Flux and current in a spherical fusion chamber
This problem is from NUCL 510, but plays off of the material in
(Lamarsh, 2002).
Neutrons are produced uniformly and isotropically throughout a spherical
chamber containing a fixture of \(\ce{^3}{H}\) and
\(\ce{^{2}H}\) gasses at \(1\times 10^{8} \unit{K}\). Calculate
the flux and current in the chamber.
We know that there are fusion reactions
\[\ce{T}\left(d, n\right) \ce{^{4}He}\]
and
\[\ce{D}\left(d, n\right)\ce{^{3}He}\]
.
First we set up the geometry. We have a sphere with its \(z\) axis
through point \(A\). \(dV\) is the volume of the skirt
\(\overline{AB}\). so we have
\[d\psi = \frac{S}{4 \pi r^{2}} dV\]
We know that \(\phi = \int_{V} d\psi\), so we integrate from
\(0\) to \(2R \cos \theta\).
\[\phi = \int_{0}^{\nicefrac{pi}{2}} \int_{0}^{2R\cos \theta} \frac{S}{4\pi r^{2}} \underbrace{r^{2} \sin \theta dr d\theta d\varphi}_{\text{spherical lagrangian}}\]
\[\phi = \frac{S}{2}\int_{0}^{1} \int_{0}^{2R\mu} drd\mu\]
\[\phi = SR\int_{0}^{1}\mu d\mu\]
\[\phi = \frac{SR}{2}\]
For current, we need to apply the solid angle
\[\vec{\Omega} = \left( \sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta \right)\]
\[\vec{J} = \int_{0}{\nicefrac{\pi}{2}} \int_{0}{2R\cos \theta} \int_{0}^{2\pi} \frac{S}{4\pi r^{2}} \vec{\Omega} r^{2}\sin \theta d\varphi dr d\theta\]
and with
\(\int_{0}^{2\pi} \cos \varphi d\varphi=\int_{0}^{2\pi} \sin \varphi d\varphi=0\),
therefore \(J_{x} = 0\) and \(J_{y}=0\)
\[J_{z} = \int_{0}{\nicefrac{\pi}{2}} \int_{0}{2R\cos \theta} \int_{0}^{2\pi} \frac{S}{4\pi } \cos \theta \sin \theta d\varphi dr d\theta\]
\[J_{z} = \frac{S}{2}\int_{0}^{1} \int_{0}^{2R\mu} \mu dr d\mu\]
\[J_{z} = \frac{SR}{3}\times \vec{e_{z}} = \frac{SR}{3}\times \vec{e_{r}}\]
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