Electrons Fermi sphere

The most famous example of the crystal structure correlating with the average number of valence electrons per atom or band filling, N, is the Hume-Rothery alloy system of noble metals with sp-valent elements, such as Zn, Al, Si, Ge, and Sn. Assuming that Cu and Ag have a valence of 1, then the fee -phase is found to extend to a value of N around 1.38, the bcc / -phase to be stabilized around 1.48, the -phase around 1.62, and the hep e-phase around 1.75, as illustrated for the specific case of Cu-Zn alloys in Fig. 6.15. In 1936 Mott and Jones pointed out that the fee and bcc electron per atom ratios correlate with the number of electrons required for a free-electron Fermi sphere first to make contact with the fee and bcc Brillouin zone faces. The corresponding values of the Fermi vector, fcF, are given by... 

Figure 8.1 Angular correlation of positrons annihilating in normal and superconducting Pb. The arrow at approximately 6 mrad indicates the position of the Fermi cutoff for a free electron Fermi sphere, and no detectable smearing is seen in the superconducting state. The triangle indicates the angular resolution. From Briscoe et al. [17],...

Part (a) shows a free-electron Fermi sphere for silicon cut by various Bragg reflection planes, reducing the area of free Fermi surface. Part (b) shows the Jones Zone, made up of (220) Bragg planes, into which all of the silicon Fermi surface has disappeared. The view is along a [HO] direction in both parts. 

The electrons that occupy the levels of a Fermi gas have energies < and may be considered as confined to a (Fermi) sphere of radius kF in k-space. For large volumes the free-electron quantum numbers may be treated as continuous variables and the number of states in a range dk = dkxdkydkz, is... 

Unit volume in k-space therefore accommodates V/4n3 electrons (two per level). The Fermi sphere accommodates... 

Following Pauli s exclusion principle, each state corresponding to a given can contain at most two electrons of opposite spin. Therefore, at the absolute zero of temperature aU the states, k, will be occupied within a sphere of radius, kF, the so-called Fermi sphere because these correspond to the states of lowest energy, as can be seen from Fig. 2.9(a). The magnitude of the Fermi wave vector, kF, may be related to the total number of valence electrons N by... 

Fig. 6.5 Scattering between filled and empty states near the Fermi surface. For the Fermi sphere of a free-electron gas the maximum number of such events occurs for q = 2kf. For a Fermi surface with flat regions the number of such events is dramatically enhanced for q = Q, the spanning wave vector.

Free-electron Fermi surface of Al [23]. The first Brillouin zone (called "first zone" in the illustration) is completely filled, because it is insidethe Fermi sphere its center is at k — 0. The second surface shown ("second zone") encloses empty levels the filled levels are between the concave surface faces shown and the Fermi sphere its center is also at k — 0. The third band ("third zone") "hot dogs" are filled states, but their origin is at the center of one of the rectangular faces of the "first zone." The fourth Brillouin zone shows small pockets of electron concentration theirorigin is again atthecenterof one of the rectangular faces of the "first zone."... 

The definition applies to electrons in metals / is the mean free path, and vF is the electron velocity on the Fermi sphere. 

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,... 

Assuming a fixed band structure (the rigid band model), a decrease in the density of states is predicted for an increase in the electron/atom ratio for a Fermi surface that contacts the zone boundary. It will be recalled that electrons are diffracted at a zone boundary into the next zone. This means that A vectors cannot terminate on a zone boundary because the associated energy value is forbidden, that is, the first BZ is a polyhedron whose faces satisfy the Laue condition for diffraction in reciprocal space. Actually, when a k vector terminates very near a BZ boundary the Fermi surface topology is perturbed by NFE effects. For k values just below a face on a zone boundary, the electron energy is lowered so that the Fermi sphere necks outwards towards the face. This happens in monovalent FCC copper, where the Fermi surface necks towards the L-point on the first BZ boundary (Fig. 4.3f ). For k values just above the zone boundary, the electron energy is increased and the Fermi surface necks down towards the face. 

Figure 4.3. For the monovalent FCC alkali metals (a) with a low electron density, the Fermi-wave vector (the radius of the Fermi sphere) lies well below the first BZ boundary. The Fermi surface is unperturbed. For monovalent FCC copper (b), the increased electron density forces the Fermi wave vector to terminate very near the L-point. The electron energy is lowered and the Fermi sphere necks outwards towards that face of the BZ boundary.

The connection follows the line of reasoning just presented. As a polyvalent metal is dissolved in a monovalent metal, the electron density increases, as does the Fermi energy and Fermi-wave vector. Eventually, the Fermi sphere touches the BZ boundary and the crystal stmcture becomes unstable with respect to alternative structures (Ra5mor, 1947 Pettifor, 2000). Subsequent work has been carried out confimiing that the structures of Hume-Rothery s alloys (alloys comprised of the noble metals with elements to the right on the periodic table) do indeed depend only on their electron per atom ratio (Stroud and Ashcroft, 1971 Pettifor and Ward, 1984 Pettifor, 2000). Unfortunately, the importance of the e/a ratio on phase equilibria is much less clear when it does not correspond precisely to BZ touching. 

This corresponds to a density of states, in wave number space, of n/(27t). The energy of each state is given by h k /2m, so the state of the gas that is of lowest energy will be obtained by putting one electron of each spin in each state within a sphere, the Fermi sphere in wave number space, with a radius kr chosen to be just large enough that all electrons are accommodated. If there are N electrons per unit volume, or a total of Nil electrons, the product of the sphere volume, times two for spin, times the density of states in wave number space, must equal NQ ... 

The states allowed by periodic boundary conditions form a fine grid in wave number space. The lowest energy state of the system is obtained if pairs of electrons are placed in the slates of smallest k, thus occupying all states within a sphere called the Fermi sphere. 

I lGURE 16-10 Owing to a vacancy or other defect, an electron in a state of wave number k on the Fermi sphere may be Scattered to some state of wave number k elsewhere on the sphere. 

A central cross-scction of the Fermi sphere for aluminum in the extended-zone scheme was shown in Fig. 16-9. By a study such as that illustrated in Fig. 16-5, you can identify two orbit types, which correspond to cross-sections of the surfaces for the second and third bands, shown in Fig. 16-8. The topology of ihe F ermi surface is the same for lead but, with four electrons per atom in lead, the sphere is larger in comparison to the zone than that for aluminum. Estimate the area of cross-sections of the surfaces for the second and third bands for lead, in units of (Inlay, with a the cube edge. 

A treatment of transport properties in terms of this surface is no more complicated in principle than that in the polyvalent metals, but there is not the simple free-clectron extended-zone scheme that made that case tractable. Friedel oscillations arise from the discontinuity in state occupation at each of these surfaces, just as they did from the Fermi sphere. When in fact there arc rather flat surfaces, as on the octahedra in Fig. 20-6, these oscillations become quite strong and directional. A related effect can occur when two rather flat surfaces are parallel, as in the electron and hole octahedra, in which the system spontaneously develops an oscillatory spin density with a wave number determined by the difference in wave number between the two surfaces, the vector q indicated in Fig. 20-5. This generally accepted explanation of the antiferromagnetism of chromium, based upon nesting of the Fermi surfaces, was first proposed by Lomer (1962). 

The concept of a spatial correlation is very rich in content, and its content is necessary and sufficient for energy assessment. It is useful to consider the spatial correlation in the approximation of the band model from all the wave functions contained in the Fermi sphere a determinant has to be formed as, also the determinants of weakly excited states a perturbation calculation has to be made with respect to 2e2/ x n — Xjn the reduction to the spatial correlation must be made the energy may be calculated. In fact the pertubation calculation has never been made for metals, while the spatial correlation was only calculated by Wigner and Seitz (1933) in the zero approximation. The result of their calculation is the well-known exchange hole, which may be represented in three-dimensional space if one electron is fixed in the coordinate origin. This may be done if the local space is homogenous i.e., has no... 

After the TCS is evaluated for a wide range of relative velocities we calculate the stopping power S = — (d /dx) by integrating electron velocities in a Fermi sphere (0 < i <, < Vp) and over the range of relative velocities (with Iv — Vgl < < v 4- Vg) [36]. This... 

Electronic transport properties are strongly influenced by a touch of the Fermi sphere with the zone boundary, in the crystalline as well as in the disordered state. Exhaustive reviews on this subject have been given by Massalski and Mizutani [5.35] and Mizutani [5.20], In the same way as sharp zone boundaries in crystalline materials are responsible for umklapp processes, in amorphous systems we can talk in terms of diffuse umklapp processes caused by the pseudo Brillouin-zone boundary. This description was first introduced by Hafner [5.36]. 

The energy of the conduction electrons is given by h2k2/2 x, where k is the wave vector number and p, is the effective mass of the electron-nucleus. In a real space of Cartesian coordinates k = [kx, ky, fcj, a Fermi sphere can be constructed with radius k = (2 lE )x 2/h. The shape of this sphere is a clearly defined by the electrical properties of the metal. The current density obeys the change in the occupancy of states near the Fermi level, which separates the unfilled orbitals in the metal from the filled ones in the linear momentum space p = hk. 

The factors determining the particular structure adopted hy an intermetallic compound or, indeed, whether such a compound exists at all as a single-phase material, have been the subject of much discussion for a considerable period of time. The Hume-Rothery rules for electron compound formation will he very familiar and are related physically to the size of the Fermi sphere in the appropriate Brillouin zone. For example, electron compounds are expected for valence electron concentrations of , fj and l for the bcc, y-brass and cph structures, respectively. The interplay of other factors such as the atomic size, solubility and crystal structure of the components on the formation of intermetallic compounds has been considered in considerable detail by many workers, including Yao (1962), who suggested that transition metal binary systems could be classified into groups according to an excess energy dE expressed as... 

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