Nuclear angular moment

FIG. 12.1 Precession of a nuclear angular moment in a magnetic field. 

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... 

We often say that an electron is a spin-1/2 particle. Many nuclei also have a corresponding internal angular momentum which we refer to as nuclear spin, and we use the symbol I to represent the vector. The nuclear spin quantum number I is not restricted to the value of 1/2 it can have both integral and halfintegral values depending on the particular isotope of a particular element. All nuclei for which 7 1 also posses a nuclear quadrupole moment. It is usually given the symbol Qn and it is related to the nuclear charge density Pn(t) in much the same way as the electric quadrupole discussed earlier ... 

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 ... 

For most nuclides the nuclear angular momentum vector L and the magnetic moment vector (L point in the same direction, i.e. they are parallel. However, in a few cases, for example, N, they are antiparallel. 

PAC atomic probes (e.g., mIn or mHf) possess a nuclear quadrupole moment and a magnetic dipole. Even if no field acts on the PAC nucleus, the successive emission of the y-photons through an intermediate state exhibits an appreciable angular anisotropy between the emission directions. If the (isolated) nucleus is then brought into a perturbing field (e.g., on a specific lattice site which is next to a vacancy), the angular anisotropy becomes time-dependent due to the precession of the nuclear spin. For example, if the PAC nucleus in the crystal is exposed to a (static) electric... 

Nuclear magnetic resonance (NMR) is based on a phenomenon that nuclei which possess both magnetic and angular moments (i.e. have odd mass number or odd atomic number) interact with an applied magnetic field B0 yielding 21 + 1 (where 1 is the nuclear spin quantum number) energy levels with separation AE ... 

Electron and nuclear magnetic moments can be regarded as arising from a property of the particles, i.e. that they possess an intrinsic angular momentum as if they were spinning. Such angular momenta are given by... 

In a rotating molecule containing one quadrupolar nucleus there is an interaction between the angular momentum J of the molecule and the nuclear spin momentum I. The operator of this interaction can be written as a scalar product of two irreducible tensor operators of second rank. The first tensor operator describes the nuclear quadrupole moment and the second describes the electrical field gradient at the position of the nucleus under investigation. 

Hyperfine interactions, that is the interaction of a nuclear magnetic moment with extranuclear magnetic fields and the interaction of a nuclear quadrupole moment with electric field gradients from extranuclear charge distributions, are measured via time differential perturbed angular correlation (TDPAC) of y-rays emitted from radioisotopes. 

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