Crystalline field Hamiltonian

In crystalline field theory, the valence electrons belong to ion A and the effect of the lattice is considered throngh the electrostahc field created by the snrronnding B ions at the A position. This electrostatic field is called the crystalline field. It is then assnmed that the valence electrons are localized in ion A and that the charge of B ions does not penetrate into the region occnpied by these valence electrons. Thns the Hamiltonian can be written as... 

Weak crystalline field //cf //so, Hq. In this case, the energy levels of the free ion A are only slightly perturbed (shifted and split) by the crystalline field. The free ion wavefunctions are then used as basis functions to apply perturbation theory, //cf being the perturbation Hamiltonian over the / states (where S and L are the spin and orbital angular momenta and. 1 = L + S). This approach is generally applied to describe the energy levels of trivalent rare earth ions, since for these ions the 4f valence electrons are screened by the outer 5s 5p electrons. These electrons partially shield the crystalline field created by the B ions (see Section 6.2). 

Table XII gives representative results for the spin Hamiltonian parameters as determined for d5 ions in different crystal fields. This table is not a complete listing of ESR results on d5 ions, since such a list would be much larger, but is rather a selection of results for various types of crystalline fields. Several facts become evident when examining the results listed in this table. The g factors are always near the free spin value of 2.0023, as would be expected for an S state. The hyperfine constant is isotropic, as would also be expected for an S-state ion. The hyperfine constant also...

In strong crystalline fields or highly distorted fields a doublet S= state becomes lowest in energy. In this situation there are generally no nearby states, so that g factors are close to 2.0023 and the ESR is observed at higher temperatures. The ESR results can be fitted to Eq. (171) and observed values for the spin Hamiltonian parameters are given in Table XVI. 

Assuming a static crystalline field the potential, Ve, of the Hamiltonian for the pure fn configuration for different symmetries may be written as follows... 

The HCp operator represents the nonspherically symmetric components of the one-electron CF interactions, i.e. the perturbation of the Ln3+ 4fN electron system by all the other ions. The states arising from the 4fN configuration are well-shielded from the oscillating crystalline field (so that spectral lines are sharp) but a static field penetrates the ion and produces a Stark splitting of energy levels. The general form of the CF Hamiltonian Hcf is given by... 

The detailed Hamiltonians appropriate to the electronic and nuclear properties of the rare earths in general have been excellently summarised elsewhere [1]. They are not given here explicitly because of the more limited depth of treatment. Note, however, that the vectorial addition of the total orbital, L, and spin angular momentum, S, denoted by J is the most useful quantum number for describing electronic states. Any crystalline field potential then acts as a perturbation to the appropriate / state. This is opposite to the situation found in Fe, where the crystalline field is the dominant term. 

From Gd Mossbauer experiments Shenoy et al. (1982) measured the quadrouple interaction in GdRli4B4 and from this they were able to deduce the crystalline electric field Hamiltonian terms for the RRh4B4 compounds with R = Gd, Tb, Dy, Ho, Er and Tm. For R = Pr, Nd, Tb, Dy, Ho, Er, Tm see also Dunlap and Niarchos (1982). 

The Mu spin Hamiltonian, with the exception of the nuclear terms, was first determined by Patterson et al. (1978). They found that a small muon hyperfine interaction axially symmetric about a (111) crystalline axis (see Table I for parameters) could explain both the field and orientation dependence of the precessional frequencies. Later /xSR measurements confirmed that the electron g-tensor is almost isotropic and close to that of a free electron (Blazey et al., 1986 Patterson, 1988). One of the difficulties in interpreting the early /xSR spectra on Mu had been that even in high field there can be up to eight frequencies, corresponding to the two possible values of Ms for each of the four inequivalent (111) axes. It is only when the external field is applied along a high symmetry direction that some of the centers are equivalent, thus reducing the number of frequencies. 

In this chapter we review the study of the solid structure of many crystalline polymers mainly with high-resolution solid-state 13C NMR, so we will briefly summarize here its principles for the convenience of the reader. For this purpose we first consider the Hamiltonian of an ensemble of nuclei possessing spin in a static field B0. The Hamiltonian to be considered of this spin system can be written as ... 

Nuclear magnetic resonance (NMR) is perhaps the simplest technique for obtaining deuterium quadrupole coupling constants in solids or in liquid crystalline solutions. In ordinary NMR experiments with a magnetic field Hq > 104 gauss, the nuclear quadrupole interaction [Eq. (6)1 for deuterium is much smaller than the Zeeman interaction and can be treated as a perturbation to the Hamiltonian... 

A Hamiltonian is an operator which operates upon a wave function. When it is applied to the wave function of a particular system, it gives the permitted energy levels of that system. A simple form of the general Hamiltonian for an ion in a crystalline environment and with an applied magnetic field may be VTitten as... 

The NO radical was originally believed [39] to be constrained to a fixed orientation in either a potassium ion or azide ion vacancy by crystalline electric-field forces. With this assumption it was necessary to explain certain features of the ESR spectrum on the basis of a 2 ground state rather than the well-established 7T ground state of NO. This led Fuller and Tarr [40] to propose that the resonances were due to NOl instead. Subsequently, studies of NO were extended to rubidium and cesium azides [41], which have the same crystal structure as potassium azide. The directions of the principal axes of the spin-Hamiltonian were reinterpreted to be consistent with a ground state. A possible rotation of the NO molecule was also suggested. 

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