X-Rays by Crystals

In a solid, the interatomic distances are of the order of an Angstrom (iA = iO i m = 100 pm). For example, the interatomic distance for a C-H bond is 1.08 A, for a single C-C bond the distance is 1.54 A, and for a metal-oxygen bond it is about 2 A, Thus, in order to distinguish two neighboring atoms, a microscope must use radiation with a wavelength of the order of 1 A. The image obtained with such a microscope obviously depends on the interaction of radiation with matter. 

X-rays are electromagnetic waves just like visible light. They interact with the electrons contained in all matter. Consequently, the image obtained with an X-ray microscope reveals the distribution of the electrons, a distribution whose maxima correspond to the atomic positions. In crystallography we use X-rays with wavelengths of the order of 0.5 to 2 A. 

According to the wave theory of elementary particles, a particle of mass m moving with a velocity v has a wavelength A = h/mr, where h is Planck s constant. Thermal (neither hot nor cold ) neutrons have a wavelength of about 1 A, Neutrons interact with matter in two different ways. On the one hand, they Interact with atomic nuclei, thus the image produced by a neutron microscope  

According to Abbe, the formation of the image of an object by means of one or more lenses takes place in two steps, diffraction of the radiation by the object followed by recombination of the diffracted radiation by means of a lens (Fig. 3,1), For example, a periodic lattice made up of lines of separation d illuminated by a plane wave with the wave vector perpendicular to the lattice plane will emit diffracted beams concentrated in specific directions given by the equation (Section 3.1,3) 

The objective in Fig. 3.1 transforms the plane waves diffracted by the lattice into spherical waves whose centers are located in the focal plane common to the two lenses. The eyepiece L2 transforms these spherical waves back into plane 

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Bijvoet, J.M., Burgers, W.G. and Hagg, G. (1972) Early Papers on Diffraction of X-rays by Crystals (Int. Union of Crystallography, Utrecht) pp. 5. 

M. von Laue (Frankfurt) discovery of the diffraction of X-rays by crystals. 

X-ray diffraction occurs in the elastic scattering of X-ray photons by atoms in a periodic lattice. The scattered monochromatic X-rays that are in phase give constructive interference. Figure 4.4 illustrates how diffraction of X-rays by crystal planes allows one to derive lattice spacings by using the Bragg relation ... 

Projectors often arise in attempts to describe experiments within the structure of Quantum Mechanics. For example, in the case of the coherent scattering of X-rays by crystals the ideal measured intensities are given by the square of the structure factors... 

Lithium Fluoride (Li + F —> LiF) is used to produce ceramics and rocket fuel and is used as welding and soldering flux and in hght-sensitive scientific instruments (e.g., X-ray diffraction, which is the scattering of X-rays by crystals that produce a specific pattern of that crystal s atoms, thus producing a technique for identifying different elements). 

A great amount of information about the structure of crystals has been obtained by use of the x-ray diffraction method. The diffraction of x-rays by crystals was discovered by Max von Laue in 1912. Shortly thereafter W. L. Bragg discovered the Bragg equation, and in 1913 he and his father, W. H. Bragg, published the first structure determinations of crystals. 

Obviously, much of the development of crystallography predates the discovery of diffraction of X-rays by crystals. Early studies of crystal structures were concerned with external features of crystals and the angles between faces. Descriptions and notations used were based on these external features of crystals. Crystallographers using X-ray diffraction are concerned with the unit cells and use the notation based on the symmetry of the 230 space groups established earlier. 

X-ray diffraction The scattering of X-rays by crystal atoms, producing a diffraction pattern that yields information about the structure of the crystal. 

One of the greatest and most surprising discoveries of our own age, that of the diffraction of X-rays by crystals (in 1912) was made by a mathematician, Max von Laue, by the sheer power of believing more concretely than anyone else in the accepted theory of crystals and X-rays. 

NOTE ON THE SELECTIVE REFLECTIONS OF X-RAYS BY CRYSTALS OF POTASSIUM BROMIDE... 

Unlike visible light, X rays cannot be focused, so we use diffraction of X rays by crystals to see molecules, but, to do this, the microscope lens must be replaced by a mathematical computation (a Fourier synthesis). 

Bragg s Law, the Bragg equation In diffraction of X rays by crystals, each diffracted beam can be considered to be reflected from a set of parallel lattice planes. If the angle between the diffracted X-ray beam (wavelength X) and the normal (perpendicular) to a set of crystal lattice planes is 90° - Ohki, and if the perpendicular spacing of the lattice planes is dhti, then ... 

Laval, J. Diffusion of X-rays by crystals in directions other than those of selective reflection. Comptes Rendues Acad. Set. (Pans) 208, 1512-1514 (1939). 

Zener, C. Theory of the effect of temperature on the reflection of X-rays by crystals. II. Anisotropic crystals. Phys. Rev. 49, 122-127 (1936). 

Max von Laue (1879-1960). German physicist who was the first to observe and explain the phenomenon of x-ray diffraction in 1912 Laue was awarded Nobel Prize in Physics in 1914 for his discovery of the diffraction of x-rays by crystals . For more information about Laue see http //wvw.nobel.se/physics/laureates/1914/... 

The reflection by a crystal of X-radiation characteristic of the atoms in the crystal itself, which Dr. G. L. Clark and the writer discovered (these Proceedings, May, 1922 and April, 1923), does not appear to be explainable in a simple manner by the theory of interference of waves. This note describes an attempt to formulate a theory of the reflection of X-rays by crystals, based on quantum ideas without reference to interference laws. 

Radiography was thus initiated without any precise understanding of the radiation used, because it was not until 1912 that the exact nature of x-rays was established. In that year the phenomenon of x-ray diffraction by crystals was discovered, and this discovery simultaneously proved the wave nature of x-rays and provided a new method for investigating the fine structure of matter. Although radiography is a very important tool in itself and has a wide field of applicability, it is ordinarily limited in the internal detail it can resolve, or disclose, to sizes of the order of 10 cm. Diffraction, on the other hand, can indirectly reveal details of internal structure of the order of 10 cm in size, and it is with this phenomenon, and its applications to metallurgical problems, that this book is concerned. The properties of x-rays and the internal structure of crystals are here described in the first two chapters as necessary preliminaries to the discussion of the diffraction of x-rays by crystals which follows. 

At first glance, the diffraction of x-rays by crystals and the reflection of visible light by mirrors appear very similar, since in both phenomena the angle of incidence is equal to the angle of reflection. It seems that we might regard the planes of atoms as little mirrors which reflect the x-rays. Diffraction and reflection, however, differ fundamentally in at least three aspects ... 

The diffraction of x-rays by crystals was discovered in 1912, and in 1913 the first determinations of the atomic arrangement in crystals were made by use of this technique by the British physicists W. H. Bragg and W. L. Bragg (father and son). Their work during this first year included the determination of the structure of diamond, as shown in the adjacent drawing. 

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