects of the physics of materials. The following is hence attributed to introduce the application of XRD in physical researches, specifically in investigating: a) the essential properties of X-rays like (i) their nature of the wave- particle duality as electromagnetic (EM) waves, (ii0 their generation characteristics, and (iii) their effects when interfering with substance; b) the concept of “diffraction” since field theory was initially used to explain the diffraction phenomena of X-rays when applied to crystals.....

CRYSTALLINE STATE

Crystalline and Amorphous States

Laue’s discovery of X-ray diffraction demonstrated that very few solids are amorphous. Materials such as the rocks and soil, metals and alloys, wood, concrete, and even textile fibers are crystalline, at least partly. Rubber becomes crystalline when being stretched, and even bones, hair, muscle fibers, and tendons of animal body are partly crystalline. The most reliable means of demonstrating this crystalline nature is the X-ray diffraction technique. Using X-ray examination, at least 95% of solid inorganic chemical substances are found to be crystalline, and more than 98% of minerals show definite crystalline structure [8-11].....

INTRODUCTION

New concept of X-ray crystalline diffractometry is presented to interpret the functions of X-ray diffraction equipment, such as the Bragg-Brentano diffractometer, Seemann-Bohlin camera (or diffractometer), and Debye back-and transmission-cameras, etc. Bragg’s law, as the fundamental theory in X-ray analysis, is suitable to interpret all the phenomena of X-ray diffraction. However, relative orientations of crystal planes observed in respect to the sample surface, which is potentially of great interest from a purely scientific and from an applicative point of view, cannot be shown by the commercial equipment itself, the term “azimuth relationship” is therefore introduced here.

The concept of two-dimension (2D) diffraction in its many aspects is important in X-ray geometric optics. Now, both Bragg’s and azimuth-angle equations are used to outline the configuration and arrangements of the X-ray equipment as a quaternity figure shown in Fig. 1 at present describing features of the multifunctional 2D/3D X-ray diffractometry.

ROUTINE QUALITATIVE IDENTIFICATION OF CRYSTALLINE POWDERS

The routine application of powder diffraction techniques for the identification of polycrystalline materials dated back to 1938 when the pioneering work of the Dow Chemical Company, was published. The simplicity and advantage of the powder diffraction method for chemical analysis were pointed out which emphasized that:

a) the powder diffraction pattern is a characteristic of the substance,

b) each substance in a mixture produces its pattern independently of the others,

c) it tells the state of chemical combination of the elements in the material,

d) only a minute amount of sample is required;

e) the method is capable of developing a quantitative analysis.....

IDENTIFICATION OF PHASES

The qualitative identification of phase depends, ultimately, upon the measurement of the precise locations and intensities of the diffraction lines. Interplanar spacings corresponding to the diffracted X-rays are to be found first, and then intensities are measured either by a diffractometer, which can give excellent results, when the Ni- or singlecrystal monochromator is used as the filter.

Principle of Superposition of X-Ray Patterns

It is well-known that X-rays diffracted by incoherent planes are according to the principle of superposition of intensities. The same principle is followed by X-ray patterns produced from different crystalline substances. Considering a system consisting of two phases denoted by capital letters, and let the same notation represent their fingerprints as peaks, then....

INTRODUCTION

Through the quantitative analytical chemistry, people can readily obtain the elemental compositions of a material, but usually to distinguish the chemical identity of various phases in a mixture and to determine the precise amounts of each phase present are of great difficulty. Powder X-ray diffraction analysis, on the other hand, is seemingly the perfect technique for the crystalline-mixture analysis, because each component of the mixture produces its characteristic pattern independently and makes the identification of various components possible. Moreover, the intensity of the pattern of each component is proportional to the amount present, so the quantitative analysis of various components is developed. Now analytical determinations of quartz in the presence of mineral silicates, mixed alloy phases of different proportions of the same elements, and the relative amounts of polymorphs in a mixture are handled routinely by the diffraction technique, while these determinations are difficult, or impossible, by using chemical methods [26-30]....

INTRODUCTION

The concept of 2D X-ray diffraction is of both fundamental and technological importance. The proper study begins with the asymmetrical Bragg reflection geometry. The term, the asymmetrical Bragg reflection geometry, means that the angles between incident and diffracted (reflected) X-ray beams, which are identical angles to the reflecting crystal planes, but different to the specimen surface. Besides the Bragg angle θ, the description needs to introduce an angular variable α, which is designated as the angle between the sample surface and the incident X-ray beam. Both angles α and θ are two independent angular variables necessary to construct the two-dimensional system in various applications of diffractometry, especially when flat-layer specimens are used to a far greater extent than the one angular variable θ.....

INTRODUCTION

Based on basic theories of 2D X-ray diffraction, which are essential and fundamental for the development of X-ray techniques, at present, we try to show how to apply these theories to explore and address problems of depth/azimuth-solved multiple-XRD patterns for nanomaterial. Since microstructure is one of the important parameters which control properties of thin films, Structure characterization of materials is therefore essential for the research and development of thin film technology [3-14].

XRD is frequently used for the characterization of thin films and modified surfaces. Although several instruments with different geometries have been devised to adapt the technique for the study of often strongly textured, thin layers among them being the most flexible tool probably use the conventional Bragg-Brentano (B-B) goniometer. Now, the presented common scan mode facilitated to have multifunction. Reliable procedures have been devised on a routine basis for phase identification, precision measurements of lattice parameters and quantitative phase analysis. The possibility of studies for thin layers is very attractive by using these STD and ADA as well as CBD, techniques.....

INTRODUCTION

General Considerations

X-ray stress measurement method takes advantage of X-ray diffraction phenomena and is used to obtain the stress in metallic materials, ceramics and polycrystalline aggregates by measuring their lattice distortions. This method has salient features: a) the measurement is non-contact and non-destructive; b) not only the added stress but also the residual stress is measurable; c) exceedingly small portions (approximate 1mm2) in a thin layer (a few μm or less) of the sample are examined. This measurement technique is especially effective, such as....

INTRODUCTION

Based on the 2D X-ray diffraction theory, for the profile analysis, the new intensity model (rather than the existing convolution and deconvolution model), is developed. In the present model, it is considered that the observed intensities are additive from those X-ray intensities scattering from small-size and distorted crystallites, as well as the other non- Bragg scattering from the instrument, cosmic radisation, etc. As a result, the observed profile is additive from the individual profiles of the size-strain profiles and the unwanted profile. On the basis of Scherrer equation, the termed Weighted-mean dimension (W-md) is first defined to evaluate the broadening due to small-size crystallites, combining with the Weighted-mean breadth (W-mb) to deal with broadening due to small-size crystallites. The effect of the instrumental broadening is effectively eliminated by using relative- intensities method in data processing.....

INTRODUCTION

All lattice imperfections, including finite sizes and strains/stress of grains or crystallites of the material, give rise to broadening of X-ray diffracted-line profiles.

The application of powder diffraction techniques for the determination of micro-strains of imperfect materials dated from 1940 when the pioneering work showed that the desired function of strain distribution can be obtained from the measurable functions of intensities by making use of Fourier series method. Luland reported the breadths of several curves of interest in the diffractometry. However, the earliest functions to be used and ones still widely employed in this field were Cauchy and Gaussian. In these cases, function was necessarily a good approximation to broaden line and there was evidence that a better representation can be obtained from the convolution of one or more Cauchy and Gaussian functions.....

INTRODUCTION

Continuing discussions on the topic of small-size and strain-broadenings, this chapter deals with the extraneous broadening due to instrument-geometrical effects. The newly developed model is of particular importance in the size and/or strain determinations, not only because it extensively meets the 2D X-ray diffraction techniques in many types of X-ray analysis, but also because it has the superiority over other existing methods for the measurement of diffraction profiles. The present series studies have greatly improved the precision of the determination of line breadth.

In order to discuss the extraneous broadening arisen from instrument geometry effects, we would like to outline briefly the basic principle on which the broadening is based and to give the formation of an actual profile by intensity data observed.....

INTRODUCTION

It is well-known that quantitative phase analysis readily gives the precise amounts of phases in a mixture, since each component of the mixture produces its own characteristic pattern independent of the others, making it identify the various components possible; on the other hand, the intensity of each component’s pattern is proportional to the amount present of the component, thus a quantitative analysis for various components is developed. The same principle is valid for the profile analysis to determine micro-strains in imperfect materials.

The broadening of X-ray diffraction line profile is caused by structural imperfections of the specimen and instrumental effects. At present, a completely new model has been developed based on a newly developed 2D X-ray diffraction theory. From investigations of principles of X-ray wave theory, it is known that the observed X-ray diffraction profile of reflection is the summation of the inherent profile and extraneous profile. A precise analysis of profiles gives a picture of microstructure changes in the materials under way. Here in the method both weightedmain breadth [replaced the Laue intensity-integral breadth and the weighted-strain breadth are combined together to solve the doublet broadening problems. Because of its rapidity and convenience, compared with the widely used Fourier series method and close function methods, the present method is a straightforward method of measurement. In addition, the broadening effects due to instrumental geometry and penetration depth can be effectively eliminated using the intensify-ratio method for the profile analysis. At present, it is assumed that the size-and strain-broadening profiles are dependent on the order of reflection and follow the cosine- and tangential- relationships respectively, but the dimension of crystallites are order-independent.....

REVIEW AND DEVELOPMENT

The texture/orientation analysis concerns the density distribution of crystallites with orientations. Orientations of crystallites affect their diffracted intensities in direction. It can be directly determined by 2D/3D X-ray techniques, based on the 2D X-ray diffraction theory.

The following text will be taken into consideration the pole-figure method. Previously, according to [1], textures in the spherical-harmonic analysis were represented by a) preparation of pole figures from X-ray data; b) expansion of the pole figure in a series of spherical harmonics (normalized by Legendre polynomials); c) three sets of coordinates required. In a sample, the orientation of every grain or crystallite can be specified by fixing a coordinate system in the sample (x, y, z) and another in the crystal (X,Y,Z) with respect to the unit cell; in addition, the orientation of a given crystallite is given by rotation angles θ, Ψ and φ which are necessary to align the two systems in a spherical coordinate system. But this is almost impossible for polycrystalline materials; because these materials consist of many crystallites with different shapes, sizes and orientations.....

INSPIRATION FROM X-RAY DIFFRACTOMETRY

XRD is a basic scientific tool capable of supplying research workers with fundamental data and information unattainable by other techniques. The complete cognition of potentialities, however, requires an elementary knowledge of crystallography and X-ray science. As mentioned in this book, people who carry out certain routine applications of X-ray diffraction demand a little familiarity with this field, but most workers need to have a deeper knowledge on theory for both theoretical and applied research. The purpose of this section is to review and develop some ideas to understand the concepts of interaction of light/EM waves and matter in XRD. Yet another and even more powerful approach is provided by EM-field theory.....

[Reprinted from Kathleen Lonsdale: Internationals for X-ray Crystallography, Vol. III p.162-165 (1962). In the Table, the numerical number under line shows the absorption edge of the element.]....

The notation used is consistent throughout the book, but does not necessarily conform to that used previously in studies of X-ray powder diffraction methods, except that some notation has to be of many meanings, which depends on the specific situation....

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