Transition probability INDEX

He points out that the variation of lifetime with glass matrix is due to at least two causes, the first being the changes in refractive index. If the wave functions of the ion remain essentially the same from host to host, the spontaneous-transition probability will increase with increasing refractive index because of the increase in density of final states. The second cause is configuration mixing of 4/ and 5d states, which must reflect the size and symmetry of the crystal field produced at the ion by the surroundings. 

Values of the radiative rate constant fcr can be estimated from the transition probability. A suggested relationship14 57 is given in equation (25), where nt is the index of refraction of the medium, emission frequency, and gi/ga is the ratio of the degeneracies in the lower and upper states. It is assumed that the absorption and emission spectra are mirror-image-like and that excited state distortion is small. The basic theory is based on a field wave mechanical model whereby emission is stimulated by the dipole field of the molecule itself. Theory, however, has not so far been of much predictive or diagnostic value. 

N is the density of states, v the frequency, n the refractive index, gj and gjj the degeneracies of the ground and excited states, and X. is the wavelength. The absorption coefficient increases with the number of states and with the transition probability, x". ... 

Values of the radiative rate constant can be estimated from the transition probability. A suggested relationship - js given in equation (25), where n is the index of refraction of the medium, is the mean of the cube of the reciprocal of the emission frequency, and gt/gu i ... 

In the case of a Markov process with state space a subset of M, and with F the associated Borel a-algebra, and with a continuous index variable t e R+, the transition probabilities are defined instead for sets Ae... 

Finally, (2J + 1) is the degeneracy of the initial state and the expression involving the refractive index n is known as Lorentz s local-field correction. Calculations of transition probabilities within file frame of JO theory are usually made assuming that all Stark sublevels within the ground level are equally populated and that the material under investigation is optically isotropic. The former hypothesis is only reasonable in some cases, e.g., when transitions initiate from non-degenerate states such as Eu( Fo). Otherwise, there is a Boltzmann distribution of the population among the crystal-field sublevels. The second assumption is not valid for uniaxial or biaxial crystals, but, of course, holds for solutions. 

Here the summations are performed over all initial and final states of bond i — 1. Division of Pi(cx, x 0) by the equilibrium probability Po(C) of bond i, yields the conformational transition probability Ci(o, x 0) from state to state a for the investigated bond i, when its conformational dynamics is assumed to be coupled to that of the preceding bond. The bond index in Cj(a, x 0) is conveniently omitted for drains with identical bonds, and the short-hand notation Ct(alOis adopted for stationary processes. 

The Hook Method. The hook method is based on the Kramer-Kroaig dispersion relation, which relates the refractive index of a gas to transition probabilities [9.151, 9.152]. At the same time as atoms in a vapour absorb at... 

We will restrict ourselves to time invariant Markov chains where the transition probabilities only depend on the states, not the time n. These are also called homogeneous Markov chains. In this case, we can leave out the time index and the transition probability matrix of the Markov chain is given by... 

This plausible choice for the transition probabilities is given by (leaving out the index / = 1, 2 for the moment)... 

In the period between 1926 and 1939 the development of our understanding of radiation processes parallels the development of the theory of atomic and molecular structure. Many of the results which had been derived by the use of the correspondence principle and the old quantum theory were re-derived in a more rigorous and satisfactory manner. The quantum-mechanical expression for the refractive index of a gas or vapour is an example of this type of progress. In hydrogenic systems the rates of radiative transitions were calculated and the theoretical lifetimes of the different excited levels were derived. In other atoms only approximate estimates of the transition probabilities were possible, but the lifetime measurements made by canal rays and fluorescence from atomic beams were not sufficiently accurate to demand more refined calculations. 

The hydration of simple ketenes (RCH=C=0—> RCH2COOH) also shows relatively constant values of oh w which are quite low (100-1000) (Tidwell, 1990 Allen et al., 1992), implying p/fj = 11 to 12 for the transition state for water attack. Corresponding to this, the Leffler index and the /3nuc are both about 0.25. Whether these low values really indicate an early transition state or arise because water and hydroxide ion react quite differently is not yet clear. However, it appears possible that water attack proceeds through a cyclic mechanism involving two (or more) water molecules (Allen et al., 1992) whereas hydroxide ion probably attacks conventionally as a nucleophile (Tidwell, 1990). Of course, any mechanism for the water reaction which is superior to simple nucleophilic attack will elevate kw and necessarily lead to low kOH/kw ratios. 

We now multiply on the left by and integrate, where m indexes die stationary state 4> for which we are interested in measuring die probability of transition. This gives... 

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