Nuclear magnetic quantum number

For a Kramers doublet the strongly allowed transitions are those in which the nuclear magnetic quantum number does not change. The frequency at which the transition occurs is not altered by the presence of the term given by eq. (14), because the value of P is the same for each component of the doublet. However, in the presence of anisotropy, transitions in which the nuclear magnetic quantum number changes by 1 or 2 may be weakly allowed if these are observable, they provide additional information from which the value of P may be derived. 

In a macroscopically isotropic sample (all molecular orientations have the same probability), the spectrum consists of contributions from aU orientations when the rotational motion is frozen on the time scale of the experiment. As ESR lines are derivative absorption lines, negative and positive contributions from neighboring orientations cancel. Powder spectra are thus dominated by contributions at the minimum and maximum resonance fields, and by contributions at resonance fields that are common to many spins. The latter contribution provides the center line in the nitroxide powder spectrum (Fig. 3b). It corresponds mainly to molecules with nuclear magnetic quantum number rrii = 0 (center line of all triplets, only g-shift). The detailed shape of this powder spectrum can be simulated, but interpretation is not easy, mainly because hyperfine and g anisotropy are of similar magnitude. 

FIGURE 5.21 Nuclear spin and nuclear magnetic quantum numbers. A nucleus with spin I has a spin angular momentum vector of magnitude K /l l + 1) and possible projections onto the Z axis of hmi, where m can vary from —I to I in increments of one. (Interestingly, all the nuclear isotopes with 1 = 2, such as C1, are radioactive.)... 

In the above equations, the order of perturbation is indicated by number n (= 0, 1, 2, 3) within brackets in the superscripts on Eum, Af and m are the electronic and nuclear magnetic quantum numbers, respectively 0 and < ) are, respectively, the polar and azimuthal angles of the principal axis of the ZFS tensor with respect to the external magnetic field and Re denotes the real part. Because of its small magnitude, only the first-order term of the nuclear Zeeman energy is taken into account. The range for rj is 0 < rj < 1/3. 

ESR spectrum of two-step acid-base-catalyzed xerogel prepared with O enrichment. Spectrum was recorded at 105 K following 15.5 Mrad x-irradialion at 77 K. Dashed curve is a computer simulation based on a NBOHC. Six 0 hyperfine lines are indicated by their nuclear magnetic quantum numbers, m, [179-180]. 

The eigenfunctions for nonaxial nuclear quadrupole interaction are mixtures of the 7, mj) basis functions and thus do not possess well-defined magnetic quantum numbers. Strictly speaking, the states should not be labeled with pure quantum numbers m/. ... 

Magnetic dipole interaction Hm (4.47) and electric quadmpole interaction //q (4.29) both depend on the magnetic quantum numbers of the nuclear spin. Therefore, their combined Hamiltonian may be difficult to evaluate. There are closed-form solutions of the problem [64], but relatively simple expressions exist only for a few special cases [65]. In Sect. 4.5.1 it will be shown which kind of information can be obtained from a perturbation treatment if one interaction of the two is much weaker than the other and will be shown below. In general, however, if the interactions are of the same order of magnitude, eQV Jl, and... 

This equation is called the Curie law and relates the equilibrium magnetization M0 to the strength of the magnetic field B0. The constants have the following meaning I is the nuclear spin quantum number (see below), y is the gyromagnetic ratio specific for a given isotope, h is Planck s constant, kB is Boltzmann s constant, N is the number of nuclei and T is the temperature. 

Equation (2.3) describes line positions correctly for spectra with small hyperfine coupling to two or more nuclei provided that the nuclei are not magnetically equivalent. When two or more nuclei are completely equivalent, i.e., both instantaneously equivalent and equivalent over a time average, then the nuclear spins should be described in terms of the total nuclear spin quantum numbers I and mT rather than the individual /, and mn. In this coupled representation , the degeneracies of some multiplet lines are lifted when second-order shifts are included. This can lead to extra lines and/or asymmetric line shapes. The effect was first observed in the spectrum of the methyl radical, CH3, produced by... 

The principal interaction experienced is the Zeeman interaction (Hz), which describes the interaction between the magnetic moment of the nucleus and the externally applied magnetic field, B0 (tesla). The nuclear magnetic moment, p (ampere meter2) is proportional to the nuclear spin quantum number (/) and the magnetogyric ratio (y, radian telsa-1 second-1) ... 

Table 9.9 Nuclear spin quantum numbers and magnetic properties of selected nuclei...

Nucleus Nuclear spin quantum number / Magnetic moment, (ampere square metre x 1027) Resonance frequency in MHz at 1.4092 TESLA Relative sensitivity at the natural isotopic abundance... 

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