Substituting the values of fractional coordinates in Eq. 8.1, we obtain structure
factor expression similar to that of diamond, that is,
F hkl
ð Þ ¼ f Ca :
exp2pi h:0 þ k:0 þ l:0
ð
Þ þ exp2pi h:
1
2 þ k:
1
2 þ l:0
À
Á þ
exp2pi h:0 þ k:
1
2 þ l:
1
2
À
Á þ exp2pi h:
1
2 þ k:0 þ l:
1
2
À
Á
"
#
þ f F :
exp2pi h:
1
4 þ k:
1
4 þ l:
1
4
À
Á þ exp2pi h:
1
4 þ k:
1
4 þ l:
3
4
À
Á þ
exp2pi h:
1
4 þ k:
3
4 þ l:
1
4
À
Á þ exp2pi h:
3
4 þ k:
1
4 þ l:
1
4
À
Á þ
exp2pi h:
3
4 þ k:
3
4 þ l:
1
4
À
Á þ exp2pi h:
3
4 þ k:
1
4 þ l:
3
4
À
Á þ
exp2pi h:
1
4 þ k:
3
4 þ l:
3
4
À
Á þ exp2pi h:
3
4 þ k:
3
4 þ l:
3
4
À
Á
2
6
6
6
4
3
7
7
7
5
¼ f Ca : 1 þ exppi h þ k
ð
Þþexppi k þ l
ð
Þþexppi l þ h
ð
Þ
½
Š
þ exp
pi h þ k þ l
ð
Þ
2
f F :
1 þ exppih þ exppik þ exppil] þ
exp
3pi hþ k þ l
ð
Þ
2
f F :½1 þ exppi Àh
ð Þþ
exppi Àk
ð Þþexppi Àl
ð Þ
2
6
4
3
7
5
Since, CaF 2 (Fluorite) is a face-centered cubic, h, k, and l must be either all odd
or all even for an observed reflection; for any other combination, F = 0. Now, let us
evaluate F and F
2 for (111) and (222) reflections.
Case I: (111) reflection:
Substituting this value in the final structure factor expression, we have
F 111
ð
Þ ¼ 4f Ca
So that; F hkl
ð Þ
j
j
2 ¼ 16f
2
Ca
Case II: (222) reflection:
Substituting this value in the final structure factor expression, we have
Fig. 8.13 CaF 2 crystal
structure
326
8 Structure Factor Calculations