Leroux's Phenomenon

At this time, important developments were in progress in the last-named subject. Since the time of Fresnel, theories of dispersion had proceeded[41] from the assumption that the radii of action of the particles of luminiferous media are so large as to be comparable with the wave-length of light.

It was generally supposed that the aether is loaded by the molecules of ponderable matter, and that the amount of dispersion depends on the ratio of the wave-length to the distance between adjacent molecules.

This hypothesis was, however, seen to be inadequate, when, in 1862, F. P. Leroux[42] found that a prism filled with the vapour of iodine refracted the red rays to a greater degree than the blue rays; for in all theories which depend on the assumption of a coarse-grained lumini. ferous medium,, the refractive index increases with the frequency of the light.

Leroux’s phenomenon was called “anomalous dispersion”.

Later investigators[43] showed that it was generally associated with “surface-colour,” i.e., the property of brilliantly reflecting incident light of some particular frequency.

Such an association seemed to indicate that the dispersive property of a substance is intimately connected with a certain frequency of vibration which is peculiar to that substance, and which, when it happens to fall within the limits of the visible spectrum, is apparent in the surface-colour.

This idea of a frequency of vibration peculiar to each kind of ponderable. matter is found in the writings of Stokes as far back as the year 1852;[44] when, discussing fluorescence, he remarked:—“Nothing seems more natural than to suppose that the incident vibrations of the luminiferous aether produce vibratory movements among the ultimate molecules of sensitive substances, and that the molecules in turn, swinging on their own account, produce vibrations in the luminiferous aether, and thus cause the sensation of light.

The periodic times of these vibrations depend on the periods in which the molecules are disposed to swing, not upon the periodic time of the incident vibrations.”

This principle of considering the molecules as dynamical systems which possess natural free periods, and which interact with the incident vibrations, lies at the basis of We may all modern theories of dispersion.

The earliest of these was devised by Maxwell, who, in the Cambridge Mathematical Tripos for 1869,[45] published the results of the following investigation:—

A model of a dispersive medium may be constituted by embedding systems which represent the atoms of ponderable matter in a medium which represents the aether. picture each atom[46] as composed of a single massive particle supported symmetrically by springs from the interior face of a massless spherical shell: if the shell be fixed, the particle will be capable of executing vibrations about the centre of the sphere, the effect of the springs being equivalent to a force on the particle proportional to its distance from the centre.

The atoms thus constituted may be supposed to occupy small spherical cavities in the aether, the outer shell of each atom being in contact with the aether at all points and partaking of its motion. An immense number of atoms is supposed to exist in each unit volume of the dispersive medium, so that the medium as a whole is fine-grained.

Suppose that the potential energy of strain of free aether per unit volume is

where η denotes the displacement and E an elastic constant; so that the equation of wave-propagation in free aether is

where ρ denotes the aethereal density.

Then if σ denote the mass of the atomic particles in unit volume, (η + ζ) the total displacement of an atomic particle at the place x at tine t, and σp2ζ the attractive force, it is evident that for the compound medium the kinetic energy per unit volume is

and the potential energy per unit volume is

.

The equations of motion, derived by the process usual in dynamics, are

Consider the propagation, through the medium thus constituted, of vibrations whose frequency is n, and whose velocity of propagation in the medium is v; so that η and ζ are harmonic functions of n(t - x/v). Substituting these values in the differential equations, we obtain

ρ/E has the value 1/c2, where c denotes the velocity of light in free aether; and c/v is the refractive index μ of the medium for vibrations of frequency n. So the equation, which may be written

determines the refractive index of the substance for vibrations of any frequency n. The same formula was independently obtained from similar considerations three years later by W. Sellmeier.[47]

If the oscillations are very slow, the incident light being in the extreme infra-red part of the spectrum, n is small, and the equation gives approximately μ2 = (ρ + σ)/ρ: for such oscillations, each atomic particle and its shell move together as a rigid body, so that the effect is the same as if the aether were simply loaded by the masses of the atomic particles, its rigidity remaining unaltered.

The dispersion of light within the limits of the visible spectrum is for most substances controlled by a natural frcquency p which corresponds to a vibration beyond the violet end of the visible spectrum: so that, n being smaller than p, we may expand the fraction in the formula of dispersion, and obtain the equation

which resembles the formula of dispersion in Cauchy’s theory[48]; indeed, we may say that Cauchy’s formula is the expansion of Maxwell’s formula in a series which, as it converges only when a has values within a limited range, fails to represent the phenomena outside that range.

The theory as given above is defective in that it becomes meaningless when the frequency n of the incident light is equal to the frequency p of the free vibrations of the atoms.

This defect may be remedied by supposing that the motion of an atomic particle relative to the shell in which it is contained is opposed by a dissipative force varying as the relative velocity: such a force suffices to prevent the forced vibration from becoming indefinitely great as the period of the incident light approaches the period of free vibration of the atoms; its introduction is justified by the fact that vibrations in this part of the spectrum suffer absorption in passing through the medium. When the incident vibration is not the same region of the spectrum as the free vibration, the absorption is not of much importance, and may be neglected.

The spectroscope shows that the atomic systems which emit and absorb radiation in actual bodies possess more than one distinct free period. The theory already given may, however, readily be extended[49] to the case in which the atoms have several natural frequencies of vibration; we have only to suppose that the external massless rigid shell is connected by springs to an interior massive rigid shell, and that this again is connected by springs to another massive shell inside it, and so on.

The corresponding extension of the equation for the refractive index is

where p1, p2, … denote the frequencies of the natural periods of vibration of the atom.

The validity of the Maxwell-Sellmeier formula of dispersion was strikingly confirmed by experimental researches in the closing years of the nineteenth century. In 1897 Rubens[50] showed that the formula represents closely the refractive indices of sylvin (potassium chloride) and rock-salt, with respect to light and radiant heat of wave-lengths between 4,240 A.U. and 223,000 A.U.

The constants in the formula being known from this comparison, it was possible to predict tho dispersion for radiations of still lower frequency; and it was found that the square of the refractive index should have a negative value (indicating complete reflexion) for wavelengths 370,000 A.U. to 550,000 A.U. in the case of rock-salt, and for wave-lengths 450,000 to 670,000 A.V. in the case of sylvin. This inference was verified experimentally in the following year.[51]

Maxwell successfully employed his electromagnetic theory to explain the propagation of light in:

• isotropic media
• crystals, and
• metals.

It may seem strange that he did not apply it to the problem of reflexion and refraction especially since the study of the optics of crystals had already revealed a close analogy between the electromagnetic theory and MacCullagh’s elastic-solid theory.

In order to explain reflexion and refraction eloctromagnetically, nothing more was necessary than to transcribe MacCullagh’s investigation of the same problem, interpreting ė (the time-flux of the displacement of MacCullagh’s aether) as the magnetic force, and curl e as the electric displacement. As in MacCullagh’s theory the difference between the contiguous media is represented by a difference of their elastic constants, 80 in the electromagnetic theory it may be represented by a difference in their specific inductive capacities.

From a letter which Maxwell wrote to Stokes in 1864, and which has been preserved,[52] it appears that the problem of reflexion and refraction was engaging Maxwell’s attention at the time when he was preparing his Royal Society memoir on the electromagnetic field; but he was not able to satisfy himself regarding the conditions which should be satisfied at the interface between the media.

He seems to have been in doubt which of the rival elastic-solid theories to take as a pattern.

It is not unlikely that he was led astray by relying too much on the analogy between the electric displacement and an elastic displacement.[53] For in the elastic-solid theory all three components of the displacement must be continuous across the interface between two contiguous media.

But Maxwell found that it was impossible to explain reflexion and refraction if all three components of the electric displacement were supposed to be continuous across the interface; and, unwilling to give up the analogy which had hitherto guided him aright, yet unable to disprove[54] the Greenian conditions at bounding surfaces, he seems to have laid aside the problem until some new light should dawn upon it.

This was not the only difficulty which beset the electromagnetic theory.

The theoretical conclusion, that the specific inductive capacity of a medium should be equal to the square of its refractive index with respect to waves of long period, was not as yet substantiated by experiment; and the theory of displacement-currents, on which everything else depended, was unfavourably received by the most distinguished of Maxwell’s contemporaries. Helmholtz indeed ultimately accepted it, but only after many years; and W. Thomson (Kelvin) seems never to have thoroughly believed it to the end of his long life.

In 1888, he referred to it as a “curious and ingenious, but not wholly tenable hypothesis,"[55] and proposed[56] to replace it by an extension of the older potential theories.

In 1896 he had some inclination[57] to speculate that alterations of electrostatic force due to rapidly-changing electrification are propagated by condensational waves in the luminiferous aether. In 1904 he admitted[58] that a bar-magnet rotating about an axis at right angles to its length is equivalent to a lamp emitting light of period equal to the period of the rotation, but gave his final judgment in the sentence[59]:—“The so-called electromagnetic theory of light has not helped us hitherto.”

Thomson appears to have based his ideas of the propagation of electric disturbance on the case which had first become familiar to him—that of the transmission of signals along a wire.

He clung to the older view that in such a disturbance the wire is the actual medium of transmission; whereas in Maxwell’s theory the function of the wire is merely to guide the disturbance, which is resident in the surrounding dielectric.