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Comment on “Hot gases: The transition from the line spectra to thermal radiation” by M. Vollmer [Am. J. Phys.73 (3), 215–223 (2005)]
1.M. Vollmer, “Hot gases: The transition from line spectra to thermal radiation,” Am. J. Phys. 73, 215–223 (2005).http://dx.doi.org/10.1119/1.1819931
2.A. Einstein, “Zur Quantum Theorie der Strahlung,” Phys. Z. 18, 121–128 (1917).
2.First printed in Mitteilungender Physikalischen Gesellschaft Zurich. No. 18, 1916. Translated into English in Van der Waerden, Sources of Quantum Mechanics (North-Holland, Amsterdam, 1967), pp. 63–77.
3.The first treatment of this transformation in a thermal gas of two level atoms was given by E. A. Milne, “Thermodynamics of the stars,” Handbuch der Astrophysik 3, Part 1, 159–164 (1930).
3.This article is reprinted in Selected Papers on the Transfer of Radiation, edited by D. H. Menzel (Dover, New York, 1966), pp. 173–178.
3.Milne’s derivation has been reproduced, almost unchanged, in several astrophysics textbooks that discuss radiative transfer. See, for example, S. Chandrasekhar, An Introduction to the Study of Stellar Structure (Dover, New York, 1958), pp. 205–207;
3.D. Mihalas, Stellar Atmospheres (Freeman, San Francisco, 1978), pp. 336–339;
3.A. Peraiah, An Introduction to Radiative Transfer (Cambridge University Press, Cambridge, 2002), pp. 15–16.
3.In all these treatments the effect of the Doppler shift on the radiation frequency due to the random motions of the atoms in a hot gas has been included only for its effect on the line width, although as we show here, it is this shift that can also give rise to thermal radiation continuous in the frequency of the radiation.
4.In our derivation of Planck’s black-body formula we required both conservation of energy and momentum in the emission and absorption of photons. In his original derivation Einstein applied only the conservation of energy, “for we have only formulated our hypothesis on emission and absorption of radiation for the case of stationary molecules.” Subsequently, he showed by a very elaborate calculation of the momentum transfer of radiation to the atoms that the mean thermal energy of the atoms satisfies the equipartition theorem, assuming that the elementary process transmits an amount of momentum . He concluded that if a radiation bundle has the effect that a molecule struck by it absorbs or emits a quantity of energy in the radiation, then a momentum is always transferred to the molecule (Ref. 2). Our derivation demonstrates the relevance of this momentum transfer to the formation of a thermal spectrum in hot gases.
5.M. Nauenberg, “The evolution of radiation towards thermal equilibrium: A soluble model which illustrates the foundations of statistical mechanics,” Am. J. Phys. 72, 313–323 (2004). In this paper I assumed that the atoms are fixed, but the extension to randomly moving atoms in a thermal gas is straightforward along the lines indicated here.http://dx.doi.org/10.1119/1.1632488
6.After obtaining Eq. (11), Vollmer writes: “Note we assumed discrete energy levels and transitions between them. But suddenly the result, Eq. (11), is interpreted as a continuous spectrum. Where do all the possible transitions arise? The answer is simple: each atom or molecule possesses at least one resonance frequency for absorption. An absorption line extends over the entire spectrum because it has a finite width, that is, each atom or molecule can absorb radiation at every wavelength, although with different cross sections…,” Ref. 1, p. 218.
7.Vollmer writes: “The justification for the assumption that the Doppler width or collisional width is not important [to explain the formation of the continuous spectrum] comes from a numerical estimate of the Doppler width and the effects of pressure broadening, ” Ref. 1, p. 216.
8.E. Fermi, “Quantum theory of radiation,” Rev. Mod. Phys. 4, 87–132 (1932).http://dx.doi.org/10.1103/RevModPhys.4.87
9.R. Loudon, The Quantum Theory of Light (Oxford University Press, New York, 2000), p. 67.
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