Dipole transition amplitude

In Eq. (12), l,m are the photoelectron partial wave angular momentum and its projection in the molecular frame and v is the projection of the photon angular momentum on the molecular frame. The presence of an alternative primed set l, m, v signifies interference terms between the primed and unprimed partial waves. The parameter ct is the Coulomb phase shift (see Appendix A). The fi are dipole transition amplitudes to the final-state partial wave I, m and contain dynamical information on the photoionization process. In contrast, the Clebsch-Gordan coefficients (CGC) provide geometric constraints that are consequent upon angular momentum considerations. 

While calculations of energies are a useful monitor of the behavior of MBPT, we wish to accurately predict a parity violating transition amplitude. For this reason it is important to calculate standard parity-conserving amplitudes and compare them to experiment. We use hyperfine splittings and oscillator strengths for this purpose, illustrating the calculations here with dipole transition amplitudes. They are determined by... 

From a theoretical point of view, the leading term contributing to CD in randomly oriented molecules is due to the interference of electric and magnetic dipole transition amplitudes, with the rotatory (R) strength taken as a measure of CD. It is expressed from dipole and angular momentum integrals between initial and final states according to the expressions... 

Table 2.11 RQ electric dipole transition amplitudes of the strongest transitions of nobelium. r is the lifetime of the upper...

K. Jankowski, L. Smentek-Mielczarek, and A. Sokolowski, Electron-correlation third-order contributions to the electric dipole transition amplitudes of rare earth ions in crystals. Molecular Physics, 59, 1165-1175 (1986). 

A particularly useful probe of remote-substituent influences is provided by optical rotatory dispersion (ORD),106 the frequency-dependent optical activity of chiral molecules. The quantum-mechanical theory of optical activity, as developed by Rosenfeld,107 establishes that the rotatory strength R0k ol a o —> k spectroscopic transition is proportional to the scalar product of electric dipole (/lei) and magnetic dipole (m,rag) transition amplitudes,... 

In the 1950s, many basic nuclear properties and phenomena were qualitatively understood in terms of single-particle and/or collective degrees of freedom. A hot topic was the study of collective excitations of nuclei such as giant dipole resonance or shape vibrations, and the state-of-the-art method was the nuclear shell model plus random phase approximation (RPA). With improved experimental precision and theoretical ambitions in the 1960s, the nuclear many-body problem was born. The importance of the ground-state correlations for the transition amplitudes to excited states was recognized. 

Fig. 20. A schematic representation of the emission of an isolated large molecule following internal conversion from the second to the first singlet, a" and 6J1 denote the amplitudes of the second singlet and quasi-degenerate vibrational levels of the first singlet, respectively, in the excited molecular state >/in. /v, and m are the corresponding electronic dipole transition matrix elements coupling < >n and as indicated.

We have considered in particular the case of multiphoton transitions, to be observed with the help of intense high frequency fields as produced by X-ray Lasers or Free-Electron Lasers (FEL). As a result of our analysis, we have shown that two-photon bound-bound transition amplitudes in high-Z hydrogenic systems are significantly affected by relativistic corrections, even for low values of the charge of the nucleus. For instance at Z = 20, the corrections amount to about 10%, a value much higher than what is observed for standard one-photon transitions in X-ray spectroscopy measurements for which the non-relativistic dipole (NRD) approximation agrees with the exact result to within 99% at comparable frequencies. 

Eq. (3.22) previously given in Section 3.2. We will now analyze in greater detail the methods of solving this equation. As is known, the product E d has to be considered as a hermitian product of E and d [140]. It is just such a product which represents the amplitude with which the vector E is contained in the vector d. As a result we have for the matrix elements of the electric dipole transition... 

A mercury atom that was ionized by a weak electron beam was captured in a miniature Paul (radio frequency) trap that has internal dimensions of rQ s 466 pm and zQ s 330 pm. The rf trapping frequency was 21.07 MHz with a peak voltage amplitude of about 730 V. The ion was laser cooled by a few microwatts of cw laser radiation that was frequency tuned below the 6s Si -6p Pi electric dipole transition near 194 nm. When the Hg+ ion was cold and the 194 nm radiation had sufficient intensity to saturate the strongly allowed S-P transition, 2 x 10 photons/s were scattered. With our collection efficiency, this corresponded to an observed peak count rate of about 10 s-1 against a background of less than 50 s— -. 

From Equation (1.35), the electric dipole transition moment Afo f may be thought of as the dipole moment of the transition density The transition density is a purely quantum mechanical quantity and cannot be inferred from classical arguments. A more pictorial representation of the electric dipole transition moment equates it to the amplitude of the oscillating dipole moment of the molecule in the transient nonstationary state that results from the mixing of the initial and the final states of the transition by the time-dependent perturbation due to the electromagnetic field, and which can be written as a linear combination Cq o + + This emphasizes the fact... 

If the dipole transition moment d is comparatively large and the value (d q)2/2E is of the order of / (recall that / (10 4 to 10 2)/cb where T 300 K), then expression (282) gives a considerable increase of the current. With this regard the stationary photocurrent for 8 < 108 V/m below room temperature is linear with the constant field 8 and quadratic with the amplitude of the variable field 8. ... 

Here Sa(t) is the dynamically imposed Stark shift of the TLS resonance frequency, 5r(t) is its random counterpart representing proper dephasing, nr = e)(g + g)(e is the dipole-transition operator, whose time-modulated form is given by S(t), with the real amplitude e(t). If the bath consists of oscillators, then... 

When Mf, = 0 we can be interpret as the induced transition amplitude of an operator A between the ground state (0) and an excited state (/) due to a perturbation B. With A as a dipole operator and spin-orbit coupling introduced as a perturbation through B we can calculate singlet-triplet transition moments (phosphorescence) [57]. For Wj, = jwy. and dipole operators is the two-photon absorption amplitude. 

The operators responsible for electric dipole transitions are molecule-fixed components of y [Eqs. (6.1.2) and (6.1.38)]. y operates exclusively on the spatial coordinates of the electrons, and hence has selection rules AS = AS = 0 and, because it is a vector operator with respect to the spatial coordinates (see Section 3.4.5), AA = Afl = 0, 1. AA = 0 transitions arise from the yz operator component. The relationship between A — A and —A — —A transition amplitudes may be derived from... 

Returning to the dipole amplitude linking the ground state with eigenfunction i /a), we will substitute the expansion of the eigenfunction given in Eq. (4) into the transition amplitude... 

Indeed, around 1980, first experimental results on atomic parity violation have been reported, in particular measurements of the optical activity of bismuth, thallium and lead vapours as well as measurements of an induced electric dipole (El) amplitude to a highly forbidden magnetic dipole transition (Ml) in caesium. These experiments have nowadays reached very high resolution so that even effects from the nuclear anapole moment, which results from weak interactions within the nucleus, have been observed in caesium. The electronic structure calculations for caesium are progressing to a sub-percent accuracy for atomic parity violating effects and the reader is referred to chapter 9 of the first part of this book [12]. 

In case of Zeeman sublevels or hfs levels, the allowed RF transitions are magnetic dipole transitions. Optimum conditions are then achieved if the sample is placed at the maximum of the magnetic field amplitude of the RF field. This can be realized, for instance, inside a coil that is fed by an RF current. For electric dipole transitions (for example, between Stark components in an external dc electric field) the electric amplitude of the RF field should be maximum in the optical pumping region. 

The transition amplitude for the dipole transition from the dark state ui)coherent to the upper state 3> is... 

In Sect. 2.4 we saw that the first-order Doppler effect can be exactly canceled for a two-photon transition if the two photons hcoi = fuo2 have opposite wave vectors, i.e., k = —k2. A combination of Doppler-free two-photon absorption and the Ramsey method therefore avoids the phase dependence (p Vx) on the transverse velocity component. In the first interaction zone the molecular dipoles are excited with the transition amplitude a and precess with their eigenfrequency (jl> 2 = ( 2 — E )/h. If the two photons come from oppositely traveling waves with frequency u), the detuning... 

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