Neutron Decay before Absorption
Dr. Yang (my professor for neutronics) considered it absolutely
essential to prove that the probability of the neutron itself decaying
was negligible for common applications of neutronics. I believe the
cross sections and half lives come from (Lamarsh, 2002), but I’m not
sure where this idea is fleshed out first.
We start with the half life of a neutron, which is \(12\unit{min}\),
and we compare that to a thermal neutron traveling through water. The
water has an absorption cross section of
\(\Sigma_{a}=0.022\unit{cm}\). So, as a general analysis point, we
want to find out the average lifetime of a neutron subject to decay, and
the average lifetime of a neutron subject to absorption.
Determination of the lifetime due to decay is straightforward. We simply
use the halflife formula to convert the half life to a lifetime.
\[T_{\nicefrac{1}{2}} = \frac{\ln \left( 2 \right)}{\lambda}\]
\[\lambda = \frac{1}{\tau_{d}}\]
\[\tau_{d} = \frac{T_{\nicefrac{1}{2}}}{\ln \left( 2 \right)}=1038.74\unit{s}\]
The calculation of the lifetime of an absorbed neutron is only slightly
more complicated. First, we’ll determine its “mean free path” (which is
denoted by \(\lambda\) in usual nomenclature like decay rate from
above, so we’ll use \(\Lambda\)), and then, using the velocity of a
thermal neutron, we’ll convert that to a lifetime.
\[\Lambda = \frac{1}{\Sigma_{a}}\]
\[v=\sqrt{\frac{E}{2m}}=\sqrt{\frac{0.0257\unit{eV}}{2\cdot 938\unit{\frac{MeV}{c^2}}}}\]
\[\tau_{a}=\frac{\Lambda}{v}=\frac{50\unit{cm}}{2200\unit{\nicefrac{cm}{s}}}=2\times 10^{-4}\unit{s}\]
So, following the above calculation, the mean lifetime of a neutron
before absorption is \(10^{7}\) shorter than that of a neutron
before its own decay. Note that the cross section is conservative, there
are larger absorbers; the speed of the neutron is also conservative,
thermal neutrons are the slowest without going to great lengths
(anything slower are called “cold” neutrons and do not happen
naturally). Therefore the probability of decay is very small, and is
negligible to our current uncertainties in neutrons.
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