5.2 Lattice Vibrations
101
X = ω max
1 − γ .
(5.21)
The group velocity is zero for optical and acoustic phonons at k = ±π/a and for optical phonons at
the point.
Usually two special cases are treated explicitly: (i) atoms with equal mass (M = M 1 = M 2 )
and different force constants [367] or (ii) atoms with unequal mass and identical force constants
C = C 1 = C 2 [368]. For the case C 1 = C 2 and M 1 = M 2 , γ = 1 and thus X = 0. Then the
dispersion relation is the same as for the monoatomic chain, except that the k space has been folded
since the actual lattice constant is now a/2.
5.2.3 Mode Patterns and Topological States
First, we chose M 1 = M 2 . In this case (Fig. 5.1c), M + = M × = M and the dispersion relation is
ω
2
± =
2C +
M
1 ±
1 −
C
2
×
C
2
+
sin
2
(k a/2)
.
(5.22)
The frequencies for the lower (acoustic) and upper (optical) branch at the zone boundary k = ±π/a
are
ω
2
± (X ) =
C 1 + C 2 ± |C 1 − C 2 |
M
,
(5.23)
or ω − (X ) =
√
2 min(C 1 , C 2 )/M and ω + (X ) =
√
2 max(C 1 , C 2 )/M. The softer spring determines the
top of the lower branch, the harder spring determines the bottom of the upper branch.
The eigenstates for the lower and upper branch (upper sign for the upper band) are given by (without
the exp[ı(k na − ω t)] periodicity in time and space),
v ± (k) =
v ±,1 (k)
v ±,2 (k)
=
1
∓
C 1 +C 2 exp(ı k a)
√
C
2
1 +C
2
2 +2 C 1 C 2 cos k a
.
(5.24)
Fig. 5.6 Dispersion
relation for a diatomic
linear chain with acoustic
(blue) and optical (green)
branch