XVIII. An Electromagnetic Illustration of the Theory of Selective Absorption of Light by a Gas. By Professor HORACE LAMB, M.A., F.R.S.

[Received 13 December 1899.]

THE calculations of this paper, so far as they are new, were undertaken with a view of obtaining a definite mathematical illustration of the theory of selective absorption of light by a gas. The current theories of selective absorption apply mainly to the case of molecules in close order, and it has not been found possible to represent the dissipation of radiant energy except vaguely by means of a frictional coefficient. It seems therefore worth while to study in detail some case where the dissipation can be exactly accounted for; and to consider in the first instance the impact of a system of plane waves on an isolated molecule.

If we assume that the molecule has a spherical boundary, then, whether we adopt the electric or the elastic theory of light, the requisite mathematical machinery is all ready to hand. It is necessary, however, for our present purpose to devise a molecule which shall have a free period of vibration, whether mechanical or electrical, of the proper order of magnitude. The mechanical analogy was in the first instance pursued, the aether being represented by an incompressible elastic medium. This enables us to illustrate many special points of interest, but for the purpose of a sustained comparison with optical phenomena the elastic-solid theory proved in the end to be unsuited from the present point of view, as well as on other well-known grounds.

As regards the electric theory, the scattering of waves by an insulating sphere has been treated by various writers*, with however the tacit assumption that the dielectric constant (K) of the sphere is not very great. In the present paper attention is specially directed to the case where K is a very large number. On this supposition free oscillations (of two types) are possible, whose wave-lengths (in the surrounding medium) are large compared with the periphery of the sphere, and whose rates of decay (owing to dissipation of energy in the form of divergent waves) are comparatively slow. And when extraneous waves whose period is coincident, or nearly coincident, with that of a free oscillation encounter the sphere, the scattered waves attain an abnormal intensity, and the original wave-system is correspondingly weakened.

* Lord Rayleigh, Phil. Mag., Feb., 1881, and April, 1899; Prof. Love, Proc. Lond. Math. Soc., t. xxx., p. 308; G. W. Walker, Quart. Journ. Math., June, 1899.

The conception of a spherical molecule with an enormous specific inductive capacity is adopted here for purposes of illustration only; and is not put forward as a definite physical hypothesis. In order to comply with current numerical estimates of molecular magnitudes, it is necessary to assume that for the substance of the sphere K has some such value as 107. This assumption may be somewhat startling; but it is not necessarily inconsistent with a very moderate value of the specific inductive capacity of a dense medium composed of such molecules arranged in fairly close order. And it may conceivably represent, in a general way, the properties of a molecule, regarded as containing a cluster of positive and negative 'electrons.' In any case the author may perhaps be allowed to state his conviction, that difficulties (such as they are) of the kind here indicated will prove to be by no means confined to the present theory.

The main result of the investigation may be briefly stated. For every free period of vibration (with a wave-length sufficiently large in comparison with the diameter of a molecule), there is a corresponding period (almost exactly, but not quite, coincident with it) of maximum dissipation for the incident waves. When the incident waves have precisely this latter period, the rate at which energy is carried outwards by the scattered waves is, in terms of the energy-flux in the primary waves,

where is the wave-length, and n is the order of the spherical-harmonic component of the incident waves which is effective. In the particular case of n = 1, this is equal to 477x2. Hence in the case of exact synchronism, each molecule of a gas would, if it acted independently, divert per unit time nearly half as much energy as in the primary waves crosses a square whose side is equal to the wave-length. Since under ordinary atmospheric conditions a cube whose side is equal to the wave-length of sodiumlight would contain something like 5 x 10 molecules, it is evident that a gaseous medium of the constitution here postulated would be practically impenetrable to radiations of the particular wave-length.

It is found, moreover, on examination that the region of abnormal absorption in the spectrum is very narrowly defined, and that an exceedingly minute change in the wave-length enormously reduces the scattering.

It may be remarked that the law expressed by the formula (1) is of a very general character, and is independent of the special nature of the conditions to be satisfied at the surface of the sphere. It presents itself in the elastic-solid theory; and again in the much simpler acoustical problem where there is synchronism between plane waves of sound and a vibrating sphere on which they impinge.

It has unfortunately not seemed possible to render this paper fairly intelligible without the preliminary recital of a number of formula which have done duty before, notably in Prof. Love's paper. The analysis has however been varied and extended in points of detail, with a view to the requirements of the present topic. In particular,

the general expression for the dissipation of energy by secondary waves, which is obtained in § 5, is found to take a very simple form, and may have other applications. Some notations which are of constant use in the sequel may be set down for reference. We write

These may be taken as the two standard solutions of the differential equation

the solution (5) being that which is finite for = 0. In the representation of waves divergent from the origin we require the combination

The functions Yn (5), Yn (5), fn() all satisfy formulæ of reduction of the types

(5).

1. The equations to be satisfied in a medium whose electric and magnetic permeabilities are K and μ may be written, as in Prof. Love's paper,

where (X, Y, Z) is the electric force, (a, B, y) the magnetic force, and c denotes the wave-velocity in the aether. Assuming a time-factor eist, we find

(V2 + h2) X=0, (+h2) Y=0, (V2+h2) Z=0

......

† See Lord Rayleigh's Sound, § 327.

..(11),

(12),

..(13).

When values of X, Y, Z satisfying these equations have been found, the corresponding values of a, B, y are given by (10). Or, we may reverse the procedure, determining the general values of a, B, y by means of equations similar to (11) and (12) and thence the values of X, Y, Z by means of (9).

The solutions of (11) and (12) subject to the condition of finiteness at the origin are of two types. In the first place we may have

where T is a spherical surface-harmonic of order n*. These make

ก

It is known that the most general solution of our equations, consistent with finiteness at the origin, can be built up from the preceding types, by giving n the values. 1, 2, 3,

....

2. Let us now suppose that a sphere of radius a, having the origin as centre, whose electric and magnetic coefficients are K and μ, is surrounded by an unlimited medium (the aether) for which K=1 and μ 1. The disturbance in this medium may be regarded as made up of two parts. We have, first, the extraneous disturbance due to sources at a distance; this is supposed to be given. Secondly, we have the waves scattered outwards by the sphere.

The general expression for the extraneous disturbance is conditioned by the fact that if the medium were uninterrupted the electric and magnetic forces at the origin would be finite. It is therefore made up of solutions of the type already given, provided we put K = 1, μ= 1, and replace h by k, where

k = σ/c

As usual, 2π/k is the wave-length of plane waves of the period 2π/σ.

..(26).

In the corresponding expressions for the divergent waves, we must further replace Yn (hr) by ƒn (kr), where fʼn is the function defined by (5). This is necessary in order that the formulæ may represent waves propagated outwards, the complete exponential factor being then eik (et—r) ̧

It is necessary to have some notation to distinguish the surface-harmonics used to represent different parts of the disturbance. Those harmonics which occur in the expression for the imposed extraneous disturbance will be denoted by Tn, Un, simply; those relating to the scattered waves by T, Un; and those relating to the inside of the sphere by Tn, Un