Hyperfine interaction Hamiltonian

The underlying physics and analysis of Mossbauer spectra have been explained in detail in Chap. 4. In that chapter, the principles of how a spectrum is parameterized in terms of spin-Hamiltonian (SH) parameters and the physical origin of these SH parameters have been clarified. Many Mossbauer studies, mainly for Fe, have been performed and there is a large body of experimental data concerning electric-and magnetic-hyperfine interactions that is accessible through the Mossbauer Effect Database. 

Finally, the spin Hamiltonian also contains contributions from the magnetic and quadrupole hyperfine interactions, Hhf and Hq where... 

However, when it comes to the simulation of NFS spectra fi om a polycrystalline paramagnetic system exposed to a magnetic field, it turns out that this is not a straightforward task, especially if no information is available from conventional Mossbauer studies. Our eyes are much better adjusted to energy-domain spectra and much less to their Fourier transform therefore, a first guess of spin-Hamiltonian and hyperfine-interaction parameters is facilitated by recording conventional Mossbauer spectra. 

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 special case of isotropic g and hyperfine interaction will now be considered. This simplification is valid when the anisotropic interactions are averaged by rapid tumbling. The quadrupole interaction will be omitted because it is purely anisotropic. The resulting simplified spin Hamiltonian is given in Equation (9). 

The spin Hamiltonian for a biradical consists of terms representing the electron Zeeman interaction, the exchange coupling of the two electron spins, and hyperfine interaction of each electron with the nuclear spins. We assume that there are two equivalent nuclei, each strongly coupled to one electron and essentially uncoupled to the other. The spin Hamiltonian is ... 

An indirect mode of anisotropic hyperfine interaction arises as a result of strong spin-orbit interaction (174)- Nuclear and electron spin magnetic moments are coupled to each other because both are coupled to the orbital magnetic moment. The Hamiltonian is... 

We now single out one of these interactions for our discussion of the integer-spin 5 = 2 system, and we defer an explanation of this deliberate choice to the end of this section. We write the spin Hamiltonian for an isolated system (i.e., no interactions between paramagnets) with 5 = 2 and 7=0 (i.e., no hyperfine interactions) as... 

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. 

The Hamiltonian which describes the magnetic hyperfine interaction between a nucleus and its associated electrons in an atom can be written (26) as... 

The weak interaction contribution to hyperfine splitting is due to Z-boson exchange between the electron and muon in Fig. 6.7. Due to the large mass of the Z-boson this exchange is effectively described by the local four-fermion interaction Hamiltonian... 

The calculation of magnetic parameters such as the hyperfine coupling constants and g-factors for oligonuclear clusters is of fundamental importance as a tool for the evaluation of spectroscopic data from EPR and ENDOR experiments. The hyperfine interaction is experimentally interpreted with the spin Hamiltonian (SH) H = S - A-1, where S is the fictitious, electron spin operator related to the ground state of the cluster, A is the hyperfine tensor, and I is the nuclear spin operator. Consequently, it is... 

S=l. For =1, the general spin Hamiltonian without hyperfine interaction is... 

J. 5=f The general spin Hamiltonian, excluding hyperfine interactions, is... 

It is sometimes possible to obtain the parameters of the spin Hamiltonian from powders or frozen solutions. This method has been used primarily for 5= systems, which is the system we shall consider here. If the system has axial symmetry with no hyperfine interaction, the magnetic field is given by the equation... 

Since all six nuclei have the same hyperfine Hamiltonian, differing only in the orientation of the principal axes, we shall calculate the hyperfine terms for atom 5 in Fig. 15. Comparing matrix elements we find hyperfine interaction parallel and perpendicular to the bond axis to be... 

For the calculation of the EPR spectra, the basis functions were extended to include the four spin functions of copper nuclei, and the following expression for the nuclear hyperfine interaction added to the Hamiltonian ... 

The unpaired electron with its spin S = 1/2 in a sample disposed into the resonator of the EPR spectrometer interacts magnetically a) with the external magnetic field H (Zeeman interaction) b) with the nuclear spin of the host atom or metal ion / (hyperfine interaction) c) with other electron spins S existing in the sample (dipole-dipole interaction). In the last case, electrons can be localized either at the same atom or ion (the so called fine interaction), for example in Ni2+, Co2+, Cr3+, high-spin Fe3+, Mn2+, etc., or others. These interac-tions are characterized energetically by the appropriate spin-Hamiltonian... 

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