Fields parity rule

It is now well established that almost all transitions within the f shell are electric dipole in nature. The breakdown of the Laporte parity rule is brought about by non-centro-symmetric terms in the crystal-field Hamiltonian V, which have the effect of mixing d and g states into the f shell. Transitions which are nominally f to f take place because of the permitted transitions f- d and f->g. An early attempt (6) to explain the hypersensitivity used the fact that for rare-earth or actinide ions at site symmetries of the types... 

The ground level electronic configuration of trivalent europium is / . Transitions within the / shell are responsible for the crystal spectra. Transitions are forbidden in a free ion by the parity rule for electric dipole transitions. In a crystal or glass, forced electric transitions become allowed as a consequence of coupling of odd electronic wave functions due to the odd parity terms in the crystal field expansion. Considering the static field approximation in the theory developed by Judd (4) and Ofelt (5), the contribution of the odd parity part of the cr5rstal field is calculated by mixing states of different parity. 

The statement that quantum electrodynamics is invariant under such a spatial inversion (parity operation) can be taken as the statement that there exist new field operators >p (x ) and A x ) expressible in terms of tji(x) and Au(x) which satisfy the same commutation rules and equations of motion in terms of s as do ift(x) and A x) written in terms of x. In fact one readily verifies that the operators... 

Transitions between states are subject to certain restrictions called selection rules. The conservation of angular momentum and the parity of the spherical harmonics limit transitions for hydrogen-like atoms to those for which A/ = 1 and for which Am = 0, 1. Thus, an observed spectral line vq in the absence of the magnetic field, given by equation (6.83), is split into three lines with wave numbers vq + (/ bB/he), vq, and vq — (HbB/he). 

Electron configuration of Bp" is (6s) (6p) yielding a Pip ground state and a crystal field split Pap excited state (Hamstra et al. 1994). Because the emission is a 6p inter-configurational transition Pap- Pip. which is confirmed by the yellow excitation band presence, it is formally parity forbidden. Since the uneven crystal-field terms mix with the (65) (75) Si/2 and the Pap and Pip states, the parity selection rule becomes partly lifted. The excitation transition -Pl/2- S 1/2 is the allowed one and it demands photons with higher energy. 

Photoluminescence of ZnS Mn occurs when the phosphor absorbs photon energy corresponding to the band gap of ZnS and relaxes to release the excess energy of the exciton (a pair of an s-p electron and a hole). Based on the selection rule of Laporte, the symmetrical field of 6-coordinated Mn(ll) does not allow the d-d transition since it is not associated with the change in the parity. The 4-coordinated Mn(lI), in contrast, allows a partial d-p hybridization, enabling the d-d transition. 

As elaborated in detail in Ref. (5) there are two principal intensity mechanisms for dimer excitations. The single-ion mechanism is based on the combined action of spin-orbit coupling and an odd-parity ligand field potential at the Cr center. It is by this mechanism that spin-forbidden transitions obtain their intensity in mononuclear complexes. The pair mechanism, on the other hand, is restricted to exchange-coupled systems. It leads to the selection rules AS = 0,... 

The first selection rule is a consequence of the fact that the transition dipole moment has negative parity. The second reflects that the quantization axis is parallel to the polarization of the electric field. The third follows from the fact that the transition dipole moment is a tensor of rank 1 (corresponding to an angular momentum with quantum number 1). 

The C are tensor operators, whose matrix elements again can be calculated exactly, whereas the crystal-field parameters Bk are regarded as adjustable parameters. The number of parameters for this potential is greatly reduced by the parity and triangular selection rules and finally by the point symmetry for the f-element ion in the crystal. Detailed information about the crystal-field potential has been given for example by Gorller-Walrand and Binnemans (1996). 

We should not leave this discussion of the intensity of rotational transitions without some mention of the parity selection rule. Electric dipole transitions involve the interaction between the oscillating electric field and the oscillating electric dipole moment of the molecule. The latter is represented in quantum mechanics by the transition moment fix(b,a) given in equation (6.300). For this transition moment to be non-zero, the integrand i/f must be totally symmetric with respect to all appropriate symme-... 

The value of k is limited to k 21, where / is the orbital angular momentum of the electrons. For f electrons k 6 k must also be even based on parity of the matrix elements involved in the crystal field potential. Thus k = 2,4 or 6 for f electrons. Allowed values of q have to follow the rule q k. Any further restrictions on q are dependent on the symmetry... 

For electric-dipole transitions the parity of the initial and final states should be different (parity selection rule). This implies that transitions within one and the same shell, for example 3d or 4/) are forbidden. This selection rule may be relaxed by the admixture of opposite-parity states due to the crystal field, or by vibrations of suitable symmetry. 

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