." The Planck Radiation Functions Russell D. Larsenl Texas Tech University, Lubbock, TX 79409 The elucidation and quantification of hlackhody radiation by Planck in 1901 ( 1 , 2 ) is considered to have been the heginning of the development of quantum theory. Many iutroductory chemistry textbooks discuss the lines of thought that lead to the emergence of ouantum theorv and the necessitv of Plauck t o for;hulate thk quantizatiod of radiation. ~ h & postulate was necessary in order to explain the observed differences in the radiation emitted by a hlackhody a t a certain temnerature comoared to the classical descri~tionof radiation coniained in the-relations known as the ~ i e f a n - ~ o l t z m a n n Law, Kirchhoff Law, Wien Displacement Law, and Rayleigh-Jeans formula ( 3 , 4 ) . Most Dhvsical chemistrv texts develop the blackbody problem in some detail eithbr as prequant&n background or as an a ~ ~ l i c a t i oofnS ~ ~ I ~ S I Imechanics CR~ ( 5 ) .T h r theory of blackbody radiation is central to spectroscopy because all bodies radiate and absorb different amounts of energy depending on their temperature. The basic features of the hlackhody radiation problem are simple enough. They are contained in the Wien displacement , = 2897.82 law which, prior to Planck, was inexplicable: AT p K. This law expresses the manner in which the color of emitted radiation varies with the temperature of the emitting source. The maximum wavelength of any spectral distrihution times the temperature of the source is a constant; moreover, as the temperature increases the entire spectrum and the maximum wavelength shift toward the violet end of the spectrum. Planck showed that the Wien constant is really a combination of fundamental physical constants, hcl 4.96511423 k. In fact, however, the Wien constant is coupled to the functional deprndc.nce chosen to express the specrral enerw .. distributtm iunctions. As a consequence, the spectral energy maximum a t a given temperature depends upon whether one is plotting frequency or wavelength and also upon the units of flux or intensity. The familiar Planck distrihution function expressed in terms of wavelength is, indeed, characterized by a certain maximum at a given temperature. However, a t that same temperature other maxima occur a t considerably different portions of the energy spectrum when frequency or wavenumber functions are used. Physical chemistrv texts eive the erroneous imoression that a unique maximum exists at a given temprrature for blackbody radiation and, by implication, that there is a single Wien relation. It is not well known, a t least among most chemists, that there are not less than 12 Planck radiation functions or hlackhody radiation formulae (see Table 1) (6, 7). These functions, while having different names and units, are really functions of different variables and so have their maxima in
different regions of the spectrum. A complete set of computer-generated plots for the 12 Planck functions is available from the author. A few of these drawings are included herein (Figs. 1-5).
Figure 1. Planck function, ur, radiam density, wavelength units.
Figure 2. Planck function, W..
radiant eminance,frequency units.
Work done, in part, at The University of Michigan, Ann Arbor Table 1. Planck Radiation Functions spectral
radiance
M i l r) MIr ) ~ ( r )4 #nl r)
spectral density
radiant
Mil r) u(4 r ) 4 r) Mnl r)
spectral radiant emitlance
Whl r ) m 4 r) mvl r ) mnl r)
Figure 3. Planck function.h,radiant intensity,wavenumber units. Volume 62 Number 3 March 1985
199
The blackbody distribution relations are very important in astronomy hecause the ahsorption and emissrm of rudlatitm are the central astruphysical properties that matter possesses \A\. The tluv or in~cnsitv(flux r,er m i t s~litlanale) of radiation with most interest being in is directly ohservab~equantk~, the emission of radiant enerev a t certain freauencies. I t is common to see the &ckbody distribution function expressed as a function of wavelength and usually given in the form of the radiant spectral density, U A :
s
Table 2. Radlanl Spectral Energy Conversion Fadors IniIial Energy Function
L may be any
Multiplier used to obtain final energy function 11
L,
L.
of Ow radiant spears1 energy functions demkd by u.
L;
W. or N. mese
relatlmalna honfwallof Ow aephadisbib~tim fumions.Fw example, 6 = P N and ~. J, = AJI.
This function is plotted in Figure 1 as a function of wavelength. The fundamental constants h , c, and k are Planck's constant, the speed of light, and Boltzmann's constant. We employ SI units throughout. The radiant density, u, at agiven point in space is the radiant energy per unit volume in the vicinity of that point (6). The 12 Planck radiation formulae all involve radiation measured in units of energy. In addition, radiation can be measured in terms of the number of ~ h o t o n emitted s which leads to an analogous number of photon distrihution relations (7).The central three radiant enerev functionsare the radiant density, u; the radiant emittancc W, and the radiance (or closely related concept of radiant intensity), N (or B). Usually, interest concerns that part of the radiant energy which lies within a spectral interval or band rather than the total radiant energy (integrated over 4a steradians). Such quantities give rise to the spectral radiant density, the spectral radiant emittance, and the spectral radiance. These three spectral radiant energy functions each can be expressed in terms of wavelength (A), wavenumber (q,frequency (u) or logarithm of wavelength (a). For example, the spectral radiant densities U A , u,, u;, and u, are different
functions which have maxima in different regions of the spectrum-they are not merely the same quantitv expressed in different units. However, these four qunntities-are~related cnoneanuther asshown in'Ihhle2. Similarly,eachof the four spectral emittances and spectral radiances are related. l'he spectral radiant density in terms of wowlength U A is defined by:
~~~~
ui=-
Uh;A+dA
dX and corresponds to that part of the total radiant density in the wavelength interval (A, A + dA). The other three spectral radiant density functions are defined by similar spectral interv a l ~In . ~addition to U Agiven by eqn. (1)the radiant emittance, W A(or F A ) , and the radiant intensity, N A(or BA),are closely related quantities which differ by a factor of a:
where
I t is common to express the Planck functions in terms of the so-called radiation constants, cl and ca, defined herein as:
and
where C2n = 1 steradian. ~here"aremany alternate definitions of c l so i t is important to check e l when the Planck functions are so expressed. For example,
Figure 4. Planck function, W,, radiant emhtanca, log wavelength uniIs.
The maxima of these three functions for which wavelength is the variable occur a t the same point in the spectrum for fixed temperature. A maximum is obtained in the usual way by equating the first derivative to zero. For example,
where
+
Figure 5. PhOton distribution function. J.. 200
emhtance, lrequency units.
Journal of Chemical Education
+
It is important to note that the spectral intervals (A. A dA) and (u, v du) are not of the same width. It is preferableto consider equal relative band widths on a logarithmic basis such that qlnu,) = d(lnv,) and d(lnA,) = d(lnA2).
and r = edAT
(10)
The photon distribution functions are of special interest t o atmospheric chemists (13).The most useful of such functions are those expressed in terms of wavelength and frequency,
Iteratively solving the resulting transcendental equation r=5-5ec'
(11)
gives a value of rh = 4.9651142317.. . so that,
Jhis the emittance of photons per cm2per s per unit X; as such i t is a spectral photon emittance. J h is related to the spectral radiant emittance by
which is the Wien displacement law for wavelength as a variable. If frequency or wavenumber instead of wavelength are variables, the Wien law has a different constant because the Planck radiation functions must be transformed using
The wavelength of maximum photon emission is found from
where $ is given as in eqn. (9). The resulting transcendental equation is now,
so that,
which is solved iteratively to give uh = 3.920690.
The maxima of these three functions also occur a t the same wavelength but a t a wavelength different from the LAfunctions. For example,
where wew ,$=-=-
W
em - 1 1 - C W
(18)
and
so that the resulting transcendental equation is now, for which w , = 2.821439. Therefore,
This value of the constant w ,also holds for the maxima of the three radiant energy functions expressed as a function of wavenumber. In this case, as E = 1/X,
In a manner similar to eqn. (2). the spectral radiant energies are related when expressed as functions of frequency, wavenumber, or logarithm of wavelength:
. .so that
I t is interesting that the maxima of the in A functions, N,, W,, and u,, also correspond to this version of the displacement law; the reason is that they, too, are inverse fourth power wavelenath functions. sever2 authors have argued that the reason that the PIanck enerrv distrihution functions should be plotted per unit loga r i t k c interval as in thew functions is because such fun;tions are more nearly symmetrical and the area under the curve is then proportional to energy (9,10). The resulting s ~ e c t r amaximum, l that given by eqn. (32), gives the maximum efficiency in producing rad~ationof'a given wavrlength ( I 1 ). The maxima using log X or log I , are the sam~%ndthus do not change when fuictional dependence on either X or u is used (12). Thus, although Wien's law is not unique in that there are a t least four versions of it, (the fourth being for the photon distrihution, J,), the maximum obtained by differentiation of Planck's formulae with respect t o log frequency, log wavelength, or log wavenumber result in a unique maximum (called Xo) corres~ondingto e m . (32). Bracewell (12) has argued thatthe 6000 K solar s p e c & n should he considered to have a natural maximum a t Xo which is located at the red end of the visible (611.5 nm). using the Wien constants given by eqns. (12) and (21), results in solar maxima a t 482.9 nm and 849.7 nm, respectively. In atmospheric chemistry the photon distrihution functions are preferred over the Planck energy distrihution functions because in solar photolysis the basic construct is photons and not watts (J s-I) (13). In terms of the foregoing prohlem of the multiplicity of Planck functions, their maxima, and corresponding Wien relations we can show that the photon distribution functions also suffer a similar nonuniqueness of spectral maxima. For example, JAhas the transcendental equation,^ = 4 -4ecT, w h e r e q = hclX-kT,x~ = 3.920690, =3669~ K. On the other hand, J, satisfies, for which A T, =9023~ K. x, = 2 - 2e-', x , = 1.5936243, so that X,,T A unique, functionally independent photon distribution function is the logarithmic derivative, J,, where J, = IdXI dlnXIJh = IdXldlnulJ, = IdXldlnEl JF= X J A . This function is given by
where, for example,
Volume 62
Number 3
March 1985
20 1
Department of Astronomy of the University of Michigan for helpful discussions; helpful discussion of the significance of photon distribution functions was provided by Donald Stedman of the Department of Chemistry of the University of Denver.
where The maximum of this function is obtained from aJ,/dw = 0, which yields the transcentental equation, u = 3 - 3ecY,where u = 2.821439. The Wien relation becomes: AT , = 5097p
K. We note in conclusion that in both the cases of the energy and photon distribution functions the functionally indepeudent Wien relations for the o functions, L,, (u,,N,, M,,J,, etc.) lie between the L h and L, functions. The optimal relation for J , which satisfies the equation, u = 3 - 3e-", is of lower magnitude than the corresponding optimal relation for W,, v = 4 - 4ecU,because the photon distribution function is related to the energy function by J = Wlhu. Acknowledgment The author wishes to thankR. Teske and C. Cowley of the
202
Journal of Chemical Education
Literature Cited (1) Planek,M.,Ann.Phyaih.4,553(1901). (2) Plan&. M. and Masius, M.. "The Theory of Heat Radiation," Mffiraw-Hill Book Co., Inc.,NewYork, 1914. (3) Richtmyer,F. K.. Kennard,E. H.,andLsuirsen, T."htduetion taMdemPhy.iea," 50, ed., McCraw~HillBwk Co.,New York, 1955. (4) Lei~hton.R. B., "Principles of Modern Physim," Mffiraw-Hill Bmk Co., bc..New York, 1959. pp. 60-65. (5) See,for example, Adamson, A. W., "A Tertbmk of Phmical Chemh.try,"A u d e m i ~ Prela. New York. 1973, pp. 791.795. (6) Pivovonky. M., and Nagel, M. R.. "Tables ofBls&body Radiation Punetions..'Msemillan. New York. NY. (7) Allen, C. W., "A~trophysicalQuantitier,l'3rd4..AtMone Presa, 1973, p. 104 (8) Aller. L.H.."Aatrophvniea." 2nd ed.. Ronald Presa, New York, 1%3,p. 141. 81 (9) Boerdijk,A. H . . P h i l ~ p s R e ~ . R ~ p . . 8 , 2(1953). (10) Rags, W.D.,J Opt. Soc Arne,., 44.770 (1954). (11) Benford, F..J Opt. Soc. A m c , 29,92 (1939). (12) B~aewell,R.N., Nature. 174,563 (1954). (13) Dickwson,R. R., Sfedman. D. H..Chameidea. W. L., Grutaen,P. *,and Fiabman, J.. Ceophya. Re*. Letters, 6,833 (1979).
