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Lattice Vibrations, Part ILattice Vibrations, Part I
Solid State PhysicsSolid State Physics
355355
IntroductionIntroduction
Unlike the Unlike the static lattice modelstatic lattice model, which deals , which deals with average positions of atoms in a crystal, with average positions of atoms in a crystal, lattice dynamics lattice dynamics extends the concept of extends the concept of crystal lattice to an array of atoms with finite crystal lattice to an array of atoms with finite masses that are capable of motion.masses that are capable of motion.
This motion is not random but is a This motion is not random but is a superposition of vibrations of atoms around superposition of vibrations of atoms around their equilibrium sites due to interactions with their equilibrium sites due to interactions with neighboring atoms.neighboring atoms.
A collective vibration of atoms in the crystal A collective vibration of atoms in the crystal forms a wave of allowed forms a wave of allowed wavelengths wavelengths and and amplitudesamplitudes. .
ApplicationsApplications
• 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 ModesNormal Modes
x1 x2 x3 x4 x5
u1 u2 u3 u4 u5
x1 x2 x3
u1 u2 u3
Consider this simplified system...
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.
)(
and
)(
12112
21121
uuCF
uuCF
)(
and
)(
32223
23232
uuCF
uuCF
Net Forces on these atoms...
)(
)()(
)(
2323
3221212
2111
uuCF
uuCuuCF
uuCF
Normal ModesNormal Modes
)(
)()(
)(
23223
2
32212122
2
21121
2
uuCdt
udm
uuCuuCdt
udm
uuCdt
udm
Mr. Newton...
To find normal mode solutions, assume that each displacement has the samesinusoidal dependence in time.
tiii euu 0
0)(
0)(
0 )(
32
222
3222
2111
2112
1
umCuC
uCumCCuC
uCumC
Normal ModesNormal Modes
0
0
0
222
22
211
12
1
mCC
CmCCC
CmC
0 3)(2 1212
21422 uCCmCCmm
Normal ModesNormal Modes
2/12/1
2122
21213
2/12/121
22
21212
1
)()(1
)()(1
0
CCCCCCm
CCCCCCm
Longitudinal Wave
q
Transverse Wave
q
22
1 12
1 1
( ) ( )
( 2 )
n n n n
n n n
d um C u u C u u
dtC u u u
tiii euu 0
1inqa iqa
nu ue e
2 ( 1) ( 1)
2
2cos
2
[ 2 ]
[ 2]
21 cos
inqa i n qa i n qa inqa
iqa iqa
qa
m ue C e e e
m C e e
Cqa
m
Traveling wave solutions
Dispersion Relation
Dispersion RelationDispersion Relation
q
mC /4
0.6
qamC 2
1sin/4
First Brillouin ZoneFirst Brillouin Zone
What range of q’s is physically significant for elastic waves?
iqan ueu
1
iqainqa
qani
n
n eue
ue
u
u
)1( 1
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 ZoneFirst 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 ZoneFirst 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.
inqan ueu
ninn ueu )1(
][sin 4
21 qa
m
C
Group VelocityGroup Velocity
The transmission velocity of a wave packet is the group velocity, defined as
)q(
or
qg
g
v
dq
dv
]cos1[ 22 qam
C
][cos21
2qa
m
Ca
dq
dvg
q
][cos21
2qa
m
Ca
dq
dvg
Group VelocityGroup Velocity
The The phase velocityphase velocity of a wave is the rate at which the phase of the 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 wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave will propagate. You could any one frequency component of the wave will propagate. You could pick one particular phase of the wave (for example the crest) and it 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 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 given in terms of the wave's angular frequency ω and wave vector kk byby
Note that the phase velocity is not necessarily the same as the Note that the phase velocity is not necessarily the same as the group velocity of the wave, which is the rate that changes in group velocity of the wave, which is the rate that changes in amplitude (known as the amplitude (known as the envelopeenvelope of the wave) will propagate. of the wave) will propagate.
Phase VelocityPhase Velocity
Pvk
Long Wavelength LimitLong Wavelength Limit
When qa << 1, we can expandso 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.
221 )(1cos qaqa
22 ][ qam
C
qv ω
Force ConstantsForce Constants
]cos1[ 22 pqaCm
pp
a
C
dqrqapqaCdqrqamp
a
ap
a
a
2
)cos(]cos1[ 2 )cos(0
2
rqacosand integrate
The integral vanishes except for p = r. So, the force constant at range pa is
for a structure that has a monatomic basis.
a
ap dqpqa
π
maC
)cos(
22
Diatomic CoupledDiatomic CoupledHarmonic OscillatorsHarmonic Oscillators
q
)2(
)2(
12
2
2
12
2
1
nnnn
nnnn
vuuCdt
vdm
uvvCdt
udm
For each q value there are two values of ω.
These “branches” are referred to as “acoustic”and “optical” branches. Only one branchbehaves like sound waves ( ω/q → const. For q→0).For the optical branch, the atoms are oscillatingin antiphase. In an ionic crystal, these chargeoscillations (magnetic dipole moment) couple toelectromagnetic radiation (optical waves).
Definition: All branches that have a frequencyat q = 0 are optical.
Diatomic CoupledDiatomic CoupledHarmonic OscillatorsHarmonic Oscillators
qammmm
mmC
mm
mmC cos1
2
21
2
21
21
21
212
q
