Case II: If h, k, l are all even unmixed indices, such as (222)
F hkl
ð Þ ¼ f:exppi: exp2pi þ exp2pi þ exp4pi þ exp4pi
ð
Þ
¼ f:ðÀ1Þ 1 þ 1 þ 1 þ 1
ð
Þ¼À 4f
and F hkl
ð Þ
j
j
2 ¼ 16f
2
Case III: If h, k, l are all odd unmixed indices, such as (111)
F hkl
ð Þ ¼ f:exp
pil
2
: exppi þ exppi þ exp2pi þ exp2pi
ð
Þ
¼ f:exp
pil
2
À1 À 1 þ 1 þ 1
ð
Þ ¼ 0
and F hkl
ð Þ
j
j
2 ¼ 0
Example 15 Determine the structure factor and intensity corresponding to the
wurtzite form of ZnS. The crystal is hexagonal and contains two ZnS molecules per
unit cell. The Zn and S ions are located at the following positions:
Zn : 0; 0; 0
ð
Þ; 1=3; 2=3; 1=2;
ð
Þ
S : 0; 0; 3=8
ð
Þ ; 1=3; 2=3; 7=8
ð
Þ
Solution Given: Hexagonal close-packed (wurtzite form of ZnS) unit cell
(Fig. 8.11b), 2 ZnS molecules in the unit cell, four fractional coordinates two each
for Zn and S, F hkl
ð Þ = ?, I = ?
Substituting the given fractional coordinates in Eq. 8.1, we obtain
F hkl
ð Þ ¼ f Zn : exp2pi h:0 þ k:0 þ l:0
ð
Þ þ exp2pi h:
1
3
þ k:
2
3
þ l:
1
2
!
þ f S : exp2pi h:0 þ k:0 þ l:
3
8
þ exp2pi h:
1
3
þ k:
2
3
þ l:
7
8
!
¼ f Zn : 1 þ exp2pi
h þ 2k
ð
Þ
3
&
'
:exp2pi:
l
2
!
þ f S : exp2pi:
3l
8
þ exp2pi
h þ 2k
ð
Þ
3
&
'
:exp2pi:
7l
8
!
¼ f Zn : 1 þ exp2pi
h þ 2k
ð
Þ
3
&
'
:exp2pi:
l
2
!
þ f S :exp2pi:
3l
8
1 þ exp2pi
h þ 2k
ð
Þ
3
&
'
:exp2pi:
l
2
!
¼ 1 þ exp2pi
h þ 2k
ð
Þ
3
&
'
:exp2pi:
l
2
f Zn þ f S :exp2pi:
3l
8
!
322
8 Structure Factor Calculations