5.2 Atomic Case
91
The displacements in the former and the latter are parallel and perpendicular to
not only the array direction but also the direction of wave propagation expressed
by the wavevector q. Emphasizing the latter facts, they are called longitudinal and
transverse modes. The sound velocity is larger for the longitudinal modes reflecting
the assumption k > k ⊥ .
5.2.2 Atomic Crystals
It is in order to proceed to cases of three-dimensional crystals. Let us start with the
simplest case for three-dimensional atomic crystals consisting of N identical atoms.
Suppose a crystal of atoms (mass m) on a simple cubic lattice. For three-dimensional
crystals, models assuming ideal springs as interatomic bonds are rather unphysical.
For example, a simple cubic lattice of point atoms connected by ideal springs is
unstable to any shear. We must consider the issue of what are force constants to
be assumed. Suppose V (R) be the potential energy of a crystal. The R is a 3N -
dimensional vector containing all information of atomic coordinates. At the static
equilibrium, the coordinates of all atoms r
◦
l satisfy
∂V
∂r α,l
eq
= 0,
(5.26)
for all l with α = x, y, or z with the subscript “eq” for the values at equilibrium. Thus,
we can expand the potential in terms of small displacements r l from the equilibrium
position and may truncate the expansion at the second-order around r
◦
l as
V ≈ V 0 +
1
2
l
l
α,β=x,y,z
∂
2 V
∂r α,l ∂r β,l
eq
r α,l r β,l
= V 0 +
1
2
l
l
α,β=x,y,z
φ αβ (l, l
)r α,l r β,l
(5.27)
where φ αβ (l, l
) are generalized force constants between the atomic pair (l, l
). Note
that φ αβ (l, l
) = φ βα (l
, l), leading to the double appearance of the same terms in
Eq. 5.27 except for l = l
.
Lagrangian L = T − V with T being the kinetic energy of the system serves a
fundamental role in the description of a mechanical system in terms of generalized
coordinate Q j and its time derivatives ˙
Q j [10]. The equation of motion is given by
˙
P j = ∂ L/∂ Q j with the generalized momentum P j = ∂ L/∂ ˙
Q j . Since the kinetic
energy in the present problem is given by
T =
1
2
m
l
d r l
dt
2
,
(5.28)
91
The displacements in the former and the latter are parallel and perpendicular to
not only the array direction but also the direction of wave propagation expressed
by the wavevector q. Emphasizing the latter facts, they are called longitudinal and
transverse modes. The sound velocity is larger for the longitudinal modes reflecting
the assumption k > k ⊥ .
5.2.2 Atomic Crystals
It is in order to proceed to cases of three-dimensional crystals. Let us start with the
simplest case for three-dimensional atomic crystals consisting of N identical atoms.
Suppose a crystal of atoms (mass m) on a simple cubic lattice. For three-dimensional
crystals, models assuming ideal springs as interatomic bonds are rather unphysical.
For example, a simple cubic lattice of point atoms connected by ideal springs is
unstable to any shear. We must consider the issue of what are force constants to
be assumed. Suppose V (R) be the potential energy of a crystal. The R is a 3N -
dimensional vector containing all information of atomic coordinates. At the static
equilibrium, the coordinates of all atoms r
◦
l satisfy
∂V
∂r α,l
eq
= 0,
(5.26)
for all l with α = x, y, or z with the subscript “eq” for the values at equilibrium. Thus,
we can expand the potential in terms of small displacements r l from the equilibrium
position and may truncate the expansion at the second-order around r
◦
l as
V ≈ V 0 +
1
2
l
l
α,β=x,y,z
∂
2 V
∂r α,l ∂r β,l
eq
r α,l r β,l
= V 0 +
1
2
l
l
α,β=x,y,z
φ αβ (l, l
)r α,l r β,l
(5.27)
where φ αβ (l, l
) are generalized force constants between the atomic pair (l, l
). Note
that φ αβ (l, l
) = φ βα (l
, l), leading to the double appearance of the same terms in
Eq. 5.27 except for l = l
.
Lagrangian L = T − V with T being the kinetic energy of the system serves a
fundamental role in the description of a mechanical system in terms of generalized
coordinate Q j and its time derivatives ˙
Q j [10]. The equation of motion is given by
˙
P j = ∂ L/∂ Q j with the generalized momentum P j = ∂ L/∂ ˙
Q j . Since the kinetic
energy in the present problem is given by
T =
1
2
m
l
d r l
dt
2
,
(5.28)