NUCLEAR PHYSICS

11. NATURE OF NUCLEAR FORCES.
    At the beginning of the twentieth century, the science of physics knew about three basic forces: gravity, electricity, and magnetism. All forces, whether they were "action at a distance" or field forces, could be always interpreted as one these three forces. And it was believed that all elastic forces, collision forces, contact forces, and chemical forces could be understood in terms of these three forces. By the end of first quarter of the twentieth century a fourth force was found at work in the universe, far mightier than any of the others and far more difficult to measure and interpret. With electrostatic, magnetic, and gravitational forces, measurement could be made in the laboratory or on a macroscopic scale and the laws governing these force were established. Even on the microscopic scale, by indirect measurement such as in the scattering experiment of Rutherford, it could be established that electrostatic or coulomb forces obey the same laws at extremely small distances and between extremely small particles as they do on the large scale. But entirely new methods were required to study the nuclear binding forces; by the 1960's the form of the laws governing these forces had not been established on a firm basis.

    Before pursuing the analysis of this nuclear force, let us summarize the conclusions concerning this strong force that binds together protons and neutrons into the nucleus.

    1. The force between neutron and neutron, neutron and proton, and proton and proton are all the same order of magnitude and are extremely powerful, being about 10³⁸ times greater than gravitational forces and 100 times greater than the electrostatic force of repulsion between proton and proton.
    2. The forces cannot be directly measured. Their magnitude can be calculated from changes in energy level. This situation is like as if we could not measure the force of gravitation on a body by weighing it, but could only calculate it from the change of the potential energy as the body moved farther or closer to the earth. Because nuclear forces are not measured directly and because energy measurements are possible only when some nuclear event occurs in which one or more particles interact, the physicist today talk about "strong" and "weak" interactions instead of forces. Energy changes, being the only measurable quantities, have largely supplanted the classical concept of force.
    3. The nuclear forces that reveal themselves through changes of energy levels occur only when particles interact at extremely short range, and not according to any simple equation, such as the inverse square law. Below the distances of about 3 × 10^-15 meters, the attractive force between nucleons is very powerful; at distances greater than that, it falls off rapidly to zero. Electrostatic forces then dominate and the nuclear forces become negligible.
    4. The functional relationship between the nuclear force of attraction and the distance between particles is not simple, and are not as other forces a simple case of a central force depending upon distance alone, but instead involve characteristics of particles that interact, such as "spin" and "magnetic moment". And furthermore, the forces acting are "saturated", that is, in a collection of several neutrons and protons, the attractive force between one nucleon and others is not merely given by addition of the forces acting on the nucleon. In other words, each force does not act independently and additively. This makes the problem of analysis of nuclear forces more complicated.
    5. Finally, all nuclear particles have a wave-particle duality, which means that they obey the statistical laws of quantum mechanics instead of the classical deterministic laws which apply to large masses. In other words, the Heisenberg's principle of uncertainty apply to the mechanics of nucleons. This means that if the position x and momentum p of a particle is measured at the same time there will be an uncertainty in the measurement of the position, which is designated by Δx, and a certain uncertainty in the measurement of the momentum, which is designated by Δp. By the uncertainty principle, when a simultaneous measurement of both position and momentum is undertaken, it is impossible to make the measurements any more accurately than the product of their uncertainty Δp and Δx, that is, the product is not below the value of Planck's constant h = 6.6 × 10^-32 joule-sec. This means that given the size of the nucleus it will be almost impossible to determine accurately what is going on mechanically in the nucleus. The immediate consequence of this principle is that the nuclear system is not a deterministic system. That is, the future events in the system, the position and momentum of a particle, cannot be predicted with absolute certainty, but only with a probability.

       The uncertainty principle can be also be stated in terms of the uncertainty of energy and uncertainty of time. This means that if there is an uncertainty in the amount of energy at a point, which is designate by ΔE, and an uncertainty in the time that a particle has at that point, which is designate by Δt, then the product of the uncertainties must be greater than Planck's constant h. This means that if a particle is within a potential well, then there is a probability that the particle may have sufficient energy at a certain time or in a time interval to escape from or "tunnel out of" the potential well. Applied to the nucleus this means that there is a probability that a nucleon can escape or "tunnel out of" the potential well of the nucleus. Thus is radioactivity and the nuclear disintegration reaction or fission explained.

    The exact form of the potential curve for the nucleus is not known, but its depth and general dimensions have been calculated from the energy changes. The highest point of the potential curve is at a point a short distance outside the nucleus (around 5 × 10^-14 meters). Outside the nucleus as a positive particle approaches the nucleus the potential curve rises steeply according to the equation
    KQq/r,
    where K is the Coulomb constant, Q is the charge on the nucleus, q is the charge on the approaching particle, and r is the distance from the center of the nucleus. If the approaching particle is an alpha particle, then
    Qq = 2Zee,
    where Z is the atomic number of the nucleus, and e is the charge on the electron.

    In the case where an alpha particle is trapped within the "potential well" formed by nucleus of a fairly heavy atom with a large Z greater than 80, then the alpha particle is in a "bound" state, and the "binding energy" is the algebraic difference between the energy that the alpha particle has within the nucleus and the ground level energy that it has outside of the nucleus. The energy of the alpha particle inside the nucleus is less than the energy that the alpha particle has outside the nucleus. If the nucleus is stable, not radioactive, then the alpha particle cannot be withdrawn from the nucleus, unless it is given sufficient additional energy both to overcome the binding energy and the potential barrier. If the nucleus is unstable, radioactive, then the energy of the alpha particle is above the outside zero ground level and the binding energy is negative. According to classical laws of physics the alpha particle can not escape, because energy is not available for the particle to climb the potential barrier even though its energy is above the outside zero ground level. However, according to quantum mechanics, given enough time the alpha particle does escape by tunneling through the barrier. The probability of the alpha particle escaping from the nucleus can be calculated by applying Schroedinger's equation to the particle, and is found to depend upon the kinetic energy of the particle and on the "radius" of the potential well. If energy is large and the radius is small, the alpha particle has a high probability of penetration, and the nucleus will disintegrate rapidly, as in the case of a radioactive material with short-life. In other cases when the energy is less and the radius somewhat larger, the probability of penetration is much smaller. In this case the nucleus has a much longer than usual lifetime, as in the case of radioactive materials with very long half-lifes.

12. NUCLEAR MODELS.
    There is much evidence that atoms and nuclei have many properties in common. For example, quantum and wave mechanics, intrinsic spin and orbital motion, and the uncertainty and exclusion principles all have a place in nuclear theory. In view of these similarities, why is the knowledge of the details of the nuclei not more certain and advanced? The answer to this question is found in the fact that the nuclei have a unique force, thousands of times stronger than the electrostatic forces governing the motion of electrons in the atom. This strong force holds the nucleus together, overcoming the replusion between the positively charged protons. The fundamental character of this force is not well known, although excellent clues as to its nature is well known (See the discussion in preceding section). The laws that govern the motion of the particles acted on by nuclear forces have not been determined. And there is no fundamental theory of nucleus with a model to explain this force.

    There have been many excellent proposals of models of the nuclei. But many of these models account for a limited set of observations. And these models appear to be contradictory, even though each have been quite successful in explaining (and predicting) certain specific features of the nuclei. Bohr, after working out the first explaination of the electronic structure of the atom, was the first to propose a specific model of the nucleus. He visualized the nucleus as much like a drop of liquid, so the behavior of the nucleons in the nucleus resembles the thermal motion of molecules in a drop of water. For this reason, Bohr's model has been called the liquid-drop model. Another model was proposed by Maria Mayer, of the University of Chicago, in which each individual nucleon moves in prescribed orbits in the nucleus, like the electrons move about and outside the nucleus in their various energy levels or shells. Thus this model have been called the nuclear-shell model. Other nuclear models have been proposed. Briefly, the most important of these models are the following.
    The alpha-particle model attempts to explain the many properties of light nuclei in terms of alpha-particle subunits.
    The Fermi gas model attempts to explain the behavior of the nucleons as analogous to the electrons in a metal, and was not unlike the liquid-drop model.
    The optical model pictures the nucleus as a "cloudy crystal ball" of varying degrees of opacity. This model uses the waves properties of matter to describe the absorption and the scattering of high-energy particles.

    1. THE LIQUID-DROP MODEL.
       The nucleus displays many qualitative features similar to those observed of a drop of an incompressible liquid. The discussion of the stability of the nuclei above has shown that the forces in the nucleus are of short range and are saturated. These motions were deduced from the fact that the average binding-energy of a nucleon in the nucleus is nearly constant, independent of the number of nucleons residing in the nucleus. And it was shown that the density of nuclear matter may be regarded as a constant, in that the volume of the nucleons depends only on the total number of nucleons. These two properties, saturation and constant density, belong also to a drop of an incompressible fluid. And there are other similarities. Like a liquid-drop, the nucleus exhibits the effect surface tension, or surface forces, which operate to hold the nucleons in bounds. The internal motions of the protons and neutrons in the nucleus are not unlike the thermal motions of molecules in a liquid. The analogy to the evaporaton of a molecule from a liquid can also be applied to the nucleus. Whenever a nucleus becomes energetic or excited, collisions between the agitated nucleons can send one of them breaking through the surface barrier. The nucleus de-excites itself by "evaporating" particles. The energy required to remove the evaporated particle (that is, binding energy) can be thought of as the heat of evaporation of the nuclear "liquid". The liquid-drop model has been particularly successful in describing several important features of excited nuclei. A direct result of the liquid-drop model is the concept of a compound nucleus. This concept helps explain the fission process of heavy nuclei. As pictured as a droplet of liquid, the nucleus takes on a dynamic rather than a static character. The nucleons appear to be moving in a far more random fashion and in this random movement; each nucleon, with great rapidity, collides with and scatters from its neighbors. Nucleons strike the surface of the nucleus as though attempting to escape. Most often, however, the barriers are too high for the particle to surmount, and it bounces back into the melée of the nucleus. Always on the move, but going nowhere, the stable nucleus confines the particles within its volume indefinitely. If the stable nucleus is bombarded by energetic neutrons (neutrons of several Mev), the neutron is able to enter the nucleus, since it is uncharged and is unaffected by the Coulomb field surrounding the nucleus. When this neutron enters the nucleus, particles coalesce, and a compound nucleus is formed. The compound nucleus is actually a new and different nuclide, the sum of the colliding particles. The resultant compound nucleus is formed in a very short time (about 10^-16 seconds). The motion of all the nucleons are so similar that the newly arrived neutron is indistinguishable from those previously resident. When a compound nucleus is formed, it has a higher energy content and the nucleons are impelled into increase random motions. The random motion will exist indefinitely unless some mechanism comes into play which reduces the energy of the system. For compound nucleus one such mechanism is the evaporation or "boiling off" of a nucleon. When a compound nucleus is formed and the motion of the nucleons have become thermalized, no individual nucleon has enough energy, on the average, to surmount the barrier of the nuclear force binding it to the nucleus (about 8 Mev). If a very high energy neutron possessing an energy of 100 Mev strikes the nucleus, some nucleons are struck so violently that they are immediately expelled from the nucleus. Left behind is a remnant of the target nucleus, which then may evaporate particles to reduce its energy of excitation.

       To understand how an excited nucleus evaporates, consider a particular nucleon that is collided in rapid succession by two, three, or four other nucleons in the nucleus. By chance, these collisons may impart to it the combined energies of the colliding particles. The energy of the nucleon may now be sufficiently great to separate it from the strong forces of the nucleus. When will a single nucleon, a neutron or proton, or an α particle, be given sufficent energy to escape from the nucleus? And when will the nucleus emit a γ ray instead of a particle. The answer to these questions depends upon the amount of energy imparted to the nucleus by the invading particle. If sufficient energy is not immediately imparted to a nucleon to separate it from the nucleus, it must await a time when by successive and fortuitous collisions, this can occur. The basic process of this concept of a compound nucleus is that the mode of decay of a compound nucleus is independent of the manner in which it was formed. How, and at what time, a nucleus attains stability can only be calculated in a statistical manner, for the compound nucleus is a system whose nature is governed by the laws of probability or chance.

       When a very heavy nucleus, such as ₉₂U²³⁵, captures a slow (thermal) neutron of 1/40 ev, a new and entirely unexpected even occurs. In 1939 Otto Hahn and Fritz Strassman, on bombarding uranium compounds with neutrons, discovered that some of the uranium nuclei split into two roughly equal parts. The nuclei had undergone fission. The mechanism of the fission can be explained in terms of the liquid-drop model. A heavy nucleus resembles, in many ways, a large drop of liquid. A small drop of liquid forms a spherical shape by reason of its surface tension. The surface area of the drop always tends to a minimum, and such a shape is a sphere. A nucleus can be visualized as a spherical agglomerate of nucleons. A nucleus is also stablized by its surface forces. When struck by a projectile, it will deform and begin to quiver, as does a liquid drop. When slightly perturbed, the deformed nucleus will always return to its original shape. The structure of the nucleus is thus stable. Now the question arises: when does the nucleus become unstable and fall apart? That is, when does a nucleus undergo fission?

       Now it is well known that a liquid drop cannot become infinitely large. Sooner or later, it will divide and form small droplets. Whenever the stability of a liquid drop becomes marginal, and slight oscillation induced into the drop pushes it "over the hill" into instability and the drop collapses into droplets. In the same manner the fission of the heavy nucleus proceeds. If a neutron is capture by an uranium 235 nucleus,
       ₀n¹ + ₉₂U²³⁵ → ₉₂U²³⁶,
       where the ₉₂U²³⁶ nucleus is formed and is in an excited state. Even the neutron that was captured had a very small kinetic energy, the very act of its being captured by the urnaium 235 nucleus, the nucleus releases about 7 Mev of energy. This amount of energy is sufficient to set off violent surface deformations and oscillations in the compound nucleus. The resultant interplay of surface tension and the replusive Coulomb forces between the positive protons rapidly leads to the fission of the nucleus. Thus a neutron of 1/40 ev energy has set off a reaction which releases some 200 Mev, and eight-billionfold of energy. An intermediate state of this fission of the ₉₂U²³⁵ is the compound nucleus, ₉₂U²³⁶. How it will decay cannot be precisely predicted, for again the laws of statistical chance prevail. The de-excitation of ₉₂U²³⁶ through fission occurs about 85 percent of the time. The principal competing process is the emission of γ rays, which occurs in the remaining 15 percent of the time. It is also possible for the nucleus, instead of fission or emitting of γ rays, to split into three fragments, including fragments as light as α particle. Fission in this mode has been observed to occur in a small fraction of 1 percent of the time. Several neutrons are emitted during the fission process, in addition to the primary fission fragments. If these neutrons are captured by other ₉₂U²³⁵ nuclei, then the fission process will be sustained. Thus a chain reaction will be generated.

    2. THE NUCLEAR-SHELL MODEL.
       Early experiments in chemistry showed that some atoms are more active chemically than others. Several kinds of atoms show no chemical activity at all. Such are the inert gases: helium, neon, argon, krypton, xenon, and radon. According the shell model of the electronic structure of the atom, it was explained that these particular elements should be chemically inactive because the electronic configration formed in those atoms were completed subshells. It was found that the atomic number Z for the inert gases followed a simple numerical series:
       Z = 2(1² + 2² + 2² + 3² + 3² + 4² + ...).
       This gives the following series for the inert gases:
       ₁He, Z = 2(1²) = 2 electrons
       ₁₀Ne, Z = 2(1² + 2²) = 10 electrons
       ₁₈A, Z = 2(1² + 2² + 2²) = 18 electrons
       ₃₆Kr, Z = 2(1² + 2² + 2² + 3²) = 36 electrons
       and similarly for ₅₄Xe and ₈₆Rn.
       It was suggested that the nuclei also had certain stabilities, but nothing as obvious and simply describable as the chemical stabilities revealed by the inert gaseous elements. Therefore, it may not be possible to develop a shell model of the nucleus of the atom with the same success as the chemical shell model, which explains the chemical behavior of the elements, as well as optical and X-ray spectra. The nucleus apparently does not have a central core about which the resident particles may orbit. And that a nucleon would have difficulty of confining itself to an orbit in the nucleus considering the density of the nucleons and the strong Coulomb forces existing between them. By 1948 experimental evidence had been accumlated on the subject to show that energy shells must exist in nuclei. Maria Mayer of the University of Chicago demontrated that nuclei clearly exhibit this effect. Mayer announced a set of "magic numbers", 2, 8, 20, 50, 82, and 126, to support her claim that shells existed in the nuclei. Any nucleus that contains a magic number of neutrons or protons is one that is particularly stable. It is called a magic number nucleus. In the magic number nuclei, there are closed shells of nucleons, similar to the closed shells of electrons in atoms of the inert gaseous elements. The Z numbers 2, 10, 18, 36, 54, and 86 of these chemical elements could be called the magic numbers of the chemical elements.

       In the place of the utter chaos of liquid-drop model, Mayer suggested that the internal motions of nucleons are in complete order. The motion of individual nucleon is much like of an electron in the atom. In its orbital motion in the nucleus, each nucleon moves independently of all others. Together they form a shell-like structures. This model, proposed by Mayer in 1949, has been called the nuclear-shell model or the independent-particle model. It has been developed into a powerful theoretical tool and has been sucessful in explaining many experimental observations. The magic numbers are correctly deduced from the model. As indicated by experiment, the magic numbers are in fact the number of nucleons required to complete successive shells.

       Although its agreement with experiment is impressive, the nucleus-shell model also poses some fundamental difficulties. Its relationship to the liquid-drop model, for instance, is far from being understood. These models are diametrically opposed to each other. In the liquid-drop model the nucleons frequently collide, rapidly transferring energy and momentum to their neighbors; in the other model, they essentially ignore each other. In concept the liquid-drop and shell models are incompatible, yet each is able to explain rather well properties of the nucleus for which it was expressly defined. Ultimately it must come to pass that these two viewpoints of nuclear structure, though seemly contradictory, will be seen to be incomplete parts of the whole. Until an comprehensive theory of the nucleus is developed, most problems relating to nuclear structure must be attacked by recourse to one or another of these models.

       Recently, the son of Niels Bohr, A. Bohr, and B. Mottelson of Denmark have developed a model that attempts to resolve the liquid-drop and nuclear-shell models dilemma. In their collective model they maintain that the shell structure of the nucleus is not quiescent as this view would seem to imply. If, as Bohr and Mattelson propose, the shell structure in the nucleus can be deform and then oscillate (in collective motion), then many of the features of the liquid-drop model can be reproduced. This collective model of the nucleus has already shown great promise in unifying some of the diverse yet valid ideas of these two models of nuclear structure.