Last Time Helmholtz Free Energy Differential of Free Energy Maxwell Relations Free Energy and the Partition Function Quantum Concentration For one atom in one box Ideal Gas Law Equipartition of Energy Today Chapter 4 (part 1) Thermal radiation and Planck Distribution Planck Blackbody Radiation Stefan-Boltzmann Law Planck Radiation Law Stars Big Bang Radiation Everything glows according to its temperature Hot things glow brighter, and at higher frequencies. Cooler things glow dimmer, at lower frequencies. Hot things radiate photons. Sspectrum of photons is determined by temperature Intensity Frequency Deviations from Wienn’s distribution • Wienn’s distribution, assumed Maxwell-Boltzman distribution of velocities: works for high frequencies (short wavelengths) • Lord Rayleigh (1900): using equipartition theorem, works for low frequencies (ultraviolet catastrophe) Rayleigh Planck Wienn Energy density, U Frequency Planck Wienn Rayleigh βν − 8πν 2kT U = αν 3e T U(ν ) = ν c3 Planck derivation, 1900 : 2 ∂ σ const ∂ 2σ const 2 = = ∂U U ∂U 2 U 2 ∂ 2σ const = ∂U 2 U (U + b) ⇐⇐⇐ Lucky guess!!! •assume that energy consists of quanta Star Spectra 1) Hot Things Glow 2) Atoms Have Specific Energy Levels Intensity Frequency The Sun “Black Body” Radiation EM radiation in thermal equilibrium… photons Energy of a photon is quantized : in a “s” tells you how many photons are in the mode, box and corresponds to the amplitude of the EM wave. E one photon two photons one photon two photons How It Equilibrates: Atoms and Light Really excited atom Absorb a Photon Excited atom Absorb a Photon Adding a photon increases the energy Unexcited atom Atoms and Light Really excited atom This sort of exchange achieves thermal equilibrium between the “box” and Release a Photon the photons inside Excited atom Release a Photon Releasing a photon decreases the energy Unexcited atom Statistical Mechanics On Photons Start with one photon mode Recognize this as: Partition function of one photon mode at energy Stat Mech on one Photon Mode Probability of finding s photons in the mode with certain energy Thermal average occupancy of the mode Stat Mech on one Photon Mode Planck Distribution Function Where we’re heading We have to quantize the EM modes to derive: The Stefan-Boltzmann Law Energy per unit volume in the box The Planck Radiation Law This peak tells you the temperature Intensity Frequency Thermal Average Energy per mode *… High temperature limit: Find a small parameter and Taylor expand. Classical Limit (High Temperature) What about the total energy? This is for one mode We’d like to know the total energy of all the photons in the box. First we need to learn how to count photon modes properly. EM Radiation in a Cavity photons L in a box EM Radiation in a Cavity Require Maxwell Equations to be satisfied The important part EM Radiation in a Cavity This means E-waves are transverse! For any given vector n (any set of nx, ny, nz), there are 2 ways to choose Eo Remember this factor of 2 for 2 slides from now when we sum over nx, ny, nz Photons in a Box Light satisfies wave equation photons L in a box ∂2 2 sin()ωt cos()() nxLx π = − ωω sin t cos () nxLx π ∂t 2 ∂2 n2π 2 x sin()ωπt cos() nxLx = − sin() ωπ t cos () nxLx ∂x2 L 2 Photons in a Box Light satisfies wave equation photons L in a box Now we know how to count the modes Photons in a Box photons L in a How to count EM modes in the box box Remember the factor of 2 for two polarizations Now we’ll convert the sum to an integral. Why is this okay? Why not just take the sum? Convert the Sum to an Integral Surface area of a sphere Photons in a Box Stefan-Boltzmann Law Is this energy, or energy density (per volume)? Photons in a Box photons L Stefan-Boltzmann in a Law box Radiant energy density in a cavity at thermal equlibrium with the walls What about Planck? This peak tells you the temperature Intensity Frequency Planck Radiation Law Energy density per frequency range “Spectral Density” n= ω L π c dn= ( Lπ cd) ω This peak tells you the temperature Intensity Frequency Estimate Surface Temperature ωeω τ ( τ ) 3− = 0 eω τ −1 ∂ ω 3 = 0 ω( τ ) ω τ 3 = ∂ω e − 1 1− e−ω τ 3 ω τ − ω τ 3ω 2 ωe ( τ ) 3− 3 e = ω τ ω τ −2 = 0 e −1 ()eω τ −1 Intensity Frequency Estimate Surface Temperature 3− 3 e−ω τ = ω τ 3− 3 e−x = x Solve numerically: ω τ ≈ 2.82 1 x=1 2.54966 aga: 2.76568 2.8112 xold=x 2.8196 This peak tells you x=3-3*exp(-x) 2.82111 the temperature if xold<>x then goto aga 2.82138 2.82143 2.82144 Converges at x = 2.82144… Intensity Frequency Estimate Surface Temperature ()ω τ 3 e−ω τ −1 Entropy of Thermal Photons Constant volume box Only temperature can vary Radiation energy flux density Black body: absorbs all radiation Energy flux density: Rate of energy emission per unit area JU = cU(τ ) / V ⋅ geometrical _ factor =4 2 4 cU (τ ) π τ (problem 15) J = = U 4V 60 3 c 2 4 JU= σ B T 24 32− 824 − − σB≡ π k B 60 c = 5.670 × 10 Wm K (Stefan-Boltzmann constant) Kirchhoff’s law The ability of a surface to emit radiation is proportional to the ability of the surface to absorb radiation 1824-1887 Black object: In equilibrium radiated energy should be equal to the absorbed energy! absorptivity Non-black body absorbs fraction a of radiation emitted by the black body, and emits fraction e of radiation emitted by BB at the same T emissivity a = e True for any wavelength! BIG BANG Proposed Big Bang Found serious evidence for it Fr. Georges Lemaitre Arno Penzias and Robert Wilson Bell Labs built a big antenna as a primitive satellite communication system using weather balloons! Arno Penzias and Robert Wilson used it as a telescope in 1965, but it had funny background noise… BIG BANG BIG BANG Universe acts like a black body, radiating at 2.9K Early universe: electrons + protons + light in thermal equilibrium at about 3000K. Then hydrogen formed, and matter stopped interacting so easily with light. Now separate thermal systems. Universe cooled by isentropic expansion (constant entropy). Expansion of universe lowered frequencies of photons, but not their occupation numbers. “Cosmic Background Radiation” is black body radiation of the early universe. Today Planck Blackbody Radiation Stefan-Boltzmann Law 4 24k 32 c Radiation energy flux density JU= σ B T σB≡ π B 60 Entropy of thermal photons: Peak frequency in spectrum tells temperature ω τ ≈ 2.82 Big Bang: background radiation from expansion of universe.