What are Positrons?

A Scientific Discussion

by R. D. Swisher, Ph.D.

WE know that positrons were discovered about five years ago by Anderson, that they are the positively charged twins of electrons, that they combine with electrons in an annihilation with complete destruction of matter and liberation of a corresponding amount of energy as gamma rays, and that gamma rays will sometimes undergo the reverse transition to give positrons and electrons. But behind these facts there is a story as romantic as the wildest flights of science-fiction, worthy of a place beside Adams’ and Leverrier’s classic prediction of the existence of the planet Neptune from irregularities in the orbit of Uranus.
	The story was really unfolded about a year before the positron was discovered, but for a more complete understanding of it we must go back to the turn of the century, when things began to happen to classical physics. Mechanics had been developed, from Newton’s three laws of motion, to their most generalized mathematical statement: the Hamiltonian equations.
	Einstein was beginning the work which would add the last touch of refinement. Relativity was the culmination of classical mechanics, and with its aid the behavior of any system of particles could be predicted, provided you had the mathematical tools and the mathematician capable of using them. But it didn’t work—not completely. You could solve problems, involving billiard balls, planets and suns, with incredible accuracy. But it wouldn’t work for things as tiny as atoms.
	So we have modern physics, eager to study the most important things left unstudied: atoms, molecules and electrons, and a mechanics that would work perfectly for billiard balls, planets and suns, but which proved quite inapplicable to these tiny particles. So quantum mechanics was developed, and is still being developed.
	Now the prime purpose of physics, and indeed of any science, is to study observable quantities, to formulate laws which will express all experimental data, and which will allow predictions of future events by means of these laws. Accordingly, the new mechanics must be one which will become identical with classical mechanics, if the particles are quite large, in addition to checking experiments made with atoms.
	One of the most important and easily studied properties of an atom is the light that it emits. Light is an electromagnetic wave, and electromagnetic waves are formed by oscillating electric particles. In an atom we have a nucleus surrounded by electrons revolving in orbits. Since electrons are electrified particles and orbits are periodic, it was thought that the revolving electrons must be the source of the light from atoms, the frequency or wave length of the light being a function of the time of revolution of the electron.
	But light emission represents energy loss, so the electron should spin in smaller and smaller orbits as it loses energy by radiation, until finally it should spiral into the nucleus. And this should take place in a very short time. Actually, each atom radiates only a few of the infinite number of possible frequencies, and those frequencies it does emit are easily measured from the spectrum. With infinite patience the spectroscopists played detective and found obscure relations between the various frequencies of the spectrum, which greatly simplified matters.

	THEN Bohr brought forth the idea that in each atom there are only relatively few orbits which electrons can follow about the nucleus. A planet can revolve about a sun in an orbit at any distance from that sun, but an electron cannot have a similar freedom in choosing its orbit. Here is one near the nucleus; here is one somewhat farther out—and so on. The electron can follow any of these, but it cannot follow intermediate ones.
	In the Bohr picture the electron does not produce electromagnetic waves during its revolution about the nucleus, but only when it falls from a larger orbit to a smaller one. The energy of the electron depends on the size of its orbit about the nucleus—the greater the radius the greater the energy—so that when the electron falls from the larger to the smaller orbit it loses energy, and that lost energy is totally accounted for by the bit of radiation emitted during the process.
	Conversely, if radiation of the proper energy impinges on the atom, an electron can absorb the energy of the radiation and jump to an outer orbit, thus gaining exactly the amount of energy absorbed from the radiation.
	The problem was then to calculate, if possible, the permissible electron orbits of the various atoms, the energies of the electrons in those orbits, then the energy differences between the various orbits. These energy differences should correspond to the energies of the various kinds of light emitted by that particular kind of atom in its spectrum.
	Hydrogen is the simplest atom, and the calculation of the possible orbits of the lone electron about its nucleus agreed almost perfectly with its observed spectrum. More complex atoms did not agree so well, but it was certain that here was a new method of approach to atomic, problems that showed signs of promise; the classical approach did not work at all.
	It soon became evident that one of the troubles with the new method was that it introduced too much unobservable detail in the atom’s workings. We spoke of electron orbits about the nucleus, and for the complex atoms even the position of the electron in its orbit must be introduced. These are things that we probably cannot hope to observe, and so are undesirable. (They are, however, very convenient in getting an approximate mental picture of things, a procedure indispensable to all of us except those geniuses who can grasp the mathematical equations directly).
	Since the orbits represent different energies of the electrons in them, we can think of the orbits as energy levels, and mentally note that the orbit is a rather crude picture of an energy level, not necessarily exact. The problem has now resolved itself into finding some sort of equations or mechanics which will enable us to calculate these energy levels of the various atoms, and the differences between these energy levels must check the observed spectra of those atoms, if the mechanics are any good.
	One answer to the problem was the wave mechanics. It was discovered that electrons exhibited certain properties of waves, i.e., they produced diffraction patterns. Later it was shown that protons likewise behaved somewhat like waves. The wave lengths of these matter waves are easily measured from the diffraction pattern, and an intimate connection was found between the wave lengths and the momentum of the particle in question. So here we have the two fundamental particles of matter exhibiting properties of waves, where the momentum of the particle corresponds to the length of the wave.
	Obviously, the next step is to use the mathematics of waves to apply to the particles, using the observed relations between momentum and wave length, and see if it works. Wave equations must be developed in which we can substitute positions and velocities of particles, as well as the force fields acting upon them, and which on solving will give positions and velocities of the particles at some later time.

	FOR VARIOUS REASONS our answers will not be exact, but—in terms of probabilities—we will find the most probable values for position and velocity of the particle at the future time. This is an inherent property of wave mathematics, and it is not too disappointing, since experiment indicates that this is actually the case with particles. So we must find some property of waves which will correspond to probability, just as wave length corresponds to momentum.
	In order for the thing to have sense, this probability function must have three properties: first, it must always be positive or zero, since a negative probability has no physical meaning; second, the integral of the probability over all space must be independent of time, which is merely a mathematical way of saying that if the particle is somewhere in space now, it must still be somewhere in space at any later time; third, this integral of the probability over all space must be invariant with change of velocity of the observer, which means that if you observe that the particle is somewhere in space, then a person in a laboratory moving with respect to yours must also observe it somewhere in space. This last condition is required if relativity is to be satisfied.
	All three of these conditions must be fulfilled if the concept of probability is to have any definite meaning, but we can get an approximate solution if only the first two are met, just as the laws of Newton are approximately correct for everyday use. Such an equation was developed by Schrödinger. It worked beautifully, except that the mathematics were so complicated as to preclude exact solutions in most cases. Approximations were necessary. But approximations can be made to almost any desired degree of accuracy, if you want to work long enough. The important part was that it did work, except for cases where relativity corrections became important.
	The problem of developing an equation or set of equations in which the probability would fulfill all three of the above conditions was finally solved by Dirac. It was shown, however, that the only possible way to meet the three conditions involved the introduction of the concept of negative energy levels, and this was very embarrassing, because the concept of negative energy was apparently foolish.
	Let us go back to energy levels for a moment. In an atom we pointed out that there are only certain possible energy levels that an electron can occupy, and that the electron tends to fall to a lower energy level when it can, i.e., when there is a lower one that is not already occupied by another electron. In so doing it emits a quantum of electromagnetic radiation (light, infra-red, ultra-violet, X rays, etc.) with the frequency of the radiation depending on the difference in energy of the two energy levels.
	But, in general, an electron does not have to be in an atom. If we have a free electron with plenty of room, its energy, or, more specifically its kinetic energy, is measured by its mass and velocity: K.E.=½MV². If the electron is standing still its V is zero, so its kinetic energy is zero. But it still has other energy, as pointed out by Einstein—the energy of its mass, which is the mass multiplied by the square of the velocity of light, so the total energy of the free particle is (approximately, at least): E=MC²+½MV², where C is the velocity of light.
	Since the velocity enters as a square, it makes no difference whether V is positive or negative (to right or to left), V² is always positive; so the energy is always positive. Also, if the electron has plenty of room to move around, it can have any velocity, and so any energy above MC². This means that the energy levels of the free electrons are not discontinuous, as in an atom, but a continuous series.
	Now, if an electron is in a negative energy level it must behave as if it has negative mass (since negative velocity squared is still positive). That means that the more you push it in one direction, the faster it goes in the other direction. Its momentum is directed opposite to its velocity. Such particles have never been observed.
	Furthermore, if you have an infinite number of these negative energy levels (just as there are an infinite number of positive energy levels) electrons should drop into them, just as in an atom they drop to unfilled lower levels, and in so doing should emit electromagnetic radiation corresponding to the energy difference. This energy difference is at least 2MC² (see Fig 1), so the radiation should be very powerful, in fact, gamma rays.
	Conversely, if we did have an electron in a negative energy level, gamma rays shining on it should lift it to a positive energy level, just as happens to atomic electrons when the proper light hits them.

	SO the Dirac theory requires the existence of the negative energy levels; yet they have never been observed. If there were such levels the electrons in ordinary positive levels should fall into them, and then exhibit the queer properties of negative mass. The difficulties are obvious. But the Dirac theory proved its worth by giving correct answers to some problems that the non-relativistic Schrödinger theory could not attack; so it must be kept, if possible.
	Dirac surmounted the obstacle with a stroke of genius, by postulating that the very necessary negative energy levels were all filled with electrons—an infinite number of electrons, since there are an infinite number of the levels—and that this infinite continuum of negative energy electrons is synonymous with what we call empty space. This avoids the difficulty of having positive energy electrons fall to the negative energy levels, since those levels are all filled.
	However, a gamma ray should be able to excite an electron in a negative level to a positive level, just as light causes atomic electrons to jump to higher levels. When this happens the excited electron becomes observable, since it has positive energy. But now we have an unfilled level in the negative energy states, and that, too, should be observable. This unfilled level, or “Dirac hole,” as it is called, is, in effect, the absence of a particle of negative mass and of negative charge; in other words, it is the presence of a particle of positive mass and positive charge. A year later such particles were observed. They are called positrons.
	The properties of the positron coincide exactly with the properties of the Dirac hole. Under proper conditions a quantum of gamma radiation sometimes disappears, and an electron and positron are observed to be created, the gamma quantum being destroyed. Conversely, when an electron and positron meet, they annihilate each other, and there is formed a quantum (or two quanta) of gamma radiation. Obviously, this corresponds to the falling of the electron from its positive energy level into the unfilled negative level, or hole, again filling it, erasing the discontinuity, and once again making “empty space.”
	Thus, in our particular region of space, at least, we have an infinite number of electrons filling the negative energy levels completely, this complete continuity making them unobservable and definable as nothingness. In addition to these, there are a number of electrons left over after the negative states are all filled, which, accordingly, must occupy various positive levels and be observable. These left-over electrons are what make up our atoms, along with protons—and neutrons, of course.

	THE question now arises: why are these extra electrons? Somehow it disturbs our sense of symmetry. Accordingly, speculations have been made that perhaps in distant regions of space there are vacancies in the negative levels, Dirac holes, or positrons—whichever you prefer—equal in number to the extra electrons in our own region. The picture is perfectly symmetrical. The positrons can be considered the actual particles and electrons the holes; it makes no difference.
	Then, in these hypothetical, far-off regions we have excess positrons. They may have negative protons, “negatrons,” “antrons,” call them what you will, with which to build up nuclei about which the positrons can form orbits to make atoms, just as our atoms are built up from electrons and nuclei made with protons. These hypothetical, inverted atoms of the far-off regions would be spectroscopically identical with our own atoms; so their presence cannot be determined if we stay on earth. However, if we should visit such regions, or if matter from there should visit us, the results would be disastrous. Our electrons would combine with their positrons, disintegrating us into a flood of gamma rays.
	The Dirac theory is not valid for heavy particles such as protons, neutrons and negative protons; but it seems quite certain that protons and negative protons do not hold the same relationship to each other as positrons and electrons; the negative proton is not the Dirac hole of a proton. Indeed, evidence indicates that protons and negative protons would actually be repelled from each other in spite of the electrical attraction due to their opposite charges. Even so, the tremendous amount of energy resulting from annihilation of the electrons and positrons of the two kinds of matter would make travel to those remote regions very dangerous indeed without the protection of something very like Dr. Richard Ballinger Seaton’s “zone of force.”