1
Theory of Solids HW-7 Solution
By Qifeng Shan
Ashcroft and Mermin’s book
6-1
Power specimen of that one sample is face-centered cubic, one is body-centered
cubic and one has the diamond structure. The approximate positions of the first four
diffraction rings in each case are (see Figure 6.13):
Values of
φ
for samples
A
B
C
42.2
o
28.8
o
42.8
o
49.2
o
41.0
o
73.2
o
72.0
o
50.8
o
89.0
o
87.3
o
59.6
o
115.0
o
(a) Identify the crystal structures of A, B, and C.
(b) If the wavelength of the incident X-ray beam is 1.5A, what is the length of the
side of the conventional cell in each other?
(c) If the diamond structure were replaced by a zincblende structure with a cubic
unit cell of the same side, at what angles would the first four rings now occur?
Solution:
(a) For the powder method, the relation for incident and diffracted wave vector is shown
in Figure 6.10:

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The diffraction vector satisfies the following relation:
2
1
sin
2
k
K
(1)
Since it is an elastic scattering,
k
is same for the incident and diffracted beam. From Eq.
(1), keeping
k
the same, the ratio of the magnitude of reciprocal lattice vector is the same
as the ration of sin (
φ
/2). Then, we have:
4
3
2
1
4
3
2
1
2
1
sin
:
2
1
sin
:
2
1
sin
:
2
1
sin
:
:
:
K
K
K
K
(2)
Thus, for A:
917
.
1
:
633
.
1
:
156
.
1
:
1
:
:
:
4
3
2
1
K
K
K
K
(3)
For B:
998
.
1
:
724
.
1
:
408
.
1
:
1
:
:
:
4
3
2
1
K
K
K
K
(4)
For C:
311
.
2
:
921
.
1
:
634
.
1
:
1
:
:
:
4
3
2
1
K
K
K
K
(5)

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