F hkl
ð Þ ¼ 4f cos
2
p
h þ 2k
ð
Þ
3
&
'
if l is even;
¼ i2f sin 2p
h þ 2k
ð
Þ
3
&
'
if l is odd
For what combination h, k, l will the structure factor vanish?
Solution Given: Graphite Structure (Fig. 8.12) with fractional coordinates:
0; 0; 0
ð
Þ; 1=3; 2=3; 0
ð
Þ ; 0; 0; 1=2
ð
Þ ; 2=3; 1=3; 1=2
ð
Þ ;
Let f be the scattering factor of the carbon atom in graphite, then the structure
factor for the given data is:
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:0
À
Á þ
exp2pi h:0 þ k:0 þ l:
1
2
À
Á þ exp2pi h:
2
3 þ k:
1
3 þ l:
1
2
À
Á
"
#
¼ f: 1 þ exp2pi
h þ 2k
ð
Þ
3
&
'
þ exp2pi
l
2
þ exp2pi
2h þ k
ð
Þ
3
&
'
:exp2pi
l
2
!
¼ f:
1 þ exp2pi
h þ 2k
ð
Þ
3
&
'
þ exp2pi l
2 þ
exp À2pi
h þ 2k
ð
Þ
3
&
'
:exp 2pi
3 h þ k
ð
Þ
3
&
'
:exp2pi l
2
2
6
6
6
4
3
7
7
7
5
Let us consider two different cases:
Case I: when l is even, then the structure factor reduces to
F hkl
ð Þ ¼ f: 1 þ exp 2pi
h þ 2k
ð
Þ
3
&
'
þ 1 þ exp À2pi
h þ 2k
ð
Þ
3
&
'
:exp2piðh þ kÞ:1
!
Fig. 8.12 Graphite structure
324
8 Structure Factor Calculations