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Total and effective formation rates
For our analysis, it is convenient to express the formation rates as a
function of the
laboratory (lab) energy, as opposed to the target
temperature given in Eq. 2.27. Specifying the initial
and final states, we then have^{}:

(47) 
where
where
is the initial rotational population distribution,
which for equilibrated targets is the standard Boltzmann distribution,
and A_{if} is a coefficient which depends on initial and final quantum
numbers. The factor
is the Doppler broadening
profile due to target motion and recoil, whose exact form is derived in
Ref. [133], but for
can be
approximated [157] as a Gaussian distribution with the width of:

(50) 
In Eq. 2.35, a
function resonance profile was
assumed (the classical Vesman model), but even with that, the formation
rate in the lab frame has a distribution with nonzero width due to the
abovementioned Doppler broadening.
At epithermal energies, the transition formation matrix elements
become very large, which means both the
formation rate and back decay width are large, resulting in a
significant probability for
not fusing but returning to the
entrance channel
.
The effective formation rate is a renormalized rate taking into account the
fusion probability, as defined in Ref. [134]:

(51) 
where
For a high density (
more than about 0.1) target such as ours,
Faifman assumes complete rotational relaxation of the K_{f} levels, hence
dropping the K_{f} dependence,
where
is the Boltzmann distribution of the K_{f} states
^{}.
The effective fusion probability for the
molecular complex is defined as [133]:

(57) 
where
is the rate for nonresonant formation,
which does not backdecay. For
,
W^{F} can be written:

(58) 
which can be understood as the average of fusion probabilities from each
state weighted by the formation rate of that state.
Figure 2.4 shows the molecular formation rates
and effective fusion probabilities W^{F} for K_{i}=0(ortho) and K_{i}=1 (para) cases, calculated by Faifman et
al. [70,71,72] for a 3 K target. Also shown are
the effective rates
.
Note that
less than about 10^{7}s^{1} is invisible in the scale plotted.
Figure 2.4:
Formation rates
for
(top) and the fusion probability W^{F}(bottom), calculated by Faifman [70,71,72] for 3
K. Also shown in dashed lines are the effective rate
.
The rates are normalized to liquid
hydrogen density.

Next: Subthreshold resonances
Up: The standard Vesman model
Previous: Decay of the molecular