Fermi hole fermion

Let us see the probability density that two identical fermions occupy the same position in space and, additionally, that they have the same spin coordinate xi,yi,zi,o-i) = (X2,) 2,Z2,o-2). We have /r(l, 1,3,4,...,Ai) = - /r(l, 1,3,4,. ..,N), hence i/r(l, 1,3,4,..., Af) = 0 and, of course, (1,1,3,4,..., N) = 0. Conclusion two electrons of the same spin coordinate (we will sometimes say "of the same spin ) avoid each other. This is called the exchange or Fermi hole around each electron. The reason for the hole is the antisymmetry of the electronic wave function, or in other words, the Pauli exclusion principle. ... 

This effect is known as exchange or Fermi correlation and is a direct consequence of the Pauli principle. The Fermi hole is in no way connected to the charge of electrons and applies equally to neutral fermions. This kind of correlation is included in the HF appro ich due to the antisymmetry of the Slater determinant [337]. The electrostatic repulsion of electrons (the l/ri2 term in the Hamiltonian) prevents the electrons from coming too close to each other and is known as Coulomb correlation. This effect is independent of the spin and is called simply electron correlation, and is completely neglected in the HF method. 

The XC-hole can be split into contributions from the exchange- or X-hole, which arises from the Fermion nature of an electron obeying the Pauli principle, and the correlation- or C-hole due to Coulomb repulsion within the pair of electrons. (The X-hole and C-hole are often referred to as Fermi hole and Coulomb hole, respectively). 

Ideally, the specific heat of conduction electrons (or holes) in a metal is a linear function of temperature C = yT, where y, known as the Sommerfeld constant, is in the range 0.001 to 0.01 J/(molK ) for normal materials. In HF compounds, y reaches values up to 10 times larger (see tables 9, 10 and 11). In the basic theory of the specific heat of itinerant electrons (free Fermi gas), y is proportional to the effective mass m of the charge carriers, and so the name heavy fermions has come to be attached to these high-y materials (see Stewart 1984). The linear relation between C and T is strictly fulfilled only in the limit of a free degenerate electron gas. In real materials, weak non-linearities show up that can be encompassed by, for example, allowing y to be temperature dependent, y T). The Sommerfeld constant of interest is then the extrapolation of y for... 

A special situation occms for some of the cerimn and ytterbium based compoimds. Here, the 4/ (Yb +) configmation is the hole analogue of 4/ (Ce +). The degree of 4/ state contribution at the Fermi energy has a drastic influence on the physical properties of these materials, leading to valence instabihties or heavy fermion behavior. 

Because of particle-hole symmetry, the PES spectrum for Yb heavy fermions, where one has only one f hole (/ vs/ " ) rather than one f electron, can be obtained from fig. la simply by applying mirror symmetry about the Fermi energy (Bickers et al. 1987). Thus in Yb heavy fermions the bulk of the KR is predicted to be occupied, which makes them ideal candidates for studying spectral weights, widths, and temperature dependencies. 

In the subsequent chapters in which we will be investigating the thermal, electrical, optical, and magnetic properties of materials, it will be necessary to be able to determine the energy distribution of electrons, holes, photons, and phonons. To do this, we need to introduce some quantum statistical mechanical concepts in order to develop the distribution fimc-tions needed for this purpose. We will develop the Bose-Einstein (B-E) distribution function that applies to all particles except electrons and holes (and other fermions) that obey the Pauli exclusion principle and show how this function becomes the Maxwell-Boltzmann (M-B) distribution in the classical limit. Also, we will show how the Planck distribution results by relaxing the requirement that particles be conserved. Next we develop the Fermi-Dirac (F-D) distribution that applies to electrons and holes and becomes the basis for imderstanding semiconductors and photonic systems. 

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