Spin-dipole hyperfine term

In addition to the isomer shift and the quadrupole splitting, it is possible to obtain the hyperfine coupling tensor from a Mossbauer experiment if a magnetic field is applied. This additional parameter describes the interactions between impaired electrons and the nuclear magnetic moment. Three terms contribute to the hyperfine coupling (i) the isotropic Fermi contact, (ii) the spin—dipole... 

These results will be able to give unambiguous determinations of the absolute values of the deuteron hyperfine constants in the upper and lower states. As a first approximation we have ignored the smaller dipole-dipole and spin-rotation constant terms in the Hamiltonian, and the differences between different rotational levels. This gives an estimate for the deuteron Fermi contact parameter b > in v=16 of llOMHz and in v=18 of 106MHz. These are close to the values calculated by Carrington and Kennedy (22). 

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)... 

An example of such a solution spectrum was given in Fig. 1. The dipole contribution to the hyperfine interaction drops out in Eq. (121) leaving the solution A to be primarily determined by the contact term which measures the amount of s character in the spin-moment distribution. 

The magnetic hyperfine splitting (MHS) depends on the nuclear spin quantum numbers and /g of the excited and ground state of the Mossbauer nucleus and on the effective magnetic field at the Mossbauer nucleus, which includes contributions from the local electronic spin, from the orbital momentum, from dipole terms, and from external fields. 

Results for dipole moments are shown in Table 29. ° Clearly, coupled with good basis sets, correlated methods can do quite well for this property. Elsewhere, we have used relaxed density-based CC and MBPT methods to study spin densities and the related hyperfine coupling constants to evaluate relativistic corrections (Darwin and mass-velocity term) when impor-tant and to evaluate highly accurate electric field gradients to extract nuclear quadruple moments. 

Anisotropic Hyperfine Interaction. The anisotropic component of the hyperfine coupling has two contributions a local anisotropy owing to spin density in p- or type orbitals on the atom of observation, and nonlocal dipolar coupling with spin on other atoms. The first type of interaction is proportioned to the orbital coefficient (squared) of the pid orbiteds. To a first approximation the second term can be considered as a classic point dipolar interaction between the nucleus and the electron spin on a nearby atom. This depends on the total electron spin density at the neighbor (p ), the distance between the spins (r,2), and the orientation of the vector between them with respect to the external magnetic field (denoted by angle 0). In the point dipole approximation,... 

For the predominating magnetic dipole term in the hyperfine interaction and for half-integer spin / > 1, as usually encountered in magnetically ordered rare-earth intermetallic compounds, 2/ -1 quadrupole satellites of the main (Zeeman) resonance (m = -I- are observed with frequency separations of... 

The first term results from the Fermi contact interaction, while the second represents the long-range dipole-dipole interaction. In the equations above, ge is the free-electron g factor, /Xe the Bohr magneton, gi the nuclear gyromagnetic ratio, and /xi the nuclear moment. Moreover, the nucleus is located at position R, and the vector r has the nuclear position as its origin. Finally, p (r) = p (r) — p (r) is the electron spin density. The only nontrivial input into these equations is precisely this last quantity, i.e. Ps(r), which can be computed in the LSDA or another DFT approximation. The resulting Hamiltonian can be used to interpret the hyperfine structure measured in experiments. A recent application to metal clusters is reported in Ref. [118]. 

The hyperfine structure of the ESR spectrum is used to identify trapped radicals. This is due to the interaction of unpaired electrons with magnetic moments of the nuclei surrounding unpaired electrons localized at the radical site. Hyperfine interaction is described by two terms. One is the isotropic interaction, a, originating from the contact interaction of the wave functions of electron- and nuclear spins, and the other is the anisotropic interaction, D, originating from the magnetic dipole-dipole interaction. The total hyperfine interaction, a -f D, is therefore essentially anisotropic and can also be expressed in the form of a tensor A. As in the case of g-tensor, A can be described by three principal values. A, Aj, and A3, corresponding to the principal axes in a proper molecular coordinate system. The magnitude of the isotropic term of hyperfine interaction depends on an overlap of the wave packets of the unpaired electrons and nuclei in the case of the hyperfine interaction of an unpaired electron with a P-proton (see Fig. 2.2), McConnell s relation [Eq. (2.3)] has been derived... 

The first term is the Fermi contact interaction and is only operable for s electrons. The second term is due to the orbital current The third term represents the dipole field due to the electron spin. These two latter terms are generally smaller than the contact term and vanish for s-state ions. For Fe in Fe " (S — 5/2, L — 0), the contact interaction gives about —60 T. For Fe in Fe " (5 = 2, L — 2), the field is somewhat smaller because of smaller spin and also appreciable positive orbital contribution. At room temperature the hyperfine magnetic field at Fe in metallic iron is —33 T and this is the reference value to determined the hyperfine magnetic field in magnetic materials using Fe Mbssbauer spectroscopy. Nuclear levels of Fe under magnetic field and the expected Mbssbauer spectrum are shown in Fig. 1.6c. 

The first term in Eq. 10 is given by Eq. 4 and the second is the nuclear Zeeman energy. I, is the nuclear spin operator for nuelear spin 1/2. The third term is the hyperfine interaction of the individual proton i, as defined by the hyperfine tensor At. S is the total electron spin operator, S = Si -I- 2. As nuclear dipole-dipole interaction can be neglected we will omit the index i in the following. 

The most important features of EPR spectra are their hyperfine structure, the splitting of individual resonance lines into components. In general in spectroscopy, the term hyperfine structure means the structure of a spectrum that can be traced to interactions of the electrons with nuclei other than as a result of the latter s point electric charge. The source of the hyperfine structure in EPR is the magnetic interaction between the electron spin and the magnetic dipole moments of the nuclei present in the radical. 

The components of the hyperfine tensor A in Eq. (12) comprise an anisotropic (dipolar) and an isotropic part as written in frequency units in Eq. (13). The term dy denotes the Kronecker delta function h in Eq. (13) is Planck s constant. The distance r and location (r ,ry) with respect to the center of spin density in the point-dipole approximation, as well as the isotropic coupling, determine the ENDOR response in the case of unit spin density. If a spin density p is distributed over various spin centres with p =, the components of the dipolar part of the tensor... 

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