We have only treated of the small effect of a thin film, and this contains the essence of the process, but it is of course necessary to discuss matter in bulk. Here the scattered wave from every atom acts on every other atom and so complicates the wave scattered by it. Nevertheless the problem proves soluble and leads to a result not very different from the simpler case. The main difference is that we now have -3 — p• in the specialcase when Np is small, n is near unity, and this reduces to as before. The general solution for oblique incidence verifies all the formulae that are given by the ordinary bulk theory, but the idea of scattering is helpful in seeing directly how reflection and refraction come about. When light falls on a thick slab, the atoms are set in motion and their scattered waves are all in definite phase relations to one another. If we consider a point outside the face of the slab, it will receive all these waves, but those from the interior will have phases spread uniformly round the 36o° and so will cancel out. Thus the reflected wave will arise from the atoms in the face where this uniformity ceases to hold. The existence of the polarizing angle becomes immedi ately obvious, as it is nothing but the rule that in a line-wave there is no emission towards its pole, which is perpendicular to the direction of the refracted wave.
The formula —4-rNp was discovered by Lorentz (by a rather different method) and becomes the foundation of the theory of dispersion. He deduced an important consequence from it. When a substance can exist in two states, p will be nearly the same for both and N will be proportional to the density, so that, if d is the density of either state, d should be the same for the two. This is verified by comparing the refractive index of a liquid and its vapour. Since their densities often differ by a factor of some hundreds, it is a very stringent test and is often fulfilled to within one or two per cent. We should hardly expect perfect agreement, because the liquid molecules are being perpetually disturbed by one another, so that there may be a small change in the value of p attached to each molecule.
The explanation of double refraction follows a very similar course. The sheet of matter must now be supposed to react dif ferently under the stimulus of light according as it is polarized along x or along y. In each case there is a line wave emitted with pole along the direction of the incident force, but the amplitudes are different and so the phase changes in the two components of the transmitted wave will be different. The detailed consider ation of the effect for matter in bulk is of course more complicated, but leads to Fresnel's normal surface and all its consequences. This part of the theory is complete, but there are difficulties in giving an atomic meaning to p, because the adjacent atoms in a crystal are not arranged isotropically and will perturb one another in a complicated manner. As a consequence, the expression
ceases to apply; but the full discussion can only be made by a detailed study of the theory of the solid state. With the help of this theory and a knowledge of the arrangement of the atoms in calcite it has been found possible to explain its high double refraction with fair numerical accuracy.
It has been mentioned that transparent substances are usually opaque for light in some parts of the spectrum, and the theory of this is fairly well understood. In these regions the phase i can take any value between o° and 180°. This theory however does not apply to metals, and it is a very interesting fact that, if we use the observed values of n and K to calculate p and n for metals, the phase change is in all cases quite small. For the very highly reflecting sodium it is only 7', and for silver little more than 1 , while, even for such a poor reflector as steel (58% at perpendicular incidence), the phase change is not io°. These facts are quite unexplained ; they suggest however that the phase change is not the typical characteristic of metals, but that their high reflection is rather to be attributed to a large scattering power ; for, if we consider a transparent substance in which Np 3 is greater than unity, we find that must be negative and there fore n is imaginary, and this means that the waves in the medium are strongly damped. From this point of view metallic reflection is more like total internal reflection than like the reflection from an absorbing substance.