98
5 Mechanical Properties
Fig. 5.3 Brillouin zone of
a one-dimensional lattice
from −π/a to +π/a and
mapping to a circle with
angles from −π over 0
(at ) to +π
Fig. 5.4 Dispersion
relation for a monoatomic
linear chain
-1
0
1
0
(k)
/a)
M
C /
4
Thus, the equation of motion (5.3) reads
Mω
2 u n = C
2 − exp(−i k a) − exp(i k a)
u n .
(5.7)
Using the identity exp(i ka) + exp(−i ka) = 2 cos(k a), we find the dispersion relation of the
monoatomic linear chain (Fig. 5.4):
ω
2
(k) =
4C
M
1 − cos(k a)
2
=
4C
M
sin
2
k a
2
.
(5.8)
The solutions describe plane waves that propagate in the crystal with a phase velocity c = ω/k and a
group velocity v g = dω/dk,
v g = ±
4C
M
a
2
cos
|k| a
2
.
(5.9)
In the vicinity of the point, i.e. k π/a the dispersion relation is linear in k
ω(k) = a
C
M
|k| .
(5.10)
5 Mechanical Properties
Fig. 5.3 Brillouin zone of
a one-dimensional lattice
from −π/a to +π/a and
mapping to a circle with
angles from −π over 0
(at ) to +π
Fig. 5.4 Dispersion
relation for a monoatomic
linear chain
-1
0
1
0
(k)
/a)
M
C /
4
Thus, the equation of motion (5.3) reads
Mω
2 u n = C
2 − exp(−i k a) − exp(i k a)
u n .
(5.7)
Using the identity exp(i ka) + exp(−i ka) = 2 cos(k a), we find the dispersion relation of the
monoatomic linear chain (Fig. 5.4):
ω
2
(k) =
4C
M
1 − cos(k a)
2
=
4C
M
sin
2
k a
2
.
(5.8)
The solutions describe plane waves that propagate in the crystal with a phase velocity c = ω/k and a
group velocity v g = dω/dk,
v g = ±
4C
M
a
2
cos
|k| a
2
.
(5.9)
In the vicinity of the point, i.e. k π/a the dispersion relation is linear in k
ω(k) = a
C
M
|k| .
(5.10)