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

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

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

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 . 

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

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

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.

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

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

    Caution:

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 :

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.

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

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

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

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

Debye Function  Debye function  

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

Debye T 3 law KCl

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 

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 :  

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

Let   Mathematica 

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

 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. 

   

P where  Mathematica 

E  Mathematica

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

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.

2nd Sound 1st sound : 2nd sound :

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 :

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

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, 

Thermodynamics of Rotons  

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

S, U, CV 

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

Phonons   Same as § 7.5

Rotons   

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

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

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