Rayleigh's Law of Extinction and the Quantum Hypothesis

Abstract

THE bearing of Rayleigh's law of gaseous extinction on some of the fundamental aspects of radiation theory does not seem to have been sufficiently emphasised in recent reports and publications on modern molecular physics. The coefficient of attenuation κ of radiation of wave-length λ travelling through a gas containing n₀ molecules per unit volume was given by Rayleigh¹ so long ago as 1871 in the form κ=⁸₄, π ³(µ²0 – I)²λ⁻⁴/n⁰, µ0 being the refractive index of the gas. It is of importance to notice that the law in question is one of the most fundamental results of molecular dynamics, its final expression being an invariant with respect to the theories of the æther or of the molecule employed², while in its derivation there is no need to draw on resources outside classical dynamics and continuous energy-flow. From the point of view of elementary electromagnetic theory, the above expression κ for is very easily derived along lines suggested in a problem set in part ii. of the “Mathematical Tripos”³; use is made of the conventional electrical doublet set into forced vibrations by a train of electromagnetic waves; by making use of the radiation formula for accelerated charges and Poynting's theorem, the flow of energy from the doublet is easily calculated in terms of the amplitude of vibration; the oscillations of the doublet contribute a term to Maxwell's displacement current, enabling the amplitude to be expressed in terms of the refractive index of the gas; by considering the depletion of energy from the original beam as a result of this scattering, and eliminating the amplitude, the above expression for κ is easily obtained. In a recent paper, Natanson⁴ has subjected the derivation of Rayleigh's law to minute criticism on the grounds of the classical electromagnetic theory, allowing for a damping term arising from the mechanical reaction due to radiation, and taking into special consideration the summation of the aggregate radiation from the random distribution of doublets which are supposed to constitute the molecules of the gas; the final result is a vindication of the above expression for the coefficient of attenuation to a very high order of accuracy. It may be noticed in passing that the same electromagnetic system forms the basis of Planck's⁵ theory of black-body radiation, the interpretation of experiment in this case, however, necessitating the hypothesis of discontinuous energy-flow, or the emission of energy by quanta.