by W Heisenberg
The first term of this equation is negative, as it must be in the case of a binding energy. On the other hand, the second term, which is considerably smaller, is positive, and therefore produces a reduction of the absolute magnitude of the negative binding energy. The latter decreases when N and Z are different from each other. Therefore, from the point of view of energy economy, the most favourable situation seems to be present when a nucleus contains exactly as many protons as neutrons.
But here we must mention a correction depending on the surface tension. The particles situated on the surface are bound together less tightly than those in the interior, and this circumstance, consequently, calls for the addition of a further positive term on the right-hand side of our equation. For since the principal term, − A, is negative, this side of the equation is essentially negative, and something positive must be added to it in order to decrease its value. The change in the quantity of binding energy due to surface tension is proportional, in every instance, to the number of the particles situated on the surface, and hence to the surface area. But the latter is proportional to the nuclear volume (or to the total number of particles) raised to the power of 2/3his portion of the binding energy can therefore be expressed by the equation E0 = C(N + Z)2/3. The portion per particle is obtained by division by N + Z, i.e,:
where C is again a constant.
Finally, another term, a result of the electrical repulsion of the protons, must be added. Here we are in the familiar territory of electrostatics. The charge on a nucleus is Ze, where e is the elementary quantum of electricity. The electric energy of a nucleus is proportional to the square of its charge, i.e. to Ze2 just as the energy of a condenser is proportional to the square of its charge. Furthermore it is inversely proportional to the radius, r, of the nucleus. There is also a numerical factor to be considered, which for a homogeneously charged sphere is 3/5 If the charge is displaced slightly toward the surface, this factor becomes smaller and will approach . Since the radius is known, anyway, not very exactly, we can retain the factor 3/5spite the fact that there is undoubtedly a certain displacement of the charge toward the surface. This energy component may therefore be written:
But since the radius r is proportional to the cube root of the volume, i.e. to the cube root of the number of particles, we can write: r = r 0 (N + Z)1/3, where r0 is a constant which would, so to speak, be the radius of a nucleus containing one single particle, but naturally must not be identified with the radius of the proton or the neutron. Thus, this portion of the binding energy per individual particle is:
We must also add this term to our equation with a positive sign, since the electric repulsion reduces the quantity of the total binding energy. The complete expression of the binding energy per particle, finally, is:
This equation contains four constants, A, B, C and r0, of which, to start with, we know only ro—and even this one rather inexactly—from the measurements of nuclei.
In order to make use of this equation, it is essential to know the four constants exactly. If we knew more about the interior nuclear forces, and about nuclear forces in particular, we could calculate them theoretically. Actually, however, only the converse method is available to us—their empirical determination on the ground of the data already known to us about the binding energies of nuclei, in other words, about mass defects. For instance, a study by Fliigge and v. Droste has resulted, by this method, in the following numerical values: A = 001574 a.m.u. ; B = 0022 a.m.u.; C = 00165 a.m.u.; = 0000646 a.m.u.
Thus the binding energies of the nuclei can actually be represented in good agreement with empirical findings. The curve representing our equation, shown in Figure 11, agrees approximately with the above-mentioned values of the constants; this curve shows the energy of the most stable elements for the atomic weights indicated. Since the binding energy is a negative quantity, the lower a point is, the greater is the binding energy of the element represented. The dots appearing along the curve are the values of binding energy per particle actually calculated from the mass defects. It will be seen that the agreement is very satisfactory.
Figure 11.—Binding energies of nuclei as functions of N + Z.
Let us now analyse this curve. The position of its lowest point is certainly determined to a dominant extent by the largest term in our equation, − A. The rise observed for small atomic weights is due to the surface tension, which naturally plays the most important part for light nuclei. The rise for heavy nuclei is due to the electric repulsion of the protons.
Furthermore, our equation indicates that the lighter nuclei mostly contain approximately as many protons as neutrons. The last term, which is due to the force of repulsion, increases with the increase in the value of Z, and plays no important part as yet in the cases of light atoms, where Z is small. In such cases, it may therefore be disregarded. The third term is determined solely by the sum total of the number of particles, and not by their ratio, as in the second term, which disappears when N = Z, in other words, when the number of protons equals the number of neutrons, and in which case the quantity of binding energy reaches the maximum. This means that from the point of view of energy, this is the most favourable situation. But this is not the case with the heavier atoms, where the last term produces a perceptible decrease of the binding energy. As regards the energy situation, it is more profitable that the second term should increase slightly, in consequence of N/Z slightly exceeding unity, in order that a very considerable diminution in the fourth term may thereby be achieved.
The binding energy is, in fact, a function of both N and Z, and we can represent it as such in a three-dimensional co-ordinate system, recording the number of protons on the horizontal axis running from left to right, the number of neutrons on the horizontal axis running from the front to the rear, and the binding energy on the vertical axis. Our equation will then produce a surface, for a particular energy value belongs to each pair of magnitudes (N, Z). The shape of the energy surface supplies all the information necessary for determining the stability of the nucleus. Figure 12 shows this surface, in the form of a map, with contour lines representing equal magnitudes of binding energy. The energy unit used is 0·001 a.m.u. Since the binding energies are negative, the surface lies below the plane of the drawing, like a valley dropping away from the lower left corner of the drawing toward the upper right corner (in other words, from the south-west to north-east, to keep the map analogy). The stable nuclei lie at the bottom of this valley, like houses along a gently winding street. Those nuclei which contain the same number of particles, always lie on straight lines, running from north-west to south-east at an angle of less than 45 deg. Among these nuclei, the one situated nearest to the bottom is always the most stable one. In so far as it is compatible with the conservation laws, one of the less stable nuclei can always change into a more stable one. The nuclei lying on the left side of the valley floor have too many neutrons, so that they must change by the emission of an electron. Those on the right side have too many protons, and would have to change by the emission of a positron. We find, however, that this very often does not happen, and there frequently exist two or three stable nuclei having the same N + Z total. These nuclei are called nuclear isobars. This situation can be explained by certain finer details of the valley bottom only, not immediately evident from our equation based on very general assumptions. Actually, the valley bottom shows small convolutions and other fine details, on which actual measurements of the binding energies will supply information. We may say, nevertheless, that the stable nuclei will lie at the valley bottom or in its most immediate neighbourhood; we can also name the most stable of those nuclei which contain a certain given number of particles.
Figure 12.—Positions of the stable nuclei and curves of constant binding energy.
Figure 13 shows a picture of the situation in a different light. In this case, the combined totals of the numbers of particles, N + Z, appear as abscissae, and the ratios of the particles, N/Z, as ordinates. The continuous straight line cor
responds to the dots along the valley floor, and the dots indicate the positions of the individual stable nuclei. We can see how they are distributed along the valley floor and in its immediate vicinity.
Figure 13.—N/Z as functions of N + Z in the case of stable nuclei.
In Figures 11, 12 and 13, only stable nuclei are shown. The complete picture is presented by Tables IVa and IVb (at the end of the book) which include all the known unstable nuclei. The abscissae show the number of protons, the ordinates show the difference between neutrons and protons, N − Z. Stable nuclei are indicated by black dots, electron emitters by upright triangles (the apex pointing upward), and positron emitters by inverted triangles. Since the electron emitters are relatively rich in neutrons, whereas the positron emitters contain relatively few, the former appear mostly in the region where N − Z is larger—i.e. on the upper periphery of the group of nuclei—while the majority of the positron emitters are found at the lower edge of the group, where N − Z is smaller. It is a striking fact that not infrequently unstable nuclei are also found interspersed among the stable ones. The reasons for this circumstance will be discussed later. Certain nuclei change by capturing an electron from the innermost electron shell (the K-shell) of their extranuclear structure. These nuclei are referred to as K-capturers, and are indicated in our tables by circles. We find also a number of squares, mostly at the end of the table, indicating nuclei which emit alpha rays.
Up to this point, we have discussed nuclear stability in its relation to transmutation by the emission of electrons or positrons. Let us now briefly discuss stability in its relation to transmutations by alpha ray emission. Studying Figure 11, we can conclude from the initial decrease and subsequent increase in the magnitude of the term E/(N + Z) that nuclear stability first increases steadily, up to about where Z + N = 40, but thereafter it gradually decreases, because of the increasing effect of electrical repulsion. Nevertheless, even in these cases, work has to be done in order to remove an individual particle from the nucleus. Let us assume that after applying the necessary work, we remove simultaneously two neutrons and two protons from the nucleus and then combine them in a helium nucleus. In this process a very large quantity of energy, 30 Mev., is liberated. If this energy is greater than that quantity of energy required in order to separate four individual particles, the net result is an actual gain in energy. Therefore, from the point of view of energetics, this process is an advantageous one and ought to be taking place spontaneously, in the form of the formation of an alpha particle in the nucleus and its subsequent emission by the latter. The probability of the occurrence of this process is bound to increase with the increase of the number of particles, for the binding energy per particle decreases as the total number of particles increases. Therefore, alpha emitters are to be looked for among the heavier nuclei, in conformity with actual experience. Actually, as we approach the heaviest nuclei, the magnitude of binding energy per particle drops—approximately at least—to a value between 6 and 7 Mev., in other words to about one-fourth of the binding energy of a helium nucleus.
When the total number of particles is high, the splitting of a nucleus into two parts, not very different in size, is advantageous from the point of view of energetics. A nucleus of mass number 230 might split into two nuclei, one of mass number 100 and another of mass number 130, for the sum total of the binding energies of these two nuclei is greater than the binding energy of the original single nucleus of mass number 230. Instances of such splitting—fission—of atoms were actually observed by Hahn and Strassmann in 1938.
As a matter of fact, we might well wonder why it is that not every one of the heavier elements is an alpha emitter or splits into two nuclei of more or less equal size, instead of holding together, as they do according to actual experience, for a remarkably long time at least. As regards fission, the lifetimes of these elements are even longer. We shall inquire into this question in our sixth lecture.
5. THE NUCLEAR FORCES
I. GENERAL PROPERTIES OF THE NUCLEAR FIELD
The cohesion of protons and neutrons within the atomic nucleus is ensured by forces, to which we have referred as nuclear forces, although we have not yet discussed their nature. The electric forces of repulsion, which are also operative in the nucleus, exert a purely disruptive effect. What can we learn today from the experiments on the nature of these nuclear forces? Let us first study briefly the form which an answer to this question must take. Assuming that we did not as yet know what electric forces were, how could we begin to explain their nature? We could begin by stating that electric charges mutually repel each other, with a force which decreases inversely as the square of the distance between them. According to the knowledge gained in the early part of the nineteenth century, we could add that a fundamental relationship exists between electric and magnetic forces—e.g.: varying electric forces always generate magnetic forces, and vice versa. Moreover, we might add that the phenomena of light, which used to be regarded as something of a special nature, are among these electromagnetic phenomena, and are simply nothing but electromagnetic waves. The next step would be to note that in certain experiments light appears not as a wave, but as flying particles, in other words, in the shape of photons, and thus we would discover a relationship between the electromagnetic field and these photons. A truly exhaustive description of electromagnetic forces can, however, be given solely with the aid of mathematical equations, expressing how electromagnetic forces change and spread. A complete picture of the ‘nature’ of these phenomena is supplied only by Maxwell’s equations in combination with the equations of the quantum theory.
But as for the nuclear forces we have not got to them yet. Nevertheless, we can already form a picture of them, which is correct qualitatively at least and includes as many details as the picture of the electromagnetic forces—except for the exact mathematical formulae.
The first question to arise here is: How does the force operating between two particles in the atomic nucleus depend on their distance from each other? Is it perhaps also in inverse proportion to the square of this distance? The simplest object available for the study of this problem is the deuteron; we inquire about the force which binds a proton and a neutron in a deuteron. Once this force is known, we have a good prospect of comprehending the cohesion of other nuclei. The force in question cannot possibly be electrical in character, if for no other reason than that the neutron carries no electric charge. Furthermore, electric forces would be much too weak to account for the considerable energies which result from the mass defects.
Figure 14.—Deflection of a neutron in the neighbourhood of a proton
We have already pointed out that when a deuteron is formed out of a proton and a neutron, the binding energy is liberated as a photon, with an energy of 2·2 Mev.—in other words, as electromagnetic energy. This means that a process is taking place in which energy is converted from one form to another—the non-electromagnetic energy of the nuclear field into the electromagnetic energy of radiation. Therefore, it follows that in common with all other types of energy, the energy of the nuclear field possesses the capacity of being transformed into other forms of energy.
We can gain some insight into the dependence of nuclear forces on the distance by observing the deflection of flying neutrons when they pass near a proton (Figure 14). Modern physics has access to sources of neutrons. All that is necessary is simply to send the neutrons through a hydrogen-containing substance, for example, a hydrocarbon, such as paraffin, or water, to cause them to be deflected from their straight paths. The magnitude of the deflection of a neutron depends, naturally, on the distance of its path from the proton. This distance is more frequently a greater than a smaller one. Instances of neutrons passing close to a proton are very rare. If the forces decrease relatively slowly with the distance—as, for instance, in the case of electric charges—even though the distance may be considerable, the neutrons will still be deflected a little. It would indeed be observed that a very large number of neutrons ar
e deflected, but the deflection is always very slight. As a matter of fact, large deflections are seldom observed. If, on the other hand, the force diminishes rapidly with the distance, the majority of the neutrons are not deflected at all. In the case of the deflected neutrons—those few which pass sufficiently close to the proton—both small and large deflections may be observed to occur with comparable frequency.
Such experiments have demonstrated that the force between neutron and proton diminishes with the distance more rapidly than is the case with electric forces of attraction and repulsion. The degree of accuracy of the measurements does not as yet permit us to formulate exactly the law of distance. Nevertheless, we can state that the force is becoming already very small at a distance of 5 × 10−13 cm. This means that the force between proton and neutron has an extremely short range, and in this respect it is very different from an electric force.
Instead of studying the force, we may base our considerations on the potential energy which a neutron has within the field of a proton (or vice versa). When the distance is a very large one, we arbitrarily assign to this energy the magnitude 0. At finite distances, it has a negative magnitude. Due to the limited range, the potential energy is practically 0 for any but a very small distance. Figure 15 shows an approximate curve of the potential energy as a function of the distance r. This energy increases rapidly from high negative magnitudes until it reaches the vicinity of 0, which it then approaches asymptotically. Dealing with small distances, the potential energy curve can be computed indirectly from the mass defect of the deuteron. In addition to its potential energy, the system has also a kinetic energy, since the proton and neutron reciprocally vibrate with reference to each other, with a continuous transformation of kinetic energy into potential energy, and vice versa. The sum total of these energies always equals the binding energy, 2·2 Mev., which is shown in Figure 15 as a horizontal line. The average magnitude of the kinetic energy can be estimated, for instance, in accordance with the uncertainty principle, from the diameter of the deuteron. The accuracy of our knowledge of the exact location of this object of finite diameter goes hand in hand with a proportionate inaccuracy of our knowledge of its velocity; and the squared velocity times one-half the mass gives us the magnitude of the average kinetic energy. Once we know both the kinetic energy and the total energy, we can calculate from them the potential energy. Figure 15 is the result of considerations of this nature.