Nuclear spin operators

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 terms in (la) and (lb) both involve sums of single nuclear spin operators Iz. In contrast, the terms in (lc) involve pairwise sums over the products of the nuclear spin operators of two different nuclei, and are thus bilinear in nuclear spin. If the two different nuclei are still of the same isotope and have the same NMR resonant frequency, then the interactions are homonuclear if not, then heteronuclear. The requirements of the former case may not be met if the two nuclei of the same isotope have different frequencies due to different chemical or Knight shifts or different anisotropic interactions, and the resulting frequency difference exceeds the strength of the terms in (lc). In this case, the interactions behave as if they were heteronuclear. The dipolar interaction is proportional to 1/r3, where r is the distance between the two nuclei. Its angular dependence is described below, after discussing the quadrupolar term. 

In quantum theory, the nuclear dipole-moment operator is proportional to the nuclear spin, i.e., nothing else but the gyromag-netic ratio yj multiplied by the nuclear spin operator Ih... 

The calculation of magnetic parameters such as the hyperfine coupling constants and g-factors for oligonuclear clusters is of fundamental importance as a tool for the evaluation of spectroscopic data from EPR and ENDOR experiments. The hyperfine interaction is experimentally interpreted with the spin Hamiltonian (SH) H = S - A-1, where S is the fictitious, electron spin operator related to the ground state of the cluster, A is the hyperfine tensor, and I is the nuclear spin operator. Consequently, it is... 

In general, a spin Hamiltonian involving only electron-spin and nuclear-spin operators can be found that will satisfactorily account for the experimental results. The spin Hamiltonian has become the crossroad for the path followed by the experimentalist and the theorist. Experimentally, the spin Hamiltonian and its constants are determined from the ESR spectra, whereas, theoretically, the spin Hamiltonian and its constants are computed from the wave function of the ion. 

Mk denotes the nuclear magnetic dipole moment operator, obtained by multiplication of the nuclear spin operator IA- by the magnetogiric factor yK. 

We assumed that the magnetic tensors appearing in the spin Hamiltonian and the diffusion tensor have the same principal axis system. The are the Wigner matrices of rank /. The nuclear spin operators are expanded over the set of (21 +1) matrices Af j, I being the nuclear angular moment, defined via... 

The explicit form of the Hamiltonian describing the spin system studied in this section and the evolution equation for the variable of interest are formally the same as those of Eqs. (3.2), (3.3), and (3.6), respectively. Here the expansion base A for the nuclear spin operators is formed by matrices of rank 4 to provide a suitable expansion for the components of the angular momentum... 

Hamiltonian operator-subscripts indicate its nature nuclear spin operator for nucleus i components of /,... 

It has been customary to use the electron spin-nuclear spin operator in its two limiting forms... 

Aa and AB are the local hyperfine tensors and IA and IB the nuclear spin operators. In Eq. (3), superhyperfine and transferred hyperfine interactions are neglected. 

The coherence level associated with a particular product operator is just the sum of the indices of the nuclear spin operators in the product operator, for example Ik,-ill,-i has a coherence level p = -2. 

The vectors and denote the Dirac 4x4 matrices for electron i (in standard representation) and the nuclear spin operator for nucleus a. The constants and Kg are nuclear parameters, while is the nuclear spin quantum number. 

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