Page 47 - Instant notes

Diffraction by solids 33

Fig. 3. (a) Diffraction by a single
crystal, with one set of lattice planes
correctly oriented for an allowed
reflection; (b) diffraction by a
crystalline powder where some
crystals are oriented for every possible
allowed reflection.

Most modern diffractometers use scintillation detectors which sweep an arc around the
angle 2θ. The detector gives a measure of X-ray intensity as a function of the angle 2θ.
The diffraction pattern which is obtained must be correlated with the unit cell of the
2
sample. By obtaining the angles for which reflections occur, the value of sin θ may be
obtained for each reflection, and these values are directly correlated to the values of h, k,
and l:
2
2
2
2
2
sin θ=(h +k +l ) (λ/2a) (Note that this is simply the square of a previous
expression)
2
2
2
2
2
As (λ/2a) is constant, the value of sin θ is directly related to the value of (h +k +l ), and
2
2
2
for a given crystal the ratios of the values gives the relative values of (h +k +l ). For
example, if X-rays of wavelength 0.1542 nm are incident on a powder sample, and the
2
2
2
angular position of the reflections is measured, the process for calculating (h +k +l ) is
given in Table 1.
Table 1. Indexing a simple powder diffraction
pattern
Angle, θ 14.30 20.45 25.33 29.61 33.53 37.24 44.33 47.83
2
Calculate sin θ 0.061 0.122 0.183 0.244 0.305 0.366 0.488 0.549
Ratio 1 2 3 4 5 6 8 9
2
2
2
The bottom row of Table 1 represents the values of (h +k +l ) corresponding to the
reflections at the angles given. The process of ascribing h, k, and l values to reflections in
a diffraction pattern is known as indexing the pattern. From the indexed pattern, it is
possible to identify the type of unit cell, which in this case can be identified as simple
2
2
2
cubic due to the presence of all possible values of (h +k +l ), with a forbidden line at
2
2
2
(h +k +l )=7.