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Nuclear Fusion Reaction Rate, Reactivity and Cross-Section

The Reaction Rate

\[ R_r(T) = n_a * n_b * \langle \sigma v \rangle (T) * V \]

Reaction Rate

Measuring the reaction rate of a fusion rection often reduces to finding a formula for the average reactivity of a reaction depending on temperature (or better, energy).

Reactivity

The averaged reactivity is defined as the integral of the cross section of the reaction over the velocity distribution of the incident particles:

\[ \langle \sigma v \rangle = \int_0^\infty \sigma(v)vf(v)\,\mathrm{d}v. \]

A Maxwellian distribution for the velocity distribution (\(f_\mathrm{MB}(v) = 4\pi v^2 \left(\frac{\mu}{2\pi k_\mathrm{B}T}\right)^{3/2}\exp\left(-\frac{\mu v_i^2}{2k_\mathrm{B}T}\right)\)) is often assumed for this purpose, resulting in the expression:

\[ \langle \sigma v \rangle = \frac{4}{\sqrt{2\pi\mu}}\frac{1}{(k_\mathrm{B}T)^{3/2}} \int_0^\infty \sigma(E)E\exp\left(-\frac{E}{k_\mathrm{B}T}\right)\,\mathrm{d}E, \]

with \(\mu = m_a m_X/(m_a+m_X)\) the reduced mass of the colliding particles.

Cross-Section

Considering a generic reaction a + X -> Y + b, the cross section \(\sigma\)(\(v\)) = \(\frac{number of reactions/nuclei X/ unit time}{number of incident particles/cm^2/unit time}\) [cm\(^2\)]

Often the particles move with a distribution of relative velocities, \(f(v)\) 1

Fusion plasma reactivity fits:

The implemented reactivity functions relations can be browsed in the API reference

Bosch and Hale, 1992

The most renowened analytical fits for fusion relevant applications were defined by Bosch and Hale in 1992 2 (check also the corrigendum of 1993). Bosch & Hale provide a 9-parameters fit for \(\sigma\)(E) and Maxwellian \(\langle \sigma \rangle\)(T) for light ions fusion reactions, based on improved evaluations using R-matrix methods and newer measurements than their predecessors.

It is implemented in both cfspopcon and PROCESS codes.

In fusdb, Bosch-Hale reactivities are available for DT, DD, and its two channels: DDn, DDp, DHe3.

NRL plasma formualry, 1972 - ongoing

The NRL plasma formulary 3 fits and tabulated data comes from the 5-parameters fit of 4 and reported also in 5. These fits are generally accepted in publications, but may give different results if compared to more modern data.

It also cites a more modern 3-parameters fit 6.

In fusdb, NRL reactivities are available for DT, DD, DHe3, TT, THe3.

ENDF/B, 1966 - ongoing

The Evaluated Nuclear Data File database includes experimental data for several nuclear reactions, including collisions of deuterons, tritons and helium-3 with other light nuclei. The latest release (as of 2026) is ENDF/B-VIII.1 7.

In fusdb, ENDF/B reactivities are available for DT, DD and its channels: DDn, DDp, DHe3, TT, He3He3, THe3 total, and branch-resolved THe3_D and THe3_np.

Other fits:

In fusdb, this currently includes Hively (1977) reactivities: DT Hively, DD Hively and its two channels DDn Hively, DDp Hively, 8 as well as Caughlan and Fowler (1988) reactivities:9 TT CF88, He3He3 CF88, and THe3 CF88 and its channels: THe3_D CF88. THe3_np CF88.

See also:

Bibliography:

  1. Donald D. Clayton. Principles of Stellar Evolution and Nucleosynthesis. University of Chicago Press, 1983. ISBN 9780226109534. URL: https://press.uchicago.edu/ucp/books/book/chicago/P/bo24324962.html. 

  2. Hans-S Bosch and George M. Hale. Improved formulas for fusion cross-sections and thermal reactivities. Nuclear Fusion, 32(4):611–631, 1992. URL: https://doi.org/10.1088/0029-5515/32/4/I07, doi:10.1088/0029-5515/32/4/I07. 

  3. Andrei Beresnyak. NRL Plasma Formulary. Naval Research Laboratory, 2023. URL: https://www.nrl.navy.mil/Portals/38/PDF%20Files/NRL_Plasma_Formulary_2023.pdf?ver=Ffbv5HssiwiNJk2ZA2Zi9g%3D%3D. 

  4. B H Duane. Fusion cross section theory. Technical Report, Originating Research Org. not identified, 01 1972. URL: https://www.osti.gov/biblio/4588400. 

  5. George H Miley, Harry Towner, and Nenad Ivich. Fusion cross sections and reactivities. Technical Report, Illinois Univ., Urbana (USA), 1974. 

  6. Xing Z. Li, Qing M. Wei, and Bin Liu. A new simple formula for fusion cross-sections of light nuclei. Nuclear Fusion, 48(12):125003, December 2008. URL: https://iopscience.iop.org/article/10.1088/0029-5515/48/12/125003 (visited on 2026-03-15), doi:10.1088/0029-5515/48/12/125003. 

  7. G. P. A. Nobre, R. Capote, M. T. Pigni, A. Trkov, C. M. Mattoon, D. Neudecker, D. A. Brown, M. B. Chadwick, A. C. Kahler, N. A. Kleedtke, M. Zerkle, A. I. Hawari, C. W. Chapman, N. C. Fleming, J. L. Wormald, K. Ramić, Y. Danon, N. A. Gibson, P. Brain, M. W. Paris, G. M. Hale, I. J. Thompson, D. P. Barry, I. Stetcu, W. Haeck, A. E. Lovell, M. R. Mumpower, G. Potel, K. Kravvaris, G. Noguere, J. D. McDonnell, A. D. Carlson, M. Dunn, T. Kawano, D. Wiarda, I. Al-Qasir, G. Arbanas, R. Arcilla, B. Beck, D. Bernard, R. Beyer, J. M. Brown, O. Cabellos, R. J. Casperson, Y. Cheng, E. V. Chimanski, R. Coles, M. Cornock, J. Cotchen, J. P. W. Crozier, D. E. Cullen, A. Daskalakis, M.-A. Descalle, D. D. DiJulio, P. Dimitriou, A. C. Dreyfuss, I. Durán, R. Ferrer, T. Gaines, V. Gillette, G. Gert, K. H. Guber, J. D. Haverkamp, M. W. Herman, J. Holmes, M. Hursin, N. Jisrawi, A. R. Junghans, K. J. Kelly, H. I. Kim, K. S. Kim, A. J. Koning, M. Koštál, B. K. Laramee, A. Lauer-Coles, L. Leal, H. Y. Lee, A. M. Lewis, J. Malec, J. I. Márquez Damián, W. J. Marshall, A. Mattera, G. Muhrer, A. Ney, W. E. Ormand, D. K. Parsons, C. M. Percher, V. G. Pronyaev, A. Qteish, S. Quaglioni, M. Rapp, J. J. Ressler, M. Rising, D. Rochman, P. K. Romano, D. Roubtsov, G. Schnabel, M. Schulc, G. J. Siemers, A. A. Sonzogni, P. Talou, J. Thompson, T. H. Trumbull, S. C. van der Marck, M. Vorabbi, C. Wemple, K. A. Wendt, M. White, and R. Q. Wright. ENDF/B-VIII.1: Updated Nuclear Reaction Data Library for Science and Applications. November 2025. arXiv:2511.03564 [physics]. URL: http://arxiv.org/abs/2511.03564 (visited on 2026-03-15), doi:10.48550/arXiv.2511.03564. 

  8. Lee M. Hively. Convenient computational forms for maxwellian reactivities. Nuclear Fusion, 17(4):873–876, 1977. URL: https://doi.org/10.1088/0029-5515/17/4/019, doi:10.1088/0029-5515/17/4/019. 

  9. George R. Caughlan and William A. Fowler. Thermonuclear reaction rates v. Atomic Data and Nuclear Data Tables, 40(2):283–334, 1988. URL: https://doi.org/10.1016/0092-640X(88)90009-5, doi:10.1016/0092-640X(88)90009-5. 

  10. Jeffrey P. Freidberg. Plasma Physics and Fusion Energy. Cambridge University Press, 2007. URL: https://doi.org/10.1017/CBO9780511755705, doi:10.1017/CBO9780511755705.