Download Phonons II

Lattice Vibrations
Part II
Solid State Physics
355
Three Dimensions
For each mode in a given propagation direction,
the dispersion relation yields acoustic and optical
branches:
• Acoustic
• Longitudinal (LA)
• Transverse (TA)
• Optical
• Longitudinal (LO)
• Transverse (TO)
If there are p atoms in the primitive cell, there
are 3p branches in the dispersion relation: 3
acoustic and 3p -3 optical.
NaCl – two atoms per
primitive cell
6 branches:
1 LA
1 LO
2 TA
2 TO
Counting
This enumeration follows from counting the number of
degrees of freedom of the atoms. For p atoms in N
primitive cells, there are pN atoms. Each atom has 3
degrees of freedom, one for each of the 3 directions x,
y, and z. This gives 3Np degrees of freedom for the
crystal.
q = ±/a
q.
q
Three Dimensions
un  ue

i ( q  r t )
Al
Ge
Quantization of Elastic Waves
The energy of an elastic mode of angular frequency 
is
E n   n  12


It is quantized, in the form of phonons, similar to the
quantization of light, as both are derived from a
discrete harmonic oscillator model.

Elastic waves in crystals are made up of phonons.
Thermal vibrations are thermally excited phonons.
Phonon Momentum
A phonon with a wavevector q will interact with particles,
like neutrons, photons, electrons, as if it had a momentum
(the crystal momentum)


p  q
• Be careful! Phonons do not carry momentum like photons do. They
can interact with particles as if they have a momentum. For example, a
neutron can hit a crystal and start a wave by transferring momentum to
the lattice.
• However, this momentum is transferred to the lattice as a whole. The
atoms themselves are not being translated permanently from their
equilibrium positions.
• The only exception occurs when q = 0, where the whole lattice
translates. This, of course, does carry momentum.
Phonon Momentum
For example, in a hydrogen
molecule the internuclear
vibrational coordinate
r1 r2 is a relative
coordinate and doesn’t
have linear momentum.
The center of mass
coordinate ½(r1 r2 )
corresponds to the uniform
mode q = 0 and can have
linear momentum.
H2
electron
r
R
r
R
Proton A
Proton B
r1
r2
O
Phonon Momentum


p  q
Earlier, we saw that the elastic scattering of x-rays from the
lattice is governed by the rule:
  
k  k  G
IfIfthe
is inelastic,
with a creation of a
thephoton
photonscattering
is absorbed,
then
phonon of wavevector q, then
   
kk
Gq
 q kkG
Phonon Scattering (Normal Process)
q1
q3 = q1 + q2
q2
q3 = q1 + q2 or q3 = q1 + q2 + G
Measuring Phonons
   
k q  k G
reciprocal lattice vector
scattered neutron
phonon wavevector
(+ for phonon created,
 for phonon absorbed)
Stokes or anti-Stokes Process
incident neutron
Measuring Phonons
q
Measuring Phonons
Measuring Phonons
Other Techniques
• Inelastic X-ray Spectroscopy
• Raman Spectroscopy (IR, near IR, and visible light)
• Microwave Ultrasonics
Heat Capacity
You may remember from your
study of thermal physics that
the specific heat is the amount
of energy per unit mass
required to raise the
temperature by one degree
Celsius. Q = mcT
Thermodynamic models give
us this definition:
 U 
CV  

 T V
Cv = yT+T3
electrons
phonons
Heat Capacity
Equipartition Theorem:
The internal energy of a system of N particles is
3
2
Nk BT
Monatomic particles have only 3 translational
degrees of freedom. They possess no rotational
or vibrational degrees of freedom. Thus the
average energy per degree of freedom is
1
2
Nk BT
It turns out that this is a general result.
The mean energy is
spread equally over all
degrees of freedom, hence
the terminology –
equipartition.
Heat Capacity
Heat Capacity
Answer: You need to use quantum
statistics to describe this properly.
 Bosons and Fermions



Bosons: particles that can be in the same energy
state (e.g. photons, phonons)
Fermions: particles that cannot be in the same energy
level (e.g. electrons)
Planck Distribution


Max Planck – first to come up
with the idea of quantum energy
worked to explain blackbody
radiation
 empty cavity at temperature
T, with which the photons are
in equilibrium
Planck Distribution
Einstein Model
1907-Einstein developed first reasonably
satisfactory theory of specific heat capacity for a
solid
 assumed a crystal lattice structure comprising N
atoms that are treated as an assembly of 3N
one-dimensional oscillators
 approximated all atoms vibrating at the same
frequency (unrealistic, but makes things easier)

Planck Distribution
number of phonons in
energy level n
total number of phonons
all possible energy levels 0, 1, 2, etc.
Planck Distribution
 n  n   nhf
Fraction of
Phonons
at energy n

Nn


N
0
n
e

  n / kT
e
0
  n / kT

e

 n  / kT
e
0
 n  / kT
small as n gets large
a constant
Planck Distribution
average
occupied
energy
level



1
1
1
 n  / k BT
n
x 
 e 
 e


  / k BT
1

e
1

e
1

x
n 0
n 0
n 0
n


x
   n n    1 

nx x  
 nx
 xx  x 


x nn00   x  1  x  1  x 2
n 0
nn
Einstein Model
average
energy
per oscillator
We have 3N such oscillators, so the total energy is
Einstein Model

k BT
dv

and

dT
k BT 2
let v 
Einstein Model
How did Einstein do?
T 
Einstein Model
How did Einstein do?
T 0K
Einstein Model
The Einstein model
failed to identically
match the behavior of
real solids, but it
showed the way.
In real solids, the
lattice can vibrate at
more than one
frequency at a time.
Answer: the Debye Model