Paramagnetic spin susceptibility

It should be noted that the equivalence of the DLM electronic structure problem to that of an Ising-like system is purely a consequence of the symmetry of the paramagnetic state, and is not the result of om imposing any restriction on the moment directions. Indeed, in the formalism for the paramagnetic spin susceptibility, which we outline now, we maintain and consider the full 3D orientational freedom of the moments. 

FIGURE 31 Paramagnetic spin susceptibilities for gadolinium at wave vectors along the [0,0,1] direction, obtained from (A) LSDA calculation and (B) LSIC-LSDA calculation. 

FIGURE 33 Normalized paramagnetic spin susceptibilities for Gd, obtained using the theoretical unit cell volume. The inset shows the susceptibility up to the zone boundary for c/a ratio 1.54. 

The behaviour of the paramagnetic spin susceptibility, x, is determined by the direct correlation function, [Eq. (100)]. This function can be fit in terms of real-space parameters ... 

FIGURE 36 Normalized paramagnetic spin susceptibilities for Gd for various WS radii, obtained using a c/a ratio of 1.54. 

The nuclei in metals sense the spin polarization of conduction electrons through the hyperfine interaction. Thus the observed nuclear resonance frequency (Wobs is shifted with respect to Eq. (3.7) by an amount that depends on the paramagnetic spin susceptibility of the electrons. This is the Knight shift K given by... 

The temperature dependence of the molar magnetic susceptibility (x) of an assembly of paramagnetic spins without interaction is characterized by the Curie behavior with x = C/T where C = /Vy2( 2.S (.S + l)/3k. It is a very common situation in the organometallic chemistry of radical species when the spin density is essentially localized on the metal atom. Since, in most cases, this atom is surrounded by various innocent ligands, intermolecular interactions are very weak and in most cases are reflected by a small contribution described by a Curie-Weiss behavior, with x = C/(T 0) where 0 is the Curie-Weiss temperature. A positive value for 0 reflects ferromagnetic interactions while a negative value — the most common situation — reflects an antiferromagnetic interaction. 

Some metals are diamagnetic because the conduction electron spin susceptibility is smaller than the induced diamagnetic susceptibility component. On the other hand, various rare earth metals display very strong paramagnetism because of unpaired / electrons that remain associated with individual atoms rather than entering into energy bands. 

The main equation for the d-electron GF in PAM coincides with the equation for the Hubbard model if the hopping matrix elements t, ) in the Hubbard model are replaced by the effective ones Athat are V2 and depend on frequency. By iteration of this equation with respect to Aij(u>) one can construct a perturbation theory near the atomic limit. A singular term in the expansions, describing the interaction of d-electrons with spin fluctuations, was found. This term leads to a resonance peak near the Fermi-level with a width of the order of the Kondo temperature. The dynamical spin susceptibility in the paramagnetic phase in the hydrodynamic limit was also calculated. 

For the paramagnetic phase it is convenient to calculate the spin susceptibility x through the memory function M [11] ... 

We applied the generating functional approach to the periodic Anderson model. Calculation of the electron GFs gdd, 9ds, 9sd and gss reduces to calculation of only the d-electron GF. For this, an exact matrix equation was derived with the variational derivatives. Iterations with respect to the effective matrix element Aij(to) allow to construct a perturbation theory near the atomic limit. Along with the self-energy, the terminal part of the GF Q is very important. The first order correction for it describes the interaction of d-electrons with spin fluctuations. In the paramagnetic phase this term contains a logarithmic singularity near the Fermi-level and thus produces a Kondo-like resonance peak in the d-electron density of states. The spin susceptibility of d-electrons... 

In compounds where both the donor and the acceptor molecules (or ions) are paramagnetic, two situations may arise (1) if the interchain interactions are negligible, the ESR spectrum is the superposition of the ESR spectra of each individual species or (2) if they are weak but there is significant fast spin exchange between the two spin systems, a single spectrum is observed with a g value that is averaged by the contributions to the spin susceptibilities of each chain [12] ... 

The intensity of ESR absorption relates to spin concentration, and spin susceptibility can also be measured using standard sample of known spin susceptibility. The g tensor depends upon the paramagnetic species present and its value is the fimction of the spin environment. For free electrons g = 2.00232. The linewidth peak to peak (A//pp) depends on the relaxation time. Homogeneously broadened lines have Lorentzian shapes, whereas inhomogeneously broadened lines have Gaussian shapes. In practice, a combination of both occurs. Normally the ratio of low field to high field is unity, but it can be greater. 

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