Chapter 4 - Estado Sólido

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SOUND WAVES LATTICE VIBRATIONS OF 1D CRYSTALS

chain of identical atom

chain of t!o t"#e of atom

LATTICE VIBRATIONS OF $D CRYSTALS

%&ONONS

&EAT CA%ACITY FRO' LATTICE VIBRATIONS

AN&AR'ONIC EFFECTS

T&ER'AL CONDUCTION BY %&ONONS



Crystal Dynamics

Concern with the spectrum of characteristics vibrations of a crystallinesolid.

Leads to;

consideration of the conditions for wave propagation in a periodic

lattice, the energy content,

the specific heat of lattice waves,

the particle aspects of quantized lattice vibrations (phonons

consequences of an harmonic coupling between atoms.



Crystal Dynamics

!hese introduces us to the concepts offorbidden and permitted frequency ranges, and

electronic spectra of solids



Crystal Dynamics

"n previous chapters we have assumed that the atoms were atrest at their equilibrium position. !his can not be entirely correct(against to the #$%; &toms vibrate about their equilibrium position atabsolute zero.

!he energy they possess as a result of zero point motion is 'nown as

zero point energy.

!he amplitude of the motion increases as the atoms gain more thermalenergy at higher temperatures.

"n this chapter we discuss the nature of atomic motions, sometimesreferred to as lattice vibrations.

"n crystal dynamics we will use the harmonic approimation , amplitudeof the lattice vibration is small. &t higher amplitude some unharmonic

effects occur .



Crystal Dynamics

)ur calculations will be restricted to lattice vibrations of smallamplitude. *ince the solid is then close to a position of stableequilibrium its motion can be calculated by a generalization of themethod used to analyse a simple harmonic oscillator.!he smallamplitude limit is 'nown as harmonic limit.

"n the linear region (the region of elastic deformation, the restoringforce on each atom is approimately proportional to its displacement (#oo'e+s Law.

!here are some effects of nonlinearity or anharmonicity+ for larger

atomic displacements.

&nharmonic effects are important for interactions between phononsand photons.



Crystal Dynamics

&tomic motions are governed by the forces eerted on atomswhen they are displaced from their equilibrium positions.

!o calculate the forces it is necessary to determine the

wavefunctions and energies of the electrons within the crystal.-ortunately many important properties of the atomic motions

can be deduced without doing these calculations.



&oo(e) La!

)ne of the properties of elasticity is that it ta'es about twice asmuch force to stretch a spring twice as far. !hat linear

dependence of displacement upon stretching force is called

#oo'es law.

xk F spring .−= ↓ F Spring constant k

It takes twice

as much force

to stretch a

spring twiceas far.

↓ F 2



!he point at which the Elatic Re*ion ends is called the inelasticlimit, or the proportional limit. "n actuality, these two points are notquite the same.

!he inelatic Limit is the point at which permanent deformationoccurs, that is, after the elastic limit, if the force is ta'en off thesample, it will not return to its original size and shape, permanentdeformation has occurred.

!he %+o#o+tional Limit is the point at which the deformation is nolonger directly proportional to the applied force (#oo'es Law no

longer holds. &lthough these two points are slightly different, wewill treat them as the same in this course.

&oo(e, La!



*)$/D 0&12*

3echanical waves are waves which propagate through amaterial medium (solid, liquid, or gas at a wave speedwhich depends on the elastic and inertial properties of thatmedium. !here are two basic types of wave motion formechanical waves4 lon*it-dinal waves and t+an.e+e waves.

Lon*it-dinal Wa.e

T+an.e+e Wa.e

5 "t corresponds to the atomic vibrations with a long 6.

5 %resence of atoms has no significance in this wavelengthlimit, since 677a, so there will no scattering due to thepresence of atoms.



*)$/D 0&12*

*ound waves propagate through solids. !his tells us thatwaveli'e lattice vibrations of wavelength long compared to theinteratomic spacing are possible. !he detailed atomic structureis unimportant for these waves and their propagation isgoverned by the macroscopic elastic properties of the crystal.

0e discuss sound waves since they must correspond to thelow frequency, long wavelength limit of the more general latticevibrations considered later in this chapter.

&t a given frequency and in a given direction in a crystal it ispossible to transmit three sound waves, differing in theirdirection of polarization and in general also in their velocity.



2lastic 0aves

& solid is composed of discrete atoms, however when thewavelength is very long, one may disregard the atomic natureand treat the solid as a continous medium. *uch vibrations arereferred to as elastic waves.

5 &t the point elastic displacement is

$( and strain e+ is defined as thechange in length per unit length.dU

edx

=

/ /0d/

A

2lastic 0ave %ropagation (longitudinal in a bar



&ccording to #oo'e+s law stress * (force per unit area is

proportional to the strain e.

!o eamine the dynamics of the bar, we choose an arbitrary

segment of length d as shown above. $sing /ewton+s second

law, we can write for the motion of this segment,

.S C e=

[ ]2

2( ) ( ) ( )

u Adx S x dx S x A

t ρ ∂ = + −

∂

/ /0d/

A

C 8 9oung modulus

'a / Accele+ation Net Fo+ce +e-ltin* f+om t+ee

Elatic Wa.e



2quation of motion

2 2

2 2

u uC

t x ρ

∂ ∂=

∂ ∂

[ ]2

2

( ) ( ) ( )u

Adx S x dx S x At

ρ ∂

= + −∂

[ ]( ) ( ) S

S x dx S x dx x

∂+ − =

∂

.S C e=du

e

dx

=

2

2

.

.

uS C

x

S uC

x x

∂=∂

∂ ∂=

∂ ∂2 2

2 2( ) u u Adx C Adxt x

ρ ∂ ∂=∂ ∂ Cancellin* common te+m of Ad/

Which i the !a.e e2n3 !ith an offe+ed

ol,n and .elocit" of o-nd !a.e

( )i kx t u Ae ω −= : ( 4 !a.e n-m5e+ 6789:; : < 4 f+e2-enc" of the !a.e : A 4 !a.e am#lit-de /

s

s

v k

v C

ω

ρ

=

=

Elatic Wa.e



sv k ω =

!he relation connecting the frequency and wave number is

'nown as the dispersion relation.

= Slo#e of the c-+.e *i.e

the .elocit" of the !a.e3

5 &t small 6 ' < (scattering occurs

5 &t long 6 ' = (no scattering

5 0hen ' increases velocitydecreases. &s ' increases further,the scattering becomes greater since

the strength of scattering increasesas the wavelength decreases, andthe velocity decreases even further.

Di#e+ion Relation



*peed of *ound 0ave

!he speed with which a longitudinal wave moves through aliquid of density > is

L C

V λν

ρ

= =C 8 2lastic bul' modulus

> 8 3ass density

5 !he velocity of sound is in general a function of the direction

of propagation in crystalline materials.

5 *olids will sustain the propagation of transverse waves, whichtravel more slowly than longitudinal waves.

5 !he larger the elastic modules and smaller the density, the

more rapidly can sound waves travel.



*peed of sound for some typical solids

51L values are comparable with direct observations of speed of sound.

5*ound speeds are of the order of ?=== m@s in typical metallic, covalent

and ionic solids.

So-nd Wa.e S#eed



5 !hey can be characterized by

: & propagation velocity, v

: 0avelength 6 or wavevector

: & frequency ν or angular frequency A8B ν

5 &n equation of motion for any displacement can be

produced by means of considering the restoring forceson displaced atoms.

& lattice vibrational wave in a crystal is a repetitive and

systematic sequence of atomic displacements of longitudinal,

transverse, or

some combination of the two

5 &s a result we can generate a dispersion relationship

between frequency and wavelength or between angular

frequency and wavevector.

So-nd Wa.e S#eed



Lattice vibrations of D crystal

Chain of identical atoms

&toms interact with a potential 1(r which can be written in!aylor+s series.

( ) 2 2

2( ) ( ) ...........

2r a

r a d V V r V a

dr =

− = + + ÷

rR

V(R)

0 r0=4

Repulsive

Attractivemin

!his equation loo's li'e as the potential energy

associated of a spring with a spring constant 4

ar dr

V d K

=

=

2

2

0e should relate E with elastic modulus C4

KaC =

( )a

ar C Force

−×= )( ar K Force −=



3onoatomic Chain

!he simplest crystal is the one dimensional chain of identical atoms.Chain consists of a very large number of identical atoms with identical

masses. &toms are separated by a distance of FaG. &toms move only in a direction parallel to the chain.

)nly nearest neighbours interact (shortHrange forces.

a a a a a a

Un>7 Un>1 Un Un01 Un07



*tart with the simplest caseof monoatomic linear chainwith only nearest neighbourinteraction

)2( 11

..

−+ +−= nnnn uuu K um

5 "f one epands the energy near the equilibrium point for the nth atom and use elastic approimation, /ewton+s equation becomes

a a

Un>1 Un Un01

'onoatomic Chain



0)2( 11

..

=−−+ +− nnnn uuu K um

!he force on the nth atom;

)( 1 nn uu K −+

5!he force to the right;

5!he force to the left;

)( 1−− nn uu K

5!he total force 8 -orce to the right : -orce to the left

'onoatomic Chain

a a

Un>1 Un Un01

Eqn’s of motion of all atoms are of this form, only the

value of ‘n’ varies

' i Ch i



&ll atoms oscillate with a same amplitude & and frequency A.

!hen we can offer a solution;

( ).

0expnn nduu i A i kx t dt

ω ω = = − −

'onoatomic Chain

( )0expn nu A i kx t ω = −

( ) ( )

2..2 2 0

2

expnn

n

d uu i A i kx t

dt ω ω

= = −

..2

nn

u uω = −

na xn =

0

nn una x +=$ndisplaced

position

Displaced

position

' t i Ch i E ti f ti f th t



'onoatomic Chain Equation of motion for nth atom

..

1 1( 2 )n nn nmu K u u u+ −= − +

( ) ( ) ( ) ( )( )0 0 0 0

1 12e e 2 e e

n n n ni kx t i kx t i kx t i kx t m K A A A A

ω ω ω ωω =

+ −− − − −− − +

kna ( 1)k n a−

( 1)k n a+kna

( ) ( ) ( ) ( )( )2

e e e 2 e e eika ikai kna t i kna t i kna t i kna t

m K A A A A ω ω ω ω ω

−− − − −− = − +

Cancel Common te+m

( )2 e 2 eika ikam K ω −− = − +

( ) ( ) ( ) ( )( )2e e 2 e e

i kna t i kna ka t i kna t i kna ka t m K A A A A

ω ω ω ωω

− + − − − −− = − +

' t i Ch i



'onoatomic Chain

( )2 e 2 eika ikam K ω −− = − +2cosix ixe e x−+ =

e e 2 cosika ika ka−+ =

( )2 2cos 2

2 (1 cos )

m K ka

K ka

ω − = −= − −

( ) 21 cos 2 sin2

x x

− = ÷ 22 4 sin

2

kam K ω

= ÷

242 sin2

K ka

mω

= ÷

4 sin2

K kam

ω = ÷

3aimum value of it is

max

4 K

mω =

' t i Ch i



A versus ' relation;

max 2

/ s

K

m

V k

ω

ω

=

=

0 л /a 2л /a–л /ak

'onoatomic Chain

5 No+mal mode f+e2-encie of a 1D chain

The #oint A? B and C co++e#ond to the ame f+e2-enc"? the+efo+e

the" all ha.e the ame intantaneo- atomic di#lacement3

The di#e+ion +elation i #e+iodic !ith a #e+iod of 789a3

k

ω

C AB

0

' t i Ch i



/ote that4

"n above equation n is cancelled out, this means that the eqn. of motionof all atoms leads to the same algebraic eqn. !his shows that our trialfunction $n is indeed a solution of the eqn. of motion of nHth atom.

0e started from the eqn. of motion of / coupled harmonic oscillators. "fone atom starts vibrating it does not continue with constant amplitude,but transfer energy to the others in a complicated way; the vibrations ofindividual atoms are not simple harmonic because of this echangeenergy among them.

)ur waveli'e solutions on the other hand are uncoupled oscillationscalled normal modes; each ' has a definite w given by above eqn. andoscillates independently of the other modes.

*o the number of modes is epected to be the same as the number ofequations /. Let+s see whether this is the case;

4sin

2

K ka

m

ω =

'onoatomic Chain



pa

Nk k p

Na p Na

π π λ λ

22=⇒==⇒=

2stablish which wavenumbers are possible for our one dimensional chain.

/ot all values are allowed because nth atom is the same as the (/Inth as

the chain is Joined on itself. !his means that the wave eqn. of

must satisfy the periodic boundary condition

which requires that there should be an integral number of wavelengths in

the length of our ring of atoms

!hus, in a range of B@a of ', there are / allowed values of '.

( )0expn nu A i kx t ω = −

λ p Na =

n N n uu +=

'onoatomic Chain



0hat is the physical significance of wave numbers

outside the range of K

2π/a

'onoatomic Chain

un

un

a

'onoatomic Chain



!his value of ' corresponds to themaimum frequency; alternate atoms

oscillate in antiphase and the waves at thisvalue of ' are standing waves.

7 2 84 7 1.1474 7

4

aa k a a aπ π π λ λ = ⇒ = ⇒ = = =

7 2 63 7 0.8573 7

3

aa k a a aπ π π λ λ = ⇒ = ⇒ = = =

22 ;a k k a

π π λ λ

= = ⇒ =

0hite line 4

reen line 4

AH' relation

'onoatomic Chain

u

n

u

n

a

'onoatomic Chain



5!he points & and C both have same

frequency and same atomic displacements

5!hey are waves moving to the left.

5!he green line corresponds to the point M

in dispersion diagram.

5!he point M has the same frequency and

displacement with that of the points & and C

with a difference.5!he point M represents a wave moving to

the right since its group velocity (dA@d'7=.

5!he points & and C are eactly

equivalent; adding any multiple ofB@a to ' does not change the

frequency and its group velocity, so

point & has no physical significance.

5'8N@a has special significance

5O8P=o

2 2nn n nk a

π π λ π = = ⇒

2 2 sin90

1

a d d a= ⇒ =1 4 2 43

Mragg reflection can be obtained at

'8 Nn@a

2 24 sin2kam K ω =

-or the whole range of ' (6

'onoatomic Chain

u

n

u

n

a

-π /a k

K V

m

K

s /

2

ω

ω

=

=

k

ω

C AB

0

AH' relation (dispertion diagram

π /a 2π /a

f



&t the beginning of the chapter,

in the long wavelength limit, the

velocity of sound waves has

been derived as

$sing elastic properties, let+s see

whether the dispersion relationleads to the same equation in the

long 6 limit.

1ka pp"f 6 is very long; so sin ka ka≈

c Ka ρ ρ

= 2 K am

m

a ρ =

K V a s k m

ω = =2 22 44

k am K ω =

cV s ρ =

'onoatomic Chain



*ince there is only one possible propagation directionand one polarization direction, the D crystal has onlyone sound velocity.

"n this calculation we only ta'e nearest neighborinteraction although this is a good approimation for theinertHgas solids, its not a good assumption for manysolids.

"f we use a model in which each atom is attached by

springs of different spring constant to neighbors atdifferent distances many of the features in abovecalculation are preserved.

5 0ave equation solution still satisfies.5 !he detailed form of the dispersion relation is changed but A

is still periodic function of ' with period B@a5 roup velocity vanishes at '8(N@a5 !here are still / distinct normal modes5 -urthermore the motion at long wavelengths corresponds to

sound waves with a velocity given by (velocity formulQ

'onoatomic Chain



Chain of two types of atom!wo different types of atoms of masses 3 and m are

connected by identical springs of spring constant E;

Un>7Un>1 Un Un01 Un07

E E E E

' 'm 'm a;

5;

6n>7; 6n>1; 6n; 6n01; 6n07;

a

5 !his is the simplest possible model of an ionic crystal.5 *ince a is the repeat distance, the nearest neighbors

separations is a@B

Chain of t!o t"#e of atom



0e will consider only the first neighbour interaction although itis a poor approimation in ionic crystals because there is along range interaction between the ions.

!he model is complicated due to the presence of two differenttypes of atoms which move in opposite directions.

)ur aim is to obtain AH' relation for diatomic lattice


!wo equations of motion must be written;

)ne for mass 3, and

)ne for mass m.





' m ' m '


2quation of motion for mass 3 (nth4

mass x acceleration = restoring force

2quation of motion for mass m (nHth4

..

1 1( ) ( )n n n n n M u K u u K u u+ −= − − −

1 1( 2 )

n n n K u u u+ −= − +

..

1 1 2( ) ( )n n n n n

mu K u u K u u− − −= − − −..

1 2( 2 )n n n nmu K u u u− −= − +

H

H





' m ' m '


0

/ 2n x na=( )0

expn nu A i kx t ω = −

)ffer a solution for the mass 3

-or the mass m;

R 4 comple number which determines the relative amplitudeand phase of the vibrational wave.

( )0expn nu A i kx t α ω = −

( )..

2 0expn nu A i kx t ω ω = − −

H




..

1 1( 2 )n n n n M u K u u u+ −= − +

( ) ( )1 1

2 22 2 22

k n a k n akna knai t i t i t i t

MAe K Ae Ae Aeω ω ω ω

ω α α

+ − − −− − ÷ ÷ ÷ ÷

÷− = − + ÷

-or nth atom (34

2 2 2 2 22 22kna kna kna knaka kai t i t i t i t i i

MAe K Ae e Ae Ae eω ω ω ω

ω α α − − − − − ÷ ÷ ÷ ÷ − = − + ÷ ÷

Cancel common terms

2 2 1 cos

2

ka M K ω α

= − ÷

2cosix ixe e x−+ =2 2 22

ka kai i

M K e eω α α − − = − + ÷





2

2 2 2 2 22 2 22

kna kna kna knaka ka kai t i t i t i t i i i

mAe e K Ae Ae e Ae eω ω ω ω

αω α

− − − −− − − ÷ ÷ ÷ ÷ − = − + ÷ ÷

..

1 1 2( 2 )n n n nmu K u u u− − −= − +( ) ( ) ( )1 1 2

2 2 22 2 2

k n a k n a k n aknai t i t i ti t

A me K Ae Ae Aeω ω ω ω

α ω α

− − − − − −− ÷ ÷ ÷ ÷

÷− = − + ÷

Cancel common terms

-or the (nHth atom (m


2 2 cos2

kam K αω α

− = − ÷

2cosix ixe e x−+ =

2 2 21 2

ka kai i

ikame K e eαω α − −

− − = − + ÷

2 2 22ka ka

i i

m K e eαω α −

− = − + ÷




/ow we have a pair of algebraic equations for R and A as afunction of '. R can be found as

& quadratic equation for AB can be obtained by crossH

multiplication

2

2

2 cos( / 2) 2

2 2 cos( / 2)

K ka K M

K m K ka

ω α

ω

−= =

−

"#

2 cos2

kam K αω α

= − ÷ for m

2 2 1 cos

2

ka M K ω α

= − ÷

for 3




The t!o +oot a+e

2 2 2 44 (1 cos ( )) 2 ( ) 02

ka K K m M Mmω ω − − + + =

2 2 2 2 44 cos ( ) 4 2 ( )2ka K K K M m Mmω ω = − + +

24 2 2 sin ( / 2)

2 ( ) 4 0m M ka

K K mM mM

ω ω +− + =

22 2 1/ 2( ) 4sin ( / 2)

[( ) ] K m M m M ka

K mM mM mM

ω + +

= −m

m

2

2

2 cos( / 2) 2

2 2 cos( / 2)

K ka K M

K m K ka

ω α

ω

−= =

−

"#

2

12

4

2

b b ac

x a

− ± −=




A versus ' relation for diatomic chain;

"#

5 No+mal mode f+e2-encie of a chain of t!o t"#e of atom3

At A? the t!o atom a+e ocillatin* in anti#hae !ith thei+ cent+eof ma at +et

at B? the li*hte+ ma m i ocillatin* and ' i at +et

at C? ' i ocillatin* and m i at +et3

5 "f the crystal contains / unit cells we would epect to findB/ normal modes of vibrations and this is the total number of

atoms and hence the total number of equations of motion for

mass 3 and m.

0 л /a 2л /a–л /a k

ωA

B

C




&s there are two values of A for each value of ', the

dispersion relation is said to have two branches;

"#

U##e+ 5+anch i d-e to the

0.e i*n of the +oot3

Lo!e+ 5+anch i d-e to the

>.e i*n of the +oot3

O#tical B+anch

Aco-tical B+anch

5 !he dispersion relation is periodic in ' with a period

B π@a 8 B π@(unit cell length.

5 !his result remains valid for a chain of containing an

arbitrary number of atoms per unit cell.

0 л /a 2л /a–л /a k

ωA

B

C




Let+s eamine the limiting solutions at =, &, M and C.

"n long wavelength region ('aS; sin('a@BT 'a@B in AH'.

( )2 2 2

21 1

2( )

K m M mM k a

mM m M ω

+ ≈ ± − ÷+

22 2 1/ 2

12( ) 4sin ( / 2)[( ) ] K m M m M ka K mM mM mM

ω + += −m

"#

( )1 2

2 2 22 4

4

K m M m M k a K

mM mM mM

ω + + ≈ ± − ÷

( )

( )

1 2

2 2

21 1

K m M mM k a

mM m M

+ = ± − ÷ ÷ +

Use Taylor expansion: for small x( ) 1 21 1 2 x x− ≈ −




Ta(in* 0.e +oot in(a@1

22

2 cos( / 2)

K M

K ka

ω α

−=

( ) 2 2 2 2

2min . 22( ) 2( )acus

K m M mMk a Kk a

mM m M m M ω + ≈ ≈ + +

1α ≈ M

mα ≈ −

6ma/ .al-e of o#tical 5+anch;

Ta(in* >.e +oot 6min .al-e of aco-tical 5+ach;

B" -5tit-tin* thee .al-e of < in 6+elati.e am#lit-de;

e2-ation and -in* co6(a97; 1 fo+ (a@1 !e find the

co++e#ondin* .al-e of a

OR

( )2

max

2opt

K m M

mM ω

+≈




1α ≈

*ubstitute into relative amplitude R

2

22 cos( / 2)

K M K ka

ω α −=ac

2 22

min!(" a )

2(m #)ω ≈

+

!his solution represents longHwavelength sound waves in the

neighborhood of point = in the graph; the two types of atoms

oscillate with same amplitude and phase, and the velocity ofsound is

k

ωA

B

C

O#tical

Aco-tical

1/ 2

2( ) s

w K v a

k m M

= = ÷

+

ac

2

min ω

0 π /a 2π /a–π /a




!his solution corresponds to point & in

dispersion graph. !his value of R shows

that the two atoms oscillate inantiphase with their center of mass at

rest.

M

mα ≈ −

22

2 cos( / 2)

K M

K ka

ω α

−=

op

2

max

2!(m #)

m#ω

+≈

*ubstitute into relative amplitude we obtain,op

2

max ω

0 π /a 2π /a–π /a

k

ωA

B

C

O#tical

Aco-tical




!he other limiting solutions of equation AB are for 'a8 π,

i.e sin('a@B8. "n this case

1/ 22

2

max

( ) 4ac

K m M M m K

Mm Mm Mmω

+ + = − ÷

m

( ) ( ) K m M K M m

Mm

+ −=

m

2

max

2

ac

K

M ω = OR

2

min

2

op

K

mω =6C; 6B;

5 &t ma.acoustical point C, 3 oscillates and m is at rest.

5 &t min.optical point M, m oscillates and 3 is at rest.



&coustic@)ptical Mranches

!he acoustic branch has this name because it gives rise tolong wavelength vibrations H speed of sound.

!he optical branch is a higher energy vibration (the frequencyis higher, and you need a certain amount of energy to ecitethis mode. !he term FopticalG comes from how these werediscovered H notice that if atom is Ive and atom B is Hve, thatthe charges are moving in opposite directions. 9ou can ecitethese modes with electromagnetic radiation (ie. !he oscillatingelectric fields generated by 23 radiation



!ransverse optical mode for

diatomic chain

&mplitude of vibration is strongly eaggeratedU



!ransverse acoustical mode fordiatomic chain



0hat is phononK

Consider the regular lattice of atoms in a uniform solidmaterial.

!here should be energy associated with the vibrations of theseatoms.

Mut they are tied together with bonds, so they cant vibrate

independently.!he vibrations ta'e the form of collective modes which

propagate through the material.

*uch propagating lattice vibrations can be considered to besound waves.

&nd their propagation speed is the speed of sound in thematerial.

%honon



!he vibrational energies of molecules are quantized and

treated as quantum harmonic oscillators.Vuantum harmonic oscillators have equally spaced

energy levels with separation W2 8 h ν.

*o the oscillators can accept or lose energy only in

discrete units of energy h ν.!he evidence on the behaviour of vibrational energy inperiodic solids is that the collective vibrational modes canaccept energy only in discrete amounts, and thesequanta of energy have been labelled XphononsX.

Li'e the photons of electromagnetic energy, they obeyMoseH2instein statistics.





s phonon

h

ν

λ =

%&ONONS

5 Vuanta of lattice vibrations

5 2nergies of phonons are

quantized

Ya=8=H=m

phonon

h p

λ =

PHOTONS

5 Quanta of electromagneticradiation

5 Energies of photons arequantized as ell

photon

hc

λ =

Y=HZm

photon

h p

λ =



2nergy of harmonic oscillator

)btained by in a classical way of considering the normal modesthat we have found are independent and harmonic.

ω ε

+=

2

1nn

5 3a'e a transition to V.3.

5 [epresents equally spaced

energy levels

ω

ω

ω

ω

$ne%&' $

2nergy levels of atoms

vibrating at a single

frequency A



"t is possible to consider as constructed by adding n ecitation

quanta each of energy to the ground state.

nε

ω

ω ε 2

10 =

& transition from a lower energy level to a higher energy level.

ω ω ε

+−

+=∆

2

1

2

112 nn

( )2 1

unit!

n nε ω ε ω ∆ = − ⇒ ∆ =h h14 2 43

absorption of phonon



!he converse transition results an emission of phonon

with an energy .

%honons are quanta of lattice vibrations with an

angular frequency of .

%honons are not localized particles."ts momentum is eact, but position can not be

determined because of the uncertainity princible.

#owever, a slightly localized wavepac'et can be

considered by combining modes of slightly differentand .

ω

ω

ω λ



ssme *a+es *i, a sp%ea o " o ; so ,is *a+epac"e, *i

e ocaie *i,in 10 ni, ces.

a10

π

is *a+epac"e, *i %ep%esen, a ai%' ocaie ponon mo+in& *i,

&%op +eoci,' .dk

d ω

onons can e ,%ea,e as "oca"i#ed partic"es *i,in some imi,s.



1 c%'s,as

ω

k #,ip' '

ω

k

$ne%&' o

ponons

%'s,a momen,m

onons a%e no, conse%+e

e' can e c%ea,e an es,%o'e %in& coisions .



!hermal energy and lattice vibrations

,oms +i%a,e ao, ,ei% eii%im posi,ion.

e' p%oce +i%a,iona *a+es.

is mo,ion is inc%ease as ,e ,empe%a,%e is

%aise.

n a soi ,e ene%&' associa,e *i, ,is vibration an pe%aps aso

*i, ,e rotation o a,oms an moeces is cae as thermal energy.

Note% &n a gas' the trans"ationa" motion o( atoms and mo"ecu"es

contribute to this energ!)



e%eo%e ,e concep, o ,e%ma ene%&' is namen,a ,o an

ne%s,anin& man' o ,e asic p%ope%,ies o sois. :e *o i"e ,o

"no*

5:a, is ,e +ae o ,is ,e%ma ene%&'<

5=o* mc is a+aiae ,o sca,,e% a conc,ion eec,%on in a me,a;

since ,is sca,,e%in& &i+es %ise ,o eec,%ica %esis,ance.5e ene%&' can e se ,o ac,i+a,e a c%'s,ao&%apic o% a ma&ne,ic

,%ansi,ion.

5=o* ,e +i%a,iona ene%&' can&es *i, ,empe%a,%e since ,is &i+es

a meas%e o ,e ea, ene%&' *ic is necessa%' ,o %aise ,e,empe%a,%e o ,e ma,e%ia.

5>eca ,a, ,e speciic ea, o% ea, capaci,' is ,e ,e%ma ene%&'

*ic is %ei%e ,o %aise ,e ,empe%a,%e o ni, mass o% 1&moe '

one !e+in.



The energy given to lattice vibrations is the dominantcontribution to ,e ea, capaci,' in most solids. "n nonHmagnetic

insulators, it is ,e on' con,%i,ion.

?,e% con,%i,ions;

5n me,as %om ,e conc,ion eec,%ons.

5n ma&ne,ic ma,e%ias %om ma&ne,in& o%e%in&.

,omic +i%a,ions eas ,o an o no%ma moe %eencies %om e%o

p ,o some maximm +ae. aca,ion o ,e a,,ice ene%&' an ea,capaci,' o a soi ,e%eo%e as in,o ,*o pa%,s

i) ,e e+aa,ion o ,e con,%i,ion o a sin&e moe an

ii) ,e smma,ion o+e% ,e %eenc' is,%i,ion o ,e moes.

#eat capacity from Lattice vibrations



n

n

n * ε ε ∑= @

+a%a&e ene%&' o a a%monicoscia,o% an ence o a a,,ice

moe o an&a% %eenc' a,

,empe%a,%e $ne%&' o oscia,o%

1

2n nε ω

= + ÷

he p%oaii,' o ,e oscia,o% ein& in ,is

e+e as &i+en ' ,e Ao,man ac,o%

exp( / )n +

k , ε −

2nergy and heat capacity of a harmonicoscillator, 2instein 3odel



0

/ 2 3 / 2 5 / 2

/ 2 / 2 /

/ 2 / 1

1exp[ ( ) ]2

.....

(1 .....

(1 )

+ + +

+ + +

+ +

n +

k , k , k ,

k , k , k ,

k , k ,

# nk ,

# e e e

# e e e

# e e

ω ω ω

ω ω ω

ω ω

ω ∞

=

− − −

− − −

− − −

= − +

= + + +

= + + += −

∑h h h

h h h

h h

h

@ 0

0

1 1exp /

2 21

exp /2

+

n

+

n

n n k ,

n k ,

ω ω

ε

ω

∞

=∞

=

+ − + ÷ ÷ =

− + ÷

∑∑

h h

h

(\

&ccording to the Minomial epansion for S where / + x k , ω = −h

n

n

n * ε ε ∑= @

2qn (\ can be written



@

/

1

2 1 +k ,

e ω

ω ε ω = +

−h

hh

B

(n ) x

x x x

∂ =∂

( )

( )

( )

@ 2 2

/ 2 @

2/

@ / 2 /2

@ /2

/

2 2 @ 2

2 2 /

1(n )

n1

n n 1

n 12

2

4 1

+

+

+ +

+

+

+

+ +

k ,

+ k ,

k , k ,

+

k ,

+

+

k , +

+ +

+ k , +

# k , k , #

# , ,

ek , , e

k , e e,

k , e, k , ,

k e

k k , k ,

k , e

ω

ω

ω ω

ω

ω

ω

ε

ε

ε

ω ε

ω

ω ε

−

−

− −

−

−

−

∂ ∂= =

∂ ∂

∂= ÷∂ − ∂ = − − ∂

∂ ∂= − − − ÷∂ ∂ = +

−

h

h

h h

h

h

h

h

h

h

( )

/

/

1

2 1

+

+

k ,

k ,

e

e

ω

ω

ω ω

−

−

= +

−

h

h

hh



@

/

1

2 1 +k ,

e

ω

ω ε ω = +

−h

hh

is is ,e mean ene%&' o ponons.e i%s, ,e%m in ,e ao+e

ea,ion is ,e e%o-poin, ene%&'. s *e a+e men,ione eo%e e+en

a, 0C! a,oms +i%a,e in ,e c%'s,a an a+e e%o-poin, ene%&'. is is

,e minimm ene%&' o ,e s's,em.

e a+a%a&e nme% o ponons is &i+en ' Aose-$ins,ein

is,%i,ion as

1

1)(

−=

, +k en

ω ω

(number of phonons) x (energy of phonon)=(second term in ) @

ε

!he second term in the mean energy is the contribution of

phonons to the energy.



ε #ean ene%&' o a

a%monic oscia,o%as a nc,ion o

low temperature limit

,

ω 2

1

, k +

, k +ω

12

1 @

−+=

, +k eω

ω ω ε

ω ε

2

1 @

= Zero point energy

*ince eponential term

gets bigger



! is independent of frequency of

oscillation.

5!his is the classical limit because the

energy steps are now small compared with

the energy of the harmonic oscillator.

5*o that is the thermal energy of the

classical D harmonic oscillator.

ε

..........D2

12

+++= x xe x

, k e +

, +k ω ω

+= 1

112

1 @

−++=

, k +

ω

ω ω ε

@

12

+k , ε ω = +h

@

+k , ε

≈

ε

high temperature limit

,

ω 2

1

, k + 3ean energy of a

harmonic oscillator as

a function of !

+k , ω h =



#eat Capacity C

=ea, capaci,' can e on ' ie%en,ia,in& ,e a+e%a&e ene%&' o

ponons o

12

1 @

−+=

, +k eω

ω ω ε

( )

( )

2

2

1

k , +

k , +

+

+

v

k e

k , d C

d, e

ω

ω

ω ω

ε

−−

= =−

h

h

hh

( )

( ) ( )

2

2 2

1

k , +

k , +

v +

+

eC k

k , e

ω

ω

ω =

−

h

h

h

k

ω θ =

h

( )

2

2

1

,

,

v +

eC k

, e

θ

θ

θ = ÷ −

Le,



( )

2

2

1

,

,

v +

eC k

, e

θ

θ

θ = ÷ −

&rea8 2

ω h

,

+k

+

k

ω h

Epeciic ea, +anises

exponen,ia' a, o* Fs an ,ens ,o

cassica +ae a, i& ,empe%a,%es.

e ea,%es a%e common ,o aan,m s's,ems; ,e ene%&' ,ens

,o ,e e%o-poin,-ene%&' a, o* Fs

an ,o ,e cassica +ae o

Ao,mann cons,an, a, i& Fs.

vC

vC

k

ω θ =

h*e%e

%lot of as a function of !



, K

3 -

!his range usually includes [!.-rom the figure it is seen that Cv is

equal to ][ at high temperatures

regardless of the substance. !his fact

is 'nown as DulongH%etit law. !his law

states that specific heat of a givennumber of atoms of any solid is

independent of temperature and is the

same for all materialsU

vC

*pecific heat at constant volume depends on temperature as shown infigure below. &t high temperatures the value of Cv is close to ][,

where [ is the universal gas constant. *ince [ is approimately B

cal@EHmole, at high temperatures Cv is app. Z cal@EHmole.

%lot of as a function of !vC

Classical theory of



Classical theory of

heat capacity of solids

e soi is one in *ic eac a,om is on ,o i,s sie 'a a%monic o%ce. :en ,e soi is ea,e ,e a,oms +i%a,e

a%on ,ei% si,es i"e a se, o a%monic oscia,o%s. e

a+e%a&e ene%&' o% a 1 oscia,o% is ". e%eo%e ,e

a+e%a&a ene%&' pe% a,om %e&a%e as a 3 oscia,o% is 3"an conseen,' ,e ene%&' pe% moe is

G

*e%e H is +a&a%oFs nme% " A is Ao,mann cons,an, an

> is ,e &as cons,an,. e ie%en,ia,ion *%, ,empe%a,%e&i+es;

3 3 + Nk , -, =

23 233 3 6.02 10 ( / ) 1.38 10 ( / )vC - atoms mo"e . K

−= = × × × ×

24.9 ;1 0.2388 6

( ) ( )

. Ca" Cv . Ca" Cv

K mo"e K mo"e

= = ⇒

− −

;

v

d C

d,

ε =

ε

2instein heat capacity of solids



p y "he theor# e$plained %# Einstein is the first quantum theor# of solids&

'e made the simplif#ing assumption that all vi%rational modes of

a * solid of atoms had the same frequenc#+ so that the holesolid had a heat capacit# times

,n this model+ the atoms are treated as independent oscillators+ %ut

the energ# of the oscillators are ta-en quantum mechanicall# as

"his refers to an isolated oscillator+ %ut the atomic oscillators in a solid

are not isolated&"he# are continuall# e$changing their energ# ith

their surrounding atoms&

Even this crude model gave the correct limit at high temperatures+ a

heat capacit# of

*ulong./etit la here R is universal gas constant&

( )

2

2

1

,

,

v +

eC k

, e

θ

θ

θ = ÷ −

3 3 + Nk -=

ω





!he Discrepancy of 2instein model

$ins,ein moe aso &a+e co%%ec,' a speciic ea, ,enin& ,o

e%o a, aso,e e%o , ,e ,empe%a,%e epenence nea% G0

i no, a&%ee *i, expe%imen,.

a"in& in,o accon, ,e ac,a is,%i,ion o +i%a,ion

%eencies in a soi ,is isc%epanc' can e accon,e sin&

one imensiona moe o monoa,omic a,,ice



Density of *tates

cco%in& ,o Ian,m #ecanics i a pa%,ice is cons,%aine;,e ene%&' o pa%,ice can on' a+e specia isc%e,e ene%&'

+aes.

i, canno, inc%ease inini,e' %om one +ae ,o ano,e%.

i, as ,o &o p in s,eps.



ese s,eps can e so sma epenin& on ,e s's,em ,a, ,e

ene%&' can e consie%e as con,inos.is is ,e case o cassica mecanics.

A, on a,omic scae ,e ene%&' can on' Jmp ' a isc%e,e

amon, %om one +ae ,o ano,e%.

Definite energy levels *teps get small 2nergy is continuous



n some cases eac pa%,ica% ene%&' e+e can e

associa,e *i, mo%e ,an one ie%en, s,a,e (o%*a+enc,ion )

is ene%&' e+e is sai ,o e e&ene%a,e.

e ensi,' o s,a,es is ,e nme% o isc%e,e s,a,es

pe% ni, ene%&' in,e%+a an so ,a, ,e nme% o s,a,es

e,*een an *i e .

( ) ρ ε

( )d ρ ε ε d ε ε +ε

e%e a%e ,*o se,s o *a+es o% so,ion;

>nnin& *a+es



>nnin& *a+es

E,anin& *a+es

0 2

π 2

π −

4

π 4

π −

6

π

k

ese ao*e " *a+enme%s co%%espons ,o ,e %nnin&

*a+es; all positive and negative values of k are allowed. A'

means o pe%ioic ona%' coni,ion

2 2 2 Na / Na p k p k p

p k Na /

π π π λ λ = = ⇒ = = ⇒ = ⇒ =

an inte*e+

Len*th of

the 1D

chain

R-nnin* !a.e

ese ao*e *a+enme%s a%e nio%m' is,i,e in " a, a

ensi,' o e,*een " an "K" . ( ) - k ρ

running waves ( )2 -

/k dk dk ρ

π =

Standin* !a.e 4π 5π 3π



n some cases i, is mo%e si,ae ,o se s,anin& *a+esi.e. cain

*i, ixe ens. e%eo%e *e *i a+e an in,e&%a nme% o a

*a+een&,s in ,e cain;

Standin* !a.e

0 /

π

2

π 6

π

2 2;

2 2

n n n / k k k

/ /

λ π π π

λ = = ⇒ = ⇒ =

3

π

7

π

k 0

π 2

π 3

π

ese a%e ,e ao*e *a+enme%s o% s,anin& *a+es; on' posi,i+e +aes a%e ao*e.

2k p

π = for

running wavesk p

/

π = for

standing waves

ese ao*e "Fs a%e nio%m' is,%i,e e,*een " an "K"



a, a ensi,' o

( )S

k dk dk ρ π =

( )2

-

/k dk dk ρ

π =

D)* of standing wave

D)* of running wave

( )S k ρ

e densit! o( standing wave states is twice that o( the running waves.

=o*e+e% in ,e case o s,anin& *a+es on' posi,i+e +aes a%e

ao*e

en ,e ,o,a nme% o s,a,es o% o, %nnin& an s,anin& *a+es*i e ,e same in a %an&e " o ,e ma&ni,e "

e s,anin& *a+es a+e ,e same ispe%sion %ea,ion as %nnin&

*a+es an o% a cain con,ainin& H a,oms ,e%e a%e exac,' H is,inc,

s,a,es *i, " +aes in ,e %an&e 0 ,o ./ aπ



modes with frequency from ω to ωId ω corresponds

modes with wavenumber from k to k Idk

!he density of states per unit frequency range g(ω4

e nme% o moes *i, %eencies ω an ωKω *i e

&(ω)ω.

&(ω) can e *%i,,en in ,e%ms o ρE(k ) an ρ> (k ).

dn

d-

;( ) ( )-dn k dk g d ρ ω ω = = ( ) ( )S dn k dk g d ρ ω ω = =



oose s,anin& *a+es ,o o,ain ( ) g ω

Le,Fs %ememe% ispe%,ion %ea,ion o% 1 monoa,omic a,,ice

2 24sin

2

K ka

mω = 2 sin

2

K ka

mω =

dk

d ω

dk

d ω

d

dk

ω 2cos2 2

a K ka

m= cos 2

K kaa m= 1

cos2

K kaa

m

1 1cos

2

mkaa K

÷

( ) ( ) - gρ S

( ) ( )S g k ω ρ =

( ) ( )S g k ω ρ =

( ) ( )S

g k ω ρ =( )

1 1

cos / 2

m

a K ka



( ) g Na

ω π

=

( )cos / 2a K ka

2cos 1 sin

2 2

ka ka = − ÷ ÷

2 2 2sin cos 1 cos 1 sin x x x x+ = ⇒ = −

( ) ( )S

g k ω ρ =2

1 1 4

41 sin

2

m

a K ka − ÷

#,i' an i+ie

( ) ( )S

g k ω ρ =2

1 2

4 4sin

2

a K K ka

m m

− ÷

( )S

k dk dk ρ

π =

Let+s remember4

( ) g

ω π

=2 2

max

2 1

a ω ω −

/ Na=

2

max

4 K

m

ω =

2 24sin

2

K ka

mω

= ÷ ( )

2 g

N ω

π = ( )

1/ 22 2

maxω ω −

−

True density of states

( )1/ 2

2 22( )

N g ω ω ω

−= −

( ) g ω



cons,an, ensi,' o s,a,es

ω

N m K π

max 2

K

mω = K

mπ

%e ensi,' o s,a,es '

means o ao+e ea,ion

( )max( ) g ω ω ω

π

%e ?E(ensi,' o s,a,es) ,ens ,o inini,' a,

since ,e &%op +eoci,' &oes ,o e%o a, ,is +ae o .

ons,an, ensi,' o s,a,es can e o,aine ' i&no%in& ,e

ispe%sion o son a, *a+een&,s compa%ae ,o a,omic spacin&.

max 2

K

mω =ω /d dk ω

e ene%&' o a,,ice +i%a,ions *i ,en e on '



&' '

in,e&%a,in& ,e ene%&' o sin&e oscia,o% o+e% ,e is,%i,ion

o +i%a,ion %eencies. s

( )/

0

1

2 1k, g d

e ω

ω ε ω ω ω

∞ = + × ÷− ∫ h

hh

( )1/ 2

2 2max2 N ω ω

π

−

−#ean ene%&' o a a%monic

oscia,o%

?ne can o,ain same exp%ession o ' means o sin&

%nnin& *a+es.

for D

, so e e,,e% ,o in 3 ?E in o%e% ,o compa%e ,e

%es,s *i, expe%imen,.

( ) g ω

]D D)*



]D D)*

Le,Fs o i, i%s, o% 2

en o% 3.

onsie% a c%'s,a in ,e sape o 2 ox *i, c%'s,a en&,s o L.

I

I

I H

H

H

L=

L

y

/

π

*tanding wave pattern for a

BD bo

Configuration in k Hspace

xk

!k

/

π

Le,Fs caca,e ,e nme% o moes *i,in a %an&e o



*a+e+ec,o% ".

E,anin& *a+es a%e coosen , %nnin& *a+es *i ea

same exp%essions.

E,anin& *a+es *i e o ,e o%m

ssmin& ,e ona%' coni,ions o

i%a,ion ampi,e so +anis a, e&es o

oosin&

( ) ( )0 sin sin

x !U U k x k !=

0; 0; ; x ! x ! = = = =

; x !

p 0k k

π π = =

positive integer

y !k



I

I

I H

H

H

L=

L

5!he allowed ' values lie on a square lattice of side inthe positive quadrant of k Hspace.

5!hese values will so be distributed uniformly with a density

of per unit area.

5 !his result can be etended to ]D.

*tanding wave pattern for

a BD bo

/

π

/

π

Configuration in k Hspace

/ /π

( ) 2

/ / π

xk

L)ctant of the crystal4



L

L

)ctant of the crystal4

','y,'z(all have positive values

!he number of standing waves;

( )3

3 3 3

3 s

/ V k d k d k d k ρ

π π

= = ÷

/ / π

k

dk

# k

!k

xk

214

8k dk π ×

( ) 3 2

3

14

8 s

V k d k k dk ρ π

π = ×

( )2

3

22 s

Vk k d k dk ρ π

=

( )2

22S

Vk k ρ

π =

5 is a new density of states defined as the2Vk



is a new density of states defined as the

number of states per unit magnitude of in ]D.!his eqn

can be obtained by using running waves as well.

5 ω(frequency space can be related to 'Hspace4

( )2

22

Vk k ρ

π =

( ) ( ) g d k dk ω ω ρ = ( ) ( ) dk

g k d

ω ρ ω

=

Let+s find C at low and high temperature by means of

using the epression of .( ) g ω

#igh and Low !emperature Limits



+

, k

ω h?

#igh and Low !emperature Limits

!his result is true only if

&t low !+s only lattice modes having low frequencies

can be ecited from their ground states;

3 + Nk , ε =2ach of the ]/ lattice

modes of a crystal

containing / atoms d C d,

ε = 3 +C Nk =

θ

k

ω

aπ 0

Low frequency long λ

sound waves

sv k ω = sv

k

ω =

( )2Vk dk

g ω =1 1k dk v

ω = ⇒ = ⇒ = and



depends on the direction and there are two transverse,one longitudinal acoustic branch4

( ) 22 g

d ω

π ω =

s

s s

vk v d vω ω

= ⇒ = ⇒ = and

( )

2

2

2

1

2

s

s

V v

g v

ω

ω π

÷ = at low T’s

sv

( ) ( )2 2

2 3 2 3 3

1 1 2

2 2 s / ,

V V g g

v v v

ω ω ω ω

π π

= ⇒ = + ÷

1elocities of sound in

longitudinal and

transverse direction

( )1

g dω

ε ω ω ω∞ = + × ÷∫

hh Zero point energy= ε



( )/

02 1k,

g d e ω

ε ω ω ω = + × ÷− ∫ hh

3

3

3

/

0 01 1

+

+

k, x

k , x

k , d dx

e eω

ω ω

∞ ∞ ÷ =

− −∫ ∫ h

hh h

h

Zero point energy= # ε

2

/ 2 3 3

0

1 1 22 1 2k,

/ ,

V d e v vω ω ω ε ω ω

π

∞

= + × + ÷ ÷− ∫ h

hh

( )

3

2 3 3 /

0

1 2

2 1 # k,

/ ,

V d

v v e ω

ω ε ε ω

π

∞ ÷= + + ÷ ÷−

∫ h

h

+

xk ,

ω =

h

+k ,

xω =h

+k ,

d dxω =h( )

4

43 3

/ 3

0 0

15

1 1

+

k, x

k , xd dx

e eω

π

ω ω

∞ ∞

=− −∫ ∫ h

h

h14 2 43

( )4 4

2 3 3 3

1 2

2 15 + #

,

k , V

v v

π ε ε π

= + + ÷ h

24 3

3 3 3

1 24

30v +

/ ,

d V C k ,

d, v v

ε π = = + ÷

h

3

2

3 3

2 1 2

15

+v +

/ ,

k , d C V k

d, v v

ε π

= = + ÷ ÷ h

at low temperatures

#ow good is the Debye approimation



g y pp

at low !K

3,

32

3 3

2 1 2

15

+

v +

/ ,

k , d C V k

d, v v

ε π = = + ÷ ÷

h

e a,,ice ea, capaci,' o sois ,s

+a%ies as a, o* ,empe%a,%es; ,is is

%ee%%e ,o as ,e e'e a*.

Mi&%e is,%a,es ,e exceen, a&&%emen,

o ,is p%eic,ion *i, expe%imen, o% a

non-ma&ne,ic insulator. The heat

capacity vanishes more slowly than theexponential behaviour of a single

harmonic oscillator because the vibration

spectrum extends down to zero

frequency.

3,

!he Debye interpolation



!he Debye interpolation

scheme e caca,ion o is a +e%' ea+' caca,ion o% 3 so i,

ms, e caca,e nme%ica'.

e'e o,aine a &oo app%oxima,ion ,o ,e %es,in& ea, capaci,' ' ne&ec,in& ,e ispe%sion o ,e acos,ic *a+es i.e. assmin&

o% a%i,%a%' *a+enme%. n a one imensiona c%'s,a ,is isei+aen, ,o ,a"in& as &i+en ' ,e %o"en ine o ensi,' os,a,es i&%e %a,e% ,an c%+e. e'eFs app%oxima,ion &i+es ,eco%%ec, ans*e% in ei,e% ,e i& an o* ,empe%a,%e imi,s an ,ean&a&e associa,e *i, i, is s,i *ie' se ,oa'.

( ) g ω

s k ω υ =

( ) g ω



1. pp%oxima,e ,e ispe%sion %ea,ion o an' %anc ' a inea%ex,%apoa,ion o ,e sma " ea+io% 4

Einstein

appro$imationto thedispersion

*e%#e

appro$imationto thedispersion

vk ω =

!he Debye approimation has two main steps4



2

3

9( )

1

N g ω ω

ω =

2 3 3 3 31 2 3 9( ) 32 / , 1 1

V N N v vπ ω ω

+ = =

Debye cutHoff frequency

B. $ns%e ,e co%%ec, nme% o moes ' imposin& a c,-o%eenc' ao+e *ic ,e%e a%e no moes. e c,-o%eenc' is cosen ,o ma"e ,e ,o,a nme% o a,,ice moesco%%ec,. Eince ,e%e a%e 3H a,,ice +i%a,ion moes in a c%'s,a

a+in& H a,oms *e coose so ,a,

0

( ) 3 1

g d N

ω

ω ω =∫ 2

2 3 3

1 2( ) ( )

2 ,

V g

v v

ω ω

π = +

2

2 3 3

0

1 2( ) 3

2

1

,

V d N

v v

ω

ω ω π

+ =∫

3

2 3 3

1 2( ) 36

1

/ ,

V N v v

ω π

+ =

1ω

1ω

1ω

2( ) / g ω ω



e a,,ice +i%a,ion ene%&' o

ecomes

an

Mi%s, ,e%m is ,e es,ima,e o ,e e%o poin, ene%&' an a epenence isin ,e secon ,e%m. e ea, capaci,' is o,aine ' ie%en,ia,in& ao+e en*%, ,empe%a,%e.

/

0

1( ) ( )

2 1 +k , g d

e ω

ω ω ω ω

∞

= +−∫ h

hh

3 3

2

/ /3 30 0 0

9 1 9( )

2 1 2 1

1 1 1

+ +k , k , 1 1

N N d d d

e e

ω ω ω

ω ω

ω ω ω

ω ω ω ω ω ω ω

= + = + − − ∫ ∫ ∫ h h

h h hh

3

/3

0

9 9

8 1

1

+ 1 k ,

1

N d N

e

ω

ω

ω ω ω

ω = +

−∫ h

hh



d

C d,

=

( )

/2 4

23 2/

0

9

1

1 +

+

k ,

1k ,

1 +

d N eC d d, k , e

ω ω

ω

ω ω ω

= =−

∫ h

h

h3

/3

0

9 9

8 1

1

+ 1 k ,

1

N d N e

ω

ω

ω ω ω ω

= +−∫ h

hh

Let+s convert this complicated integral into an epression for

the specific heat changing variables to

and define the Debye temperature

x

1Θ

+

xk ,

ω =

h

1 1

+k

ω Θ =

h

!he heat capacity is

d k,

dx

ω =h

k, xω =

h



( )

4 /2 4

23 2

0

9

1

1 , x

+ +

1 x

1 +

k , k , d N x eC dx

d, k , eω

Θ = = ÷ ÷ −

∫ h

h h

( )

3

/ 4

2

0

91

1 , x

1 + x

1

, x eC Nk dx

e

Θ = ÷Θ −∫

!he Debye prediction for lattice specific heat

1 1

+k ω Θ =

hwhere



3 /

2

0

9 3

1 ,

1 1 + +

1

, , C Nk x dx Nk

Θ Θ ⇒ ≅ = ÷Θ ∫ ?

#ow does limit at high and low temperaturesK

#igh temperature

1C

1, Θ?

2 3

12D 3D

x x xe x= + + + +

( ) ( )

4 4 4 2

2 2 2(1 ) (1 )

1 11

x

x

x e x x x x x x xe

+ += = =+ −−

N is a*a's sma



1, Θ=

( )

3 / 4

2

0

91

1 , x

1 1 + x

1

, x e, C Nk dx

e

Θ Θ ⇒ ≅ ÷Θ −

∫ =

3412

5

+ 1

1

Nk , C

π ≅ ÷Θ

#ow does limit at high and low temperaturesK

Low temperature

44 / 15π

1C

Mo% o* ,empe%a,%e ,e ppe% imi, o ,e in,e&%a is inini,e; ,e

in,e&%a is ,en a "no*n in,e&%a o .

:e o,ain ,e e'e a* in ,e o%m3,

Lattice heat capacity due to Debye interpolation

scheme



scheme

Mi&%e so*s ,e ea, capaci,'

e,*een ,e ,*o imi,s o i& an o* as p%eic,e ' ,e e'ein,e%poa,ion o%ma.

3 +

C

Nk ,

Aecase i, is exac, in o, i& an o*

imi,s ,e e'e o%ma &i+es i,e a &oo

%ep%esen,a,ion o ,e ea, capaci,' o mos, sois

e+en ,o& ,e actua" phonon2densit! o( states

curve ma! di((er app%ecia' %om ,e e'e

assmp,ion.e'e %eenc' an e'e ,empe%a,%e scae *i, ,e +eoci,' o son in

,e soi. Eo sois *i, o* ensi,ies an a%&e eas,ic moi a+e i& . aes oo% +a%ios sois is &i+en in ,ae. e'e ene%&' can e se ,o

es,ima,e

,e maximm ponon ene%&' in a soi.

Lattice heat capacity of a solid as

predicted by the Debye interpolation

scheme

( )

3 / 4

2

0

91

1 , x

1 + x

1

, x eC Nk dx

e

Θ = ÷Θ −

∫

/ 1, Θ

*olid &r /a Cs -e Cu %b C ECl

P] ?^ ]^ _?` ]_] =? BB]= B]?( ) 1 K Θ

1Θ 1Θ

1ω h

&nharmonic 2ffects



&nharmonic 2ffects

n' %ea c%'s,a %esis,s comp%ession ,o a smae% +ome ,an i,s eii%im +aemo%e s,%on&' ,an expansion e ,o a a%&e% +ome.

is is e ,o ,e sape o ,e in,e%a,omic po,en,ia c%+e. is is a epa%,%e %om =oo"eFs a* since a%monic appica,ion oes no, p%oce

,is p%ope%,'. is is an ana%monic eec, e ,o ,e i&e% o%e% ,e%ms in po,en,ia *ic a%e

i&no%e in a%monic app%oxima,ion.

e%ma expansion is an exampe ,o ,e ana%monic eec,.

n a%monic app%oxima,ion ponons o no, in,e%ac, *i, eac o,e% in ,e asenceo ona%ies a,,ice eec,s an imp%i,ies (*ic aso sca,,e% ,e ponons) ,e,e%ma conc,i+i,' is inini,e.

n ana%monic eec, ponons coie *i, eac o,e% an ,ese coisions imi,,e%ma conc,i+i,' *ic is e ,o ,e o* o ponons.

( ) 2

2

2( ) ( ) ....................

2r a

r a d V V r V a

dr =

− = + + ÷

%honon phonon collisions



%hononHphonon collisions

e copin& o no%ma moes ' ,e na%monic ,e%ms in ,ein,e%a,omic o%ces can e pic,%e as coisions e,*een ,e ponons

associa,e *i, ,e moes. ,'pica coision p%ocess o

phonon

phonon!

1 1 k ω

2 2 k ω

3 3 k ω

&fter collision another phonon is

produced

3 1 2k k k = +

3 1 2k k k = +h h h

3 1 2ω ω ω = +3 1 2ω ω ω = +h h h

and

conservation of energy

conservation of momentum

onons a%e %ep%esen,e ' *a+enme%s *i,



"honon# has k

a

π p $ "honon# has and "honon#="honon#’ k

a

π f

12

k

ε

aπ − 0

aπ

3 B

%mklapp process

&due to anharmonic effects'

3

12

k

ε

aπ − 0

aπ

(ormal process

3)ongitudinal

Transverse

0n = ⇒ 0n ≠ ⇒

k a a

π π − ≤ ≤

ies o,sie ,is %an&e a a si,ae m,ie o ,o

%in&

i, ac" *i,in ,e %an&e o . en

ecomes

3k

3 1 2

2nk k k

a

π ± = +

*e%e an a%e a in ,e ao+e %an&e.

2

a

π

3 1 2k k k = +k a a

π π − ≤ ≤

2k 3k 1k

,his phonon is indistinguishab"e

(rom a phonon with wavevector 3k

!hermal conduction by phonons



!hermal conduction by phonons

A flow of heat takes place from a hotter region to a cooler regionwhen there is a temperature gradient in a solid.

The most important contribution to thermal conduction comes fromthe flow of phonons in an electrically insulating solid.

Transport property is an example of thermal conduction.

Transport property is the process in which the flow of somequantity occurs.

Thermal conductivity is a transport coefficient and it describes theflow.

The thermal conductivity of a phonon gas in a solid will becalculated by means of the elementary kinetic theory of the transportcoefficients of gases.

Einetic theory



Einetic theory

n ,e eemen,a%' "ine,ic ,eo%' o &ases ,e s,ea' s,a,e x o a p%ope%,' in ,e i%ec,ion is

* @ 1

3

d* ("ux "

d# υ =

#ean %ee pa,

n&a% a+e%a&e

ons,an, a+e%a&e spee o%

moecesn ,e simpes, case *e%e is ,e nme% ensi,' o pa%,ices ,e ,%anspo%,

coeicien, o,aine %om ao+e en. is ,e diffusion coefficient .

is ,e ene%&' ensi,' ,en ,e x : is ,e ea, o* pe% ni, a%ea so

,a, @ @

1 13 3

d d d, 3 " " d# d, d#

υ υ = =

Ho* is ,e speciic ea, pe% ni, +ome so ,a, ,e ,e%ma

conc,i+i,'; @ 1

3 K " C υ =

* @ 1

3 1 " υ =

*

/d d, C

:o%"s *e o% a ponon &as

=ea, conc,ion in a ponon an %ea &ase essen,ia ie%ences e,*een ,e p%ocesses o ea,



conc,ion in a ponon an %ea &as;

%honon gas [eal gas

Epee is app%oxima,e' cons,an,.

Ao, ,e nme% ensi,' an ene%&'

ensi,' is &%ea,e% a, ,e o, en.

=ea, o* is p%ima%i' e ,o ponono* *i, ponons ein& created a, ,e

o, en an estro!ed a, ,e co en

Ho o* o pa%,ices

+e%a&e +eoci,' an "ine,ic ene%&' pe%

pa%,ice a%e &%ea,e% a, ,e o, en , ,e

nme% ensi,' is &%ea,e% a, ,e co en

an ,e ene%&' ensi,' is nio%m e ,o ,e

nio%m p%ess%e.

=ea, o* is soe' ' ,%anse% o "ine,ic

ene%&' %om one pa%,ice ,o ano,e% in

coisions *ic is a mino% eec, in ponon

case.hot cold

hot cold



empe%a,%e epenence o ,e%ma conc,i+i,' !

@ 1

3 K " C υ = &pproimately equal to

velocity of sound and so

temperature independent.anises exponen,ia' a,

o* Fs an ,ens ,o cassica

+ae a, i& Fs +k

K

empe%a,%e epenence o ponon mean %ee en&, is e,e%mine '

ponon-ponon coisions a, o* ,empe%a,%es

Eince ,e ea, o* is associa,e *i, a o* o ponons ,e mos, eec,i+e

coisions o% imi,in& ,e o* a%e ,ose in *ic ,e ponon &%op +eoci,'

is %e+e%se. , is ,e Om"app p%ocesses ,a, a+e ,is p%ope%,' an ,ese a%e

impo%,an, in imi,in& ,e ,e%ma conc,i+i,'

onc,ion a, i& ,empe%a,%es



onc,ion a, i& ,empe%a,%es

, ,empe%a,%es mc &%ea,e% ,en ,e e'e ,empe%a,%e ,e ea, capaci,' is&i+en ' ,empe%a,%e-inepenen, cassica %es, o

e %a,e o coisions o ,*o ponons ponon ensi,'.

coisions in+o+in& a%&e% nme% o ponons a%e impo%,an, o*e+e% ,en ,esca,,e%in& %a,e *i inc%ease mo%e %api' ,an ,is *i, ponon ensi,'.

, i& ,empe%a,%es ,e a+e%a&e ponon ensi,' is cons,an, an

,e ,o,a a,,ice ene%&' ; ponon nme% so

Eca,,e%in& %a,e an mean %ee en&,

en ,e ,e%ma conc,i+i,' o .

1,

−

µ

1Θ

3 +C Nk =

µµ µ

@ 1

3 K " C υ = µ 1, −

µ

$xpe%imen,a %es,s o ,en ,o*a%s ,is ea+io% a, i& ,empe%a,%es

i i ( )



as so*n in i&%e (a).

1

,

? = B= ?= ==

=

=

=H

( ), K

1

1

(

)

K 3 c m K −

−

B ? = B= ?= ==

( ), K

=

=

=H

1

1

(

)

K 3

c m

K −

−

3,

(a)e%ma conc,i+i,' o a a%,

c%'s,a

()e%ma conc,i+i,' o a%,iicia

sappi%e %os o ie%en, iame,e%s

onc,ion a, in,e%meia,e ,empe%a,%es



onc,ion a, in,e%meia,e ,empe%a,%es

>ee%%in& i&%e a, P ; ,e conc,i+i,' %ises mo%e s,eep' *i, ain& ,empe%a,%e a,o&

,e ea, capaci,' is ain& in ,is %e&ion. :'<

is is e ,o ,e ac, ,a, Om"app p%ocesses *ic *i on' occ% i ,e%e a%e ponons o sicien, ene%&' ,o c%ea,e a ponon *i, . Eo

$ne%&' o ponon ms, e ,e e'e ene%&' ( )

e ene%&' o %ee+an, ponons is ,s no, sa%p' eine , ,ei% nme% isexpec,e ,o +a%' %o&' as

*en

*e%e is a nme% o o%e% ni,' 2 o% 3. en

is exponen,ia ac,o% omina,es an' o* po*e% o in ,e%ma conc,i+i,'

sc as a ac,o% o %om ,e ea, capaci,'.

1θ

3 /k aπ >

1k θ

/ 1

b, e

θ −

1, θ =

/ 1 b, " eθ µ3,

µ

onc,ion a, o* ,empe%a,%es



onc,ion a, o* ,empe%a,%es

o% ponon-ponon coisions ecomes +e%' on& a, o* Fs an e+en,a'excees ,e sie o ,e soi ecase

nme% o i& ene%&' ponons necessa%' o% Om"app p%ocesses eca' exponen,ia'as

is ,en imi,e ' coisions *i, ,e specimen s%ace i.e.

Epecimen iame,e%

epenence o ! comes %om *ic oe's a* in ,is %e&ion

empe%a,%e epenence o omina,es.

"

/ 1 b, e

θ −

3

,

" µ

"

vC

3412

5

+ 1

1

Nk , C

π ≅ ÷Θ

vC

*ize effect



:en ,e mean %ee pa, ecomes compa%ae ,o ,e imensions o ,e sampe,%anspo%, coeicien, epens on ,e sape an sie o ,e c%'s,a. is is "no*n as asize effect.

,e specimen is no, a pe%ec, c%'s,a an con,ains impe%ec,ions sc as isoca,ions&%ain ona%ies an imp%i,ies ,en ,ese *i aso sca,,e% ponons. , ,e +e%'o*es, Fs ,e ominan, ponon *a+een&, ecomes so on& ,a, ,ese impe%ec,ions

a%e no, eec,i+e sca,,e%e%s so; ,e ,e%ma conc,i+i,' as a epenence a, ,ese ,empe%a,%es.

e maximm conc,i+i,' e,*een an %e&ion is con,%oe 'impe%ec,ions.

Mo% an imp%e o% po'c%'s,aine specimen ,e maximm can e %oa an o* [i&%e(a) on p& 59] *e%eas o% a ca%e' p%epa%e sin&e c%'s,a as is,%a,e in i&%e()on p& 59 ,e maximm is i,e sa%p an conc,i+i,' %eaces a +e%' i& +ae o,e o%e% ,a, o ,e me,aic coppe% in *ic ,e conc,i+i,' is p%eominan,' e ,o

3,

3,

/ 1 b, e

θ

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Chapter 4 - Estado Sólido