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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.
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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
