1. Blackbody Radiation
Wien’s displacement law :
Stefan-Boltzmann law :

2. 7.3. Thermodynamics of the Blackbody Radiation
2 equivalent point of views on radiations in cavity :
Planck : Assembly of distinguishable harmonic oscillators
with quantized energy
2. Einstein : Gas of indistinguishable photons with energy

3. Planck’s Version
Oscillators : distinguishable  MB statistics with  quantized
From § 3.8 :
Rayleigh expression 
= density of modes within ( ,  + d ) 
= energy density within ( ,  + d )
Planck’s formula

4. Einstein’s Version
Bose : Probability of level s ( energy = s ) occupied by ns photons is
Boltzmannian  
(av. energy of level s )
= volume in phase space for
photons within ( ,  + d )
Einstein : Photons are indistinguishable ( see § 6.1 with N not fixed so that  = 0 )
Oscillator in state ns with E = ns  s .
= ns photons occupy level s of  =  s . 

5.  
Dimensionless
Long wavelength limit ( ) : 
Rayleigh-Jeans’ law
Short wavelength limit ( ) :
Wiens’ (distribution) law
[ dispacement law + S-B law ]

6. Blackbody Radiation Laws
Planck’s law
Wiens’ law
Rayleigh-Jeans’ law
Wiens’ displacement law

7. Stefan-Boltzmann law  From § 6.4 , p’cle flux thru hole on cavity is
Radiated power per surface area is obtained by setting
so that
Stefan-Boltzmann law
Stefan const.

8. Grand Potential
Bose gas with z = 1 or  = 0 ( N  const ) :  

9. Thermodynamic Quantities 
Adiabatic process ( S = const ) 
 For adiabats : or

10.    
Caution:

11. 7.4. The Field of Sound Waves
2 equivalent ways to treat vibrations in solid :
Set of non-interacting oscillators (normal modes).
Gas of phonons.
N atoms in classical solid :
“ 0 ” denotes equilibrium position. 
Harmonic approximation :

12. Normal Modes
Using { i } as basis, H is a symmetric matrix  always diagonalizable.
Using the eigenvectors { qi } as basis, H is diagonal.
 = characteristic frequency
of normal mode .
System = 3N non-interacting oscillators.
Oscillator  is a sound wave of frequency  in the solid.
Quantum mechanics :
System = Ideal Bose gas of {n } phonons with energies {   }.
Phonon with energy   is a sound wave of frequency  in the solid.

13. U, CV
Difference between photons & phonons is the # of modes ( infinite vs finite )
# of phonons not conserved   = 0 
Note: N is NOT the # of phonons; nor is it a thermodynamic variable.  
Einstein function

14. Einstein Model Einstein model :  High T ( x << 1 ) :
( Classical value )
Mathematica
Low T ( x >> 1 ) :
Drops too fast.

15. Debye Model Debye model : = speed of sound
Polarization of accoustic modes in solid : 1 longitudinal, 2 transverse.   

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

17. Debye Function 
Debye function  

18. Mathematica
T >> D ( xD << 1 ) :
T << D ( xD >> 1 ) :
Debye T3 law

19. Debye T 3 law
KCl

20. Liquids & the T 3 law
Solids: T 3 law obeyed  Thermal excitation due solely to phonons.
Liquids:
No shear stress  no transverse modes.
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.
Longitudinal modes only 
Specific heat (per unit mass)
Mathematica 

21. 7.5. Inertial Density of the Sound Field
Low T l-He4 : Phonon gas in mass (collective) motion ( P , E = const )
From §6.1 :
with
 extremize
Bose gas :  

22. Occupation Number Let and = drift velocity For phonons :
c = speed of sound
Phonon velocity 

23. Let  
Mathematica 

24. Galilean Transformation
General form of travelling wave is :
Galilean transformation to frame moving with v :   or
where

25. 
In rest frame of gas :
( v = 0 )
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.  
B-E distribution is derived in rest frame of gas. 

26.    

27. P
where 
Mathematica 

28. E 
Mathematica

29.  
Inertial Mass density
For phonons,   
l-He4 : 

30. n / 
rotons
T 5.6
phonons
◦ Andronikashvili viscosimeter,
• Second-sound measurements
Second-sound measurements
Ref: C. Enss, S. Hunklinger, “Low-Temperature Physics”,Springer-Verlag, 2005.

31. 2nd Sound
1st sound :
2nd sound :

32. 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 :
Good for T < 2K
At T < T :
At T  T :

33. Neutron Scattering
Excitation of energy  = p c created by neutron scattering. f i
Energy conservation :  p
Momentum conservation :
Roton near
Speed of sound = 238 m/s

34. Rotons Excitation spectrum near k = 1.92 A1 : with
c ~ 237 m/s
Landau thought this was related to rotations and called the related quanta rotons.
Bose gas with N  const 
Predicted by Pitaevskii
For T ≤ 2K, 

35. Thermodynamics of Rotons 

36. F, A
For T ≤ 2K 
Mathematica 
 = 0 

37. S, U, CV 

38.  From § 7.5, Ideal gas with drift v :  By definition of rest frame : 
Good for any spectrum & statistics

39. Phonons  
Same as § 7.5

40. Rotons   

41. mrot 0.3K 0.6K 1K Phonons | both | Rotons ~ normal fluid At T = 0.3K,
Mathematica
Assume TC is given by 
c.f. 
Landau : 

42. vC
Consider an object of mass M falling with v in superfluid & creates excitation (e , p) .  
for M large 
i.e., no excitation can be created if
Landau criteria
vC = critical velocity of superflow
Exp: vC depends on geometry ( larger when restricted ) ; vC  0.1 – 70 cm/s

43. Ideal gas : 
( No superflow )
Superflow is caused by non-ideal gas behavior.
E.g., Ideal Bose gas cannot be a superfluid.
Phonon : 
for l-He
Roton :   
c.f. observed vC  0.1 – 70 cm/s
Correct excitations are vortex rings with