Fermi contact hyperfine term

More subtle are the spin-orbit-induced heavy-atom chemical shifts at the atom nearest to the heavy one, or at more remote nuclei. If one uses the same Hamiltonians as Ramsey, one must go to third-order perturbation theory, with one Zeeman, one hyperfine, and one spin-orbit matrix element [22]. For a recent discussion on the nature of this shift, see Ref. [23]. It was also noted that an analogous effect, a heavy-atom shift on the heavy atom can occur, for instance on the Pb(II) nucleus in PbR compounds. The early semiempirical calculations suggested that the Zeeman-SO-Fermi contact cross term, zero in Ramsey s theory, could then become the dominant contribution to the Pb chemical shift [24]. 

The hyperfine structure is described by the isotropic Fermi contact interaction term (75), which for the negative ion of naphthalene can be written in the form ... 

The calculated Fermi contact hyperfine field, Bhf,c, calculated for 1 ML of Fe on W(llO) was decomposed into core-Bhfcp and conduction-Bhf,ce electron contributions. The core electrons contribute to Bhf.cp vvith a large negative value of — 30.6T, which scales exactly with the magnetic moment. The conduction electrons contribute to Bhf,ce with a positive value of 15.8 T (because of direct polarization) and greatly reduce the magnitude of the total Fermi contact term. 

It is well-known that the hyperfine interaction for a given nucleus A consists of three contributions (a) the isotropic Fermi contact term, (b) the spin-dipolar interaction, and (c) the spin-orbit correction. One finds for the three parts of the magnetic hyperfine coupling (HFC), the following expressions [3, 9] ... 

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... 

The hyperfine coupling tensor (A) describes the interaction between the electronic spin density and the nuclear magnetic momentum, and can be split into two terms. The first term, usually referred to as Fermi contact interaction, is an isotropic contribution also known as hyperfine coupling constant (HCC), and is related to the spin density at the corresponding nucleus n by [25]... 

This hyperfine coupling is of two kinds An isotropic interaction arises from the possibility that the electronic wave-function, x , be non-zero at the nucleus, N. This is the Fermi contact term and the hyperfine coupling constant is given by ... 

The hyperfine coupling frequency in the hydrogen ground state is given to the leading term by the Fermi contact interaction, yielding... 

This interaction leads to hyperfine splittings in atomic and molecular spectra. One particular term for this interaction is the Fermi contact term, which dominates chemical shifts in nuclear magnetic spectra and splittings in electron paramagnetic spectra ... 

Chemical Shifts in NMR. The first effect is very useful in chemical analysis The nuclear spin transitions are affected by the "hyperfine" I S coupling between electron spin S (for any single electron in the molecule that has density at the nuclear position) and nuclear spin I this is due to the isotropic Fermi contact term ... 

The remaining important interactions which can occur for a 2 or3X molecule involve the presence of nuclear spin. Interactions between the electron spin and nuclear spin magnetic moments are called hyperfine interactions, and there are two important ones. The first is called the Fermi contact interaction, and if both nuclei have non-zero spin, each interaction is represented by the Hamiltonian term... 

The first-order contribution of these hyperfine interactions to the effective electronic Hamiltonian involves the diagonal matrix elements of the individual operator terms over the electronic wave function, see equation (7.43). As before, we factorise out those terms which involve the electronic spin and spatial coordinates. For example, for the Fermi contact term we need to evaluate matrix elements of the type ... 

There are three separate contributions to the total magnetic hyperfine interaction, namely, the Fermi contact term, the orbital hyperfine term, and the electron spin-nuclear spin dipolar term ... 

The magnetic hyperfine interaction is represented by the sum of two terms, representing the Fermi contact interaction, and 3Qiip representing the electron spin-nuclear spin dipolar interaction. They are written as follows ... 

We come now to the magnetic hyperfine interaction terms. Again we make use of the results derived earlier for ortho-H2 in the c 3 nu state. For the Fermi contact interaction, from equation (8.220) ... 

It is instructive to consider in quantitative detail the analysis of a particular hy-perfine transition a simple example would seem to be the F = 2 <+ 1 transition in the level It = 1, N = 0, J = 1, which is observed at a frequency of 20.846 MHz for the v = 0 level. The expressions for the matrix elements of the magnetic and electric hy-perfine terms, (8.258), (8.259) and (8.270), show that for N =0 only the Fermi contact interaction is non-zero and the energies of the hyperfine levels are... 

The next term in the magnetic hyperfine Hamiltonian (8.351) describes the Fermi contact interaction and the calculation of its matrix elements proceeds in a manner similar to that just described for the orbital hyperfine term, as follows ... 

Freund, Herbst, Mariella and Klemperer [112] expressed their magnetic hyperfine constants in the form originally given by Frosch and Foley [117]. As discussed elsewhere in this book, particularly in chapters 9, 10 and 11, we prefer to separate the different physical interactions, particularly the Fermi contact and dipolar interactions, in our effective Hamiltonian. This separation is usually made by other authors even when the effective Hamiltonian is expressed in terms of Frosch and Foley constants, because it is the natural route if the molecular physics of a problem is to be understood. Nevertheless since so many authors, particularly of the earlier papers, use the magnetic hyperfine theory presented by Frosch and Foley, we present in appendix 8.5 a detailed comparison of their effective Hamiltonian with that adopted in this book. The merit of the Frosch and Foley parameters is that they form the linear combination of parameters which is best determined (i.e. with least correlation) for a molecule which conforms to Hund s case (a) coupling. The values of the constants determined experimentally from the 7 LiO spectrum were therefore, in our notation (in MHz) ... 

The constant b therefore contains contributions from two quite different magnetic interactions, the Fermi contact and the electron-nuclear dipolar interactions. Interpretation of the magnitudes of these constants in terms of electronic structure theory always involves the separate assessment of these different effects, so that we prefer to use an effective Hamiltonian which separates them at the outset. Consequently the effective magnetic hyperfine Hamiltonian used throughout this book is... 

The orbital hyperfine constant a is the same in both (8.509) and (8.510), but the second term in (8.510) describes only the Fermi contact interaction, the constant hp being equal to the first part of b in (8.509). It is clear from a comparison of the two equations that... 

We turn now to the magnetic hyperftne Hamiltonian in (9.30) which may be written as the sum of three terms representing the orbital, Fermi contact and dipolar hyperfine... 

The axial component of the magnetic hyperfine interaction for the 2 n 3/2 component is designated //3/2 in terms of the original Frosch and Foley constants [25] h n is equal to a + (1 /2)(b + c), and in terms of our preferred hyperfine constants it is a + (l/2)(fa + 21), the latter constants describing the orbital, Fermi contact and dipolar hyperfine interactions separately. Specifically, our constants are given by,... 

The magnetic hyperfine terms are now familiar, representing the Fermi contact interaction and axial dipolar interaction ... 

As we have shown in Appendix 8.5, and elsewhere, to is the axial component of the dipolar interaction obtained from the fourth term in equation (11.2). The large value of the Fermi contact constant is consistent with a model in which the unpaired electron occupies a a-type molecular orbital which has 45% N atom, v character. Radford produced convincing arguments to show that the model is also consistent with the small dipolar hyperfine constant, and also the electric quadrupole coupling constant. 

Note that, as in other cases, complete separation of the Fermi contact and dipolar hyperfine terms cannot be achieved because only one of the fine-structure states, the 2n1/2, could be studied thoroughly. 

The fourth and last terms in (11.49) are spin-orbit distortion corrections to the spin-rotation and Fermi contact interactions. The hyperfine and quadrupole terms in this Hamiltonian refer to the 14N nucleus. 

The contact interaction is also referred to as the Fermi contact term. In a given atomic orbital basis -0, the isotropic hyperfine coupling constant (hfcc) for a particular nucleus N,, is given by the expression,... 

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