Fermi momentum

Figure 7. The occupation number densities as functions of wave vector for Na. The thick curves labeled (100), (110) and (111) represent the three principal directions within the first Brillouin zone, obtained by the FLAPW-GWA. The thin solid curve is obtained from an interacting electron-gas model [27]. The dash-dotted line represents the Fermi momentum. 

GWA leads to a good agreement between the theory and experiment. This finding can be interpreted as being mainly due to a different behavior between N GWA) and A(gas) around the Fermi momentum as seen in Figure 7. 

As was mentioned in Section 2, there exists a variety of different theoretical approaches to calculate the local field factor g q). Following Farid et al. [7], the behavior of g q) for large q is connected to the size z of the step in the occupation number function n(k) fork = kF, kF being the Fermi-momentum (see Figure 8). This... 

One can clearly see the large positive anisotropy in the [111] direction near the boundary of the first Brillouin zone (BZB). It is caused by the [111] high momentum component, which produces a continuous distribution of the momentum density across the BZB, as the Fermi surface has contact with the BZB in this direction. In the other directions, especially in [100], calculations show a steep decrease of the momentum density at the Fermi momentum and therefore a negative deviation from the spherical mean value. 

It is well known that the energy profiles of Compton scattered X-rays in solids provide a lot of important information about the electronic structures [1], The application of the Compton scattering method to high pressure has attracted a lot of attention since the extremely intense X-rays was obtained from a synchrotron radiation (SR) source. Lithium with three electrons per atom (one conduction electron and two core electrons) is the most elementary metal available for both theoretical and experimental studies. Until now there have been a lot of works not only at ambient pressure but also at high pressure because its electronic state is approximated by free electron model (FEM) [2, 3]. In the present work we report the result of the measurement of the Compton profile of Li at high pressure and pressure dependence of the Fermi momentum by using SR. 

Here z is the number of valence electrons per atom and qF is the Fermi momentum given by... 

When thermal energy is negligible, cells in phase space are uniformly occupied up to the Fermi momentum pp, given by... 

Note that the Casimir calculation under the presence of fermionic on-relativistic) matter fields simplifies enormously since the presence of the second scale, the chemical potential p,=h2k2F/2rn or the Fermi momentum kF, provides for a natural UV-cxAoii, Any = p, and kuv = kF- Therefore the Casimir energy for fermions between two impenetrable (parallel) planes at a distance L is simply given as... 

In the standard choice BHF the self-consistency requirement (5) is restricted to hole states (k < kF, the Fermi momentum) only, while the free spectrum is kept for particle states k > kF- The resulting gap in the s.p. spectrum at k = kF is avoided in the continuous-choice BHF (ccBHF), where Eq. (5) is used for both hole and particle states. The continuous choice for the s.p. spectrum is closer in spirit to many-body Green s function perturbation theory (see below). Moreover, recent results indicate [6, 7] that the contribution of higher-order terms in the hole-line expansion is considerably smaller if the continuous choice is used. 

Recently the density dependence of the symmetry energy has been computed in chiral perturbation effective field theory, described by pions plus one cutoff parameter, A, to simulate the short distance behavior [23]. The nuclear matter calculations have been performed up to three-loop order the density dependence comes from the replacement of the free nucleon propagator by the in-medium one, specified by the Fermi momentum ItF... 

The resulting EoS is expressed as an expansion in powers of k/, and the value of A 0.65 GeV is adjusted to the empirical binding energy per nucleon. In its present form the validity of this approach is clearly confined to relatively small values of the Fermi momentum, i.e. rather low densities. Remarkably for SNM the calculation appears to be able to reproduce the microscopic EoS up to p 0.5 fm-3. As for the SE the value obtained in this approach for 4 = 33 MeV is in reasonable agreement with the empirical one however, at higher densities (p > 0.2 fm-3) a downward bending is predicted (see Fig. 4) which is not present in other approaches. 

At low energy, the typical momentum transfer by quarks near the Fermi surface is much smaller than the Fermi momentum. Therefore, similarly to the heavy quark effective theory, we may decompose the momentum of quarks near the Fermi surface as... 

Figure 8. The dependence of the free energy of the combined DFS and LOFF phases on the center-of-mass momentum of the pairs Q (in units of Fermi momentum pf) and the relative deformations Se for a fixed density asymmetry [18].

Let us start with a single massive fermion ( neutron , N). Since the minimal nonzero fifth momentum component is p5 = h/Rc, then the extra direction of the phase space is not populated until the Fermi-momentum pp < h/Rc. However, at the threshold both // = h/Rc states appear. They mimic another... 

At some Fermi momentum the phase space opens up in the 5th direction. Henceforth pp increases slower, which will be reflected by the M(R) relation. However note that 5th dimensional effects can mimic particle excitations. 

The relativistic formulation of Thomas-Fermi theory started at the same time as the original non-relativistic one, the first work being of Vallarta and Rosen [9] in 1932. The result they arrived at can be found by replacing the kinetic energy fimctional by the result of the integration of the relativistic kinetic energy in terms of the momentum p times the number of electrons with a given momentum p from /i = 0 to the Fermi momentum p = Pp. ... 

As already remarked, the idea underlying the Thomas-Fermi (TF) statistical theory is to treat the electrons around a point r in the electron cloud as though they were a completely degenerate electron gas. Then the lowest states in momentum space are all doubly occupied by electrons with opposed spins, out to the Fermi sphere radius corresponding to a maximum or Fermi momentum pt(r) at this position r. Therefore if we consider a volume dr of configuration space around r, the volume of occupied phase space is simply the product dr 47ipf(r)/3. However, we know that two electrons can occupy each cell of phase space of volume h3 and hence we may write for the number of electrons per unit volume at r,... 

One can now eliminate the Fermi momentum pt(r) between equations (1) and (2) to obtain the density-potential relation of the TF statistical theory... 

Pf is the Fermi momentum. To get rid of the momentum dependence, as it is usual, an average is performed in the momentum space ... 

Fig. 11. Differential transition rate AF/Ak (in atomic units) as a function of the initial momentum of the electron (normalized to the Fermi momentum) fej /fef for an Auger capture process from a free electron gas of = 2 to the 3p state of an Ar ion. Solid lines include the Ar in the calculation of the response function. Dashed lines are the unperturbed free electron gas results. Thick lines are calculated with the self-consistent response function while thin lines show the results using the Hartree response xq (the latter are multiplied by 0.5 before being plotted).

Here we consider a uniform gas of interacting electrons, the electron density n being equal to the average density of valence electrons in aluminum metal (r = 2.07), for which the Fermi momentum = (3tt m) / ] and... 

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