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ro = ——s =3.82 x 10ˆ°”m, (32.11) |
Now let us actually calculate the Q of an atom that is emitting light—let us |
say a sodium atom. Eor a sodium atom, the wavelength is roughly 6000 angstroms, |
in the yellow part of the visible spectrum, and this is a typical wavelength. Thus |
=——#x~5x10 32.12 |
9= | (32.12) |
so the Q of an atom is of the order 10. 'This means that an atomic oscillator |
will oscillate for 10 radians or about 107 oseillations, before its energy falls by a |
factor 1/e. The Írequency of oscillation of light corresponding to 6000 angstroms, |
= c/À, is on the order of 1015 cyeles/sec, and therefore the lifetime, the time |
1t takes for the energy oŸ a radiating atom to die out by a factor l/e, is on the |
order of 10” sec. In ordinary cireumstances, freely emitting atoms usually take |
about this long to radiate. 'This ¡is valid only for atoms which are in empty space, |
not being disturbed in any way. If the electron is in a solid and it has to hit |
other atoms or other electrons, then there are additional resistances and diferent |
damping. |
The efective resistance term + in the resistance law for the oscillator can |
be found from the relation 1/Q = +/œo, and we remember that the size oŸ + |
determines how wide the resonance curve is (Fig. 23-2). Thus we have just |
computed the œidths oƒ spectral lines for freely radiating atomsl Since À = 27/0, |
we fnd that |
AA = 2me Au/uŸ = 2mcy/uạ = 2xc/Quo |
= À/Q = 4mro/3 = 1.18 x 10”! m. (32.13) |
32-4 Independent sources |
In preparation for our second topic, the scattering of light, we must now |
discuss a certain feature of the phenomenon of interference that we neglected to |
discuss previously. 'Phis is the question of when interference does øø occur. lf |
we have two sources 5 and %2, with amplitudes 4¡ and 4a, and we make an |
--- Trang 561 --- |
observation in a certain direction in which the phases of arrival of the two signals |
are ôi and óa (a combination oŸ the acbual tỉme oŸ oscillation and the delayed |
tỉme, depending on the position of observation), then the energy that we receive |
can be found by compounding the §wo complex number vectors 4 and 4a, one |
at angle ở and the other at angle ó2 (as we did in Chapter 29) and we find that |
the resultant energy is proportional to |
A? = 4? + A +2AiAa cos (ới — 9a). (32.14) |
Now If the cross term 24 4a cos (ở — da) were not there, then the total energy |
that would be received in a given direction would simply be the sum of the |
energies, 4Ý + 43, that would be liberated by each source separately, which is |
what we usually expect. 'That is, the combined intensity of light shining on |
something om two sources is the sum of the intensities of the two lights. On the |
other hand, if we have things set just right and we have a cross term, it is not |
such a sum, because there is also some interference. lf there are circumstances |
in which this term is of no importance, then we would say the interference 1s |
apparently lost. Of course, in nature it is always there, but we may not be able |
to detect it. |
Let us consider some examples. Suppose, first, that the two sources are |
7,000,000,000 wavelengths apart, not an impossible arrangement. 'Then in a given |
direction 1È is true that there is a very defñnite value of these phase differences. |
But, on the other hand, if we move just a haïr in one direction, a few wavelengths, |
which is no distance at all (our eye already has a hole in it that is so large that we |
are averaging the efects over a range very wide compared with one wavelength) |
then we change the relative phase, and the cosine changes very rapidly. If we |
take the øuerage of the intensity over a little region, then the cosine, which øgoes |
plus, minus, plus, minus, as we move around, averages tO zero. |
So iÝ we average over regions where the phase varies very rapidly with position, |
we get no interference. |
Another example. Suppose that the Ewo sources are two independent radio |
oscillators—not a single oscillator being fed by two wires, which guarantees that |
the phases are kept together, but two independent sources—and that they are not |
precisclu tuned at the same frequency (it is very hard to make them at exactly |
the same frequency without actually wiring them together). In this case we have |
what we call two zndependen‡ sources. Of course, since the frequencies are not |
exactly equal, although they started in phase, one of them begins to get a little |
--- Trang 562 --- |
ahead of the other, and pretty soon they are out of phase, and then it gets still |
further ahead, and pretty soon they are in phase again. So the phase diference |
between the two is gradually drifting with time, but if our observation is so crude |
that we cannot see that little time, if we average over a much longer time, then |
althouph the intensity swells and falls like what we call “beats” in sound, if these |
swellings and fallings are too rapid for our equipment to follow, then again this |
term averages Out. |
In other words, in any cireumstance in which the phase shift averages out, we |
get no interferencel |
One fnds many books which say that two distinct light sources never interfere. |
This is not a statement of physics, but is merely a statement of the degree of |
sensitivity of the technique of the experiments at the time the book was written. |
'What happens in a light source is that first one atom radiates, then another atom |
radiates, and so forth, and we have just seen that atoms radiate a train of waves |
only for about 10~Š sec; after 10” sec, some atom has probably taken over, then |
another atom takes over, and so on. So the phases can really only stay the same |
for about 10~Ẻ sec. Therefore, if we average for very much more than 10” sec, |
we do not see an interference from two diferent sources, because they cannot hold |
their phases steady for longer than 10” sec. With photocells, very high-speed |
detection is possible, and one can show that there is an interference which varies |
with time, up and down, in about 10~Š sec. But most detection equipment, of |
course, does not look at such fine time intervals, and thus sees no interference. |
Certainly with the eye, which has a tenth-of-a-second averaging time, there is no |
chance whatever of seeing an interference between two diferent ordinary sources. |
Recently ít has become possible to make light sources which get around this |
efect by making all the atoms emit fogether in time. The device which does this |
1s a very complicated thing, and has to be understood in a quantum-mechanical |
way. It is called a laser, and it is possible to produce from a laser a source in |
which the time during which the phase is kept constant, is very much longer than |
10 sec. It can be of the order of a hundredth, a tenth, or even one second, and |
so, with ordinary photocells, one can pick up the frequency between ©wo diferent |
lasers. One can easily detect the pulsing of the beats between two laser sources. |
Soon, no doubt, someone will be able to demonstrate two sources shining on a |
wall, in which the beats are so slow that one can see the wall get bright and darkl |
Another case in which the interference averages out is that in which, instead |
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