= Neutron flux
{title2=$\phi(E)=v(E)n(E)$}
{wiki}
Let $n(E)\,dE$ be the number of <neutrons> per unit volume in an <energy> interval. The scalar <neutron flux> per unit <energy> is $\phi(E)=v(E)n(E)$, with $v(E)$ the <speed>. Its integral has units of inverse area per unit <time>. It counts path length traversed per volume per <time>, including all directions; it is not the signed current through an oriented surface.
For a small uniform specimen containing $N_T$ target <atomic nuclei>, a reaction-specific <collision cross-section> $\sigma_a(E)$ gives the expected production rate
$$
R=N_T\int_0^\infty\sigma_a(E)\phi(E)\,dE.
$$
Assume the incident field is little changed within the specimen, target depletion is negligible and each counted reaction creates the product of interest. If that product has a production branching fraction below one, include it in $\sigma_a$ or as an additional factor. A single <neutron activation> measurement supplies one weighted integral of an <energy> distribution, not a unique reconstruction of that distribution.
For an original two-group example, stipulate response <collision cross-sections> $\sigma_s=10\sigma_f$. With the same total scalar <neutron flux> $\Phi$, changing from all fast <neutrons> to half slow and half fast changes $R/(N_T\sigma_f\Phi)$ from $1$ to $5.5$. No additional <neutrons> need be emitted by the source. The chosen factor ten is not a historical <silver> measurement.
For some capture reactions over a limited low-<energy> interval, $\sigma_a\propto1/v$ is a useful approximation. At fixed <neutron flux>, it gives increasing reaction probability per path length as <speed> decreases. At fixed <neutron> number density, $\sigma_a v$ is constant in this approximation, so the rate per target is unchanged. A real <neutron moderator> changes the local density, <energy> distribution, escape and absorption together. Neither all <collision cross-sections> nor all <neutron activation> rates vary monotonically as $1/v$.
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