hydro_vs_rates.cpp

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/* This file is part of Cloudy and is copyright (C)1978-2017 by Gary J. Ferland and
 * others.  For conditions of distribution and use see copyright notice in license.txt */
/* CS_VS80 compute thermally averaged collision strength for collisional deexcitation 
 * of hydrogenic atoms, from
 * >>refer      H1      collision       Vriens, L., & Smeets, A.H.M. 1980, Phys Rev A 22, 940
 *hydro_vs_ioniz generate hydrogenic collisional ionization rate coefficients */
/*Hion_coll_ioniz_ratecoef calculate hydrogenic ionization rates for ions 
 * with all n, and Z*/
#include "cddefines.h"
#include "dense.h"
#include "phycon.h"
#include "iso.h"
#include "hydro_vs_rates.h"
#include "thirdparty.h"
#include "lines_service.h"
#include "integrate.h"

STATIC double hydro_vs_coll_str( long nHi, long gHi, double IP_Ryd_Hi, long nLo, long gLo, double IP_Ryd_Lo, double Aul, long nelem, long Collider, double energy );

namespace {
        class my_Integrand : public std::unary_function<double, double> 
        {
        public:
                long nelem, Collider;
                double Aul;
                double temp;
                long nHi, gHi, nLo, gLo;
                double IP_Ryd_Hi, IP_Ryd_Lo;
                
                double operator() (double EOverKT) const
                        {
                                double col_str = hydro_vs_coll_str( nHi, gHi, IP_Ryd_Hi, nLo, gLo, IP_Ryd_Lo, 
                                        Aul, nelem, Collider, EOverKT * EVRYD * temp/TE1RYD );
                                return exp( -1.*EOverKT ) * col_str;
                        }
        };
}

/*
 Neither of these rates can be modified to account for impact by non-electrons because they 
 are fits to thermally-averaged rates for electron impact...it can not be unravelled to 
 expose a projectile energy that can then be scaled to account for other projectiles.
 Instead, we have included the original cross section formula (eq 14) from 
 Vriens & Smeets (1980) below.
*/

/* VS80 stands for Vriens and Smeets 1980 */
/* This routine calculates thermally-averaged collision strengths. */
double CS_VS80( long nHi, long gHi, double IP_Ryd_Hi, long nLo, long gLo, double IP_Ryd_Lo, double Aul, long nelem, long Collider, double temp )
{
        double coll_str;

        if( Collider == ipELECTRON )
        {
                coll_str = hydro_vs_deexcit( nHi, gHi, IP_Ryd_Hi, nLo, gLo, IP_Ryd_Lo, Aul );
        }
        else
        {
                /* evaluate collision strength, two options,
                 * do thermal average (very slow) if set with
                 * SET COLLISION STRENGTH AVERAGE command,
                 * default just do point at kT
                 * tests show no impact on test suite, much faster */
                if( iso_ctrl.lgCollStrenThermAver )
                {
                        my_Integrand func;

                        func.nHi = nHi;
                        func.gHi = gHi;
                        func.IP_Ryd_Hi = IP_Ryd_Hi;
                        func.nLo = nLo;
                        func.gLo = gLo;
                        func.IP_Ryd_Lo = IP_Ryd_Lo;
                        func.nelem = nelem;
                        func.temp = temp;
                        func.Collider = Collider;
                        func.Aul = Aul;

                        Integrator<my_Integrand,Gaussian32> VS80( func );
                        /* do average over Maxwellian */
                        coll_str = VS80.sum( 0., 1. );
                        coll_str += VS80.sum( 1., 10. );
                }
                else
                {
                        /* the argument to the function is equivalent to evaluating the function at
                         * hnu = kT */
                        coll_str = hydro_vs_coll_str( nHi, gHi, IP_Ryd_Hi, nLo, gLo, IP_Ryd_Lo, 
                                        Aul, nelem, Collider, EVRYD*temp/TE1RYD );
                }
        }

        ASSERT( coll_str >= 0. );
        return coll_str;
}

/*hydro_vs_coll_str compute collision strength for collisional deexcitation for hydrogen atom, 
 * from Vriens and Smeets */
STATIC double hydro_vs_coll_str( long nHi, long gHi, double IP_Ryd_Hi, long nLo, long gLo, double IP_Ryd_Lo, double Aul, long nelem, long Collider, double energy )
{
        DEBUG_ENTRY( "hydro_vs_coll_str()" );

        // number of colliders is 4 in def, should be macro
        ASSERT( Collider >= 0.&& Collider <4 );
        double reduced_mass = dense.AtomicWeight[nelem]*colliders.list[Collider].mass_amu/
                (dense.AtomicWeight[nelem]+colliders.list[Collider].mass_amu)*ATOMIC_MASS_UNIT;

        /* This comes from equations 14,15, and 16 of Vriens and Smeets. */
        /* >>refer he-like cs Vriens, L. \& Smeets, A. H. M. Phys. Rev. A, 22, 940 */ 
        /* Computes the Vriens and Smeets
         * rate coeff. for collisional dexcitation between any two levels of H.
         * valid for all nhigh, nlow
         * at end converts this excitation rate to collision strength   */

        double n = (double)nHi;
        double p = (double)nLo;

        double g_n = (double)gHi;
        double g_p = (double)gLo;

        double ryd = EVRYD;
        double s = fabs(n-p);
        ASSERT( s > 0. );

        double Epn = EVRYD * (IP_Ryd_Lo - IP_Ryd_Hi);
        double Epi = EVRYD * IP_Ryd_Lo;

        /* This is an absorption oscillator strength. */
        double abs_osc_str = GetGF( Aul, Epn*RYD_INF/EVRYD, g_n)/g_p;
        double rEpn = 1./Epn;
        double Apn = 2.*ryd*rEpn*abs_osc_str;

        double rp = 1./p;
        double bp = rp*(1.4*log(p) - .7 + rp*(- .51 + rp*(1.16*rp - 0.55*rp)));

        double Bpn = 4.*ryd*ryd/pow3(n)*rEpn*rEpn*(1. + Epi*rEpn*(4./3. + bp*Epi*rEpn));

        double delta = exp(-Bpn/Apn) - 0.4*Epn/ryd;


        /* Scale the energy to get an equivalent electron energy. */
        energy *= colliders.list[ipELECTRON].mass_amu/colliders.list[Collider].mass_amu;

        double cross_section;
        /* cross section in units of PI*a_o^2 */
        if( energy/2./ryd+delta <= 0 /*|| energy < Epn*/ )
        {
                cross_section = 0.;
        }
        else
        {
                double gamma = ryd*(8. + 23.*POW2(s*rp))*s*s/
                        ( s*s*(8. + 1.1*n*s) + 0.8 + 0.4*powpq(n*s,3,2)*fabs(s-1.0) );
                cross_section = 2.*ryd/(energy + gamma)*(Apn*log(energy/2./ryd+delta) + Bpn);
                cross_section = MAX2( cross_section, 0. );
        }

        /* convert to collision strength */
        double coll_str = ConvCrossSect2CollStr( cross_section * PI*BOHR_RADIUS_CM*BOHR_RADIUS_CM, g_n, energy/EVRYD, reduced_mass );

        ASSERT( coll_str >= 0. );

        return( coll_str );
}

/*hydro_vs_coll_recomb generate hydrogenic collisional recombination rate coefficients */
double hydro_vs_coll_recomb( double ionization_energy_Ryd, double Te, double stat_level, double stat_ion )
{
        double coef, 
          denom, 
          epi, 
          t_eV;

        DEBUG_ENTRY( "hydro_vs_coll_recomb()" );

        /* This is equation 9 of
         * >>refer      H1      collision recomb        Vriens, L., & Smeets, A.H.M. 1980, Phys Rev A 22, 940 */

        /* this is kT in eV */
        t_eV = Te/EVDEGK;

        /* this is the ionization energy relative to kT, dimensionless */
        epi = ionization_energy_Ryd * EVRYD / t_eV;

        /* this is the denominator of equation 8 of VS80. */
        denom = pow(epi,2.33) + 4.38*pow(epi,1.72) + 1.32*epi;

        /* this is equation 9 of VS80 */
        coef = 3.17e-27 / pow3(t_eV) * stat_level / stat_ion / denom;

        ASSERT( coef >= 0. );
        return( coef );
}


/*hydro_vs_ioniz generate hydrogenic collisional ionization rate coefficients */
double hydro_vs_ioniz( double ionization_energy_Ryd, double Te )
{
        double coef, 
          denom, 
          epi, 
          t_eV;

        DEBUG_ENTRY( "hydro_vs_ioniz()" );

        /* a function written to calculate the rate coefficients 
         * for hydrogen collisional ionizations from
         * Jason Ferguson, summer 94
         * valid for all n
         * >>refer      H1      collision       Vriens, L., & Smeets, A.H.M. 1980, Phys Rev A 22, 940
         * */

        /* this is kT in eV */
        t_eV = Te/EVDEGK;

        /* this is the ionization energy relative to kT, dimensionless */
        epi = ionization_energy_Ryd * EVRYD / t_eV;

        /* this is the denominator of equation 8 of VS80. */
        denom = pow(epi,2.33) + 4.38*pow(epi,1.72) + 1.32*epi;

        /* this is equation 8 of VS80 */
        coef = 9.56e-6 / powpq(t_eV,3,2) * dsexp( epi ) / denom;

        ASSERT( coef >= 0. );
        return( coef );
}

/*Hion_coll_ioniz_ratecoef calculate hydrogenic ionization rates for all n, and Z*/
double Hion_coll_ioniz_ratecoef(
                /* the isoelectronic sequence */
                long int ipISO ,
                /* element, >=1 since only used for ions 
                 * nelem = 1 is helium the least possible charge */
                long int nelem,
                /* principal quantum number, > 1
                 * since only used for excited states */
                long int n,
                double ionization_energy_Ryd,
                double Te )
{
        double H, 
          HydColIon_v, 
          Rnp, 
          charge,
          chim, 
          eone, 
          etwo, 
          efour,
          g, 
          rate, 
          rate2, 
          boltz,
          t1, 
          t2, 
          t3, 
          t4, 
          tev, 
          xn, 
          y;
        static const double arrH[4] = {1.48,3.64,5.93,8.32};
        static const double arrRnp[8] = {2.20,1.90,1.73,1.65,1.60,1.56,1.54,1.52};
        static const double arrg[10] = {0.8675,0.932,0.952,0.960,0.965,0.969,0.972,0.975,0.978,0.981};

        static double small = 0.;

        DEBUG_ENTRY( "Hion_coll_ioniz_ratecoef()" );
        /*calculate hydrogenic ionization rates for all n, and Z
         * >>refer      HI      cs      Allen 1973, Astro. Quan. for low Te.
         * >>refer      HI      cs      Sampson and Zhang 1988, ApJ, 335, 516 for High Te.
         * */

        charge = nelem - ipISO;
        /* this routine only for ions, nelem=0 is H, nelem=1 he, etc */
        ASSERT( charge > 0);
        ASSERT( n>1 );

        if( n > 4 )
        {
                H = 2.15*n;
        }
        else
        {
                H = arrH[n-1];
        }

        if( n > 8 )
        {
                Rnp = 1.52;
        }
        else
        {
                Rnp = arrRnp[n-1];
        }

        if( n > 10 )
        {
                g = arrg[9];
        }
        else
        {
                g = arrg[n-1];
        }

        tev = EVRYD/TE1RYD*Te;
        xn = (double)n;
        chim = EVRYD * ionization_energy_Ryd;
        y = chim/tev;
        boltz = dsexp( chim/tev );

        eone = e1(y);
        etwo = e2(y);
        efour = expn(4,y);

        t1 = 1./xn*eone;
        t2 = 1./xn*efour;
        t3 = 3.*H/xn/(3. - Rnp)*(y*etwo - 2.*y*eone + boltz);
        t4 = 3.36*y*(eone - etwo);
        rate = 7.69415e-9*sqrt(Te)*9.28278e-3*powi(xn/(charge+1),4)*g*y;
        rate *= t1 - t2 + t3 + t4;
        rate2 = 2.1e-8*sqrt(Te)/chim/chim*dsexp(2.302585*5040.*chim/Te);

        /* don't let the rates go negative */
        rate = MAX2(rate,small);
        rate2 = MAX2(rate2,small);

        /* Take the lowest of the two, they fit nicely together... */
        /* >>chng 10 Sept 02, sometimes one of these is zero and the other is positive.
         * in that case take the bigger one.    */
        if( rate==0. || rate2==0. )
                HydColIon_v = MAX2(rate,rate2);
        else
                HydColIon_v = MIN2(rate,rate2);

        ASSERT( HydColIon_v >= 0. );
        return( HydColIon_v );
}

/*hydro_vs_deexcit compute collisional deexcitation rates for hydrogen atom, 
 * from Vriens and Smeets 1980 */
double hydro_vs_deexcit( long nHi, long gHi, double IP_Ryd_Hi, long nLo, long gLo, double IP_Ryd_Lo, double Aul )
{
        DEBUG_ENTRY( "hydro_vs_deexcit()" );

        double kT_eV = EVRYD * phycon.te / TE1RYD;

        /* This comes from equations 24 of Vriens and Smeets. */
        /* >>refer he-like cs Vriens, L. \& Smeets, A. H. M. Phys. Rev. A, 22, 940 */ 
        /* Computes the Vriens and Smeets
         * rate coeff. for collisional dexcitation between any two levels of H.
         * valid for all nhigh, nlow
         * at end converts this excitation rate to collision strength   */

        double n = (double)nLo;
        double p = (double)nHi;

        ASSERT( n!=p );

        double g_n = (double)gLo;
        double g_p = (double)gHi;

        double ryd = EVRYD;
        double s = fabs(n-p);

        double Enp = EVRYD * (IP_Ryd_Lo - IP_Ryd_Hi);
        double rEnp = 1./Enp;
        double Eni = EVRYD * IP_Ryd_Hi;

        ASSERT( Enp > 0. );

        /* This is an absorption oscillator strength. */
        double abs_osc_str = GetGF( Aul, Enp*RYD_INF/EVRYD, g_p)/g_n;
        double Anp = 2.*ryd*rEnp*abs_osc_str;

        double rn = 1./n;
        double bn = rn*( 1.4*log(n) - .7 +rn*(- .51 + rn*(1.16 - 0.55*rn)));
        
        double Bnp = 4.*ryd*ryd/pow3(p)*rEnp*rEnp*(1. + Eni*rEnp*(4./3. + bn*Eni*rEnp));

        double delta_np = exp(-Bnp/Anp) + 0.06 * s*s / (p*n*n);

        double rate;

        if( 0.3*kT_eV/ryd+delta_np <= 0 )
        {
                rate = 0.;
        }
        else
        {
                double Gamma_np = ryd*log(1. + n*n*n*kT_eV/ryd) * (3. + 11.*s*s*rn*rn) * s * s /
                        ( s*s* (6. + 1.6*p*s) + 0.3 + 0.8*powpq(p*s,3,2)*fabs(s-0.6) );

                rate = 1.6E-7 * sqrt(kT_eV) * g_n / (g_p * ( kT_eV + Gamma_np ) ) *
                        ( Anp * log(0.3*kT_eV/ryd + delta_np) + Bnp );
        }

        double col_str = rate / COLL_CONST * phycon.sqrte * g_p;

        return col_str ;
}