Energy and momenta balances¶

Understanding energy and momentum balances is crucial to understanding of ion transport through matter. This material is reasonably common and spread out, but I’m referencing [Knoll2000] and ({Nastasi, et. al. 1996)

Consider the reaction \(\ce{_{4}^{9}Be\left(p,\alpha\right)}\) with \(T_{p}\) and \(T_{\alpha}\) given, and \(\alpha\) emerges at an angle of \(90^{\circ}\). Find \(T_{R}\) for the recoil.

First, we want to figure out the recoil. The recoil will have a \(Z=4+1-2=3\) and an \(A=9+1-4=6\) and is therefore \(\ce{^{6}Li}\). Then we move on to our energy conservation with

\[\cancelto{0}{T_{Be}}+m_{Be}c^{2}+T_{p}+m_{p}c^{2}=T_{R}+m_{R}c^{2}+T_{\alpha}+m_{\alpha}c^{2}T_{R}=T_{\alpha}-T_{p}+\left(m_{R}+m_{\alpha}-m_{p}-m_{Be}\right)c^{2}\]

And we follow this up with a momentum conservation

\[ \begin{align}\begin{aligned}\cancelto{0}{p_{Be}}+p_{p}=p_{R}+p_{\alpha}\frac{1}{c}\sqrt{T_{p}\left(T_{p}+2m_{p}c^{2}\right)}=\frac{1}{c}\sqrt{T_{R}\left(T_{R}+2m_{R}c^{2}\right)}+\frac{1}{c}\sqrt{T_{\alpha}\left(T_{\alpha}+2m_{\alpha}c^{2}\right)}\\and solve this, for :math:`T_{R}`.\end{aligned}\end{align} \]

Energy and momenta balances¶

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