solid state physics by kettel chapter 4

7/30/2019 solid state physics by kettel chapter 4

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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




11/17

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




14/17

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

solid state physics by kettel chapter 4

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