Fermi motion

In the previous chapters we saw that in so far as deep inelastic lepton-nucleon scattering was concerned the nucleon could be visualized as a bound system of constituent quark-partons, with which the lepton interacted as if they were free particles. Our aim now is to try to give some sort of justification for such a picture, and to derive more reliable results for deep inelastic scattering in which allowance is made for the internal longitudinal and transverse (Fermi) motion of the quark-partons. The approach also allows us to evaluate the forward hadronic matrix elements of currents which appear in the sum rules discussed in Chapters 16 and 17 in terms of parton number densities. 

Finally we note that the internal Fermi motion in a nucleus can be treated by the above approach. This will be discussed in relation to the so-called EMC effect in Section 17.1.3. 

On the theoretical side, an overabundance of mechanisms has been advocated to explain the data and this makes the whole matter somewhat inconclusive. The present prevailing theoretical attitude can be summarized by saying that in the very small x region (x < 0.1) a number of non-perturbative effects (shadowing, sea quarks and gluons) dominate in the intermediate x domain (0.2 nuclear binding and Fermi motion in a nuclear physics approach, and/or, in a parton-QCD approach, to a partial quark deconfinement within the hadronic boundary which affects the basic properties of the hadrons. A nucleon bound in a nucleus appears somewhat larger and somewhat less massive than a free nucleon. 

The basic formula to take Fermi motion into account can be derived along lines very similar to those followed in the appendix to Chapter 16 [see eqn (16.9.19)]. The difference, of course, is that now V represents the momentum distribution of a nucleon in the nucleus. The following convolution formula emerges either using the techniques of the operator product expansion, or more simply, by considering the kinematics of Fig. 17.11 which shows a nucleus of atomic number A in a reference frame in which it is moving very fast with momentum P along OZ. A nucleon i inside the nucleus has -component of momentum pz = zP/A and a parton. 

The idea of having two distinct quasi-Fermi levels or chemical potentials within the same volume of material, first emphasized by Shockley (1), has deeper implications than the somewhat similar concept of two distinct effective temperatures in the same block of material. The latter can occur, for example, when nuclear spins are weakly coupled to atomic motion (see Magnetic spin resonance). Quasi-Fermi level separations are often labeled as Im p Fermi s name spelled backwards. 

Since two electrons of the same spin have a zero probability of occupying the same position in space simultaneously, and since t / is continuous, there is only a small probability of finding two electrons of the same spin close to each other in space, and an increasing probability of finding them an increasingly far apart. In other words the Pauli principle requires electrons with the same spin to keep apart. So the motions of two electrons of the same spin are not independent, but rather are correlated, a phenomenon known as Fermi correlation. Fermi correlation is not to be confused with the Coulombic correlation sometimes referred to without its qualifier simply as correlation . Coulombic correlation results from the Coulombic repulsion between any two electrons, regardless of spin, with the consequent loss of independence of their motion. The Fermi correlation is in most cases much more important than the Coulomb correlation in determining the electron density. 

Nevertheless, encouraging results have been obtained in recent years (see reviews(Bonetto et al, 1961) and the references therein). For example it is now known that in one dimensional systems of the Fermi-Pasta-Ulam (FPU) type (Lepri et al, 1998), heat conduction is anomalous and the coefficient of thermal conductivity k diverges with the system size L as k L2/5 (when the transverse motion is considered n L1/3... 

We report on a new force that acts on cavities (literally empty regions of space) when they are immersed in a background of non-interacting fermionic matter fields. The interaction follows from the obstructions to the (quantum mechanical) motions of the fermions in the Fermi sea caused by the presence of bubbles or other (heavy) particles immersed in the latter, as, for example, nuclei in the neutron sea in the inner crust of a neutron star. 

When it comes to polyatomic molecules, there are two problems that complicate the issue, as already discussed in Note 1 of Chapter 3. One is the separation of the overall rotation of the molecule (Jellinek and Li, 1989). The other is that, depending on the choice of internal coordinates, certain coupling terms can be assigned to be kinetic or potential terms. A simple and familiar case is a linear triatomic, when one uses bond coordinates versus Jacobi coordinates. The case for Fermi coupling for a bending motion is discussed in Sibert, Hynes, and Reinhardt (1983). 

We presented fully self-consistent separable random-phase-approximation (SRPA) method for description of linear dynamics of different finite Fermi-systems. The method is very general, physically transparent, convenient for the analysis and treatment of the results. SRPA drastically simplifies the calculations. It allows to get a high numerical accuracy with a minimal computational effort. The method is especially effective for systems with a number of particles 10 — 10, where quantum-shell effects in the spectra and responses are significant. In such systems, the familiar macroscopic methods are too rough while the full-scale microscopic methods are too expensive. SRPA seems to be here the best compromise between quality of the results and the computational effort. As the most involved methods, SRPA describes the Landau damping, one of the most important characteristics of the collective motion. SRPA results can be obtained in terms of both separate RPA states and the strength function (linear response to external fields). 

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