**Nuclear Models**

The structure of atoms is now well understood: quantum physics governs all; the electromagnetic force is the main force; each atom contains a massive force center (the nucleus) that tends to dominate the physics. However, things are not in such a happy state for the nucleus. Quantum mechanics still governs its behavior, but the forces are complicated and cannot, in fact, be written down explicitly in full detail. We are dealing with a many-body problem of great complexity.

So, in the absence of a comprehensive nuclear theory, we turn to the construction of nuclear models. A nuclear model is simply a way of looking at the nucleus that gives a physical insight into as wide a range of its properties as possible. The usefulness of a model is tested by its ability to provide predictions that can be verified experimentaly in the laboratory.

Two models of the nucleus have proved useful: the liquid drop model and the independent particle model. Although based on assumptions that seem flatly to exclude each other, each accounts very well for a selected group of nuclear properties. After describing them separately, we shall see how these two models may be combined to form a single coherent picture of the atomic nucleus known as the collective model.

**The Liquid Drop Model**

In the liquid drop model, formulated by Niels Bohr, the nucleons are imagined to interact strongly with each other, like the molecules in a drop of liquid. A given nucleon collides frequently with other nucleons in the nuclear interior, its mean free path as it moves about being substantially less than the nuclear radius. This constant "jiggling around" reminds us of the thermal agitation of the molecules in a drop of liquid. The liquid drop model permits us to correlate many facts about nuclear masses and binding energies; it is useful in explaining nuclear fission. It also provides a useful model for understanding a large class of nuclear reactions.

If we consider the sum of the following three types of energies, then the picture of a nucleus as a drop of liquid accounts for the observed variation of binding energy per nucleon with mass number:

Volume Energy - Because each bond energy is shared by two nucleons, each has a binding energy of one-half that. When an assembly of spheres of the same size is packed together into the smallest volume, as we suppose is the case of nucleons within a nucleus, each interior sphere has 12 other spheres in contact with it. So, this energy is proportional to the volume.

Surface Energy - A nucleon at the surface of a nucleus interacts with fewer other nucleons that one in the interior of the nucleus and hence its binding energy is less. This surface energy takes that into account and is therefore negative.

Coulomb Energy - The electric repulsion between each pair of protons in a nucleus also contributes toward decreasing its binding energy. The coulomb energy of a nucleus is equal to the work that must be done to bring together the protons from infinity into a spherical aggregate the size of the nucleus. The coulomb energy is negative because it arises from an effect that opposes nuclear stability.

Compare the following two graphs (adapted from Beiser): One of the actual binding energy per nucleon and one with the liquid drop model.

The above binding-energy graph can be improved by taking into account two other energies that do not fit into the simple liquid-drop model but which are readily explainable in terms of a model that provides for nuclear energy levels:

Asymmetry Energy - an energy associated needed as a correction when the number of neutrons is greater than the number of protons.

Pairing Energy - an energy which is a correction term that arises from the tendency of proton pairs and neutron pairs to occur.

**The Independent Particle Model**

In the liquid drop model, we assume that the nucleons move around at random and bump into each other frequently. The independent particle model, however, is based on just the opposite assumption, namely, that each nucleon moves in a well-defined orbit within the nucleus and hardly makes any collisions at all! A nucleon in a nucleus, like an electron in an atom, has a set of quantum numbers that defines its state of motion. also, nucleons obey the Pauli exclusion principle, just as electrons do.That is, no two nucleons may occupy the same state at the same time. In this regard, the neutrons and the protons are treated separately, each having its own array of available quantized states.

The fact that nucleons obey the Pauli exclusion principle helps us to understand the relative stability of nucleon states. If two nucleons within the nucleus are to collide, the energy of each of them after the collision must correspond to the energy of an unoccupied state. If these states are filled, the collision simply cannot take place. In time, any given nucleon will undergo a possible collision, but meanwhile it will have made enough revolutions in its orbit to give meaning to the notion of a nucleon state with a quantized energy.

In the atomic realm, the repetitions of physical and chemical properties that we find in the periodic table are associated with the fact that the atomic electrons arrange themselves in shells that have a special stability when fully occupied. We can take the atomic numbers of the noble gases, 2, 10, 18, 36, 54, 86, ... as magic electron numbers that mark the completion of such shells. The nuclear realm also shows such closed shell effects associated with certain magic nucleon numbers:

2, 8, 20, 28, 50, 82, 126, ...

Any nuclide whose proton number or neutron number has one of these values turns out to have a special stability that may be made apparent in a variety of ways. For example, an alpha particle is exceptionally stable because its proton number and neutron number are both equal to 2, a magic number. An alpha particle is therefore said to be doubly magic because they contain filled shells of both protons and neutrons.

The central idea of a closed shell is that a single particle outside a closed shell can be relatively easily removed but that considerably more energy must be expended to remove a particle from the shell itself. There is much additional experimental evidence that the nucleons in a nucleus form closed shells and that these shells exhibit stable properties.

We have seen that wave mechanics can account beautifully for the magic electron numbers, that is, for the populations of the orbitals into which atomic electrons are grouped. It turns out that, by making certain reasonable assumptions, wave mechanics can account equally well for the magic nucleon numbers!

**The Collective Model**

Consider a nucleus in which a small number of neutrons (or protons) orbit outside a core of closed shells that contains a magic number of neutrons (or protons). The "extra" nucleons move in quantized orbits, in a potential well established by the central core, thus preserving the central feature of the independent particle model. These extra nucleons also interact with the core, deforming it and setting up "tidal wave" motions of rotation or vibration within it. These "liquid drop" motions of the core preserve the central feature of that model. This collective model of nuclear structure thus succeeds in combining the seemingly irreconcilable points of view of the liquid drop and independent particle models. It has been remarkably successful and is currently our best theory. Perhaps it represents the limits of what we can hope for in nuclear physics, given the absence of a theory.

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