Conduction electrons density

We now switch on an external potential U(r) that is slowly varying over the dimensions of the metallic box. This makes the conduction electron density... 

The simplest model of metals is the Sommerfeld theory of free-electron metals (Ashcroft and Mermin 1985, Chapter 2), where a metal is described by a single parameter, the conduction electron density n. A widely used measure of... 

The efficient screening approximation means essentially that the final state of the core, containing a hole, is a completely relaxed state relative to its immediate surround-ing In the neighbourhood of the photoemission site, the conduction electron density of charge redistributes in such a way to suit the introduction of a core in which (differently from the normal ion cores of the metal) there is one hole in a deep bound state, and one valence electron more. The effect of a deep core hole (relative to the outer electrons), may be easily described as the addition of a positive nuclear charge (as, e.g. in P-radioactive decay). Therefore, the excited core can be described as an impurity in the metal. If the normal ion core has Z nuclear charges (Z atomic number) and v outer electrons (v metallic valence) the excited core is similar to an impurity having atomic number (Z + 1) and metalhc valence (v + 1) (e.g., for La ion core in lanthanum metal, the excited core is similar to a Ce impurity). 

The most recent calculations, however, of the photoemission final state multiplet intensity for the 5 f initial state show also an intensity distribution different from the measured one. This may be partially corrected by accounting for the spectrometer transmission and the varying energy resolution of 0.12, 0.17, 0.17 and 1,3 eV for 21.2, 40.8, 48.4, and 1253.6 eV excitation. However, the UPS spectra are additionally distorted by a much stronger contribution of secondary electrons and the 5 f emission is superimposed upon the (6d7s) conduction electron density of states, background intensity of which was not considered in the calculated spectrum In the calculations, furthermore, in order to account for the excitation of electron-hole pairs, and in order to simulate instrumental resolution, the multiplet lines were broadened by a convolution with Doniach-Sunjic line shapes (for the first effect) and Gaussian profiles (for the second effect). The same parameters as in the case of the calculations for lanthanide metals were used for the asymmetry and the halfwidths ... 

It was noted earlier that the charge density of a narrow resonance band lies within the atoms rather than in the interstitial regions of the crystal in contrast to the main conduction electron density. In this sense it is sometimes said to be localized. However, the charge density from each state in the band is divided among many atoms and it is only when all states up to the Fermi level have contributed that the correct average number of electrons per atom is produced. In a rare earth such as terbium the 8 4f electrons are essentially in atomic 4f states and the number of 4f electrons per atom is fixed without reference to the Fermi level. In this case the f-states are also said to be locaUzed but in a very different sense. Unfortunately the two senses are often confused in literature on the actinides and, in order not to do so here, we shall refer to resonant states and Mott-localized states specifically. 

In conducting solids, the conduction electron density is spatially modulated, forming charge density waves (CDW) the periodic distortion accompanying the CDW (due to interaction between the conduction electron and the lattice) is responsible for the incommensurate phase (Overhauser, 1962 Di Salvo Rice, 1979 Riste, 1977). The occurrence of CDW and the periodic distortion can be understood in terms of the model proposed by Peierls and Frdhlich for one-dimensional metals. Let us consider a row of uniformly spaced chain of ions (spacing = a) associated with conduction electrons of energy E k) and a wave vector k. At 0 K, all the states are filled up to the Fermi energy, = E(kp). If the electron density is sinusoidally modulated as in Fig. 4.15 such that... 

Here, an is the Bohr orbit radius of the isolated center and nc is the critical carrier density at the M-NM transition. Another way of viewing the transition is that of an electronic instability which ensues when the trapping of an electron into a localized level also removes one electron from the Fermi gas of electrons. This must clearly lead to a further reduction in the screening properties (which are themselves directly related to the conduction electron density) and a catastrophic situation then ensures the localization of electrons from the previously metallic electron gas. 

A CDW is a periodic modulation of the conduction electron density within a material. It is brought about when an applied electric field induces a symmetry-lowering lattice modulation in which the ions cluster periodically. The modulation mechanism involves the coupling of degenerate electron states to a vibrational normal mode of the atom chain, which causes a concomitant modulation in the electron density that lowers the total electronic energy. In one-dimensional systems, this is the classic Peierls distortion (Peierls, 1930, 1955). It is analogous to the JT distortion observed in molecules. 

The Dy " " ion has been detected in CaFj doped with Dy " " and reduced electrolytically [13 The chemical isomer shift is about —7 mm s relative to Dy + in Dyp3, and this value was used to estimate the conduction-electron density in Dy metal. 

These results give an Immediate explanation for the observed (19) coexistence of the high T. superconductivity and magnetic ordering In the RBa2Cu307- structures. The lack of conduction electron density around the R-slte (J. Yu and A. J. Freeman, to be published) means that the unpaired rare-earth f-electrons are decoupled from the Cooper pairs (i.e., magnetic Isolation) and so cannot pair-break. 

These mixed-valenee systems have been discussed by several authors with respect to final state effects in their core level spectra [12,35]. A distinction is generally made between the cases where the extra electrons occur as itinerant conduction band electrons (metallic case) or whether they are completely localized to single sites. In the former case, the core ionization of one site will in itself lead to the creation of localized levels (by pulling down from the conduction band), whose occupancy in the final state of the ionization process will depend statistically on the conduction electron density. Thus, the final state localized level may be either filled or empty. The net result is a loss of direct correlation between observed relative peak intensities and the number of inserted electrons per formula unit (x value), since the population of the two possible core-hole states is considered to be entirely a final state phenomenon. On the other hand, for the case of complete single-site localization (non-metaUic case) the relative intensities are expected to truly represent the relative number of the two possible valence states before ionization and will not be affected by effects due to the final state of the core ionization process. 

Fig. 15. The Lorentz construction. The case shown refers to a sample fully magnetized by an externally applied field. Symbols in the Lorentz sphere solid circle and arrow, paramagnetic ion open circle and arrow, implanted muon on interstitial site broken lines, dipolar field lines dots, conduction electron density.

As outlined before, once a low-energy positive muon has been deposited in the sample of interest, it slows down to thermal velocities in a time of order 10 s (for solids), with no loss of original polarization. It may emerge from thermalization as an apparently bare (in metals) or in a muonium-like state (sometimes, in semiconductors and insulators). Muonium-like states will not be discussed further here. A bare muon in a metal is in contact hyperfine interaction with the conduction electron density at its location, which results in a muonic Knight shift (usually small in nonmagnetic materials). A bare x+ in an insulator is a diamagnetic center, and is quite likely to be bonded to the most electronegative species present. 

Here i/ (z) = d In T(z) /dz is the digamma function and W is the band width of the Lorentzian conduction electron density of states. Furthermore, T) is the effective temperature-dependent coupling strength of the longitudinal sound waves to quasiparticles. It may be written as... 

The values obtained for fip are very small compared with the full conduction electron density (fipi 2 eV), suggesting that only a small fraction (Mfree/ cond) of the conduction electrons are delocalized macroscopically. This small fraction of delocalized carriers is consistent with the small number of percolation paths that occur close to the percolation threshold in composite systems. The fraction of the carriers that are delocalized can be estimated by comparing the plasma frequency of free electrons (fip) with the full conduction electron plasma frequency (fipi). 

---