Magnetic hyperfine Hamiltonian

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 ... 

Appendix 8.5. Magnetic hyperfine Hamiltonian and hyperfine constants... 

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... 

Finally we add the magnetic hyperfine Hamiltonian which is required to analyse the proton hyperfine structure. It is given, in the molecule-fixed axis system, as [76]... 

The main additions to the theory for ortho-H2 are, of course, the magnetic hyperfine terms. These were discussed with particular reference to the c3 flu state of ortho-H2 in chapter 8, the work of Jette and Cahill [30] being particularly important. The magnetic hyperfine Hamiltonian is usually written as the sum of three terms,... 

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

The leading term in T nuc is usually the magnetic hyperfine coupling IAS which connects the electron spin S and the nuclear spin 1. It is parameterized by the hyperfine coupling tensor A. The /-dependent nuclear Zeeman interaction and the electric quadrupole interaction are included as 2nd and 3rd terms. Their detailed description for Fe is provided in Sects. 4.3 and 4.4. The total spin Hamiltonian for electronic and nuclear spin variables is then ... 

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. 

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

Fd n can be studied in two oxidation states. In the oxidized state the cluster has electronic spin S = H. This spin results fiom antiferromagnetic coupling of three high-spin ferric (Si = 2 = S3 = 5H) iron sites. The magnetic hyperfine parameters obtained from an analysis of the low tempo ture MSssbauer spectra have been analyzed (18) in the frmiework of the Heisenberg Hamiltonian. 

Even in fairly small applied magnetic fields, say B = 20 mT, the terms of He are much larger than the hyperfine terms. This implies that the expectation value of the electronic spin, (S, alSIS, a) = (S), is determined by He. Under these circumstances, we can replace the spin operator S in the magnetic hyperfine term by its expectation value, (S), obtaining from Hm the nuclear Hamiltonian Hn... 

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 most important examples of 2S states to be described in this book are CO+, where there is no nuclear hyperfine coupling in the main isotopomer, CN, which has 14N hyperfine interaction, and the Hj ion. A number of different 3E states are described, with and without hyperfine coupling. A particularly important and interesting example is N2 in its A 3ZU excited state, studied by De Santis, Lurio, Miller and Freund [19] using molecular beam magnetic resonance. The details are described in chapter 8 the only aspect to be mentioned here is that in a homonuclear molecule like N2, the individual nuclear spins (1 = 1 for 14N) are coupled to form a total spin, It, which in this case takes the values 2, 1 and 0. The hyperfine Hamiltonian terms are then written in terms of the appropriate value of h As we have already mentioned, the presence of one or more quadrupolar nuclei will give rise to electric quadrupole hyperfine interaction the theory is essentially the same as that already presented for1 + states. 

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 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. 

Terms describing the Zeeman and magnetic hyperfine interactions must be added to (9.172). The complete Zeeman Hamiltonian for a molecule in a 25+1A state is given by Nelis, Beaton, Evenson and Brown [76] as... 

The theory of the magnetic hyperfine interactions in NCI is essentially the same as that already described for the PF radical in the previous section, except that the nuclear spins / are 1 for 14N and 3/2 for 35C1. The form of the effective Hamiltonian for the quadrupole interaction and its matrix elements for two different quadrupolar nuclei was described in some detail in chapter 8 when we discussed the electric resonance spectra of CsF and LiBr. We now use the same case (b) hyperfine-coupled basis set as was used for PF. The quadrupole Hamiltonian for the two nuclei can be written as the sum of two independent terms as follows ... 

In addition, the93 Nb magnetic hyperfine coupling was large enough to be readily observed, even in the electronic spectrum. It was fitted to the normal Frosch and Foley Hamiltonian for a L state,... 

Figure 11.47 is based upon the final analysis, to which we will return. The effective Hamiltonian for the magnetic hyperfine and spin-rotation interactions may be written as usual, except that we adopt the alternative formulation of the dipolar interactions which is more appropriate for our coupling scheme ... 

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