1
Theory of Solids HW7 Solution
By Qifeng Shan
Ashcroft and Mermin’s book
61
Power specimen of that one sample is facecentered cubic, one is bodycentered
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 Xray 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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 Fall '09
 Shan
 Power, Solid State Physics, Diffraction, Cubic crystal system, Reciprocal lattice, Bragg, diamond structure

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