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 4f C : 1 þ exp
pi h þ k þ l
ð
Þ
2
 4f C : 1 þ exp
Àpi h þ k þ l
ð
Þ
2
¼ 16f
2
C : 1 þ exp
pi h þ k þ l
ð
Þ
2
þ 1 þ exp
Àpi h þ k þ l
ð
Þ
2
&
'
But, we know that e
ix
þ e
Àix
¼ 2cosx
Therefore,
F hkl
ð Þ
j
j
2 ¼ 16 f
2
C : 2 þ 2cos
p h þ k þ l
ð
Þ
2
&
'
¼ 32 f
2
C : 1 þ cos
p h þ k + l
ð
Þ
2
&
'
¼ 32 f
2
C when h þ k þ l
ð
Þ is an odd integer
¼ 64 f
2
C when h þ k þ l
ð
Þ is an even multiple of 2
¼ 0 when h þ k þ l
ð
Þ is an odd multiple of 2
Example 10 Determine the general form of structure factor and intensity corresponding to an hcp (hexagonal close-packed) unit cell.
Solution Given: Hexagonal close-packed (hcp) unit cell, F hkl
ð Þ ¼ ?; I ¼ ?
A simple calculation will give us two atoms in an hcp unit cell (Fig. 8.8). The
fractional coordinates of atom 1 is (0, 0, 0) and atom 2 is (1/3, 2/3, 1/2), respectively. Substituting these values in Eq. 8.1, we obtain
F hkl
ð Þ ¼ f: exp2pi h:0 þ k:0 þ l:0
ð
Þ þ exp2pi h:
1
3
þ k:
2
3
þ l:
1
2
!
¼ f: 1 þ exp2pi h:
1
3
þ k:
2
3
þ l:
1
2
!
In order to solve this, let us put
h þ 2k
ð
Þ
3
þ
l
2
!
¼ p
8.2 Determination of Phase Angle, Amplitude …
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