Stoner factor

In the above paragraphs, we have already introduced several approximations in the description of the shift and relaxation rates in transition metals, the most severe being the introduction of the three densities of states Dsp E ),Dt2g(E ), and Deg E ). The advantage is that these values can be supplied by band structure calculations and that the J-like hyperfine field can sometimes be found from experiment. We have no reliable means to calculate the effective Stoner factors ai that appear in Eq. (2), and the disenhancement factors ki in the expression for the relaxation rate, Eq. (4), are also unknown. It is often assumed that k/ can be calculated from some /-independent function of the Stoner parameter k (x), thus k/ = k((X/). A few models exist to derive the relation k((x), all of them for simple metals [62-65]. For want of something better they have sometimes been applied to transition metals as well [66-69]. We have used the Shaw-Warren result [64], which can be fitted to a simple polynomial in rx. There is little fundamental justification for doing so, but it leads to a satisfactory description of, e.g., the data for bulk Pt and Pd. 

Clearly, the critical parameter that must be evaluated is simply the Modified Stoner Factor N(Ef)Ixc - if this exceeds unity then the magnetic energy will become negative and spontaneous magnetisation will become favourable. This is the so-called Stoner Criterion . Janak [6] has calculated l c for several elements, providing an insightful commentary on trends in this quantity across the periodic table. 

As was mentioned in Section 4.V.2.2, the Pauli susceptibility of bulk Pd is exchange-enhanced by a Stoner factor S = 9.4. Its temperature-dependence is a result of marked energy-dependence in the density of states. The susceptibility of the clusters is indeed indicative of enhancement of the Pauli susceptibility, but one which is reduced. Temperature-dependence is also observed, again reduced relative to that of the bulk. The size-dependence was described with a model which assumes a reduction of the density of states at the cluster surface as a result of the ligand bonding, by analogy with similar effects on nickel surfaces, as described above. Accordingly, the susceptibility without enhancement effects was taken to depend on... 

Interpretation of the x(T) data begins with a distinction between Stoner and mass enhancements. The electronic specific-heat parameter y increases with the mass enhancement [14]. By measuring both /(T) and y at low temperatures, Sreedhar et al. [80] determined a Stoner factor S = 0.58, well below the S = 1 for a ferromagnetic instability. These nickel oxides are clearly on the itinerant-electron side of the transition from localized to itinerant... 

In the spin-fluctuation model the tendency towards magnetism is determined by the strength of the effective exchange interaction between electrons in a narrow band. The presence of this exchange interaction leads to an enhanced susceptibility over the Pauli value, predicted for a free-electron gas. At T = 0 K this enhaneement factor, known as the Stoner factor, is given by... 

Li J, Zhang D, Stoner GD and Huang C. 2008. Differential effects of black raspberry and strawberry extracts on BaPDE-induced activation of transcription factors and their target genes. Mol Carcinogenesis 47 286-294. 

Stoner enhancement factor). This solution is evidently unstable when ... 

In the Stoner product (30), we find as a factor N((Xf), which can be measured independently. The other factor I is an important parameter which we shall discuss briefly. 

However, the intra-atomic Coulomb interaction Uf.f affects the dynamics of f spin and f charge in different ways while the spin fluctuation propagator x(q, co) is enhanced by a factor (1 - U fX°(q, co)) which may exhibit a phase transition as Uy is increased, the charge fluctuation propagator C(q, co) is depressed by a factor (1 -H UffC°(q, co)) In the case of light actinide materials no evidence of charge fluctuation has been found. Most of the theoretical effort for the concentrated case (by opposition to the dilute one-impurity limit) has been done within the Fermi hquid theory Main practical results are a T term in electrical resistivity, scaled to order T/T f where T f is the characteristic spin fluctuation temperature (which is of the order - Tp/S where S is the Stoner enhancement factor (S = 1/1 — IN((iF)) and Tp A/ks is the Fermi temperature of the narrow band). 

Xo, Xs-o and Xs are orbital, spin-orbit and spin contributions, S is the Stoner enhancement factor (see Sect. Ill of this chapter). Xs, Xo and Xs-o are given by ... 

Table 7. Stoner enhancement factor S determination for light actinide metals...

To summarize, we again want to make a sharp distinction between the enhancement in nearly ferromagnetic metals like Ni3Al, where the Stoner enhancement factor can increase x without limit but not dx/dT or y, and the enhancement in nearly antiferromagnetic metals to be discussed in Chapter 4, where x> V and dx/dT are all enhanced. In the former case, all models involve either a non-integral number of electrons in the band or degenerate orbitals. In the latter, two electrons on the same atom necessarily have antiparallel spins. 

This critical field called coercivity ff. or switching field Ff., is also equal to FF. If a field is applied in between 0 and 90° the coercivity varies from maximum to zero. In the case of this special example the applied field Ha = Hs = Hc = Hk. Based on the classical theory, Stoner-Wohlfarth (33) considered the rotation unison for noninteracted, randomly oriented, elongated particles. The anisotropic axis can be due to the shape anisotropy (depending on the size and shape of the particle) or to the crystalline anisotropy. In the prolate ellipsoids b is the short axis and a the longest axis. The demagnetizing factors are IV (in the easy direction) and The demagnetizing fields can then be calculated by Hda = — Na Ms, and Hdb = — Nb Ms. The shape anisotropy field is Hd = (Na — Nb)Ms. Then the switching field Hs = Hd = (Na — Nb)Ms. 

In the weak-coupling limit (t U, V), the effect of electron-electron interactions measured in terms of an effective, /eff, lead to a Stoner-type enhancement factor, which can slightly reduce the deviation from the expected Pauli susceptibility [5] ... 

The factors that must be taken into account in the selection of the equipment and, consequently, for efficient waste separation, are waste characteristics (particle size, waste stream composition, waste components density, moisture content, tendency for aggregation, etc.), products specifications, design parameters, and space requirements. The various types of air separators (air concentrators, Stoners) are the most widely used equipment. Heavy media separators are also used in some cases. 

Here S is the so-called Stoner enhancement factor, usually written as... 

---