Nuclear Physics

by W Heisenberg

  III. THE SATURATION OF NUCLEAR FORCES

  The above-mentioned analogy with valency forces suggests a conclusion which, even though we cannot prove it here, we can at least make plausible. An essential difference—among others—between valency and electric forces is that the former are capable of saturation. The chemist provides the symbol of every atom with a certain number of valency bonds, the number of which corresponds to the valency of the element. In every structural formula of a saturated chemical compound, such valency bonds are represented by lines issuing forth from the atomic symbol and terminating in another atomic symbol, from which, in turn, as many such lines issue forth as the numerical value of its valency number. For instance, carbon dioxide, O=C=O, the compound of the tetravalent carbon atom and two divalent oxygen atoms. The characteristic feature in this case is that an atom, all the valency bond lines of which are taken up, is saturated, in other words, it has used up its valency. Thus, for instance, in the case of the water molecule, H—O—H, the two valencies of the oxygen atom are saturated by hydrogen atoms, and no further hydrogen atoms can be bound. A more or less normal molecule of the OH3 type is non-existent.

  Now the nuclear forces or at least a large part of them have a completely similar property of saturation. A neutron can bind to itself not more than two protons, and a proton not more than two neutrons. If we wished to express this fact in the form of valency lines, the proton would have to be written with two such lines, which could connect with neutrons only, and similarly, a neutron would have two valency lines, which could connect with protons only. This is, of course, not quite correct, since we are disregarding the forces which act between any two protons or any two neutrons. Nevertheless, this statement furnishes a preliminary understanding of the actual conditions. This property of saturation of nuclear forces explains why—as already discussed—the binding energy of the individual building blocks of nuclei is independent of the size of the nucleus. If a particle becomes lodged in a nucleus, owing to the extremely short range of the nuclear forces it enters into interaction with its immediate neighbours only—quite unlike the case of electric forces—and secondly, owing to the saturation of the forces, this particle is capable of linking itself to two of these immediate neighbours only. This is a further justification of the analogy between nuclear matter and a liquid. For the same applies basically to the atoms in a liquid, owing to the very similar properties of the forces responsible for their cohesion.

  IV. THE STABILITY OF NUCLEI

  All that we have set forth above points to the important conclusion that the number 2—and in general, the even numbers—must have a preferred position in atomic nuclei. Consequently, we may expect those nuclei in which both the number of protons and the number of neutrons are even numbers to be particularly stable. For since every proton is able to bind two neutrons, and conversely, every neutron is able to bind two protons, in this case—and only in this case—all valencies can be utilized. This situation is, naturally, more advantageous from the point of view of energetics than one where the valency of an odd proton or neutron is not utilized.

  But there is still another reason for the preferred position of the number 2: namely, Pauli’s exclusion principle, which we discussed when dealing with the extranuclear structure of the atom. Stated in general terms, this principle asserts that in a stationary state (which with reference to the extranuclear atomic structure means a certain definite electronic orbit, or expressed in terms of the wave aspect, a certain constant vibration with a certain given direction of the electronic spin) there is room for one single particle only at any one time. Taking into consideration the fact that the spin of an electron may be either positive or negative—clockwise or counterclockwise—this can be expressed in the following form: No more than two electrons (with opposite spins) can occupy the same stationary orbit of the extranuclear structure of the atom. The same applies to the protons and neutrons in the nuclei, which likewise possess a spin momentum. It follows, therefore, that not more than two neutrons or two protons can occupy the same stationary orbit in a nucleus. From the point of view of energetics, it is of course more advantageous to have the possibility thus allowed fully utilized. This fact again results in a preferred position for nuclei containing an even number of neutrons and an even number of protons.

  This preferred status of the number 2 is especially obvious in the common helium nucleus 2He4, which consists of two neutrons and two protons. This is an especially stable structure, and so is its extranuclear electron structure, which consists of two electrons. This is demonstrated by the fact that helium is an inert gas and does not enter into any chemical combination whatsoever. Actually, the binding energy of the helium nucleus is extraordinarily high: roughly 30 Mev. On the other hand, the binding energy of the deuteron, composed of one neutron and one proton, is only 2·2 Mev., as already mentioned. In the deuteron, only one valency of the proton and one of the neutron are utilized, whereas in the helium nucleus all valencies are saturated.

  It is therefore to be expected in general that the preferred nature of even numbers will manifest itself in a particularly high stability of the nuclei which contain both an even number of protons and an even number of neutrons, which we shall call ‘doubly even nuclei’. Those nuclei in which either N or Z is an even number while the other one is odd, are less stable; those in which both N and Z are odd, are still less stable. There is a general empirical proof of this statement: It might be expected that the more stable a certain nucleus is, the more frequently it would occur in nature, since at the time of the original formation of nuclei from their building blocks, the most stable nuclei would have a preferred status both as to frequency of formation and the chance of remaining intact. Many years ago, Harkins attempted to determine the empirical relationships between odd or even numbers on the one hand and the natural abundance of elements on the other. He found that the by far most abundant atomic species were, in fact, those which today we know to have ‘doubly even’ nuclei. Nuclei in which either N or Z is an odd number—‘odd nuclei’—are much rarer, and rarest of all are the nuclei in which both N and Z are odd numbers, which we shall call ‘doubly odd’ nuclei.

  ‘Doubly even’ oxygen, 8O16, is one of the most common elements. The ‘odd’ lithium, 3Li7, is much rarer. Finally, we must note the fact that only a very few stable ‘doubly odd’ nuclei exist at all. The simplest one of these is the deuteron, the nucleus of deuterium, 1D2. The only others are: the lithium nucleus 3Li6, the boron nucleus 5B10, and the nitrogen nucleus 7N14. All the others of this kind are radioactive and change (‘decay’) by the emission of electrons or positrons.

  On the ground of the foregoing considerations, let us now analyse the question of nuclear stability in greater detail. We have already described the energy surface (Figure 12). This is a very steeply slanting surface with a channel or groove, at the bottom of which the stable nuclei lie. We shall now make a cut through the surface, from the upper left to the lower right, at an angle of 45 deg. to the axes, diagonally to the channel, so giving us a cross-section of the surface (Figure 18). As a result of the nature of this cut, this cross-section will contain only nuclei with the same N + Z total, i.e. with the same mass number. We arrange this cross-section first of all so that N + Z is an odd number. The result is a curve, the lowest point of which is on the valley floor. Only those nuclei lie on this curve which are capable of changing into each other by the emission of an electron or positron; only the lowest nucleus, the nucleus possessing the greatest binding energy, is expected to be stable. In this diagram, the neutron numbers (their absolute magnitudes are of no importance here) are recorded on the abscissa in ascending order, while the binding energy is shown by the ordinate. Since N + Z is constant, an increase in the magnitude of (N − Z) means an increasing N and a diminishing Z. Those nuclei which lie to the right of the lowest point, have a greater number of neutrons and a smaller number of protons than the stable nucleus, and will gradually change into the stable nucleus by the emiss
ion of one or more electrons. On the other hand, the nuclei situated to the left, which contain more protons and fewer neutrons than the stable nucleus, will effect their change by the emission of positrons or by the capture of an electron from the extranuclear atomic structure (K-capture), as shown by the arrows in Figure 18. All this is in full agreement with experimental evidence. Figures 18 and 19 show the conditions for the specific mass numbers 91 and 92.

  Figure 18.—Binding energy of uneven nuclei.

  The situation is totally different in the case of ‘even’ atoms, i.e. where N + Z is an even number. Here one of the finer details of the energy surface (not reflected in the equation formulated in the fourth chapter of this book) shows up, namely: the difference already discussed in the stability, and consequently, in the binding energies, of the ‘doubly even’ and ‘doubly odd’ nuclei. As already pointed out, the former possess a considerably greater degree of stability than the latter. Therefore, in order to represent both nuclear types, we must draw two different curves; the first one, representing the ‘doubly even’—and therefore the more stable—nuclei lies below the other one which represents the ‘doubly odd’ nuclei. The change of one nucleus into another can take place by discrete steps only, by the emission of one electron or positron (plus the necessary neutrino) at one time and never by a simultaneous emission of two electrons or two positrons. While it is still possible, for instance, for the nucleus in which N − Z = 6 (e.g. N = 49 and Z = 43—a ‘doubly odd’ nucleus) to change into the ‘doubly even’ Mo nucleus in which N − Z = 8 (N = 50 and Z = 42), by the emission of a positron, as shown by the arrow in Figure 19, for energy is liberated in this process. But in order to change further into the most stable Zr nucleus, in which N − Z = 12 (N = 52 and Z = 40), the Mo nucleus with an N − Z value of 8 would first have to change, by the emission of another positron, into the Nb nucleus with an N − Z value of 10—in other words, once again into a ‘doubly odd’ nucleus. This is, however, impossible from the point of view of energetics, for it would require an expenditure of energy. It is much easier for the Nb nucleus with an N − Z value of 10 to change, conversely, into the Mo nucleus where N − Z = 8, by the emission of an electron, or into the most stable Zr nucleus where N − Z = 12, by the emission of a positron. Thus, a study of Figure 19 will show that in addition to the most stable nucleus, Zr, situated on the valley floor, other ‘doubly even’ nuclei with the same mass number, situated a little higher, may also be stable, whereas all the ‘doubly odd’ nuclei, situated on the upper curve, are unstable. In Figure 19, too, the possibilities of change are indicated by arrows. The arrows pointing to the lower right indicate positron emission (in certain cases, a K-radiation), while those pointing to the left indicate electron emission. Thus, the unstable nuclei situated on the upper curve may change by either process, if there are appropriate stable nuclei on the lower curve. An example is the potassium nucleus 19K40, which can change by electron emission into the calcium nucleus 20Ca40, or by positron emission into the argon nucleus 18A40.

  Figure 19.—Binding energy of the even nuclei.

  On the ground of the evidence of these curves, we formulate therefore the following rules, to be verified by actual experiments:

  1. When the mass number is an odd figure—in other words, in the case of ‘odd’ nuclei—there is only one stable nucleus for any given mass number. All the others are unstable and emit either electrons or positrons (or decay by K-capture).

  2. When the mass number as well as the number of neutrons and the number of protons are all even figures—in other words, in the case of ‘doubly even’ nuclei—usually there are several but not many (say, two or three) stable nuclei with the same mass number.

  3. Stable nuclei which have an even mass number but an odd number of protons and an odd number of neutrons—the ‘doubly odd’ nuclei—are likely to be non-existent as a rule.

  Exceptions to the third rule mentioned, however, occur among the very lightest nuclei. These can be accounted for by a particularly sharp flexure of the two curves, the result of which is that the most stable nucleus, situated approximately at the lowest point of the upper curve, lies below the nearest nuclei of the lower curve.

  Otherwise, all the above conclusions are borne out almost completely by actual experience, as will be evident by reference to Table IV (at the end of the book), in which stable nuclei are indicated by black dots, the unstable ones by upright triangles (electron emitters) or inverted triangles (positron emitters). A careful study of this table will show that, in fact, except for a very few instances, there is never more than one stable nucleus for any given mass number. This statement is known as Mattauch’s rule. In this table, too, atoms having the same mass number are situated on a straight line, ascending to the left at an angle of 45 deg. to the abscissa. Thus, for instance, the palladium nucleus 46Pd111, the silver nucleus 47Ag111, the cadmium nucleus 48Cd111 and the indium nucleus 49In111 are all situated on such a straight line. Of these, only the 48Cd111 is a stable one; the 46Pd111 and 47Ag111 nuclei are electron emitters, while the 49In111 nucleus is a positron emitter. Thus this table fully confirms the fact that on the same straight line with a stable nucleus of an odd mass number all other nuclei of the same mass number are unstable. There is an exception to this rule, in the case of the mass number 113. In addition to the stable indium nucleus 49In113 there is also a stable cadmium nucleus, 48Cd113. This exception can probably be accounted for by the fact that both these nuclei happen to be situated at an approximately equal height on both sides of the energy surface, and the difference in energy between them is too small to permit the formation of an electron and a neutrino, or to admit of any measurable probability of the capture of an electron, so that a change of one of these nuclei into the other cannot occur. There may be also other exceptions, but their existence has not so far been confirmed by experience.

  When the mass number is even, the occurrence of several stable nuclei having the same mass number is the general rule. Such nuclei are called nuclear isobars. As a result of this privileged character of even numbers, for every nucleus containing an even number of protons (having an even atomic number) there exist more or fewer stable isotopes—i.e. stable nuclei having the same atomic number and the same chemical properties, all of which are varieties of the same element. On the other hand, elements with an odd atomic number have far fewer stable isotopes. Thus, the element titanium, with its even number of protons (22), has as many as five stable isotopes, while its neighbour, vanadium, with its 23 protons, has only one single stable isotope. The next element, chromium, has again four isotopes, whereas the immediately following one, manganese, has just one. Cadmium, with its 48 protons, has as many as eight stable isotopes, whereas silver (Z = 47), which immediately precedes it, has only two. And so on throughout the entire periodic system.

  The assumptions already discussed concerning the nature of nuclear forces, in particular their short range and their capacity for saturation, are thus definitely confirmed by actual experience in the form of conclusions based on them regarding the relative abundance and stability of the various nuclear species.

  This disclosure of the existence of isotopes of the various elements furnishes an explanation of the fact that Prout’s hypothesis, based on the assumption that all the known atomic weights are integers—which was in some degree a movement in the right direction,—remained consigned to the limbo of oblivion for nearly a whole century. Subsequent measurements of the atomic weights of the heavier elements proved most of them to be very far from integers, or even approximately integers. But there is a simple explanation of this observation: the existence of isotopes. Every element made chemically pure is a mixture of isotopes (unless it has no more than isotope). The mass numbers of the individual isotopes are, in fact, always approximately integers. But chemical processes only give the average mass of the atoms in a mixture of isotopes depending on the proportions in which the isotopes are present, and which therefore may assume all possible non-integral values.


  6. THE NUCLEAR REACTIONS

  I. ALPHA RADIATION

  Much has been said in the preceding lectures about the changes of atomic nuclei. In such changes, one chemical element turns into another, and in this respect modern nuclear physics has realized, to a certain extent, the hopes which inspired the alchemists of past ages. Let us now consider these nuclear transmutations more closely. The following two questions arise here: What elements can be changed into each other? Under what conditions is such a transmutation at all possible? In order to answer these questions, we will classify the transmutation processes into two groups: Firstly, the reactions which occur spontaneously, and secondly, those produced by external agencies.

  The spontaneous change of an element is called radioactivity, for the process is accompanied by an emission of radiation. The radioactive elements can be further subdivided into two classes: Firstly, those which emit alpha radiation, and secondly, those which emit beta radiation (either electrons or positrons). Gamma radiation may occur simultaneously, too. In addition to these, there are a few other processes, which will be discussed later.

  We shall begin with those transmutation processes in which alpha rays are emitted. Alpha radiation consists of helium nuclei, each composed of two neutrons and two protons. We have already discussed when an emission of alpha radiation may be expected to occur. Approximately, from the element zinc upwards, the binding energy per particle decreases with the increase of the number of particles, due to the increasing intensity of the electrostatic forces of repulsion, and consequently, from the point of view of energetics, the emission of an alpha particle by the heavier elements may be advantageous under certain circumstances. When this occurs, the energy and range of the alpha particle are determined by the difference in mass defect between the original nucleus and the nucleus which is the product of the process. Thus, all alpha particles which are the product of a specific decay process have the same range (Figure 3). Actually, most of these alpha emitters are at the end of the periodic table of elements. Radium and uranium are the best-known ones.