TwoProductUtil.H

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/* Copyright 2022 Remi Lehe
 *
 * This file is part of WarpX.
 *
 * License: BSD-3-Clause-LBNL
 */
 
#ifndef WARPX_TWO_PRODUCT_UTIL_H
#define WARPX_TWO_PRODUCT_UTIL_H
 
#include "Utils/ParticleUtils.H"
#include "Utils/WarpXConst.H"
 
#include <AMReX_Math.H>
#include <AMReX_Random.H>
#include <AMReX_REAL.H>
 
#include <cmath>
#include <limits>
 
namespace {
    AMREX_GPU_HOST_DEVICE AMREX_INLINE
    void TwoProductComputeProductMomenta (
                            const amrex::ParticleReal& u1x_in,
                            const amrex::ParticleReal& u1y_in,
                            const amrex::ParticleReal& u1z_in,
                            const amrex::ParticleReal& m1_in,
                            const amrex::ParticleReal& u2x_in,
                            const amrex::ParticleReal& u2y_in,
                            const amrex::ParticleReal& u2z_in,
                            const amrex::ParticleReal& m2_in,
                            amrex::ParticleReal& u1x_out,
                            amrex::ParticleReal& u1y_out,
                            amrex::ParticleReal& u1z_out,
                            const amrex::ParticleReal& m1_out,
                            amrex::ParticleReal& u2x_out,
                            amrex::ParticleReal& u2y_out,
                            amrex::ParticleReal& u2z_out,
                            const amrex::ParticleReal& m2_out,
                            const amrex::ParticleReal& E_reaction,
                            const amrex::RandomEngine& engine )
    {
        using namespace amrex::literals;
        using namespace amrex::Math;
 
        constexpr amrex::ParticleReal c_sq = PhysConst::c * PhysConst::c;
        constexpr amrex::ParticleReal inv_csq = 1._prt / ( c_sq );
        // Rest energy of incident particles
        const amrex::ParticleReal E_rest_in = (m1_in + m2_in)*c_sq;
        // Rest energy of products
        const amrex::ParticleReal E_rest_out = (m1_out + m2_out)*c_sq;
 
        // Compute Lorentz factor gamma in the lab frame
        const amrex::ParticleReal g1_in = std::sqrt( 1._prt
            + (u1x_in*u1x_in + u1y_in*u1y_in + u1z_in*u1z_in)*inv_csq );
        const amrex::ParticleReal g2_in = std::sqrt( 1._prt
            + (u2x_in*u2x_in + u2y_in*u2y_in + u2z_in*u2z_in)*inv_csq );
 
        // Compute momenta
        const amrex::ParticleReal p1x_in = u1x_in * m1_in;
        const amrex::ParticleReal p1y_in = u1y_in * m1_in;
        const amrex::ParticleReal p1z_in = u1z_in * m1_in;
        const amrex::ParticleReal p2x_in = u2x_in * m2_in;
        const amrex::ParticleReal p2y_in = u2y_in * m2_in;
        const amrex::ParticleReal p2z_in = u2z_in * m2_in;
        // Square norm of the total (sum between the two particles) momenta in the lab frame
        const amrex::ParticleReal p_total_sq =  powi<2>(p1x_in+p2x_in) +
                                                powi<2>(p1y_in+p2y_in) +
                                                powi<2>(p1z_in+p2z_in);
 
        // Total energy of incident macroparticles in the lab frame
        const amrex::ParticleReal E_lab = (m1_in * g1_in + m2_in * g2_in) * c_sq;
        // Total energy squared of the reactants in the center of mass frame, calculated using the
        // Lorentz invariance of the four-momentum norm
        const amrex::ParticleReal E_star_sq = E_lab*E_lab - c_sq*p_total_sq;
        // Total energy squared of the products in the center of mass frame
        // In principle, the term - E_rest_in + E_rest_out + E_reaction is not needed and equal to
        // zero (i.e. the energy liberated during the reaction is equal to the mass difference). However,
        // due to possible inconsistencies in how the mass is defined in the code, it is
        // probably more robust to subtract the rest masses and to add the reaction energy to the
        // total kinetic energy.
        const amrex::ParticleReal E_star_f_sq = powi<2>(std::sqrt(E_star_sq)
                                                         - E_rest_in + E_rest_out + E_reaction);
 
        // Square of the norm of the momentum of the products in the center of mass frame
        // Formula obtained by inverting E^2 = p^2*c^2 + m^2*c^4 in the COM frame for each particle
        // The expression below is specifically written in a form that avoids returning
        // small negative numbers due to machine precision errors, for low-energy particles
        const amrex::ParticleReal E_ratio = std::sqrt(E_star_f_sq)/((m1_out + m2_out)*c_sq);
        const amrex::ParticleReal p_star_f_sq = m1_out*m2_out*c_sq * ( powi<2>(E_ratio) - 1._prt )
                + powi<2>(m1_out - m2_out)*c_sq*0.25_prt * powi<2>( E_ratio - 1._prt/E_ratio );
 
        // Compute momentum of first product in the center of mass frame, assuming isotropic
        // distribution
        amrex::ParticleReal px_star, py_star, pz_star;
        ParticleUtils::RandomizeVelocity(px_star, py_star, pz_star, std::sqrt(p_star_f_sq),
                                         engine);
 
        // Next step is to convert momenta to lab frame
        amrex::ParticleReal p1x_out, p1y_out, p1z_out;
        // Preliminary calculation: compute center of mass velocity vc
        const amrex::ParticleReal mass_g = m1_in * g1_in + m2_in * g2_in;
        const amrex::ParticleReal vcx    = (p1x_in+p2x_in) / mass_g;
        const amrex::ParticleReal vcy    = (p1y_in+p2y_in) / mass_g;
        const amrex::ParticleReal vcz    = (p1z_in+p2z_in) / mass_g;
        const amrex::ParticleReal vc_sq   = vcx*vcx + vcy*vcy + vcz*vcz;
 
        // Convert momentum of first product to lab frame, using equation (13) of F. Perez et al.,
        // Phys.Plasmas.19.083104 (2012)
        if ( vc_sq > std::numeric_limits<amrex::ParticleReal>::min() )
        {
            const amrex::ParticleReal gc = 1._prt / std::sqrt( 1._prt - vc_sq*inv_csq );
            const amrex::ParticleReal g_star = std::sqrt(1._prt + p_star_f_sq / (m1_out*m1_out*c_sq));
            const amrex::ParticleReal vcDps = vcx*px_star + vcy*py_star + vcz*pz_star;
            const amrex::ParticleReal factor0 = (gc-1._prt)/vc_sq;
            const amrex::ParticleReal factor = factor0*vcDps + m1_out*g_star*gc;
            p1x_out = px_star + vcx * factor;
            p1y_out = py_star + vcy * factor;
            p1z_out = pz_star + vcz * factor;
        }
        else // If center of mass velocity is zero, we are already in the lab frame
        {
            p1x_out = px_star;
            p1y_out = py_star;
            p1z_out = pz_star;
        }
 
        // Compute momentum of the second product in lab frame, using total momentum conservation
        const amrex::ParticleReal p2x_out = p1x_in + p2x_in - p1x_out;
        const amrex::ParticleReal p2y_out = p1y_in + p2y_in - p1y_out;
        const amrex::ParticleReal p2z_out = p1z_in + p2z_in - p1z_out;
 
        // Compute the momentum of the product macroparticles
        u1x_out = p1x_out/m1_out;
        u1y_out = p1y_out/m1_out;
        u1z_out = p1z_out/m1_out;
        u2x_out = p2x_out/m2_out;
        u2y_out = p2y_out/m2_out;
        u2z_out = p2z_out/m2_out;
    }
}
 
#endif // WARPX_TWO_PRODUCT_UTIL_H