Operator Zeeman electronic

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)... 

We now notice that we could write a Hamiltonian operator that would give the same matrix elements we have here, but as a first-order result. Including the electron Zeeman interaction term, we have the resulting spin Hamiltonian ... 

Lso is the commutator with the electron spin Zeeman Hamiltonian (assuming isotropic g tensor, Hso = gS- Bo), Lrs = Lzfs (the sub-script RS stands for coupling of the rotational and spin parts of the composite lattice) is the commutator with the ZFS Hamiltonian and Lr = —ir, where is a stationary Markov operator describing the conditional probability distribution, P(QolQ, t), of the orientational degrees of freedom through ... 

The summation index n has the same meaning as in Eq. (31), i.e., it enumerates the components of the interaction between the nuclear spin I and the remainder of the system (which thus contains both the electron spin and the thermal bath), expressed as spherical tensors. are components of the hyperfine Hamiltonian, in angular frequency units, expressed in the interaction representation (18,19), with the electron Zeeman and the ZFS in the zeroth order Hamiltonian Hq. The operator H (t) is evaluated as ... 

The stabilized temperature platform furnace (STPF) concept was first devised by Slavin et al. It is a collection of recommendations to be followed to enable determinations to be as free from interferences as possible. These recommendations include (i) isothermal operation (ii) the use of a matrix modifier (iii) an integrated absorbance signal rather than peak height measurements (iv) a rapid heating rate during atomization (v) fast electronic circuits to follow the transient signal and (vi) the use of a powerful background correction system such as the Zeeman effect. Most or all of these recommendations are incorporated into virtually all analytical protocols nowadays and this, in conjunction with the transversely heated tubes, has decreased the interference effects observed considerably. 

Level-6. The most complete treatment utilizes the basis set of all free-atom terms v,L,Ml,S,Ms) for the given electronic configuration dn, and the calculation of energy levels is performed by involving the operators of the electron repulsion, CF, spin-orbit interaction, orbital-Zeeman and spin-Zeeman terms ... 

If the total angular momentum is derived only from an isotropic electron spin, that is, J=S, the Zeeman operator can be rewritten as... 

The first term is the Zeeman interaction depending upon the g(RS OW, q ) tensor, external magnetic field B0 and electron spin momentum operator S the second term is the hyperfine interaction of the th nucleus and the unpaired electron, defined in terms hyperfine tensor A (Rsklw, qj) and nuclear spin momentum operator n. The following terms do not affect directly the magnetic properties and account for probe-solvent [tfprobe—solvent (Rsiow, qJ)l ld solvent-solvent //solvent ( qj)] interactions. An explicit... 

Sx, Sy, and Sz are the three components of the spin operator, D and E are the anisotropy constants which were determined via high-frequency electron paramagnetic resonance (D/kB 0.275 K and E/kB 0.046 K [10]), and the last term of the Hamiltonian describes the Zeeman energy associated with an applied field H. 

This term describes the electronic contribution to the rotational g-factor. The contribution (7.223) represents the interaction between the applied magnetic field By and the magnetic moment produced by the electrons in the molecule as it rotates in laboratory space. It has an operator form identical to that of the first-order nuclear orbital (i.e. rotational) Zeeman interaction... 

The spin-Hamiltonian operator can be written as the sum of a large number of terms for example, see Equation (A3) in Appendix 3. As may be seen therein, the convention for the two Zeeman terms (electron spin and nuclear spin) is that their signs differ, due to the difference in charge of the two most basic particles to be dealt with the electron and the proton. Hence the g values of these particles are taken to be both positive. 

The presence of a magnetic field introduces two new terms, being linear and quadratic in the field. The B-L term is the orbital analogue of the Zeeman effect for the electron spin, discussed in section 8.1 (eq. (8.19)). The second-order property is the magnetizability which according to eqs. (10.17) and (10.18)/(10.27) contains contributions from both linear and quadratic perturbation operators. The operator is (half) the angular moment L, while the P operator may be written as... 

The point to emphasise concerning this example is the reduced overlap between the ground vibronic wavefunctions associated with the two electronic states, on account of the displacement of the potential energy minima. The expectation value of any electronic operator connecting the two orbital electronic states will be similarly affected. Consider the orbital Zeeman interaction about the z axis, given by... 

Simplifications. In the form we give to tp, the use of half integers, which is a complication, is avoided. Only the integers m G Z that appear in the associated Legendre polynomials and P 1 1 are employed. Half integers m = m + 1/2 appear, for example, in the formula implying the total angular momentum operator of the electron (see Appendix C) and will be introduced in the Zeeman-effect (Chap. 14). 

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