Electric quadrupole transition operator

In the optical activity arising from higher-order cross-terms, the effects are in most cases expected to be orientation-dependent. Pseudoscalar terms are the only ones which survive in random orientation (molecules in solution or liquid phase). At the same order of perturbation as El-Ml there is a product of the electric dipole and electric quadrupole transition operators (E1-E2). Since the latter product involves tensors of unequal rank, the result cannot be a pseudoscalar and this term would not, therefore, contribute in random orientation but can be significant for oriented systems with quadrupole-allowed transitions. The E1-E2 mechanism was developed by Buckingham and Dunn and recognized by Barron" as a potential contribution to the visible CD in oriented crystals containing the [Co(en)3] " ion. 

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. 

Our final remark about transitions in atoms concerns the matrix elements for quadrupole transitions. The foregoing selection rules are deduced on the assumption that the electric dipole operator belongs to the representation The electric quadrupole moment operator... 

As stated in an earlier paragraph, the sharp emission and absorption lines observed in the trivalent rare earths correspond to/->/transitions, that is, between free ion states of the same parity. Since the electric-dipole operator has odd parity,/->/matrix elements of it are identically zero in the free ion. On the other hand, however, because the magnetic-dipole operator has even parity, its matrix elements may connect states of the same parity. It is also easily shown that electric quadrupole, and other higher multipole transitions are possible. 

A) with respect to the operation of inversion about the origin of the system. The electric dipole operator is antisymmetric (A) with respect to inversion at a point of symmetry. The electric quadrupole operator is inversion symmetric (S). A transition is allowed if the product function in the expression for transition moment is symmetric for electric dipole radiation and antisymmetric for electric quadrupole radiation. 

The probability of a transition being induced by interaction with electromagnetic radiation is proportional to the square of the modulus of a matrix element of the form where the state function that describes the initial state transforms as F, that describing the final state transforms as Tk, and the operator (which depends on the type of transition being considered) transforms as F. The strongest transitions are the El transitions, which occur when Q is the electric dipole moment operator, — er. These transitions are therefore often called electric dipole transitions. The components of the electric dipole operator transform like x, y, and z. Next in importance are the Ml transitions, for which Q is the magnetic dipole operator, which transforms like Rx, Ry, Rz. The weakest transitions are the E2 transitions, which occur when Q is the electric quadrupole operator which, transforms like binary products of x, v, and z. 

The spin selection rule is a consequence of the fact that the electric dipole and quadrupole moment operators do not operate on spin. Integration over the spin variables then always yields zero if the spin functions of the two states 0 and are different, and an electronic transition is spin allowed only if the multiplicities of the two states involved are identical. As a result, singlet-triplet absorptions are practically inobservable in the absorption spectra of hydrocarbons, or for that matter, other organic compounds without heavy atoms. Singlet-triplet excitations are readily observed in electron energy loss spectroscopy (EELS), which obeys different selection rules (Kuppermann et al., 1979). 

Obviously, the 4f - 5 p crossing does not correspond to an electric dipole transition, but to an electric quadrupole moment. It has no clear-cut effect in photo-electron spectra of tungsten, rhenium and osmium, suggesting that the non-diagonal elements of the effective one-electron operator are smaller than 0.3 eV. It would be worthwhile to study volatile molecules such as WF6 or 0s04 under conditions of high resolution using either the continuous spectrum emitted by a synchrotron, or 132.3 eV photons from an yttrium anti-cathode (20, 21). 

The second-rank tensor P,y (m) depends on the velocity dipole operator, while Mif(co) depends on the velocity dipole operator and on the magnetic dipole operator and finally T (co) on the velocity dipole operator and the velocity form of the electric quadrupole operator, respectively. Their mathematical expressions are reported and described in detail in Chapter 2. Once more, like we did for TPA, invoking the BO approximation and integrating over the electronic coordinates, the TPCD intensity between vibronic states can be written in terms of elements of electronic transition tensors Pej/it, rj, co), Me,f x, rj, co), and T yr(x, rf, co) between the vibrational states and Z5(/)) associated with the initial and final electronic states 0,) and 0/), respectively. 

Hamiltonian = matrix element of the Hamiltonian H I = nuclear spin I = nuclear spin operator /r( ), /m( ) = energy distributions of Mossbauer y-rays = Boltzmann constant k = wave vector L(E) = Lorentzian line M = mass of nucleus Ml = magnetic dipole transition m = spin projection onto the quantization axes = 1 — a — i/3 = the complex index of refraction p = vector of electric dipole moment P = probability of a nuclear transition = tensor of the electric quadrupole q = eZ = nuclear charge R = reflectivity = radius-vector of the pth proton = mean-square radi-S = electronic spin T = temperature v =... 

Electric quadmpole transitions are also possible despite the even parity of the electric quadrupole operator, the intensity of these transitions is low, and they are less relevant for the lanthanide ions. They will therefore not be addressed here [29]. 

For example, the singlet-triplet transitions in ethylenic compounds generally have tmax <3C 1. The fact that spin-forbidden transitions can be observed at all shows that the transition moment, f electric dipole operator. This operator also contains small terms such as quadrupole operators and spin-orbit operators. The latter is the part of any dynamical operator which couples orbital and spin angular moments this term is responsible for the appearance of weak triplet — singlet absorption spectra. 

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