Statistical distributions Porter-Thomas

The fission process is less well understood than the others, and there is not a good theoretical argument for a particular statistical distribution. Porter and Thomas 42) predict that it should be a fairly broad distribution corresponding to a small value of v (say, in the range 1 to 4) rather than a very high value as is the case with capture widths. One is forced to rely on the experimental data in spite of the poor statistics. Fischer 32) analyzed 12 resonances of and 19 resonances of Pu , both by the moments method (79) and the maximum likelihood method (87), with the results shown in Table V. 

If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

The coimection between the Porter-Thomas P(lc) distribution and RRKM theory is made tln-ough the parameters j -and v. Waite and Miller [99] have studied the relationship between the average of the statistical... 

Figure 2. Statistics of current for the transmission through the Sinai billiard for T 0. The upper left panel shows the computed distribution for p = 2 together with the Porter-Thomas distribution P(p) (solid curve). In the inset in the same panel the computed wave function statistics f(p) for the real part of ip is compared with a random Gaussian distribution (solid curve). 

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]. 

Thus, for state-specific decay and the most statistical (or nonseparable) case, a micro-canonical ensemble does not decay exponentially as predicted by RRKM theory. It is worthwhile noting that when v/2 becomes very large, the right-hand side of Eq. (8.24) approaches exp -kt) (Miller, 1988), since lim (1 + xln) " = exp (-x), when n-> °o. Other distributions for P(k), besides the Porter-Thomas distribution, have been considered and all give M(f, E) expressions which are nonexponential (Lu and Hase, 1989b). 

The connection between the Porter-Thomas nonexponential N(r, E) distribution and RRKM theory is made through the parameters k and v. The average of the statistical state-specific rate constants k is expected to be similar to the RRKM rate constant k(E). This can be illustrated (Waite and Miller, 1980) by considering a separable (uncoupled) two-dimensional Hamilton H = + Hy whose decomposition path is... 

Three-dimensional quantum mechanical calculations have been performed to determine the unimolecular rate constants for the resonances in HOj H + O2 dissociation (Dobbyn et al., 1995). The resonances are not assignable and the fluctuations in the resonance rate constants can be represented by the Porter-Thomas distribution. Equation 8.17. Thus, the unimolecular dissociation of HO2 is apparently statistical... 

For the high-energy unresolved resonances, statistical methods have to be applied. The radiation width can be assumed to be constant, but the neutron widths show a Porter-Thomas [22] distribution. The probability for Tn to be in an interval dFn is... 

The general class of functions used to describe partial widths are the chi-squared distributions with v degrees of freedom, sometimes called the Porter-Thomas distributions when used to describe resonance widths. Porter and Thomas (42) have shown that the statistical model of the nucleus leads to this type of distribution for the partial widths. The chi-squared distribution for the statistical quantity x can be written as... 

Porter and Thomas (42) have given strong theoretical arguments that the statistical frequency functions for all the partial resonance level widths should be chi-squared distributions with v degrees of freedom as given by (75). In this case, the number of degrees of freedom is the number of channels open for decay of the compound nucleus by the process to be... 

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