4. Chapter Phonons
Lecture 9
1/25/11
Last Lecture: Chapter 4 Phonons
Phonons:
Quantization
Modes
Momentum
Scattering
This Lecture Chapter 5
More on Phonons:
Density of states
Heat capacity
Debye approximation
Reminder: Homework #4
due Feb. 2 Tuesday
5.1, 5.2, 5.4
See end of lecture
Homework
Describe the neutron scattering experiments referenced
in Chapter 4
Experimental set-up
Neutron emission direction
How is k[100], k[110], k[111] generated and
collected.
Why do these plots differ in mode #?
Discuss momentum and energy
conservation.
Describe Raman scattering
e What is physical basis?
xperiment
What can be learned from these
experiments?
Contrast Raman with neturon scattering in
approach and resulting data.
Derive Density of States 1D
L
Real space
Fixed
us
s=0
a
Fixed
1
2
s=10
10a
2
10a
10
10a
Reciprocal
space
0
K
Each normal vibrational mode of polarization p has the
form of a standing wave.
Displacement of the particle us = u (0) exp(i K , p t ) sin sKa
s:
K,p is related to K by the appropriate dispersion relation.
In class work:
L
Real space
Fixed
us
s=0
1
2
a
Fixed
s=10
Define primitive lattice cell.
What is size (length of primitive lattice cell?
Draw reciprocal space lattice.
Reciprocal
space
0
10a
2
10a
Draw 1D Brillouin zone. K
What is Kmax?
Define the size or length of one BZ.
How many modes in each BZ?
If us (s=0)=us (s=10)=0, write a time
independent equation for us(s).
10
10a
Quantized Phonon Modes
Energy of the lattice vibration is
quantized exist only at specific values,
i.e. energy can
proportional to number of phonons in a mode.
kinetic energy
K
= (nK + 1 )hK
2
K : oscillation frequency of the mode
nK : number of phonons in the mode
Each phonon has K
nK refers to number of phonons, nK=0,
1, 2, mode n=0 still has energy? 0,K hK
=1
Empty
2
Quantized Phonon Modes
contd.
Phonon mode us(t), fixed
u0
us
Mode amplitude u0, K
nK
At specific
x
t
K
u0
K = 2 / K
Some force (photon collision)
starts the oscillation at t=0, excites
K mode amplitude
Oscillation
us(t=0)=us,max=u0 varies with time
Kinetic energy is max as us=0
Potential energy is max as
us=umax=u0 shows no loss of
This picture
energy from phonon mode
us
t
Damped oscillation mode is
losing energy
This picture shows energy loss
from this particular mode
Quantized Modes Phonon
contd.
A mode is defined by oscillation frequency K, energy
of the mode depends on number of phonons at that
frequency.
Total system
energy:
1
Total =
(n
K
K
+ 2 )hK = K
K
This energy is potential and kinetic energy when
averaged over time.
u
x
Kinetic Energy
Esystem
1
1
= .. + ..
2
2
Kinetic energy density (energy per
volume):
12
1 2
lik E = ma
K .E . = (
)
2
2
e
: mass density (mass/V)
For the diatomic system considered here,
usk = u0 K cos Kx cos K t
Standing wave version us = u exp(isKa ) exp( it )
of:
usK is the displacement of a volume element from its
equilibrium position at x in the crystal
Kinetic Energy contd.
Kinetic energy
(instantaneous) 1
EK (t ) = V K 2u0 K 2 sin 2 K t
4
1
dtEK (t ) = V K 2u0 K 2
8
0
1
1
1
2
2
V K u0 K = (n + )hK
8
2
2
1
4(nK + ) h
2
u0 K 2 =
(29)
V K
amplitude of oscillation
based on phonon
occupation n
(instantaneou
s)
(time
average)
1
E = (n + )h
(from
2
)
(27)
Kinetic Energy contd.
u0,n+1
u0,n
u0,n-1
u0(t)
n+1
n
n-1
t
For mode of
K
u0,n
u0(t)
t
K
For mode of
K <
<
Take time
Complete math between (27) and (29)
Rewrite (27) with expression in (29). This
shows relationship between energy and
mode occupation.
Phonon Momentum
p momentum
p hK
Wave
vector
Note: a phonon interacts with other particles as if it
has momentum K. However, a phonon does not have
physical momentum.
Why?
We can use momentum conservation and selection
rules to describe phonon creation.
Phonon Momentum contd.
Recall selection rule for elastic scattering of a photon
v
with crystal lattice:
v
v
vv
k ' k +G
k'
v
k
G
v
k incident wave vector
v
k ' reflected wave vector
v
G general reciprocal lattice vector
Elastic scattering no net energy is exchanged between
photon and lattice
Inelastic scattering net energy is exchanged between
photon and lattice
Phonon Momentum contd.
Elastic scattering
Ephoton
t
Elattice
t
Interaction
time
Ephoton+Elattice=Esystem
Esystem is
constant and
conserved.
Inelastic Photon Scattering
Photon loses energy to lattice
phonon
v
k
v
k'
v
Phonon, K
Inelastic Photon Scattering
contd.
v
K
v
k'
v
G
v
k
vvvv
k '+ K = k + G
v
k'
v
k
v
G
v
K
Esystem = E photon + Elattice
inelastic selection
rule
Phonon creation
Phonon annihilation
A phonon is absorbed in
lattice
Energy is conserved.