Thermal properties of Solids:phonons

a

n-1 n n+1m

)2(sin 2 :relation sion Disper

, )2(sin4 ) )cos(1)(cos(2 )cos(

))cos(2))1(cos())1(cos(()cos(

-)cos( : vector wave with waveeprogressiv be can solution

)2( :motion of Eqn

tensionmequilibriu theis andconstant spring is

)( & )(

: mass withatom

mequilibriu from t displasmen allongitudinconsider Lets

222

002

0

11

0

10)1(10)1(

ka/m

ka/mkaknatknatm

knatanktanktuknatum

knatuuk

uuuum

F

uuFFuuFF

m

u

n

nnnn

nnnnnnnn

n

ka a0 )2( is )largest (or the shortest the

2 is wavelength

2 isfrequency highest the

/aak

/km

2m

Brillouin zone

naktunaa

lktuu

akm

aa

lk

m

la

lkk

n 110120

1112

12

cos)2

(cos

2/sin22/)2

(sin2

integer is where2

When

ka a0

k1 k2

k1

k2

Displacement of lattice pointFirst Brillouin zone is range of wave vectors k:

-ak a

dk

dVg

is packet) waveof velocity velocity(Group

a

l2

Longitudinal vibration of a linear chain of two masses

a/2

2n 2n+1 2n+22n-1

M Mm m

a/2

mM

ka

MmMm

Mka

kam

Mka

kam

kantunkatu

uuuuM

uuuum

nn

nnnn

nnnn

2/sin41111

, 0)2(2/cos2

2/cos2)2(

whenexists solution trivial-nona 0)2(2/cos2

02/cos2)2(

2/)12(cos ,2/2cos :solution Trial

)2(

)2(

:motion of Eqn

222

2

2

2

2

122

2122212

122122

Vibration of a linear chain of two masses

k

acoustic

optical

2/a0-2/a

optical /2, /2

acoustic /2 , /2

/At

112

optical )(2/11

2

acoustic )(2/

smallAt

22

12

3

222

222

mm

MM

ak

Mm

MmakMm

Mmak

k

1st critical frequency: heavy masses only moving, light masses at nodes

2st critical frequency: light masses only moving, heavy masses at nodes 3st critical frequency: light and heavy masses are moving in opposite directions

M2/21

m2/22

)/1/1(23 Mm

Quantized lattice vibrations Quantized model of the crystal vibrations: there is set of 3N independent linear oscillators( modes) with energy E=(n()+1/2)

The mean number of phonons in the mode with is

1exp

1

Tk

n

B

Debye’s frequency D: the largest frequency of vibration in crystal assuming

linear dispersion: v k. Debye’s temperature D/kB

Typical acoustic phonon frequency is Hz, optical phonon frequency

~ Hz, Deby’s temperature: diamond -3000 K, Cu -320K, Pb -90K

Inelastic neutron scattering

Neutrons can be scattered from the crystal away from Bragg’s lawwhen absorbs or emits phonon:

p' ,'* pnn Kkrk

phonon offrequency and vector wave, lattice, reciprocal

ofvector - * neutron, scattered andincident of vectorswave- ',

pp

nn

ωK

rkk

Infrared absorption in ionic

crystals

Transmitted light in infrared range, ~ Hz (~40-100m)

are absorbed by ionic crystals with optical mode of phonons

Cl ClCl Na Na

Ions of Cl and Na are moving in opposite directions

50 60 70 (m)T

rans

mitt

ance 100%

Transmittance through thin film (0.17m) of NaCl