Interaction Hamiltonian electric quadrupole

The effective Hamiltonian contained terms for both nuclei describing the orbital hy-perfine interaction, the electric quadrupole interaction and the nuclear spin-rotation interaction ... 

The Hamiltonian operator for the electric quadrupole interaction, 7/q, given in (4.29), coimects the spin of the nucleus with quantum number I with the EFG. In the simplest case, when the EFG is axial (y = Vyy, i.e. rf = 0), the Schrddinger equation can be solved on the basis of the spin functions I,mi), with magnetic quantum numbers m/ = 7, 7—1,. .., —7. The Hamilton matrix is diagonal, because... 

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 low-temperature Mossbauer spectra of the spinel type oxides, NiCr204 [14,18] (Fig. 7.6b) and NiFe204 [3, 18], have been found to exhibit combined magnetic dipole and electric quadrupole interaction (Fig. 7.7). For the evaluation of these spectra, the authors have assumed a small quadrupolar perturbation and a large magnetic interaction, as depicted in Fig. 7.3 and represented by the Hamiltonian [3]... 

The nuclear spins give rise to additional terms in the Breit-Pauli Hamiltonian due to the interaction of the electrons with the magnetic moment of the nuclei and the electrostatic interaction with the electric quadrupole interaction of the nuclei. The magnetic interaction term of the spins with the nuclei is of the same type as the spin-spin interaction and following Abragam and Pryce (61) can be written as... 

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. 

We recall from chapter 4 that there are many individual types of interaction which involve the nuclear magnetic dipole and electric quadrupole moments. Let us take just three of these to exemplify how the effective Hamiltonian is constructed. They are as follows ... 

The form of the nuclear electric quadrupole interaction in the effective Hamiltonian for a diatomic molecule is given in equations (7.158) and (7.161), with the latter applying only to molecules in n electronic states. The two parameters which can be determined from a fit of the experimental data are eqo Q and et/i Q respectively. Since the electric quadrupole moment eQ is known for most nuclei, an experimental observation gives information on q0 (and perhaps qi), the electric field gradient at the nucleus. This quantity depends on the electronic structure of the molecule according to the expression... 

The CN radical in its 21 ground state shows fine and hyperfine structure of the rotational levels which is more conventional than that of CO+, in that the largest interaction is the electron spin rotation coupling../ is once more a good quantum number, and the effective Hamiltonian is that given in equation (10.45), with the addition of the nuclear electric quadrupole term given in chapter 9. The matrix elements in the conventional hyperfine-coupled case (b) basis set were derived in detail in chapter 9,... 

In addition to the interactions which broaden the sohd-state NMR resonances of spin-1/2 nuclei, the electric quadrupole interaction considerably decreases the resolution of the spectra of quadrupolar nuclei (spin />l/2). These nuclei have a non-spherically symmetrical distribution of nuclear charge and possess an electric quadrupole moment, which interacts strongly with any electric field gradient, created by the surrounding electron cloud, at the site of the nucleus. In the tensorial Cartesian form the quadrupolar Hamiltonian is... 

Since atomic nuclei are not perfectly spherical their spin leads to an electric quadrupole moment if I>1 which interacts with the gradient of the electric field due to all surrounding electrons. The Hamiltonian of the nuclear quadrupole interactions can be written as tensorial coupling of the nuclear spin with itself... 

The nuclear interactions observed in ENDOR are three the nuclear Zeeman, electron-nuclear hyperfine, and (for I > i) the nuclear electric quadrupole interactions. The Hamiltonian H including these nuclear interactions in an external magnetic field B is given as... 

The Pr ground state nuclear spin levels consist of three levels split by the second-order hyperfine interaction and the relatively small nuclear electric quadrupole interaction. In non-zero magnetic fields each of these levels is additionally split due to the enhanced Zeeman interaction (see Fig. 4), The Hamiltonian of this system has been given by Teplov. ... 

Hyperfine levels for the hyperfine field Hamiltonian of a uniaxial magnetic field with a collinear axial electric quadrupole interaction. 

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