Thermonuclear Fusion: The Basic Process — Question 6

Question 6

Pair counting distinguishes identical-reactant and mixed-reactant rates

For a homogeneous gas use reaction rate density $R_{ab}=n_an_b\langle\sigma v\rangle$ for distinct species and $R_{aa}=\tfrac12n_a^2\langle\sigma v\rangle$ for identical species. Take total ion density $n=10^{20}\,\mathrm{m^{-3}}$ and the same illustrative reactivity $10^{-22}\,\mathrm{m^3/s}$ for both comparisons.

1. Compare event rate densities for a 50:50 distinct-species mixture and a single-species gas.

2. Compare ion-consumption rates and explain the factor one-half for identical pairs.

Question 6 - Solution

Pair counting distinguishes identical-reactant and mixed-reactant rates

(a) Method and calculation

$$\boxed{R_{ab}=(n/2)^2\langle\sigma v\rangle=2.50\times10^{17}\,\mathrm{m^{-3}s^{-1}}}.$$$$\boxed{R_{aa}=\tfrac12n^2\langle\sigma v\rangle=5.00\times10^{17}\,\mathrm{m^{-3}s^{-1}}}.$$

(b) Interpretation and further calculation

$$\boxed{|\dot n|_{ab}=2R_{ab}=5.00\times10^{17},\quad|\dot n|_{aa}=2R_{aa}=10^{18}\,\mathrm{m^{-3}s^{-1}}}.$$$$\boxed{N_{\mathrm{unordered\,pairs}}=N(N-1)/2}\quad\text{avoids counting each identical pair twice}.$$

Physical check

The equal reactivities are a supplied comparison, not a statement about real D-D and D-T cross sections. Both reactions consume two ions per event.