BoschHaleFusionCrossSection.H

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/* Copyright 2022 Remi Lehe
 *
 * This file is part of WarpX.
 *
 * License: BSD-3-Clause-LBNL
 */
 
#ifndef WARPX_BOSCH_HALE_FUSION_CROSS_SECTION_H
#define WARPX_BOSCH_HALE_FUSION_CROSS_SECTION_H
 
#include "Particles/Collision/BinaryCollision/BinaryCollisionUtils.H"
#include "Utils/WarpXConst.H"
 
#include <AMReX_REAL.H>
 
#include <cmath>

AMREX_GPU_HOST_DEVICE AMREX_INLINE
amrex::ParticleReal BoschHaleFusionCrossSection (
    const amrex::ParticleReal& E_kin_star,
    const NuclearFusionType& fusion_type,
    const amrex::ParticleReal& m1,
    const amrex::ParticleReal& m2 )
{
    using namespace amrex::literals;
 
    constexpr amrex::ParticleReal joule_to_keV = 1.e-3_prt/PhysConst::q_e;
    const amrex::ParticleReal E_keV = E_kin_star*joule_to_keV;
 
    // The Bosch-Hale fusion cross-section parameterization is only valid above a
    // lower energy threshold: 0.3 keV for He3(d,p)He4 and 0.5 keV for all other
    // reactions. Since the fusion cross section is negligible below these thresholds,
    // return zero. See H.-S. Bosch and G.M. Hale 1992 Nucl. Fusion 32 611, Table IV.
    const amrex::ParticleReal E_cutoff_keV =
    (fusion_type == NuclearFusionType::DeuteriumHeliumToProtonHelium) ? 0.3_prt : 0.5_prt;
    if (E_keV < E_cutoff_keV) {return 0._prt;}
 
    // Compute the Gamow constant B_G (in keV^{1/2})
    // (See Eq. 3 in H.-S. Bosch and G.M. Hale 1992 Nucl. Fusion 32 611)
    const amrex::ParticleReal m_reduced = m1 / (1._prt + m1/m2);
    // The formula for `B_G` below assumes that both reactants have Z=1
    // When one of the reactants is helium (Z=2), this formula is corrected further below.
    amrex::ParticleReal B_G = MathConst::pi * PhysConst::alpha
         * std::sqrt( 2._prt*m_reduced*PhysConst::c2 * joule_to_keV );
    if (fusion_type == NuclearFusionType::DeuteriumHeliumToProtonHelium) {
        // Take into account the fact that Z=2 for one of the reactant (helium) in this case
        B_G *= 2;
    }
 
    // Compute astrophysical_factor
    // (See Eq. 9 and Table IV in H.-S. Bosch and G.M. Hale 1992 Nucl. Fusion 32 611)
    amrex::ParticleReal A1=0_prt, A2=0_prt, A3=0_prt, A4=0_prt, A5=0_prt, B1=0_prt, B2=0_prt, B3=0_prt, B4=0_prt;
    if (fusion_type == NuclearFusionType::DeuteriumTritiumToNeutronHelium) {
        A1 = 6.927e4_prt;
        A2 = 7.454e8_prt;
        A3 = 2.050e6_prt;
        A4 = 5.2002e4_prt;
        A5 = 0_prt;
        B1 = 6.38e1_prt;
        B2 = -9.95e-1_prt;
        B3 = 6.981e-5_prt;
        B4 = 1.728e-4_prt;
    }
    else if (fusion_type == NuclearFusionType::DeuteriumDeuteriumToProtonTritium) {
        A1 = 5.5576e4_prt;
        A2 = 2.1054e2_prt;
        A3 = -3.2638e-2_prt;
        A4 = 1.4987e-6_prt;
        A5 = 1.8181e-10_prt;
        B1 = 0_prt;
        B2 = 0_prt;
        B3 = 0_prt;
        B4 = 0_prt;
    }
    else if (fusion_type == NuclearFusionType::DeuteriumDeuteriumToNeutronHelium) {
        A1 = 5.3701e4_prt;
        A2 = 3.3027e2_prt;
        A3 = -1.2706e-1_prt;
        A4 = 2.9327e-5_prt;
        A5 = -2.5151e-9_prt;
        B1 = 0_prt;
        B2 = 0_prt;
        B3 = 0_prt;
        B4 = 0_prt;
    }
    else if (fusion_type == NuclearFusionType::DeuteriumHeliumToProtonHelium) {
        A1 = 5.7501e6_prt;
        A2 = 2.5226e3_prt;
        A3 = 4.5566e1_prt;
        A4 = 0_prt;
        A5 = 0_prt;
        B1 = -3.1995e-3_prt;
        B2 = -8.5530e-6_prt;
        B3 = 5.9014e-8_prt;
        B4 = 0_prt;
    }
 
    const amrex::ParticleReal astrophysical_factor =
        (A1 + E_keV*(A2 + E_keV*(A3 + E_keV*(A4 + E_keV*A5)))) /
        (1_prt + E_keV*(B1 + E_keV*(B2 + E_keV*(B3 + E_keV*B4))));
 
    // Compute cross-section in SI units
    // (See Eq. 8 in H.-S. Bosch and G.M. Hale 1992 Nucl. Fusion 32 611)
    constexpr amrex::ParticleReal millibarn_to_sqm = 1.e-31_prt;
    return millibarn_to_sqm * astrophysical_factor/E_keV * std::exp(-B_G/std::sqrt(E_keV));
}
 
#endif // WARPX_BOSCH_HALE_FUSION_CROSS_SECTION_H