Nuclear structure

Understanding the structure of the atomic nucleus is one of the central challenges in nuclear physics. Understanding the structure of the atomic nucleus is one of the central challenges in nuclear physics. The liquid drop model is one of the first models of nuclear structure, proposed by Carl Friedrich von Weizsäcker in 1935. It describes the nucleus as a semiclassical fluid made up of neutrons and protons, with an internal repulsive electrostatic force proportional to the number of protons. The quantum mechanical nature of these particles appears via the Pauli exclusion principle, which states that no two nucleons of the same kind can be at the same state. Thus the fluid is actually what is known as a Fermi liquid.In this model, the binding energy of a nucleus with Z {displaystyle Z} protons and N {displaystyle N} neutrons is given by where A = Z + N {displaystyle A=Z+N} is the total number of nucleons (Mass Number). The terms proportional to A {displaystyle A} and A 2 / 3 {displaystyle A^{2/3}} represent the volume and surface energy of the liquid drop, the term proportional to Z 2 {displaystyle Z^{2}} represents the electrostatic energy, the term proportional to ( N − Z ) 2 {displaystyle (N-Z)^{2}} represents the Pauli exclusion principle and the last term δ ( A , Z ) {displaystyle delta (A,Z)} is the pairing term, which lowers the energy for even numbers of protons or neutrons.The coefficients a V , a S , a C , a A {displaystyle a_{V},a_{S},a_{C},a_{A}} and the strength of the pairing term may be estimated theoretically, or fit to data.This simple model reproduces the main features of the binding energy of nuclei. The assumption of nucleus as a drop of Fermi liquid is still widely used in the form of Finite Range Droplet Model (FRDM), due to the possible good reproduction of nuclear binding energy on the whole chart, with the necessary accuracy for predictions of unknown nuclei. The expression 'shell model' is ambiguous in that it refers to two different eras in the state of the art. It was previously used to describe the existence of nucleon shells in the nucleus according to an approach closer to what is now called mean field theory.Nowadays, it refers to a formalism analogous to the configuration interaction formalism used in quantum chemistry. We shall introduce the latter here. Systematic measurements of the binding energy of atomic nuclei show systematic deviations with respect to those estimated from the liquid drop model. In particular, some nuclei having certain values for the number of protons and/or neutrons are bound more tightly together than predicted by the liquid drop model. These nuclei are called singly/doubly magic. This observation led scientists to assume the existence of a shell structure of nucleons (protons and neutrons) within the nucleus, like that of electrons within atoms. Indeed, nucleons are quantum objects. Strictly speaking, one should not speak of energies of individual nucleons, because they are all correlated with each other. However, as an approximation one may envision an average nucleus, within which nucleons propagate individually. Owing to their quantum character, they may only occupy discrete energy levels. These levels are by no means uniformly distributed; some intervals of energy are crowded, and some are empty, generating a gap in possible energies. A shell is such a set of levels separated from the other ones by a wide empty gap. The energy levels are found by solving the Schrödinger equation for a single nucleon moving in the average potential generated by all other nucleons. Each level may be occupied by a nucleon, or empty. Some levels accommodate several different quantum states with the same energy; they are said to be degenerate. This occurs in particular if the average nucleus has some symmetry. The concept of shells allows one to understand why some nuclei are bound more tightly than others. This is because two nucleons of the same kind cannot be in the same state (Pauli exclusion principle). So the lowest-energy state of the nucleus is one where nucleons fill all energy levels from the bottom up to some level. A nucleus with full shells is exceptionally stable, as will be explained.