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Formulation of the formation process

Calculation of the resonant molecular formation rates is difficult since it is a rearrangement process involving six bodies. Several different groups have published the calculations  [127,140,141,139,142,143,133,,71,144,145,146,45], but each work is often criticized by other groups, which creates for the experimentalists a situation that is confusing (and sometimes frustrating!).

For the conventional homogeneous target experiments, the resonant molecular formation rate $\lambda ^{mf}$, as a function of the target temperature, is written as [147,4]:

\lambda ^{mf} (T) = N \sum _{f} \int d\epsilon 2\pi
...vert f>\vert^2 f(\epsilon, T)I(\epsilon - \epsilon _{if}, T),
\end{displaymath} (41)

where N is the target density, <i|H'|f> is the matrix element for molecular formation, $f(\epsilon, T)$is the kinetic energy distribution in the center of mass for the collision at the temperature T (e.g., the Maxwellian distribution if $\mu t$is thermalized), $\epsilon _{if}$ is the resonant energy, and $I(\Delta,
T)$ is the resonance intensity profile.

The popular choice of the interaction operator has been:

H'={\bf d}\cdot {\bf E},
\end{displaymath} (42)

where ${\bf d}$ is the dipole operator of $d\mu t$ and ${\bf E}$is the electric field at the center of mass of the $d\mu t$ due both to the spectator nucleus and electrons in the molecular complex. The importance of including the field of the electrons was pointed out by Cohen and Martin [142] and independently by Menshikov and Faifman [143]; it screens the nuclear field, substantially reducing the matrix elements.

The use of the dipole operator in H' was criticized by Petrov et al. [144,145,146] and by Scrinzi [148], who each proposed alternative forms of the operators, but the accuracies of simplifications adopted were in turn questioned by Faifman et al. [134,71]. Faifman and his colleagues recently included the quadrupole corrections in the interaction operator in Ref. [71], and those are what we used in our analysis of the experimental data. We note that the perturbative formulation used by Faifman as well as by Petrov and by Scrinzi were criticized by Lane [141] and Wallenius [45,46]. Lane [141] also criticized the formulation of earlier work by Cohen and Leon [14]. In fact, Wallenius in his thesis goes as far as calling Menshikov and Faifman's justification for the Born approximation ``nonsense.'' Wallenius concludes, however, that different ``working formulae'' in the literature actually end up to be equivalent (presumably due to cancellation of what he calls mistakes), and would yield the same cross sections, if no further approximations were made [46].

Thus theorists seem to agree, though maybe for different reasons, at least on the form of the starting formula they use to calculate the matrix elements. For actual evaluations, however, further approximations are necessary in the interaction operators, such as the multipole expansion as mentioned above, or in the wave functions as discussed next.

next up previous contents
Next: Wave functions Up: The standard Vesman model Previous: The standard Vesman model