Reactivity in nuclear fusion In nuclear fusion, the graph of reactivity is shown as below.
How can we explain that by increasing the temperature after a certain value, the reactivity decreases?

 A: The main reason for the peak in the reactivities is a nuclear resonance effect. To show this, it's worth looking into the rich physics of cross-sections (see Bosch & Hale, Nuclear Fusion, 1992).

There are three main factors that affect fusion cross-sections (plotted above with data from Bosch & Hale),
\begin{equation}
\sigma = S(E) \frac{1}{E}\exp{\left(-\frac{B_G}{\sqrt{E}}\right)}\,.
\end{equation}

*

*The factor $\frac{1}{E}$ derives from the effective size of a particle due to its de Broglie wavelength,
\begin{equation} 
\text{effective size} \sim \pi \lambda_{\mathrm{dB}}^2  = \pi\frac{h^2}{p^2} =\pi \frac{h^2}{2mE} \propto \frac{1}{E}\,.
\end{equation}

*The exponential factor derives from the tunneling probability through the potential barrier created by the Coulomb repulsion between the reactants, \begin{equation} 
\text{tunneling probability} \propto  \exp{\left(-\frac{\pi \alpha Z_1 Z_2  \sqrt{2m_rc^2}}{\sqrt{E}}\right)} \equiv \exp{\left(-\frac{B_G}{\sqrt{E}}\right)}\,,
\end{equation}
where $Z_1$ and $Z_2$ are the atomic numbers of the particles, $m_r$ is the reduced mass of the system, $\alpha={k_e e^2}/{\hbar c}$ is the fine structure constant, and $B_G$ is known as the Gamow constant (well-explained in this wiki entry).

*The factor $S(E)$ is the so-called S-function and was introduced by astrophysicists to capture the remaining, relatively slowly varying nuclear physics contribution to the cross-section. In fact, the S-function varies so slowly that it can be plotted on linear scales (plotted below with data from Bosch & Hale). Hence it is often more useful to compare the S-functions of similar reactions, rather than the cross sections. (For more information about the astrophysical origin of the S-function see this review article by Margaret Burbidge et al., Rev. Mod. Phys., 1957.)

The peaks in the S-functions are due to resonances, which arise only at certain energies when the relative phase and amplitude of the internal bound wavefunction and external traveling wavefunction of the quasi-particle match well and facilitate tunneling. This causes the cross sections and reactivities to peak for DT and $\mathrm{D}\,^3\mathrm{He}$, while the DD reaction is far from resonance in the plotted energy range.

A: At low temperatures the reactants in each of the reactions will not have enough energy on average to overcome electromagnetic repulsion and get close enough so that they can interact via the strong force. But if the temperature is too high then the reactants will have so much energy on average that the strong force cannot bind them together and they will just go past each other. Also, collisions with fast moving particles will break apart reaction products faster than they can be formed. So for each of the reactions there is an ideal temperature - not too hot, not too cold - at which the reaction rate is maximised.
