Interference of X-rays

Max von Laue, 1879-1960. German physicist who in 1912 discovered the interference of X-rays diffracted by crystals, measured the wave lengths of X rays, and studied the structure of crystals. In 1914 he was awarded the Nobel Prize for physics. 

FIGURE 4.20 Constructive interference of x-rays scattered by atoms in lattice planes. Three beams of x-rays, scattered by atoms in three successive layers of a crystal are shown. Note that the phases of the waves are the same along the line CH, indicating constructive interference at this scattering angle 26. 

Figure 6.13 Potential interference of X-ray detection due to low take-off angle in the SEM.

Interference of X-rays supplies a powerful weapon for investigating the structure of crystals. For this purpose we do not even require large pieces of the crystal, but can use it in the form of powder (Debye-Scherrer, 1915 Hull, 1917). The interference figures in the lattcT case are rings round the direction of the incident beam. Indeed, the pow(l( r grains may be of molecular size even. What is more, it is found that... 

The adsorbed layer is seldom of sufficient thickness to produce interference of X-rays, hence this tool sheds little light on the status of the adsorbed substance. Studies of optical and magnetic properties reveal but little. 

FIGURE 316. William H. Bragg and his son William L. Bragg reversed von Laue s experiment and used x-rays to measure the distances between ions or atoms in crystals (see text), (a) Depicts the conditions for constructive interference of x-rays termed Braggs Law (b) schematic of Braggs s x-ray apparatus (WH. Bragg and W.L. Bragg, X-Rays And Crystal Structure, 4th ed., London, 1924). 

T 0 understand at which values of 28 one may see peaks in a diffraction pattern, it is often convenient to treat the phenomenon of diffraction from a polycrystalline sample in terms of the interference of x-rays reflected from different planes within the unit cell of a crystal. In this way, Bragg s law, which reiates the 20 angle at which diffraction peaks occur to distances between planes in the unit cell (d ki), can be readily understood (see Fig. 2). Planes are described by Miller indices, hkL which are related reciprocally to the point at which the plane intercepts the cell edges, a. h, and c. 

Diffraction of X-rays is the basic technique for obtaining information on the atomic structure of crystalline soHds and is one of the key standard laboratory techniques. XRD is based on the interference of X-ray waves elastically scattered by a series of atoms orientated along a particular direction in a crystal characterized by a vector A. The waves scattered by two atoms a and b interfere constructively with each other when the path difference PQR is equal to an integer number of wavelengths PQR = W. This condition is valid for orientations K of the scattered waves which satisfy the Laue condition ... 

Notice that while F is a complex number, only the absolute square, F, appears in eq. (9). This gives rise to the well-known phase problem of X-ray crystallography. The exponential in eq. (9), the Debye-Waller factor, involves the terms hor and u, which are the mean-square atomic displacements in the horizontal and vertical directions, respectively. The factor 7 (ap) describes interference of X-rays diffracted upwards with X-rays diffracted down and subsequently reflected back up by the interface. The factor ITCap) equals unity except near ap= ttc where it peaks sharply (the Yoneda-Vineyard peak) [65, 66]. It is convenient for deducing the zero-point of the f scale, which often covers a range of 0 to 10 , but otherwise... 

Constructive interference of X-rays reflected at two planes separated by a distance d (Bragg s Law). 

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