1.1 Equations of Motion and Dispersion Law
19
In mechanics the dependence of frequency on the wave number is called the disper-
sion law or dispersion relation. Thus, (1.1.6) gives the dispersion relation
ω
2
= ω
2
(k)
for the lattice vibrations.
From (1.1.6) we note that the dispersion law determines the frequency as a periodic
function of the quasi-wave num ber with a period of a reciprocal lattice
ω(k)=ω(k + G), G =
2π
a
.
This pe riodicity is the basic distinction between the dispersion law of crystal vi-
brations and that of continuous medium vibrations, since the monotonic wave-vector
dependence of the frequency is typical for the latter. The difference between the quasi-
wave number k and the ordinary wave number is also observed in the fact that only
number k values ly ing inside one unit cell of a reciprocal lattice (−π/a < k < π/a)
correspond to physically nonequivalent states of a crystal.
When the lattice period a tends to zero, the Brillouin zone dimension becomes
infinitely large and we return to the concept of momentum and its eigenfunctions in
the form of plane waves.
To clarify the available restrictions on the region of physically nonequivalent k
values we note that k = 2π/λ always, where λ is the corresponding wavelength.
We consider, for simplicity, a o ne-dimension al crystal (a linear chain) with a period
a for which the recip ro cal lattice “vector” G = 2π/a . Choose the interval −π/a ≤
k ≤ π/a as the r eciprocal lattice unit cell. The limiting value of the quasi-wave
number k = π/a will then respond to the wavelength λ = 2a. It fo llows from the
physical meaning of wave motion that this wavelength is the minimum in the crystal,
since we can observe the substance motion only at points where material particles
are located. A wave of this length is shown as a solid curve in Fig. 1.1 (the dark
points are th e equilibrium positio ns of particles, the light ones are their positions at a
certain moment of the motion) . A wave with wave number larger than the limiting one
reciprocal lattice period namely, k = π/a + 2π/a = 3π/a, is shown as a dashed
line. Both waves correctly reproduce the crystal motion but the introduction of the
wavelength λ = 2a/3, carrying no additional information on the particle mo tion, is
not justified physically.
We now propose a short analysis of the dispersion relation of the one-dimensional
crystal. According to (1.1.6), possible vibration frequencies fill band (0, ω
m
)where
ω
m
is the upper boundary of the band of possible vibration frequencies. To continue
the analysis one needs to know a spectrum of quasi-wave number values inside the
Brillouin zone. In order to define such a spectrum consider the one-dimensional crys-
tal containing N atoms (N 1) and having the length L (L = Na). The spectrum
mentioned depends on the boundary conditions at the crystal ends. We formulate the
simplest boundary conditions supposing that the atomic chain is closed up into a ring