by W Heisenberg
A nucleus is built of protons and neutrons. Its building blocks interact firstly, through an electric field which results from the charge carried by the protons, and secondly, through a nuclear field, which in some as yet undisclosed way ensures the internal cohesion of the nucleus. The electric field is responsible for the emission of gamma-ray photons, and the nuclear field is responsible for the emission of electrons, positrons and neutrinos.
Whether the connection between the nuclear field and the particles correlated with it here is as simple as that between the electric field and the photons, is, of course, open to doubt. As we shall see in due course, this relationship is probably a more complex one. But generally speaking, it is permissible to apply the analogy outlined above.
Thus we have obtained a fairly clear picture of the nucleus. We may interpret it as meaning that a sufficiently powerful super-microscope would show nuclei to be such as we have described them to be—built of two kinds of component parts, protons and neutrons. Therefore, every nucleus can be identified extremely simply, by two numbers: the number of its protons, and the number of its neutrons.
The mass of a nucleus is (to be sure, not quite exactly) equal to the combined number of its protons and neutrons; both particles have a mass of approximately 1 mass unit. On the other hand, the nuclear charge is possessed by the protons alone, each of which carries one elementary electric charge.
From this simple picture of the nucleus it follows, in general, automatically that different nuclear mass numbers can be associated with a given nuclear charge (atomic number). In other words, there are different varieties of atomic nuclei for the same chemical element; these different varieties are called the isotopes of the element.
The mass number of the nucleus indicates the sum total of the number of its protons and neutrons, whereas the atomic number shows the number of protons only. Hence, the number of the neutrons is simply the difference between the mass number and the atomic number. These two numbers, the essential characteristics of the atom, have already been introduced as a superscript and a subscript t6 the chemical symbol of the element. For instance, the notation 7N14 means that this nitrogen nucleus consists of 7 protons and 14 − 7 = 7 neutrons.
Let us now examine some of the simpler nuclei more closely. Disregarding the neutron, which is not usually grouped with the elements in the proper sense of the word, the simplest nucleus of an element is the proton itself. The proton is a hydrogen nucleus. Its symbol is 1H1, expressing that it consists of 1 proton and 1 − 1 = 0 neutron. The chart below shows schematic representations of the simplest nuclei; black dots stand for protons, circles for neutrons. Thus, the hydrogen nucleus 1H1 is represented simply by a black dot. But as Urey discovered in 1932, there exists another nucleus, a heavy isotope of hydrogen, which consists of one proton and one neutron. This isotope occurs in natural hydrogen to the very small extent of 0·02%. This variety of hydrogen is called deuterium, and its nucleus a deuteron. Since it differs chemically in certain respects from ordinary hydrogen, it is usually designated by the symbol D, or more exactly, 1D2. It may be written also as 1H2. This is the so-called ‘heavy hydrogen’. A third variety of hydrogen was discovered later; its nucleus consists of 1 proton and 2 neutrons. It is called tritium, and its nucleus is referred to as a triton. Its symbol is 1T3 or 1H3. This nucleus is not stable; it is radioactive, with a very long half-life (probably about thirty-one years), and is an emitter of electrons. Therefore, tritium does not occur in nature, but is a product of nuclear transmutation processes.
The next simplest element is helium, the nucleus of which contains two protons. Helium occurs also in the form of several isotopes, each with a different number of neutrons in the nucleus. The lightest helium nucleus consists of 2 protons and 1 neutron, and its symbol, therefore, is 2He3. It became known as the product of the radioactive transmutation of the triton, 1T3, although it occurs also in natural helium, in very minute, quantities. (This nucleus and the nuclei of the other helium isotopes, are shown in the above chart.) The next nucleus is that of ordinary helium, consisting of 2 protons and 2 neutrons; its symbol, as we already know, is 2He4. This is an especially stable structure. There are two other helium nuclei, neither of which is stable. They contain, respectively, 3 and 4 neutrons, and are therefore written 2He5 and 2He6. They are not present in natural gaseous helium.
Continuing to add further protons and neutrons, we come to nuclei which are increasingly complex in structure. We could make a chart of all the existing nuclei, recording their atomic numbers—in other words, the number of their protons—designated by the symbol Z as abscissa, and the numbers of their neutrons, designated by the symbol N, as ordinate. It is however more convenient for the printer to adopt a different system in which the abscissae, Z, represent the number of protons and the ordinates, N−Z, the amount by which the number of neutrons exceeds that of the protons. This is the method used in our Tables IVa and IVb. The order of the individual elements also corresponds to their sequence in the periodic system. In the tables, the nuclei are also distinguished according to their stability or radiation characteristics. Stable nuclei are represented by black dots. One of these is the nucleus 2He4, which appears where Z = 2 and N−Z = 0. Triangles indicate the radioactive nuclei which emit beta rays. When the apex of the triangle points upward, it indicates an emitter of electrons; a triangle with the apex pointing downward indicates an emitter of positrons. The electron emitters always appear in the uppermost row—e.g., 2He6 or 3Li8—and the positron emitters mostly in the lowest row, e.g., 6C11. The radioactive nuclei which emit alpha particles, are indicated by small squares. Finally, there are unstable nuclei which change by the capture of an electron from the innermost extranuclear electron shell and thus reduce their nuclear charge number by one unit. In our tables, these nuclei are indicated by small circles. Those nuclei which can emit both electrons and protons, are marked by two triangles superimposed on each other in the shape of a star. Thus, our tables furnish a simple survey of all the existing nuclei, their structures and properties.
As we see, the neutron excess, N−Z, is negative in just a few species of atoms only; in all the rest, it is a positive magnitude, and never a very big one. The nuclei of the lighter elements contain, throughout, practically as many neutrons as they contain protons, and the neutron excess becomes more or less considerable in the heavier elements only.
We have now described, in general outline, the structure of individual nuclei, and this discussion brings up a great many further questions: What is holding together a nucleus which consists of protons and neutrons? What is the nature of the forces that bind these particles together? Why do lighter nuclei contain approximately as many protons as neutrons, while the heavier ones show a slight neutron excess—in other words, why does the magnitude of the neutron excess increase with the increase of the number of unit charges on the nucleus? Why is there only a limited number of nuclei? Why are many of these radioactive, and why do they emit those very particles which we observe to be emitted by them? In the following lectures, we shall discuss these problems.
4. THE NORMAL STATES OF ATOMIC NUCLEI
I. THE BINDING ENERGY OF THE NUCLEI
Of the questions mentioned at the conclusion of the previous lecture, let us discuss first the one concerning the forces which are operative between the building blocks of the nucleus and hold these particles together. This question may be formulated as follows: What physical magnitude or what property of the atom determines its stability? One might be inclined to think, at first, that this question is a difficult one, and that in order to answer it, we must be familiar with the entire mechanical system represented by an atomic nucleus, in all its details. However, fortunately, this is not the case. There are a few fundamental laws which enable us to discuss the stability and other general properties of a system even though we may be ignorant of the nature of the forces acting within it, as well as of the details of its structure. These are the well known laws of the conservation of mass, energy,
etc., which state that these things can neither be created out of nothing nor be annihilated. Those laws which are of primary significance here, are the laws of the conservation of energy, of electric charge, and of angular momentum.
We shall begin with the law of the conservation of energy. Let us assume that it is possible to employ certain forces so as actually to remove a particle from a nucleus (which latter consists of protons and neutrons). We may make this theoretically possible by the naive assumption that we can actually seize this particle and transport it to a point far away from the nucleus. Since the particle was originally firmly bound to the nucleus, it is now attracted by it, so that a certain amount of work is required in order to remove it; in other words, energy must be introduced into the system. Now, according to the energy laws, this work, this imported energy, is quite independent of the method employed to remove the particle. Therefore it follows that every particle is bound within the nucleus by a definite quantity of energy, and this quantity of energy can be calculated if it is somehow possible to ascertain the quantity of energy of the system before and after the removal of the particle. Let us now proceed to define the concept of the binding energy of a nucleus. This term stands for that change in nuclear energy which takes place when the constituent parts of the nucleus—originally far away from each other—are combined to form that nucleus. Since the reverse of this process, the breaking up of the nucleus, requires an expenditure of energy—in other words, energy must be applied to the nucleus—the nucleus most lose energy in the process of its formation. Therefore, the binding energy of an atomic nucleus is always a negative quantity, by definition. Naturally, the more stable a nucleus is, the more difficult it is to split it into its constituent parts, and the greater is the quantity of work that is necessary for this purpose. Therefore, stability increases with the absolute magnitude of the (negative) binding energy, and thus, strictly mathematically, the smaller the binding energy, the greater is the stability of the nucleus. For this reason, when speaking of a greater or smaller binding energy of a nucleus, we usually refer to its absolute magnitude. In this sense, the greater is the binding energy of a nucleus, the greater is its stability.
Since we are still unfamiliar with the details of nuclear structure, we are unable to calculate the binding energies from the nuclear properties. We must therefore attempt, conversely, to ascertain the magnitudes of binding energies by other methods, in order to use them as a basis for drawing conclusions about the properties of the nucleus.
The simplest example of a nucleus containing more than one particle is the deuteron, the hydrogen nucleus of mass number 2, which consists of 1 proton and 1 neutron. When such a nucleus is formed from these two component parts, the same quantity of energy must be liberated which would be necessary to break it up. Thus we start out from a state in which the proton and the neutron are still in a state of rest, far away from each other, and exert practically no force on each other. Let the quantity of energy present, under these conditions, within the system constituted by these two particles, be called 0. (The choice of zero for the potential energy of a system is arbitrary, and this choice may always be made in the most expedient manner.) As soon as the particles have combined in a deuteron, the energy of the system has decreased by the absolute magnitude of its binding energy. If we manage to measure in some way the energy content of a deuteron, we can ascertain the magnitude of its binding energy from the difference in energy before and after the joining of its constituent parts, and we can use it as a basis for reaching a conclusion concerning the stability of this nucleus.
We can proceed by the same method and add, let us say, another proton. This gives us the binding energy of the helium nucleus 2He3. And we could continue in the same manner, step by step, to ascertain the magnitude of the binding energy of every nucleus.
The physicist customarily measures energy in ergs (the erg is a unit of the centimetre-gramme-second system), whereas the engineer measures it in foot-pounds or, e.g. kilowatt-hours. For thermal energies he uses the calorie as a unit. Thus, the different branches of physics and technical science use those units, which are most suitable for the field in question because they are designed for the orders of magnitude of energy generally encountered in that particular field, so that the energy units are neither so small nor so big as to be unwieldy or inconvenient to work with. The same principle applies to atomic physics. Atomic physicists often make use of charged particles, electrons, accelerated by high voltage, in order to measure the binding energies of electrons. Therefore, in this case the unit of energy is that quantity of energy which is gained by an electron—or in general, by any particle which carries one elementary quantum of electric charge—in passing through a potential drop of 1 volt. This energy unit is called the electron-volt (ev). It is a very convenient unit to use when dealing with extranuclear atomic structure, for it is more or less of the same order of magnitude as the binding energies of this structure. But the binding energy of the nuclear particles is about one million times greater. Therefore, in nuclear physics it is customary to use a million times this unit—1 million ev = 1 Mev. One Mev is the energy which a particle carrying one elementary quantum of electricity gains on dropping through a potential difference of 1 million volts. However, it is still a very small quantity when compared with 1 erg, viz:
1 Mev= 1·6 • 10−6 erg.
We have just mentioned the energy which is liberated when a proton and a neutron combine in a deuteron. As a matter of fact, it is possible to measure this energy very well by actually producing this phenomenon. A source of neutrons is required for this purpose. Modern experimental physics has such sources available. Of course, it is true that during the process of their production, neutrons move at very high velocities and must first be slowed down in order to be brought nearly to rest, so as to be capable of combining with hydrogen atoms from this state of rest. We therefore shoot them through some substance which contains hydrogen. The neutrons collide with very many hydrogen atoms, and as a result of these collisions, they gradually lose most of their kinetic energy, except for a very small quantity which corresponds to the temperature of that particular substance. They end up with a so-called thermal velocity. In this state they are caused to combine with protons.
In this process, the binding energy of the deuteron is liberated, and in conformity with the law of the conservation of energy, it must still exist somewhere—in other words, it must go somewhere, in some form. The most logical assumption is that it leaves in the form of an electromagnetic radiation, namely, as gamma radiation of an extremely short wavelength. It is therefore to be expected that the formation of each deuteron, from one proton and one neutron, is associated with the emission of a gamma-ray photon very rich in energy. The relationship of the energy E of this photon to its frequency v is determined by Planck’s formula, E = hv, where h is Planck’s constant. The energy E is thus identical with the amount of the binding energy of the deuteron. This gamma radiation can actually be observed. Since we know how to measure the frequency v, it is actually possible to measure the binding energy of a deuteron. It amounts to 23 Mev. One might surmise that this energy is radiated in the form of several photons, rather than as a single one. However, it can be proved that such a process is actually far more improbable than the emission of one single photon.
There is, however, another, simpler method available for acquiring knowledge concerning the binding energy of atomic nuclei. According to Einstein’s theory of relativity, there is a simple relationship between the mass of a body and its energy content. A relationship of this kind was known, in a special form, in pre-relativistic days, too; it follows from the electro-dynamics of moving objects. Hasenöhrl had already pointed out that the radiation enclosed in a cavity had a seemingly inert mass m, proportional to the energy of the system, m being proportional to E/c2, where c is the velocity of light. But he failed to compute the proportionality factor correctly. The relationship between energy and mass was made clear by the theory of relati
vity, which—this is the decisive point—raised it to the status of a universal natural law, applicable not only to the theory of radiation, but to every other branch of physics as well. In other words, the following relation is true universally:
The meaning of this equation is: Any system having the energy content E has a mass m commensurate with this energy content, and the quantity of this mass is E/c2. This conclusion has rather strange consequences. For instance, when a clock is being wound, it must become slightly heavier because energy is being stored in its spring. But the quantities of energy in this case are so small as to make the increase in the mass of the clock too minute to be capable of being demonstrated. The mass E/c2 is much too small in comparison with the mass of the clock itself.
But this relationship between energy and mass can be put to practical use in nuclear physics, where the stored-up energy is of appreciable magnitude compared with nuclear masses. Expressed in the form E = mc2, this relationship enables us to calculate the energy content, E, of a system from its mass, m. The velocity of light is known; it is almost exactly 300,000 km. per second, i.e. 3 X 1010 cm. per second. In atomic nuclei, the orders of magnitude are totally different from those met with in the above example of winding up a clock. It is true that the binding energies of nuclei are very small, but so are their masses too. Therefore, the mass m = E/c2 is no longer negligibly small in comparison with the masses of the nuclei themselves. Consequently, any change taking place in the nuclei as a result of changes in energy content, can be measured with a considerable degree of accuracy. The application of the above formula to atomic nuclei confirms at the same time the important relationship it expresses.
