Process Edit

Nuclear fusion in stars Edit

Requirements Edit

Methods for achieving fusion Edit

Important reactions Edit

Mathematical description of cross section Edit

Fusion under classical physics Edit 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: E thresh = 1 4 π ϵ 0 Z 1 Z 2 r → 2 protons 1 4 π ϵ 0 e 2 1 fm ≈ 1.4 MeV {\displaystyle E_{\text{thresh}}={\frac {1}{4\pi \epsilon _{0}}}{\frac {Z_{1}Z_{2}}{r}}{\xrightarrow {\text{2 protons}}}{\frac {1}{4\pi \epsilon _{0}}}{\frac {e^{2}}{1{\text{ fm}}}}\approx 1.4{\text{ MeV}}} 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 10 − 290 {\displaystyle 10^{-290}} , that is, fusion would never occur. However, fusion in the sun does occur due to quantum mechanics. Parameterization of cross section Edit The probability that fusion occurs is greatly increased compared to the classical picture, thanks to the smearing of the effective radius as the DeBroglie wavelength as well as quantum tunnelling through the potential barrier. To determine the rate of fusion reactions, the value of most interest is the cross section, which describes the probability that particle will fuse by giving a characteristic area of interaction. An estimation of the fusion cross sectional area is often broken into three pieces: σ ≈ σ g e o m e t r y × T × R {\displaystyle \sigma \approx \sigma _{geometry}\times T\times R} Where σ g e o m e t r y {\displaystyle \sigma _{geometry}} is the geometric cross section, T is the barrier transparency and R is the reaction characteristics of the reaction. σ g e o m e t r y {\displaystyle \sigma _{geometry}} is of the order of the square of the de-Broglie wavelength σ g e o m e t r y ≈ λ 2 = ( ℏ 2 m r v ) 2 ∝ 1 ϵ {\displaystyle \sigma _{geometry}\approx \lambda ^{2}={\bigg (}{\frac {\hbar ^{2}}{m_{r}v}}{\bigg )}^{2}\propto {\frac {1}{\epsilon }}} where m r {\displaystyle m_{r}} is the reduced mass of the system and ϵ {\displaystyle \epsilon } is the center of mass energy of the system. T can be approximated by the Gamow transparency, which has the form: T ≈ e − ϵ G / ϵ {\displaystyle T\approx e^{-{\sqrt {\epsilon _{G}/\epsilon }}}} where ϵ G = ( π α Z 1 Z 2 ) 2 × 2 m r c 2 {\displaystyle \epsilon _{G}=(\pi \alpha Z_{1}Z_{2})^{2}\times 2m_{r}c^{2}} is the Gamow factor and comes from estimating the quantum tunneling probability through the potential barrier. 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 R ( ϵ ) {\displaystyle R(\epsilon )} is small compared to the variation from the Gamow factor and so is approximated by a function called the Astrophysical S-factor, S ( ϵ ) {\displaystyle S(\epsilon )} , 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: σ ( ϵ ) ≈ S ( ϵ ) ϵ e − ϵ G / ϵ {\displaystyle \sigma (\epsilon )\approx {\frac {S(\epsilon )}{\epsilon }}e^{-{\sqrt {\epsilon _{G}/\epsilon }}}} More detailed forms of the cross section can be derived through nuclear physics based models and R matrix theory. Formulas of fusion cross sections Edit The Naval Research Lab's plasma physics formulary[31] gives the total cross section in barns as a function of the energy (in keV) of the incident particle towards a target ion at rest fit by the formula: σ N R L ( ϵ ) = A 5 + ( ( A 4 − A 3 ϵ ) 2 + 1 ) ) − 1 A 2 ϵ ( e A 1 ϵ − 1 / 2 − 1 ) {\displaystyle \sigma _{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)}}} with the following coefficient values: NRL Formulary Cross Section Coefficients DT(1) DD(2i) DD(2ii) DHe3(3) TT(4) THe3(6) A1 45.95 46.097 47.88 89.27 38.39 123.1 A2 50200 372 482 25900 448 11250 A3 1.368e-4 4.36e-4 3.08e-4 3.98e-3 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[32] also reports a R-matrix calculated cross sections fitting observation data with Padé approximants. With energy in units of keV and cross sections in units of millibarn, the astrophysical factor has the form: S Bosch-Hale ( ϵ ) = A 1 + ϵ ( A 2 + ϵ ( A 3 + ϵ ( A 4 + ϵ A 5 ) ) ) 1 + ϵ ( B 1 + ϵ ( B 2 + ϵ ( B 3 + ϵ B 4 ) ) ) {\displaystyle 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 )}}}} , with the coefficient values: Bosch-Hale Astrophysical Cross Section Coefficients DT(1) DD(2ii) DHe3(3) THe4 ϵ G {\displaystyle \epsilon _{G}} 31.3970 68.7508 31.3970 34.3827 A1 5.5576e4 5.7501e6 5.3701e4 6.927e4 A2 2.1054e2 2.5226e3 3.3027e2 7.454e8 A3 -3.2638e-2 4.5566e1 -1.2706e-1 2.050e6 A4 1.4987e-6 0 2.9327e-5 5.2002e4 A5 1.8181e-10 0 -2.5151e-9 0 B1 0 -3.1995e-3 0 6.38e1 B2 0 -8.5530e-6 0 -9.95e-1 B3 0 5.9014e-8 0 6.981e-5 B4 0 0 0 1.728e-4 Applicable Energy Range [keV] 0.5-5000 0.3-900 0.5-4900 0.5-550 ( Δ S ) max % {\displaystyle (\Delta S)_{\text{max}}\%} 2.0 2.2 2.5 1.9

See also Edit

References Edit