Fermi-gas model

Atomic units will be used throughout. The explicit density functionals representing the different contributions to the energy from the different terms of the hamiltonian are found performing expectation values taking Slater determinants of local plane waves as in the standard Fermi gas model. Those representing the first relativistic corrections are calculated in the Appendix. 

Figure 6.19 A schematic version of the potential energy well derived from the Fermi gas model. The highest filled energy levels reach up to the Fermi level of approximately 32 MeV. The nucleons are hound hy approximately 8 MeV, so the potential energy minimum is relatively shallow.

In this equation, jjl is the reduced mass of the system, and o-inv is the cross section for the inverse process in which the particle b is captured by the nucleus B where b has an energy, Eb. The symbols p(E B) and p( c) refer to the level density in the nucleus B excited to an excitation energy E% and the level density in the compound nucleus C excited to an excitation energy, . The inverse cross section can be calculated using the same formulas used to calculate the compound nucleus formation cross section. Using the Fermi gas model, we can calculate the level densities of the excited nucleus as... 

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... 

SCALING THEORY OF A ONE-DIMENSIONAL FERMI GAS MODEL WITH TWO CHARACTERISTIC ENERGIES... 

The properties of a one-dimensional Fermi gas model with two characteristic energies, two bandwidth cut-offs, is studied. The direct electron-electron coupling and the phonon mediated effective coupling are cut off at energies Ea and <0,, respectively, where E is the bandwidth of the electron energy band and to is the Debye frequency. The model is treated in the framework of renormalization group approach. It is shown that this model can be mapped on the usual one cut-off model and the results obtained for that model can be applied. 

The one-dimensional Fermi gas model has been intensively studied recently by several authors (1-6) using different methods, such as parquet diagram summation, the renormalization group and the bosonisation transformation. 

There are two usual ways to introduce a cut-off in the one-dimensional Fermi gas model. One is to restrict the momenta or energies of all the electrons participating in a scattering process to be within a range of width 2kc or 2(0 = 2 rkcaround the Fermi points. Here v is the Fermi velocity. This type of cut-off is called bandwidth cut-off. Another possibility is to put a... 

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. 

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. 

In O Fig. 5.9 the experimentally observed 0 level distances in the transmission resonance groups of Pu (fiiU circles) are compared to calculations of the level density in the first minimum according to different mcxlels. The soUd fine on the left side of the figure corresponds to the level distances in the first minimum calculated in the framework of the back-shifted Fermi gas model in the parameterization by Rauscher et al. (1997). In order to reproduce the experimentally observed 0 level distances, this curve has had to be shifted by 2.25 MeV, corresponding to the ground-state energy in the second minimum of Pu. For comparison, also shown are the level distances in the first minimum calculated with the Bethe formula and within the constant temperature formalism in a parameterization by von Egidy et al. (1988). 

The first quantum mechanical calculations for nonadiabatic electrochemical electron transfer reactions at metals and semiconductors were performed by Dogonadze, Chizmadzhev, and Kuznetsov(1962-1964). In Ref. 27 the totally degenerated Fermi gas model was used to describe the state of the electrons in the electrode, and in Ref. 28 an integration over the energy spectrum was performed, taking account of the Fermi distribution of the electrons over a range of energy. Later that theory was extended to other processes at semiconductors and thin semiconductor films. 

Hale in 1968 calculated the limiting electric current for an electron transfer reaction associated with the activationless process. The degenerate Fermi gas model was used to describe the electrons in the metal. 

In Eq. (69), Aq = 47reokBnie/ 2 rrhf is a dimensionless function which distinguishes between the behavior of electrons in the metal, described by the ideal Fermi gas model, for which = 1. 

Here d is the space dimension, p is the number density, and ef, the Fermi energy, is the energy of the highest occupied level at zero temperature. We omit the calculation of this formula, which may be found in the context of the Fermi gas model of electrons in solids in most solid state textbooks. Obviously the pressure this time satisfies Eq. (5.159). But what about the Bosons The answer is contained in Fig. 5.15. At finite density P a and thus dP/dT v 0 for T 0. Again Nernst s theorem is satisfied. Let us therefore look at the relation between the Nernst theorem and quantum theory. 

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