The Black Body Radiation = Chapter 4 of Kittel and Kroemer The Planck distribution Derivation Black Body Radiation Cosmic Microwave Background The genius of Max Planck Other derivations Stefan Boltzmann law Flux => Stefan- Boltzmann Example of application: star diameter Detailed Balance: Kirchhoff laws Another example: Phonons in a solid Examples of applications Study of Cosmic Microwave Background Phys 112 (S09) 6 Black Body Radiation short 1
The Planck Distribution*** Photons in a cavity Mode characterized by Number of photons s in a mode => energy Frequency ν Angular frequency ω = 2πν ε = shν = s ω 1st Quantification Wave like Massive Particles Discrete states Photons Obvious Occupation number 2nd Quantification Discrete particles Radiation energy density between ω and ω+dω? s = 1 exp ω τ 2 different polarizations ω 3 dω u ω dω = π 2 c 3 exp ω τ 1 Obvious 1 = u ν dν = Planck # photons / mode c 3 ε ω = 8πhv 3 dv exp hν τ 1 ω ( exp ω ) τ 1 Phys 112 (S09) 6 Black Body Radiation short 2
Cosmic Microwave Radiation Big Bang=> very high temperatures! When T 3000K, p+e recombine =>H and universe becomes transparent Redshifted by expansion of the universe: 2.73K Phys 112 (S09) 6 Black Body Radiation short 3
WMAP 5 years 3/5/08 Power Spectrum of temperature fluctuations TT TE EE EE Fitted Cosmological Parameters Phys 112 (S09) 6 Black Body Radiation short 4 Correlation Power Spectra of temperature fluctuations and E polarization
Other derivations Grand canonical method ( ) What is µ? s = Microcanonical method N = τ logz µ = 1 exp ω µ 1 τ γ + e e µ γ + µ e µ e = 0 µ γ = 0 To compute mean number photons in one mode, consider ensemble of N oscillators at same temperature and compute total energy U We have to compute the multiplicity g(n,n) of number of states with energy U= number of combinations of N positive integers such that their sum is n 1 τ = σ U = 1 N ω log 1 + = 1 ω U ω log 1 + 1 s s = 1 exp ω τ 1 Phys 112 (S09) 6 Black Body Radiation short 5
Fluxes Energy density traveling in a certain direction So far energy density integrated over solid angle. If we are interested in energy density traveling traveling in a certain direction, isotropy implies u ω ω ( 3 dωdω θ,ϕ )dωdω = 4π 3 c 3 exp ω 1 τ 2hν ( 3 dνdω θ,ϕ )dνdω = c 3 exp hν 1 τ Note if we use ν instead of ω Flux density in a certain direction=brightness (Energy /unit time, area, solid angle,frequency) opening is perpendicular to direction I ν dνdadω = 1 dt dν u cdtda ν dω = 2hν 3 dνdadω energy in c 2 exp hν 1 cylinder cdt τ Flux density through a fixed opening (Energy /unit time, area,frequency) J ν dνda = 1 dt dν 2π dϕ 1 u ν cdt cosθda d cosθ = 0 0 Phys 112 (S09) 6 Black Body Radiation short 6 u ν energy in cylinder! c 2 2πhν 3 dνda exp hν = c 4 u ν dvda 1 τ cdt da da θ n
Total Energy Density Integrate on ω Stefan-Boltzmann Law u = π 2 15 3 c 3 τ 4 = a B T 4 with a B = π2 k 4 B 15 3 c 3 Total flux through a fixed aperture*** multiply above result by c/4 π 2 J = 60 3 c 2 τ 4 = σ B T 4 with σ B = π2 k 4 B 60 3 c 2 = 5.67 10 8 W/m 2 /K 4 Stefan-Boltzmann constant! Phys 112 (S09) 6 Black Body Radiation short 7
Detailed Balance: Kirchhoff law Definition: A body is black if it absorbs all electromagnetic radiation incident on it Usually true only in a range of frequency e.g. A cavity with a small hole appears black to the outside Detailed balance: in thermal equilibrium, power emitted by a system=power received by this system! Otherwise temperature would change! Consequence: The spectrum of radiation emitted by a black body is the black body spectrum calculated before u BB ( ω)dω = u cavity ( ω)dω Absorptivity, Emissivity: Absorptivity =fraction of radiation absorbed by body Black Body Cavity Black Body Emissivity = ratio of emitted spectral density to black body spectral density. Grey Body Kirchhoff: Emissivity=Absorptivity a( ω) = e( ω) Phys 112 (S09) 6 Black Body Radiation short 8
Entropy, Number of photons Entropy σ = σ ω d 3 x ω 2 dω π 2 c 3 σ σ = 4 ω = p s log p s 3 s u τ V = 4 3 a B k B VT 3 Number of photons N γ = V2ζ ( 3) π 2 c 3 3 τ 3 = 30ζ ( 3) a B π 4 VT 3 0.37 a B VT 3 k B k B Proportional σ 3.6N γ Phys 112 (S09) 6 Black Body Radiation short 9
Spectrum at low frequency log I v d Rayleigh-Jeans region (= low frequency) A e Ω e Ω r A r if ω << τ ε u ω dω τ ω 2 dω u π 2 c 3 ν dν τ 8πν 2 ω τ dν c 3 Power / unit area/solid angle/unit frequency =Brightness 2hν 3 dνdadω I ν dνdadω = c c 3 exp hν τ 2ν 2 dνdadω = τ 2dνdAdΩ c 2 λ 2 Power emitted / unit (fixed) area τ 1 J ω dω = J ν dν τ ω2 dω 4π 2 c 2 = τ 2πdν λ 2 Power received from a diffuse source A e = Ω r d 2 Ω e = A r d 2 If telescope is diffraction limited: Ω r A r = λ 2 Detected Power (1 polarization) Antenna temperature Phys 112 (S09) 6 Black Body Radiation short 10 log v dp dν = I νω e A e = 2τΩ ea e λ 2 dp dν = 2τ = 2τΩ ra r λ 2 dp dν (1polarization) = τ T A = 1 dp k B dν (1polarization)
Applications Many! e.g. Star angular diameter Approximately black body and spherical! Spectroscopy =>Effective temperature Apparent luminosity l =power received per unit area A r Ω e d Ω e = A r d 2 l = LΩ e 4π 1 A r = L 4πd 2 L = 4πr 2 4 σ B T eff But power output l = r 2 d 2 σ 4 BT eff angular diameter = r d = l 4 σ B T eff => Baade-Wesserlink distance measurements of varying stars Oscillating stars (Cepheids, RR Lyrae) Supernova assuming spherical expansion Phys 112 (S09) 6 Black Body Radiation short 11
Phonons: Quantized vibration of a crystal described in same way as photons If s phonons in a mode Same as for photons but 3 modes Phonons in a solid ε = ω p = k = ω ε s = s ω 1 c s s = exp ω 1 τ Maximum energy (minimum wavelength/finite # degrees of freedom) = Debye Energy Debye approximation: ω ω D = isotropic k= ω c s or p= ε c s with velocity c s = constant Introducing k the Debye temperature T D = θ = c s U = 3 5 π4 k B N T 4 3 C T V = 12 D 5 π4 k B N T T D 3 k B 6π 2 N V 1/3 = ω D k B σ = 12 15 π4 k B N T T D Velocity 3 Phys 112 (S09) 6 Black Body Radiation short 12
Applications Calorimetry: Measure energy deposition by temperature rise ΔE ΔT = ΔE C need small C Heat Capacity Bolometry: Measure energy flux F by temperature rise ΔT = ΔF G Sky Load Chopping e.g., between sky and calibration load ΔF need small G but time constant= C limited by stability small heat capacity C G Heat Conductivity Very sensitive! Heat capacity goes to zero at low temperature Study of cosmic microwave background Search for dark matter particles C T 3 Phys 112 (S09) 6 Black Body Radiation short 13