Free Vibrations of the Atom.—Let us suppose that we have a virtual electron of mass m vibrating along the axis of x, under a force Kx towards the origin. It will obey the equation If left entirely to itself such an electron would vibrate with constant amplitude and frequency
for ever; but it cannot be regarded as left alone, because the moving electricity will be perpetually changing the electric forces everywhere in the manner that we have described as a line-wave. This wave will carry away energy, which is supplied at the expense of the elec tron, so that its amplitude must decrease. The electron is always linked with the aether, and it may be shown that the reaction of the aether on it can be represented by introducing a damping term so that the vibration is expressed by where e is the charge and c as usual the velocity of light. The new term is very small (except for penetrating X-ray frequencies), and the equation may be solved by approximation, and gives
COS 2
This represents a vibration which decreases during each vibration by a fraction
or //X where 1 is
; for an or 3 me 3 me - dinary electron it is 3.7 X r
and, as the charge of a virtual electron is usually smaller, the approximation is evidently justi fiable for ordinary light of wave-lengths about
cm. The line wave emitted by this electron will also be damped with the con sequence that, in Michelson's interferometer, when the two light paths differ by a considerable amount the interference will be come imperfect. This is in fact observed, but usually for path differences somewhat shorter than might be expected. If we imagine the light to be passed through a grating we observe the same phenomenon in a slightly different way; the damping factor gives the line a finite breadth, and in fact spectral lines are nearly always somewhat broader than is indicated by the theory of pure electromagnetic damping. A variety of causes contribute to this broadening; such as the frequent collisions between the radiating atoms and air molecules, for these collisions will change the phases of the emitted waves, and the Doppler effect of the motion of the atoms, which slightly alters the frequency. These effects can both be observed directly by alterations of pressure and temperature, but it is uncertain whether they are sufficient causes of the broadening. The whole question is getting very near to the point where the classical conception of a virtual atom fails, and also to the limits of experimental technique, and all that can be said is that under the most favourable conditions it seems that the damping is not far from the value predicted by electromagnetic theory. We can avoid raising the question by introducing a fictitious damping factor which replaces the electro magnetic and may write as our equation:— so that the electron's motion is resisted by a small force ma
and as long as a is small we do not need to enquire into its origin.
is necessary next to suppose that the atom contains a num ber of virtual electrons with different frequencies, and so to adjust their properties that the spectrum lines will occur in the correct relative strengths. Suppose that the first electron, with charge
frequency
etc., is vibrating with amplitude al. Then it emits a line-wave, and, at distance r in the equator, we have seen that the intensity will be proportional to
In order to compare this intensity with that given by the second electron e, we have to consider what their respective amplitudes will be. This requires an assumption, and the appropriate as sumption to make (subject to some conditions which we shall not discuss) is the equiparation of energy, that on the average each virtual electron has the same energy. Now the energy of tional to
We therefore conclude that the intensity of the first line will depend on
It happens that the quan tity which is most important in spectrum theory is not the in tensity itself, but the intensity divided by the fourth power of the frequency. This quantity has no name, but is always re
ferred to as "Einstein's B" by which letter we shall therefore denote it. Then, for the line of frequency
we so choose
and
that
is proportional to B., and the intensity is pro portional to &J.'.
When the virtual atom is exposed to light there will be an additional force acting on it. The equation of motion of the first electron will now be where F is the amplitude of the incident electric force E and v is its frequency; the direction of the wave-front does not matter. This is the equation which gives the ordinary phenomenon of resonance, and we must consider the form that its solution takes for different frequencies of the incident light. In the first place, a plays practically no part in the solution when it is small, unless v is very near v,. Excluding that case we have If v< vi,
is in phase with the electric force E and its amplitude increases strongly as v approaches v
If v > vi the phases are opposite ; the amplitude is small for large values of v, but as
is approached from above it be comes large (see fig. 20). There is thus a transitional stage when v passes through
and in this stage the amplitude of
has to pass from a large positive to a large negative value. In order to see how it does so we must include the damping term in our equation. There is now a phase difference between xi and E, and the solution is 7 is practically zero, and continues so up to values differing from v, by a not very large multiple of a. From this point on the phase grows rapidly and becomes go° at v = v, and then increases further, so that, at an equal distance on the other side of
it is practically 580°. The amplitude follows the course of fig. 20 nearly up to vi, but instead of becoming infinite it attains a very large maximum at v =
of amount
F , and then mi 27101, decreases. When well beyond vi it has the same course as in fig. 20, but with changed sign now because the negative value is allowed for by the phase. Fig. 21 shows the general features, but it has been necessary to take a comparatively large value of a in order to show the form clearly. In the figure a is 27r2/1/5, whereas for actual spectra it is of the order of
X
so that in any diagram which showed the maximum the rest of the figure would be quite invisible. It thus appears that, except for fre quencies very near vi, the damping can be disregarded, and we shall for the most part take advantage of this simplification and so make use of the curve of fig. 20 instead of fig. 21.
The motion of the electron, forced in this way, will cause a line-wave to be emitted of which the magnitude is given by multiplying the amplitude of the electron's motion by its charge. Omitting the damping factor, we thus say that the electron has a scattering constant which is proportional to
The other virtual elec trons will give similar effects and they are all to be superposed. Thus the virtual atom will have scattering constant proportional to This is the best form for theoreti cal purposes, but experimental work more frequently makes use of wave-lengths. Re-writing the equation we have as the dispersion formula, which should apply for any wave length not too close to
etc. The relative magnitudes of the terms are derivable from the intensities of the associated lines in emission spectra. Their absolute values can also be given by carrying the theory somewhat deeper, and we shall touch on this below.