Editing Nuclear fusion (section)

== Mathematical description of cross section ==

=== Fusion under classical physics ===
{{unreferenced section|date=August 2023}}
In a classical picture, nuclei can be understood as hard spheres that repel each other through the Coulomb force but fuse once the two spheres come close enough for contact. Estimating the radius of an atomic nuclei as about one femtometer, the energy needed for fusion of two hydrogen is:

:<math chem>E_{\ce{thresh}}= \frac{1}{4\pi \epsilon_0} \frac{Z_1 Z_2}{r} \ce{->[\text{2 protons}]} \frac{1}{4\pi \epsilon_0} \frac{e^2}{1\ \ce{fm}} \approx 1.4\ \ce{MeV}</math>

This would imply that for the core of the sun, which has a [[Boltzmann distribution]] with a temperature of around 1.4 keV, the probability hydrogen would reach the threshold is <math>10^{-290}</math>, that is, fusion would never occur. However, fusion in the sun does occur due to quantum mechanics.

=== 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.

=== Formulas of fusion cross sections ===
The Naval Research Lab's plasma physics formulary<ref>{{Cite web|url=https://library.psfc.mit.edu/catalog/online_pubs/NRL_FORMULARY_13.pdf|title=NRL PLASMA FORMULARY|last=Huba|first=J.|year=2003|website=MIT Catalog|access-date=11 November 2018|archive-date=17 April 2018|archive-url=https://web.archive.org/web/20180417231213/http://library.psfc.mit.edu/catalog/online_pubs/NRL_FORMULARY_13.pdf|url-status=live}}</ref> gives the total cross section in [[Barn (unit)|barns]] as a function of the energy (in keV) of the incident particle towards a target ion at rest fit by the formula:

:<math>\sigma^\text{NRL}(\epsilon) = \frac{A_5 + \big( (A_4 - A_3 \epsilon)^2 + 1 \big)^{-1} A_2}{ \epsilon (e^{A_1 \epsilon^{-1/2} }-1)}</math> with the following coefficient values:
{| class="wikitable" style="text-align:right;"
|+NRL Formulary Cross Section Coefficients
!
!DT(1)
!DD(2i)
!DD(2ii)
!DHe<sup>3</sup>(3)
!TT(4)
!The<sup>3</sup>(6)
|-
|A1
|45.95
|46.097
|47.88
|89.27
|38.39
|123.1
|-
|A2
|50200
|372
|482
|25900
|448
|11250
|-
|A3
|{{val|1.368e-2}}
|{{val|4.36e-4}}
|{{val|3.08e-4}}
|{{val|3.98e-3}}
|{{val|1.02e-3}}
|0
|-
|A4
|1.076
|1.22
|1.177
|1.297
|2.09
|0
|-
|A5
|409
|0
|0
|647
|0
|0
|}
Bosch-Hale<ref>{{Cite journal|title=Improved formulas for fusion cross-sections and thermal reactivities|journal=Nuclear Fusion|volume=32|issue=4|pages=611–631|last=Bosch|first=H. S|year=1993|doi=10.1088/0029-5515/32/4/I07|s2cid=55303621}}</ref> also reports a R-matrix calculated cross sections fitting observation data with [[Padé approximant|Padé rational approximating coefficient]]s. With energy in units of keV and cross sections in units of millibarn, the factor has the form:

:<math>S^{\text{Bosch-Hale}}(\epsilon) = \frac{A_1 + \epsilon\bigg( A_2 + \epsilon \big(A_3 + \epsilon(A_4 + \epsilon A_5)\big)\bigg)}{1 + \epsilon\bigg(B_1 + \epsilon \big(B_2 + \epsilon (B_3 +\epsilon B_4)\big) \bigg)}</math>, with the coefficient values: 
{| class="wikitable" style="text-align:right;"
|+Bosch-Hale coefficients for the fusion cross section
!
!DT(1)
!DD(2ii)
!DHe<sup>3</sup>(3)
!The<sup>4</sup>
|-
|<math>\epsilon_G</math>
|31.3970
|68.7508
|31.3970
|34.3827
|-
|A1
|{{val|5.5576e4}}
|{{val|5.7501e6}}
|{{val|5.3701e4}}
|{{val|6.927e4}}
|-
|A2
|{{val|2.1054e2}}
|{{val|2.5226e3}}
|{{val|3.3027e2}}
|{{val|7.454e8}}
|-
|A3
| {{val|-3.2638e-2}}
| {{val|4.5566e1}}
| {{val|-1.2706e-1}}
|{{val|2.050e6}}
|-
|A4
|{{val|1.4987e-6}}
|0
|{{val|2.9327e-5}}
|{{val|5.2002e4}}
|-
|A5
|{{val|1.8181e-10}}
| 0
| {{val|-2.5151e-9}}
|0
|-
|B1
|0
| {{val|-3.1995e-3}}
|0
|{{val|6.38e1}}
|-
|B2
|0
| {{val|-8.5530e-6}}
| 0
| {{val|-9.95e-1}}
|-
|B3
|0
|{{val|5.9014e-8}}
|0
|{{val|6.981e-5}}
|-
|B4
|0
|0
|0
|{{val|1.728e-4}}
|-
|Applicable Energy Range [keV]
|0.5–5000 
|0.3–900
|0.5–4900
|0.5–550
|-
|<math>(\Delta S)_{\text{max}}\%</math>
|2.0
|2.2
|2.5
|1.9
|}
where <math>\sigma^{\text{Bosch-Hale}}(\epsilon) = \frac{S^{\text{Bosch-Hale}}(\epsilon)}{\epsilon \exp(\epsilon_G/\sqrt{\epsilon})}</math>

===Maxwell-averaged nuclear cross sections===
{{unreferenced section|date=August 2023}}
In fusion systems that are in thermal equilibrium, the particles are in a [[Maxwell–Boltzmann distribution]], meaning the particles have a range of energies centered around the plasma temperature. The sun, magnetically confined plasmas and inertial confinement fusion systems are well modeled to be in thermal equilibrium. In these cases, the value of interest is the fusion cross-section averaged across the Maxwell–Boltzmann distribution. The Naval Research Lab's plasma physics formulary tabulates Maxwell averaged fusion cross sections reactivities in <math>\mathrm{cm^3/s}</math>.
{| class="wikitable" style="text-align:right;"
|+NRL Formulary fusion reaction rates averaged over Maxwellian distributions
!Temperature [keV]
!DT(1)
!DD(2ii)
!DHe<sup>3</sup>(3)
!TT(4)
!The<sup>3</sup>(6)
|-
|1
|{{val|5.5e-21}}
|{{val|1.5e-22}}
|{{val|1.0e-26}}
|{{val|3.3e-22}}
|{{val|1.0e-28}}
|-
|2
|{{val|2.6e-19}}
|{{val|5.4e-21}}
|{{val|1.4e-23}}
|{{val|7.1e-21}}
|{{val|1.0e-25}}
|-
|5
|{{val|1.3e-17}}
|{{val|1.8e-19}}
|{{val|6.7e-21}}
|{{val|1.4e-19}}
|{{val|2.1e-22}}
|-
|10
| {{val|1.1e-16}}
| {{val|1.2e-18}}
| {{val|2.3e-19}}
|{{val|7.2e-19}}
|{{val|1.2e-20}}
|-
|20
|{{val|4.2e-16}}
|{{val|5.2e-18}}
|{{val|3.8e-18}}
|{{val|2.5e-18}}
|{{val|2.6e-19}}
|-
|50
|{{val|8.7e-16}}
| {{val|2.1e-17}}
| {{val|5.4e-17}}
|{{val|8.7e-18}}
|{{val|5.3e-18}}
|-
|100
|{{val|8.5e-16}}
| {{val|4.5e-17}}
|{{val|1.6e-16}}
|{{val|1.9e-17}}
|{{val|2.7e-17}}
|-
|200
|{{val|6.3e-16}}
| {{val|8.8e-17}}
| {{val|2.4e-16}}
| {{val|4.2e-17}}
|{{val|9.2e-17}}
|-
|500
|{{val|3.7e-16}}
|{{val|1.8e-16}}
|{{val|2.3e-16}}
|{{val|8.4e-17}}
|{{val|2.9e-16}}
|-
|1000
|{{val|2.7e-16}}
|{{val|2.2e-16}}
|{{val|1.8e-16}}
|{{val|8.0e-17}}
|{{val|5.2e-16}}
|}
For energies <math>T \le 25 \text{ keV}</math> the data can be represented by:

:<math>(\overline{\sigma v})_{DD} = 2.33\times 10^{-14}\cdot T^{-2/3} \cdot e^{-18.76\ T^{-1/3}} \mathrm{~{cm}^3/s}</math>

:<math>(\overline{\sigma v})_{DT} = 3.68\times 10^{-12}\cdot T^{-2/3} \cdot e^{-19.94\ T^{-1/3}} \mathrm{~{cm}^3/s}</math>
with {{mvar|T}} in units of keV.