Laue method In the Laue method, a stationary single crystal is

5.3.3.1 Laue method In the Laue method, a stationary single crystal is

bathed in a beam of ‘white’ radiation. Then, because the specimen is a fixed single crystal, the variable nec- essary to ensure that the Bragg law is satisfied for all the planes in the crystal has to be provided by the range of wavelengths in the beam, i.e. each set of crys-

spectrum to give a Bragg reflection. Radiation from a target metal having a high atomic number (e.g. tung- sten) is often used, but almost any form of ‘white’ radiation is suitable. In the experimental arrangement shown in Figure 5.10a, either a transmission photo- graph or a back-reflection photograph may be taken, and the pattern of spots which are produced lie on ellipses in the transmission case or hyperbolae in the back-reflection case. All spots on any ellipse or hyper- bola are reflections from planes of a single zone (i.e. where all the lattice planes are parallel to a common direction, the zone axis) and, consequently, the Laue pattern is able to indicate the symmetry of the crystal. For example, if the beam is directed along a [1 1 1] or [1 0 0] direction in the crystal, the Laue pattern will show three- or fourfold symmetry, respectively. The Laue method is used extensively for the determina- tion of the orientation of single crystals and, while charts are available to facilitate this determination, the

method consists essentially of plotting the zones taken Figure 5.10 (a) Laue method of X-ray diffraction. (b) Asterisms on a Laue transmission photograph of from the film on to a stereogram, and comparing the

deformed zinc (after Cahn, 1949) . angles between them with a standard projection of that crystal structure. In recent years the use of the Laue technique has been extended to the study of imper-

the successive diffracted cones, which consist of rays fections resulting from crystal growth or deformation,

from hundreds of grains, intersect the film to produce because it is found that the Laue spots from perfect

concentric curves around the entrance and exit holes. crystals are sharp, while those from deformed crystals

Figure 5.12 shows examples of patterns from bcc and are, as shown in Figure 5.10b, elongated. This elon-

fcc materials, respectively.

gated appearance of the diffraction spots is known as Precise measurement of the pattern of diffraction asterism and it arises in an analogous way to the reflec-

lines is required for many applications of the powder tion of light from curved mirrors.

method, but a good deal of information can readily be obtained merely by inspection. One example of this

5.3.3.2 Powder method is in the study of deformed metals, since after defor- mation the individual spots on the diffraction rings are

The powder method, devised independently by Debye blurred so much that line-broadening occurs, especially and Scherrer, is probably the most generally useful of

all the X-ray techniques. It employs monochromatic radiation and a finely-powdered, or fine-grained poly-

able, since the collection of randomly-oriented crystals will contain sufficient particles with the correct orien- tation to allow reflection from each of the possible reflecting planes, i.e. the powder pattern results from

a series of superimposed rotating crystal patterns. The angle between the direct X-ray beam and the reflected

reflecting planes producing the cone. Thus, if a film is placed around the specimen, as shown in Figure 5.11,

Figure 5.11 Powder method of X-ray diffraction .

136 M

oder n

Physical

M etallur

g y and M

ater ials E ngineer

ing

Figure 5.12 Powder photographs taken in a Philips camera (114 mm radius) of (a) iron with cobalt radiation using an iron filter and (b) aluminium with copper radiation using

1 2 are observable. .

The characterization of materials 137 at high Bragg angles. On low-temperature annealing,

standards. Yet a third use of the powder method as an the cold-worked material will tend to recover and this

inspection technique is in the detection of a preferred is indicated on the photograph by a sharpening of the

orientation of the grains of a polycrystalline aggregate. broad diffraction lines. At higher annealing tempera-

This is because a random orientation of the grains will tures the metal will completely regain its softness by a

produce a uniformly intense diffraction ring, while a process known as recrystallization (see Chapter 7) and

preferred orientation, or texture, will concentrate the this phenomenon is accompanied by the completion

intensity at certain positions on the ring. The details of the line-sharpening process. With continued anneal-

of the texture require considerable interpretation and ing, the grains absorb each other to produce a structure

are discussed in Chapter 7.

with an overall coarser grain size and, because fewer reflections are available to contribute to the diffrac-

5.3.3.3 X-ray diffractometry tion cones, the lines on the powder photograph take

In addition to photographic recording, the diffracted on a spotty appearance. This latter behaviour is some-

X-ray beam may be detected directly using a counter times used as a means of determining the grain size of

tube (either Geiger, proportional or scintillation type)

a polycrystalline sample. In practice, an X-ray photo- with associated electrical circuitry. The geometrical graph is taken for each of a series of known grain sizes

arrangement of such an X-ray diffractometer is shown to form a set of standards, and with them an unknown

in Figure 5.13a. A divergent beam of filtered or grain size can be determined quite quickly by com-

monochromatized radiation impinges on the flat face paring the corresponding photograph with the set of

of a powder specimen. This specimen is rotated at

Figure 5.13 Geometry of (a) conventional diffractometer and (b) small-angle scattering diffractometer, (c) chart record of diffraction pattern from aluminium powder with copper radiation using nickel filter .

138 Modern Physical Metallurgy and Materials Engineering precisely one-half of the angular speed of the receiving

recording the diffraction image, but subsequent magni- slit so that a constant angle between the incident and

fication of up to 500 times may be achieved with high- reflected beams is maintained. The receiving slit is

resolution X-ray emulsions. Large areas of the crystal mounted in front of the counter on the counter tube

to thicknesses of 10–100 µ m can be mapped using arm, and behind it is usually fixed a scatter slit to

scanning techniques provided the dislocation density ensure that the counter receives radiation only from

is not too high ⊲6>10 10 m ⊳ . the portion of the specimen illuminated by the primary

The X-ray method of detecting lattice defects suf- beam. The intensity diffracted at the various angles is

fers from the general limitations that the resolution recorded automatically on a chart of the form shown

is low and exposure times are long (12 h) although in Figure 5.13c, and this can quickly be analysed for

very high intensity X-ray sources are now avail- able from synchrotrons and are being used increas-

The technique is widely used in routine chemical ingly with very short exposure times (¾minutes). By analysis, since accurate intensity measurements allow

comparison, the thin-film electron microscopy method

a quantitative estimate of the various elements in the (see Section 5.4.2) is capable of revealing dislocations sample to be made. In research, the technique has

with a much higher resolution because the disloca- been applied to problems such as the degree of order

tion image width is 10 nm or less and magnifications in alloys, the density of stacking faults in deformed

up to 100 000 times are possible. The X-ray method alloys, elastic constant determination, the study of

does, however, have the great advantage of being able imperfections and preferred orientation.

to reveal dislocations in crystals which are compara- tively thick (¾1 mm, cf. 0.1 µ m in foils suitable for transmission electron microscopy). The technique has

5.3.3.4 X-ray topography been used for studying in detail the nature of dis- With X-rays it is possible to study individual crys-

locations in thick single crystals with very low dis- tal defects by detecting the differences in intensity

location densities, such as found in semiconducting diffracted by regions of the crystal near dislocations,

materials; Figure 5.14b shows an example of an X-ray for example, and more nearly perfect regions of the

topograph revealing dislocations in magnesium by this crystal. Figure 5.14a shows the experimental arrange-

technique.

ment schematically in which collimated monochro- matic K˛-radiation and photographic recording is

5.3.4 Typical interpretative procedures for

used.

diffraction patterns

Any imperfections give rise to local changes in diffracted or transmitted X-ray intensities and, con-