A simple calculation will give us four atoms in a face-centered cubic unit cell
(Fig. 8.5). The fractional coordinates are: (0, 0, 0), (1/2, 1/2, 0), (0, 1/2, 1/2), (1/2,
0, 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
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
"
#
¼ f: 1 þ exp pi h þ k
ð
Þþ exp pi k þ l
ð
Þþ exp pi l þ h
ð
Þ
½
Š
It can be seen that If the reflecting plane indices h, k, l are all odd or all even (i.e.,
unmixed) then the sum (h + k), (k + l) and (l + h) are all even integers and the
structure factor of fcc will be
F hkl
ð Þ = f 1 þ 1 þ 1 þ 1
½
Š¼4f and I / F(hkl)
j
j
2 = 16f
2
Further, if indices h, k, l are mixed, it can be seen that
F hkl
ð Þ¼ 0; and I / F(hkl)
j
j
2 ¼ 0
The structure factor for Cu is simplified by substituting N = 4, coordinates of
four atoms: (0, 0, 0), (1/2, 1/2, 0), (0, 1/2, 1/2), (1/2, 0, 1/2), we obtain
F hkl
ð Þ¼
X 4
i¼1
f Cu
1
2d hkl
e
2piH hkl
ð Þ:r j
= f Cu
1
2d hkl
1 þ exppi h þ k
ð
Þþexppi k þ l
ð
Þþexppi l þ h
ð
Þ
½
Š
Now, substituting different hkl values along with Cromer–Mann coefficients in
Eq. 8.5, the required structure factor can be calculated. The calculation for the third
value of hkl = 020, when the indices h, k, l are all even, is
Fig. 8.5 Face-centered cubic
structure of copper (Cu)
310
8 Structure Factor Calculations