Operators Magnetic dipole transition

Let us now proceed to E2- and M 1-transitions. Formulas of the sort (30.7) hold for 2-transitions, too. The magnetic dipole transition operator Omi has no radial part. Since the orthogonal radial orbitals are used in calculations, the operators and do not contribute to the... 

Usually, the magnetic dipole transition operator is found in the literature in a form that reads (neglecting constants)... 

Transitions between the electron and nuclear spin sublevels of a triplet state are predominantly magnetic dipole in nature. Thus, if //,(t) is an alternating magnetic field with components in the xyz coordinate system, and is the magnetic dipole transition operator defined by... 

Using the transition moment operator for dipole and quadrupole transitions (discussed above) and the magnetic dipole transition operator L 2S, the triplet-singlet transition... 

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)... 

In terms of transition matrix elements of the electric and magnetic dipole moment operators, the transition dipole moments are... 

Suppose one first considers electric-dipole and magnetic-dipole transitions. As is now well recognized, these are the major contributors to rare-earth absorption and emission spectra. We know that the electric-dipole operator transforms as a polar vector, that is, just as the coordinates (23, 24). This means that it has odd parity under an inversion operation. On the other hand, the magnetic-dipole operator transforms as an axial vector or pseudovector and of course must have even parity (23, 24). 

What about parity in electric-quadrupole and magnetic-dipole transitions The quantities (3.58) are even functions. Hence for electric-quadrupole transitions, parity remains the same. Magnetic-dipole transitions involve angular momentum operators. For example, consider Lz = -ih(xd/dy — yd/dx). Inversion of coordinates leaves this operator unchanged. Hence for magnetic-dipole transitions, parity remains the same. 

The relation between the spherical components AJ0( ) of a general tensor A of rank 2 and the cartesian components A, ( ) are given in Appendix 4. Equations (3.36) will form the basis for derivation of selection rules for rotation-internal motion transitions of SRMs presented in the next section. They also may serve for derivation of the transformation properties of the electric and magnetic dipole moment operators referred to the laboratory system (VH G... 

Most electronic transitions between different states of the f-electrons are dominated by electric dipole transitions. Only in exceptional cases like Eu(III), magnetic dipole transitions are found to be as strong as electric dipole transitions. However, in the case of an f element, electric-dipole transitions between the 4fw states are forbidden because the parity of initial and final state is conserved. Only when the f element is embedded in a crystal providing a point group symmetry that does not contain the inversion operation, these transitions can be observed readily. 

Here i//0 is the ground vibrational wave function and ij/ is the wavefunction corresponding to the first excited vibrational state of the th normal mode /< is the electric dipole moment operator Qj is the normal coordinate for the /th vibrational mode the subscript 0 at derivative indicates that the term is evaluated at the equilibrium geometry. The related rotational strength or VCD intensity is determined by the dot product between the electric dipole and magnetic dipole transition moment vectors, as given in (2) ... 

Here E(PJ) - E(j8 J ) = AE 0, the operators in general are defined by (4.12) and (4.13), AE and submatrix elements are presented in atomic units. Then WEk is in s-1. Let us recall that the oscillator strengths are dimensionless quantities. The relevant expressions for magnetic dipole transitions are given by formulas (27.8) and (27.9), respectively. 

Here, m is the magnetic moment operator, and p, is the linear momentum of electron i. (If Po and Pq are real wave functions, as is generally the case, the magnetic dipole transition moment is an imaginary quantity,... 

Magnetic-dipole transitions also are allowed between states of the 4P configuration. The magnetic-dipole operator is... 

The problem is with the magnetic dipole transition moment, p m 2p), which vanishes in the zeroth approximation. The magnetic dipole selection rule A/ =0, allows the transition from 2p to the np and continuum ep states but, since m is a pure angular operator it cannot connect states which are radially orthogonal. This results in the A =0 selection rule for bound states and also clearly forbids 2p —> ep except via core-hole relaxation. 

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