Transition amplitude tensors

The TPA cross-section for an excitation from the ground state 0)0 to a final state 0)/ encountered in experimental measurements is defined in terms of the normalized line shape function, g(tol, + tofl), and the TPA transition amplitude tensor, T°V"V-/ 115,116 ... 

The cartesian components of the TPA transition amplitude tensor, T f, between the ground state 0)0 and the excited state 0)/ of an isolated molecular system are defined as... 

The tensorial form of each product of the unit tensor operators in the terms contributing to the transition amplitude (10.11) can be simplified by coupling the objects to a tensorial product following the relation. 

This is in fact the main point of the derivation of the expression for the transition amplitude of the standard Judd-Ofelt formulation. The procedure described here is applied to both terms in equation (10.11). As mentioned above, in general these two terms differ by the order of operators in the matrix elements and energy denominators. However they both are expressed by the same effective tensor operator, yet each is associated with a different energy denominator. This is the reason that the following approximations/assumptions are introduced in order to make the final expression even simpler ... 

The symmetry properties of the 6j- symbol limit the even values of rank A to 2,4,6. In addition, due to the triangular conditions for the non-vanishing 3j- symbols that determine the reduced matrix elements of spherical tensors in equation (10.18), the rank t of the crystal field potential operator has to be odd. Indeed, since is even, and/ = 3 for 4/ electrons, the odd parts of the crystal field potential contribute to the transition amplitude, while the terms with even values of t contribute to the energy. 

When the intermediate coupling scheme is used for identification of the energy levels of the lanthanide ion, the matrix element of unit tensor operator U > in the expression for the transition amplitude (10.17) (or the line strength) exists only if the triangular condition is satisfied, namely... 

From a physical point of view this relativistic model is also based on the perturbation approach, and at the second order, similarly as in the case of the standard J-O Theory, the crystal field potential plays the role of a mechanism that forces the electric dipole/ t—>f transitions. The only difference is that now the transition amplitude is in effectively relativistic form, as determined by the double tensor operator, but still of one particle nature. Furthermore, the same partitioning of space as in non-relativistic approach is valid here. The same requirements about the parity of the excited configurations are expected to be satisfied. As a final step of derivation of the effective operators, the coupling of double inter-shell tensor operators has to be performed. This procedure is based on the same rules of Racah algebra as presented in the case of the standard J-O theory. However, the coupling of the inter-shell double tensor operators consists of two steps, for spin and orbital parts separately. Thus, the rules presented in equations (10.15) and (10.16) have to be applied twice for orbital and spin momenta couplings, resulting in two 3j— and two 6j— coefficients. 

We can call the terms shown explicitly in Equation (A3) quadratic terms other (higher-order) terms can exist [Ref. 7, Section 6.7]. The above spin-Hamiltonian is used to obtain the energies (and hence also transition energies) of the spin system considered. Another, similar spin-Hamiltonian (with the excitation magnetic-field amplitude vector B, replacing B in Equation (A3)), yields the transition relative intensities. The line positions and intensities obtained are expressed in terms of the scalar ( tensors of zeroth rank) parameters g, gn, D, P,A,..., derived as projections from the matrices g, gn, D, P and A.12... 

The dependence on the m indices of the amplitude for the transition between states JM) and J M ) of total angular momentum due to a tensor operator Tq has a remarkably simple form in which the indices M, M and Q all appear in a single 3-j symbol. It is given by the Wigner—Eckart theorem. 

However, for all of these interactions it is important to have the dynamics effectively modeled. Librational dynamics of significant amplitude can average the tensor elements. Such motions are present even in polycrystalline samples [18]. Shown in Fig. 6.4.5 are powder pattern spectra at 276 K, below the gel to liquid crystalline phase transition that quenches global dynamics, but retains local dynamics [19] and at 143 K, well below the temperature that quenches most librational motions [20]. 

More recent interest has focused upon the interpretation of the relative intensities of the electronic Raman transitions. The theory of electronic Raman spectroscopy has been well-summarized elsewhere [63, 202], and the electronic Raman scattering amplitude from an initial i/rf) to a final jfj) vibronic state (where the phonon states are the same, and usually zero-phonon (i.e. electronic) states) is given by (i/ r czpCF In this expression, the cartesian polarizations of the incident photon (hcv) and the scattered photon (hcvs) are a and p, respectively. The Cartesian electronic Raman scattering tensor is written as... 

ENDOR and FT ESEEM spectra differ mainly in the intensities of the lines, which in ESEEM are given by a factor related to the ESR transition probabilities. A necessary prerequisite for modulations in the time domain spectrum is that the allowed Ami = 0 and forbidden Ami = 1 hyperfine lines have appreciable intensities in ESR. The zero ESEEM amplitude thus predicted with the field along the principal axes of the hyperfine coupling tensor is of relevance for the analysis of powder spectra. Analytical expressions describing the modulations have been obtained for nuclear spins I = V2 and / = 1 [54, 57] by quantum mechanical treatments that take into account the mixing of nuclear states under those conditions. Formulae are reproduced in Appendix A3.4. 

There exist several possibilities to generate such turbulent fluctuations at the inflow boundary [19]. Batten et al. [3] reformulated on the ideas of Kraichnan [14] and Smirnov et al. [21] for wall bounded flows. The velocity signal is generated by a sum of sines and cosines with random phases and amplitudes. The wave numbers are calculated from a three-dimensional spectrum and are scaled by the values of the Reynolds-stress tensor. A special wall treatment was applied to elongate near-wall structures. A transition length to physical turbulence of about ten channel half heights was obtained at low Reynolds number channel flow. 

Amplitudes of molecular optical events are proportional to off-diagonal matrix elements of interaction operators between the wavefunctions of the initial, final, and possibly also intermediate states of the molecule, 0), f), and /) respectively. These operators are projections of molecular transition vector and tensor operators onto the polarization directions of photons created or annihilated in the event (Table 1). The amplitudes depend on the wavenumber of the light used and can be real or complex. The probability of an optical event W is proportional to the square of the absolute value of its amplitude (Table 2). The proportionality constant is of no... 

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