Dipole transition probability

The electric dipole transition probability (expression (5.10)) can be ronghly approximated by... 

In this paper we investigate the consequences of this addition to the original Montroll-Shuler equation. To keep it simple we retain the harmonic oscillator potential and the simple dipole-transition probabilities in the linear part of the equation and in the nonlinear part we restrict ourselves to the simple resonance transition... 

We now consider radiative processes. Let [ S 0]) be a singlet state lying well below the (I) curve, as depicted in Figure 4. The dipole transition probability is proportional to... 

Let us consider the intercombination transitions. Then, we shall retain only the corrections containing the spin operator in the expansion. To find the form of the operator describing the electric multipole intercombination transitions and absorbing the main relativistic corrections, one has to retain in the corresponding expansion the terms containing spin operator S = a and to take into account, for the quantities of order v/c, the first retardation corrections, whereas, for the quantities of order v2/c2 one must neglect the retardation effects. Then the velocity form of the electric dipole transition probability may be written as follows ... 

The general definition of the electron transition probability is given by (4.1). More concrete expressions for the probabilities of electric and magnetic multipole transitions with regard to non-relativistic operators and wave functions are presented by formulas (4.10), (4.11) and (4.15). Their relativistic counterparts are defined by (4.3), (4.4) and (4.8). They all are expressed in terms of the squared matrix elements of the respective electron transition operators. There are also presented in Chapter 4 the expressions for electric dipole transition probabilities, when the corresponding operator accounts for the relativistic corrections of order a2. If the wave functions are characterized by the quantum numbers LJ, L J, then the right sides of the formulas for transition probabilities must be divided by the multiplier 2J + 1. 

In a multipole expansion of the interaction of a molecule with a radiation field, the contribution of the magnetic dipole is in general much smaller than that of the electric dipole. The prefactor for a magnetic dipole transition probability differs from the one for an electric dipole by a2/4 1.3 x 1 () 5. Magnetic dipoles may play an important role, however, when electric dipole transitions are symmetry-forbidden as, e.g., in homonuclear diatomics. 

In practice values of B are also often quoted in cm-1. For the simple rigid rotor the rotational quantum number J takes integral values, J = 0, 1, 2, etc. The rotational energy levels therefore have energies 0, 2B, 6B, 12B, etc. Elsewhere in this book we will describe the theory of electric dipole transition probabilities and will show that for a diatomic molecule possessing a permanent electric dipole moment, transitions between the rotational levels obey the simple selection rule A J = 1. The rotational spectrum of the simple rigid rotor therefore consists of a series of equidistant absorption lines with frequencies 2B, 4B, 6B, etc. 

One other important difference between electric and magnetic dipole transition probabilities involves the inversion symmetry of all spatial coordinates (i.e. parity). A magnetic dipole moment is an axial vector that does not change sign under inversion, unlike an electric dipole moment. Consequently magnetic dipole transitions occur only between states of the same parity. 

These two functions do not have definite parities but the symmetric and antisymmetric combinations of them do we use these combinations to calculate both the Stark effect and the electric dipole transition probabilities. 

Two further aspects need to be considered in order to understand the magnetic resonance spectrum, namely, the effects of an applied magnetic field, and the electric dipole transition probabilities. The effective Hamiltonian describing the interactions with an applied magnetic field, expressed in the molecule-fixed axis system q, is ... 

Many of the observed levels have measured g-factors which are closer to the pure case (c) values than to any alternative pure coupling case. However there is extensive rotational electronic coupling which, in many instances, mixes the case (c) states case (e) is then a better limiting basis, as we shall see in due course. First we investigate the electric dipole transition probabilities for the Zeeman components, so that we can understand the pattern of lines illustrated in figure 10.73. 

The electronic state calculation by discrete variational (DV) Xa molecular orbital method is introduced to demonstrate the usefulness for theoretical analysis of electron and x-ray spectroscopies, as well as electron energy loss spectroscopy. For the evaluation of peak energy. Slater s transition state calculation is very efficient to include the orbital relaxation effect. The effects of spin polarization and of relativity are argued and are shown to be important in some cases. For the estimation of peak intensity, the first-principles calculation of dipole transition probability can easily be performed by the use of DV numerical integration scheme, to provide very good correspondence with experiment. The total density of states (DOS) or partial DOS is also useful for a rough estimation of the peak intensity. In addition, it is necessary lo use the realistic model cluster for the quantitative analysis. The... 

This is the title of the paper [107], where for the first time such properties, as well as electric dipole transition probabilities, were reported for the bound excited states of the negative ion of Be, namely the Be ls 2s2p P and ls 2p S°, whose existence had previously been predicted theoretically. The paper is cited here in order to provide evidence of the flexibility and efficiency of methods in the framework of the SPSA (Breit-Pauli Hamiltonian) in producing correlated wavefunctions of excited states that are usable toward the calculation of other properties, such as fine and hyperfine structures and oscillator strengths [107]. Note that the computational facilities available to us in the early 1980s were far from optimal. 

In his calculation of the induced dipole-dipole and dipole-quadrupole processes of energy transfer Kushida (17) made use of the Judd-Ofelt1) expression for the forced electric dipole transition probability in the rare earths incorporated in solids. 

The sum rule can also be derived by elementary quantum mechanics from the definition of the dipole transition probability and its relation to the / value. 

The ratio between dipole-quadrupole and dipole-dipole transition probabilities is given in Ref. 66) as... 

The theoretical dipole-dipole transition probabilities calculated from Dexter s formula are 0.6 sec-i for 3% Gd and 3% Tb. Comparing this result with the experimentally determined energy transfer probability (817 sec i) given in Table 13 we see that the experimental results do not agree with the theory of the resonant energy transfer process. 

The theoretical dipole-dipole transition probabilities for energy trcuis-fer from Tm 1D2 to Er (2G7/2, Kis/s) calculated from Dexter... 

In all XNCD measured so far, it has been found that the predominant contribution to X-ray optical activity is from the E1-E2 mechanism. The reason for this is that the El-Ml contribution depends on the possibility of a significant magnetic dipole transition probability and this is strongly forbidden in core excitations due to the radial orthogonality of core with valence and continuum states. This orthogonality is partially removed due to relaxation of the core-hole excited state, but this is not very effective and in the cases studied so far there is no definite evidence of pseudoscalar XNCD. 

Expressions have been deduced also for reduced magnetic dipole transition probabilities B (Ml) and moments (p) (see, e.g., Bohr and Mottelson 1974 Nathan and Nilsson 1965). 

The first factor is associated with the electronic dipole transition probability between the electronic states the second factor is associated between vibrational levels of the lower state v" and the excited state V, and is commonly known as the Franck-Condon factor, the third factor stems from the rotational levels involved in the transition, J" and /, the rotational line-strength factor (often termed the Honl-London factor). In particular, the Franck-Condon information from the spectrum allows one to gain access to the relative equilibrium positions of the molecular energy potentials. Then, with a full set of the spectroscopic constants that are used to approximate the energy-level structure (see Equations (2.1) and (2.2)) and which can be extracted from the spectra, full potential energy curves can be constructed. 

Average ratio of the electric-quadrupole to electric-dipole transition probabilities over states up to 40 eV above Ep for different core levels (adapted from Muller and Wilkins 1984). 

The shorter lifetime in the glass ceramic is probably due to the effect of the oxide matrix incorporating the nanoparticles. Indeed, several studies have proved that the oxide glassy matrix interacted with the rare-earth ions situated inside the nanosized crystallites and influenced their spectroscopic properties [65, 66]. Indeed, those Er ions close to the nanocrystallite/glass interface are in distorted sites. As the distortions lower the symmetry, this could result in an increase in the electric dipole transition probability and consequently decrease the radiative lifetime. Moreover, those Er " " ions close to the surface of the crystallites can be sensitive to the presence of oxide ions in their coordination polyhedron, inducing multiphonon nonradiative contribution to the Er " " de-excitation and lowering the lifetime. 

K. Jankowski and L. Smentek-Mielczarek, Effect of electron correlation on the forced electric dipole transition probabilities in/ systems. A General Effective Operator Formulation, Molecular Physics, 38, 1445-1457 (1979) Electron correlation effects on transition probabilities of LaCf Pr, ibid 38, 1459-1465 (1979) Effect of electron correlation on the forced electric dipole transition probabilities in/ systems. II. A model study of Pr LaCh and Eu LaCh, ibid 43, 371-382 (1981) Effect of electron correlation on the forced electric-dipole transition probabilities in/ systems. III. Some aspects of the mechanism of hypersensitive transitions. International Journal of Quantum Chemistry, 24(sl7), 339-346 (1983). 

By contrast in low-current gas discharges, the equilibrium population is determined by a balance between excitation by electrons ancollisional processes. In this case detailed information is required about electron collision cross-sections and electric dipole transition probabilities before Nj can be calculated. However, the population in excited levels rarely exceeds 10 of... 

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