Transition probabilities electric dipole

According to Equations (5.14) and (5.15), we see that the probability of a particular transition depends on the electric dipole matrix element /x, given by Equation (5.12). These transitions, which are induced by interactions of the electric dipole element with the electric field of the incident radiation, are called electric dipole transitions. Therefore, electric dipole transitions are allowed when p- 0. 

The transition probability R is related to selection mles in spectroscopy it is zero for a forbidden transition and non-zero for an allowed transition. By forbidden or allowed we shall mostly be referring to electric dipole selection mles (i.e. to transitions occurring through interaction with the electric vector of the radiation). 

Fig. 7.3 Effect of magnetic dipole interaction (7/m), electric quadmpole interaction (Hq), and combined interaction// = Hu + //q, Em> q on the Mossbauernuclear levels of Ni. The larger spacings between the sublevels of the ground state are due to the somewhat larger magnetic dipole moment of the nuclear ground state as compared to the excited state. The relative transition probabilities for a powder sample as well as the relative positions of the transition lines are indicated by the stick spectra below...

There are also electric quadrupole E2 terms of order —exiXjdFi/dxj Fpr/X and magnetic dipole Mi terms of order (r x v).Be/c Fpv/c Fpr/X since B0 = F0. These provide smaller transition probabilities by factors of the order of (r/X)2 10-8 in the optical region. However, when the dipole vanishes, they can give rise to forbidden lines indicated by square brackets, e.g. [O ill]. Still higher orders of transition are sometimes significant for nuclear y-rays. 

Application of the F-D theorem produced [122] several significant results. Apart from the Nyquist formula these include the correct formulation of Brownian motion, electric dipole and acoustic radiation resistance, and a rationalization of spontaneous transition probabilities for an isolated excited atom. 

The average energy flux in the evanescent wave is given by the real part of the Poynting vector S = (c/47t)ExH. However, the probability of absorption of energy per unit time from the evanescent wave by an electric dipole-allowed transition of moment pa in a fluorophore is proportional to lnfl - El2. Note that Re S and pa E 2 are not proportional to each other they have a different dependence on 0. 

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

For electric multipolar interactions, the energy transfer mechanism can be classified into several types, according to the character of the involved transitions of the donor (D) and acceptor (A) centers. Electric dipole-dipole (d-d) interactions occur when the transitions in D and A are both of electric dipole character. These processes correspond, in general, to the longest range order and the transfer probability varies with l/R, where R is the separation between D and A. Other electric multipolar interactions are only relevant at shorter distances dipole-quadrupole (d-q) interaction varies as l/R, while quadrupole-quadrupole interaction varies as l/R °. 

The intensity of an electric dipole transition in absorption or emission depends, on one hand, on factors particular to the experiment measuring the intensity, e.g., the number density of molecules in the initial state of the transition and, for absorption experiments, the absorption path length and the intensity of the incident light. On the other hand, the intensity involves a factor independent of the experimental parameters. This factor, the line strength 5(f <— i), determines the probability that a molecule in the initial state i of the transition f <— i will end up in the final state f within unit time. 

As the electron approaches the molecule, an electric field is established that is described in terms of a Coulomb potential, (()(-. It is assumed that when the Coulomb potential reaches the electron transition energy (the ionization potential, Eq) the orbital electron involved in the transition absorbs energy from the field, the efficiency of the ionization depending on the transition probability, F, .. When the electron-induced dipole contribution is neglected, a cross section, which will be an underestimate, can be calculated from the interparticle separation when (()(- = Eq. In order to deduce the maximum ionization cross section, a., the transition probability P,. must be taken into account ... 

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. 

The other model is also principally possible. Europium initially enters the barite lattice as Eu" ", which oxidizes to Eu + at 700 °C. The relatively small difference between the Ba " and the Eu" ionic radii (1.5 and 1.7 A) makes this substitution possible. The luminescence of Eu was still not observed in minerals, but is known in luminofors (Gorobets et al. 1968). Eu has 6s electron configuration and the mostly probable are electric-dipole electron transitions 6s -6p, taking place between uneven and even 2,3,4 Ps,4,5 terms. 

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... 

If incident radiation with a frequency equal to one of the fundamental frequencies falls on a molecule, it may make a transition from the ground state to the appropriate fundamental level. These normal frequencies usually occur in the infra-red spectral region. The probability of such a transition occurring, however, depends on the relationship between the molecule s electric dipole moment (as a function of the nuclear coordinates) and the wavefunctions of the ground state and of the fundamental level. 

Quantum mechanics tells us that the probability of an electric dipole induced transition occurring between the states described by tpn and tpm is proportional to... 

We assume that the absorbing gas is of a uniform composition and in thermal equilibrium. The absorption coefficient, which is defined by Lambert s law, Eq. 3.1, is expressed in terms of the probabilities of transitions between the stationary states of the supermolecular system, in response to the incident radiation. Assuming the interaction of radiation and matter may be approximated by electric dipole interaction, i.e., assuming the wavelengths of the radiation are large compared with the dimensions of molecular complexes, the transition probability between the initial and... 

According to the results of the last section, if the integrals in (3.48) all vanish, then the probability for a transition between states m and n is zero. Actually, (3.47) is the result of several approximations, and even if the electric dipole-moment matrix elements vanish, there still might be some probability for the transition to occur. 

The irregularity of the spectrum has consequences on the properties of the matrix elements of observables like the electric dipole moment and, thus, on the radiative transition probabilities. For radiative transitions, a single channel is open and the statistics of the intensities follow a Porter-Thomas or x2 distribution with parameter v = 1, as observed in NO2 [5, 6]. 

So far, only transitions involving unperturbed molecules have been considered. If collisions between molecules are allowed, then electric dipole transitions of otherwise forbidden systems become more probable. From the measurements of the absorption in oxygen, Badger et al.13 predict that the effective first-order transition probability, A, for the O-st Aj, -> 3S5") system, will be given by... 

Thus the probability for internal conversion from the first excited singlet (J = 1) to the ground state (i = 0, S( = C) was predicted to be low since electric dipole transitions to high vibrational levels of the ground state were of low probability. 

The Einstein coefficients are related to the most fundamental quantity which describes the transition probability, known as the transition moment. During an electronic transition for instance, an electron jumps from one orbital to another. Its distance from the nucleus changes, so there is a change in the instantaneous dipole moment. The greater this change, the more probable the transition because it is the interaction between this transition dipole and the electric vector of light. 

The observed cross sections for the 18s (0,0) collisional resonance with v E and v 1 E are shown in Fig. 14.12. The approximately Lorentzian shape for v 1 E and the double peaked shape for v E are quite evident. Given the existence of two experimental effects, field inhomogeneties and collision velocities not parallel to the field, both of which obscure the predicted zero in the v E cross section, the observation of a clear dip in the center of the observed v E cross section supports the theoretical description of intracollisional interference given earlier. It is also interesting to note that the observed v E cross section of Fig. 14.12(a) is clearly asymmetric, in agreement with the transition probability calculated with the permanent electric dipole moments taken into account, as shown by Fig. 14.6. 

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