User:WalkingRadiance/Gamow Peak

Reaction rate
The reaction rate density between species A and B, having number densities nA,B is given by:
 * $$r = n_A \, n_B \, k $$

where k is the reaction rate constant of each single elementary binary reaction composing the nuclear fusion process:
 * $$k = \langle \sigma(v)\,v \rangle$$

here, σ(v) is the cross-section at relative velocity v, and averaging is performed over all velocities.

Semi-classically, the cross section is proportional to $$\pi\,\lambda^2$$, where $$\lambda = h/p$$ is the de Broglie wavelength. Thus semi-classically the cross section is proportional to $\frac{m}{E}$.

However, since the reaction involves quantum tunneling, there is an exponential damping at low energies that depends on Gamow factor EG, giving an Arrhenius equation:
 * $$\sigma(E) = \frac{S(E)}{E} e^{-\sqrt{\frac{E_\text{G}}{E}}}$$

where S(E) depends on the details of the nuclear interaction, and has the dimension of an energy multiplied for a cross section.

One then integrates over all energies to get the total reaction rate, using the Maxwell–Boltzmann distribution and the relation :
 * $$\frac{r}{V} = n_A n_B \int_0^{\infty}\frac{S(E)}{E} \, e^{-\sqrt{\frac{E_\text{G}}{E}}} 2\sqrt{\frac{E}{\pi(kT)^3}} e^{-\frac{E}{kT}} \,\sqrt{\frac{2E}{m_\text{R}}}dE$$

where $$m_\text{R} = \frac{m_1 m_2}{m_1 + m_2}$$ is the reduced mass.

Since this integration has an exponential damping at high energies of the form $$\sim e^{-\frac{E}{kT}}$$ and at low energies from the Gamow factor, the integral almost vanished everywhere except around the peak, called Gamow peak, at E0, where:
 * $$\frac{\partial}{\partial E} \left( -\sqrt{\frac{E_\text{G}}{E}} - \frac{E}{kT}\right) \, = \, 0$$

Thus:
 * $$E_0 = \left(\frac{1}{2}kT \sqrt{E_\text{G}}\right)^\frac{2}{3}$$

The exponent can then be approximated around E0 as:
 * $$e^{-\frac{E}{kT} - \sqrt{\frac{E_\text{G}}{E}}} \approx e^{-\frac{3E_0}{kT}} \exp\left(-\frac{(E - E_0)^2}{\frac{4}{3} E_0 kT}\right)$$

And the reaction rate is approximated as:
 * $$\frac{r}{V} \approx n_A \, n_B \, \frac{4\sqrt{2}}{\sqrt{3 m_\text{R}}}\, \sqrt{E_0} \frac{S(E_0)}{kT} e^{-\frac{3E_0}{kT}} $$

Values of S(E0) are typically 10−3 – 103 keV·b, but are damped by a huge factor when involving a beta decay, due to the relation between the intermediate bound state (e.g. diproton) half-life and the beta decay half-life, as in the proton–proton chain reaction. Note that typical core temperatures in main-sequence stars give kT of the order of keV.

Thus, the limiting reaction in the CNO cycle, proton capture by, has S(E0) ~ S(0) = 3.5keV·b, while the limiting reaction in the proton–proton chain reaction, the creation of deuterium from two protons, has a much lower S(E0) ~ S(0) = 4×10−22keV·b. Incidentally, since the former reaction has a much higher Gamow factor, and due to the relative abundance of elements in typical stars, the two reaction rates are equal at a temperature value that is within the core temperature ranges of main-sequence stars.