96
5 Lattice Dynamics of Molecular Crystals
φ
ab
αβ ( p, p
) =
∂
2 V
∂x α, p ∂x β, p
eq
,
(5.59)
where superscripts a and b are “T” or “R”, x
a
α is the α component of r with a = T or
θ with a = R. Thus, the equations of motion are
m p
d
2 r α, p
dt 2 = −
p
β=x,y,z
φ
TT
αβ ( p, p
)r β, p
+ φ
TR
αβ ( p, p
)θ β, p
,
(5.60)
i p
d
2
θ p
dt 2 = −
p
β=x,y,z
⎛
⎜
⎝
φ
RT
xβ ( p, p
)r β, p
+ φ
RR
xβ ( p, p
)θ β, p
φ
RT
yβ ( p, p
)r β, p
+ φ
RR
yβ ( p, p
)θ β, p
φ
RT
zβ ( p, p
)r β, p
+ φ
RR
zβ ( p, p
)θ β, p
⎞
⎟
⎠ . (5.61)
Now, we assume the following forms for solutions
r l (t) = r
0
j exp[i(−ωt + r
◦
l q)],
(5.62)
θ l (t) = θ
0
j exp[i(−ωt + r
◦
l q)],
(5.63)
where j is the index distinguishing molecules in a unit cell, and r
◦
l is a vector
specifying the unit cell, to which the lth molecule belongs. Putting the above solutions
into the equations of motion yields the following conditions for solutions,
ω
2 Mu(t) = Fu(t),
(5.64)
where
M =
m
T 0
0 m
R
,
(5.65)
m
T
=
⎛
⎜
⎝
m 1 . . . 0
. . . m j
. . .
0 . . . m n
⎞
⎟
⎠ ,
(5.66)
m j =
⎛
⎝
m j 0 0
0 m j 0
0 0 m j
⎞
⎠ ,
(5.67)
m
R
=
⎛
⎜
⎝
i 1 . . . 0
. . . i j
. . .
0 . . . i n
⎞
⎟
⎠ ,
(5.68)