Zeeman-quadrupole Hamiltonian

The interpretation of a complex Mossbauer spectrum will obviously be simplified if the relative intensities of the various components are known. Once the energy levels of the Zeeman/quadrupole Hamiltonian have been calculated, and the spin quantum numbers for each state assigned (or appropriate linear combinations if the states are mixed), it is possible to calculate the intensities from the theory of the coupling of two angular momentum states [32, 33]. 

When the Zeeman interaction is much larger than the quadrupole interaction, nonsecular terms may be discarded to give a relatively simple expression for the quadrupole Hamiltonian ... 

If the electric quadrupole splitting of the 7 = 3/2 nuclear state of Fe is larger than the magnetic perturbation, as shown in Fig. 4.13, the nij = l/2) and 3/2) states can be treated as independent doublets and their Zeeman splitting can be described independently by effective nuclear g factors and two effective spins 7 = 1/2, one for each doublet [67]. The approach corresponds exactly to the spin-Hamiltonian concept for electronic spins (see Sect. 4.7.1). The nuclear spin Hamiltonian for each of the two Kramers doublets of the Fe nucleus is ... 

The leading term in T nuc is usually the magnetic hyperfine coupling IAS which connects the electron spin S and the nuclear spin 1. It is parameterized by the hyperfine coupling tensor A. The /-dependent nuclear Zeeman interaction and the electric quadrupole interaction are included as 2nd and 3rd terms. Their detailed description for Fe is provided in Sects. 4.3 and 4.4. The total spin Hamiltonian for electronic and nuclear spin variables is then ... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

The nuclear spin hamiltonian (H) for the Zeeman (Hz) and the quadrupole (Hq) interactions may be written... 

In the limit where the nuclear Zeeman term in the nuclear spin hamiltonian is much larger than the quadrupole interaction, it is only the secular part of Hq that contributes to the time-independent hamiltonian, H0. 

In the absence of an external magnetic field the Zeeman Hamiltonian provides zero energy and all the 11, Mi) levels (termed as / manifold) have the same energy. However, this may not be true for nuclei with / > V2. In this case, the non-spherical distribution of the charge causes the presence of a quadrupole moment. Whereas a dipole can be described by a vector with two polarities, a quadrupole can be visualized by two dipoles as in Fig. 1.11. 

The spin Hamiltonian used to model the deuterium ligand hyperfine interaction consisted of nuclear Zeeman, electron-nuclear hyperfine and nuclear quadrupole terms. 

In the high field limit, where the quadrupole interaction acts as a perturbation of the Zeeman states, the terms of this Hamiltonian which commute with L lead to the perturbation of first-order... 

It is desirable to apply fields of strong enough amplitude so that dominates all other interaction Hamiltonians except for the Zeeman interaction. The rf pulses can then be treated as infinitely short delta pulses, and the analysis of the experimental spectra becomes comparatively simple. However, arcing in the probe limits useful amplitudes to the order of 200 kHz, so that in solid-state NMR the delta-pulse approximation must be treated with care for the dipole-dipole interaction among protons, and it breaks down for the quadrupole interaction. 

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... 

In the following we will demonstrate how the effective Hamiltonian, Eq. (1.7), which will be discussed in more detail in the final Chapter, is used in practical spectroscopy. For this purpose we will discuss in detail the analysis of the Zeeman multiplets of an asymmetric top molecule with subsequent shorter sections on symmetric top molecules, linear molecules and molecules containing quadrupole nuclei. 

The nuclear interactions observed in ENDOR are three the nuclear Zeeman, electron-nuclear hyperfine, and (for I > i) the nuclear electric quadrupole interactions. The Hamiltonian H including these nuclear interactions in an external magnetic field B is given as... 

Summarizing, foiu different magnetic interactions may occiu, which influence the behavior of electrons in a magnetic field (a) the Zeeman interaction, Hu (b) the nuclear hyperfine interaction. Hup, (c) the dectrostatic quadrupole interaction, Hq and (d) the zero-field spHtting if S > V2> Hps- The sum of these interactions results in the total spin Hamiltonian, Hf. 

In Fig. 8 typical high resolution spectra of Cu + in dehydrated zeolites are shown. The corresponding ESR parameters (Table 2) are obtained by simulation with an axial spin Hamiltonian, including the Zeeman, the hyperfine and quadrupole interactions, applying second order perturbation. The spectra of Cu-ZSM-5 were obtained from the Hterature [25-29] and the parameters obtained by visual... 

Xemr can calculate (ESR) EPR transitions using the first order simulation or the solution of fully numerical spin Hamiltonian. In the latter case the numerical transition moments can also be calculated. The first order simulation is restricted to 5 = Vi and to electron Zeeman and hyperfine interaction whereas the numerical method can handle electron and nuclear Zeeman, hyperfine interaction, electron-electron interaction, and nuclear quadrupole interaction. The latter method can simulate both (ESR) EPR and ENDOR spectra. In addition a simple 1st order ENDOR simulation is also possible, so that the parameters can be extracted from the ENDOR spectra with better accuracy. 

The first three terms are usually the ones of relevance for the ESR analysis, where D and A are the zero-field (or fine structure) and hyperfine coupling tensors. They are represented by 3-3 symmetric matrices and specified by three principal values and three principal directions as for the -tensor. The remaining nuclear Zeeman and quadrupole (/ > Vi) terms do not affect the ESR spectra, unless they are of comparable magnitude to the hyperfine coupling, but must be taken into account in the analysis of ENDOR and ESEEM spectra. The spin Hamiltonian formalism introduced by M.H.L. Pryce and A. Abragam [79] is used explicitly or implicitly in the ESR literature as a convenient way to summarise resonance parameters. 

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