Blackbody Radiation

Wien’s displacement law : max

b

T

Stefan-Boltzmann law :4*j T

7.3. Thermodynamics of the Blackbody Radiation

2 equivalent point of views on radiations in cavity :

s s sn

1. Planck : Assembly of distinguishable harmonic oscillators

with quantized energy

0,1,2,sn

2. Einstein : Gas of indistinguishable photons with energy s s

Planck’s Version

Oscillators : distinguishable MB statistics with quantized

/ 1s

ss k Te

From § 3.8 :

3

3, 2

2pol

Vd k

k

0

V d

22

0

Vd k k

22 3

0

Vd

c

2

2 3c

= density of modes within ( , + d ) Rayleigh expression

ck

u

3

2 3 / 1k Tu

c e

= energy density within ( , + d )

Planck’s formula

0 0

Ud d u

V

Einstein’s Version

Bose : Probability of level s ( energy = s ) occupied by ns photons is

s

s s sn

n n p n

1s sn

sp n eZ

0

s s

s

n

n

Z e

/ 1s

sk Te

1

1 se

1 ln

s

Z

1

s

s

e

e

1

1se

s

s s s sn

n p n s sn (av. energy of level s )

2

2 3g d V d

c

= volume in phase space for photons within ( , + d )

Boltzmannian

Einstein : Photons are indistinguishable ( see § 6.1 with N not fixed so that = 0 )

1

1n

e

Oscillator in state ns with E = ns s .

= ns photons occupy level s of = s .

3

2 3 / 1k Tu d d

c e

xkT

3

1x

xu

e

4 3

2 3 1x

kT xd x

c e

u x d x

4 3

2 3 1x

kT xu x

c e

4

2 3

kTu x

c

Dimensionless

Long wavelength limit ( ) : 2u x1h c

xkT

Short wavelength limit ( ) : 3 xu x e1h c

xkT

22 3

kTu

c

Rayleigh-Jeans’ law

u u xkT

3

2 3

kTu x

c

Wiens’ (distribution) law[ dispacement law + S-B law ]

3 /2 3

k Tu ec

kT

Blackbody Radiation Laws

3

1x

xu

e

2u x3 xu x e Rayleigh-

Jeans’ lawWiens’

lawPlanck’s

law

41

4 6.493! 15

1

10

1

1x

xg z d x

z e

3! 6 0

d x u

max

b

T

Wiens’ displacement law

32.8977685 10 m Kb

Stefan-Boltzmann law

0

Ud x u x

V

4

2 3

kTu x u x

c

2 44

3 315

kT

c

Radiated power per surface area is obtained by setting

2 44

2 3

1

4 60U

U kR c T

V c

2 48 2 4

2 35.670 10

60

kWm K

c

Stefan-Boltzmann law

Stefan const.

From § 6.4 , p’cle flux thru hole on cavity is 1

4R n u

3

1x

xu

e

4

0 15d x u

&U

n u cV

4Tso that

44UT

V c

Grand Potential

Bose gas with z = 1 or = 0 ( N const ) :

/, ln ln 1 k TF T V kT kT e PV

Z

/

1

1 k Te

Z

2

3, 0

2 42pol

Vd k k

k

22 3 3

0

Vd

c

2 /2 3 3

0

ln 1 k TV kTF d e

c

3 /

3 /2 3 3 /0

0

1 1ln 1

3 1

k Tk T

k T

V kT ee d

c kT e

3

2 3 3 /03 1k T

Vd

c e

4 3

2 3 303 1x

V kT xd x

c e

4

2 3 33! 4

3

V kT

c

1

10

1

1x

xg z d x

z e

4 2

2 3 345

V kT

c

2 44

3 315

U kT

V c

1

3U

3

UP

V

kc

Thermodynamic Quantities

3 3U F PV

A F N

F PV

A F PV

A U T S U AS

T

1

3U

4

3

U

T

44UT

V c

316

3V T

c 3V T

,

V

V N

SC T

T

3 S

Adiabatic process ( S = const ) 3V T const

41 4

3 3

UP T

V c 4T For adiabats :

1

3

UP

V

3/4V P const

4/3V P constor

3U PV

4H U PV PV

4P

P P

H VC P

T T

44

3P T

c

4P T d P

T dT P

3/41/4

4

P P

V VP

T P

PC

4P

T P

3/4

1/4

4P

P

1

1n

e

N n

2

2 3 30 1

Vd

c e

22 3 3

0

Vd

c

1

10

1

1x

xg z d x

z e

3 2

2 3 30 1x

V kT xd x

c e

3

2 3 32! 3

V kT

c

3T 2

TN

Caution:

T

7.4. The Field of Sound Waves

2 equivalent ways to treat vibrations in solid :

1.Set of non-interacting oscillators (normal modes).

2.Gas of phonons.

N atoms in classical solid :

2

1

1

2

N

II

K m

x3

2

1

1

2

N

ii

m x

3i I ax with i I a x

32

1

1

2

N

ii

m

0i i ix x

“ 0 ” denotes equilibrium position.

0I I I x x ξ

2

0,

0

1

2i i ji j i j

xx x

0

0ix

Harmonic approximation : 2

,

1 1

2 2i i j i ji i j

H m k 2

0

i ji j

kx x

0 0

Normal Modes2

,

1 1

2 2i i j i ji i j

H m k

Using { i } as basis, H is a symmetric matrix always diagonalizable.

Using the eigenvectors { qi } as basis, H is diagonal.

3

2 2 2

1

1

2

N

H m q q

= characteristic frequency of normal mode .

System = 3N non-interacting oscillators.

Oscillator is a sound wave of frequency in the solid.

Quantum mechanics :

232 2

1

1

2 2

N pH m q

m

3

1

1

2

N

E n n

System = Ideal Bose gas of {n } phonons with energies { }.

Phonon with energy is a sound wave of frequency in the solid.

U, CV

3

1

1

2

N

E n n

Difference between photons & phonons is the # of modes ( infinite vs finite )

3

1

1

2

N

E n

/ 23

1 1

N e

e

Z

lnU

Z

# of phonons not conserved = 0

3 3

1 1

1

2 1

N N

e

lnF kT Z 3

1

1ln 1

2

N

kT e

3 3

1 1

1ln 1

2

N N

kT e

Note: N is NOT the # of phonons; nor is it a thermodynamic variable.

1

1n

e

V

V

UC

T

3

2 21 1

N e

kTe

23

21 1

xN

x

x ek

e

xkT

3

1

N

VC k E x

2

21

x

x

x eE x

e

Einstein function

Einstein Model

3

1

N

VC k E x

2

21

x

x

x eE x

e

Einstein model : E

xkT

3V EC N k E x EEx

kT

E

T

High T ( x << 1 ) :

2

2

1x xE x

x

1

3VC N k ( Classical value )

Low T ( x >> 1 ) : 2 xE x x e

2

3 expE EVC N k

kT kT

Drops too fast.

Mathematica

Debye Model 3

1

N

VC k E x

2

21

x

x

x eE x

e

Debye model : 3 3

1 1 1 0

3DN N

polpol pol

d g N

xkT

= speed of sound

max

23

0

42

kV

d k k

k

max

22 3

02

Vd

c

Polarization of accoustic modes in solid : 1 longitudinal, 2 transverse.

22 3 3

0

1 23

2

D

L T

Vd N

c c

ck

32 3 3

1 2 1

2 3 DL T

V

c c

1

3 23 3

1 218D

L T

N

V c c

2 2

2 3 3 3

1 2 9

2

0

DL T D

D

V N

g c c

23

9D

D

Ng

1 0

0 0

xx

x

Refinements

22 3 3

0

1 23

2

D

L T

Vd N

c c

can be improved with

,

22 3

0

1

2

D L

L

Vd N

c

3,2

1

2 3 D L

V

3 2 3, 6D L L

Nc

V

,

22 3

0

22

2

D T

T

Vd N

c

3,2

2

2 3 D T

V

3 2 3, 6D T T

Nc

V

1/32

max 6N

kV

1/3

min

4

3

V

N

Optical modes ( with more than 1 atom in unit cell )can be incorporated using the Einstein model.

Al

Debye Function 3

1

N

VC k E x

2

21

x

x

x eE x

e

xkT

23

9D

D

Ng

3

1 0

DN

d g

3 4

20

91

Dx x

V xD

kT x eC N k d x

e

3 DN k D x

4

230

3

1

Dx x

D xD

x eD x d x

x e

Debye function

2

1

1 1

x

x x

d e

d x e e

4 3

300

34

1 1

D Dx x

D x xD

x xD x d x

x e e

43

0 0lim lim 0

1xx x

xx

e

4 3

30

34

1 1

D

D

x

DD x x

D

x xD x d x

x e e

4 4 3

23 30 0

3 34

1 11

D D

D

x xxD

D x xxD D

x e x xD x d x d x

x x e ee

3V DC N k D x

D DDx

kT T

Mathematica

DD k

T >> D ( xD << 1 ) :

4 6

23

0

3

12 240

Dx

DD

x xD x d x x

x

2

3 3

12 43! 4

5D Dx x

2 4

120 480

D Dx x

T << D ( xD >> 1 ) : 3

30

12

1D xD

xD x d x

x e

324

5 D

T

3212

5VD

TC N k

3

1 146.89D

Tcal mole K

Debye T3 law

AN N

Debye T 3 law

KCl

233 3D K

Liquids & the T 3 law

Solids: T 3 law obeyed Thermal excitation due solely to phonons.

Liquids:

1. No shear stress no transverse modes.

2. Equilibrium points not stationary

vortex flow / turbulence / rotons ( l-He4 ),....

3. He3 is a Fermion so that CV ~ T ( see § 8.1 ).

l-He4 is the only liquid that exhibits T 3 behavior.

321 12

3 5VD

TC N k

32

,

4

5 D L

kTN k

Longitudinal modes only

3 2 3, 6D L

Nc

V

43

3 3

2

15V

V

C kc T

M c

Specific heat (per unit mass)

43

3 3

2

15

kV T

c

30.1455 / , 238 /g cm c m s

3 1 1

3 1 1, exp

0.02079

0.0204 0.0004

V

V

c T joule g K

c T joule g K

Mathematica

7.5. Inertial Density of the Sound Field

Low T l-He4 : Phonon gas in mass (collective) motion ( P , E = const )

lnL W n n n E

γ p P

n E

*lnS k W n From §6.1 : with n

p P

extremize

1 !ln ln

! 1 !

n gW n

n g

Bose gas :

ln 1 ln 1g n

n gn g

ln 1 0g

n

γ p

1

1

n

g e

γ p 1

1n

e γ p

p p

Occupation Number 1

1n

e γ p

p

Let ˆPP z and ˆvv z = drift velocity

Phonon velocity p p

kc pc

ˆc pku

For phonons : p k c = speed of sound

ˆ cosv c u P

cosn

cN

p

p N np

p

22

30 0 0

sin2

Vd d d p p

p

ˆγ z 1

22 3 cos

1 0

1cos

4 1p p c

VN d d p p

e

12

2 3 cos1 0

coscos

4 1p p c

c Vv N d d p p

e

12

2 3 cos1 0

1cos

4 1p p c

VN d d p p

e

12

2 3 cos1 0

coscos

4 1p p c

c Vv N d d p p

e

Let cos 1p pc

cos 1d p d pc

3

2 2cos

0 0

1 1cos 1

1 1p p c c pd p p d p p

e c e

13

11

3

1

d c

v c

d c

cos

22 2 2

22 2 2

2

2

cc

c c

Mathematica

ˆ ˆv γ z z v

Galilean Transformation

t k xGeneral form of travelling wave is :

t x x vGalilean transformation to frame moving with v : t t

t k x k v

t t k x v

k v

or p v

p kwhere

k k

t t k x k x

ˆ ˆv γ v z z 1

1n

e γ p

p

1

1n

e

p v pp In rest frame of gas :

( v = 0 ) 00 0

1

1n

e p

p

In lab ( x ) frame : phonon gas moves with av. velocity v.

Dispersion (k) is specified in the lab frame where solid is at rest.

Rest frame ( x ) of phonons moves with v wrt x-frame.

rest lab k k v rest p p v

0n np p1

1reste

1

1e

p v p

B-E distribution is derived in rest frame of gas.

1

1n

e v p

p

0 p v p

1 cos 0v

p pc

cos

cv

v c

0n p

cosp c v p ˆvv z

P 1

1n

e v p

p

np

P p p

22

3 cos0 0 0

sin cos , sin sin , cossin

12v p

pVd d d p p

e

ˆvv z

1v

p pc

0,0, zP

1 3

31 0

0 , 0 , 2

12v p

pVd d p

e

1 3

2 31 04 1

z v p

V pP d d p

e

where

41 3

2 31 0

14 1z c p

V v pP d d p

c e

32 4

2 3 2 4 4

81

4 3 15z

V v vP

c c c

32 2

3 4 4 2

21

45

v vV

c c c

Mathematica

cos

32 2

3 4 4 2

21

45

vV

c c c

vP

E

E n pcp

p

22

3 cos0 0 0

sin12

v p

V pcd d d p p

e

ˆvv z

1

1n

e v p

p

1 3

2 31 04 1v p

V c pd d p

e

cos

1v

p pc

41 3

2 31 0

14 1c p

V c v pd d p

c e

2

2 4

32 3 4 42

2

2 13

4 151

vcV c

cvc

2

2 2

33 3 4 2

2

13

301

vE cV c v

c

Mathematica

32 2

3 4 4 2

21

45

vV

c c c

vP

M v

32 2

3 5 4 2

21

45

M v

V c c

Inertial Mass density

For phonons,2 4

43 5

2

45ph

kT

c

v c

2

2 2

33 3 4 2

2

13

301

vE cV c v

c

2

3 3 430

E

V c

2

4

3ph

E

c V

30.1455 / , 238 /g cm c m s l-He4 :4 41.22 10ph

He

T

normal

He

n /

◦ Andronikashvili viscosimeter, • Second-sound measurements

Second-sound measurements

phonons

rotons

T 5.6

Ref: C. Enss, S. Hunklinger, “Low-Temperature Physics”,Springer-Verlag, 2005.

2nd Sound

1st sound :

0

0

n s

S

T

v v2nd sound :

22

0n n s s

s

n

Tv S

S

v v1

S

pv

7.6. Elementary Excitations in Liquid Helium II

Landau’s ( elementary excitation ) theory for l-He II :

Background ( ground state ) = superfluid.

Low excited states = normal fluid

Bose gas of elementary excitation.

At T = 0 : 0n s He

At T < T : s He nT T

At T T :

0s T n HeT

pc Good for T < 2K

2.19T K

Neutron Scattering

Energy conservation :

2

2 2

1 1

2 n i f

h

m

Excitation of energy = p c created by neutron scattering.

Momentum conservation :2 2

2 2

1 1 12 cos

i f i f

p h

if

p

239 5 /c m sp

Speed of sound = 238 m/s

1max 14 1.11p

K at Ak

11.92p

A

Roton near

Rotons

Excitation spectrum near k = 1.92 A1 :

2

0

1

2p p

p with

10

8.65 0.04

1.92 0.01

0.16 0.01 He

Kkp

A

m

Landau thought this was related to rotations and called the related quanta rotons.

For T ≤ 2K,

76k Te

n e pp

1

1n

e p

pBose gas with N const

Predicted by Pitaevskii

c ~ 237 m/s

Thermodynamics of Rotons 2

0

1

2p p

p

n e pp

1

1 e p

p

Z

lnF kT Z ln 1kT e p

p

kT e p

p kT n

p

p

0

F kT N U TS PV

PVN

kT

22

030

14 exp

22

Vd p p p p

2

0

3/2 2

02 32

2x

x

Ve kT d x x x e

0

2

p px

kT

00

2

px

kT

F, A

For T ≤ 2K 0 8.4x 2

21/2 2

02 30

2 12

xPV V xe kT p d x e

kT x

2

0

3/2 2

02 32

2x

x

PV Ve kT d x x x e

kT

0

02

px

kT

2

0

0 1

/ 2 2

n x

n

d x x e n

n

1/2/ 202 3 2

0

12 1

2 2k TPV V

e kT pkT x

2

1/2/02 3

22

k TPV V pN e kT

kT

Mathematica

= 0 A F N F PV N kT 3/2 / k TT e

S, U, CV

V

AS

T

2

3

2A

T kT

3

2N k

kT

1

2U A TS N kT

kT

1

2kT N

2

1 1 1+

2 2 2V

V

UC k N kT N

T kT T

3/2 / k TA N kT T e

2

1/2/02 3

22

k TPV V pN e kT

kT

2

1

2V

NN

T kT T

23

4N k

kT kT

From § 7.5, Ideal gas with drift v : n n vp p v p

n M vp

P p p v

1

4 23

1 0

2

2

nd d p p

ˆ ˆ cos p v

1M

V V

P

v 1

ˆnv V

vp

p p v

0 0

1ˆlim

vn

v V

p

v p p v 0

1ˆlim

v

nn

v V

p

v p p v

By definition of rest frame : 0 0n p

P p p

2

0

1ˆ

n

V

p

v p

40 2 3

0

1

6

nd p p

Good for any spectrum & statistics

Phonons 4

0 2 30

1

6

nd p p

1

1p cn n p

e

40 2 3

0

1

6

n pd p p

c p

4 32 3 0

0

14

6p n p d p p n p

c

2 30

2 3

2 2

3

E

c c V

02

4

3

E

c V

Same as § 7.5

p c

Rotons 4

0 2 30

1

6

nd p p

n e

40 2 3

06d p p n p

n

n

2 32

2 3

2

6N p

V

21

3

Np

kT V

20

1

3

Np

kT V

2

1/2/02 3

22

k TV pN e kT

1/24

/00 2 3

2

6k T

rotons

pe

kT

mrot

0.3K 0.6K 1KPhonons | both | Rotons ~ normal fluid

Assume TC is given by rot He 2.81T K 2.19T K c.f.

MathematicaAt T = 0.3K, 11 6 5, , 10 ,10 ,10ph rot rot

He He ph

20 0

1

3

Np

kT V

20

1

3rot

Mm p

N kT

15 1

10 1.5rot He

rot He

m m at T K

m m at T K

Landau :

0

2

00~

2

c k k k

k kk k

3

3/2 /V kT

T T TC

T e T T

0.16 Hem

vC

Consider an object of mass M falling with v in superfluid & creates excitation ( , p) .

M M v v p2 21 1

2 2M v M v

2

2

p

M v p

cos

vp

p

i.e., no excitation can be created if minCv vp

Landau criteria

vC = critical velocity of superflow

Exp: vC depends on geometry ( larger when restricted ) ; vC 0.1 – 70 cm/s

v p for M large

2

2m

pp min min 0

2C

pv

p m

( No superflow )

Superflow is caused by non-ideal gas behavior.

E.g., Ideal Bose gas cannot be a superfluid.

Phonon :

Ideal gas :

p c p Cv c

Roton : 2

0

1

2p p

p

42.4 10 /cm s for l-He

20

2 2

11 0

2

d p

dp p p p

20 2Cp p 2

0 0

12Cv p p

0p

36.3 10 /cm s

c.f. observed vC 0.1 – 70 cm/s

Correct excitations are vortex rings with p