by W Heisenberg
One might be inclined to assume that a nucleus in which the principles of energetics admit the possibility of the occurrence of alpha decay, would disintegrate immediately or at least within a very short period of time. However, that this is not the case is proved by the fact that large quantities of uranium still exist in the world, and that these actually decay only very slowly. In fact, their atoms have existed in an unchanged form for several thousand millions of years. The same applies to thorium and actinium. As a matter of fact, if these three long-lived radioactive elements did not exist, radioactivity would have disappeared from the world a long time ago, except for a few of the lighter elements. For the majority of the other natural radioactive substances originate from these, and are much shorter-lived. The products of the decay of a substance are designated collectively as a radioactive series. There are three natural radioactive series: the uranium series, the thorium series, and the actinium series.
It was stated in the third lecture that the half-life is used as the measure of the life span of a radioactive substance; the half-life is the period of time during which one-half of the number of atoms present at the start of the period disintegrates. The half-life periods of the various alpha emitters show extraordinary differences in order of magnitude. The half-life of uranium, for instance, is 4,600 million years, whereas the half-life of one of its daughter elements, radium C’, is just about one one-millionth of a second. Between these two extremes there are all conceivable intermediate values; for instance, the half-life of radium is 1,580 years. This raises the question as to the cause of these vast differences.
In this connection, it is a very important fact that there exists a simple relationship between the energy of the alpha particles and the half-life of the substance in question. This relationship was discovered, relatively early, by Geiger and Nuttall. These two scientists found that, on the whole, there exists a linear relationship between the logarithm of the decay probability (the reciprocal of the average life of the atom) and the energy of the alpha particles. The latter, as indicated by their equal range (Figure 3), is a magnitude characteristic of every specific radioactive substance. The greater is the quantity of energy available for the decay process, and the greater is the energy of the particle, the more quickly will the decay occur, on the average. If λ designates the decay probability, discussed in the second chapter of this book, and E the energy of the alpha particle, we can write the Geiger-Nuttall relationship in the following form:
logλ = A + BE
where A and B are constants, to be determined experimentally. The value of the decay coefficient can be obtained from the already mentioned equation N = N0e−λt, which indicates the number of the atoms not yet decayed during the time t. The energy E can be computed, from the range, by means of a law also discovered by Geiger and Nuttall.
Figure 20 shows the relationship between logλ and E for all alpha emitters, in the form of a diagram based on measurements. We see here three adjacent curves, one for each radioactive series—the uranium series, the thorium series and the actinium series. Although these lines are not perfectly straight, as the Geiger-Nuttall relationship would require, they are at least not too strongly curved. The very fact that we get three different, approximately parallel curves, shows that although the constant B is the same for all the three radioactive series, the values of the constant A differ slightly. At the very bottom we find uranium, with the smallest decay probability, and at the very top is radium C’ with the highest such probability.
The point now is to explain this regularity, and above all, the fact—very astonishing at first glance—that such a small change in the energy, E, produces such an enormous change in the decay probability, λ. In this entire range, the energy varies only between 6 × 10−6 and 13 × 10−6 ergs—a ratio of 1 to 2. On the other hand, the decay probability varies between the orders of magnitude of 10−18 and 106 per second—a ratio of 1 to 1024.
Figure 20.—Illustrating the Geiger-Nuttall law.
This fact was explained by Gamow, Condon and Gurney, in 1928. In order to understand this theory, let us first discuss an imaginary experiment. Let us imagine that we have captured an alpha particle which has just emerged from the nucleus and have restored it to the place from which it came, and investigate what forces are acting upon this alpha particle, and what work must be applied. The conditions which we encounter are similar to those which prevail in the proton-neutron relationship (Figure 15). As long as the alpha particle is at a considerable distance from the nucleus, it is subject solely and only to the action of the field, by the positive charge of which it is repelled.
Figure 21.—Potential between heavy nucleus and alpha particle.
Therefore, work has to be applied to the particle in order to bring it closer to the nucleus. This means that as our particle approaches the nucleus, its potential energy increases at first. But when it has come sufficiently close to the nucleus, the short-range nuclear forces of attraction become operative, and they finally overcome the electric repulsion. After a certain potential barrier has been overcome, the force of repulsion changes into a force of attraction, and from there on, the potential energy drops steeply in the direction of the interior of the nucleus. This change in potential energy is shown by the curve in Figure 21.
When the alpha particle is hurled out of the nucleus, it also passes through this potential range, in this case proceeding from inside outwards. Since it reaches a considerable distance from the nucleus with a considerable quantity of kinetic energy, its total energy at that point is a positive magnitude, since its potential energy vanishes. This fact is indicated in Figure 21 by two straight horizontal lines, one of which corresponds to the slow alpha particles of long-lived uranium, the other to the faster alpha particles of the vastly shorter-lived thorium C’. Since the alpha particle carries this energy along out of the nucleus, it must have been in possession of it while still in there. Therefore, we extend the straight line of the uranium right into the interior of the nucleus. The kinetic energy in the interior of the nucleus was obviously still greater; its value at any given point is indicated by the separation of the horizontal straight line from the potential energy curve. The illustration suggests that the particle vibrates to and fro in the interior and bounces backwards and forwards, one might say, from one side of the ‘potential container’ to the other. The energy in the interior, shown by the straight line, is always the sum total of its kinetic and potential energies.
At first glance it seems to be impossible to see clearly how this particle can emerge from the nucleus at all. For according to the concepts of ordinary mechanics, it would not be able to move any further outward than the sides of the ‘potential container’, for at the point where the straight line crosses the potential curve, the kinetic energy vanishes—the particle comes to rest. The energy at its disposal does not enable it to traverse the potential barrier which separates the interior from the outside space. Thus according to classical mechanics, a decay reaction could not take place at all. The potential barrier would guarantee the stability of the nucleus. It could, of course, be assumed that the nucleus contains other particles as well which are likewise in motion, and that these particles might transfer energy to the alpha particle to help the latter cross the potential barrier. But this can occur only when the nucleus is in an excited state—when it has a surplus of energy. In the normal state of the atom, no such free energy is available. For the energy which the particles still possess is—according to the uncertainty principle—zero-point energy, and this energy cannot be utilized, nor can it be transferred to other particles.
This is where wave mechanics comes to the rescue. The movement of alpha particles is governed by the laws of wave mechanics and quantum mechanics, not by the laws of ordinary mechanics. On the ground of the repeatedly mentioned wave-particle duality, we can think, instead of a particle vibrating back and forth within the nucleus, of a wave which is reflected back and forth from the walls of the ‘potential container
’, and we can even imagine it as a stationary wave. But now we must explain how such a wave can be reflected from these walls at all. Obviously, this phenomenon is based on a process for which we can find an analogy in the total reflection of light at the surface separating two transparent refractive substances, such as, for instance, the boundary plane between glass and air. This phenomenon occurs, for instance, in the prisms of a Zeiss field glass. If light falls perpendicularly on one of the short sides of a right-angled triangular glass prism, it will strike the surface of its hypotenuse at an angle of 45 deg. However, according to the law of refraction, it cannot emerge from this surface by refraction, but will be totally reflected in the same way as it would be by a perfectly reflecting mirror. (Figure 22 (a).)
Figure 22.—Total reflection in a glass prism.
If a second prism is placed near the first one (Figure 22 (b)), no change will take place so long as the distance between them is sufficiently great. But as soon as this distance becomes very small, a little light can pass into the second prism, too. For in total reflection, a small quantity of light energy always seeps through the reflecting surface, but only for a very short distance, of the order of magnitude of the wavelength. If the second prism is brought sufficiently close, the light which has seeped through can penetrate into the second prism and can travel further in the ordinary manner. The closer the two surfaces are to each other, the greater is the quantity of light that goes through, and if the surfaces are pressed firmly together, no more total reflection occurs at all.
Something quite analogous to this phenomenon takes place in the case of the matter waves of the alpha particles. In this analogy, the interior is to be regarded as the equivalent of one prism, and the outside space as the equivalent of the other one, while the potential barrier between them is the equivalent of the layer of air separating the two prisms. Some waves will always get through the barrier, and the thinner is this barrier, the larger proportion of the waves will escape into the outer space. In this connection, ‘barrier’ means that part of the potential curve which rises above the horizontal energy level of the particle. From this it follows automatically that the higher is this level—and hence, the greater is the energy of the alpha particle—the more transparent will be the barrier to the waves of the alpha particle, for the potential barrier that must be overcome, will be proportionately narrower. Thus, assuming that originally the waves were present in the interior only, as more and more time elapses, we will find an increasingly larger portion of them outside, too, and will find that this portion increases with the increase of the energy of the particle.
Now this description in terms of the wave aspect must be translated back into the language of the particle aspect. In doing so, we must bear in mind that in the course of our discussion of the conditions prevailing within the extranuclear atomic structure we stated that the local density of the matter waves constitutes a measure of the probability of encountering alpha particles at the point under consideration. But this density outside of the nucleus, and hence the probability of finding alpha particles outside of it, increases rapidly when we deal with a particle which is rich in energy, and more slowly when our particle is less rich in energy. The more energy a particle has, the greater is the probability that it will no longer be found inside, but outside of the nucleus—in other words, that it is hurled out within a very short time. This is where we find the explanation of the fact that in the case of alpha particles rich in energy the decay probability is very much greater than in the case of those less rich in energy. The exact mathematical expression of this idea leads to a very satisfactory agreement with the Geiger-Nuttall law.
If we wish to give a quite summary description, in terms of the particle aspect, of the effect just discussed, we may state that contrary to all expectations based on the law of the conservation of energy, after a certain period of time (the length of which is a matter of chance) the particle is able to break through the potential barrier as though through a tunnel. It is customary therefore to speak of a tunnel effect.
We have already mentioned that although all the elements above zinc in the periodic system might be expected to be prone to alpha decay, this decay can, nevertheless, be observed in the heaviest elements only. It would be logical to assume that no sharp dividing line existed between the well-known alpha emitters and those elements which would seem capable, from the point of view of energetics, of emitting alpha radiation, but are never observed to do so. It may well be possible that a major number of the elements above zinc in the periodic table do actually emit alpha particles, but both the energy and range of these particles are very small. Both these factors would account amply for our practical inability to observe them. However, such radioactivity would be imperceptible mainly because the low magnitude of energy corresponds to an extremely small decay probability. For even though such an element might emit an alpha particle at infrequent intervals, it is nevertheless stable for all practical purposes, even when measured by cosmical time standards.
II. THE BETA EMITTERS
Let us now turn our attention to the second type of spontaneous nuclear transmutation, which takes place with beta radiation—the emission of electrons or positrons, accompanied by a neutrino. Such transmutations take place when they are compatible with the conservation laws, and thus in particular when energy is liberated from the nucleus in the process. Here, too, there arises the question, why a nucleus in which all the prerequisites of a transmutation are given, does not change immediately.
The differences in the lifetimes of the various beta emitters are smaller than those among the alpha emitters. The half-lives of beta emitters vary from a few seconds to a few years. Only very few of them have a much longer half-life.
We cannot account for beta radiation by the same arguments which we used to explain the properties of alpha emitters; at any rate, these arguments are not applicable to the electron emitters, for every electron carries a negative charge, and consequently, when outside of the nucleus, it is attracted by the latter. Therefore, in this case, the potential barrier is non-existent. However, since there is no essential difference between electron emitters and positron emitters, we cannot suppose that there is an analogy between positron emitters and alpha emitters either. Furthermore, there is the fact that the electrons and pcsitrons, together with their accompanying neutrinos, do not constitute integral parts of the nucleus as do the alpha particles, but are created from the nuclear field only at the instant when the nuclear transmutation actually occurs. An analogy with the emission of photons from the extranuclear atomic structure seems to be a more logical one.
Empirically, it is generally true that the decay probability of the beta emitters also increases with the energy of the beta particle. It must be borne in mind that every electron and positron is accompanied by a neutrino, and as already pointed out, the reaction or decay energy is shared by this pair according to statistical laws. Therefore, the energy of the fastest beta particle, the neutrino of which happened not to receive any share of the decay energy, is decisive for the decay energy.
Now, in order to understand, at least qualitatively, the relationship between energy and decay probability, let us borrow from the theory of electric waves. For a beta radiation should actually be likened to the emission of light from the extranuclear atomic structure. We are thinking now in terms of the wave aspect, and therefore will speak of an electron or a positron, plus a neutrino, as a wave emanating from the atomic nucleus. We will liken this wave to the electric wave sent out by a radio aerial.
Planck’s law, E = hv, holds good for these matter waves, too; it correlates their energy, E, with their frequency, ν. In this case, E is the energy of the beta particle, or more exactly, that portion of the decay energy which falls to the share of the electron. If, then, the decay probability increases as the energy, the higher is the magnitude ν, the greater will be the decay probability, and therefore the shorter the wavelength of the radiation. If the dipole moment of a vibrating ae
rial is kept constant which can be achieved by maintaining the peak voltage between the condenser plates of the oscillatory circuit constant, then the radiation will be all the stronger the higher the frequency in the circuit. The intensity of the waves at a point is proportional to the fourth power of the frequency. The situation with respect to the matter waves of beta radiation is quite analogous, with the sole difference that their energy is proportional not to the fourth power of the frequency, but—as shown by a closer theoretical analysis by Fermi—to its sixth power. It follows, therefore, that the decay probability (the energy emitted per second divided by the energy of the individual decay) is proportional to the fifth power of the energy available for the decay process.
The above considerations represent the actual conditions in their general outlines only, but not quantitatively correctly. Before we can gain a complete understanding of the subject, we must discuss still another consideration. We have assumed the dipole moment of the aerial to be constant. However, this cannot be expected to be the case for every one of the various nuclear types. On the contrary, considerable individual differences must be expected to exist, and do in fact exist. One would therefore expect the decay probability to be the product of two factors, the first being determined by the dipole moment (and as the latter is determined quantitatively by the size of the nucleus in question, it may vary widely from one nucleus to another), and the second one being proportional to the fifth power of the decay energy. Figure 23 shows a summary of the lifetimes, and hence of the decay probabilities and energies, of the beta emitters. Whenever a power of a quantity is supposed to be proportional to a power of another quantity, it is most convenient to record on the co-ordinate axes not these powers themselves but their logarithms, for the logarithms must be linearly related to each other. Thus, Figure 23 shows log T, the logarithm of the half-life, T, measured in seconds, as a function of log E; the scales on the two axes differ by the factor 5. If the fifth-power law is valid, all the beta emitters must lie on a straight line forming an angle of 135 deg. with the axes. Actually, the empirical points lie, mostly, between two such straight lines; the distance between these two straight lines provides a measure of the fluctuation of the dipole moment from one nucleus to another.