Unfolding procedures

The application of statistical theories of chemical reaction to ion-molecule reactions by Light resulted in an unsatisfactory situation where there were no experimental excitation functions available for comparison with the predictions of the theory. This deficiency led Light to develop an unfolding procedure to utilize existing mass-spectrometric rate constant data to generate an experimental excitation function. 

For an idealized mass-spectrometer ion-source experiment, the familiar thin-target formula 

Q is an energy-averaged cross section, an average obtained over the range of collision energies 0 E Ef of the excitation function a(E) according 

As already noted, there are severe practical limitations on the accuracy to which a E) may be derived by this means. Nevertheless, it has been employed by several workers and the outcome of these applications is duly discussed in Section 3.6.3. It is worth noting that the unfolding may also be performed electronically as part of the detector system of the mass spectrometer by differentiation of H Ej) according to Eq. (8).  

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One should also note that the unfolding procedure makes the average density of levels unifomi over the entire energy range. Thus, the difference... 

In order to calculate S, one has to find the nearest configuration of Pi points that is G-symmetric. In the majority of cases, S is the distance of a chiral object from a reflection mirror. The fblding/unfolding procedure was developed for finding that configuration. 

Computationally the super-CI method is more complicated to work with than the Newton-Raphson approach. The major reason is that the matrix d is more complicated than the Hessian matrix c. Some of the matrix elements of d will contain up to fourth order density matrix elements for a general MCSCF wave function. In the CASSCF case only third order term remain, since rotations between the active orbitals can be excluded. Besides, if an unfolded procedure is used, where the Cl problem is solved to convergence in each iteration, the highest order terms cancel out. In this case up to third order density matrix elements will be present in the matrix elements of d in the general case. Thus super-CI does not represent any simplification compared to the Newton-Raphson method. 

A cube of data sized (mxnxp) can be unfolded in three different directions along the row space, along the column space, and along the third direction of the cube, also called the tube space. The three unfolding procedures give a row-wise... 

Fig. 4 also shows the spectrum of primary protons, which has been derived from the flux of unaccompanied hadrons measured with the KASCADE experiment Antoni et al. 2004c. The spectrum is compatible with the proton flux as obtained from the unfolding procedure when using the QGSJET model. The EAS-TOP experiment published two sets of spectra with different assumptions about the contribution of protons and helium nuclei derived from the measurements of the electromagnetic and muonic shower components Navarra et al. 

If the three assumptions are shown to be valid, it follows rigorously that experimental rate constants may only be compared if Vf is identical in both cases. The same restriction applies to comparisons of experimental phenomenological cross sections. If this is not so, such comparisons can only be made quantitatively if the excitation function has been measured, either by unfolding procedures (Section 2.2) or by direct measurement. 

In conclusion, the method is powerful for obtaining thermal rate constants whose accuracy is probably good to 10-20%. The velocity distribution is not strictly Maxwellian and thus attempts to study the temperature dependence of this thermal rate constant would not be fruitful. Certainly no unfolding procedure could be applied to reveal the excitation function at low energy at best, the technique gives a phenomenological cross section relevant to thermal energies. 

Minimize the 5 value by repeating the folding-unfolding procedure (steps 4-7) for all orderings and all orientations of the group elements. This step is equivalent to finding the best cluster of folded points. (In the present case, due to the cyclic connectivity, minimization is reduced to the two... 

Thus unfolding Pi with any one of the elements in C2 will form P2, with any of the elements in C3 will form F3, etc., creating a symmetric tetrahedron. Again, as in the previous cases, we use the construction of a symmetric shape as a guideline for the folding-unfolding procedure for evaluating the Td symmetry of any four vertices, as follows. 

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