Light waves and their uses/Lecture IV

by Albert Abraham Michelson

 

LECTURE IV

 

THE APPLICATION OF INTERFERENCE METHODS TO SPECTROSCOPY

 

Doubtless most of us, at some time or other, have looked through an old-fashioned prismatic chandelier pendant and observed that when held horizontally it produces the very curious effect of making objects appear to slope downward as though going down hill; and certainly you have all noticed the colored border which such a pendant produces at the edge of luminous objects. This experiment was made first under proper conditions by Newton, who allowed a small beam of sunlight to pass through a narrow aperture into a dark room and then through a glass prism. He observed that the sun's image was drawn out into what we call a spectrum, i. e., into a band of colors which succeed one another in the well-known sequence—red, orange, yellow, green, blue, violet; the red being least refracted and the violet most.

If Newton had made his aperture sufficiently narrow and, in addition, had introduced a lens in such a way that a distinct image of the slit through which the sunlight passed was formed on the opposite wall, he would have found that the spectrum of the sun was crossed by a number of very fine lines at right angles to the direction in which the colors extended. These lines, called after the discoverer Fraunhofer's lines, have this very important characteristic, that they always appear at certain definite positions in the spectrum; and hence they were used for a considerable time for describing the location of the different colors of the spectrum. We shall endeavor roughly to present this experiment. Not having sunlight, however, we shall take an electric arc and produce a spectrum. It will be noticed that this spectrum is not crossed by black lines, but that it is, at least for our purpose, practically continuous, as shown on Plate III, No. 1. Instead of using the electric light, let us try a source which emits but a single color. For this purpose we shall introduce into the electric arc a piece of sodium glass. Instead of a spectrum of many colors, we have one consisting mainly of one color, namely, of one yellow band. This yellow band in reality consists of two images of the slit, which are very close together, as can be shown by making the slit narrower, for then the two lines will also become narrower in proportion. If, instead of sodium glass, we introduce a rod of zinc, then, instead of one bright yellow line, the spectrum consists of lines in the red, green, and violet—two or three in the violet, one in the green, and one in the red. If we were to introduce copper, the spectrum would consist of quite a number of lines in the green; and if other substances were used, other lines would appear in the spectrum (cf. Plate III, Nos. 3 and 4).

Now, the lines produced by any one substance are found to occur always at a particular place in the spectrum, and are thus characteristic of the substance which produces them. If, instead of the electric light, we had used sunlight, we should find, as Fraunhofer did, that the spectrum of the sun is crossed by a number of fine, dark lines, perhaps as many as one hundred thousand, distributed throughout the spectrum. Some of the more important of these lines are shown in Fig. 54. The red end of the spectrum is at the bottom. Only the visible portion of the spectrum of the sun is shown in the figure. The pair of dark lines marked D coincide in position with the bright lines which are produced by sodium, as shown on Plate III, Nos. 2 and 3, and is an indication of the presence of sodium in the sun's atmosphere.

FIG. 54As was remarked above, this sodium line is double, i. e., is really made up of two lines close together. The distance between these two lines is a convenient standard of measurement for our subsequent work. This distance is so small that a single prism scarcely shows that the line is double. As we increase the number of prisms, the lines are separated more and more widely. If, instead of a prism, we use one of the best grating spectroscopes, the two lines are separated so far that we might count sixty or eighty lines between; and this fact gives a fair idea of the resolving power of these instruments. If we have two lines so close together as to be separated by only one-hundredth of the distance between these two sodium lines, the best spectroscope will hardly be able to separate them; i. e., its limit of resolution has been reached.

The difference in the character of the lines from different substances is illustrated in Fig. 55. The spectrum that you have just seen is a photograph from a drawing, not a photograph from a spectrum. These are from spectra. On the right is a portion of the spectrum of iron, the other the corresponding portion of that of zinc. The enormous diversity in the appearance of the lines will be noted. Some are exceedingly fine—so fine that they are not visible at all; others are so broad that they cover ten or twenty times the distance between two sodium lines. This width of the lines depends somewhat upon the conditions under which the different substances are burned. If the incandescent vapor which sends out the lines is very dense, then the lines are very broad; if it is very rare, then the lines are exceedingly narrow.FIG. 55 Some of the lines are double, some triple, and some are very complex in their character; and it is this complexity of character or structure to which I wish particularly to draw your attention.

This complexity of the character of the lines indicates a corresponding complexity in the molecules whose vibrations cause the light which produces these lines; hence the very considerable interest in studying the structure of the lines themselves. In very many cases—indeed, I may say, in most cases—this structure is so fine that even with the most powerful spectroscope it is impossible to see it all. If this order of complexity, or order of fineness, or closeness of the component lines is something like one-hundredth of the distance we have adopted as our standard, it is practically just beyond the range of the best spectroscopes. It therefore becomes interesting to attempt to discover the structure by means of interference methods.

In order to understand how interference can be made use of, let us consider the nature of the interference phenomena which would be produced by an absolutely homogeneous train of waves, i. e., one which consisted of only one definite simple harmonic vibration. If such a train of waves were sent into an interferometer, it would produce a definite set of fringes, and if the mirror C (Fig. 39) of the interferometer were moved so as to increase the difference in path between the two interfering beams, then, as was explained above on p. 58, these interference fringes would move across the field of view. Now, in this case, since the light which we are using consists of waves of a single period only, there will be but one set of fringes formed, and consequently the difference of path between the two interfering beams can be increased indefinitely without destroying the ability of the beams to produce interference. It is perhaps needless to say that this ideal case of homogeneous waves is never practically realized in nature.

What will be the effect on the interference phenomena if our source of light sends out two homogeneous trains of waves of slightly different periods? It is evident that each train will independently produce its own set of interference fringes. These two sets of fringes will coincide with each other when the difference in the lengths of the two optical paths in the interferometer is zero. When, however, this difference in path is increased, the two sets of fringes move across the field of view with different velocities, because they are due to waves of different periods. Hence, one set must sooner or later overtake the other by one-half a fringe, i. e., the two systems must come to overlap in such a way that a bright band of one coincides with a dark band of the other. When this occurs the interference fringes disappear. It is further evident that the difference of path which must be introduced to bring about this result depends entirely on the difference in the periods of the two trains of waves, i. e., on the difference in the wave lengths, and that this disappearance of the fringes takes place when the difference of path contains half a wave more of the shorter waves than of the longer. Hence we see that it is possible to determine the difference in the lengths of two waves by observing the distance through which the mirror C must be moved in passing from one position in which the fringes disappear to the next.

If the two homogeneous trains of waves have the same intensity, then the two sets of fringes will be of the same brightness, and when the bright fringe of one falls on the dark fringe of the other, the fringes disappear entirely. If, however, the two trains have different intensities, one set of fringes will be brighter than the other, and the fringes will not entirely disappear when one set has gained half a fringe on the other. In this case the fringes will merely pass through a minimum of distinctness. We see then that, if our source of light is double, i. e., sends out light of two different wave lengths, we should expect to see the clearness or visibility of the fringes vary as the difference of path between the two interfering beams was increased.

If we invert this process and observe the interference fringes as the difference in path is increased, and find this variation in the clearness or visibility of the fringes, it is proved with absolute certainty that we are dealing with a double line. This is found to be the case with sodium light, and, therefore, by measuring the distance between the positions of the mirror at which the fringes disappear, we find that we actually can determine accurately the difference between the wave lengths of the two sodium lines. In order to carry the analysis a step farther, suppose that we magnify one of these two sodium lines. It would probably appear somewhat like a broad, hazy band. For the sake of simplicity, however, we will suppose that it looks like a broad ribbon of light with sharp edges. The distance between these edges, i. e., the width of this one line, if the sodium vapor in the flame is not too dense, is something like one-fiftieth, or, perhaps, in some cases as small as one-hundredth, of the unit we have adopted—the distance between the sodium lines.

This is proved by noting the greatest difference in path which can be introduced before the fringes disappear entirely. This distance is different for different substances, and the greater it is the narrower the line, i. e., the more nearly does it approach the ideal case of a source which emits waves of one period only. Now, experiment shows that the fringes formed by one sodium line will overtake those formed by the other in a distance of about five hundred waves, corresponding to about one-third of a millimeter, and that we can observe interference fringes with sodium light, under proper conditions, until the difference in path between the two interfering beams is approximately thirty millimeters. This means that the width of the band is something like one-hundredth of the distance between the two bands. The width of a single line can be appreciated in the ordinary spectroscope when the sodium vapor is dense, and under these conditions the fringes vanish when the difference in path is only one-half inch, or even less. When we try to make the source bright by increasing the temperature and density of the sodium vapor in the flame, the band broadens out to such an extent that the difference in path over which interference can be observed may be less than one-hundredth of an inch.

The above discussion of the case of the two sodium lines may easily be extended to include lines of greater complexity, and it will be found that, whatever the nature of the source, the clearness or visibility of the fringes will vary as the difference in path between the two interfering beams is increased. It may also be shown that each particular complex source will show variations in the visibility of the fringes which are peculiar to it.

Inversely it is evident that by the observation of the character of the curve which expresses the relation between the clearness of the fringes and the difference of path—the FIG. 56 visibility curve, as it may be termed—we can draw conclusions as to the character of the radiations which cause the interference phenomena, even when such investigation is beyond the power of the best spectroscopes. In order to make the method (it may perhaps be called the method of light-wave analysis) an accurate process, it is necessary, in the first place, to produce a number of visibility curves from known sources. Thus, for example, we may take two lines corresponding to the sodium lines, and produce their visibility curve, as we did before, by adding up the separate fringes and obtaining the resultant; we may then take three or four or any number of lines, and determine the corresponding visibility curves. Each of these, instead of being a single line, may have an appreciable breadth, and the brightness of the line may be distributed in various ways within the breadth.

Now, the process of adding up such a series of simple harmonic curves (for the interference fringes are represented by simple harmonic curves) is very laborious. Hence the instrument shown in Fig. 56, called a harmonic analyzer, was devised to perform this work mechanically. It looks very complex; in reality it is very simple, the apparent complexity arising from the considerable number of elements required. A single element is shown in Fig. 57. A curved lever which is pivoted at o is represented at B. One cud of this lever is attached to the collar of the eccentric A. When this eccentric revolves, it therefore transmits to the lever B a motion which is very nearly simple harmonic. The amount of the motion which is communicated to the writing lever u is regulated by the distance of the connecting rod R from the axis o. When the connecting rod is on one side of the axis the motion is positive; when on the other side the motion would be negative. The end of this lever is connected to another lever x, and the farther FIG. 57 end of this lever is connected with a small helical spring s. There are eighty such elements arranged in a row, as shown in Fig. 56. In order to add the force of all of the springs, they are connected with the drum C, which can turn about its axis, and counterbalanced by a very much larger spring S connected to the other side of the drum. This gives us the means of adding forces which are proportional to the amount of displacement of the lever below, and hence the sum of the forces of these eighty springs is in direct proportion (at any rate to a close degree of approximation) to the sum of the motions themselves. We have thus a mechanical device for adding simple harmonic motions.

To illustrate this addition of simple harmonic motions by means of our machine, one of the connecting rods is first moved out to the extreme end of the lever. We shall then have but one simple harmonic motion to deal with, and this corresponds to an absolutely homogeneous source. The resulting curve is the first one in Fig. 58. Each one of the oscillations corresponds to an interference fringe, and there would be an infinite number of such if the difference in path were indefinitely increased. Now we will take the case of two simple harmonic motions. At b, curve 2, the fringes have disappeared completely. One series of fringes has just overtaken the other by one-half a fringe, and, therefore, they neutralize each other. At c the fringes have begun to appear again, and at d they have attained a maximum visibility or clearness. They then disappear and reappear again, and so on indefinitely.

Curve 3 represents the case of the two sodium lines, each of which is supposed to be double. It will be observed that in this case there are two periods; one, the same as that of curve 2, which corresponds to the double sodium line, and the other a longer period whose first minimum occurs at e and which corresponds to the shorter distance between the two components of each line. The conclusion which can be drawn from observation of such a curve as this is that theFIG. 58 source which was used in obtaining it was a double line, each of whose components was double.

Curve 4 represents the visibility curve of two lines, one of which is very much brighter than the other, but whose distance apart is the same as that of the lines of curve 2. The period of the visibility curve is the same as that of 2, but instead of going to zero it merely goes to a minimum at f. Inversely, when we get such a curve as this we know that one of the lines is brighter than the other—just how much brighter can be learned from the ratio of the maximum and minimum ordinates.

Curve 5 is that due to a single broad source of uniform intensity throughout. It will be noted how quickly the fringes lose their distinctness. Curve 6 is that due to a broad source which is brighter in the middle than at the edges. The distribution in this case is supposed to follow the exponential law. The corresponding visibility curve does not exhibit maxima and minima, but gradually dies out and remains at zero. Curve 7 corresponds to a double source each of whose components is brighter in the middle. Curve 8 represents a triple source each of whose components is a simple harmonic train of waves of the same intensity. Curve 9 represents the visibility due to a triple source in which the outer components are much fainter than the middle one.

We might go on indefinitely constructing on the machine the visibility curves which correspond to any assumed distribution of the light in the source. The curves presented will suffice to make clear the fact that there is a close connection between the distribution of light in any source and the visibility curve which can be obtained with the use of that source. It is, however, the inverse problem, i. e., that of determining the nature of the source from observation of the visibility curve, in which the greatest interest lies.

In order to determine by this method the character of the source with which we are dealing, we must find our visibility curve by turning the micrometer screw of the interferometer and noting the clearness of the fringes as the difference of the path varies. We then construct a curve which shall represent this variation of visibility on a more or less arbitrary scale, and compare it with one of the known forms, such as those shown in Fig. 58. There is, however, a more direct process. The explanationFIG. 59 of this process involves so much mathematics that I shall not undertake it here. It will be sufficient to state that the harmonic analyzer cannot only be used as has been described, but is also capable of analyzing such visibility curves. Thus, if we introduce into the instrument the curve corresponding to the visibility curve, by making the distances of the connecting rods from the axis proportional to the ordinates of the visibility curve, and then turn the machine, it produces directly a very close approximation to the character of the source. For example, take curve 2 of Fig. 58. By its derivation we know that it corresponds to a double source each of whose components is absolutely homogeneous. If we introduce this curve, or rather the envelope of it, into the machine, it will give a resultant which represents the character of the source to a close degree of approximation. The actual result is shown in Fig. 59, in which the ordinates represent the intensity of the light. We thus see that the machine can operate in both ways, i. e., that it can add up a series of simple harmonic curves and give the resultant, which in the case before us is the visibility curve, and that it can take the resultant curve and analyze it into its components, which here represent the distribution of the light in the source.

Now the question naturally arises as to how the observations by which the visibility curve is determined are conducted; also as to what units to adopt, and what scale of measurement. It is apparently something very indefinite. The visibility is not a quantity that can be measured, as we can a distance or an angle—unless, to be sure, we first define it. After defining it properly, we can produce, in accordance with that definition, interference fringes that shall have any desired visibility. By the use of fringes which have a known visibility we can educate the eye in estimating visibility, or we may have these standard fringes before us for comparison at the time of observation, and may then determine when the two systems are of the same clearness; and when they are of the same clearness, we say that the desired visibility is the same as that whose value is known from our formula. This is the more accurate method, and is the one which was finally adopted; but long before its adoption it was found that fairly accurate visibility curves could be obtained by merely agreeing to call the visibility 100 when it was perfect, 75 when good, and 50 when fair. Then 25 would be rather poor, 10 would be bad, and at zero the fringes would vanish. Of course, there would be a greater or less difference in what we should agree to call good, but in general we can tell where the fringes were half as clear as their perfect value, provided, of course, we had this perfect value given, etc.

As a matter of fact, however, it is not of the utmost importance to determine the visibility with great accuracy. We know that we can measure a minimum or a maximum independently of any scale, and these points are the really important ones. For example, a curve may come to zero gradually or abruptly—in both cases the distance between the two lines which produced the curve would be exactly the same. The two pairs might differ in character in other ways, but the distance between the two components of each pair would be the same. So, even without an absolute scale that we have tested, and even without any very great amount of experience in observation, we can get a very fair visibility curve, and from that a very fair conception of the nature of the spectrum of the particular source we are examining, by merely determining the points of maximum and minimum clearness.FIG. 60

Before discussing some of the visibility curves that have been obtained, I should like to say a word concerning the source of light. When the source is under ordinary conditions, i. e., under atmospheric pressure, the molecules are not vibrating freely, and disturbing causes come in to make the oscillations not perfectly homogeneous. Hence the light from such a source, instead of being a definite, sharp line, is a more or less diffuse band. In order to obtain the character of the line under the extreme conditions, i. e., under as small pressure as possible, the substance must be placed in a vacuum tube. The tube is then connected to an air pump and exhausted until the pressure in it is reduced to a few thousandths of an atmosphere.

When the exhaustion has become sufficient—the time depending on the particular degree of exhaustion required by the substance which we wish to examine—the tube is heated to drive off the remaining water vapor, sealed up, and is then ready for use. The residual gas is made luminous by the spark from an induction coil. In some cases the substance is sufficiently volatile to show the spectrum at ordinary temperatures; e. g., that of mercury appears after slight heating. In the case of such substances as cadmium and zinc the tube is placed in a brass box, as illustrated in Fig. 60, and heated until the substance is volatilized, a thermometer giving us an idea of the temperature reached.

Fig. 61 illustrates the arrangement of the apparatus as it is actually used. An ordinary prism spectroscope gives a preliminary analysis of the light from the source.FIG. 61 This is necessary because the spectra of most substances consist of numerous lines. For example, the spectrum of mercury contains two yellow lines, a very brilliant green line, and a less brilliant violet line. If we pass all the light together into the interferometer, we have a combination of all four. It is usually better to separate the various radiations before they enter the interferometer. Accordingly, the light from the vacuum tube at a passes through an ordinary spectroscope bcd, and the light from only one of the lines in the spectrum thus formed is allowed to pass through the slit d into the interferometer.

As explained above, the light divides at the plate e, part going to the mirror f, which is movable, and part passing through to the mirror g. The first ray returns on the path feh. The second returns to e, is reflected, and passes into the telescope h. If the two paths are exactly equal, we have interference phenomena in white light; but for monochromatic light the difference of path (from the point e to the mirror f, and from the same point to the mirror g) may be very considerable. Indeed, in some cases interference can be obtained when the difference in the two paths amounts to over half a million waves.

It is rather important to note that the surface of the mirror g must be so set by means of the adjusting screws at its back that its image in the mirror e shall be parallel with the surface of the movable mirror f. When this is the case the fringes, instead of being straight lines, as in the case of the fringes in white light, are concentric circles very similar in appearance to Newton's rings. Having thus adjusted the interferometer so that the fringes are circles, the difference in path is increased by turning the micrometer screw a definite amount, say half a millimeter at a time. At every half millimeter an observation is taken of the visibility, and then these readings are plotted on co-ordinate paper as ordinates, the corresponding difference of path serving as abscissæ. The ends of these ordinates trace out the visibility curve. This curve is then set on the harmonic analyzer, as described above, and the machine turns out the curve corresponding to the distribution of the light in the line examined.

In this way the radiations of many substances were analyzed, and in almost every case it was found that the line was not produced by homogeneous vibrations, but was double, treble, or even more complex. The distances between the components of these compound lines are so small that it is practically impossible, except in a few cases, to observe them in the ordinary spectroscope.

The following diagrams (Figs. 62–8) present a number of these visibility curves. Thus Fig. 62 represents that obtained from the red radiation of hydrogen. The curve to the right represents the visibility curve, while on the left the corresponding distribution of the light is drawn. Beginning at a difference of path zero, the visibility wasFIG. 62 100, and at one millimeter it was somewhat less, and so on, until at about seventeen millimeters we find a minimum. As the difference in path increases, we find that there is a maximum at twenty-three millimeters. After that the curve slopes down, and at about thirty-five millimeters it disappears entirely. Since the curve is periodic, we may be pretty sure that this red line of hydrogen is a double line. This fact, I believe, has never yet been observed, though the distance between the two components is not beyond the range of a good spectroscope, being about one-fortieth or one-fiftieth of the distance between sodium lines.^[1]

Fig. 63 represents the curve which was obtained from sodium vapor in a vacuum tube. When we burn sodium at atmospheric pressure—as, for example, when we place sodium glass in a Bunsen flame—the visibility curve due to its radiations diminishes so rapidly that it reaches zero when the difference of path is about forty millimeters; it is practicallyFIG. 63 impossible to go farther than this. It is seen that the curve is periodic, which would indicate that each one of the sodium lines is a double line. The intensity curve at the left represents one of the sodium lines only. The other, on the same scale, would be distant about half a meter. We can from this get some idea of the relative sensitiveness of this process of light-wave analysis, as compared with that of ordinary spectrum analysis. It will be observed that the intensity curve shows still another small component which corresponds to still another longer period, but the existence of these short companion lines is not absolutely certain.

FIG. 64

Fig. 64 represents the curve of thallium. The oscillation shows that it is a double line, and not very close. The distance between the components is about one-sixtieth of the distance between the sodium lines. We have also a longer oscillation which shows that each one of the components is double. The distance between these small components and the larger ones is something like one-thousandth of the distance between sodium lines, corresponding to a separation of lines far beyond the possible limit of the most powerful spectroscope.

The curve of the green radiation of mercury is shown in Fig. 65. This curve is really so complicated that the character of the source is still a little in doubt. The machine has not quite enough elements to resolve it satisfactorily, having but eighty when it ought to have eight hundred. The curveFIG. 65 looks almost as though it were the exceptional result of this particular series of measurements, and we might imagine that another series of measurements would give quite a different curve. But I have actually made over one hundred such measurements, and each time obtained practically the same results, even to the minutest details of secondary waves. The nearest interpretation I can make as to the character of the spectral source is given at the left of this diagram. It will be noticed that the width of the whole structure is, roughly speaking, one-sixtieth of the distance between the sodium lines. The distance between the close components of the brighter line is of the order of one-thousandth of the distance between the sodium lines. The fringes in this case remain visible up to a difference of path of 400 millimeters, and they have actually been observed up to 480 millimeters, or nearly one-half meter's difference in path—corresponding to something like 780,000 waves.

In the curve of Fig. 66 we have quite a contrast to the preceding. Here we have a radiation almost ideally homogeneous. Instead of having numerous maxima and minima like the curves we have been considering, this visibility curve diminishes very gradually according to a very simple mathematical law, which tells us that the source of light is a single line of extremely small breadth, the breadth being of the order of one eight-hundredth to one-thousandth of theFIG. 66 distance between the sodium lines. It is impossible to indicate exactly the width of the line, because the distribution of intensity throughout it is not uniform. The important point to which I wish to call attention, however, is that this curve is of such a simple character that for a difference of path of over 200 millimeters, or 400,000 light waves, we can obtain interference fringes. This indicates that the waves from this source are almost perfectly homogeneous. It is therefore possible to use these light waves as a standard of length, as will be shown in a subsequent lecture. The curve corresponds to the red radiation from cadmium vapor in a vacuum tube. In using this red cadmium wave as a standard of length it is very important to have other radiations by which we can check our observations. The cadmium has two other lines, which serve as a control or check to the result obtained by the first.

Fig. 67 represents the green radiation of cadmium. This curve is not quite so simple as that of the red, but extends almost to 200 millimeters. The corresponding line is shown to be a close double.

The curve corresponding to the violet light of cadmium is shown in Fig. 68, and is seen to be comparatively simple.FIG. 67

We have thus shown that spectral lines are complex distributions of light, whose resolution in general is beyond the power of the spectroscope. This complexity of the spectral lines is particularly interesting because it indicates corresponding complexity of the molecules which cause the vibrations which give rise to the corresponding spectral lines.FIG. 68 This complexity may be likened to the complexity of a solar system; and while this may bring dismay to the Keplers and Newtons who may hope to unravel the mysteries of this pigmy world, it certainly increases the interest in the problem.

 

SUMMARY

 

1. The spectrum of the light emitted by incandescent gases is not continuous, but is made up of a number of bright lines whose position in the spectrum is very definite, and which are characteristic of the elements which produce them.

2. These "lines" are not such in a mathematical sense, but have an appreciable width and a varying distribution of light, and in some cases are highly complex.

3. This variation in distribution is, however, restricted to such narrow limits that in most cases it is impossible to investigate it by the best spectroscopes; but by the method of visibility curves the lines may be resolved into their elements.

4. An important auxiliary for the interpretation of the visibility curves is the harmonic analyzer—an instrument which sums up any number of simple harmonic motions, and which also analyzes any complex motion into its simple harmonic elements.

 

1. This prediction has since been amply confirmed by direct observation.