Stoner parameter

In Chap. A, we have seen that, in the Stoner model, (ferromagnetic) spin-polarization of electrons originates two electron states E+ and E from each electron state E of a non-spin-polarized electron band, the difference between the two being (E+ - E ) = Im, where I is the Stoner parameter and m = n+ - n is the magnetization density. 

Table 1. 5 f bandwidth Wf intra-atomic Coulomb correlation U " and Stoner parameter time the density of states I x N(Ep) for light actinide metals... 

The f-band width was found to be about 5 eV in Ac, about 3 eV for Th-Np and around 2 eV for Pu. In Am it is down to 1 eV. The Stoner parameter, was calculated to be about 0.5 eV and almost constant throughout the series. At Am, however, the product I N(Ef) of the Stoner parameter and the f-density of states at the Fermi level exceeds one and spontaneous spin polarization occurs in the band calculation. Since Am has about 6.2 f-electrons and the moment saturates, this leads to an almost filled spin-up band and an empty spin-down band. The result is that the f-pressure all but vanishes leading to a large jump in atomic volume - in agreement with experiment. This has been interpreted as Mott-localization of the f-electrons at Am and the f-electrons of all actinides heavier than Am are Mott-localized. The trend in their atomic volumes is then similar to those of the rare earths. 

Table 4. The product of the Stoner parameters and the density of states per spin at the Fermi level. INsfEp) > 1 for ferromagnetism, UN is antiferromagnetic...

TABLE 8.6. Stoner Parameters for Several Elements in Rydbergs (1 Ry = 13.6 eV). Obtained by the Korringa-Kohn-Rostoker (Green Function) Nonspin Polarized Density-Functional Calculations in the Local-Spin Density Approximation... 

Source Zeller, R. (2006) and the spin-polarized exchange-correlation integral, I c, calculated by the local spin density approximation. The Stoner parameter is, to a first approximation, element-specific and independent of the atom s local environment. 

From Table 8.6, it can be seen that the Stoner parameter for Pd is 0.025 Ry, or 0.34 eV. The DOS at the Fermi energy is given as N(Ef) = 2.28 eV. Their product is 0.7752, which is less than unity and, therefore, does not satisfy the criterion for magnetic behavior. 

Figure 4. Stoner parameter I(M) and loss of kinetic energy D(M) (dashed line) for Co as functions of magnetization. - full correlation calculation, I2...

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. 

The stronger enhancement at low densities could be attributed phenomenologically to an increase in the Stoner parameter a. This approach leads to an incorrect description of the physics as may be seen by the following argument (Warren, 1984 Chapman and March, 1988). The plot in Fig. 3.2 shows that the enhancement of the measured susceptibility is limited near the Curie values calculated for cesium on the coexistence curve. Between the peak and the critical point, the liquid-state susceptibility tends to follow the Curie law (x oc p/T). Stoner enhancement, however, can increase without limit and the susceptibility actually diverges at the transition point of a metallic ferromagnet. In contrast. Curie law behavior is the limit expected from Eq. (3.3) if the enhancement is due to a very high density of states. 

The Stoner criterion provides some insight into the differences in magnetic properties that are expected to exist in thin films and surfaces. The Stoner parameter and the DOS at the Fermi level depend on the chemical elements and the system s dimensionality and can thus be modified. The DOS at the Fermi level depends on the width of the d-band, which can be roughly described as... 

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