300
5 Solid State Physics
5.2 Volume of the unit cell = a
3 . Since there are two atoms per unit cell, 8 × 1/8
for the corner atoms and 1 × 1 for the centre atom,
Volume = 2 ×
4
3
πr
3
Since the body diagonal atoms touch one another,
4r = a
√
3
Volume of atoms in terms of a is
2 ×
4
3
πr
3
= 2 ×
4
3
π[a
√
3/4]
3
=
√
3πa
3
/8
Or the fraction of the volume occupied by the body-centred cubic structure is
√
3π/8.
5.3
2d sin θ = nλ
d 1 =
1.λ
2 sin θ
=
0.1
2 sin 4 ◦ = 0.717 nm
d 2 =
0.1
2 sin 8 ◦ = 0.359 nm
5.4 nλ =
2a
(h 2 + k 2 + l 2 ) 1/2 sin θ =
2 × 0.4
(1 2 + 1 2 + 1 2 ) 1/2 sin θ
sin θ =
0.3
√
3
0.8
= 0.6495
θ = 40.5
◦
5.5 2d sin θ = nλ
d =
1.λ
2 sin θ
=
0.16
2 sin 30 ◦ = 0.136 nm
For n = 2,
sin θ =
2 × 0.16
2 × 0.136
= 1.176
a value which is not possible. Thus higher order reflections are not possible.
5.6 The de Broglie wavelength for electrons is calculated from
λ =
150
V
=
150
54
= 1.66 ˚
A
Bragg’s equation will be satisfied for neutrons of the same wavelength.
λ =
0.286
√
E
˚
A