Chapter 15. Thermodynamics of Radiation Introduction

Transcription

1 A new scientific truth does not triumph by convincing its opponents and maing them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it science advances one funeral at a time. Chapter 15 Max Planc, Scientific Autobiography and Other Papers, trans. F. Gaynor (New Yor, 1949), pp Thermodynamics of Radiation 15.1 Introduction In 19 Max Planc discovered that the temperature dependant law of radiating bodies could not be derived solely from Maxwellian electrodynamics according to which the energy of an electromagnetic field is 1 E = 1 dx [E (x, t) + B (x, t) ]. (15.1) Instead, Planc was able to arrive at results consistent with the relevant electromagnetic experiments by treating radiation of a given frequency ν as though it consisted of pacets of energy photons each with energy hν, expressing a corresponding electromagnetic field energy as E P lanc = nhν (15.) where n =, 1,,... is the number of photons in the pacet and h is Planc s universal constant. Planc s hypothesis was the initial lin in the chain of th century discoveries that is quantum physics. 1 In rationalized c.g.s. units. Surely inspired by Botzmann s concept of microstates. 1

2 CHAPTER 15. THERMODYNAMICS OF RADIATION 15. Electromagnetic Eigen-energies Proceeding directly to the thermodynamics of radiation, Eq.15. is more usefully replaced by the electromagnetic quantum eigen-energies for photons of a single mode having polarization λ E,λ = hν,λ [n (, λ) + 1/] (15.3) where n (, λ) =, 1,,... is the number of photons. and λ are quantum numbers for a single photon with ν,λ the frequency of that photon. 3 Photons are regarded as particle-lie excitations of the electromagnetic field. They are fundamental particles with zero mass and are their own antiparticle. They can be created and absorbed (destroyed) without numerical conservation and have, similar to phonons, the elastic excitations of Chapter 1, zero chemical potential. They appear in an almost limitless variety of atomic, molecular or nuclear processes as well as continuous radiation (synchrotron radiation) associated with inematic acceleration of particles. The photon carries spin angular momentum ± h corresponding to right and left circular polarization. The Maxwellian total electromagnetic energy [see Eq.15.1] is similarly replaced by the quantum result H EM = λ=1, hν,λ [ n op (, λ) + 1/] (15.4) where H EM is the electromagnetic hamiltonian, with n op (, λ) the photon number operator (with integer eigenvalues, 1,,...) for the mode labeled by wave vector, and polarization λ. n op (, λ) is the average number of photons in that mode. The wave vector is the direction of propagation of the photon with three components x, y, z. The polarization λ is the vector direction of the concurrent electric field E. It is perpendicular (transverse) to with only components λ λ 1, λ. This follows from the free space Maxwell equation E = (for which there is no analog in elastic equations of motion). 3 Deriving Planc s result from Maxwell s equations is the realm of quantum field theories, the details of which are well beyond the scope of this boo. Nevertheless, for completeness an outline of the method is discussed in Appendix H.

3 15.3. THERMODYNAMICS OF ELECTROMAGNETISM 3 For electromagnetic radiation in free space, the energy of a photon associated with a particular mode (, λ) is hν,λ = h c (15.5) π which depends only on the magnitude and is independent of the polarization λ. Here c is the speed of light in vacuum. As we will see, in free space n op (, λ), the average number of photons in the mode (, λ), also depends only on and is independent of the polarization λ Thermodynamics of Electromagnetism Based on macroscopic electromagnetic eigen-energies (see Appendix H) given by Eq.15.3, a least bias Electromagnetic Lagrangian L EM is constructed L EM = B n,λ P (n,λ ) ln P (n,λ ) λ n,λ P (n,λ ) (15.6) λ 1 P (n,λ ) {hν,λ (n,λ + 1/)} n,λ λ=1, where P (n,λ ) is the least biased probability that n,λ photons are in the mode with polarization λ. The sum (15.7) n,λ ranges over n,λ =, 1,,... for each mode and both polarizations (λ = 1, ). The Lagrange multipliers are λ, λ 1 with, as before, λ 1 = 1/T. Following by now familiar procedures, we find from Eq.15.6 P (n,λ ) = exp β hν,λ (n,λ + 1/) λ=1, exp β hν,λ (n,λ + 1/) n,λ λ=1, (15.8)

4 4 CHAPTER 15. THERMODYNAMICS OF RADIATION where the normalizing denominator is the thermal equilibrium electromagnetic partition function, Z EM = exp β n,λ λ=1, hν,λ (n,λ + 1/). (15.9) Thermodynamics For a Single Photon Mode To simplify evaluation of Eq.15.9 we first examine the partition function for a single mode α Z α EM = n α,λ =,1,,3... exp β λ=1, hν α,λ (n α,λ + 1/) (15.1) where n α,λ =,1,,3... (15.11) is the sum over all integer photon numbers for a single mode ( α, λ). Explicitly summing over both polarizations in the exponent of Eq.15.1 ZEM α = e βhν α exp [ βhν α (n α,1)] exp [ βhν α (n α,)] n a,1 n a, (15.1) = e βhν α exp [ βhν α (n α,1)] n a,1 (15.13) where the factor e βhν α results from the vacuum radiation term in the eigenenergies. The sum over photon numbers n α,1 =, 1,,... is just a geometric series, which gives for the single mode α and both polarizations ZEM α = e βhν α 1 1 e βhν α (15.14)

5 15.3. THERMODYNAMICS OF ELECTROMAGNETISM Average Photon Number From Eq.15.8 the average number of photons in the single mode α with a single polarization λ = 1, say n α,1, is This is identical with n α,1 = n α,1 n α,1 exp { βhν α,1 n α,1} n α,1 exp { βhν α,1 n α,1}. (15.15) 1 n α,1 = [ hν a,1 β ln ( Zα EM)] 1/ (15.16) 1 = e βhν a,1 1, (15.17) which is called the Planc distribution function Helmholtz Potential The Helmholtz potential F α for a single mode α, is F α = 1 β log Zα EM (15.18) = hν α + β log (1 e βhν α ) (15.19) Internal Energy Similarly, for a single mode α (counting both polarizations) the the radiation field internal energy U α is U α = β ln Zα EM (15.) = hν α ( eβhν α + 1 e βhν α 1 ) (15.1) = hν α ( e βhν α 1 ) (15.) = hν α ( 1 + n α ) (15.3)

6 6 CHAPTER 15. THERMODYNAMICS OF RADIATION Radiation Field Thermodynamics The thermal electromagnetic field assembles all photon modes with both transverse polarizations λ = 1,, requiring the full evaluation of Eq Rewriting Z EM as Z EM = e βhν 1 (n 1 +1/) e βhν (n +1/)... (15.4) = n 1 =,1,,... [e βhν 1 ( 1 e βhν n =,1,,... ) ], (15.5) in analogy with Eqs and 15.3 the Helmholtz potential F EM is F EM = 1 β and the internal energy U EM is ln [e βhν 1 ( 1 e βhν ) ] (15.6) = [hν + β ln (1 e βhν )] (15.7) U EM = hν ( 1 + n ). (15.8) Stefan-Boltzmann, Planc, Rayleigh-Jeans Radiation Laws In the limit of macroscopic volume V the sum over in Eq.15.8 is replaced by (see Appendix H) V (π) 3 d x d y so that ignoring the divergent, constant vacuum energy U EM = V (π) 3 d x d y d z, (15.9) hν d z ( ). (15.3) e βhν 1 But in vacuum we have Eq.15.5 so that the integrals can be carried out in spherical coordinates, in which case d x d y d z = π dϕ π dθ sin θ d (15.31)

7 15.3. THERMODYNAMICS OF ELECTROMAGNETISM 7 and the internal energy U EM is (with h = π h) U EM = hcv π d 3 1 ( e β hc 1 ). (15.3) The integral can be brought into a more standard form with the substitution = x (15.33) β hc so that hcv U EM = π (β hc) 4 dx x 3 1 ( e x 1 ). (15.34) The integral in Eq is one of several similar integrals that appear in thermal radiation theory. They are somewhat subtle to evaluate but they can be found in comprehensive tables or computed with Mathematica with the result dx x 3 1 ( e x 1 ) = π4 15 (15.35) so that the radiation energy density is U EM V = π 15β 4 ( hc) 3. (15.36) This is called the Stefan-Boltzmann Radiation Law. If in Eq we substitute x = βhν, the radiation energy density may be written where U EM V = u (ν) = 8πh c 3 ν 3 e βhν 1 dν u (ν) (15.37) (15.38) which is the frequency distribution of thermal radiation at any temperature T. It is called the spectral density or Planc s Radiation Law. In the high temperature limit, βhν << 1, the Planc Law becomes the classical Rayleigh-Jeans Law. [See Fig 15.1.] u classical (ν) = 8π BT c 3 ν. (15.39) The cancellation of Planc s constant eliminates any quantum reference.

8 8 CHAPTER 15. THERMODYNAMICS OF RADIATION U EM Raleigh-Jeans Planc hν B T Figure 15.1: Plan Radiation Law and Rayleigh-Jeans approximation Example: What is the average number density of thermal photons all modes, both polarizations at temperature T. Beginning with Eq the average photon number is n = λ=1, n,λ (15.4) = V 1 (π) 3 d. e βhν 1 (15.41) In spherical coordinates this becomes (with ν = c π ) n = V (π) 3 = V π π dφ π d e β hc 1 dθ sin θ d e β hc 1 (15.4) (15.43)

9 15.3. THERMODYNAMICS OF ELECTROMAGNETISM 9 which after the change of variable β hc = x the number density is n V = 1 π (β hc) 3 x dx e x 1. (15.44) This integral resembles Eq.15.35, but unlie that integral this one has no result in terms of familiar constants. But it can be estimated by the following series of steps. x dx e x 1 = = = = s= s= = dx x e x 1 e x (15.45) dxx e x e sx (15.46) s= s =1 dxx e x(s+1) (15.47) (s + 1) 3 (15.48) 1 s 3 (15.49) The last sum is called the Riemann ζ-function. 4 In this case we have ζ (3) which can only be evaluated by summing, term-by-term, to any desired accuracy. In this case ζ (3) = Therefore the photon number density is n V = ζ (3) π (β hc) 3 (15.5) Wien s Law In stellar astronomy two revealing parameters are a stars surface temperature and its luminosity. The surface temperature is found from its color, which corresponds to the 4 ζ (z) = 1. For even integer values of z (but not for odd values) the ζ-function can be found n n=1 z in closed form. In general, ζ is complex.

10 1 CHAPTER 15. THERMODYNAMICS OF RADIATION frequency pea in the spectral density curve of Eq These two pieces of information determine the star s location on the empirical but important Hertzsprung-Russel diagram [which you can read about in any introductory astronomy text.] From this it is possible to accurately determine the chemistry of the star, its age and its stage of evolution. The spectral pea is determined from Eq by differentiation, i.e. which gives This can be solved graphically [See Fig 15.] W (βhν) = d dν ( ν 3 e βhν 1 ) = (15.51) W (βhν) = 3e βhν 3 βhν e βhν =. (15.5) hν W ( B T) hν T B to find Figure 15.: Graphical determination of Wien s Law constant. hν max =.8 B T. (15.53) The frequency ν max at which the pea of Planc s radiation curve is located is proportional to the absolute temperature of the radiating source. This is called Wien s Law.

11 15.3. THERMODYNAMICS OF ELECTROMAGNETISM Entropy of Thermal Radiation The entropy of thermal radiation is S EM = B where P (n,λ ) was found in Eq This is equivalent to P (n,λ ) ln P (n,λ ) (15.54) n,λ =,1,,... S EM = B β β ln Z EM (15.55) = B β ( F EM β ) V (15.56) where Z EM is the partition function [see Eq,15.9] and F EM is the electromagnetic Helmholtz potential [see, e.g. Eq.15.18]. Completing the full radiation field calculation of Eq requires summing over all modes. Substituting x = βhν into that equation and ignoring the vacuum field contribution (it does not depend on β), the Helmholtz potential is the integral Integrating by parts gives F EM = F EM = 8πV β 4 (hc) 3 8πV 3β 4 (hc) 3 dx x ln (1 e x ). (15.57) x 3 dx (e x 1) (15.58) = V π 45 h 3 c 3 β 4 (15.59) where the integral Eq has been used. Finally, applying Eq.15.55, the entropy is which is proportional to T 3. S EM = 4π B V 45 (β hc) 3 (15.6)

12 1 CHAPTER 15. THERMODYNAMICS OF RADIATION Stephan-Boltzmann Radiation Law An object at temperature T radiates electromagnetic energy from its surface [see Fig 15.3]. The energy radiated per unit area per unit time (energy current density) is the Poynting Vector S, which in the classical theory is S classical = 1 8π V dx E B. (15.61) θ Figure 15.3: Photon leaving the surface of a radiating body. The quantum Poynting vector for a single photon, λ is S,λ = V 1 hν n,λ c ˆ (15.6) where ˆ is the unit vector in the direction of propagation of the mode, λ and [See Eq.15.16]. n,λ = 1 e βhν 1 (15.63) The differential radiation flux density dφ,λ of the mode with polarization λ from the element of area da [see Fig 15.3] is dφ,λ = S,λ da (15.64)

13 15.3. THERMODYNAMICS OF ELECTROMAGNETISM 13 with the total radiation flux differential dφ dφ = V 1 where ˆn is the unit vector normal to the surface element total radiation flux density per unit area is therefore hν n,λ c ˆ ˆn da (15.65) λ=1, da [see Fig 15.3]. The dφ da = V 1 hν n,λ c ˆ ˆn (15.66) λ=1, With ˆ ˆn = cos θ the total outward ( < θ < π/) radiation flux density per unit area is dφ da = (π) 3 π dφ π/ dθ sin θ cos θ d chν e βhν 1 (15.67) which is integrated to give or where dφ da = π ( B T ) 4 (15.68) 6 h 3 c dφ da = σ B T 4, (15.69) σ B = π 4 B 6 h 3 c (15.7) = W atts m K 4 (15.71) is called the Stefan-Boltzmann constant Example: Surface Temperature of Gliese 481 d. A recently discovered exoplanet (7), assigned the name Gliese 581 d, orbits at an approximate distance m from its host star Gliese 581. Gliese 581 is a red

14 14 CHAPTER 15. THERMODYNAMICS OF RADIATION dwarf star with an approximate radius of 1 5 m and a surface temperature about 348K. Assuming the star and the exoplanet behave as perfect radiators, estimate the temperature at the surface of Gliese 581 d. Assuming radiative equilibrium, 4πrpσ B Tp 4 energy rate reradiated from planet fraction intercepted by planetary dis = 4πR σ B T 4 energy rate radiated from star { πr p } (15.7) 4πR p where r p is the exoplanet radius, T p is the exoplanet temperature, R is the star radius and R p is the exoplanet s orbital radius. After simplification we have so that T p = ( R 1/4 ) T 4R p (15.73) T p =19 K. (15.74) Radiation Momentum Density The momentum p of a photon (a massless particle) is related, by special relativity, to its energy Eγ by p = E γ c, (15.75) therefore the radiated momentum flux density Π per unit area is dπ da = V 1 hν n,λ ˆ ˆn (15.76) λ=1, which is evaluated, as in Eqs , to give dπ da = π ( B T ) 4 6 h 3 c 3 (15.77)

15 15.3. THERMODYNAMICS OF ELECTROMAGNETISM Example: Star Dust Blow-out When stars are formed by gravitational collapse they are initially embedded in cocoons of tiny silicate dust particles, a dust so dense that at this stage the star can be detected only from secondary radiation emitted by the heated dust. When the newly formed star reaches a sufficiently high temperature the radiation from the star begins blowing out the dust cocoon allowing the star to become visible by direct observation. Assuming that the radiation absorption cross-section of a dust particle is approximately 1/ of the physical cross-section and the dust particles have radii a < 1 7 m, at what stellar temperature will all cocoon dust particles be blown away from the star? The total momentum density per unit time radiated by the star is Π = π ( B T ) 4 6 h 3 c 3 4πR (15.78) where T is the star s temperature and R is its radius. Therefore the momentum density per unit time intercepted by a dust particle of radius a at a distance R dust from the star is Π dust = π ( B T ) 4 4πR 6 h 3 c 3 ( πa ). (15.79) 4πRdust Taing into account the 5% absorption cross-section, the momentum density per unit time absorbed by a dust particle is Π A dust = π ( B T ) 4 4πR 6 h 3 c 3 ( πa ) (1/). (15.8) 4πRdust On the other hand, the force of gravity acting on the particle is, by Newton s Law of gravitation, F grav = GM ( 4πa3 Rdust 3 ρ dust) (15.81) where G is the gravitational constant and ρ dust = g m 3 is the approximate density of silicate dust.

16 16 CHAPTER 15. THERMODYNAMICS OF RADIATION For the radiation-momentum absorbing dust particle to be in gravitational equilibrium π ( B T ) 4 4πR 6 h 3 c 3 ( πa ) (1/) = GM ( 4πa3 4πRdust Rdust 3 ρ dust), (15.8) where M is the mass of the star, so that T 4 = 8a 3 cgm ρ dust σ B R (15.83) where σ B is the Stefan-Boltzmann constant [see Eq.15.7]. Assuming, for numerical specificity, that the newly formed star is sun-lie with 1 solar mass and 1 solar radius, i.e. M = g (15.84) R = m (15.85) G = N m g (15.86) the dust blow-out temperature is T 6K, independent of the distance from the star to the dust particles, R dust.