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## Blackbody Radiation

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**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**Wiens’ law Planck’s 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. He3is a Fermion so that CV ~ T ( see § 8.1 ). l-He4is 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 Tl-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 : where or**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 TT :**Neutron Scattering**Excitation of energy = pc 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 A1 : with c ~ 237 m/s Landau thought this was related to rotations and called the related quanta rotons. Bose gas with N const For T ≤ 2K, Predicted by Pitaevskii **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**0.3K0.6K1K 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: vCdepends 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