aa r X i v : . [ phy s i c s . c l a ss - ph ] N ov A blackbody is not a blackbox
Matteo Smerlak ∗ Centre de Physique Th´eoriqueCampus de Luminy, case 90713288 Marseille cedex 9, France (Dated: November 9, 2010)We discuss carefully the blackbody approximation , stressing what it is (a limit case of radiativetransfer), and what it is not (the assumption that the body is perfectly absorbing, i.e. black ).Furthermore, we derive the Planck spectrum without enclosing the field in a box, as is done in mosttextbooks. Athough convenient, this trick conceals the nature of the idealization expressed in theconcept of a blackbody: first, the most obvious examples of approximate blackbodies, stars, aredefinitely not enclosed in boxes; second, the Planck spectrum is continuous, while the stationarymodes of radiation in a box are discrete. Our derivation, although technically less elementary, isconceptually more consistent, and brings the opportunity to introduce to students the importantconcept of local density of states , via the resolvent formalism. “No colors anymore, I want them to turn black [ ... ] I wanna see the sun blotted out from the skyI wanna see it painted, painted, painted, painted blackYeah!”
The Rolling Stones
I. INTRODUCTION
Planck’s law for blackbody radiation is celebrated as alandmark in the history of physics, being the first physi-cal law conflicting patently with classical mechanics. De-rived in 1900, it is usually regarded as the igniter of thecentury of quantum mechanics. But its relevance goesbeyond quantum mechanics. Blackbody radiation is astriking example of universality in statistical mechanics– the light radiated by a blackbody does not depend onits constitution, only on its temperature –, a topic whichcame to the foreground with the later studies of criti-cal phenomena. More recently, Planck’s distribution hasappeared to fit with an astounding accuracy the cosmicmicrowave background (deviations are at most 50 permillion ), thus opening the era of precision cosmology.Because of its seminal importance, one feels thatthe derivation of Planck’s law presented to studentsshould be as lucid as possible. In most textbookpresentations , however, one apparently innocuous stepundermines the understanding of its applicability: theradiation field is assumed to be enclosed in a box. Thisprompts students to think that the Planck spectrum isthe one radiated by a black box . This is true, of course,but also very misleading. If the blackbody radiation isreally the blackbox radiation, why should stars, whichare not enclosed in boxes, have a Planckian spectrum?Why should hot metal bars, bulb filaments, incandescentlava or the background of the universe have a Planckianspectrum?Since the physics of thermal radiation generally does not involve a box, we feel that the discussion of Planck’slaw should not either. In this note, we propose such adiscussion. We introduce the blackbody approximation in the framework of radiative transfer theory, and derivethe Planck spectrum using the notion of local density ofstates. We do not claim originality in the method used,the resolvent formalism, which is standard in condensedmatter physics and scattering theory. Rather, our aim isto promote an approach we believe to be pedagogicallymore transparent. (See also Ref. for a more advanceddiscussion of Planck’s law from the many body theoryperspective.)The paper is organized as follows. In the next section,we discuss the blackbody approximation, and its relationwith Kirchhoff’s law. In section III, we define the localdensity of states, and derive Planck’s law without thebox, using the resolvent formalism. Section IV presentsour conclusion. II. WHAT IS A BLACKBODY?A. The common definition
A blackbody is usually defined as “a body which com-pletely absorbs all radiation incident on it”. Landau andLifschitz addSuch a body can be realised in the formof a cavity with highly absorbing internalwalls and a small aperture. Any ray enter-ing through the aperture can return to it andleave the cavity only after repeated reflectionfrom the walls of the cavity. When the aper-ture is sufficiently small, therefore, the cavitywill absorb practically all the radiation inci-dent on the aperture, and so the surface ofthe aperture will be a black body.Clearly, this picture makes perfect experimental sense.Theoretically, on the other hand, the nature of the ide-alization it is meant to express remains clouded. If ablackbody emits light, why is it called “black”? And ifit is an ideal emitter, why is it defined as an ideal “ab-sorber”? What is the role of the cavity? More impor-tantly, what would an imperfect blackbody be? A cavitywhich does not absorb all incident radiation? One witha bigger aperture? B. A continuous spectrum
Many objects emit light like imperfect blackbodies.The most prominent among them is, of course, our Sun(see Fig. 1). Its spectrum, first measured in the earlynineteenth century by Wollaston and Frauhofer, is wellapproximated by that of a blackbody of temperaturebetween 5500 and 6000 K. It does displays deviationsfrom it, the so-called Fraunhofer lines (corresponding tothe absorption of certain frequencies by the solar atmo-sphere), but it is indisputable that, athough very dissim-ilar from an absorbing cavity with a tiny hole, the Sunqualifies as an imperfect blackbody. On the other hand,the spectroscopy of monoatomic gases shows clearly thattheir emission spectrum is not
Planckian. They emit onlycertain discrete frequencies (Fig. 2), determined by theelectronic structure of the atoms. This observation hints at what a blackbody really is: abody with a rich energy spectrum, capable of exciting allfrequencies of light by thermalization . From this perspective, the box definition appears para-doxical. As is well-known, a closed box selects certainslight frequencies, through the condition ω i = nπc/L i ,where L i is the dimension of the box in the i -direction.Thus, instead of permitting a wide range of thermallyexcited frequencies, the box restricts the emission spec-trum, even making it discrete. Of course, one could arguethat the volume of the box can be made arbitrarily large,and therefore that this quantization of frequencies is notphysically relevant. But this is precisely our point: as faras the frequencies of light are concerned, the box is notphysically relevant. C. Kirchhoff ’s law
In its standard definition, a blackbody is one that “ab-sorbs all incident light” – a black body. This fact aloneshould disturb the mindful student: how can a body be black , and yet emit a colourful spectrum of thermal light?In any case, what should the absorptive power of a bodyhave anything to do with the caracteristics of its thermal emission ?The answer lies in an experimental observation whichplayed a key role in the nineteenth century developmentswhich led to Planck’s successful analysis of thermal radi-ation, and which is too seldom mentioned in undergrad-uate discussions of thermal radiation – Kirchoff’s law. The radiative properties of a body are characterized byits emissivity and absorptivity (and scattering, which canusually be neglected). These can be defined by the fol-lowing schematic model for the propagation of radiation
FIG. 1: The Sun is an imperfect blackbody, monoatomic gasesare not (top: hydrogen; bottom: iron). within a medium. As the (monochromatic) beam trav-els trough the medium, the variation of its energy density u ( l, ω ) receives two contributions: a positive one, corre-sponding to emission, and a negative one, correspondingto absorption: dudl ( l, ω ) = ε ( ω ) − α ( ω ) u ( l, ω ) . (1)The coefficients ε ( ω ) and α ( ω ) are the emissivity and ab-sorptivity of the body. Now, Kirchoff’s law states that,although ε ( ω ) and α ( ω ) largely depend on the constitu-tion of the material, at thermal equilibrium, their ratio J T ( ω ) ≡ ε ( ω ) /α ( ω ) is universal ; it depends on temper-ature and frequency only. A good absorber ( α ( ω ) large)at a certain frequency is also a good emitter ( ε ( ω ) large)at that frequency, and vice versa.At this point, our mindful student’s worries should al-ready be eased: if Kirchhoff’s law is right, then black-bodies, which are by definition excellent absorbers, mustalso be excellent emitters. But further reflection shouldreveal a caveat in this line of thought: emissivity andabsorptivity usually depend on the actual material used,while the blackbody radiation it emits does not. Why isthat?Some insight into this question is provided by the fol-lowing consideration, due to Einstein. The interactionbetween matter and radiation boils down to transitionsbetween energy levels: given two levels a and b , withrespective energies E a < E b , emission corresponds toan upgoing transition a → b , while absorption corre-spond to a downgoing transition b → a . The rate ofthese transitions, Γ a → b and Γ b → a , is what controls at themicroscopic level the absorptivity and emissivity of thebody. Now, the condition of thermal equilibrium fixesthe probabilites of each levels, through the Gibbs distri-bution p a,b ∝ e − Ea,bkBT . But, and this is the key point, ithas no bearing on the transition rates Γ a → b and Γ b → a themselves, but only on their ratio . Indeed, in this mi-croscopic perspective, thermal equilibrium translates intothe condition of detailed balance , according to which theprobability flux between microstates cancel exactly: p a Γ a → b = p b Γ b → a . (2)This leads to Γ a → b Γ b → a ∝ e − ~ ωabkBT , (3)in which the RHS is a function of the transition frequency ω ab and temperature only. Kirchhoff’s law is a conse-quence of this constraint on the transition rates imposedby detailed balance. Combining the phenomenological radiative transferequation (1) with Einstein’s microscopic model, we un-derstand that thermal equilibrium of the material source,through the condition of detailed balance, constrain theratio of emission to absorption – Kirchhoff’s law – butnot their respective rates independently: this is why dif-ferent materials have different thermal emission and ab-sorption properties. It is only under a further assumption that a universal function describing blackbody emission– Planck’s function – can be obtained.What is this further assumption? Is it that the body“absorb all light incident on it”, as in the standard defini-tion? In other words, that the body be perfectly absorb-ing (‘black’) on the whole spectrum? No! Such a materialdoes not exist. That is, the condition of ‘blackness’ en-tering the standard definition of a blackbody cannot bethe idealization underlying blackbody radiation. Thenwhat is it?
D. Optical thickness
The answer should be obvious by now: the additionalcondition is that the radiation field itself should be atthermal equilibrium. This means that the random pro-cesses of emission and absorption of light by the hot bodyshould have reached their stationary state. Solving (1)with the initial condition u ( l = 0 , ω ) = 0, we find u ( l, ω ) = (1 − e − α ( ω ) l ) J T ( ω ) (4)In particular, we see that if α ( ω ) l ≫
1, the so-called ‘op-tically thick’ limit, the observed energy density is givenby J T ( ω ), which is nothing but the Planck spectrum. Inother words, the coefficient α ( ω ) measures the rate ofconvergence of the emitted light to its equilibrium value J T ( ω ) – the blackbody approximation is just the condi-tion that α ( ω ) l ≫
1. This is achieved not only for good absorbers ( α ( ω ) large), but also for objects involving longoptical paths ( l large).This is precisely what a closed cavity does: it providesthe conditions for this convergence process to take place,whatever the intrinsic properties ε and α of the materialwithin the cavity. The box, in other words, is a useful ex-pedient for the actual production of thermal radiation inthe laboratory when the hot body is not black : it permitsto reach the optically thick limit even if the absorptivityis low, because it generates very long paths l withing thematerial. But the presence of a box is of course not anecessary condition for the thermalization of light: starstoo are optically thick – and this is why their spectrumlooks Planckian.
III. PLANCK’S LAW WITHOUT THE BOXA. Partition of energy
From the previous discussion, we know that the equilib-rium spectral energy density coincides with the function J T in Kirchoff’s law: the energy contained in thermal ra-diation within an infinitesimal volume d x about a point x in the frequency range dω is dE ( x, ω ) = J T ( x, ω ) d xdω (5)How can one compute this function? The answers tothis question is obvious when one recalls that the totalenergy of the field is distributed over its modes, and thatmodes with the same frequency ω are degenerate, i.e.store the same energy. Hence: dE ( x, ω ) = (energy of each ω -mode) × (occupation number of each ω -mode) × (number of ω -modes accessible at x ) (6)From quantum mechanics, we know that the energystored in a mode with frequency ω is E ( ω ) = ~ ω . More-over, the occupation number n T ( ω ) of each such mode atthermal equilibrium is given by the Bose-Einstein distri-bution, n T ( ω ) = 1 e ~ ωkBT − . (7)It follows that the thermal energy density J T ( x, ω ) isgiven by J T ( x, ω ) = ~ ω e ~ ωkBT − ! ρ ( x, ω ) , (8)where ρ ( x, ω ) denotes the density of modes with fre-quency ω around the point x . Indeed, for an observerlocalized at x , certain modes might be inaccessible, oronly partly accessible: ρ ( x, ω ) is the space-resolved den-sity of modes, often called (using quantum mechanicalparlance) the local density of states (LDOS). Of course,if the medium is homogeneous, this quantity does notdepend on x . However, in more general situations (suchthose described at the end of the next section), it does.All in all, the Planck spectrum can be described as theoutcome of the balance between the energy per modeand the LDOS, which tend to grow as ω gets large,and the Bose-Einstein distribution, which favors the low-frequency modes. This is illustrated in Fig. 2. Planck spectrumBose1Einstein distributionLDOS Energy per photon FrequencyMaximum at ν=2.82kT/h
FIG. 2: Balance between the LDOS, the energy per mode andthe Bose-Einstein distribution.
B. A hint of spectral theory
The reason why the evaluation of the LDOS is non-trivial mathematically is because the light frequenciesspan a continuous spectrum. Had it been discrete, theLDOS would just have been the degeneracy of ω , under-stood as an eigenvalue of the wave operator. In the con-tinuous case, however, the notion of degeneracy is sub-tler, for it involves the definition of a measure on thespectrum. This is analogous to the ambiguity one faceswhen trying to extend a discrete sum to an integral: theappropriate weighting of the integration variable is notobvious anymore. The case of the standard measure onthe sphere, which involves a factor sin θ , is an obviousexample of such subtleties. Of course, in this case, theappropriate measure is determined by the condition of ro-tational invariance. What determines the spectral mea-sure of an operator?A powerful tool to answer such a question is the ‘resol-vent formalism’. Given a (self-adjoint) operator A , theresolvent is the operator-valued function of the complexvariable z defined by R ( z ) = ( A − z ) − . (9) By construction, R ( z ) is analytic in C \ σ ( A ), the com-plement of the spectrum of A . The discrete part of σ ( A )corresponds to isolated poles of R ( z ), while the contin-uous part of σ ( A ) generates a branch cut along the realaxis. In other words, the spectrum of A is encoded in theanalytic structure of R ( z ).Given any two vectors ψ and φ , it follows from thespectral theorem that h ϕ | R ( z ) ψ i = Z σ ( A ) dµ ϕ,ψ ( λ ) λ − z (10)= X λ ∈ σ d ( A ) g ( λ ) λ − z + Z σ c ( A ) ρ ϕ,ψ ( λ ) dλλ − z where dµ ϕ,ψ is the spectral measure associated to ϕ and ψ , and in the second equality the spectrum is decomposedinto its discrete σ d ( A ) and continuous σ c ( A ) parts. Thespectral density ρ ϕ,ψ ( λ ) is therefore defined as the den-sity of the continuous part of the spectral measure dµ ϕ,ψ with respect to the Lebesgue measure dλ .When A is a wave operator, and ( ϕ, ψ ) = ( x, y )are position (generalized) eigenvectors, the quantity G ( x, y ; z ) = h y | R ( z ) x i is called in the physics literaturethe ‘Green function’. The spectral decomposition of itsdiagonal elements G ( x, x ; z ) reads G ( x, x ; z ) = X λ ∈ σ d ( A ) g ( λ ) λ − z + Z σ c ( A ) ρ x ( λ ) dλλ − z . (11)This expression provides the spectral density ρ x ( ω ) withthe following interpretation: the generalized eigenvalue λ , when analyzed through a state localized at the point x , comes with a weight ρ x ( λ ). This weight is the overlapof the density of λ -eigenmodes with the local state | x i –in other words, the LDOS ρ ( x, λ ) of A .Moreover, the formula (11) indicates a procedure toevaluate ρ x ( λ ). Indeed, just as the degeneracy g ( λ ) ofan isolated eigenvalue λ can be read off as the residueof h y | R ( z ) x i at z = λ , the LDOS is given by the dis-continuity of G ( x, x ; z ) along the branch-cut singularity.This is the so-called Stieltjes-Perron inversion formula(Appendix A): ρ ( x, λ ) = 12 iπ lim ǫ → ∆ ǫ G ( x, x ; λ ) , (12)with∆ ǫ G ( x, x ; λ ) = G ( x, x ; λ + iǫ ) − G ( x, x ; λ − iǫ ) . (13) C. Resolvent of the Laplace operator
Let us apply this resolvent formalism to the case ofelectromagnetism. For the sake of simplicity, we shallconsider here the scalar Helmholtz equation – the twoindependent polarizations of the field will be taken intoaccount a posteriori , by multiplying the LDOS thus ob-tained by a factor of 2.The Helmholtz equation in vacuum for a monochro-matic wave φ ω reads − ∆ φ ω ( x ) = ω c φ ω ( x ) , (14)which is an eigenvalue equation for the Laplacian, witheigenvalue λ = ω /c . We introduce the resolvent( − ∆ − z ) − , and evaluate the corresponding Green func-tion G ( x, y ; z ) by Fourier transform G ( x, y ; z ) = Z d k (2 π ) e ik. ( x − y ) k − z . (15) D. The vacuum LDOS
The two integrals G ( x, y ; λ ± iǫ ) are readily evaluatedby residue calculus (Appendix B), yieldinglim ǫ → ∆ ǫ G ( x, x ; λ ) = i √ λ π = iω πc . (16)Using (12) and the fact that dλ = 2 ω/c dω , we obtainthe electromagnetic vacuum LDOS (with a factor of 2 forthe two polarizations): ρ ( x, ω ) = ω π c . (17)Note that, as expected, the translational invarianceof the vacuum translates into the independence of theLDOS on the space point x . Plugging this value into (8)immediately yields Planck’s law, J T ( ω ) = ~ ω π c e ~ ωkBT − . (18) E. Surface effects
We conclude this section by an illustration of thestrength of the LDOS method for a finer analysis of ther-mal radiation, in particular close to a metallic surface.At a distance to the hot body comparable to the thermalwavelength, the vacuum approximation breaks down, asthe field reveals evanescent modes and polariton excita-tions. Such local excitations have been shown to modifysignificantly the LDOS, and hence the hot body spec-trum, notably by enhancing monochromaticity (Fig. 3),spatial coherence, and directivity.
This shows an-other facet of the blackbody approximation which is notusually emphasized: it is a far-field approximation. D en s i t y o f s t a t e s
14 2 3 4 5 6
15 2 3 4 5 6
16 2 3 4 5 6 ω (rad s -1 ) z=1nm z=10nm z=100nm z=1 µ m FIG. 3: LDOS versus frequency at different heights above asemi-infinite sample of aluminium, from Joulain et al. Theresonance around 10 rad/s corresponds to the excitation ofsurface-plasmon polaritons. IV. CONCLUSION
In this paper, we have tried to disentangle two aspectsof the standard picture of a blackbody as a cavity with asmall aperture. The first one relates to the efficiency ofthe thermalization of light through its almost everlastinginteraction with the walls of the cavity: this is the oneinvoked in Landau and Lifschitz’s definition. Although itdoes not make the conditions of thermalization of radia-tion explicit, the cavity picture is useful to demonstratehow a blackbody can be realized in the laboratory. Thesecond aspect relates to the selection of certain frequen-cies, and the subsequent Fourier space mode-counting ar-gument. Unlike the former, this aspect does not merelyserve illustrative purposes, but usually enters the actualderivation of the Planck spectrum. As such, it appearsto students as an important feature of thermal radiation.We have argued that it is not, and that it actually con-tradicts a key feature of blackbody radiation, namely thefact that all wavelengths are emitted.Our approach, focused on local properties of the elec-tromagnetic field, allows to derive Planck’s law withoutusing a blackbox, and provides tools for subtler consider-ations, such as the near-field regime of thermal radiation.With this perspective, we believe that students are lesslikely to miss the point of the blackbody approximation,which expresses an idealization of the interactions be-tween the electromagnetic field and heated materials –and not of the materials themselves.
Acknowledgments
I am grateful to the African Institute for the Mathe-matical Sciences (where this work was started) for giv-ing me, and many others, the opportunity to do sciencein Africa. I also thank Mohammed Suleiman HusseinSuleiman for discussions on Planck’s law and much more,and all the students of the 2007 AIMS promotion for theirenthusiasm and generosity.
Appendix A: The Stieltjes-Perron formula
Let a complex function f be defined as f ( z ) = Z I dt ρ ( t ) t − z (A1)where ρ is a continuous density on the open interval I .We consider the values f ± ≡ f ( λ ± iǫ ), for λ ∈ I , anduse the well-known identity (the Sokhatsky-Weierstrasstheorem)lim ǫ → t − λ ∓ iǫ = P ( 1 t − λ ) ± iπδ ( t − λ ) (A2)where P ( t − λ ) is the Cauchy principal value. It followsthat lim ǫ → ( f + − f − ) = Z I dt ρ ( t )2 iπδ ( t − λ ) (A3)= 2 iπρ ( λ ) , (A4)which is the Stieltjes-Perron inversion formula. Appendix B: Evaluation of G ( x, y ; λ ± iǫ ) Consider the two integrals G ( x, y ; λ ± iǫ ) = Z d k (2 π ) e ik. ( x − y ) k − λ ∓ iǫ , (B1)denoted G ± for simplicity. The angular integration isstraightforward, and yields G ± = 2 π (2 π ) | x − y | Z ∞ dk k sin( k | x − y | ) k − λ ∓ iǫ . (B2)The remaining integral is evaluated using the residue the-orem, considering as integration contours the standardhalf-circles γ ± , enclosing the poles k ± = √ λ ± iǫ respec-tively. We obtain G ± = ± i π sin( k ± | x − y | ) | x − y | , (B3)The discontinuity across the cut is then given bylim ǫ → ( G + − G − ) = i π sin( √ λ | x − y | ) | x − y | , (B4)from which (16) follows by setting y = x . ∗ Electronic address: [email protected] D. J. Fixsen, E. S. Cheng, J. M. Gales, J. C. Mather, R. A.Shafer, and E. L. Wright, The Astrophysical Journal ,576 (1996). L. D. Landau and E. M. Lifschitz,
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For the emissivity to not depend on the frequency, Thom-son scattering should be the dominant process in thematter-radiation interaction, and this cannot be the casenear thermal equilibrium. This explains Landau and Lifschitz’s otherwise cryptic ob-servation that the blackbody is not the cavity itself, butthe “surface of [its] aperture”.26