BACKGROUND

In man's efforts to understand his physical environment, a problem encountered perhaps more frequently than any other is that of determining the composition of objects or materials under study. This particular problem is the whole object of analytical chemistry, and in later courses you will become aware of many elegant methods which chemists are able to employ in their work.

The chemist must, of course, have on hand an adequate amount of the material he wishes to investigate, although "adequate" can mean an impressively small quantity in some instances. If an actual sample of the unknown substance cannot be brought into the laboratory, then chemistry fails us, and except for one very fortunate circumstance, we might be out of luck in discovering the composition of the sun and stars, for example.

The "circumstance" alluded to above is what will concern us in this experiment. It is the fact that each of the hundred or so elemental substances which occur in nature (or are made artificially in a few cases) identifies itself uniquely by the light it emits when it is in a very hot environment.

Atomic Spectra

You are certainly aware by now that atoms consist of heavy, positively-charged nuclei, surrounded by relatively large and tenuous electron "clouds." The overall size of an atom is so small on a conventional size scale, however, that the behavior of this system of electrons, protons, and neutrons can only be described by means of Quantum Mechanics. Many of the results of quantum mechanical analysis of atomic systems seem to violate our "common sense" notions of how things should behave; it is found, among other things, that the total energy possessed by any one of the electrons orbiting the nucleus can have certain discrete values and no others. We are familiar with the fact that the energy of a larger orbiting body, e.g., a satellite of the earth, can be changed continuously by applying thrust from a rocket. In an atomic system, however, nature simply does not allow this. If any change upward or downward in an electron's energy is to occur, it must take place in instantaneous jumps between energy levels which are characteristic of the particular kind of atom.

These transitions between energy levels may be induced in a number of ways, among them collisions between atoms and irradiation of an atom with electromagnetic energy. In the case of a downward transition of an electron's energy, the change may occur spontaneously, without external stimuli being involved.

In all of these changes, however, one law already familiar to us from classical, or large-scale mechanics, is still rigorously obeyed: energy must be conserved. When a downward transition of electron energy from, say, a level W_l to another energy W₂ occurs, the difference W₁-W₂ must show up somewhere else. This "somewhere else" is almost always the radiation of a burst of electromagnetic waves. What is the nature of this burst of waves? Classical theory is of no help to us here, and the discovery of the actual relationship between the transition energy and the wavelength of the radiation by Max Planck, Albert Einstein, and R. A. Millikan near the turn of the century was the "breakthrough" that led to the development of quantum theory.

The quantum theory, then, relates the total energy, W_rad, of the burst to its frequency v, or wavelength , by the equation

Here, c is the velocity of light, and h is a proportionality constant called Planck's constant.

Now, let us assemble several important facts:

1. Each species of atom is characterized by a certain array of permitted energy levels W_l, W₂, .... W_n. No two elements are alike in their energy level arrangements, but every member of a species is exactly the same as the species in a galaxy two million light years away.
2. Since atoms are identified by their permitted energy levels, it follows that the various differences between levels in an atom are also a set unique to that atom, and hence, the combination of possible radiation wavelengths one can obtain from one kind of atom is absolutely unique to it.
3. If a very great number of atoms are in an environment (usually very hot) which permits very large numbers of transitions back and forth between energy levels, there will be a steady flow of radiation from that region, and by carefully measuring all the wavelengths present, one may completely identify the various species present there, even though the observer may be astronomical distances away from the source. It is by this means alone that we know the composition of the outer layers of the stars.
4. The radiation emitted by atoms covers a very large range of wavelengths. At the short end of the scale are x-rays, which are invisible and have the ability to penetrate matter quite easily. In contrast to this, atoms can also generate radiation which has sufficiently long wavelength to be classed as radio microwave energy. This range covers more than seven orders of magnitude, from wavelengths of less than one angstrom unit (1 = 10 cm) up to the order of 10⁷ , i.e., 1 mm. In the middle of this range is a band of wavelengths between about 4000 and 7000 , which constitute visible light. The various atomic emissions in this range then appear to us different colors.

The study of atomic properties through analysis of emission of visible light is called opticalspectroscopy. The principal instrument employed in this work is the spectrograph, or spectrometer. It has the function of dispersing an incoming beam of light from the source into separated beams of different colors, so that the various wavelength constituents of the light fall along different positions on some calibrated scale. This linear array of light components from a source is referred to as its spectrum.

In the present experiment, you will employ an elementary spectrometer, which will enable you to identify some elements through observations of their spectra.

The Diffraction Grating

A convenient mental image to employ in thinking about E.M. (electromagnetic) waves is that of a succession of flat "sheets" of electric field energy, all moving in a direction perpendicular to their planes, and carrying electric fields of alternating direction.

Figure 1

The sheets may be considered as being of infinite width, at least in comparison to their spacing. The separation between two successive sheets of the same sign is the wavelength, . One sees, in the graph in Fig. 1, that the actual variation of E along the propagation direction z is sinusoidal, i.e.,

.

The process of most importance to us now is that which occurs when a small obstruction is placed in the way of this train of waves. It is called diffraction, and while we cannot discuss this quite complicated subject in very great detail here, it is sufficient to note that the obstruction effectively extracts energy from the plane wave incident upon it, and re-radiates the energy in all directions from its own position.

Thus, we have two systems of waves--the incident plane waves and the "scattered" or diffracted waves (Fig. 2).

Figure 2

Suppose now that we place several such obstructions in the path of the incident waves; let all of these sites be spaced at a uniform interval, d, along a line. We will first let this line be parallel to the wave fronts, so we get the situation shown in Fig. 3.

Figure 3

Now, let us concern ourselves with the cumulative effect of the diffracted waves from the scatterers. At most positions in the neighborhood of the array, the local electric field, which is a sum of components coming from the various obstacles, is made up of a mixture of positive (upward) and negative (downward) components, which tend to average out to a zero total amplitude, leaving only the original incident waves. However, there are particular scattering angles where the contributions from all the scatterers re-enforce each other, and add together to make new plane waves moving in a different direction from the incident train. In Fig. 4, we see that the third wave crest from center #1, the second from #2, and the first from #3 are coalescing to produce a new plane wave moving away at angle from the initial direction. Each curve in the figure is a positive crest, and so, these are a distance apart. One can see from the geometrical construction in the figure that this condition is satisfied if

.

It is, in fact, satisfied if d sin is any multiple number of wavelengths, i.e., if

,

where n is any integer.

Figure 4

A unique relation between the emerging angle of the diffracted wave and wavelength also exists if the incident wave fronts are not parallel to the line of scatterers. The result, which can be easily derived in a manner similar to that used in Fig, 4, is simply

, (The grating equation)

where now is the angle between the incident wave fronts and the line of scatterers.

These results immediately suggest a way of dispersing a multi-colored beam of light into its separate colors. If we could build or obtain a scattering array whose members were spaced by a distance d comparable to a wavelength of light, it would be possible to send each component off in a different direction. By measurement of these angles, one might then determine wavelengths, and hence, identify the atomic species producing the light.

Such diffraction gratings can indeed be manufactured; in fact, the majority of all spectrographs produce dispersed spectra through the use of gratings. The production of an optical diffraction grating of good quality is probably the most difficult of all mechanical operations. A typical grating consists of a blank of glass (or a reflecting metal surface) upon which are engraved as many as 25,000 accurately parallel and evenly spaced grooves per inch of surface. Some gratings are as wide as 10 inches, although these are rare.

In any event, the "ruling" of a grating, and the manufacture of a "ruling engine" are enormously expensive projects. Fortunately, it is not usually necessary to purchase an original grating in order to produce excellent spectra; "replica" gratings, made by molding of collodion or other plastic material against a good original are at best nearly comparable in performance, and often low enough in cost to be available to almost any experimenter. You will employ in your experiment a small replica grating of 25,000 lines per inch.