Editing Nuclear fusion (section)

=== Parameterization of cross section ===
{{unreferenced section|date=August 2023}}
The probability that fusion occurs is greatly increased compared to the classical picture, thanks to the smearing of the effective radius as the [[Matter wave|de Broglie wavelength]] as well as [[quantum tunneling]] through the potential barrier. To determine the rate of fusion reactions, the value of most interest is the [[Cross section (physics)|cross section]], which describes the probability that particles will fuse by giving a characteristic area of interaction. An estimation of the fusion cross-sectional area is often broken into three pieces:
:<math>\sigma \approx \sigma_\text{geometry} \times T \times R, </math>
where <math>\sigma_\text{geometry} </math> is the geometric cross section, {{mvar|T}} is the barrier transparency and {{mvar|R}} is the reaction characteristics of the reaction.

<math>\sigma_\text{geometry} </math> is of the order of the square of the de Broglie wavelength <math>\sigma_\text{geometry} \approx \lambda^2 = \bigg( \frac{\hbar}{m_r v} \bigg)^2 \propto \frac{1}{\epsilon} </math> where <math>m_r </math> is the reduced mass of the system and <math>\epsilon  </math> is the center of mass energy of the system.

{{mvar|T}} can be approximated by the Gamow transparency, which has the form: <math>T \approx e^ {- \sqrt{\epsilon_G /\epsilon} } </math> where <math>\epsilon_G = (\pi \alpha Z_1 Z_2)^2 \times 2 m_r c^2 </math> is the [[Gamow factor]] and comes from estimating the quantum tunneling probability through the potential barrier.

{{mvar|R}} contains all the nuclear physics of the specific reaction and takes very different values depending on the nature of the interaction. However, for most reactions, the variation of <math>R(\epsilon)</math> is small compared to the variation from the Gamow factor and so is approximated by a function called the astrophysical [[S-factor]], <math>S(\epsilon)</math>, which is weakly varying in energy. Putting these dependencies together, one approximation for the fusion cross section as a function of energy takes the form:

:<math>\sigma(\epsilon) \approx \frac{S(\epsilon)}{\epsilon} e^{ - \sqrt{\epsilon_G / \epsilon}}</math>

More detailed forms of the cross-section can be derived through nuclear physics-based models and [[R-matrix]] theory.