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Sodium Doublet

1  Introduction

A sodium lamp contains vapor of the volatile metal sodium. A current excites the sodium atoms, and they emit a surprisingly simple spectrum, almost monochromatic. You can be sure of this because if you look at this light with a Michelson interferometer, you see nice yellow interference fringes. You can measure the wavelength of this light in the same way you measured the wavelength of the HeNe laser light, by counting fringes as the moveable mirror traverses a measurably long distance. Part of the experiment below is to measure the wavelength of this yellow light.

As you do this, you might notice a more interesting and subtle feature of the fringes. Sometimes they are clear and easy to see, alternately yellow and black, like tiger stripes. Other times they look pale and washed out. This is not a problem with your eyes or with the alignment of the interferometer, but a real effect. There are actually TWO wavelengths present here with comparable intensity. Each of them forms a pattern of fringes. What you explore with the interferometer is two fringe patterns overlapping. Where they happen to be aligned, with dark fringes coinciding and bright areas coinciding, you see clear fringes. But where the bright parts of one fringe pattern overlap the dark parts of the other fringe pattern, and vice versa, you don't see much contrast. The applet below shows the intensity pattern obtained by adding two fringe intensity patterns with slightly different periods. (Start the applet).

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The distance you have to move along the pattern from one washout to the next is a measure of the difference in the two spatial periods. The spectrum of a sodium lamp is often called "the sodium doublet" because in a spectrometer capable of resolving these nearby wavelengths, you see two bright yellow lines next to each other, a doublet. The Michelson interferometer sees the doublet rather indirectly, by the overlapping of the two fringe patterns.

Here is a more formal treatment of what is going on. Suppose we think about just one wavelength for the moment, and consider adding the two contributions from the two arms of the interferometer, imagining that the light travels a distance L in each arm, except that the moveable mirror has been moved an extra distance x. Then the two amplitudes together at some fixed observation point are
E1=cos(k1L-w1t)+cos(k1(L+2x)-w1t) = 2cos(k1(L+x)-w1t)cos(k1x)
(1)
Squaring and averaging over time gives the intensity at this point,
I1=2cos2(k1x)=1+cos(2k1x)
(2)
We check that if x, the moveable mirror position, changes by l1/2, the intensity oscillates through one full period, i.e., we see one fringe go by. Now suppose we are actually looking at both contributions to the intensity,
I1+I2=2+cos(2k1x)+cos(2k2x)=2+2cos((k1+k2)x)cos((k1-k2)x)
(3)
The rapid modulation of the intensity with x (i.e., the fringe pattern) is at spatial frequency k1+k2, but an envelope modulates the contrast with spatial frequency k1-k2. This is what we see in the figure above. The distance d you must move the mirror to go from one minimal contrast to the next satisfies (k1-k2)d=p. Therefore p/d=Dk=D(2p/l) -2pDl/l2, so in magnitude Dl/l l/2d.

2  Experiment

Use the Michelson interferometer to measure l and Dl for sodium light. Be sure to think about turning the screw which controls the mirror position always in the same direction when you are doing the measurement.

To find the distance d between "washouts", you don't need to count fringes, only to count "washouts." You will get better accuracy if you find the distance to go through a large number of them than if you only try to measure the distance between one and the next, since they are not easy to locate precisely.




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On 23 Aug 2001, 15:23.