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