Hamiltonian, nuclear quadrupole

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

The spin Hamiltonian for the nuclear quadrupole interaction has the form2,149,150) ... 

The expressions for the various parts of the Hamiltonian (equation 1) are well documented and for our purpose and the following discussion it suffices to summarize the results for axially symmetric situations in angular frequency units with the equations 2-6, where and Ashielding tensor and the shielding anisotropy, respectively, D is the dipole coupling, eq or V is the electric field gradient at the nucleus, eQ is the nuclear quadrupole moment and the other symbols have their usual meaning ... 

The interaction between a nuclear quadrupole moment eQ and the electric field gradient q at that nucleus gives a term in the Hamiltonian... 

With pure quadrupole resonance experiments one observes the transitions between the energy levels of the nuclear quadrupole coupling directly. These energy levels may be obtained from the quadrupolar hamiltonian ... 

The most important terms in the effective hyperftne Hamiltonian are those which describe the nuclear quadrupole and nuclear spin-rotation interactions ... 

The spin Hamiltonian used to model the deuterium ligand hyperfine interaction consisted of nuclear Zeeman, electron-nuclear hyperfine and nuclear quadrupole terms. 

In microwave spectroscopy, pure rotational transitions are studied. If one or more nuclei in the molecule has a nuclear quadrupole moment, the quadrupole Hamiltonian [Eq. (6)] has to be included in the quantum mechanical treatment because the field gradient q [Eq. (5)] is dependent on the rotational wave function. The nuclear quadrupole interaction, which causes the rotational transitions to split into hyperfine structure, can usually be treated as a perturbation to the rotational Hamiltonian. 

Nuclear magnetic resonance (NMR) is perhaps the simplest technique for obtaining deuterium quadrupole coupling constants in solids or in liquid crystalline solutions. In ordinary NMR experiments with a magnetic field Hq > 104 gauss, the nuclear quadrupole interaction [Eq. (6)1 for deuterium is much smaller than the Zeeman interaction and can be treated as a perturbation to the Hamiltonian... 

The quadrupolar nature of deuterium is due to the nonspherical charge distribution at the nucleus, caused by the presence of the neutron next to the proton. The quadrupolar Hamiltonian Hq arises from the electrostatic interaction of the nuclear quadrupole moment with the electric field gradient... 

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... 

In this section, we attempt to review the solid-state Co NMR data published up to the end of 1998. The data on chemical shift tensors and quadrupole coupling constants are summarized in Tables 1-4. Data obtained by nuclear quadrupole resonance (NQR) are included wherever appropriate. Readers interested in the spin-spin coupling between spin-1 /2 and Co nuclei are referred to an article by Wasylishen and colleagues and the references cited therein. The NMR conventions employed in this review are defined in the next section. This is followed by a discussion of the techniques that have been applied to solid-state Co NMR studies with emphasis on what has been applied to Co systems. A detailed discussion on the techniques and Hamiltonians for quadrupolar nuclei is referred to the comprehensive review by Freude and Haase. 

The Hamiltonian which describes the interaction between the nuclear quadrupole moment and an electric field gradient is given by (7, 54, 58, 60)... 

Xemr can calculate (ESR) EPR transitions using the first order simulation or the solution of fully numerical spin Hamiltonian. In the latter case the numerical transition moments can also be calculated. The first order simulation is restricted to 5 = Vi and to electron Zeeman and hyperfine interaction whereas the numerical method can handle electron and nuclear Zeeman, hyperfine interaction, electron-electron interaction, and nuclear quadrupole interaction. The latter method can simulate both (ESR) EPR and ENDOR spectra. In addition a simple 1st order ENDOR simulation is also possible, so that the parameters can be extracted from the ENDOR spectra with better accuracy. 

A more general spin Hamiltonian of the form (3.9) is applied when zero-field splittings (S > V2), anisotropic hyperfine interactions (/ 0), and nuclear quadrupole couplings (/ > 1) occur. 

We defer writing the explicit form of Eq. (26) for specific cases until later. However, it is important to note the similarity of Eqs. (23)-(26) to Eqs. (11)-(13). The nuclear quadrupole Hamiltonian is identical in form to the... 

The interaction between a nonzero nuclear quadrupole moment and a surrounding nonspherical distribution of electric charges, as measured by the electric field gradient at the nucleus, gives rise to a quadrupole interaction. This hyperfine interaction, which also depends upon a nuclear and an electronic factor, is described by the Hamiltonian... 

EFG (a property of a sample) is called the qnadrupole interaction. The nuclear quadrupole coupling is expressed by the Hamiltonian... 

---