# What is the probability of fusion reaction happening with energy less than the coulomb barrier (probability of reacting nuclei quantum tunneling)?

$$E_{out} = 4 \pi n_1 n_2 (m_0-m_a) c^2{(\frac {m}{2k_B T})}^{3/2} \int_0^ \infty \sigma(v) v^2 e^{\frac {-mv^2}{2k_B T}} dv$$

$$m_0$$ is the sum masses of the reacting nuclei, $$m_a$$ is the sum masses of the product nuclei, $$n_1$$ is the number density of one of the reacting nucleus and $$n_2$$ is the number density of the other reacting nucleus, $$\sigma$$ is the probability of fusion reaction happening, if the energy is less than the coulomb barrier, it is the probability of quantum tunneling through the coulomb barrier. If we could find that probability we could calculate the total energy output of a nuclear fusion reaction.

• physics.stackexchange.com/a/450292 – user191954 Dec 25 '18 at 13:37
• For many reactions one can look up the cross sections. For ‘easy’ reactions such as D-T one needs to include resonances with the compound nucleus to get an accurate description of the cross section. – Jon Custer Dec 25 '18 at 16:20
• I am trying to derive a formula that works for any light nuclei – Film Lee Dec 26 '18 at 15:51