100
5 Mechanical Properties
M 1 ¨
u
1
n = −C 1
u
1
n − u
2
n
− C 2
u
1
n − u
2
n−1
(5.13a)
M 2 ¨
u
2
n = −C 1
u
2
n − u
1
n
− C 2
u
2
n − u
1
n+1
.
(5.13b)
With the plane-wave ansatz u
1
n (x, t) = v 1 exp(i(kna − ωt)) and u
2
n (x, t) = v 2 exp(i(kna − ωt)) and
periodic boundary conditions we find
0 = −M 1 ω
2 v 1 + C 1 (v 1 − v 2 ) + C 2 [v 1 − v 2 exp(−ika)]
(5.14a)
0 = −M 2 ω
2 v 2 + C 1 (v 2 − v 1 ) + C 2 [v 2 − v 1 exp(ika)] .
(5.14b)
These equations for v 1 and v 2 can only be solved nontrivially if the determinant vanishes, i.e.
0 =
M 1 ω
2
− (C 1 + C 2 ) C 1 + e
−ika C 2
C 1 + e
ika C 2
M 2 ω
2
− (C 1 + C 2 )
(5.15)
= M 1 M 2 ω
4
− (M 1 + M 2 )(C 1 + C 2 ) ω
2
+ 2C 1 C 2 [1 − cos(k a)] .
Using the substitutions C + = (C 1 + C 2 )/2, C × =
√
C 1 C 2 , the arithmetic and geometrical averages,
and accordingly for M + and M × , the solution is
ω
2
± (k) =
ω
2
max
2
1 ±
1 − γ 2 sin
2
(k a/2)
,
(5.16)
with
0 < γ =
C × M ×
C + M +
≤ 1 ,
(5.17)
and the maximum frequency ω max (for the upper branch (‘+’ sign in (5.16)) at the zone center (k = 0),
ω max =
4C ×
γ M ×
= 2
C + M +
M
2
×
.
(5.18)
The dispersion relation, as shown in Fig. 5.6, now has (for each longitudinal and transverse mode) two
branches. The lower branch (‘−’ sign in (5.16)) is related to the acoustic mode; neighboring atoms
have the same phase for k = 0 (Fig. 5.5). For the acoustic mode ω = 0 at the point and the frequency
increases towards the zone boundary.
The upper branch is called the optical mode (since it can interact strongly with light, see Sect. 9.10)
and neighboring atoms have opposite phase at k = 0. In the vicinity of the point, the dispersion of
optical phonons is parabolic with negative curvature:
ω(k) ∼ = ω max
1 −
1
2
γ
4
2 (k a)
2
.
(5.19)
Thus, four different vibrations exist that are labeled TA, LA, TO, and LO. Both the TA and TO branches
are degenerate.
At the zone boundary (X point) a frequency gap exists (for γ = 1). The gap center is at
ω X =
ω max
√
2
1 + γ
2
,
(5.20)
and the total width of the gap is
5 Mechanical Properties
M 1 ¨
u
1
n = −C 1
u
1
n − u
2
n
− C 2
u
1
n − u
2
n−1
(5.13a)
M 2 ¨
u
2
n = −C 1
u
2
n − u
1
n
− C 2
u
2
n − u
1
n+1
.
(5.13b)
With the plane-wave ansatz u
1
n (x, t) = v 1 exp(i(kna − ωt)) and u
2
n (x, t) = v 2 exp(i(kna − ωt)) and
periodic boundary conditions we find
0 = −M 1 ω
2 v 1 + C 1 (v 1 − v 2 ) + C 2 [v 1 − v 2 exp(−ika)]
(5.14a)
0 = −M 2 ω
2 v 2 + C 1 (v 2 − v 1 ) + C 2 [v 2 − v 1 exp(ika)] .
(5.14b)
These equations for v 1 and v 2 can only be solved nontrivially if the determinant vanishes, i.e.
0 =
M 1 ω
2
− (C 1 + C 2 ) C 1 + e
−ika C 2
C 1 + e
ika C 2
M 2 ω
2
− (C 1 + C 2 )
(5.15)
= M 1 M 2 ω
4
− (M 1 + M 2 )(C 1 + C 2 ) ω
2
+ 2C 1 C 2 [1 − cos(k a)] .
Using the substitutions C + = (C 1 + C 2 )/2, C × =
√
C 1 C 2 , the arithmetic and geometrical averages,
and accordingly for M + and M × , the solution is
ω
2
± (k) =
ω
2
max
2
1 ±
1 − γ 2 sin
2
(k a/2)
,
(5.16)
with
0 < γ =
C × M ×
C + M +
≤ 1 ,
(5.17)
and the maximum frequency ω max (for the upper branch (‘+’ sign in (5.16)) at the zone center (k = 0),
ω max =
4C ×
γ M ×
= 2
C + M +
M
2
×
.
(5.18)
The dispersion relation, as shown in Fig. 5.6, now has (for each longitudinal and transverse mode) two
branches. The lower branch (‘−’ sign in (5.16)) is related to the acoustic mode; neighboring atoms
have the same phase for k = 0 (Fig. 5.5). For the acoustic mode ω = 0 at the point and the frequency
increases towards the zone boundary.
The upper branch is called the optical mode (since it can interact strongly with light, see Sect. 9.10)
and neighboring atoms have opposite phase at k = 0. In the vicinity of the point, the dispersion of
optical phonons is parabolic with negative curvature:
ω(k) ∼ = ω max
1 −
1
2
γ
4
2 (k a)
2
.
(5.19)
Thus, four different vibrations exist that are labeled TA, LA, TO, and LO. Both the TA and TO branches
are degenerate.
At the zone boundary (X point) a frequency gap exists (for γ = 1). The gap center is at
ω X =
ω max
√
2
1 + γ
2
,
(5.20)
and the total width of the gap is