Intensity of diffraction sequently, dislocations show up as bands of contrast,

5.3.4.1 Intensity of diffraction sequently, dislocations show up as bands of contrast,

Many applications of the powder method depend on some 5 to 50 µ m wide. No magnification is used in

the accurate measurement of either line position or

Figure 5.14 (a) Geometry of X-ray topographic technique, (b) topograph from a magnesium single crystal showing dislocation loops, g D 0 1 1 0 (after Vale and Smallman, 1977).

The characterization of materials 139 line intensity. The arrangement of the diffraction lines

symmetry, which includes most metals, the expression in any pattern is characteristic of the material being

is much simpler because the sine terms vanish. examined and, consequently, an important practical

This equation may be applied to any structure, use of the method is in the identification of unknown

but to illustrate its use let us examine a pure metal phases. Thus, it will be evident that equation (5.9)

crystallizing in the bcc structure. From Figure 2.11c can indicate the position of the reflected beams,

it is clear that the structure has identical atoms (i.e. as determined by the size and shape of the unit

f 1 Df 2 ) at the coordinates ⊲0 0 0⊳ and f 1 1 2 1 2 2 g so that cell, but not the intensities of the reflected beams.

equation (5.10) becomes:

These are determined not by the size of the unit

2 cell but by the distribution of atoms within it, 2 I/f and while cubic lattices give reflections for every

Df

possible value of ⊲h 2 Ck 2

Cl 2 ⊳

all other structures

give characteristic absences. Studying the indices of It then follows that I is equal to zero for every the ‘absent’ reflections enables different structures to

reflection having ⊲h C k C l⊳ an odd number. The

be distinguished. significance of this is made clear if we consider In calculating the intensity scattered by a given

in a qualitative way the 1 0 0 reflection shown in atomic structure, we have first to consider the inten-

Figure 5.15a. To describe a reflection as the first-order sity scattered by one atom, and then go on to consider

reflection from ⊲1 0 0⊳ planes implies that there is the contribution from all the other atoms in the par- ticular arrangement which make up that structure. The

planes A and those reflected from planes A 0 . However, efficiency of an atom in scattering X-rays is usually

the reflection from the plane B situated half-way denoted by f, the atomic scattering factor, which is

between A and A 0

from plane A, so that complete cancellation of the 1 0 0 by a single electron A e . If atoms were merely points,

the ratio of amplitude scattered by an atom A a to that

reflected ray will occur. The 1 0 0 reflection is therefore their scattering factors would be equal to the number

absent, which agrees with the prediction made from of electrons they contain, i.e. to their atomic numbers,

equation (5.11) that the reflection is missing when

⊲h C k C l⊳ is an odd number. A similar analysis shows is proportional to the square of amplitude. However,

and the relation I a DZ 2 .I e would hold since intensity

that the 2 0 0 reflection will be present (Figure 5.15b), because the size of the atom is comparable to the wave-

length of X-rays, scattering from different parts of the of phase with the rays from A and A 0 . In consequence,

if a diffraction pattern is taken from a material having The scattering factor, therefore, depends both on angle

atom is not in phase, and the result is that I a 2 .I e .

a bcc structure, because of the rule governing the sum and on the wavelength of X-rays used, as shown in

of the indices, the film will show diffraction lines Figure 5.9, because the path difference for the individ-

almost equally spaced with indices N D 2, ⊲1 1 0⊳; ual waves scattered from the various electrons in the

4, ⊲2 0 0⊳; 6, ⊲2 1 1⊳; 8, ⊲2 2 0⊳; . . ., as shown in Figure 5.12a. Application of equation (5.10) to a pure

. Thus, to consider the intensity scattered by a given structure, it is necessary to sum up the waves which come from all the atoms of one unit cell of that struc- ture, since each wave has a different amplitude and

a different phase angle due to the fact that it comes from a different part of the structure. The square of the amplitude of the resultant wave, F, then gives the intensity, and this may be calculated by using the f- values and the atomic coordinates of each atom in the unit cell. It can be shown that a general formula for the intensity is

of those atoms having scattering factors f Figure 5.15 (a) 1 0 0 reflection from bcc cell showing

where x 1 ,y 1 ,z 1 ;x 2 ,y 2 ,z 2 , etc., are the coordinates

interference of diffracted rays, (b) 2 0 0 reflection showing respectively, and hkl are the indices of the reflection

1 ,f 2 , etc.,

reinforcement (after Barrett, 1952; courtesy of being computed. For structures having a centre of

McGraw-Hill) .

140 Modern Physical Metallurgy and Materials Engineering metal with fcc structure shows that ‘absent’ reflections

will occur when the indices of that reflection are mixed, i.e. when they are neither all odd nor all even. Thus, the corresponding diffraction pattern will contain lines according to N D 3, 4, 8, 11, 12, 16, 19, 20, etc; and the characteristic feature of the arrangement is

a sequence of two lines close together and one line separated, as shown in Figure 5.12b. Equation (5.10) is the basic equation used for deter- mining unknown structures, since the determination of the atomic positions in a crystal is based on this rela- tion between the coordinates of an atom in a unit cell and the intensity with which it will scatter X-rays.