Diffraction phenomena

10-1 Uniform and Gaussian beams 10-2 Light containment by a fiber 10-3 Applicability of geometric optics 

10-5 Transit time for the lateral shift 10-6 Lateral shift in planar waveguides 10-7 Preferred ray directions 

The transform of the Gaussian beam is Gaussian in shape, and the transform of the uniform beam is the cylindrical polar analogue of sin x/x. It is also useful 

Conversely, a Gaussian beam with a = 0.75r has nearly the same diffraction intensity pattern as the uniform beam of radius r. The situation is illustrated in Fig. 10-1. Clearly A (u) is also approximately Gaussian for various smoothed-out beam profiles intermediate between the uniform and Gaussian cases, e.g. the profiles given by Eq. (15-9), so the results and conclusions for the Gaussian beam have wide applicability. 

It is helpful to describe the diffraction intensity pattern by a characteristic small angular spread in 0. We relate this spread to the corresponding value of the transform variable when the diffraction pattern amplitude has fallen to 1/e of its peak value, i.e. = 1/e. This gives 

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Figure 31 BiPbSr2YCusOfi Several types of satellites are observed in different areas of the same crystals ([001]) (a) satellites set up along a direction roughly parallel to [120] (q-6.25) (b) bidimens-ional system of satellites resulting from the existence of misori-ented areas and double diffraction phenomena, the angle between both systems is close to 100 (c) satellites along [110] with q 3.65 (d) previous satellites have disappeared but streaks remain along (II0 (e) multisplitting of the spots and first satellites they are indications of distortions and microtwins.

Figure 33 Tin tSr CaCugO (001J E D. pattern. numerous extra spots are observed whkh van be interpreted as the result of the mnoraentation of two lamellae and double diffraction phenomena.

Equation (21) is called the Bragg equation after the father-and-son team of W. H. and W. L. Bragg (Nobel Prize, 1915) it is the underlying relationship for all diffraction phenomena. We encounter the Bragg equation again in Chapter 9 when we discuss the diffraction of low-energy electrons by surface atoms (see Section 9.8b). 

A comparative study has been made by optical and electron microscopy of the anisotropic texture of several cokes from caking coals and pitches carbonized near their resolidification temperature. A simple technique made it possible to examine, by both methods, the same area of each sample and to identify the corresponding zones of the two very similar images. The anisotropy observed in polarized light appears in electron microscopy as differences in contrast resulting not from inequalities in electron absorption, but, as revealed by microdiffraction and dark Reid examinations, from diffraction phenomena depending on the general orientation of the carbon layers within each anisotropic area. 

The crystal symmetry changes that accompany order-disorder transitions, discussed in Section 17.1.2, give rise to diffraction phenomena that allow the transitions to be studied quantitatively. In particular, the loss of symmetry is accompanied by the appearance of additional Bragg peaks, called superlattice reflections, and their intensities can be used to measure the evolution of order parameters. 

Fraunhofer diffraction phenomena are observed when both the source and the point of observation are effectively at infinite distance from the diffracting ohject, obstacle, or aperture. This condition is sometimes brought about by passing the light from the source through a collimator before it is diffracted, and then focusing the parallel diffracted rays at the point of observation. 

ELECTRON DIFFRACTION. Beams of high-speed electrons exhibit diffraction phenomena analogous to those obtained with light, thus showing the wavc-like character of electron beams. Such patterns are useful in the interpretation of the structure of matter. 

Site populations of less abundant cations estimated from X-ray diffraction measurements have limited accuracies particularly when iron is present, since diffraction phenomena involve the cooperative scattering effects of many atoms in the unit cell. It is very difficult, and sometimes impossible in X-ray structure refinements, to distinguish between cations of different valences, such as Fe2+ and Fe3+, Mn2+ and Mn3+ or Ti3+ and Ti4+, and between neighbouring elements in the periodic table with similar scattering factors, such as iron... 

In electron microscopy as in any field of optics the overall contrast is due to differential absorption of photons or particles (amplitude contrast) or diffraction phenomena (phase contrast). The method provides identification of phases and structural information on crystals, direct images of surfaces and elemental composition and distribution (see Section H below). Routine applications, however, may be hampered by complexities of image interpretation and by constraints on the type and preparation of specimens and on the environment within the microscope. 

Alternatively, images can be formed with electrons scattered at high angles (HAADF mode, for High Angle Annular Dark Field mode). An annular detector is placed below the sample and the focused electron beam is scanned over the sample. Interestingly, the contrast is not altered by diffraction phenomena and depends both the thickness and the composition (25). 

The controversy was settled in favour of the undulatory theory because the latter could immediately give a reasonable explanation of interference and diffraction phenomena. Again the observation (Foucault) that the velocity of propagation of light in a medium is less than in a vacuum was in contradiction to the emission theory. 

It was only in 1927 from the experiments of Davisson and Germer and slightly later from those of G. P. Thomson that it was found that these electron beams exhibit exactly the same diffraction phenomena as those which Von Laue, Friedrich and Knipping had observed with X-rays in 1912. For X-rays this result was in agreement with the prevailing conception of the nature of these rays. With electrons, however, this wave character appeared to be completely in conflict with the ideas which had been supported for more than 50 years, nevertheless only three years before De Broglie had published his fundamental hypothesis on the wave nature of electrons in his thesis (1924). 

What would have been the development of physics if Hittorf, Goldstein or Lenard has just put a gold foil in the path of the cathode rays instead of a massive cross Would not they have regarded the diffraction phenomena as an incontestable proof of the wave nature of the cathode rays ... 

In general, there is no principal difference in the diffraction phenomena using the synchrotron and conventional x-ray sources, except for the presence of several highly intense peaks with fixed wavelengths in the conventionally obtained x-ray spectrum and their absence, i.e. the continuous distribution of photon energies when using synchrotron sources. Here and throughout the book, the x-rays from conventional sources are of concern unless noted otherwise. 

Finally, we see from the Fourier transform equations, for the structure factor Fhu and the electron density p x, y, z), that any change in real space (e.g., the repositioning of an atom) affects the amplitude and phase of every reflection in diffraction space. Conversely, any change in the intensities or phases in reciprocal space (e.g., the inclusion of new reflections) affects all of the atomic positions and properties in real space. There is no point-to-point correspondence between real and reciprocal space. With the Fourier transform and diffraction phenomena, it is One for all, and all for one (Dumas, The Three Musketeers, 1844). 

Data on the ordered structures found in particular alloys are given by Barrett and Massalski [G.25] and Pearson [G.16]. The theory of the diffraction phenomena involved is treated by Warren [G.30] and Guinier [G.21]. 

Although the reciprocal lattice may at first appear rather abstract or artificial, the time spent in grasping its essential features is time well spent, because the reciprocal-lattice theory of diffraction, being general, is applicable to all diffraction phenomena from the simplest to the most intricate. Familiarity with the reciprocal lattice will therefore not only provide the student with the necessary key to complex diffraction effects but will deepen his understanding of even the simplest. 

The reciprocal lattice was invented by crystallographers as a simple and convenient representation of the physics of diffraction by a crystal. It is an extremely useful tool for describing all kinds of diffraction phenomena occurring in powder diffraction (Figure 1.5). 

Further experiments by G. P. Thomson, Rupp and others showed that when beams of electrons are made to pass tlirough thin foils (metals, mica), diffraction phenomena are obtained, of the same kind... 

The deduction of the uncertainty relation, with the help of diffraction phenomena and other processes capable of being visualized, gives only a result specifying an order of magnitude. To obtain an exact inequality, we must call upon the formalism of quantum mechanics. Of this a short exposition will now be given, and the various theorems will be illustrated by examples in the preceding Appendices. 

There are technical causes characteristic to the instrument and collectively known as instrumental parameters width of entrance slit, quality of the optics, focal distance, diffraction phenomena through narrow orifices. However, there are also causes due to quantum mechanics, which ensure that the spectral transitions have a natural width . The radiations emitted by the atoms are not quite monochromatic. In particular with the plasmas, a medium in which the collision frequency is high (this reduces enormously the lifetimes of the excited states), Heisenberg s uncertainty principle plays a large role (Figure 14.7). Moreover, the elevated temperatures increase the speed of the atoms, enlarging line widths by the Doppler effect. Finally for all of these reasons, the width of the lines at 6000 K reaches several picometres. 

Photonic crystals are natural or artificial solids that are able to manipulate light in a predetermined fashion, rather as X-rays are manipulated by ordinary crystals. For this to be possible they must contain an array of scattering centres analogous to the atoms in ordinary crystals. Perhaps surprisingly, the diffraction phenomena are little different than that described for X-ray, electron and neutron diffraction, and the equations given above in this Chapter apply to photonic crystals as well as X-rays. 

In this chapter we shall show how the observed phenomena may be explained by means of elementary catastrophe theory. In principle, the discussion will be confined to examination of non-chemical systems. However, some of the discussed problems, such as a stability of soap films, a phase transition in the liquid-vapour system, diffraction phenomena or even non-linear recurrent equations, are closely related to chemical problems. This topic will be dealt with in some detail in the last section. The discussion of catastrophes (static and dynamic) occurring in chemical systems is postponed to Chapters 5, 6 these will be preceded by Chapter 4, where the elements of chemical kinetics necessary for our purposes will be discussed. 

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