Operators, angular momenta magnetic moment

The magnetic dipole operator describes the magnetic moment induced by the angular momentum of the electrons arising from the orbital motion, in contrast to the magnetic moment arising from the electron spin which we discuss when we consider the spin-Zeeman interaction. 

A particle possesses an intrinsic angular momentum S and an associated magnetic moment Mg. This spin angular momentum is represented by a hermitian operator S which obeys the relation S X S = i S. Each type of partiele has a fixed spin quantum number or spin s from the set of values 5 = 0, i, 1,, 2,. .. The spin s for the electron, the proton, or the neutron has a value The spin magnetie moment for the electron is given by Mg = —eS/ nie. 

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)... 

At variance with the nucleus, the electron is associated with an orbital, i.e. a wavefunction which is related to the distribution in space of the electron cloud, and which displays an angular momentum L and a magnetic moment ill- In analogy with the spin operators (see Eqs. (1.7)), the following relations hold... 

Dirac interpreted this as the electron having a spin angular momentum, s = (h/2) ff, that has to be added to the orbital angular momentum I to get a constant of the motion. It is the same matrix/operator vector a that fixes the direction of s and that of the magnetic moment p. derived from Eq. (2.21), and this justifies the simple... 

The calculation of the magnetic transition dipoles requires a preamble. The magnetic moment was already defined in Eq. (4.128) of Chap. 4. By explicitly writing the angular momentum operator in terms of the linear momentum operator sr x p one obtains ... 

For 4Jp + 2Jf = 0, the matrix splits into two separate 2x2 blocks, which have the same eigenvalues. The splitting pattern is thus as in the central panel of Fig. 7.5. Such a case can occur for a state. The orbital part of this state has no angular momentum, since the corresponding operator is not included in the direct square Ti E x E. As result, the magnetic moment of such a state is due only to the doublet spin part. Such a state behaves as a pseudo-doublet. 

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