104
5 Mechanical Properties
ω
2
± =
2 C M +
M
2
×
1 ±
1 −
M
2
×
M
2
+
sin
2
(k a/2)
.
(5.31)
At the zone boundary the frequencies for the acoustic and the optical branch are, assuming M 2 < M 1 ,
ω X,1 =
√
2C/M 1 with v 2 = 0 (oscillation of the larger mass) and ω X,2 =
√
2C/M 2 with v 1 = 0
(oscillation of the smaller mass), respectively. In the vibrations at the X-point thus only one atom
species oscillates, the other does not move. Close to the point, the atoms are in phase in the acoustic
branch with v 2 = v 1 . For the optical branch, the frequency at the point is given by ω =
√
2C/M r
(with the reduced mass M
−1
r
= M
−1
1 + M
−1
2 = 2M + /M
2
× ) and the amplitude ratio is given by the mass
ratio: v 2 /v 1 = −M 1 /M 2 , i.e. the atoms are out of phase and the heavier atom has the smaller amplitude.
It should be repeated that the relative phase of v 2 and v 1 , as defined in (5.27), is the same for all
masses (dotted line in Fig. 5.7). The situation C 1 = C 2 is therefore for all masses topologically the
same as the gapless state and the gap has different character when due to different spring constants or
due to different masses.
Finally, we consider the symmetry classification looking at Fig. 5.1 again. The monoatomic chain
has two mirror-like symmetries, indicated by vertical dashed lines; it is symmetric with respect to
exchange X of the A- and B-atoms and to the exchange S of the springs. The cases Fig. 5.1b, c
preserve only one such symmetry while the general diatomic chain Fig. 5.1d preserves none of them
individually but still possesses the so-called chiral symmetry which is the combination X S. In terms
of the ‘periodic system’ of the CAZ classification of topological systems the diatomic linear chain is
a BDI-type system [371] and the topological invariant is, derived from (5.30), an integer of the form,
γ /2π ∈ Z.
5.2.4 Lattice Vibrations of a Three-Dimensional Crystal
When calculations are executed for a three-dimensional crystal with a monoatomic base, there are 3N
equations of motion. These are transformed to normal coordinates and represent 3 acoustic branches
(1 LA phonon mode and 2 TA phonon modes) of the dispersion relation. In a crystal with a base with p
Fig. 5.9 Phonon
dispersion in Si,
experimental data and
theory (solid lines: bond
charge model, dashed lines:
valence force field model).
Adapted from [164]
5 Mechanical Properties
ω
2
± =
2 C M +
M
2
×
1 ±
1 −
M
2
×
M
2
+
sin
2
(k a/2)
.
(5.31)
At the zone boundary the frequencies for the acoustic and the optical branch are, assuming M 2 < M 1 ,
ω X,1 =
√
2C/M 1 with v 2 = 0 (oscillation of the larger mass) and ω X,2 =
√
2C/M 2 with v 1 = 0
(oscillation of the smaller mass), respectively. In the vibrations at the X-point thus only one atom
species oscillates, the other does not move. Close to the point, the atoms are in phase in the acoustic
branch with v 2 = v 1 . For the optical branch, the frequency at the point is given by ω =
√
2C/M r
(with the reduced mass M
−1
r
= M
−1
1 + M
−1
2 = 2M + /M
2
× ) and the amplitude ratio is given by the mass
ratio: v 2 /v 1 = −M 1 /M 2 , i.e. the atoms are out of phase and the heavier atom has the smaller amplitude.
It should be repeated that the relative phase of v 2 and v 1 , as defined in (5.27), is the same for all
masses (dotted line in Fig. 5.7). The situation C 1 = C 2 is therefore for all masses topologically the
same as the gapless state and the gap has different character when due to different spring constants or
due to different masses.
Finally, we consider the symmetry classification looking at Fig. 5.1 again. The monoatomic chain
has two mirror-like symmetries, indicated by vertical dashed lines; it is symmetric with respect to
exchange X of the A- and B-atoms and to the exchange S of the springs. The cases Fig. 5.1b, c
preserve only one such symmetry while the general diatomic chain Fig. 5.1d preserves none of them
individually but still possesses the so-called chiral symmetry which is the combination X S. In terms
of the ‘periodic system’ of the CAZ classification of topological systems the diatomic linear chain is
a BDI-type system [371] and the topological invariant is, derived from (5.30), an integer of the form,
γ /2π ∈ Z.
5.2.4 Lattice Vibrations of a Three-Dimensional Crystal
When calculations are executed for a three-dimensional crystal with a monoatomic base, there are 3N
equations of motion. These are transformed to normal coordinates and represent 3 acoustic branches
(1 LA phonon mode and 2 TA phonon modes) of the dispersion relation. In a crystal with a base with p
Fig. 5.9 Phonon
dispersion in Si,
experimental data and
theory (solid lines: bond
charge model, dashed lines:
valence force field model).
Adapted from [164]