Nuclear quadrupole terms

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

The small random crystal strains can be quantified and this is also given in Table 5. It was found that the sttain can be described by a Gaussian distribution characterised by a mean value, 8, of zero and a half width of 8a = 2cm . The analysis also differed from that of previous workers in both the hyperfine values and the requirement of a nuclear quadrupole term. The transitions within the lowest excited singlet could also be observed directly [31]. It can be concluded that the Cu(ll)/MgO system can be described as an almost pure dynamic Jahn-Teller case. 

A. Fine-Structure Terms Jfps Electron Zeeman Terms Electron-Nuclear Hyperfine Terms jP f Nuclear Quadrupole Terms J q Nuclear Zeeman Terms III. Zero-Field Experiments A. Selection Rules... 

It should be noticed that there are no nuclear quadrupole terms in the above formulae, as these arise only when the nucleus is considered as a structured particle. Such terms are important in, for example, the consideration of relaxation phenomena in solids. Their forms are discussed elsewhere (e.g. Abragam, 1%1 Slichter, 1964). Briefly, the extra term required is... 

Terms up to order 1/c are normally sufficient for explaining experimental data. There is one exception, however, namely the interaction of the nuclear quadrupole moment with the electric field gradient, which is of order 1/c. Although nuclei often are modelled as point charges in quantum chemistry, they do in fact have a finite size. The internal structure of the nucleus leads to a quadrupole moment for nuclei with spin larger than 1/2 (the dipole and octopole moments vanish by symmetry). As discussed in section 10.1.1, this leads to an interaction term which is the product of the quadrupole moment with the field gradient (F = VF) created by the electron distribution. 

On the basis of the point-charge model formalism, applied on the experimental nuclear quadrupole splitting rationalization, I Agxp I, the results obtained were interpreted in terms of strong complex formation by either Me2Sn(OH)2 or Me3Sn(0H)(H20) with (n = 1 or 2, obtained in phosphate buffer) and... 

In contrast, the second term in (4.6) comprises the full orientation dependence of the nuclear charge distribution in 2nd power. Interestingly, the expression has the appearance of an irreducible (3 x 3) second-rank tensor. Such tensors are particularly convenient for rotational transformations (as will be used later when nuclear spin operators are considered). The term here is called the nuclear quadrupole moment Q. Because of its inherent symmetry and the specific cylindrical charge distribution of nuclei, the quadrupole moment can be represented by a single scalar, Q (vide infra). 

The shape of the nucleus is best described by a power series, the relevant term of which yields the nuclear quadrupole moment. In Cartesian coordinates, this is represented by a set of intricate integrals of the type J p (r)(3x,x, — 6-jr )Ax, where x, = x, y, z, and pfifi) is the nuclear charge distribution (4.12). The evaluation of Pn(r) for any real nucleus would be very challenging. 

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)... 

In the first row of (3.1) the terms denote the electron Zeeman (2 EZ), the hf (2 hft), the nuclear Zeeman (XNZ) and the nuclear quadrupole interaction (CXQ) of the central (metal) ion. The second row represents the hf, the nuclear Zeeman and the nuclear quadrupole interactions for sets of magnetically equivalent ligand nuclei. Each particular set is denoted by the index k, the individual nuclei of set k by kx. 

The nuclear quadrupole tensor of the Cu(II) ion is discussed in terms of the formalism outlined in Sect. 5.2.1. Kita et al.158) treated the Stemheimer antishielding factor (1 - as an adjustable parameter. The rhombicity of the measured quadrupole tensor was found to originate from the charge distribution on the ligands, the field gradient due to the copper valence electrons being nearly axially symmetric. 

The measured Q.S. values for the surface sites of AU55 give nuclear splittings of 21 mK, 13 mK, and 4 mK for the PPhj coordinated sites, the Cl coordinated sites, arid the bare surface sites respectively. From these values, and the known site occupations, the nuclear quadrupole contribution to the zero field specific heat by these three two-level systems [143] has been calculated directly [144], This value is 5 times as large as that experimentally observed [54]. The maximum value of a linear term in the specific heat of AU55 has been estimate to be no more than one fifth of the bulk value [144]. 

The coordination numbers based on this structure work extremely well for describing the microscopic physical properties of this material, including the Mossbauer I.S.s of the surface sites and of the specific heat of the clusters below about 65 K. No linear electronic term in the specific heat is seen down to 60 mK, due to the still significant T contribution from the center-of-mass motion still present at this temperature. The Schottky tail which develops below 300 mK in magnetic fields above 0.4 T has been quantitatively explained by nuclear quadrupole contributions. 

For high accuracy, it is necessary to add the nuclear-Zeeman and electric-quadrupole terms... 

Bonding in linear dihalogenocuprate(I) (1, 2, 47-49) and di-halogenoargentate(I) (47-49) ions has been discussed in terms of ds hybridization of the metal atom. A theoretical study of the bonding and nuclear quadrupole coupling in [CuCl2] and [CuBr2]" has demon-... 

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

The quantity pv C is the unpaired w-electron spin density at the carbon atom to which the hydrogen atom in question is bonded p c is defined as 1 times the fractional number of unpaired it electrons on the carbon atom, with the sign being determined by whether the net unpaired spin at the carbon atom is in the same or opposite direction as the spin vector of the molecule. The term -electron spin density is somewhat misleading in that pn is not an electron probability density (which is measured in electrons/cm3), but rather is a pure number. The semiempirical constant Q (no connection with nuclear quadrupole moments) is approximately —23 G. 

In the limit where the nuclear Zeeman term in the nuclear spin hamiltonian is much larger than the quadrupole interaction, it is only the secular part of Hq that contributes to the time-independent hamiltonian, H0. 

Nuclear hyperfine splittings in the rotational spectra of dimers have been observed in the molecular beam electric resonance experiments and the Fourier transform microwave experiments. In most cases, the coupling constants are interpreted with the simplified expression given in Eqn. (6) for axially symmetric molecules in the K=0 rotational manifold. Thus both the nuclear quadrupole coupling term and the... 

The second term in Eq. (II. 1) is the crucial one in discussing NQR investigations in the frame of chemical bonding in the solid state. In a general form, we may write the nuclear quadrupole interaction energy as... 

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