Laue spots

Switzerland). The short wave-length limit of X-radiation present in the incident beam was 0,24 A values of nX calculated for Laue spots on the basis of the unit obtained by putting m = —n3 =4 were found often... 

The presence of all observed Laue spots, produced by planes belonging to over two hundred different forms, is accounted for by this unit. (Representative Laue data are recorded in Table II.) No evidence was found for the existence of a larger unit hence this unit may be accepted as the true one. All indices used in this paper (including Table II) are referred to the axes of this unit of structure. The unit contains 8 Ti02 the density calculated from the X-ray data is 4,12, within the range 4,03—4,22 of the experimental determinations reported in Groth,... 

On account of the lack of definite knowledge of the F-curves no attempt was made to account for the intensities of Laue spots. 

When indices were assigned to the Laue spots on a basis of this unit, values of n X as low as 0.13 A. were found. As the minimum wave length... 

The theoretically obtained electron densities of ions may be used for the calculation of the so-called F curves, which give the effective reflecting power of the ion as a function of the angle of reflection and the wave-length of X-rays, and which are of use in the determination of crystal structures. It may be mentioned that the high maximum value of the electron density at the nucleus given by our calculations provides considerable justification for the method of determining crystal structures with the aid of the relative intensities of Laue spots produced by crystal planes with complicated indices. 

Fig. 3-7 Location of Laue spots (a) on ellipses in transmission method and (b) on hyperbolas in back-reflection method. (C = crystal, F = film, Z.A. = zone axis.)...

The position of any Laue spot is unaltered by a change in plane spacing, since the only effect of such a change is to alter the wavelength of the diffracted beam. It follows that two crystals of the same orientation and crystal structure, but of different lattice parameter, will produce identical Laue patterns. 

The Bragg angle 0 corresponding to any transmission Laue spot is found very simply from the relation... 

If necessary, the intensity of a Laue spot may be increased by means of an intensifying screen, as used in radiography. This resembles a fluorescent screen in having an active material coated on an inert backing such as cardboard, the active material having the ability to fluoresce in the visible region under the action of... 

An intensifying screen should not be used if it is important to record fine detail in the Laue spots, as in some studies of crystal distortion, since the presence of the screen will cause the spots to become more diffuse than they would ordinarily be. Each particle of the screen which is struck by x-rays emits light in all directions and therefore blackens the film outside the region blackened by the diffracted beam itself, as suggested in Fig. 5-5. This effect is aggravated by the fact that most x-ray film is double-coated, the two layers of emulsion being separated by an appreciable thickness of film base. Even when an intensifying screen is not used, double-coated film causes the size of a diffraction spot formed by an obliquely incident beam to... 

Fig. 5-6 Effect of double-coated film on appearance of Laue spot (a) section through diffracted beam and film (b) front view of doubled spot on film.

We will see later that Laue spots become smeared out if the reflecting crystal is distorted. Here, however, we are concerned with the shapes of spots obtained from perfect, undistorted crystals. These shapes are greatly influenced by the nature of the incident beam, i.e., by its convergence or divergence, and it is important to realize this fact, or Laue spots of unusual shape may be erroneously taken as evidence of crystal distortion. 

Fig. 5-11 Shapje of transmission Laue spots as a function of position.

In back reflection, no focusing occurs and a divergent incident beam continues to diverge in all directions after diffraction. Back-reflection Laue spots are therefore more or less circular near the center of the pattern, and they become increasingly elliptical toward the edge, due to the oblique incidence of the rays on the film, the major axes of the ellipses being approximately radial. Figure 3-6(b) is typical. 

A transmission Laue pattern is made of an aluminum crystal with 40-kV tungsten radiation. The film is 5 cm from the crystal. How close to the center of the pattern can Laue spots be formed by reflecting planes of maximum spacing, namely (111), and those of next largest spacing, namely (200) ... 

In the absence of a Greninger chart, the pole corresponding to any observed Laue spot may be plotted by means of an easily constructed stereographic ruler. The construction of the ruler is based on the relations shown in Fig. 8-10. This drawing is a... 

Fig. 8-24 Effect of lattice distortion on the shape of a transmission Laue spot. CN is the normal to the reflecting plane.

Since the asterism on the back-reflection pattern chiefly follows zone lines, the major portion of planes inclined at large angles to the incident beam are bent about a single axis. However, the existence of Debye arcs shows that there are latent Laue spots of considerable area superimposed on the visible elongated spots, and that a Simall portion of the planes referred to are therefore bent about a number of axes. 

The sensitivity of the ordinary Laue method in the detection of crystal disorientation may be estimated as follows. Suppose the crystal-to-film distance is 5 cm, and assume that the minimum detectible broadening of a Laue spot is 1 mm. Then the diffracted beam has diverged by about 1/50 radian or 1°. This divergence corresponds to a disorientation of the reflecting planes of about 0.5°. This disorientation applies only to the area irradiated by the incident beam, which is typically 1 mm in diameter the irradiated area is therefore only 10" cm. ... 

Fig. 8-28 Enlarged transmission Laue spots from a thin crystal of silicon ferrite ( -iron containing 3.3 percent silicon in solid solution) (a) as bent to a radius of 9 mm., (b) after annealing 10 min at 950°C, (c) after annealing 4 hr at 1300°C. Dunn and Daniels [8.9].

If the crystal is bent or polygonized, as in Fig. 8-29(b), the reflected beam will broaden and the Laue spot on the film will be a rectangle of width w. (The dashed lines in this drawing will be discussed later.) The disorientation of the crystal may be calculated from the magnitude of w and the other dimensions involved. Disorientations as small as 5 x 10 degree are detectible, and this value pertains to an irradiated crystal area of 1-2 cm. Fig. 8-30(b) shows a broadened reflection of this kind. 

A modification of the Guinier-Tennevin method can reveal additional information about a deformed crystal [8.12, 8.13]. If a Seller slit with horizontal plates is placed in the incident beam and the crystal is twisted about a vertical axis, the original vertical-line Laue spot broadens into a striated region composed of fine inclined lines, and the inclination of these lines is a measure of the torsional strain in the crystal (Fig. 8-30(c)). 

Fig. 8-30 Single Laue spots obtained by the Guinier-Tennevin method on a film placed at the focusing position Fof Fig. 8-29(a). Spots (a), (b), and (c) are from the transverse planes of a quartz crystal plate, 37 x 13 x 0.5 mm (a) unstrained, magnification 2X, (b) elastically bent, 2X, (c) elastically twisted, 5X. Spot (d) is from an aluminum crystal after plastic deformation, 4X. Julien et al. [8.12-8.14].

A back-reflection Laue photograph is made of an aluminum crystal with a crystal-to-film distance of 3 cm. When viewed from the x-ray source, the Laue spots have the... 

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