Transition probability strength

Figure 24. Electron-transfer rate versus electronic coupling strength. The temperature is T = 500 K. Solid line with circle-present results from Eq. (126) with the transition probability averaged over the seam surface. Solid line with square-present results with the transition probability taken at the minimum energy crossing point (MECP). Dashed line-Bixon-Jortner theory Ref. [109]. Dotted line-Marcus s high temperature theory. Taken from Ref. [28].

The dipole oscillator strength is the dominant factor in dipole-allowed transitions, as in photoabsorption. Bethe (1930) showed that for charged-particle impact, the transition probability is proportional to the matrix elements of the operator exp(ik r), where ftk is the momentum transfer. Thus, in collision with fast charged particles where k r is small, the process is again controlled by dipole oscillator strength (see Sects. 2.3.4 and 4.5). 

The literature on transition probabilities (or oscillator strengths) is vast, rapidly growing and difficult to summarize. A small selection for atoms and molecules is given in Allen, AQ and larger selections in... 

The last assumption is very fundamental. It results in time-independent transition probabilities and makes a clean theory possible. It requires that the product of the time scale of the decay time for the tcf (called the correlation time and denoted x ) and the strength of the perturbation (in angular frequency units) has to be much smaller than unity (17-20). This range is sometimes denoted as the Redfield limit or the perturbation regime. 

Relativistic Quantum Defect Orbital (RQDO) calculations, with and without explicit account for core-valence correlation, have been performed on several electronic transitions in halogen atoms, for which transition probability data are particularly scarce. For the atomic species iodine, we supply the only available oscillator strengths at the moment. In our calculations of /-values we have followed either the LS or I coupling schemes. 

Spectral lines are often characterized by their wavelength and intensity. The line intensity is a source-dependent quantity, but it is related to an atomic constant, the transition probability or oscillator strength. Transition probabilities are known much less accurately than wavelengths. This imbalance is mainly due to the complexity of both theoretical and experimental approaches to determine transition probability data. Detailed descriptions of the spectra of the halogens have been made by Radziemski and Kaufman [5] for Cl I, by Tech [3] for BrIwA by Minnhagen [6] for II. However, the existing data on /-values for those atomic systems are extremely sparse. 

The relativistic quantum defect orbitals lead to closed-form analytical expressions for the transition integrals. This allows us to calculate transition probabilities and oscillator strengths by simple algebra and with little computational effort. 

In Tables -A, we report oscillator strengths for some fine structure transitions in neutral fluorine, chlorine, bromine and iodine, respectively. Two sets of RQDO/-values are shown, those computed with the standard dipole length operator g(r) = r, and those where core-valence correlation has been explicitly introduced, Eq. (10). As comparative data, we have included in the tables /-values taken from critical compilations [15,18], results of length and velocity /-values by Ojha and Hibbert [17], who used large configuration expansions in the atomic structure code CIVS, and absolute transition probabilities measured through a gas-driven shock tube by Bengtson et al. converted... 

The crystal field model may also provide a calciflation scheme for the transition probabilities between levels perturbed by the crystal field. It is so called weak crystal field approximation. In this case the crystal field has little effect on the total Hamiltonian and it is regarded as a perturbation of the energy levels of the free ion. Judd and Ofelt, who showed that the odd terms in the crystal field expansion might connect the 4/ configuration with the 5d and 5g configurations, made such calculations. The result of the calculation for the oscillator strength, due to a forced electric dipole transition between the two states makes it possible to calculate the intensities of the lines due to forced electric dipole transitions. 

Fig. 17. A comparision of the temperature dependence of the line-shape function (G) of the transition probability for the multimode case (solid line) as against a single mode approximation (dashed line). Here the phonon frequency spectrum (A) is assumed to be of Gaussian form, A a>) = 2 2) 1,2exp [—(to — cu0)2/2

To estimate the oscillator strengths or transition probabilities for electric-dipole transitions is a very difficult problem. For such a transition, the operator is of the form... 

The strength of an electronic transition is generally expressed in terms of a quantity called oscillator strength f. It is defined as the ratio of the experimental transition probability to that of the ideal case of a harmonic oscillator, that is... 

For a one-electron transition the ideal value of unity is obtained when all the molecules are transferred to the higher energy state, i.e. when the transition probability is unity. The oscillator strength can also be expressed as... 

The critical separation distance calculated from the quenching data was found to be 13 A which is of the same order as the van der Waals separation. The critical separation distance remained unchanged when halogen substituted naphthalenes were used. The halo-substitution is expected to increase T1A SoA transition probability in naphthalenes. Since oscillator strengths / (naphthalenes /(iodonaphthalenes) is as 1 1000, no increase in transfer efficiency is clear indication of the lack of dependence on the oscillator strength. 

Arnold et al.24 have calculated radiative lifetimes for the various collision complexes of singlet molecular oxygen on the basis of a collision time of 10"13 sec. The data for wavelengths and transition probabilities are presented in Table III. A recent paper25 describes the theory of double electronic transitions, and gives calculated oscillator strengths for the oxygen systems. 

I 5.1 Einstein Transition Probabilities, 23 I 5.2 Absorplion Intensity of Atoms, 24 I 5.3 Oscillator Strength, 25 1-6 Resonance Absorption and Emission by Atoms, 27... 

Fig. 15.10 Cross sections as a function of rf field strength for the first four orders of sideband resonances of the K 29s + K 27d radiative collisions in a 4 MHz rf field, (a) The zero-photon resonant collision cross section, (b) the +1 sideband resonance, (c) the —2 sideband resonance, and (d) the +3 sideband resonance. The solid line shows the experimental data, the bold line indicates the prediction the Floquet theory, and the dashed fine is the result of numerical integration of the transition probability (from ref. 17).

Fig. 15.11 The K 29s + 27d resonance in the presence of a low frequency rf field. In zero rf field (a), the FWHM is 1.6 MHz. In (b)-(d), a 1.0 MHz field of strength 0.05 V/cm, O.lV/cm, and 0.2 V/cm respectively is present. The solid line in (b) is a numerical integration of the transition probability, and the bold line is the convolution of a Lorentzian lineshape with a sinusoidal shift from resonance. In (e), the rf frequency is 0.5 MHz and its strength is 0.2 V/cm. For these low frequencies, the features are no long frequency dependent but rather are field strength dependent (from ref. 18).

The energy levels and eigenfunctions, obtained in one or other semi-empirical approach, may be successfully used further on to find fairly accurate values of the oscillator strengths, electron transition probabilities, lifetimes of excited states, etc., of atoms and ions [18, 141-144]. 

Oscillator strength, transition probability, lifetime and line intensity... 

Line and multiplet strengths are useful theoretical characteristics of electronic transitions, because they are symmetric, additive and do not depend on the energy parameters. However, they are far from the experimentally measured quantities. In this respect it is much more convenient to utilize the concepts of oscillator strengths and transition probabilities, already directly connected with the quantities measured experimentally (e.g. line intensities). Oscillator strength fk of electric or magnetic electronic transition aJ — a J of multipolarity k is defined as follows ... 

Oscillator strength or transition probability is the individual characteristic of a separate atom or ion. However, in reality we usually have to deal with a large number of them, where, depending on the specific physical situation, various elementary processes of excitation, ionization, recombination, etc. may take place. Real spectral lines are characterized by the intensity of radiation, defined in the conditions of natural isotropic excitation as... 

Unlike line or oscillator strengths and transition probabilities, line intensities are directly measured quantities. 

Certain sum rules are known for oscillator strengths and transition probabilities as well. So, we can define the oscillator strength and probability of the transition between terms, respectively ... 

In order to be able to calculate oscillator strengths or transition probabilities of M/c-radiation, we need the corresponding expressions for the submatrix elements of appropriate operators (4.9) and (4.16). The one-electron submatrix element of relativistic operator (4.9) is equal to... 

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