Thermonuclear Fusion: The Basic Process — Question 5

Question 5

The Gamow peak balances thermal rarity against tunneling suppression

Approximate a nonresonant fusion integrand by $I(E)\propto e^{-\phi(E)}$, where $\phi(E)=E/(kT)+\sqrt{E_G/E}$. Ignore slowly varying prefactors. Let kT=1.00 keV and $E_G=1000$ keV.

1. Minimize the exponent to find the peak energy $E_0$.

2. Evaluate $E_0$ and the exponent there, and compare with kT.

Question 5 - Solution

The Gamow peak balances thermal rarity against tunneling suppression

(a) Method and calculation

$$\phi\prime(E)=\frac1{kT}-\frac{\sqrt{E_G}}{2E^{3/2}}=0.$$$$\boxed{E_0=\left[\frac{E_G(kT)^2}{4}\right]^{1/3}}.$$

(b) Interpretation and further calculation

$$\boxed{E_0=250^{1/3}=6.29961\,\mathrm{keV},\quad E_0/(kT)=6.29961}.$$$$\sqrt{E_G/E_0}=2E_0/(kT)\Rightarrow\boxed{\phi(E_0)=3E_0/(kT)=18.8988}.$$

Physical check

The dominant reacting particles lie above the thermal energy scale. This leading-exponential peak assumes a smooth nonresonant cross-section factor.