Lattice Vibrations, Part I

1
Lattice Vibrations, Part I

• Solid State Physics
• 355

2
Introduction

• Unlike the static lattice model, which deals with
  average positions of atoms in a crystal, lattice
  dynamics extends the concept of crystal lattice
  to an array of atoms with finite masses that are
  capable of motion.
• This motion is not random but is a superposition
  of vibrations of atoms around their equilibrium
  sites due to interactions with neighboring atoms.
• A collective vibration of atoms in the crystal
  forms a wave of allowed wavelengths and
  amplitudes.

3
Applications

• Lattice contribution to specific heat
• Lattice contribution to thermal conductivity
• Elastic properties
• Structural phase transitions
• Particle Scattering Effects electrons, photons,
  neutrons, etc.
• BCS theory of superconductivity

4
Normal Modes
5
Consider this simplified system...
x1
x2
x3
u1
u2
u3
Suppose that only nearest-neighbor interactions
are significant, then the force of atom 2 on atom
1 is proportional to the difference in the
displacements of those atoms from their
equilibrium positions.
Net Forces on these atoms...
6
Normal Modes
Mr. Newton...
To find normal mode solutions, assume that each
displacement has the same sinusoidal dependence
in time.
7
Normal Modes
8
Normal Modes
9
Longitudinal Wave
10
Transverse Wave
11
Traveling wave solutions
Dispersion Relation
12
Dispersion Relation
13
First Brillouin Zone
What range of qs is physically significant for
elastic waves?
The range ?? to ? for the phase qa covers all
possible values of the exponential. So, only
values in the first Brillouin zone are
significant.
14
First Brillouin Zone
There is no point in saying that two adjacent
atoms are out of phase by more than ?. A
relative phase of 1.2 ? is physically the same as
a phase of ?0.8 ?.
15
First Brillouin Zone
At the boundaries q ?/a, the solution Does
not represent a traveling wave, but rather a
standing wave. At the zone boundaries, we
have Alternate atoms oscillate in opposite
phases and the wave can move neither left nor
right.
16
Group Velocity
The transmission velocity of a wave packet is the
group velocity, defined as
17
Group Velocity
18
Phase Velocity

• The phase velocity of a wave is the rate at which
  the phase of the wave propagates in space. This
  is the velocity at which the phase of any one
  frequency component of the wave will propagate.
  You could pick one particular phase of the wave
  (for example the crest) and it would appear to
  travel at the phase velocity. The phase velocity
  is given in terms of the wave's angular frequency
  ? and wave vector k by
• Note that the phase velocity is not necessarily
  the same as the group velocity of the wave, which
  is the rate that changes in amplitude (known as
  the envelope of the wave) will propagate.

19
Long Wavelength Limit
When qa ltlt 1, we can expand so the dispersion
relation becomes The result is that the
frequency is directly proportional to the
wavevector in the long wavelength limit. This
means that the velocity of sound in the solid is
independent of frequency.
20
Force Constants
21
Diatomic CoupledHarmonic Oscillators
22
Diatomic CoupledHarmonic Oscillators
For each q value there are two values of
?. These branches are referred to as
acoustic and optical branches. Only one
branch behaves like sound waves ( ?/q ? const.
For q?0). For the optical branch, the atoms are
oscillating in antiphase. In an ionic crystal,
these charge oscillations (magnetic dipole
moment) couple to electromagnetic radiation
(optical waves). Definition All branches that
have a frequency at q 0 are optical.
?
q