Nucleus as a Fermi Gas

The preceding discussion of nuclear structure and models was mostly aimed at explaining the detailed properties of the ground states and small excitations of 

The number of different combinations of the positive integer quantum numbers that fulfill this equality is given by the volume of one-octant of a sphere  

Remember that the Pauli principle allows us to put particles with two spins (up/own) into each level, and if the nucleons are all in their lowest possible states, the number of filled states can be assumed to be equal to the number of each type of nucleon. Thus, the Fermi wavenumber for protons is 

If the number of neutrons is greater than the number of protons, as in heavy nuclei, then the Fenni energies will be slightly different for the two kinds of particles. An approximate representation of the Fermi energy for protons and neutrons is 

The average kinetic energy of the nucleons in the well can be shown to be / Ej, or approximately 20 MeV. Notice that the nucleons are moving rapidly inside the potential well but not extremely fast. 

---

The experimental observables ascribed to the preequilibrium mechanism have usually been interpreted in the context of the exciton model (Griffin et al. 1966 Blann et al. 1975). In the basic model, the nucleus is treated as a Fermi gas in which the projectile initiates a series of sequential N-N collisions, generating unstable particle-hole states, or excitons exciton = a particle-hole pair). The number of excitons is thus proportional to the degree of thermalization. 

On the other hand the Thomas-Fermi method, which treats the electrons around the nucleus as a perfectly homogeneous electron gas, yields a mathematical solution that is universal, meaning that it can be solved once and for all. This feature already represents an improvement over the method which seeks to solve Schrodinger equation for every atom separately. This was one of the features that made people go back to the Thomas-Fermi approach in the hope of... 

The experimental data have been successfully reproduced using a calculation technique known as the Monte Carlo method and assuming a Fermi gas model for the nucleus. This model treats the nucleons like molecules of a very cold ideal gas in a potential well. The nucleons do not follow the Pauli exclusion principle and fill all vacant orbitals. 

In 1926 the physicist Llewellyn Thomas proposed treating the electrons in an atom by analogy to a statistical gas of particles. No electron-shells are envisaged in this model which was independently rediscovered by Italian physicist Enrico Fermi two years later, and is now called the Thomas-Fermi method. For many years it was regarded as a mathematical curiosity without much hope of application since the results it yielded were inferior to those obtained by the method based on electron orbitals. The Thomas-Fermi method treats the electrons around the nucleus as a perfectly homogeneous electron gas. The mathematical solution for the Thomas-Fermi model is universal , which means that it can be solved once and for all. This should represent an improvement over the method that seeks to solve Schrodinger equation for every atom separately. Gradually the Thomas-Fermi method, or density functional theories, as its modem descendants are known, have become as powerful as methods based on orbitals and wavefunctions and in many cases can outstrip the wavefunction approaches in terms of computational accuracy. 

Therefore the main assumptions of this approach, provided that such matching radius rg exists, can be summarized as i) for r > rg the system is adequately described as a local relativistic Fermi gas, ii) for r< rg the main contribution is due to the U singleparticle state, and iii) the potential near the nucleus is approximated to - )/r. In addition to this, exchange effects were not considered. 

The idea of calculating atomic and molecular properties from electron density appears to have arisen from calculations made independently by Enrico Fermi and P.A.M. Dirac in the 1920s on an ideal electron gas, work now well-known as the Fermi-Dirac statistics [19]. In independent work by Fermi [20] and Thomas [21], atoms were modelled as systems with a positive potential (the nucleus) located in a uniform (homogeneous) electron gas. This obviously unrealistic idealization, the Thomas-Fermi model [22], or with embellishments by Dirac the Thomas-Fermi-Dirac model [22], gave surprisingly good results for atoms, but failed completely for molecules it predicted all molecules to be unstable toward dissociation into their atoms (indeed, this is a theorem in Thomas-Fermi theory). 

While Eq. (12) represents the correct prediction of the non-relativistic Schro-dinger equation as Z —> oo, in the range of the Periodic Table corrections are needed. One is due to the fact that Eq. (11) near the point atomic nucleus assumed, shows that pit) diverges as r 3/2 and this is due to neglecting density gradients in the Fermi gas model employed. This, as was shown by Scott [12], corrects [13,14] Eq. (12) with a term (1/2)Z2. Earlier Dirac had introduced the exchange energy A into the Thomas-Fermi atom, with the result... 

This section is divided into several parts that focus on the near-barrier domain. The following section addresses N-N dominated reactions. One final comment must be made in an attempt to be forward looking. While the partition of reactions into low and high energy (mean-field dominated or not) has been common in the past, future work must move beyond this mental partition. As mentioned above, due to correlations in the nucleus, the ground state is replete with high-energy nucleons, well above what would be expected from the Fermi-gas model. It is actually this aspect, the correlations that exist in nuclei (for example as a function of n/p asymmetry), that will be the focus of many reaction studies in the future. 

---