5. Lattice Vibrations
Solid State Physics
355
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.

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
Normal Modes
x1
x2
u1
x3
u2
x4
u3
x5
u4
u5
Check this out...
http://www-users.york.ac.uk/~dmh8/cymatic/PhysicalModelling.htm
Consider this simplified system...
x1
x2
u1
x3
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.
F21  C1 (u1  u2 )
and
F12  C1 (u2  u1 )
F32  C2 (u3  u2 )
and
F23  C2 (u2  u3 )
Net Forces on these atoms...
F1  C1 (u1  u2 )
F2  C1 (u2  u1 )  C2 (u2  u3 )
F3  C2 (u3  u2 )
Normal Modes
Mr. Newton...
m
d 2u1
m
dt
d 2 u2
m
dt
d 2u3
dt
2
 C1 (u1  u2 )
2
 C1 (u2  u1 )  C2 (u2  u3 )
2
 C2 (u3  u2 )
To find normal mode solutions, assume that each displacement has the same
sinusoidal dependence in time.
ui  ui 0e
it
Normal Modes
(C1  m 2 )u1  C1u2
0
 C1u1  (C1  C2  m 2 )u2  C2u3  0
 C2u2  (C2  m 2 )u3  0
C1  m 2
 C1
0

 C1
C1  C2  m 2
 C2
0
 C2
0
C2  m 2

m 2 m2 4  2(C1  C2 )m 2  3C1C2 u1  0
Normal Modes
1  0




1
2
2
1/ 2 1/ 2
2  (C1  C2 )  (C1  C2  C1C2 )
m
1
2
2
1/ 2 1/ 2
3  (C1  C2 )  (C1  C2  C1C2 )
m
Brillouin Zones
The value of the Brillouin
Zone is that it gives a vivid
geometrical interpretation of
the diffraction condition:
D
k2
2 k G  G 2
k1
1G
2 D
O
1G
2 C
We divide both sides by 4 to
get,
C
k 12 G
2
1


 2G
Brillouin Zones
The first Brillouin zone is defined to be the
Wigner-Seitz primitive cell of the reciprocal
lattice.
 Or, it could be defined as the set of points in k
space that can be reached from the origin
without crossing any Bragg plane.
 The second Brillouin zone is the set of points
that can be reached from the first zone by
crossing only one Bragg plane.
 The n th Brillouin zone can be defined as the set
of points that can be reached from the origin by
crossing n - 1 Bragg planes, but no fewer.

Wigner-Seitz Primitive Cell
How can we choose primitive cells?
One algorithm is the Wigner-Seitz cell.
Steps:
(1) Draw lines to connect a given lattice
point to all nearby lattice points
(2) At the midpoint and normal to these
lines, draw new lines
(or planes in 3D)
(3) The smallest area (or volume in 3D)
enclosed in this way is the Wigner-Seitz
primitive cell
Wigner-Seitz Primitive Cells
Brillouin Zones
BCC
Brillouin Zones
Brillouin Zones
Back to Normal Modes
q
Longitudinal Wave
q
Transverse Wave
Back to Normal Modes
m
d 2 u2
dt
2
 C (un 1  un )  C (un 1  un )
 C (un 1  un 1  2un )
ui  ui 0e
it
inqa  iqa
un1  ue
e
Traveling wave
solutions
m 2ueinqa  C [ei ( n 1) qa  ei ( n 1) qa  2einqa ]
m 2  C [eiqa  e iqa  2]
2cos qa
2C
 
1  cos qa 
m
2
Dispersion
Relation
Dispersion Relation

0.6

4C / m
4C / m
q
 sin 12 qa
First Brillouin Zone
What range of q’s is physically significant for elastic waves?
un1  ue
 iqa
i ( n 1) qa
un 1 ue
iqa


e
inqa
un
ue
The range  to + for the phase qa covers all
possible values of the exponential. So, only values in
the first Brillouin zone are significant.
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 .
First Brillouin Zone
At the boundaries q = ± /a, the solution
u n  ue
inqa
Does not represent a traveling wave, but rather a
standing wave. At the zone boundaries, we have
un  ue
 in
 (1)
n
Alternate atoms oscillate in opposite phases and the
wave can move neither left nor right.
Group Velocity
The transmission velocity of a wave packet is the
group velocity, defined as
2C
 
[1  cos qa]
m
2

d
vg 
dq
or
v g   q (q)
d
Ca 2
vg 

[cos 12 qa]
4C
dq
m
1
[sin qa]
m
2
Group Velocity
d
Ca 2
vg 

[cos 12 qa]
dq
m
q
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 ω and wave vector k
by
vP 


k
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.
Long Wavelength Limit
1 ( qa) 2
cos
qa

1

When qa << 1, we can expand
2
so the dispersion relation becomes
C
2
  [qa]
m
2
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.
v  ωq
Force Constants
2
 
m
2
 cos rqa
m
a
a 
2
 C p [1  cos pqa]
p
cos(rqa ) dq  2
a
 C p a [1  cos pqa] cos(rqa) dq
p 0
and integrate
C
 2
a
The integral vanishes except for p = r. So, the force constant at range pa is
ma
Cp  
2π
a
a
 2 cos( pqa) dq
for a structure that has a monatomic basis.
Diatomic Coupled
Harmonic Oscillators

q
m1
m2
d 2 un
2
 C ( vn  vn 1  2un )
2
 C (un  un 1  2vn )
dt
d 2 vn
dt
Diatomic Coupled
Harmonic Oscillators
m  m2
2  C 1
C
m1m2
2
 m1  m2 
2

 
1  cos qa 
m1m2
 m1m2 
For each q value there are two values of ω.
These “branches” are referred to as “acoustic”

and “optical” branches. Only one branch
behaves like sound waves ( ω/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