Understanding Nuclear Structure and Forces

Nuclear structure

A nucleus is identified by its atomic number Z (i.e., the number of protons), the neutron number, N, and the mass number, A, where A = Z + N Size, shape, binding energy, angular momentum, and (if it is unstable) half-life, atomic number and mass number Nucleus characterization One of the best ways to determine the size of a nucleus is to scatter high-energy electrons from it. The angular distribution of the scattered electrons depends on the proton distribution. Size average radius Radii range from 1-10 x10-15 m The nuclei of some atoms are spherical, while others are stretched or flattened into deformed shapes.

The binding energy of a nucleus is the energy holding a nucleus together. This energy varies from nucleus to nucleus and increases as A increases. Because of variations in binding energy, some nuclei are unstable and decay into other ones. The rate of decay is related to the mean lifetime of the decaying nucleus. The time required for half of a population of unstable nuclei to decay is called the half-life. Half-lives vary from tiny fractions of a second to billions of years. The two types of particle which are nueleons differ mainly in their electrical properties, the proton (p) having a positive charge, while the neutron (n) is neutral. They have the same spin , and their masses are almost equal ( 1836.1 and 1838.6 electron masses respectively). Despite the difference in electrical properties, the proton and the neutron are very similar particles, and this similarity is of fundamental importance.

Charge symmetry of nuclear forces Apart from E.F - the forces of interaction between two protons are very similar to those between two neutrons. Two protons {pp) and two neutrons (nn) - Same - Obey FD statistics Symmetry of the wave functions (r1, 1; r2, 2) are permissible for the pp and nn systems Antisymmetrical with respect to a simultaneous interchange of the coordinates and spins of the particles. Charge symmetry Physical similarity between protons and neutrons, known as isotopic invariance. Existence of an analogy not only between pp and nn systems ----- analogy pn system, which consists of different particles. There cannot be a complete analogy - since the possible states of the pn system, in which the particles are non identical, are certainly not restricted to those with antisymmetrical wave functions.

It is found, however, that among the possible states of the pn system there are some whose properties are almost exactly the same as those of systems of two identical nucleons. These states are, of course, described by antisymmetrical wave functions (the remaining states of the pn system are described by symmetrical wave functions and do not occur in the pp and nn systems). Isotopic invariance, like charge symmetry, is valid only if electromagnetic interactions are neglected. Another reason why isotopic invariance is only approximately true is the slight mass difference between the neutron and the proton ; if there were exact symmetry between neutrons and protons, their masses would of course be identical also

It follows naturally from the fact that isotopic invariance is equivalent to the possibility of classifying the states of a system of nucl ons with respect to the symmetry of its coordinate-spin wave functions , independent of the types of nucl ons concerned. The required formalism must therefore enable us to define for the description of the states of the system a new quantum number which uniquely determines the symmetry of the functions . A similar situation has already been encountered in connection with the properties of a system of particles with spin 1/2. If the total spin S of such a system is specified, then the symmetry of its coordinate wave function is uniquely determined, regardless of which of the two possible values ( 1/2) is taken by the component of the spin of each particle. Nuclear Force In the non-relativistic limit we can say that nuclear forces are independent of the velocities of the nucl ons and have a potential ; the velocities of the nucl ons in the nucleus are about one-quarter of the velocity of light .The potential energy U of the interaction of two nucl ons depends not only on the distance r between them but also quite strongly on their spins.f The precise dependence on r could, of course, be established only by a consistent theory of nuclear forces. The nature of the spin dependence, however, can be found from simple considerations based on the properties of spin operators.

Shell Model In this model each nucl on in the nucleus is regarded as moving in a self-consistent field due to all the other nucl ons ; owing to the small range of action of nuclear forces, this field decreases rapidly outside the volume bounded by the "surface" of the nucleus. Accordingly the state of the nucleus as a whole is described by specifying the states of the individual nucl ons. The self-consistent field is spherically symmetrical, and the centre of symmetry is, of course, the centre of mass of the nucleus. The following difficulty arises here, however. In the self- consistent field method, the wave function of the system is constructed as the product (or the appropriately symmetrized sum of the products) of the wave functions of the individual particles. Such a function, however, does not keep the centre of mass fixed: although the mean velocity of the centre of mass calculated from this function is zero, it gives a finite probability of non-zero values of the velocity.