88
5 Lattice Dynamics of Molecular Crystals
5.2.1.2 Alternate Array
Now, we proceed to an alternate array of two kinds of atoms, from which something
can be learned on regular lattices containing plural particles. N atoms of another
type (mass being M) are inserted between adjacent atoms in the previous treatment.
Neighboring atoms are connected by a spring with the spring constant k and natural
length a/2. Label the new atom by l between old atoms labeled as l and l + 1. The
instantaneous coordinate of the new lth atom is denoted as X l (t). Note that this array
has the same periodicity a as in the previous case. The range of the wavevector q
remains the same, accordingly.
By defining
R l (t) = X l (t) −
l +
1
2
a,
(5.10)
the equations of motion are written as
m
d
2 r l (t)
dt 2 = −k[r l (t) − R l−1 (t)] − k[r l (t) − R l (t)],
(5.11)
M
d
2 R l (t)
dt 2 = −k[R l (t) − r l (t)] − k[R l (t) − r l+1 (t)].
(5.12)
Assuming the following solution, beside Eq. 5.3,
R l (t) = R
0 exp[i(−ωt + laq)],
(5.13)
the equations of motion become
− mω
2 r l (t) = −2kr l (t) + k(1 + e
−iqa
)R l (t),
(5.14)
−Mω
2 R l (t) = k(1 + e
iqa
)r l (t) − 2k R l (t),
(5.15)
or, in a matrix form,
− ω
2
m 0
0 M
r l (t)
R l (t)
=
−2k
(1 + e
−iqa
)k
(1 + e
iqa
)k
−2k
r l (t)
R l (t)
.
(5.16)
This form means that ω
2 are eigenvalues of the matrix product D(q), often called a
dynamical matrix,
D(q) =
m
−1
0
0 M
−1
2k
−(1 + e
−iqa
)k
−(1 + e
iqa
)k
2k
.
(5.17)
Eigenvalues of this dynamical matrix are real. In reality, they are given explicitly as
ω ± (q)
2
=
k
m M
m + M ±
m 2 + M 2 + 2m M cos qa
.
(5.18)