We may now consider the march of events when a substance is illuminated by light of increasingly shorter wave-length. We will suppose X > X2>X3 . We saw that the refractive index depends on the product of p and the number of atoms in a cubic centimetre, but, as we have omitted a factor of proportionality already, we may also omit the factor for the number of atoms, and may equate p above to 0/(0+2). For very long waves we shall have practically 2) = Bid-B2+ . If is deduced from this it should agree with the dielectric constant determined by purely electrical means. Though there is little doubt that this would be verified, optical observations are usually lacking for sufficiently long waves. As X de creases the first term grows in comparison to the others, because gets more rapidly smaller, and just before this term entirely dominates the refraction. At there will be a region of absorption and on the other side the first term will be negative, the rest still positive. As X decreases further the first term, still negative, shrinks in importance and the second grows until near X2 when it becomes dominant. Passing beyond it changes sign and later shrinks in impor tance, its place being taken by and so on. Finally when all the lines have been passed, 2) will be negative and proportional to the square of the wave-length. For a strict test of the theoretical dispersion against experiment, we should have to work out the value of n from that of 2) ; but for a rough comparison this is not necessary. Where is large so is n, when the former is nega tive n is less than unity, and, if n is plotted against frequency, the curve will have the same general characteristics as has 2). An example of the general course of the de pendence of n on frequency is given without any numerical accuracy in fig. 22. There are supposed to be three lines of which the middle one is the strongest. In the neighbourhood of each line the refraction cannot be observed on account of absorption. As each line is passed there is a strong decrease in the refractive index, associated with the change of phase of the corresponding virtual electron by i8o°. Whether the refractive index actually becomes less than unity will depend on the breadth of the un observable region and on the strength of this line compared to the others. At frequencies higher than the highest frequency of the atom the refractive index will always be less than unity, and such values have been found for X-rays, though, strictly speaking, even here the atom has higher frequencies still.
Much the most striking example of dispersion is the phenomenon called anomalous dispersion, though in fact it is not at all anomalous. There are dyes which strongly absorb certain colours in the spectrum while being transparent to the others; thus if the green is absorbed, the dye will look purple by transmitted light. In consequence of the very strong green absorption there is a reversal in the usual order of refraction. The blue light, which lies on the short wave-length side of the green, is much less re fracted than the yellow which lies on the side of long wave-lengths. Fig. 23 shows with some exaggeration the way in which a prism made of such a colour would fold the spectrum back on itself. A similar effect is shown in Plate, fig. 8 for sodium vapour. Sodium has two very strong lines close together in the yellow, and both of them show anomalous dispersion. A flame containing the vapour is given the form of a prism, and white light is sent through it. This light is then sent through a spectroscope so as
to spread out the colours in a direction at right angles to the dispersion of the prism. The colours on opposite sides of either of the sodium lines will be deflected in opposite directions by the prism, so that their images on the photographic plate are shifted up and down in Plate, fig. 8. The form of the dispersion curve is thus made directly evident.
On account of the brilliance and fineness of the sodium lines, sodium vapour has been used with more success than any other medium in the study of dispersion, and several important results have emerged. It has been found possible to make direct measures of the absorption of light in the immediate neighbourhood of the two lines and so to evaluate the damping factor a; this was found to be in quite good agreement with the electromagnetic damping factor. Another important experiment consists in finding the absolute value of the scattering constants for these two lines of the sodium atom. The theory of this depends on quantum principles, and cannot be given in detail; but, loosely speaking, the two virtual electrons together correspond to a single actual electron, so that, if we can make experiments in which the two lines are indistinguishable, the scattering constant should be given by the use of the ordinary values of e and m. The straight torward way of doing this would be to observe the refraction with light of a very different colour, because then the difference be tween the influences of two such close lines would be insensible, but this is useless because the refraction itself becomes too small. Indirect means depending on magnetic gyration have been used, and have entirely supported the theoretical prediction that both lines are due to a single electron.
The most accurate measures of refractive index have been made with transparent substances, substances in which the ab sorption only occurs far in the infra-red or ultra-violet. To an alyze the dispersion the usual procedure is based on the fact that for the infra-red terms, can be expanded in powers of X', while for the ultra-violet terms it can be expanded in inverse powers of V. A formula is therefore constructed of the type A , and A, C, E are fitted to the observed values of 2). The term in C corresponds to the infra-red lines, and A,E,F to the ultra violet. The actual positions of the lines are then found by trial. The infra-red lines, or, more usually, bands, can often be fixed with fair accuracy, both by experiments with rest-rays and by observing the refraction near them; but the ultra-violet are more troublesome, because it is often not possible to get observations very close to the lines. Indeed it is usually found that wave lengths and B's can be chosen with considerable latitude, and can yet give all the observed results with a high degree of ac curacy. The whole process is very laborious and has only been carried out for a few substances, such as rock-salt and potassium chloride. It is found possible to represent the refraction by one term for the infra-red and perhaps two in the ultra-violet, to gether with a constant term which must correspond to wave-lengths so short that does not change per ceptibly in value in the whole region accessible to observation. The ultra-violet lines are attributed to electron vibrations of some kind, but those in the infra-red arise through the motions of whole atoms, and have been fitted satisfactorily into the general dynamical theory of the crystalline state.