Electrons 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. 

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... 

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. ... 

Figure 2 Various scattering processes in one-dimensional conductors. The linearized electronic energy spectrum is shown around the Fermi surface (k = kF). The processes are depicted by trajectories showing the transfer of electrons in momentum space, i refers to backward (/ = 1), forward (/ = 2, 4), and umklapp (i = 3) processes.

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,... 

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... 

The Fermi momentum implies that the de Broglie wavelength of the most energetic electrons in a degenerate gas is comparable with nj 1,/ 3, the average distance between electrons. 

The solid (dashed) lines represent electrons with momentum near + kp (-kp) and OC is the spin index. The energy spectrum of the electrons is usually approximated by a linear dispersion relation with Fermi velocity U p. 

If all the carriers are concentrated in one Landau band then the Fermi momentum will be connected with the electron density by the relation... 

The parameter which measures the importance of relativistic effects is the ratio P of the Fermi momentum and the momentum of a non-relativistic electron travelling at the speed of light... 

The value kp is called the Fermi momentum. It depends on the electron density. It is seen from (5.17) that for large N the occupied region in fe-space is indistinguishable from a sphere. 

The band electrons have a continuous distribution of momentum up to the Fermi momentum, and only the distribution of one component of the momentum is measured. So the final angular distribution of photons contains information on an average property of the Fermi surface. The information is not specific enough to help map out the Fermi surface, but it can be used to check theoretical models. The interpretation of the data is usually based on the independent particle model. The angular distribution N(0) is given by... 

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