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Phonons I: Crystal vibrations
one dimensional vibration
one dimensional vibration for crystals with basis
three dimensional vibration
quantum theory of vibration
M.C. Chang
Dept of Phys
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One dimensional vibration
consider only the longitudinal motion
consider only the NN coupling
analyzed by Newtons law of motion (classical)
Assume where
then we' ll get
u Ae X na
M e e e e
n
i kX t
n
ikna ikna ik n a ik n a
n= =
=
+
( )
( ) ( )
, ,
( ) ,
2 1 1
2
dispersion relation ( )
which leads to
( ) = k ka M M Msin( / ) , / 2 2=
2
1 12( ) ( )n n n n n
d uM u u u u
dt + =
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Dispersion curve ( ) =k kaM sin( / )2
The waves with wave numbers kand k+2p/adescribe the
same atomic displacement
Therefore, we can restrict kto within the first BZ [-p/a,p/a]
k
/a-/a
2a > 2a 2a M1)
Patterns of vibration
similar
See a very nice demo at http://dept.kent.edu/projects/ksuviz/leeviz/phonon/phonon.html
ad
b
c
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Crystal vibrations
one dimensionalvibration one dimensional vibration for crystals with basis
three dimensional vibration
quantum theory of vibration
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Three dimensional vibrationAlong a given direction of propagation, there are
one longitudinal wave and 2 transverse waves,
each may have different velocities
Sodium
(bcc)
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Crystal with atom basis
Rules of thumb
p ,
N
N normal modes
3pN(= total DOF of this system)
3 acoustic branches,
3(p-1) optical branches
cm-1
FCC lattice with 2-atom basis
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Crystal vibrations
one dimensionalvibration
one dimensional vibration for crystals with basis
three dimensional vibration
quantum theory of vibration
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After quantization, the energy becomes discrete
is the energy of an energy quantum
The numbernof energy quanta depends on the amplitude of the
oscillator.
Review:
Quantization of a1-dim simple harmonic oscillator
(DOF=1)
Hp
mx= +
22
2 2
Classically, it oscillates with a single freq w=(a/M)1/2
The energy e can be continuously changed.
= +FHG
IKJ
n1
2h
h
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Classically, for a given k, it vibrates with a single frequency w(k).
The amplitude ( and hence energy e) can be continuously changed.
Quantization of a1-dim vibrating lattice (DOF=N)
1( ) ( )
2kk n k
= +
h
After quantization, the vibrational energy of the lattice becomes
discrete (see Appendix C)
For a given k(or wave length), the energy quantum isand the number of energy quanta (called phonons) is nk.
There are no interaction between phonons, so the vibrating lattice can
be treated as a free phonon gas.
Total vibrational energy Eof the lattice = summation of phonon energies.
h ( )k
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In 1-dim, if there are two atoms in a unit cell, then there are
two types of phonons with the following energies:
A k
A
A
O k
O
O
k n k
k n k
( ) ( / )
( ) ( / )
= +
= +
1 2
1 2
h
h
( ) for an acoustic mode
( ) for an optic mode
The total vibrational energy of the crystal is
E k n k nAk
k
A
O
k
k
O= +FHG
IKJ
+ +FHG
IKJ h h ( ) ( )
1
2
1
2
In general, for a crystal with more branches of dispersioncurves, just add in more summations.
k=2pm/L, (L=Na, m=1N).If we write nkas nm, then themicroscopic vibrational state of the whole crystal (state of the
phonon gas) is characterized by
{ 1 2, , ... each n is a non-negative integerNn n n
At 0o K, there are no phonons being excited. The hotter thecrystal, the more the number of phonons.
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uniform translation of the crystal
no center-of-mass motion
A k-mode phonon acts as if it has momentumhk
in the scattering process(for a math proof, see Ashcroft and Mermin, App. M)
Elastic scattering ofphoton: hk = hk+ hG (chap 2)
Inelastic scattering ofphoton: hk = hkhkphonon + hG
(Raman scattering)
P Mdu
dtu Ae
MA i e e
MA i e
e
e
k m Na k
n
n
n
i kX t
i t ikna
n
N
i tikNa
ika
n= =
=
=
= =
=
,
( )
( )
/
( )
0
1
1
1
0 2since ( 0 ONLY when = 0)
However, the physical momentum of a vibrating crystal with wave
vector k is zero
Therefore, we call hka crystal momentum (of the phonon), in order
not to be confused with the usual physical momentum.
Recoil momentum of
the crystal