The Black Body Radiation

Transcription

1 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 Search for Dark Matter Phys 2 (S06) 5 Black Body Radiation short

2 The Planck Distribution*** Photons in a cavity Mode characterized by Occupation number Number of photons s in a mode => energy st Quantification Wave like 2nd Quantification Discrete particles Frequency! Angular frequency! = 2"# Radiation energy density between ω and ω+dω? 2 different polarizations s = # exp% h! $ " Massive Particles Discrete states Obvious & ( ) ' Photons Obvious Planck # photons /mode! = sh" = sh#! " = h" ( exp h" ) # $ u! d! = h! 3 d! $ " 2 c 3 $ exp h! % & % & # ' ( ) * ' ( ) = u + d+ = 8"hv 3 dv $ c 3 $ exp h+ ' ' % & # ( ) * % & ( ) Phys 2 (S06) 5 Black Body Radiation short 2

3 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 2 (S06) 5 Black Body Radiation short 3

4 Grand canonical method Other derivations s = N =! ( " logz ) =!µ % exp h# $ µ ( ' * $ & " ) What is µ?! + e " e µ! + µ e " µ e = 0 # µ! = 0 Microcanonical method 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! = "# "U = % log' Nh$ ( + * = h$ & U ) h$ log % ' + & s ( * ) s = # exp% h! $ " & ( ) ' Phys 2 (S06) 5 Black Body Radiation short 4

5 Phys 2 (S06) 5 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 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! d!dad" = u! (",# )d!d$ = u! (",# )d!d$ = h! 3 d!d$ ' ' 4% 3 c 3 exp h! * * ) ),-, ( ( & + + 2h! 3 d!d$ & & c 3 exp h! ) ) ( ( +,+ ' ' % * * dt d! u cdtda! d" = 2h! 3 d!dad" $ $ energy in c 2 exp h! ' ' & & )*) cylinder % % # ( ( Flux density through a fixed opening (Energy /unit time, area,frequency) J! d!da = dt d! 2% d" $ $ u! cdt cos#da d cos# = 0 0 energy in cylinder! cdt 2%h! 3 d!da c 2 exp h! = c ' ' * * 4 u dvda! ) ),-, ( ( & + + cdt da da θ n

6 Stefan-Boltzmann Law Total Energy Density Integrate on ω u =! 2 5h 3 c 3 " 4 = a B T 4 with a B =!2 k 4 B 5h 3 c 3 Total flux through a fixed aperture*** multiply above result by c/4 J =! 2 60h 3 c 2 " 4 = # B T 4 with # B =!2 k B 4 60h 3 c 2 = $8 W/m 2 /K 4 Stefan-Boltzmann constant! Phys 2 (S06) 5 Black Body Radiation short 6

7 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 (!)d! = u cavity (! )d! Absorptivity, Emissivity: Absorptivity =fraction of radiation absorbed by body Emissivity = ratio of emitted spectral density to black body spectral Black Body Cavity Detailed Balance: Kirchhoff law density. Grey Body Black Body Kirchhoff: Emissivity=Absorptivity a (!) = e (!) Phys 2 (S06) 5 Black Body Radiation short 7

8 Entropy, Number of photons Entropy! = #! " d 3 x " 2 d"! " = # $ p s log p s s $ 2 c 3! = 4 3 u " V = 4 a B VT 3 3 k B Number of photons Proportional N! = V2" ( 3 ) # 2 c 3 h 3 $ 3 = 30" ( 3 ) a B # 4 VT 3 % 0.37 a B VT 3 k B k B! " 3.6N # Phys 2 (S06) 5 Black Body Radiation short 8

9 Spectrum at low frequency log I v Rayleigh-Jeans region (= low frequency) Power / unit area/solid angle/unit frequency =Brightness Power emitted / unit (fixed) area A e =! r d 2 Power received from a diffuse source d A e! e! r! e = A r d 2 If diffraction limited:! r A r = " 2 if h! << " #! $ " Detected Power ( polarization) Antenna temperature log v u! d! " #! 2 d! u $ 2 c 3 % d% " # 8$% 2 d% c 3 I! d!dad" # $ 2! 2 d!dad" c 2 J! d! = J " d" # $!2 d! 4% 2 c 2 = $ 2%d" & 2 = $ 2d!dAd" % 2 dp d! = I!" e A e = 2#" ea e $ 2 = 2#" r A r $ 2 = 2# dp d! (polarization) = " T A = k B dp d! (polarization) A r Phys 2 (S06) 5 Black Body Radiation short 9

10 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! e d! e = A r d 2 But power output l = L! e 4"! l = r 2 d 2 " 4 BT eff angular diameter = r! d = A r = => Baade-Wesserlink distance measurements of varying stars Oscillating stars (Cepheids, RR Lyrae) Supernova assuming spherical expansion L 4"d 2 L = 4!r 2 4 " B T eff l 4 " B T eff Phys 2 (S06) 5 Black Body Radiation short 0

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! = h" p = hk = h"! s = sh" c s s = # exp% h! & ( ) $ " ' Maximum energy (minimum wavelength/finite # degrees of freedom) = Debye Energy Debye approximation:!! D = isotropic k=! c s or p= " c s!with velocity c s = constant Introducing k the Debye temperature T D =! = hc s U = 3 5!4 k B N T 4 3 C T V = 2 " D 5!4 k B N T % $ ' # T D & 3 k B # $ % 6" 2 N V /3 & ' ( = h) D k B! = 2 # 5 "4 k B N T & % ( $ T D ' Velocity 3 Phys 2 (S06) 5 Black Body Radiation short

12 Applications Calorimetry: Measure energy deposition by temperature rise!e!t =!E "!need small C C Heat Capacity Bolometry: Measure energy flux F by temperature rise!t =!F G Sky Load Chopping e.g., between sky and calibration load!f "!need small G but time constant= C G Heat Conductivity Very sensitive! Heat capacity goes to zero at low temperature Study of cosmic microwave background Search for dark matter particles limited by stability " small heat capacity C C! T 3 Phys 2 (S06) 5 Black Body Radiation short 2