Thermonuclear Fusion: The Basic Process — Question 7

Question 7

An unequal fuel mixture reduces fusion rate at fixed total density

Let $n_D=xn$ and $n_T=(1-x)n$ at fixed total ion density n and fixed temperature. Reaction rate density is $R=n_Dn_T\langle\sigma v\rangle$.

1. Find the mixture fraction x that maximizes R and show it is a maximum.

2. For x=.700 find the fraction of the maximum rate, and explain why changing total density is a different comparison.

Question 7 - Solution

An unequal fuel mixture reduces fusion rate at fixed total density

(a) Method and calculation

$$R=x(1-x)n^2\langle\sigma v\rangle,\quad\frac{dR}{dx}=(1-2x)n^2\langle\sigma v\rangle.$$$$\boxed{x_{\max}=1/2,\quad R_{\max}=\frac14n^2\langle\sigma v\rangle,\quad R^{\prime\prime}<0}.$$

(b) Interpretation and further calculation

$$\boxed{R(.700)/R_{\max}=4(.700)(.300)=.840}.$$$$\boxed{R\propto n^2}\quad\text{at fixed composition and temperature}.$$

Physical check

The optimization applies only to a uniform two-species mixture at fixed total density and reactivity. Fuel evolution and impurities can change those constraints.