Class 11: Thermal radiation

Transcription

1 Class : Thermal radiation By analyzing the results from a number of eperiments, Planck found the energy density of the radiation emitted by a black body in wavelength interval (, d + was well described by the formula hc u ( d = d, 5 hc ( kt e (. where T is the temperature of the black body and h and k are constants. Planck, a short time later, developed a theory that eplained this result. Before considering Planck s theory, it is useful to see what classical physics predicts for black body radiation. As a preliminary we need a result from statistical mechanics due to Boltzmann. The Boltzmann factor Consider a physical system that can be in a number of states. Associated with each state is an energy. Boltzmann showed that when the system is thermal equilibrium with a heat reservoir of temperature T, the probability, p, that the system is in a particular state of energy, E, is E ( kt =, (. p Ae where k is a constant, now known as the Boltzmann constant, and A is a constant normalization factor E ( kt that is determined by the requirement that the probabilities sum to unity. The term e is called the Boltzmann factor. As an eample of application of the Boltzmann factor consider a system that has two states: a ground state of energy and an ecited state of energy E. The probabilities that the system is one of the two states are E ( kt A for the ground state and Ae for the ecited state. By summing the probabilities, we find A =. (.3 + E ( kt We see that at low temperature ( kt E, the probability that the system is in its ground state is close to, and the ecited state has a very low probability. At high temperatures( kt E, the Boltzmann factors are equal and the two states have equal probability. The average energy of the system is e ( kt E + Ee E E = =. E ( kt E ( kt + e e + (.

2 For a system with many states, it is likely that some states have the same energy. The number of states with a particular energy is called the statistical weight if the energies are discrete. For a continuous energy distribution, the corresponding quantity is called the density of states. The Rayleigh-Jeans law To develop an epression for the energy density of black body radiation, Rayleigh considered the radiation emitted through a small hole in a cubical cavity with walls kept at temperature, T. The walls contain classical harmonic oscillators that can absorb and emit radiation. This provides the coupling between the system, i.e. the radiation in the cavity, and the heat reservoir, i.e. the cavity walls, needed for the system to be in thermal equilibrium, so that we can use the Boltzmann factor. The first step in the calculation is determine the density of states. To do this, it is assumed that the walls of the cavity are perfect electrical conductors. This means that the electrical field associated with the electromagnetic waves in the cavity must be zero on the cavity walls to avoid generating currents of infinite magnitude. The electromagnetic waves are then standing waves with nodes at the cavity walls. A D analogy is standing waves on a string of length L with fied ends. The allowed wavelengths for these waves are where n is a positive integer. Treating n as a continuous variable, so that the number of waves with wavelengths in the interval (, d L n =, (.5 n L n =, (.6 + is L dn = d. (.7 Note that dn is taken to be positive. We see that the density of states is higher at shorter wavelengths. The generalization of equation (.5 to three dimensions is = ( L ( n + ny + nz, (.8 where n, n y, and n z are independent positive integers. To determine the density of states, we treat the integers as continuous variables and note that once we restrict to positive integers the volume element in n-space is ( n π dn, where n = n + n + n. Because there are two independent polarizations of y z light, the number of waves in this volume element is

3 where V is the volume of the cavity. ( π L L V = π = (.9 n dn d d, The second step is to consider the mean energy of a wave of wavelength. In classical physics, the energy of the wave is proportional to the square of the amplitude of the wave, and there are no other state variables, i.e. the statistical weight is. The mean energy can be found by employing the Boltzmann factor: E = = kt. Ee e E kt E kt de de (. Combining these results, we find that the energy density per unit wavelength is u = kt. (. This is the Rayleigh-Jeans law. Integration over all wavelengths gives the troubling result that the cavity should contain an infinity amount of energy! Since the epression for the energy density diverges as the wavelength becomes shorter, this result was called the ultraviolet catastrophe. Note that at long wavelengths (low frequencies, the Planck formula gives u hc hc = = kt, 5 5 e hc + + kt ( hc ( kt (. which agrees with the Rayleigh-Jeans result. Note that the Rayleigh-Jeans law is independent of h. Planck s theory Planck s approach to avoiding the ultraviolet catastrophe was to find a way to make the average energy of a high frequency oscillator less than kt, yet leave the energy of a low frequency oscillator close to kt to get correct low frequency limit. He postulated that the oscillators could emit or absorb energy in discrete packets, with the energy in a packet being an integer multiple of a constant times the frequency En = nhf, (.3 where h is Planck s constant, and n =,,,. The mean energy of a wave is now 3

4 E = n= n= nhfe e nhf kt nhf kt. (. To perform the summations, let = hf kt. The denominator is The numerator is Hence nhf kt n n Z = e = e = ( e = e n= n= n= nhf kt n d n d e nhfe = hf ne = hf e = hf = hf d d e e n= n= n=. (. E e = =. kt e e (.5 We see that when = hf kt, the mean energy is close to kt, but when, the eponential term in the denominator dominates and the mean energy is much less than kt. Also note that the classical result is recovered if Planck s constant is replaced by. Using equation (.5 in place of equation (., we find that the energy density per unit wavelength is u hf hc, hf kt hc kt e e ( = = 5 (.6 which is the Planck result. Planck s postulate that the oscillator could absorb or emit only in discrete packets of energy was revolutionary, and marks the beginning of quantum physics. Radiation intensity The radiation from a small hole in a cavity wall is emitted not in a single direction but over one hemisphere of directions (i.e. into solid angle π steradians. By averaging over all possible directions, it is found that ¼ of the radiation flu is directed outward perpendicular to the wall. The radiation intensity (or flu per unit wavelength is then Integrating over all wavelengths gives the total flu I =. (.7 ( cu (

5 hc F = c u ( d = c d. 5 hc kt e (.8 Making the substitution = hc ( kt, we have ( kt 3 ( hc 3 π F = c d. (.9 e Hence the radiative flu is proportional to the th power of the temperature of the black body, in agreement with Stefan s law (also called the Stefan-Boltzmann law: On evaluating the integral, we find that the Stefan-Boltzmann constant is F = σt. (. 5 π k σ =. (. 3 5 c h The Wien displacement law The figure below shows the Planck function plotted against wavelength for a temperature equal to the effective temperature of the Sun. 3. B (erg cm -3 s - sr K blackbody Rayleigh - Jeans law UV Visible Infra-red (nm 5

6 We see that, by no coincidence, the peak of the black body radiation lies in the visible part of the electromagnetic spectrum, near a wavelength of 5 nm. The figure below shows the Planck function for three temperatures. We see the wavelength of the peak 6 T / B (erg cm -3 s - sr nm 5 nm nm T T depends inversely on the temperature of the black body. This is Wien s displacement law, which can also be found by differentiation of the energy density given in equation (.6. Again let = hc ( kt, so that where (nm ( ( kt ( hc 5 u = H, (. ( ( H 5 =. e (.3 The function H( has a maimum at =.965, and so the peak occurs at wavelength peak 6 hc.898 =. = nm K. (. kt T 6