F hkl
ð Þ ¼ 4: f Zn þ f S :exp
pi h þ k þ l
ð
Þ
2
Now, multiplying the right hand term with its complex conjugate, we can obtain
the F hkl
ð Þ
j
j
2 , that is,
F hkl
ð Þ
j
j
2 ¼ 4: f Zn þ f S :exp
pi h þ k þ l
ð
Þ
2
 4: f Zn þ f S :exp
Àpi h þ k þ l
ð
Þ
2
On simplification, we obtain
F hkl
ð Þ
j
j
2 ¼ 16: f
2
Zn þ f
2
S þ 2f Zn :f S cos
p h þ k þ l
ð
Þ
2
&
'
¼ 16: f
2
Zn þ f
2
S
Â
Ã
if h þ k þ l
ð
Þ is an odd integer
¼ 16: f
2
Zn À f
2
S
Â
à 2 if h þ k þ l
ð
Þ is an odd multiple of 2
¼ 16: f
2
Zn þ f
2
S
Â
à 2 if h þ k þ l
ð
Þ is an even multiple of 2
Example 14 A unit cell of tetragonal crystal has its atoms positioned at the following locations:
0; 1=2; 1=4
ð
Þ ; 1=2; 0; 1=4
ð
Þ ; 1=2; 0; 3=4
ð
Þ ; 0; 1=2; 3=4
ð
Þ
Determine the corresponding structure factor and intensity.
Solution Given: The fractional coordinates (0, 1/2, 1/4), (1/2, 0, 1/4), (1/2, 0, 3/4),
(0, 1/2, 3/4), F hkl
ð Þ ¼ ?; I ¼ ?
Let f be the scattering factor of the given atom, then the structure factor for the
given data is:
F hkl
ð Þ ¼ f:
exp2pi h:0 þ k:
1
2 þ l:
1
4
À
Á þ exp2pi h:
1
2 þ k:0 þ l:
1
4
À
Á þ
exp2pi h:
1
2 þ k:0 þ l:
3
4
À
Á þ exp2pi h:0 þ k:
1
2 þ l:
3
4
À
Á
"
#
¼ f:exp
pil
2
exppik þ exppih þ exppi h þ l
ð
Þþexppi k þ l
ð
Þ
½
Š
Let us consider different cases:
Case I: If h, k, l are mixed indices, such as (110)
F hkl
ð Þ ¼ f:exp
pil
2
: exppi þ exppi þ exppi þ exppi
ð
Þ
¼ f: À1 À 1 À 1 À 1
ð
Þ ¼ À 4f
and F hkl
ð Þ
j
j
2 ¼ 16f
2
8.2 Determination of Phase Angle, Amplitude …
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