Chapter 5Crystal Thermodynamics · Chapter 5Crystal Thermodynamics. 2 5.1 phonon heat capacity I. Density of states Consider an atomic chain consisting of N lattice sites: l L l Na

理学院物理系沈嵘

1

5.1 Phonon heat capacity

5.2 Experimental Determination of Phonon

Spectrum

5.3 Anharmonic Effects

Chapter 5 Crystal Thermodynamics

2

5.1 phonon heat capacityI. Density of states

Consider an atomic chain consisting of N lattice sites:

lL

lNa

q pp 22 ==

where l is an integer

NaL = —the length of the chain

the q-points are regularly arranged in a 1d chain, with

distance to each other.Lp2

5.1 phonon heat capacity

3

For large L, the distance becomes

small, and the q-points are nearly

continuous; each q-value (a single

point) represents a vabration

mode.

For an arbitrary distance dq，number

of modes dn in between is:

qLn d2

dp

=

q is related to frequence w via the dispersion relation, we can

thus get the # of vibration modes in the range of .( )www d, +


4

1. DefinitionDensity of modes (states): # of vibration modes per unit frequency interval

( )w

wddng =

2. D.O.S. of an atomic chain

( ) ng dd =ww

qLn d2

dp

=

( ) qLg d2

dp

ww =


5

( ) ÷ø

öçè

æ=q

Lgdd

2w

pw

for a given w value, there exist two symmetric q vlues:

( ) ÷ø

öçè

æ=q

Lgddw

pw

dispersion：2

2 qaM

sin÷øö

çèæ=bw

222

dd qaa

mqcos÷

øö

çèæ=bw

22qaa

m cosw=

Mmbw 4=in which,

Mb4


6

( )

2

2qaa

Lgm coswp

w =

2

cos2m

Nqapw

=

2 2

2

m

Np w w

=-

is finite,0=•w ( )wg

,mww =• ( ) ¥®wg,mww >• ( ) 0=wg


7

3. Diatomic chain

for each branch ( )q+w ( )q-w

corresponds： ( ) ( ),g gw w+ -

( ) ( ) ( )www -+ += ggg

4. General crystalsq-points with the same frequencies

consititutes closed contours, or a

isosurface.


8

( ) ww =qj!

( )d dj q q! !w w w+ = +

Count # of all modes between these two isosurfaces: jnd

Equal. ( ) ww djg Þ determine ( )wjg

Sum over all branches, one gets the total density of states:

( ) ( )å=j

jgg ww

For a given branch j

two isoenergetic contours/isosurfaces


9

Example：A cube with length L

According to periodic boundary condition,

ii

ii aN

lq p2=L

lip2

=

volume per q-point:32÷øö

çèæ

Lp

A unit volume in q-point space contains:

3

3

82 ppVL

=÷øö

çèæ

allowed q

V——volume of the crystall


10

Isosurfaces are spheres, with radius q. The # of modes in this sphere:

33 2

34

÷øö

çèæ=

Lqn pp

( )3

3 34

2qV p

p=

For q, modes between q ~ q+dq:

( )qqVn d3

34

2d 23

pp

=

( )qqV d4

22

3 pp=


11

( )( )

qqVg jj d42d 23 ppww =

( ) 23d

4d2( )

jj

Vg qqw

w pp

æ ö= ç ÷

è ø

qj

ddw

can be obtained from dispersion relation

( )( ) ( )33

33

22

11

221111

ppr VNN

bN

bN

bN

q dr

=W=W

=÷ø

öçè

æ´×

=!!!

For arbitrary isosurface, and any branch, vibration modes are

uniformly distributed in space. Corresponding DOS is:q!


12

( )( )3d d2

Vn w w wp

= ´ +频率和等频面间的体积

( )3d d d

2 S

Vn S qp

^= ò

dS--surface element,

dq^--distance between two surfaces

( )qq jqj ww Ñ= ^dd

( )qjqwÑ ——gradient along direction normal to the surface

(volume between and ) w ww d+


13

( ) ( )3dd d

2j

S q j

V Snq

wwp

=Ñò

( )( ) ( )3

d dd 2

jj S q j

n V Sgq

ww wp

= =Ñò

Total Density of States：

( ) ( )å=j

jgg ww

( ) PNg 3d =ò wwwhich satifies：


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II. Heat Capacity1. lattice heat capacity

Specifici heat of solid is defined: V

V TC ÷

øö

çèæ¶¶

=e

: average intermal energy of solid, including energies of

lattice vibration and electron motions

e

According to classical theory, every degree of freedom

corresponds to ave. energy , with kinetic energy

and potential energy; suppose there are N atoms, the

total energy is .

TkB2TkB

2TkB

TNkB3=e


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suppose N=NA (1 mole of atoms)

therefore mole capacity is:

3V A BV

C N kTe¶æ ö= =ç ÷¶è ø

which has nothing to do with specific material or any temperature dependence.——Dulong-Petit’s law


16

Experiments show，CV vialates the Dulong-Petit’s

law.

This demonstrates that

the classical theory of

equipartition of energy is

no longer valid, and the

quantum theory of crystal

vibration is needed.


17

lattice vibrations are quantized, at temperature T, the

virbration with frequency w has energy:

we !÷øö

çèæ +=

21n

average phonon #： ( ) ( ) 11

-= Tkqi Bie

qn !"w

Ignore the zero point energy:12w!

D.O.S , satisfy：( )g w

In which, w m----max angular frequency, N----# of primitive unit cell


18

Average Energy：

( )ò -=m

Bg

e Tkw

w wwwe

0d

1!!

=÷øö

çèæ¶¶

=V

V TC e

( )2

20d

( 1)

Bm

B

k T

B k TB

ek gk T e

ww

w

w w wæ öç ÷ -è ø

ò!

!

!

Heat Capacity：

Þ calculate the phonon heat capacity( )g w


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2. Einstein ModelSuppose all the vibration modes are with the same frequency wE .

Average energy of the crystal:

where N is # of primitive cells (PN is # of atoms)

heat capacity:

=÷øö

çèæ¶¶

=V

V TC e ( )2

2

13

-÷ø

öçè

æTk

Tk

B

EB

BE

BE

ee

TkNPk

w

ww!

!!


20

Einstein Temperture: EΘ EBE Θk=w!

( )22

13

-÷øö

çèæ=

TΘ

TΘE

BVE

E

ee

TΘPNkC

Determination of ：Select a proper value, such that the

calculated specific heat agrees with the experimental data,

in an extensive range of temperatures.

EΘ

,EΘT >>• at high temperature

( ) ( )22Θ221

1 TTΘTΘTΘ

EEE

E

eeee

--=

-

2

2

22

1÷ø

öçè

æ=

÷øö

çèæ

»EE Θ

T

TΘ


21

BE

EBV PNkΘ

TTΘPNkC 33

22

=÷ø

öçè

æ÷øö

çèæ=

agrees with Dulong-Petit’s law.

,EΘT TΘEe TΘTΘ EE ee »-1

TΘEBV

EeTΘPNkC -÷

øö

çèæ=

2

3

E

Bk TEB

B

PNk ek T

2

3ww -æ ö

= ç ÷è ø

!

!

low temperature


22

0®T

0®VC

asymptotic behavior (from experiments)：

3T

Einstein model oversimplified the difference in frequenceof various lattice waves (vibration modes)


23

3. Debye model

Regard Braivais lattices as isotropic continuous media, and the

crystal vibration an elastic wave with longitudinal and

transverse waves presumably moving in the same velocity.

Volume of the crystal V, for elastic wave in an isotropic media

quw = qdd uw =

In each branch, # of vibration modes in the range of q~q+ dq

( )qqVn d4

2d 23 pp

=


24

# of vibration modes in the range of w ~ w + dw :

( )w

up

pd4

2d

2

3qVn = w

uw

pd

2 32

2V=

For each q, there corresponds 3 elastic waves (1 longitudinal，

2 transverse modes).

Therefore, # of modes in w ~ w + dw :

wwup

d2

3d 232Vn =

( ) wwup

ww d2

3dd 232Vng ==


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( ) 23223 wup

w Vg =

ò ×-=m

B

Ve Tk

w

w wwupwe

0

232 d2

31!

!

ò -=m

BTkeV w

w ww

up 0

3

32 d123

!

!

( )ò -÷øö

çè

æ×=

m

B

B

Tk

Tk

BBV

ee

TkkVC

w

w

w

wwwup 0 2

22

3 d123

!

!!

Average Energy:

Heat Capacity:

Density of States:


26

For N atoms, 3N canonical frequencies:

( )òm g

www

0d NVm 3d

23 2

320== ò wwup

w

upw31

26÷ø

öçè

æ=V

Nm

Tkx

B

w!=

Tkx

B

mm

w!=

( )ò -÷øö

çè

æ=mx

x

xB

BV xe

xeTkVkC0 2

43

2 d123

up !

set


27

upw31

26÷ø

öçè

æ===QV

NTkTk

xT BB

mm

D !!

Debye Temperature： DΘB

mD k

Θ w!=

xe

xΘTTNk

TΘ

xD

BD d

19

0

33

ò -÷øö

çè

æ=e

ò -×÷øö

çè

æ= m

x

x

x

DBV xe

xeΘT

VNVkC

0 2

432

2 d)1(6

23 p

p

ò -×÷øö

çè

æ=

TΘ

x

x

DBV

D xe

xeΘTNkC

0 2

43

d)1(

9


28

Debye temperature is a characteristic temperature of the system

DT Q>>• xe x +» 1

TNkT

TTNk BDD

B 3319

33

=÷øö

çèæQ÷

ø

öçè

æQ

=e

BV

V NkTC 3=÷

øö

çèæ¶¶

=e

tends to classical limit


29

DΘT

301.from elastic const. and wave velocity2.from specific heat dataD

Θ ofion determinat


31

4. Comparison of Einstein and Debye Models

optical branch---Einstein model

acoustic branch---Debye modelCrystal Vibration


32

Einstein Model:

Approximate all branches as a straight line, ignore the

effects of low energy phonons, and treat all lattice waves as

optical modes.

Debye Model:

For long wavelength acoustic waves, effective model at low

temperatures.


33

We also adopt the combination of the two models—— hybridized model

( ) ( ) ( )www DE ggg +=


34

5.2 Experimental Determination of Phonon Spectrum

The dispersion relation between frequency and wave vector

——vibration sepctrum of crystal, phonon spectrum

When photons or neutrons are shot into crystals, they can

exchange energies with the lattice, and excite/annihilate

phonons—inelastic scattering.

Inelastic X-Ray scattering, neutron scattering, light scattering...

5.2 experimental determination of phonon spectrum

light scattered by acoustic phonon——Brillouin Scattering

light scattered by optic phonon——Raman Scattering.

Raman spectrum is very important tool for investigating

microstructure in condensed matter.

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36

90K, Na crystal, determined dispersion relation along three directions


37

5.3 Anharmonic Effects

Cannot convey energy, nor establish thermal equilibrium.

Harmonic Approximation:

Force on the atom is proportional to its displacement, potential

energy is kept up to d 2.

Crystal vibration can be discribed as a series of

independent harmonic oscillators, which do not interact

with each other, nor exchange energy with each other.

5.3 Anharmonic effects

38

Realistic Crystal:

ü Strictly, the interaction between atoms are not completely

harmonic, the crystal vibration modes are not completely

independent, but can instead talk to each other.

ü Phonon-phonon interaction exists, which enable them to

exchange energies. A phonon with a specific frequency could

be transfered into another phonon with different frequency,

and the distribution of phonons achieves thermal equilibrium

after a period of time.

ü Anharmonic terms accounts for the existence of thermal

equilibrium of crystals.


39

I. Lattice Free Energy and Equation of State

lattice free energy ：

1. thermodynamic relationsTSUF -=

TVFP ÷øö

çèæ¶¶

-=

VTFS ÷øö

çèæ¶¶

-=

VVV T

FTTSTC ÷

ø

öçè

æ¶¶

-=÷øö

çèæ¶¶

= 22

( )VT

FTFTSFTU ÷øö

çèæ¶¶

-=+=

pressure：

entropy：

energy：

heat capacity：


40

2. Free Energy( ) ZTkVUF B ln-=

Z —phonon partition function :

å -= TkBieZ e

lattice free energy:

U(V) —crystal cohesive energy at T = 0 , a function of volume V, independent of temperature T

For lattice wave wi： å¥

=

÷øö

çèæ +-

=0

21

i

B

ii

n

Tkn

i eZw!

i

B

i

B

k T

k T

e

e

12

1

w

w

-

-=

-

!

!

Partition functionfor all vibration modes： ÕÕ -

-

-==

iTk

Tk

ii Bi

Bi

eeZZ w

w

!

!

1

2


phonon contribution

41

( ) ( )i Bk TiBi B

F U V k T ek T

1 ln 12

ww -ì üé ù= + - - - -í ýê úë ûî þ

å !!

( ) ( )å úûù

êëé -++= -

i

TkBi

BieTkVU ww !! 1ln21

Sum over all branches and values (modes).q!

3. Equation of StatesNonlinear vibration changes the volume and frequency wi, wi is a function of V.

TVFP ÷øö

çèæ¶¶

-=Ve

eVU i

iTk

Tk

Tbi

Bi

dd

121 w

w

w

å úû

ùêë

é-

+-÷øö

çèæ¶¶

-= --

!

!!

!


42

VeVU i

iTki

iT

bi ddln

121 www wå úû

ùêëé

-+-÷

øö

çèæ¶¶

-=!

!!

VVVU i

ii

T lndlnd1 weå-÷

øö

çèæ¶¶

-=

121

-+= Tk

iii Bie w

wwe!

!!in which：

Vi

lndlnd wg -=set ——Gruneisen parameter,

independent of wi, related with nonlinear vibration.

VVUP eg+-=

dd


43

II. Thermal ExpansionVolume changes with T, under a

fixed pressure.

1. Qualitative analysisGiven a symmetric potential

around equilibrium position, then

Realistic potential curve is not strictly a parabola, steeper on the

left and smoother on the right side. Atoms tend to move

rightwards when vibration amplitudes increase.

Þ equilibrium position independent of amplituide of vibration or

temperature.


44

Anharmonic potential is non-symmetric, inducing some

“repulsive” interactions when atoms vibrate, accounting for

the themal expansion phenomenon.

2. quantitative analysisvolume expansion

coefficient： VP TP

KTV

V ÷øö

çèæ¶¶

=÷øö

çèæ¶¶

=11

0

a

linear expansion

coefficient： Pl T

ll ÷ø

öçèæ¶¶

=0

1a

isotropic cubic system:V

l TP

K ÷øö

çèæ¶¶

==313aa

K is bulk modulus


TVPVK ÷øö

çèæ¶¶

= 0-

45

According to equation of state:

VVUP eg+-=

dd

For most solids, volume change is small, expand around

equilibrium V0: VU

dd

( ) !+÷ø

öçè

æ-+÷

øö

çèæ=

00

2

2

0 dd

dd

dd

VV VUVV

VU

VU

First term vanishes, retain up to the second term:

úúû

ù

êêë

é÷ø

öçè

æ-=

0

2

2

00

0

dd

dd

VVUV

VVV

VU

0

0

VVVK -=


46

VVVVKP eg+--=0

0

0=PVKV

VV eg=

-

0

0

thermal expansion is the relation between V and T at P = 0,

0d VVV -=

VC

KTV

VV

P

ga =÷øö

çèæ¶¶

=0

1VC

KV

l 3ga =

Grüneisen's law, thermal expansion of solids is proportional to the specific heat (per volume).


47

III. Thermal Conductivity

1. phonon scattering

Lattice waves are no longer independent due to anharmonic

effects, leading to phonon scattering.

The precess of phonon collision obeys energy conservation and

quasi-momentum conservation：

321 www !!! =+

hGqqq!!!! +=+ 321


48

(1) Normal process

Wave vectors , are relatively small, resulting is still in

the 1st BZ, thus called normal process.

Total energy/wave vector keeps unchanged, only the energy

and quasi-momentum of the two phonons are tranfered to a

3rd one, net heat flow does not decreases, nor does its direction

change.

1q 2q 3q

If all phonon collisions are normal processes, the lattice

thermal conductivity diverges and resistivity is zero, i.e.,

normal process contributes nothing to thermal resistivity.


49

(2) Umklapp process

1q

2q

3q

exceeds 1st BZ, due to the

peroidicity of lattice, wave vector

describes the same vibration state

as . can be moved

back to 1st BZ by adding some

reciprocal lattice vector.

q q1 2+! !

q!

hq G+!!

ü Such process is called umklapp process, a large angle

scattering. The direction of phonon movement is changed

greatly, making a shorter mean free path, and give rise to

some finite thermal resistivity.


q q1 2+! !

50

2. Thermal ConductivitySuppose different temperatures at two ends of the rod: T1, T2

12 TT >Heat flow, following the temperature gradient, move from

the hot end to the cold one, with energy density proportional

to the temperature gradient.

xTJq d

dk-=

where k denotes thermal conductivity coefficient.

Thermal conductance: electrons (metal), phonon (insulator)


Fourier's law

T2 T1

l Atoms are vibrating more

strongly in the hot (left) end

than those near the cold (right)

end, leading to a larger

density of phonons.

l Phonons move from left to

right, carrying heat, and flow

along the opposite direction of

temperature gradient.

ü Apply the gas molecule

dynamic theory to the phonon

gas, thermal conductivity

reads：

lCVuk 31

=

u —phonon velocity; l —mean free path, mean distance a phonon

moves between two successive collisions with other phonons.


52

Three mechanisms:

Phonon-phonon scattering due to anharmonic couplings,

especially important at high tempertures.

(1) Collision between phonons

Mean phonon number:

High T: DΘT >>

TTke

nq

BTkq Bq

µ»-

=ww !! 1

1

Collision probability proportional to # of phonons,

Corresponding mean free path thus is inverse proportional to

temperatures:T

l 1~


53

(2) Scattering of phonon with crystal defects

(3) scattering with the boundaries of specimen

Impurities and defects also scatter phonons, since they partially

break the lattice periodicity. The larger the mass difference and

density of impurities are, the stronger the scattering is, and l

becomes shorter.

At very low temperatures, (1,2) collisions are scarce.

Ø only few phonons exist, dilute phonon gas

Ø phonons with long wavelength at low temperatures, which

cannot be effectively scattered by objects like impurities of much

smaller size.


54

At low temperatures, the main mechanism is the boundary

scattering. It cause some geometric effects, since the phonon has

very long wavelength, which is comparable to the size of

specimen. Mean free path l = L, independent of temperatures.

lCVuk 31

=Remenber the thermal conductivity:

At low T, thermal conductivity determined by heat capacity3~ Tk

At high T, thermal conductivity determined by mean free path l

T1~k


Documents

Chapter 5Crystal Thermodynamics · Chapter 5Crystal Thermodynamics. 2 5.1 phonon heat capacity I. Density of states Consider an atomic chain consisting of N lattice sites: l L l Na