User:Programmatically/Gamow Peak

The reaction rate density between species A and B, having number densities n_A,B is given by:

${\displaystyle r=n_{A}\,n_{B}\,k}$

where k is the reaction rate constant of each single elementary binary reaction composing the nuclear fusion process:

${\displaystyle k=\langle \sigma (v)\,v\rangle }$

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

However, since the reaction involves quantum tunneling, there is an exponential damping at low energies that depends on Gamow factor E_G, giving an Arrhenius equation:

${\displaystyle \sigma (E)={\frac {S(E)}{E}}e^{-{\sqrt {\frac {E_{\text{G}}}{E}}}}}$

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 :

${\displaystyle {\frac {r}{V}}=n_{A}n_{B}\int _{0}^{\infty }{\frac {S(E)}{E}}\,e^{-{\sqrt {\frac {E_{\text{G}}}{E}}}}2{\sqrt {\frac {E}{\pi (kT)^{3}}}}e^{-{\frac {E}{kT}}}\,{\sqrt {\frac {2E}{m_{\text{R}}}}}dE}$

where ${\displaystyle m_{\text{R}}={\frac {m_{1}m_{2}}{m_{1}+m_{2}}}}$ is the reduced mass.

Since this integration has an exponential damping at high energies of the form ${\displaystyle \sim e^{-{\frac {E}{kT}}}}$ and at low energies from the Gamow factor, the integral almost vanished everywhere except around the peak, called Gamow peak,^[1]: 185 at E₀, where:

${\displaystyle {\frac {\partial }{\partial E}}\left(-{\sqrt {\frac {E_{\text{G}}}{E}}}-{\frac {E}{kT}}\right)\,=\,0}$

Thus:

${\displaystyle E_{0}=\left({\frac {1}{2}}kT{\sqrt {E_{\text{G}}}}\right)^{\frac {2}{3}}}$

The exponent can then be approximated around E₀ as:

${\displaystyle e^{-{\frac {E}{kT}}-{\sqrt {\frac {E_{\text{G}}}{E}}}}\approx e^{-{\frac {3E_{0}}{kT}}}\exp \left(-{\frac {(E-E_{0})^{2}}{{\frac {4}{3}}E_{0}kT}}\right)}$

And the reaction rate is approximated as:^[2]

${\displaystyle {\frac {r}{V}}\approx n_{A}\,n_{B}\,{\frac {4{\sqrt {2}}}{\sqrt {3m_{\text{R}}}}}\,{\sqrt {E_{0}}}{\frac {S(E_{0})}{kT}}e^{-{\frac {3E_{0}}{kT}}}}$

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.^{[3]: ch. 3}

Thus, the limiting reaction in the CNO cycle, proton capture by ¹⁴
₇N, has S(E₀) ~ S(0) = 3.5 keV·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) = 4×10⁻²² keV·b.^[4][5] 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.