5.2 Atomic Case
93
where
v cell = a(b × c)
(5.38)
is the volume of a unit cell. Note that asterisks are used to indicate not complex
conjugate but reciprocal lattice vectors, according to the tradition. It is easy to verify
the following identities,
2π = a
∗ a = b
∗ b = c
∗ c
(5.39)
0 = a
∗ b = a
∗ c = b
∗ c = b
∗ a = c
∗ a = c
∗ b
(5.40)
If q and q
differ by a sum of integer multiples of reciprocal lattice vectors, the
relation
exp(i qa ll ) = exp(i q
a ll )
(5.41)
holds because a ll is a lattice vector (Eq. 5.33). The most straightforward choice of
the region, therefore, is a parallelepiped corresponding to a unit cell of reciprocal
space. However, it is usual to adopt as the region the first Brillouin zone, which is the
region consisting of points, of which the closest reciprocal lattice point is the origin
q = 0.
The other point to be generalized is the possibility of plural atoms in a unit cell.
Representatives are simple salts such as rock salt (NaCl). Suppose n atoms exist in
a cell. An additional index, j, distinguishing plural atoms, is necessary. Then, the
coordinate of an atom is specified like r jl , and the force constant becomes
φ αβ ( j, j
; l, l
) =
∂
2 V
∂r α,l ∂r β,l
eq
.
(5.42)
Using these force constants, the equation of motion is given by
m j
d
2 r jα.l (t)
dt 2
= −
j
l
β=x,y,z
φ αβ ( j, j
; l, l
)r β,l (t).
(5.43)
The dynamical matrix is a (3n × 3n)-matrix D(q) of the form
D(q) = M
−1 F(q)
(5.44)
where M
−1 and F(q) are composed of (n × n) small matrices of the size (3 × 3) as
M
−1
=
⎛
⎜
⎝
m
−1
1
. . . 0
. . . m
−1
j
. . .
0 . . . m
−1
n
⎞
⎟
⎠
(5.45)
93
where
v cell = a(b × c)
(5.38)
is the volume of a unit cell. Note that asterisks are used to indicate not complex
conjugate but reciprocal lattice vectors, according to the tradition. It is easy to verify
the following identities,
2π = a
∗ a = b
∗ b = c
∗ c
(5.39)
0 = a
∗ b = a
∗ c = b
∗ c = b
∗ a = c
∗ a = c
∗ b
(5.40)
If q and q
differ by a sum of integer multiples of reciprocal lattice vectors, the
relation
exp(i qa ll ) = exp(i q
a ll )
(5.41)
holds because a ll is a lattice vector (Eq. 5.33). The most straightforward choice of
the region, therefore, is a parallelepiped corresponding to a unit cell of reciprocal
space. However, it is usual to adopt as the region the first Brillouin zone, which is the
region consisting of points, of which the closest reciprocal lattice point is the origin
q = 0.
The other point to be generalized is the possibility of plural atoms in a unit cell.
Representatives are simple salts such as rock salt (NaCl). Suppose n atoms exist in
a cell. An additional index, j, distinguishing plural atoms, is necessary. Then, the
coordinate of an atom is specified like r jl , and the force constant becomes
φ αβ ( j, j
; l, l
) =
∂
2 V
∂r α,l ∂r β,l
eq
.
(5.42)
Using these force constants, the equation of motion is given by
m j
d
2 r jα.l (t)
dt 2
= −
j
l
β=x,y,z
φ αβ ( j, j
; l, l
)r β,l (t).
(5.43)
The dynamical matrix is a (3n × 3n)-matrix D(q) of the form
D(q) = M
−1 F(q)
(5.44)
where M
−1 and F(q) are composed of (n × n) small matrices of the size (3 × 3) as
M
−1
=
⎛
⎜
⎝
m
−1
1
. . . 0
. . . m
−1
j
. . .
0 . . . m
−1
n
⎞
⎟
⎠
(5.45)