Also, for any cubic structure,
d hkl ¼
a
h
2
þ k
2
þ l
2
À
Á 1=2
Therefore,
sinh hkl
k
¼
1
2d hkl
¼
h
2 + k
2 + l
2
À
Á 1=2
2a
and
f
sinh hkl
k
¼ f
1
2d hkl
¼ f
h
2
þ k
2
þ l
2
À
Á 1=2
2a
"
#
This shows that
sinh hkl
k
À
Á
is a function only of hkl values. The structure factor for
Po is simplified by substituting N = 1 and coordinates of the atom
x 1 ; y 1 ; z 1 ¼ 0; 0; 0, we obtain
F hkl
ð Þ ¼
X N
i¼1
f Po
1
2d hkl
e
2piH hkl
ð Þ:r j ¼ f Po
1
2d hkl
e
2pi hx 1 þ ky 1 þ lz 1
ð
Þ = f Po
1
2d hkl
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 first
hkl = 001 is
F 001
ð Þ ¼ f 0:1489
ð
Þ¼
X 4
i¼1
a i e
Àbi 0:1489
ð
Þ
2 þ c
¼ a 1 e
Àb1 0:1489
ð
Þ
2 þ a 2 e
Àb2 0:1489
ð
Þ
2 þ a 3 e
Àb3 0:1489
ð
Þ
2 þ a 4 e
Àb4 0:1489
ð
Þ
2 þ c
¼ 34:6726 e
À0:700999 0:1489
ð
Þ
2 þ 15:4733 e
À3:55078 0:1489
ð
Þ
2 þ 13:1138 e
À9:55642 0:1489
ð
Þ
2 þ
7:02588 e
À47:0045 0:1489
ð
Þ
2 þ 13:677
¼ 34:13788 þ 14:30188 þ 10:6099 þ 2:47800 þ 13:677 ¼ 75:20
Calculations for other reflections can be made similarly. They are shown below
in tabulated form for the initial 13 hkl values. The atomic scattering curve for Po
(z = 84) with 001 and 011 values is shown in Fig. 8.2.
8.2 Determination of Phase Angle, Amplitude …
305