Therefore the structure factor becomes
F hkl
ð Þ ¼ f:½1 þ exp2pipŠ
The exponent containing the term p may take fractional values and the
expression therefore will remain complex. To overcome this difficulty, let us again
make use of the basic quantum mechanical concept related to the wave function and
the amplitude of the wave. Accordingly, for a wave Ae
iu and its complex conjugate
Ae
Àiu the intensity (square of the amplitude) is given by
A
2
¼ Ae
iu
 Ae
Àiu
Therefore,
F hkl
ð Þ
j
j
2 ¼ f
2
: 1 þ exp 2pip
ð
Þ
½
Š : 1 þ exp À2pip
ð
Þ
½
Š
¼ f
2
: 2 þ exp 2pip
ð
Þþexp( À 2pip)
½
Š
But, we know that e
ix
þ e
Àix
¼ 2 cos x
F hkl
ð Þ
j
j
2 ¼ f
2
: 2 þ 2cos2pp
½
Š
¼ f
2
: 2 þ 2 2cos
2
pp À 1
À
Á
Â
à ¼ f
2
: 4 cos
2
pp
Â
Ã
= 4f
2 cos
2
p:
h þ 2k
ð
Þ
3
þ
l
2
!
¼ 0 when h þ 2k
ð
Þis a multiple of 3 and l is odd:
This indicates that the reflections such as 11.1 (11 2 1), 11.3 (11 2 3), 22.1 (22 4
1), 22.3 (22 4 3), etc. are absent in hcp structure. Further, if we assume that
(h + 2 k) is a multiple of 3 but l is even, then
Fig. 8.8 Hexagonal closed
packed structure
316
8 Structure Factor Calculations