31-3 Dispersion Notice that in the above process we have obtained ng ve For we have not only a number for the index of refraction which can be computed F from the basic atomic quantities, but we have also learned how the index of refraction should vary with the frequency w of the light. This is something we would never understand from the simple statement that " light travels slower in a transparent material. "We still have the problem, of course, of knowing how many atoms per unit volume there are, and what is their natural frequency wo. We do not know this just yet, because it is different for every different material, and we cannot get a general theory of that now. Formulation of a general theory of the properties of different substances-their natural frequencies, and so on-i possible only with quantum atomic mechanics. Also, different materials have different properties and different indexes, so we cannot expect, anyway, to get a general formula for the index which will apply to all substances However, we shall discuss the formula we have obtained, in various possible circumstances. First of all, for most ordinary gases(for instance, for air, most colorless gases, hydrogen, helium, and so on) the natural frequencies of the electron oscillators correspond to ultraviolet light. These frequencies are higher than the frequencies of visible light, that is, wo is much larger than w of visible light, and to a first approximation, we can disregard w 2 in comparison with a3. Then we find that the index is nearly constant. So for a gas, the index is nearly constant. Thi is also true for most other transparent substances, like glass. If we look at our expression a little more closely, however, we notice that as w rises, taking a little bit more away from the denominator, the index also rises. So n rises slowly with frequency. The index is higher for blue light than for red light. That is the reason why a prism bends the light more in the blue than in the red The phenomenon that the index depends upon the frequency is called the phenomenon of dispersion, because it is the basis of the fact that light is "dispersed by a prism into a spectrum. The equation for the index of refraction as a function of frequency is called a dispersion equation. So we have obtained a dispersion equa- tion.(In the past few years"dispersion equations"have been finding a new use in the theory of elementary particles. Our dispersion equation suggests other interesting effects. If we have a natural frequency wo which lies in the visible region, or if we measure the index of refraction of a material like glass in the ultraviolet, where w gets near wo, we see that at frequencies very close to the natural frequency the index can get enor mously large, because the denominator can go to zero. Next, suppose that w is greater than wo. This would occur, for example, if we take a material li say,and shine x-ray radiation on it. In fact, since many materials which are to visible light, like graphite for instance, are transparent to X-rays, we can also Ik about the index of refraction of carbon for All the natural fre of the carbon atoms would be much lower than the frequency we are using in the X-rays, since x-ray radiation has a very high frequency. The index of refraction is that given by our dispersion equation if we set wo equal to zero(we neglect wl comparison with A similar situation would occur if we beam radiowaves(or light) ree electrons. In the upper atmosphere electrons are liberated from their atoms by ultraviolet light from the sun and they sit up there as free electrons. For free electrons wo=o(there is no elastic restoring force). Setting sion equation yields the correct formula for the index of refraction for radiowaves in the stratosphere, where N is now to represent the density of free electrons(num r per unit volume)in the stratosphere. But let us look again at the equation, if we beam x-rays on matter, or radiowaves(or any electric waves)on free electrons ) becomes negative, and we obtain the result that n is less than one. That means that the effective speed of the waves in the substance is faster than c! Can that be c It is correct. In spite of the fact that it is said that you cannot send signals any faster than the speed of light, it is nevertheless true that the index of refraction of materials at a particular frequency can be either greater or less than 1. This