102
5 Lattice Dynamics of Molecular Crystals
M
II
=
μ 0
0 μ
,
(5.100)
where μ is the reduced mass of two atoms constituting the molecule. It is important
to remember that the intramolecular vibrations are involved in F
II . Indeed, F
II is
decomposed into two parts,
F
II
= F
II(0)
+ F
II(1)
=
φ in 0
0 φ in
+
F
II
11 F
II
12
F
II
21 F
II
22
.
(5.101)
with
F
II(1)
i j
=
p
= p
φ
II
i j ( p, p
) exp(−i d l q).
(5.102)
Here, φ in is a force constant for the stretching vibration of a molecule. The presence
of φ in is one of the reasons for the absence of sum rules for the “II” sector.
It is easy to extend this treatment for general molecules having more internal
vibrations. The inclusion of many modes is unnecessary in most cases because of
generally weak interaction between lattice and molecular vibrations. However, there
exist cases where a limited number of intramolecular modes having frequencies similar to those of lattice vibrations are crucially important for understanding properties
of molecular crystals, as will be exemplified in the next section.
Note that F
II is not a diagonal matrix. Intramolecular vibrations of n mol translationally inequivalent molecules are not equivalent to one another but split into n mol even
for a common wavevector q. The splitting also occurs at q = 0, which is responsible
for most optical spectroscopy as discussed in the case of alternate array of atoms.
This splitting is an example of a phenomenon called the factor group splitting, also
known as the Davydov splitting.
5.5 Related Issues and Examples
5.5.1 Heat Capacity and Debye Temperature
The number of states as a function of energy is generally called as the density of states.
A density of states of lattice vibrations is a phonon density of states. The phonon
density of states is fundamental for the thermodynamic properties of non-metallic
crystals because lattice vibrations in the harmonic treatment described in this chapter
are mutually independent of each other. That is, we can calculate thermodynamic
functions by adding those of harmonic oscillators. For example, the heat capacity of
a crystal is expressed using a phonon density of state, g(ω) as
5 Lattice Dynamics of Molecular Crystals
M
II
=
μ 0
0 μ
,
(5.100)
where μ is the reduced mass of two atoms constituting the molecule. It is important
to remember that the intramolecular vibrations are involved in F
II . Indeed, F
II is
decomposed into two parts,
F
II
= F
II(0)
+ F
II(1)
=
φ in 0
0 φ in
+
F
II
11 F
II
12
F
II
21 F
II
22
.
(5.101)
with
F
II(1)
i j
=
p
= p
φ
II
i j ( p, p
) exp(−i d l q).
(5.102)
Here, φ in is a force constant for the stretching vibration of a molecule. The presence
of φ in is one of the reasons for the absence of sum rules for the “II” sector.
It is easy to extend this treatment for general molecules having more internal
vibrations. The inclusion of many modes is unnecessary in most cases because of
generally weak interaction between lattice and molecular vibrations. However, there
exist cases where a limited number of intramolecular modes having frequencies similar to those of lattice vibrations are crucially important for understanding properties
of molecular crystals, as will be exemplified in the next section.
Note that F
II is not a diagonal matrix. Intramolecular vibrations of n mol translationally inequivalent molecules are not equivalent to one another but split into n mol even
for a common wavevector q. The splitting also occurs at q = 0, which is responsible
for most optical spectroscopy as discussed in the case of alternate array of atoms.
This splitting is an example of a phenomenon called the factor group splitting, also
known as the Davydov splitting.
5.5 Related Issues and Examples
5.5.1 Heat Capacity and Debye Temperature
The number of states as a function of energy is generally called as the density of states.
A density of states of lattice vibrations is a phonon density of states. The phonon
density of states is fundamental for the thermodynamic properties of non-metallic
crystals because lattice vibrations in the harmonic treatment described in this chapter
are mutually independent of each other. That is, we can calculate thermodynamic
functions by adding those of harmonic oscillators. For example, the heat capacity of
a crystal is expressed using a phonon density of state, g(ω) as