Hamiltonian crystal-field term

The total Hamiltonian of a system thus consists of a crystal field term and an exchange (with or without external) field and may be expressed as... 

A satisfactory description of the magnetic anisotropy in R2Fe14B requires a Hamiltonian which consists of crystal-field terms as well as of exchange terms. For a given rare earth atom this Hamiltonian may be expressed as... 

The intermolecular Hamiltonian (2.1) is supplemented with a one-particle external out-of-plane crystal-field term representing the interaction of each molecule with the underlying substrate... 

Substitution (101) appears to be the simplest way in which spin-dependent phenomena can be incorporated into the theory. The operator (100) commutes with S and is thus consistent with effects produced by such spin-independent interactions as the Coulomb and crystal-field terms in the Hamiltonian. Moreover, the effect of (101) turns out to be equivalent to adding to each reduced matrix element of F " a part (proportional to c,) that involves the reduced matrix element of the double tensor where = s (Judd 1977b). Such double tensors are straightforward to evaluate furthermore the proportionality = /4 holds for all terms of... 

To analyze the influence of high hydrostatic pressure on the emission of lanthanide ions, the semiempirical crystal-field approach is used where the total Hamiltonian of the 4f system is given by the sum of a free ion and the crystal-field terms as follows [113]. 

The magnetic anisotropy of RRh2Si2 (R = Nd, Tb, Ho, Er) which has an API type magnetic structure can be explained by using a R ion Hamiltonian which includes the crystal field term and the isotropic exchange term under the two-dimensional molecular field approximation, when the appropriate crystal field parameters are adopted. The crystal field parameters calculated are listed in table 20 (Takano et al. 1987a). 

We note that three spin-allowed electronic transitions should be observed in the d-d spectrum in each case. We have, thus, arrived at the same point established in Section 3.5. This time, however, we have used the so-called weak-field approach. Recall that the adjectives strong-field and weak-field refer to the magnitude of the crystal-field effect compared with the interelectron repulsion energies represented by the Coulomb term in the crystal-field Hamiltonian,... 

Here, Hcf is the crystal field spin Hamiltonian, which can be expressed by (neglecting the fourth order axial term)... 

The origin of zero-field terms in the spin Hamiltonian is rooted in crystal-field theory, in which coordination complexes are represented as geometric structures of point charges (Stevens 1997). The crystal-field potential of these point charges is... 

The crystal field model may also provide a calciflation scheme for the transition probabilities between levels perturbed by the crystal field. It is so called weak crystal field approximation. In this case the crystal field has little effect on the total Hamiltonian and it is regarded as a perturbation of the energy levels of the free ion. Judd and Ofelt, who showed that the odd terms in the crystal field expansion might connect the 4/ configuration with the 5d and 5g configurations, made such calculations. The result of the calculation for the oscillator strength, due to a forced electric dipole transition between the two states makes it possible to calculate the intensities of the lines due to forced electric dipole transitions. 

The direct dipole-dipole interaction between electron spins given in Eq. (14) can also contribute to D and E in the spin Hamiltonian. Various estimates of its contribution have shown it to be much smaller than the spin-orbit terms for transition-metal ions. For systems in which the crystal field is greatly distorted, this term can become large, however, and it is found to be the major source of D in the spin Hamiltonian of organic triplet-state molecules, where the spin-orbit terms are small as a result of the small size of the spin-orbit coupling parameter. 

The terms in the Hamiltonian that represent the non-spherical part of the interaction with the crystal are modeled using the so-called crystal-field Hamiltonian. It is important to recognize that this Hamiltonian is not restricted to electrostatic effects, which form a minor part of the total crystal-field effect (Ng and Newman, 1987). When the parameters are fitted to experimental energies, their values reflect all one-electron non-spherical interactions. The crystal-field Hamiltonian is expressed in Wyboume (1965) notation as,... 

The one-electron crystal field Hamiltonian does not take into account electron correlation effects. For some systems, it has been useful to augment the crystal field Hamiltonian with additional terms representing the two-electron, correlated crystal field. The additional terms most commonly used (see, for example, Peijzel et al., 2005b Wegh et al., 2003) are from the simplified delta-function correlation crystal field model first proposed by Judd (1978) that assumes electron interaction takes place only when two electrons are located at the same position (hence the name delta-function ). This simplified model, developed by Lo and Reid (1993), adds additional terms, given as,... 

The most common approach to the interpretation of EPR and Mossbauer spectra of siderophores is the spin Hamiltonian formalism. The wavefimctions are parameterized in terms of a few coupling constants that arise in the spin Hamiltonian description of the electronic states. In this approach, the crystal field potential is generally described by a series of spherical harmonics. The corresponding operators are tabulated. ... 

The energies of each of the (2/-i-1) crystal-field levels within a multiplet may be calculated by diagonalization of the full SL/Mj) matrix of a semiempirical effective Hamiltonian, H, when this is expanded to include both CF and CCF terms as in Eq. (28) [36,37] ... 

Finally, it has to be recalled that internal strains or crystal packing forces have to be present, in order to stabilize one of several equivalent minima so that the system is seen as statically distorted. It has been stated that even small low-symmetry perturbations can be amplified by vibronic coupling A mechanism of this kind may be more effective in lifting electronic degeneracies than low-symmetry ligand field components. The important point is whether JT contributions are greater or comparable with the low-symmetry ones if it is so, JT terms must be included in to the Hamiltonian and have to be considered before or together with the static low-symmetry crystal field. 

Another orbital triplet " Ti, which appears in the energy level schemes of the chosen ions, represents somewhat more sophisticated case. The matter is that there are two " Ti states arising from two different LS terms (" F and " P) of the d (d ) electron configurations. These states are mixed up by both crystal field Hamiltonian... 

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