dispersion equation by imagining that there are several different kinds of oscil- lators, but that each oscillator acts separately, and so we simply add the contri- butions of all the oscillators. Let us say that there are Nk electrons per unit of lume whose natural frequency is wk and whose damping factor is tk. We would then have for our dispersion equation n=1 N w"+iRa We have, finally, a complete expression which describes the index of refraction that is observed for many substances. The index described by this formula varies with frequency roughly like the curve shown in Fig.31-5 You will note that so long as w is not too close to one of the resonant frequen- cies, the slope of the curve is positive. Such a positive slope is called"normal" ec dispersion(because it is clearly the most common occurrence). Very near the resonant frequencies, however, there is a small range of o' s for which the slope is negative. Such a negative slope is often referred to as"anomalous"(meaning abnormal) dispersion, because it seemed unusual when it was first observed, long Fig. 31-5. The index of refraction as before anyone even knew there were such things as electrons. From our point of a function of frequency. view both slopes are quite"normal""! 31-4 Absorption Perhaps you have noticed something a little strange about the last form Eq. 31. 20)we obtained for our dispersion equation. Because of the term i we put in to take account of damping, the index of refraction is now a comple number!What does that mean? By working out what the real and imaginary parts of n are we could write (31.21) where n'and n"are real numbers. (We use the minus sign in front of the in because then n"will turn out to be a positive number, as you can show for yourself.) We can see what such a complex index means by going back to Eq. (31.6) which is the equation of the wave after it goes through a plate of material with an index n. If we put our complex n into this equation, and do some rearranging, we n"△x/e。-i(n'-1)△zlep,i(t-2/c) (31.22) The last factors, marked B in Eq.(31. 22), are just the form we had before, and again describe a wave whose phase has been delayed by the angle w(n'-1)4z/c in traversing the material. The first term(A)is new and is an exponential factor with a real exponent, because there were two is that cancelled. Also, the exponent is negative, so the factor is a real number less than one. It describes a decrease in the magnitude of the field and, as we should expect by an amount which is more the larger 4z is. As the wave goes through the material, it is weakened. The material is ""absorbing"part of the wave. The wave comes out the other side with less energy. We should not be surprised at this, because the damping we put in for the oscillators is indeed a friction force and must be expected to cause a loss of energy. We see that the imaginary part n"of a complex index of refraction represents an absorption(or"attenuation")of the wave. In fact, n"is sometimes eferred to as the" absorption index We may also point out that an imaginary part to the index n corresponds to bending the arrow Ea in Fig. 31-3 toward the origin. It is clear why the transmitted field is then decreased Actually, although in quantum mechanics Eg. (34.20)is still valid hydrogen, has several resonant frequencies. Therefore N is not really the number of electrons having the frequency wk, but is replaced instead by Nfk, where N is the number of atoms per unit volume and fk(called the oscillator strengthis a factor that tells how strongly the atom exhibits each of its resonant fr