Hyperfine splitting Hamiltonian

In the Breit Hamiltonian in (3.2) we have omitted all terms which depend on spin variables of the heavy particle. As a result the corrections to the energy levels in (3.4) do not depend on the relative orientation of the spins of the heavy and light particles (in other words they do not describe hyperfine splitting). Moreover, almost all contributions in (3.4) are independent not only of the mutual orientation of spins of the heavy and light particles but also of the magnitude of the spin of the heavy particle. The only exception is the small contribution proportional to the term Sio, called the Darwin-Foldy contribution. This term arises in the matrix element of the Breit Hamiltonian only for the spin one-half nucleus and should be omitted for spinless or spin one nuclei. This contribution combines naturally with the nuclear size correction, and we postpone its discussion to Subsect. 6.1.2 dealing with the nuclear size contribution. 

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 nuclear Zeeman term describes the interaction of the nuclear spins with the external magnetic field. Just as the hyperfine splitting, this term is not incorporated in the original purely electronic Breit-Pauli Hamiltonian as presented in Eqs. (59) and (60) but becomes relevant for ESR spectroscopy. 

The operators so and ss are compound tensor operators of rank zero (scalars) composed of vector (first-rank tensor) operators and matrix (second-rank tensor) operators. We will make use of this tensorial structure when it comes to selection rules for the magnetic interaction Hamiltonians and symmetry relations between their matrix elements. Similar considerations apply to the molecular rotation and hyperfine splitting interaction... 

The resolution of the molecular beam experiments is high enough to observe even rather small nuclear hyperfine interactions such as the spin-spin and spin-rotation interactions as well as the larger quadrupole coupling interactions. The largest terms in the Hamiltonian for the hyperfine splittings are given below 66) ... 

The LMR spectra of this class of molecules provide accurate measurements of some rotational intervals and some hyperfine splittings. Independent measurements of the molecules in the 2 If 1/2 component are needed to provide a complete determination of the parameters in the effective Hamiltonian, such as the magnetic hyperfine parameters. 

Six rotational transitions in the v" = 0 level and three in the v" = 1 level were observed, the microwave frequency spanning the range 29 to 60 GHz. Although 19F hyperfine splittings were expected, the particular transitions necessary to determine the interaction constants were not observable. The experimental results were therefore fitted to the usual effective Hamiltonian,... 

Hyperfine interactions between the electron and any magnetic nuclei (7>0) present (such as a proton, for example) produces hyperfine splitting, as illustrated in a very simple example in Fig. 2.51. This hyperfine interaction may be divided into an isotropic and an anisotropic component. The isotropic part arises from unpaired electron density at the nucleus and can only be nonzero for i -type orbitals. The anisotropic term corresponds to the classical part of the magnetic dipole interaction for which the Hamiltonian is ... 

In zero magnetic field the two-pulse photon-echo decay exhibits a beat which is shown in Fig. 32. Such beat patterns may be expected for radicals where the hyperfine splitting in the ground and excited state is different, in other words, the spin Hamiltonians in these states do not commute. The spectrum was not further analyzed but it was concluded that the beat was... 

Extraction of the isotropic hyperfine splitting from an experimental spectrum, of an organic free radical in solution, is normally quite straightforward with the aid of the first order Hamiltonian,... 

The fine structure of atomic line spectra and the hyperfine splittings of electronic Zeeman spectra are non-symmetric for those atomic nuclei whose spin equals or exceeds unity, / > 1. The terms of the spin Hamiltonian so far mentioned, that is, the nuclear Zeeman, contact interaction, and the electron-nuclear dipolar interaction, each symmetrically displace the energy, and the observed deviation from symmetry therefore suggests that another form of interaction between the atomic nucleus and electrons is extant. Like the electronic orbitals, nuclei assume states that are defined by the total angular momentum of the nucleons, and the nuclear orbitals may deviate from spherical symmetry. Such non-symmetric nuclei possess a quadrupole moment that is influenced by the motion of the surrounding electronic charge distribution and is manifest in the hyperfine spectrum (Kopfer-mann, 1958). 

Nowik et al. (1972) solved the full hamiltonian for an ion in a cubic crystal field with an applied (external) magnetic field. This is done in the manner outlined in section 3.2 for the ESR of lanthanide ions, and the temperature dependence of the magnetic hyperfine field is then calculated from a Boltzmann average over all states as was done for the quadrupole hyperfine splitting in section 2.1.3. For YbPdj, the cubic crystal field parameters are found to be A4 = - 12 cm and = 0.6 cm . These values imply that the ground state is a r, doublet with the Fg quartet and Fg doublet at 29 cm" and 39 cm" above the ground state, respectively. 

Zeeman interaction Hamiltonian High frequency Hyperfine splitting High-pressure oxidative induction time... 

If magnetic and quadrupole interactions are simultaneously present, the treatment of hyperfine splitting may become quite complex since the choice of quantization axis is not a priori clear. One is forced to diagonalize the full Hamiltonian containing the sum of the two interactions. Clearly, m, is then no longer a good quantum number. The situation is characterized by the appearance of additional hyperfine transitions, which are forbidden according to the selection rules for pure ttij states. 

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