Question 5.6: Suppose that we want to subdivide the thermal neutron flux ϕ......
Suppose that we want to subdivide the thermal neutron flux ϕ(E), which follows a Maxwell–Boltzmann probability distribution, into 10 equally spaced intervals between 0.0 and 0.1 eV. If the relative magnitude of the neutron flux in each interval is given by the entries in the third row of the table below and the absorption cross section of a particular material is given by the entries in the fourth row, what is the value of the thermal absorption cross section that would have to be used to conserve reaction rates if we defined the thermal energy group to begin at 0.0 eV and to end at 0.10 eV?
To find the average value of the absorption cross section <σ_a> between 0.0 and 0.01 eV, we must multiply the value of the absorption cross section in each energy range by the value of the neutron flux in that energy range and divide by the total flux between 0 and 0.10 eV. Mathematically, this is equivalent to saying
\lt \sigma_{\mathrm{a}}\gt \,=\,\frac{\left(\sigma_{1}\phi_{1}+\sigma_{2}\phi_{2}+\sigma_{3}\phi_{3}+\sigma_{4}\phi_{4}+\sigma_{5}\phi_{5}+\sigma_{6}\phi_{6}+\sigma_{7}\phi_{7}+\sigma_{8}\phi_{8}+\sigma_{9}\phi_{9} + \sigma_{10}\phi_{10}\right)}{\left(\phi_{1}+\phi_{2}+\phi_{3}+\phi_{4}+\phi_{5}+\phi_{6}+\phi_{7}+\phi_{8} +\phi_{9} +\phi_{10}\right)}
\lt S_{a}\gt = \frac{(100 × 0.12 + 90 × 0.25+ 80 × 0.15 + 70 × 0.13 + 60 × 0.10 + 50 × 0.08 + 40 × 0.07 + 30 × 0.05 + 20 × 0.03 + 10 × 0.02 )}{(0.12 + 0.25 + 0.15 + 0.13 + 0.10 + 0.08 + 0.07 + 0.05 + 0.03 + 0.02)}
={\frac{\left(12+22.5+12+9.1+6+4+2.8+1.5+.6+.2\right)}{1.0}}
= 70.7 barns