Blackbody Radiation Thomas Wedgwood 1792 objects in kilns all become red at same temperature independent of composition Blackbody Radiation various observers in mid-1800s spectrum is continuous in contrast to discrete spectrum of heated gases infrared: melting ice cube visible: spectrum of Hg vapor Blackbody Radiation Gustav Kirchhoff 1859 for an object in thermal equilibrium with radiation its emitted radiation is equal to the radiation it absorbs • a good absorber is a good emitter P universal function = Rtot = eJ (T ) for all objects: as A yet unknown power per area • emissivity e = 1 for perfect absorber (blackbody) Blackbody Radiation Josef Stefan 1879 discovers universal function • Stefan’s Law – empirically determined P = R = eσT 4 A tot W σ = 5.67 × 10−8 m2K 4 • Ludwig Boltzmann derives this later Blackbody Radiation blackbody radiation distribution • continuous • well-defined peak wavelength • long-wavelength tail, asymmetric • area under curve proportional to power P R(λ) power per area per wavelength interval, such that: R = R λ dλ tot ∫ ( ) λ Blackbody Radiation Wilhelm Wien 1893 • relates peak wavelength to object temperature • Wien’s Displacement Law – empirically determined −3 λpeakT = 2.898 × 10 mK but... what physical mechanism is responsible for R ( λ ) ? Blackbody Radiation classical Rayleigh-Jeans Equation • model: radiation is composed of standing waves in a cavity 1 R(λ) = cEaven(λ) 4 number density of modes: how many ways per unit 8π volume can radiation of a n(λ) = 4 given wavelength oscillate λ in cavity 2πc but... what is the average R(λ) = 4 Eave energy per mode of λ oscillation? Blackbody Radiation Maxwell-Boltzmann energy distribution statistical thermodynamics: given a fixed amount of energy and a fixed number of particles, how are particles arranged in available energy states? Boltzmann factor: probability E − k T of occupying state of energy P(E) ∝ e B E what happens as E gets large compared to kT? Blackbody Radiation ... so the average energy per mode of oscillation is... ∞ EP E dE average = value*probability ∫ ( ) N.B. energy is a 0 continuous Eave = ∞ quantity in ∫ P(E)dE must normalize classical physics 0 ∞ − E ∫ Ee kBT dE 0 = ∞ = ... = kBT − E ∫ e kBT dE 0 Blackbody Radiation Rayleigh-Jeans Equation 2πc 2πckBT R(λ) = Eave = (whew!) λ 4 λ 4 The Ultraviolet R(λ) Catastrophe (oh no!) λ Blackbody Radiation Max Planck 1895 • model: cavity resonators (oscillating electric charges) emit and absorb only discrete energies n is an integer En = nhf f is frequency of oscillation h is a constant of proportionality The Quantum Hypothesis Blackbody Radiation Maxwell-Boltzmann energy distribution statistical thermodynamics: given a fixed amount of energy and a fixed number of particles, how are particles arranged in available energy states? Boltzmann factor: probability E nhf − n − of occupying state of energy kBT kBT Pn (En ) ∝ e = e E=nhf N.B. energy is quantized Blackbody Radiation ... so the average energy per mode of oscillation is... ∞ average = value*probability ∑EnPn (En ) n=0 Eave = ∞ ∑Pn (En ) must normalize n=0 ∞ nhf − ∑nhfe kBT n=0 hc λ = ∞ nhf = hc − λk T ∑e kBT e B −1 n=0 Blackbody Radiation Planck Equation 2 5 2πc 2πhc λ where h = 6.63×10−34 J ⋅s R(λ) = 4 Eave = hc λ e λkBT −1 at long wavelengths: hc hc e λkBT −1 ≈ 1+ −1 so λk T B agrees with 2 5 2 5 2πhc λ 2πhc λ 2πckBT Rayleigh-Jeans R(λ) = hc ≈ = 4 λkBT hc k T equation at e −1 λ B λ long wavelengths Blackbody Radiation Planck Equation 2 5 2πc 2πhc λ h = 6.63×10−34 J ⋅s R(λ) = 4 Eave = hc where λ e λkBT −1 at short wavelengths: hc hc 2πhc2 λkBT λkBT e −1 ≈ e so R(λ) = hc λ 5e λkBT.