Zeeman factors

A compensation of the magnetic moments may be expected in a ladderlike motif for a value of the ratio gNd/Scu close to 2, and being the Zeeman factors for the ground Kramers doublets of Nd(III) and Cu(II), respectively (55). 

Zeeman effects, second-order, cobalt, 320 Zeeman factors, 213, 224-225... 

In the assumption of dominant exchange, i.e., if the total spin S, is a good quantum number, then the individual spins can be easily projected on the total spin, and the related observables in the total spin multiplets are linear combinations of the analogous observables of the individual spins. The relations between the Zeeman factor, the hyperfine coupling constants, and the ZFS parameters of the coupled pair and those of the individual ions are given below ... 

The number of energy levels found to date, with the aid of the Zeeman effect and the isotope shift data, is 605 even and 586 odd levels for Pu I and 252 even and 746 odd for Pu II. The quantum number J has been determined for all these levels, the Lande g-factor for most of them, and the isotope shift for almost all of the Pu I levels and for half of those of Pu II. Over 31000 lines have been observed of which 52% have been classified as transitions between pairs of the above levels. These represent 23 distinct electron configurations. 

Thus both the (/-factor and the Upvalues affect Tq-S mixing. In general, the two radicals comprising the radical pair will have different gr-values and therefore different Zeeman energies [the first term in equation (12)]. Likewise the upvalues and hyperfine energies [the second term in equation (12)] will differ. The difference between the LPFs for two radicals is then given by equation (13), where gx and are the electronic y-factors... 

Ml = 3/2 than for mj = 1/2. Moreover, the ground state experiences pure Zeeman splitting A M.g 4s given by (4.48) (recall, the nuclear g factor of the 7g = 1/2 ground state is different from that of the L = 3/2 excited state). 

If the electric quadrupole splitting of the 7 = 3/2 nuclear state of Fe is larger than the magnetic perturbation, as shown in Fig. 4.13, the nij = l/2) and 3/2) states can be treated as independent doublets and their Zeeman splitting can be described independently by effective nuclear g factors and two effective spins 7 = 1/2, one for each doublet [67]. The approach corresponds exactly to the spin-Hamiltonian concept for electronic spins (see Sect. 4.7.1). The nuclear spin Hamiltonian for each of the two Kramers doublets of the Fe nucleus is ... 

Fig. 4.15 Effective nuclear g values for the excited I = 3/2 state of Fe in units of the corresponding nuclear g factor (g e = —0.10317). The left panel shows the Zeeman splitting of the 7 = 3/2 manifold with large quadrupole slitting under the influence of a weak field, and the two panels on the right show the 77-dependence of the corresponding effective nuclear g values for the I m/ = 1/2 and m/ = 3/2) doublets with the field oriented along the x, y, and z principal axes of the EFG...

From (4.56) and Table 4.3, we derive the relative intensity ratios 3 2 1 1 2 3 for the hyperfine components of a Zeeman pattern of a powder sample. The transition probability for the case of the polar angle 6 = Oq can readly be calculated by integrating (4.56) only over the azimuthal angle (j). One obtains a factor (1 + cos 0o)/2 and sin 0o for m = 1 and m = 0, respectively, which are multiplied by the square of the Clebsch-Gordan coefficients. As a consequence of the angular correlation of the transition probabilities the second and fifth hyperfine components (Fig. 4.17) disappear if the direction k of the y-rays and the magnetic field H are parallel (0q = 0). 

The anisotropic/factor may also manifest itself in the relative line intensities of Zeeman split hyperfine spectra in a poly crystalline absorber. Expanding f(0) in a power series... 

In [49, 76], the line intensities for electric quadrupole and Zeeman (magnetic dipole) splitting and including the anisotropy of the /-factor are also given for / = 2 <-> 7g = 0 transitions (even-even isotopes, e.g., in the rare earth region or in W, Os). 

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)... 

For / = 0 there is obviously no first-order Zeeman splitting however, application of a magnetic field can result in second-order splitting. As such, it is necessary to evaluate the corresponding factor, which isg(j = 2 + 7,(2 + S). 

In this framework hazard assessment is mainly based on toxicity testing in clean laboratory conditions. Findings of laboratory studies are then extrapolated to higher levels of natural system hierarchy (from organisms to communities and even ecosystems) using various factors (Smrchek and Zeeman, 1998). 

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