Lecture 27 Lecture 27 ——The Planck Distribution The Planck Distribution Chapter 8, Friday March 21Chapter 8, Friday March 21stst•Quick review of exam 2 •Black-body radiation•Before Planck: Wien and Rayleigh-Jeans •The ultraviolet catastrophe•The Planck distributionReading: Reading: All of chapter 8 (pages 161 All of chapter 8 (pages 161 --186)186)Homework 8 not due until Mon. Mar. 31stHomework 8 not due until Mon. Mar. 31stAssignment will be handed out on MondayAssignment will be handed out on MondayExam 2 Exam 2 ––question 1question 1()()123 1 2 3,, 3/2nnn n n nεω=+++=Exam 2 Exam 2 ––question 2question 2The Planck DistributionThe Planck DistributionA. A. Michelson (late 1900s): “The grand underlying principles (of physics) have been firmly established... ...the future truths ofphysics are to be looked for in the sixth place of decimals.”Planck credited with the birth of quantum mechanics (1900)Planck credited with the birth of quantum mechanics (1900)--developed the modern theory of blackdeveloped the modern theory of black--body radiationbody radiationQuantum nature of radiationQuantum nature of radiation1st evidence from spectrum emitted by a black1st evidence from spectrum emitted by a black--bodybodyWhat is a blackWhat is a black--body?body?An object that absorbs all incident radiation, An object that absorbs all incident radiation, i.e.i.e.no reflectionno reflectionA small hole cut into a cavity is the most popular and realistic example. ⇒None of the incident radiation escapesWhat happens to this radiation?What happens to this radiation?•The radiation is absorbed in the walls of the cavity•This causes a heating of the cavity walls•Atoms in the walls of the cavity will vibrate at frequencies characteristic of the temperature of the walls•These atoms then re-radiate the energy at this new characteristic frequencyThe emitted "thermal" radiation characterizes the The emitted "thermal" radiation characterizes the equilibrium temperature of the blackequilibrium temperature of the black--bodybodyBlackBlack--body spectrumbody spectrumBlackBlack--body spectrumbody spectrum•Black-bodies do not "reflect" any incident radiationThey may re-radiate, but the emission characterizes the black-body only•The emission from a black-body depends only on its temperatureWe (at 300 K) radiate in the infraredObjects at 600 - 700 K start to glowAt high T, objects may become white hotWien's displacement LawWien's displacement LawλλmmTT= constant = 2.898 = constant = 2.898 ××1010−−33m.K, or m.K, or λλmm∝∝TT−−11Found empirically by Joseph Stefan (1879); later calculated by Boltzmannσ= 5.6705 × 10−8W.m−2.K−4.A black-body reaches thermal equilibrium when the incident radiation power is balanced by the power re-radiated, i.e. if you expose a black-body to radiation, its temperature rises until the incident and radiated powers balance.StefanStefan--Boltzmann LawBoltzmann LawPower per unit area radiated by blackPower per unit area radiated by black--bodybodyRR= = σ σ TT44RayleighRayleigh--Jeans equationJeans equationConsider the cavity as it emits blackbody radiationThe power emitted from the blackbody is proportional to the radiation energy density in the cavity. One can define a spectral energy distribution such that u(λ)dλ is the fraction of energy per unit volume in the cavity with wavelengths in the range λ to λ + dλ.Then, the power emitted at a given wavelength, Then, the power emitted at a given wavelength, RR((λλ) ) ∝∝uu((λλ))uu((λλ) may be calculated in a straightforward way from classical ) may be calculated in a straightforward way from classical statistical physics.statistical physics.uu((λλ))ddλλ= (# modes in cavity in range = (# modes in cavity in range ddλλ) ) ××(average energy of modes)(average energy of modes)# of modes in cavity in range # of modes in cavity in range ddλ, λ, nn(λ)(λ)ddλλ= 8πλ= 8πλ−−4 4 ddλλAverage energy per mode is Average energy per mode is kkBBTT, according to kinetic theory, according to kinetic theory⇒⇒uu((λλ) = ) = kkBBTTnn(λ) = 8π(λ) = 8πkkBBTTλλ−−44Wien, RayleighWien, Rayleigh--Jeans and Planck distributionsJeans and Planck distributions() () ()()/RJ W P/45588;;1BTBhc k TkT e hcuuueβλλππλλλλλλ−=∝=−Wilhelm Carl Werner Otto Fritz Franz WienThe ultraviolet catastropheThe ultraviolet catastropheThere are serious flaws in the reasoning by Rayleigh and JeansThere are serious flaws in the reasoning by Rayleigh and JeansFurthermore, the result does not agree with experimentFurthermore, the result does not agree with experimentEven worse, it predicts an infinite energy density as Even worse, it predicts an infinite energy density as λλ→→0!0!(This was termed the ultraviolet catastrophe at the time by Paul(This was termed the ultraviolet catastrophe at the time by PaulEhrenfestEhrenfest))Agreement between theory and experiment is only to be found at very long wavelengths.The problem is that The problem is that statistics predict an infinite statistics predict an infinite number of modes as number of modes as λλ→→0; 0; classical kinetic theory classical kinetic theory ascribes an energy ascribes an energy kkBBT to T to each of these modes!each of these modes!Planck's law (quantization of light energy)Planck's law (quantization of light energy)In fact, no classical physical law could have accounted for measured blackbody spectraThe problem is clearly connected with The problem is clearly connected with uu((λλ) ) →→∞∞, as , as λλ→→00Planck found an empirical formula that fit the data, and then maPlanck found an empirical formula that fit the data, and then made de appropriate changes to the classical calculation so as to obtainappropriate changes to the classical calculation so as to obtainthe the desired result, which was nondesired result, which was non--classical. classical. Max Planck, and others, had no way of knowing whether the Max Planck, and others, had no way of knowing whether the calculation of the number of modes in the cavity, or the averagecalculation of the number of modes in the cavity, or the averageenergy per mode (energy per mode (i.e.i.e.kinetic theory), was the problem. It turned kinetic theory), was the problem. It turned out to be the latter.out to be the latter.The problem boils down to the fact that there is no connection bThe problem boils down to the fact that there is no connection