Partial reaction cross-sections

The formation probability of each product corresponds to a partial reaction cross-section. The total reaction cross-section is the sum of all the partial cross-sections and measures the probability that the projectile causes a nuclear reaction indqjendent of the products formed. Thus, the decrease in intoisity of the particle flux is proportional to oT. The amount of an individual product formed is proportional to where a,- corresponds to the partial reaction cross-section for the formation of the th product. 

The reaction cross-section depends on the projectile energy as shown in Figure 14.2. The curves obtained for the partial reaction cross-section as a function of projectile energy are known as excitation functions or excitation curves. 

Aj is referred to as the disappearance constant of nuclide i, while K is the partial formation constant, and a and X the partial reaction cross-sections and decay constants for nuclide i + 1 a, a, X and X may be equal to zero in some cases. The exact meaning of A and A is explained below with examples. Introducing these abbreviations for the formation of the Xj j species of the constant (A + l)-chain, and dropping the second index, which is = 1 in the left vertical row in Figure 15.2, we obtain the equation ... 

The technologically most important isotope, Pu, has been produced in large quantities since 1944 from natural or partially enriched uranium in production reactors. This isotope is characterized by a high fission reaction cross section and is useful for fission weapons, as trigger for thermonuclear weapons, and as fuel for breeder reactors. A large future source of plutonium may be from fast-neutron breeder reactors. 

For a diatom-diatom reaction, the reaction cross-section from a specific initial state (vi, V2, ji, ji) is obtained by summing the reaction probabilities over all the partial waves with total angular momentum J46 ... 

A number of techniques have been used previously for the study of state-selected ion-molecule reactions. In particular, the use of resonance-enhanced multiphoton ionization (REMPI) [21] and threshold photoelectron photoion coincidence (TPEPICO) [22] has allowed the detailed study of effects of vibrational state selection of ions on reaction cross sections. Neither of these methods, however, are intrinsically capable of complete selection of the rotational states of the molecular ions. The TPEPICO technique or related methods do not have sufficient electron energy resolution to achieve this, while REMPI methods are dependent on the selection rules for angular momentum transfer when a well-selected intermediate rotational state is ionized in the most favorable cases only a partial selection of a few ionic rotational states is achieved [23], There can also be problems in REMPI state-selective experiments with vibrational contamination, because the vibrational selectivity is dependent on a combination of energetic restrictions and Franck-Condon factors. 

In the partial-wave approximation, the reaction cross-section is equal to... 

Several methods have been used to solve this problem partially. They can be divided into two major types. In the first a size-dependent physical separation of the clusters is made. In the second the expansion conditions are varied and so cause a change in the composition of the mixture. This is monitored by the change in product formation. If the dependence of the composition on the stagnation pressure can be modeled, this method may provide information on the reaction cross section of each complex. 

Elander and collaborators, as well as other researchers, have shown that the eigenvalues and eigenfunctions of resonant states can be used to compute partial wave cross sections at and thereby also reaction rates in terms of a Laurent type expansion of the partial wave S-matrix[19, 28, 29, 30, 31]... 

The reaction cross-section is closely related to the excited energy states of the con und nucleus. Four such levels are shown for Be in Figure 14.3. To the left of the figure the (p,7), (p,n) and (p,a) partial cross-sections (excitation functions) are shown as function of... 

At energies higher than about 1 eV, the reaction cross sections exhibit huge variations with the incident neutron energy and are called resonances. Up to about 0.1 MeV, however, the differential flux, E), of partially thermalized neutrons is approximately proportional to 1/E. Hence, an analogous expression can be derived for epithermal neutron activation ... 

A review of the differential data used by Conley indi> cates that the bias can only partially be accounted for on the basis of the reaction cross sections Involved. After correcting for the effects of deviant reaction cross sections and caleulatlonal approximations, the remaining bias was used to establish the mean energy c the u equilibrium fission neuron spectrum to be 2.012 ... 

Even though the reaction probability provides distinctly different characteristics for the above two types of reactive resonance, the experimental observables of the reaction cross sections are often smeared out due to the summation of all possible Pj(Ec), which is inevitable because of the existence of many partial. The interpretation of the experimental observables and further understanding of the effects of reactive resonance require the intense interplay between theory and experiment. For example, the transition state region is often tight, and only a small range of partial waves contributing to the DCS in certain direction, which is amenable to quantum mechanical calculations. We shall be able to focus on reaction probabilities of a certain range of partial waves, from which the different kinds of reactive resonance manifest themselves with distinct features. 

The reaction cross section is obtained from partial wave expansion and the capture approximation discussed earlier that is, the reaction takes place if there is enough energy to surmount the centrifugal barrier. It follows that... 

The major problem in method (a) is that in ion-molecule interchange, considerable momentum in the direction of travel of the incident ion is imparted to both final products. Hence, in a perpendicular type apparatus only transfer of low weight particles can be observed at all and only at very low velocities of the incident ions (1, 9, 10, 11, 12, 13, 19, 20, 23, 27). Cross-sections cannot be measured. The value of these investigations is that some ion-molecule reactions—e.g., proton transfer and hydride ion transfer—can be identified. The energetics and the competition between charge exchange and ion-molecule reactions can be discussed, and by using partially deuterated compounds, one can obtain a detailed picture of the reaction. 

Fig. 4. Computed partial cross-sections in A2 for the F + HD (v = 0, j = 0) —> HF + D reaction as a function of the total angular momentum quantum number, J, up to collision energies of 3 kcal/mol.

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