Kondo model

Figure 1. Potential profile ij/(x) across the cell wall (x < 0) according to the simplified expression (6) from the Ohshima and Kondo model [15]. The profile within the solution (x > 0) is assumed to follow the usual GC decay expression ij/ = ij/Dm exp ( — kx) (equation (2)). Parameters T — 298 K, z—+1, = 78.5, thickness of the Donnan layer = 10 nm, p — 1447Cdm Curves correspond with c — 0.2, 0.1, 0.05, 0.02 and...

In the absence of a well-defined maximum in C, the description of the dominant microscopic mechanism can become rather difficult. We can quote as an example the similar CjT versus T behaviours of Ce24Con, CeCu3Al2 and CeCu3Ga2- The first compound was described in sect. 7 as a spin fluctuator, while the other two were recently described within the spin-j single-impurity Kondo model (Kohlmann et al. 1990). 

The 5f-electron level in UBe 13 is close to the Fermi level as seen from photoemission results. Therefore the HF-behaviour which sets in below a fluctuation temperature of T 8-25 K does not correspond well to the Kondo picture which requires 5f states to lie sufficiently removed from the Fermi energy. Nevertheless the Kondo model in its single and multichannel... 

The susceptibility x(T)/x(0) with x(0) from eq. (12) is shown in fig. 7. It is again similar for VF and HF systems, apart from a peak near 7], which is more pronounced for VF systems. This peak depends strongly on the 4f-level degeneracy N and vanishes in the Kondo model with N = 2 or effective spin S = j. Figure 8 shows this behavior for the susceptibility in the Coqblin-Schrieffer model, see eq. (3), (in which charge fluctuations are ignored) for various values of the total angular momentum j. 

Another remarkable feature measured in CeCu is the temperature dependence of the plasma resonance cUp of the heavy quasiparticles, which has been found near 150meV at 5 K and from which the quasiparticle concentration and the eflTective mass has been deduced (with the help of the y value). We show in fig. 143 the plasma frequencies obtained from the optical data at different temperatures. The value at 1.4 K (the asterisk) has been reported from millimeter-wave data from Beyermann et al. (1988) and has a value of about 200 meV, which agrees quite reasonably with our 150 meV It is remarkable that the plasma frequency preserves practically the same small value up to a temperature of about 20 K. This is in contrast with what one expects from a simple Kondo model where the characteristic temperature for the formation of a resonant state has been estimated to be between 3 and 6K (Satoh et al. 1985, Steglich et al. 1985, Onuki et al. 1984). Up to room temperature the plasma frequency continuously shifts to 2-3 eV, which means that... 

Fig. 22, The specific-heat curve C(T) of the Kondo model for a range of integral and half-odd integral spin values S from j to j (Rajan 1983).

Fig. 26. The zero temperature magnetore-sistance for the Kondo model (solid curve) in comparison with data for a dilute alloy of Ce (Schlottmann 1989).

We summarize the results of some l/N calculations in a set of figures. In figs. 30 and 31 we compare the susceptibility and specific heat results of the l/N calculation with those of the Kondo model. The two sets of results, for N = 6 and 4 respectively, show that the spin fluctuation model virtually reproduces the predictions of the Kondo model. In fig. 32 the specific heat of a dilute alloy of Ce in LaBg is compared... 

Fig. 30. The magnetic susceptibility (upper curves) and specific heat (lower curves) for the single Ce impurity as predicted by the spin fluctuation resonance model for JV, = 6 and n, = 0.97. The itf = I curves are the prediction of the Kondo model (Bickers et al. 1987).

The evolution of theoretical models has been driven by experiment. To date, the f component of the one-electron addition/removal spectra of alloys containing Ce ions, as well as ordered Ce compounds, are qualitatively similar. Due to the difficulty of separating out the f component of the spectrum, the only concentration dependence that has been identified is the linear dependence of the overall intensity. This unsatisfactory state of affairs has caused theorists to focus their attention primarily on single-impmity models that have the same qualitative features as observed in experiments. Thereby, effects that are present in ordered compotmds have been, hitherto, largely ignored. We shall categorize these theories into two classes, namely Kondo models and screening models. 

Kondo models are based on the single-impurity Anderson (1961) model, which describes an f shell that is bathed in a Fermi sea of conduction electrons. The Hamiltonian, H, can be written as the sum of three terms. 

As outlined above, the physics associated with the Kondo model can be ascribed to the local moment inducing a compensating spin polarization cloud in the gas of conduction electrons, at low temperatures. The models considered in the next section are categorized as screening models, in which considerations of electrical charge neutrality in the unit cell are brought into play. 

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