Angular momentum nuclei interaction terms

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

In both of the above treatments, spherical tensor and cartesian, we have factored the quadrupole interaction into the product of two terms, one of which operates only on functions of proton coordinates within the nucleus and the other only on functions of coordinates of electrons and protons outside the nucleus. We shall see in subsequent chapters that the spherical tensor form is rather more convenient for the calculation of matrix elements of 3Cq. However, we shall find this easier to appreciate once we have considered some of the theory of angular momentum in chapter 5 so we defer discussion until later. 

In this Hamiltonian (5) corresponds to the orbital angular momentum interacting with the external magnetic field, (6) represents the diamagnetic (second-order) response of the electrons to the magnetic field, (7) represents the interaction of the nuclear dipole with the electronic orbital motion, (8) is the electronic-nuclear Zeeman correction, the two terms in (9) represent direct nuclear dipole-dipole and electron coupled nuclear spin-spin interactions. The terms in (10) are responsible for spin-orbit and spin-other-orbit interactions and the terms in (11) are spin-orbit Zeeman gauge corrections. Finally, the terms in (12) correspond to Fermi contact and dipole-dipole interactions between the spin magnetic moments of nucleus N and an electron. Since... 

For many electron systems, QED corrections must also include many-body contributions. For the time being only a limited number of results, besides semi-empirical extrapolations, are available for heavy elements where a perturbative Za approach (in terms of the electron nucleus interaction) is irrelevant. The reason is not only that the most precise numerical methods developed for the one-electron contributions [34] encounter serious numerical accuracy problems for high angular momentum values but also that, even for two-electron atoms or ions, the standard QED prescription [35] is unable to deal with quasi-degenerate levels. Recent developments [36-37] open new perspectives for getting accurate estimates in two-electron systems without any restriction on the nuclear charge. 

In term Hs the spin-spin relates to the electron-electron interaction because the helium nucleus has spin angular momentum of 0. 

The presence of unquenched orbital and rotational components of angular momentum in a free radical necessitates the introduction of several additional terms into the Hamiltonian for the energy of the molecule in a magnetic field. In the case of diatomic free radicals, Frosch and Foley have shown that four magnetic hyperfine coupling constants a, b, c and d are needed to describe the interaction between a nucleus and the electron, where... 

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). 

P is the Bohr magnetone, g is the electronic g-factor tensor and A is the tensor which accounts both for the isotropic and the anisotropic hyperfine interaction of the unpaired electron spin with the nucleus i. and are the operators of the electronic and the nuclear angular momentums, respectively. The vectors andl are most conveniently expressed in terms of a space fixed coordinate system U V, W (Fig. 13). 

The mass-velocity term is therefore the lowest-order term from the relativistic Hamiltonian that comes from the variation of the mass with the velocity. The second relativistic term in the Pauli Hamiltonian is called the Darwin operator, and has no classical analogue. Due to the presence of the Dirac delta function, the only contributions for an atom come from s functions. The third term is the spin-orbit term, resulting from the interaction of the spin of the electron with its orbital angular momentum around the nucleus. This operator is identical to the spin-orbit operator of the modified Dirac equation. 

This table contains the data obtained from the magnetic hyperfine structure and the Zeeman effect for molecules in a Z state or more generally in a state with Q = 0, i.e., the projection of the angular momentum onto the molecular axis is zero. For the magnetic hyperfine stracture one usually considers four terms the spin-rotation interaction for each nucleus and the scalar and tensorial spin-spin interaction of the two nuclear spins. For the Zeeman effect one takes into account the rotational Zeeman effect, the nuclear Zeeman effect with the scalar and tensorial shielding, and the scalar and tensorial magnetic susceptibility. The hamiltonian of these interactions can be written with the concept of spherical tensor operators [57Edm]... 

A part of the relativistic corrections due to a heavy neighboring atom can be approximated by third-order perturbation theory with nonrelativistic functions. The contribution to shielding comes from the cross term involving the spin-orbit coupling, I S interaction, and the external field-orbit interaction. In nonrelativistic terms, the shielding mechanism is that the external field induces an orbital angular momentum on the electrons of the heavy atom (Br or I), this produces a polarization in the electron spin by spin-orbit coupling, and the spin polarization is transferred to the resonant nucleus by a Fermi contact and by a nuclear spin-electron spin dipolar... 

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