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Fusion Cross-Sections for Deuterium Cycle Reactors

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Fusion Cross-Sections for Deuterium Cycle Reactors Fusion cross-sections for deuterium cycle fusion reactors (D-cycle): an analysis of geometric, Gamow- Sommerfeld and astrophysical S-factors Abstract Fusion reactions in the deuterium cycle (D+D, D+T and D+3He) are the main nucleus-nucleus interactions which occur in tokamaks and stellerators. These reactions are the limiting case between the Woods-Saxon potential field at nuclear distances and the Coulomb electrostatic potential (scattering) at longer distances. In this paper several fusion cross-sections, geometric, Gamow- Sommerfeld and astrophysical S-factors have been reviewed and compared with experimental data from the last ENDF/B-VIII.0 cross-section library. The XDC-fusion code has been developed to calculate fusion cross-sections, geometric, Gamow-Sommerfeld and S-factors of the deuterium-cycle (D-cycle), including resonance parameters (energy and partial width). The software estimates also fusion reaction heat (Q) and Woods-Saxon/Coulomb proximity potentials. Although relative differences between fusion cross-sections are lower than 5 %, S-factors present considerable differences between the energies and partial width (FWHM) of the single-level Breit-Wigner (SLBW) resonances. The energy at which is placed the maximum fusion cross-section is also different between cases. In conclusion, fusion reaction models for light nuclei (deuterium, tritium and helium) should be reviewed in order to apply fusion to energy production in safety conditions. Keywords: Fusion reaction, fusion cross-section, astrophysical S-factor, Gamow-Sommerfeld factor 1. Introduction Fusion reactions in the deuterium cycle (D+D, D+T and D+3He) are the main nucleus-nucleus interactions which occur in the deuterium cycle (D-cycle) in tokamaks and stellerators fusion reactors [Smith and Cowley 2010, Donné 2018, Zohm 2019]. Fusion reactions are the limiting case between the Woods-Saxon potential field at nuclear distances and the Coulomb electrostatic potential (scattering) at longer distances. In order to overpass the Coulomb electrostatic barrier and achieve nuclear distances to allow fusion reactions, ionized plasma is confined by a magnetic field inside tokamaks and stellerators [Knoepfel 1978, Woods 2006] Fusion reactions are characterized by their fusion cross-sections and described by three energy dependent functions: geometric, Gamow-Sommerfeld and astrophysical S-factor. These factors retain the physical characteristics of fusion reaction: a Coulomb electrostatic potential barrier penetration, in a form of an exponential attenuation, G(E), and a strong force interaction after tunneling effect in the nuclear potential well (Woods-Saxon), at the boundary conditions, in a form of an astrophysical S(E) factor or resonance function [Miley et al 1974, Li 2002, Chen 2016]. Fusion cross section is expressed in a simplified form as f (E) GS where ϕ is the geometric factor, a term inversely proportional with incident particle energy, S is a strong-energy correlated parameter S-factor, called also astrophysical or S-factor, which depends on the boundary conditions between nuclear well and Coulomb electrostatic field and G(E), a tunneling or penetration factor, called Gamow-Sommerfeld factor. In particular, the astrophysical S-factor is a resonance function, based on the classical formula of Breit-Wigner of single-level resonances [Breit and Wigner 1936]. Fusion cross-section of light nuclei were firstly estimated by the Naval Research Laboratory, who published a 5 parameter formula for D+D, D+T and D+3He light nuclei fusion reactions, also known as Duane 5-parameter formula, based on accelerator beam-target experiments [NRL 2019]. Corresponding author: [email protected] Further studies, such as those carried out by Bosch and Hale in 1992, included a 9 parameter formula for Gamow and S-factor, based on the R-matrix of thousand experiments, and an inverse energy dependent geometric factor. However, in the case of D(d,p)T fusion reaction, no resonance peak was included [Bosch and Hale 1992]. In the last decades, fusion cross-sections have been modeled with a complex phase-shift of the incident wavefunction at the boundaries of Coulomb electrostatic potential and Woods-Saxon potential for the nuclear well, including SLBW resonance peaks for all target nuclei (D, T or 3He) [Li et al 2006,2008, Singh et al 2019]. In order to estimate fusion cross-section, geometric, Gamow-Sommerfeld, S-factors and SLBW resonance parameters in the D-cycle, the XDC-fusion code has been developed. Fusion cross-sections equations estimated through different expression have been compared with ENDF/B-VIII.0 cross- section libraries: Duane 5-parameter formula, Bosch-Hale 9-parameter formula and a combined geometric factor based on wavenumber and a complex phase-shift S-factor. 2. The D-cycle The deuterium cycle of fusion is composed of several interdependent nuclear reactions [Atzeni and Meyer-ter-Vehn 2001, Ball 2019]: 2 2 3 1 1 D + 1 D 1T (1.01 MeV) + 1 H ( 3.02 MeV) 2 2 3 1 1 D + 1 D 2 He (0.82 MeV) + 0 n ( 2.45 MeV) 2 3 4 1 1 D + 1T 2 He (3.52 MeV) + 0 n (14.07 MeV) 2 3 4 1 1 D + 2 He 2 He (3.67 MeV) + 1 H (14.68 MeV) which can be summarized in 2 4 1 1 6 1 D 2 He (3.52 MeV) + 1 H (3.02 MeV) + 0 n (14.07 MeV) 4 1 1 2 He (3.67 MeV) + 1 H (14.68 MeV) + 0 n (2.45 MeV) + 1.83 MeV Furthermore, protons produced by fusion reactions can also react with low probability as 1 1 2 1 H + 1 H 1 D + e + + 0.93 MeV (99.76 %) 1 1 2 1 H + 1 H + e 1 D + + 1.95 MeV (0.24 %) Other fusion reactions of the deuterium cycle with a low branching ratio are [Robouch et al 1993, Bystritsky et al 1998] 2 2 4 1 D + 1 D 2 He (0.08 MeV) + γ0(23.77 MeV) and with tritium as target nuclei 2 3 4 1 1 D + 1T 2 He (3.52 MeV) + 0 n (14.07 MeV) 2 3 5 * 1 D + 1T 2 He (0.05 MeV) + γ0(16.65 MeV) 2 3 5 * 1 D + 1T 2 He (3.2 MeV) + γ1(13.50 MeV) There are other fusion reactions in which the reaction heat is positive, which have not been considered, such as 2 3 3 1 1 D + 1T 2 He + 2 0 n 2 3 3 1 1 1 D + 2 He 2 He + 1 H + 0 n Table 1 summarizes the main characteristics of the D-cycle fusion: branching ratios (BR), reaction heat (Q) and emergent particle/gamma energies. 3. Geometric, Gamow-Sommerfeld and S-factor 3.1. Geometric factor (ϕ) In most of the fusion cross-section expressions, geometric factor is expressed as 1/ E . In the last decades, geometric factor has been expressed as the area subtended by a de Broglie wavelength or wavenumber k, which depends on energy [Li et al 2006, 2008] (E) 2 k 2 with k 2E / where μ is the reduced mass of the system, m1m2 / m1 m2 The geometric factor ϕ(E) can also be expressed with two independent terms, 2 1 (E) 2 E The first term is a conversion factor which is included in some expressions in the Gamow-Sommerfeld or in the S-factor, with units of keV.b in the SI. 3.2. Gamow-Sommerfeld factor 3.2.1 Introduction The Gamow-Sommerfeld factor G(E) is a penetration factor of the Coulomb barrier by the incident particle and characterize the probability of incident/target nuclei of reaching a fusion reaction. It is expressed with an exponential attenuation function as [Brennan and Coyne 1964, Humblet et al 1987] 1 G(E) exp 2 2 l with ηl the Sommerfeld parameter, evaluated as r 1 tp k (r) dr l l rn with kl the wave number inside the nuclear well 1/ 2 2 k (r) E U(r) l 2 where U(r) is the potential field, rn is the classical distance of closest approach (nuclear square-well radius) and rtp is the classical turning point, that is, the distance in which the energy of the incident particle energy is E U (rtp ) [Lee and Jung 2017] 3.2.2 Case 1: Duane parametrization Duane parametrization included an interpolation of the Gamow factor as the Mott’s form of the Coulomb barrier penetrability [Mott and Massey 1965, NRL 2019] 1 G(E) exp A1/ E 1 3.2.3 Case 2 Let us consider that the potential field U(r) can be approximated as 2ll 1 U (r) U (r) C 2 r 2 where UC is the Coulomb potential and the second term is the spin-orbit potential, with l the magnetic quantum number. The Coulomb electrostatic potential is expressed as Z Z e2 U (r) k 1 2 r r C e r n with ke 1/ 4 0 the Coulomb electrostatic constant and rn is estimated as 1/3 1/3 rn r0 A1 A2 with r0 the specific radius per nucleon of fusion reaction and the classical turning point rtp as 2 rtp keZ1Z2e / E The centrifugal term is zero at the boundaries of the nuclear potential well (l=0), then the integral is calculated as [Atzeni 2004] 1 B E E E g arccos 1 0 2 E U U U 0 0 0 where rtp U0 U (rn ) E rn and 1/ 2 2 2 Bg g 2 keZ1Z2e / 2 is the Gamow constant. The η0 can be also expressed in term of the lengths rn and rtp as 1 B r r r g n n n 0 arccos 1 2 E r r r tp tp tp 1/ 2 The arccosine can be expanded in powers of E /U 0 , thus 1/ 2 3/ 2 5/ 2 1 Bg E 1 E 3 E E E ..
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