kg y medio entre los tres.

The story of modern nuclear structure studies of the atomic nucleus can be con- sidered to have a beginning in the turn of the 20th century, where the works of Becquerel, Rutherford, Geiger, Chadwick and many more gave us our first glimpse of what lay at the core of the atom. By the 1930’s this core had come to be seen as analogous to a drop of liquid with the Liquid Drop Model (LDM)[34] gaining favour among the early few models that had been developed by then. One of the principle observations behind the development of the LDM was that the that the nuclear density saturates, with the binding energy per nucleon initially increasing rapidly up to A ≈ 10-20. After this point this binding energy as a function of atomic number levels out, becoming approximately constant despite any addition of more nucleons. Owing to this saturation it was found that past A ≈ 20 the charge radius of the nucleus can be described in terms of its atomic mass, A, by the relationship

R≈r0A1/3, (3.1) where r0 is an empirically derived constant: 1.2 fm.

This saturation arises from the limited range of the nuclear force (RN ∼ 1 fm) restricting the nucleon interactions via this force to only those that are closest to one another.

For nucleons on the surface this finite force distance will mean that they experi- ence the interaction with their neighbouring nucleons in a manner that is different to a nucleon that is completely surrounded, a difference not unlike the surface tension of a drop of liquid. Continuing this liquid drop analogy, the nucleons within the nucleus are considered to be mobile and colliding with one another with a temperature dependant frequency.

The LDM was later refined into the semi-empirical mass formula (otherwise known as the Bethe-Weizs¨acker formula or shortened to SEMF)[35] which re- mains useful for the purpose of explaining particle evaporation, and expresses the nuclear binding energy, EB, in the form

EB =αVA−αSA2/3−αC

Z2

A1/3 −αA

(A−2Z)2

where A and Z have their usual meaning. The right hand side comprises of five terms, the first four of which possess coefficients αV, αS, αC and αA that are derived by fitting to the empirically determined masses of the nuclei.

The first term, αVA, is known as the volume term and describes the binding energy contribution from the nuclear force interactions of the nucleons. Were every nucleon to interact with one another the total number of pairs available to A particles would be A(A₂−1), however owing to the very limited RN the number of interacting pairs is more proportional to A.

Since the volume term does not account for the differing situation between nucle- ons on the surface of the nucleus as opposed to within, a surface term, αSA2/3, is introduced to correct for this. Given that the volume term is proportional to A the radius would be proportional to A1/3, therefore giving A2/3 as an appropriate representation of the nuclides surface.

The third term,αC Z

2

A1/3, reproduces the repulsive Coulomb force between the pro-

tons present. Unlike the situation involving the short ranged nuclear force, there is no limit to range of the Coulomb interaction between charged particles and thus in the case of Z protons present, Z(Z-1) (proportional to Z2_{) may be paired} although the strength of the interaction is inversely proportional to the range over which it occurs, which may be represented by the radius of the nuclei, or A1/3.

An asymmetry term, αA(N−Z)

2

A , is included to account for the fermion nature of protons and neutrons, which means they are subject to the Pauli principle. If we consider the protons and neutrons as independently filling two separate energy ”wells”, it becomes apparent that the stablest configuration for a nucleus is when the number of protons and neutrons is equal. A surplus of one type of fermion will mean that some of those fermions will be higher in energy than the other type, introducing an imbalance that reduces the overall binding energy. This term is most relevant in the heavier mass nuclei where the Coulomb force drives a trend towards neutron surplus that gradually increases with increasing Z. Lastly there is a pairing term, δ, which is related to spin coupling. This is the ten- dency of protons and neutrons to form proton-proton and neutron-neutron pairs. The term reflects the empirically observed changes in binding energy between nuclei that have an even number of protons and neutrons and nuclei that do not.

The trend is that even-even nuclei are the most tightly bound and odd-odd are the least. This phenomena arises from the attractive short ranged component of the nuclear force working in concert with the Pauli exclusion principle, which will be discussed in more detail in 3.1.6.

While the SEMF provided a good description of the overall trends of the nu- cleus and was successfully employed to explain fission it was increasingly found to be inadequate to fully describe the quantal behaviours observed, such as the existence of especially stable proton and neutron configurations corresponding to what came to be called the ”magic numbers”. These numbers correspond to quantities of protons or neutrons, namely N or Z = 2, 8, 20 ,28, 40, 50, 82 and 126, where the the overall binding energy per nucleon experiences an increase inexplicable to the SEMF alone.