helike_recom.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 */
/*HeRecom - do recomb coef for He, called by HeLike */
/*cross_section - calculates the photoionization cross_section for a given level and photon energy*/
/*radrecomb - calculates radiative recombination coefficients. */
/*He_cross_section returns cross section (cm^-2), 
 * given EgammaRyd, the photon energy in Ryd,
 * ipLevel, the index of the level, 0 is ground, 3 within 2 3P,
 * nelem is charge, equal to 1 for Helium 
 * this is a wrapper for cross_section */
/*Recomb_Seaton59 - find recombination for given n,using Seaton 59 approximation.
 * The following three are needed by Recomb_Seaton59:
 *     ExponentialInt
 *     X1Int
 *     X2Int    */

#include "cddefines.h" 
#include "helike_recom.h"
#include "hydro_bauman.h"
#include "iso.h"
#include "thirdparty.h"
#include "atmdat.h"
#include "freebound.h"
#include "integrate.h"

/* The three of these are used in the computation of Recomb_Seaton59    */
STATIC double ExponentialInt( double v );
STATIC double X1Int( double u );
STATIC double X2Int( double u );

STATIC double cross_section(double EgammaRyd, double EthRyd, long nelem, long n, long l, long s);
STATIC double GetHS98CrossSection( long n, long l, long s, double EgammaRyd );

static double Xn_S59; 

/*He_cross_section returns cross section (cm^-2), 
 * given EgammaRyd, the photon energy in Ryd,
 * ipLevel, the index of the level, 0 is ground, 3 within 2 3P,
 * nelem is charge, equal to 1 for Helium 
 * this is a wrapper for cross_section */
double He_cross_section( double EgammaRyd , double EthRyd, long n, long l, long S, long nelem )
{
        // get cross section in megabarns
        double cs = cross_section( EgammaRyd, EthRyd, nelem, n, l, S );

        // rescale low-lying He values to Hummer & Storey 98, Table 1 Extrapolated
        if( nelem==ipHELIUM && n <=5 && l<=2 )
        {
                static const double rescaled[31] = {
                        7.394, 
                        5.485,  9.219, 15.985, 15.985, 15.985, 13.504,
                        8.018, 14.417, 28.501, 18.486, 18.132, 27.009,
                        10.721, 20.235, 41.568, 36.717, 35.766, -1.000, -1.000, 41.787, // };
                        13.527, 26.539, 55.692, 55.010, 53.514, -1.000, -1.000, -1.000, -1.000, 58.120 };
                long ipLev = iso_sp[ipHE_LIKE][nelem].QuantumNumbers2Index[n][l][S]; 
                ASSERT( rescaled[ipLev] > 0. ); 
                cs *= rescaled[ipLev]/cross_section( EthRyd, EthRyd, nelem, n, l, S ); 
        }

        // convert to cm^-2
        return cs * (1.e-18);
}

/*cross_section calculates the photoionization cross_section for a given level and photon energy
 * this routine returns megabarns */
STATIC double cross_section(double EgammaRyd, double EthRyd, long nelem, long n, long l, long S)
{
        /* These fit parameters (E0, sigma, y_a, P, y_w, yzero, and yone) all come from the following work: */
        /* >>refer      He      pcs     Verner, D. A., Verner, E. M., \& Ferland , G. J. 1996,
         * >>refercon   Atomic Data and Nuclear Data Tables, Vol. 64, p.1 */
        static const double E0[29] = {
        1.36E+01,2.01E+01,1.76E+01,3.34E+01,4.62E+01,6.94E+01,8.71E+01,1.13E+02,1.59E+02,2.27E+02,
        2.04E+02,2.74E+02,2.75E+02,3.38E+02,4.39E+02,4.17E+02,4.47E+02,5.18E+02,6.30E+02,6.27E+02,
        8.66E+02,7.67E+02,9.70E+02,9.66E+02,1.06E+03,1.25E+03,1.35E+03,1.43E+03,1.56E+03};
        static const double sigma[29] = {
        9.49E+02,3.20E+02,5.46E+02,2.85E+02,2.34E+02,1.52E+02,1.33E+02,1.04E+02,6.70E+01,4.00E+01,
        6.14E+01,4.04E+01,4.75E+01,3.65E+01,2.45E+01,3.14E+01,3.11E+01,2.59E+01,1.94E+01,2.18E+01,
        1.23E+01,1.76E+01,1.19E+01,1.31E+01,1.20E+01,9.05E+00,8.38E+00,8.06E+00,7.17E+00};
        static const double y_a[29] = {
        1.47E+00,7.39E+00,1.72E+01,2.16E+01,2.18E+01,2.63E+01,2.54E+01,2.66E+01,3.35E+01,5.32E+01,
        2.78E+01,3.57E+01,2.85E+01,3.25E+01,4.41E+01,3.16E+01,3.04E+01,3.28E+01,3.92E+01,3.45E+01,
        5.89E+01,3.88E+01,5.35E+01,4.83E+01,5.77E+01,6.79E+01,7.43E+01,7.91E+01,9.10E+01};
        static const double P[29] = {
        3.19E+00,2.92E+00,3.16E+00,2.62E+00,2.58E+00,2.32E+00,2.34E+00,2.26E+00,2.00E+00,1.68E+00,
        2.16E+00,1.92E+00,2.14E+00,2.00E+00,1.77E+00,2.04E+00,2.09E+00,2.02E+00,1.86E+00,2.00E+00,
        1.62E+00,1.93E+00,1.70E+00,1.79E+00,1.72E+00,1.61E+00,1.59E+00,1.58E+00,1.54E+00};
        static const double y_w[29] = 
        {2.039,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0};
        static const double yzero[29] =
        {0.4434,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0};
        static const double yone[29] =
        {2.136,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0};

        double pcs,Egamma,y,F,x;
        double rel_photon_energy;
        
        Egamma = EgammaRyd * EVRYD;

        /* >>chng 02 apr 24, more protection against calling with too small an energy */
        /* evaluating H-like photo cs at He energies, may be below threshold -
         * prevent this from happening */
        rel_photon_energy = EgammaRyd / EthRyd;
        rel_photon_energy = MAX2( rel_photon_energy , 1. + FLT_EPSILON*2. );

        long s=0;//portland group 11.2 trips without =0, does not recognized that TotalInsanity does not return
        if( S == 1 )
                s=0;
        else if( S == 3 )
                s=1;
        else
                TotalInsanity();

        if( nelem==ipHELIUM && n<=25 && l<=4 )
        {
                // use Hummer & Storey 1998 cross-sections
                pcs = GetHS98CrossSection( n, l, s, EgammaRyd );
        }
        else if( nelem==ipHELIUM && n>25 && l<=2 )
        {
                static const double scale[3][2] = {
                        {1.4673,1.3671},
                        {1.5458,1.5011},
                        {1.4912,1.5144}};

                long ipLev = iso_sp[ipHE_LIKE][nelem].QuantumNumbers2Index[25][l][S];
                double EthRyd_25 = iso_sp[ipHE_LIKE][nelem].fb[ipLev].xIsoLevNIonRyd;
                pcs = GetHS98CrossSection( 25, l, s, EthRyd_25 * rel_photon_energy );
                pcs *= pow((double)n/25., scale[l][s]);
        }
        else if( n==1 )
        {
                /* >>refer Helike       PCS     Verner, D.A., Ferland, G.J., Korista, K.T., & Yakovlev, D.G.
                 * >>refercon   1996a, ApJ 465,487      */
                /* All helike (but not Helium itself) ground state cross-sections calculated here.      */
                x = Egamma/E0[nelem-1] - yzero[nelem-1];
                y = sqrt(x*x + yone[nelem-1]*yone[nelem-1]);
                F = ((x-1)*(x-1)+y_w[nelem-1]*y_w[nelem-1])
                        * pow(y,0.5*P[nelem-1]-5.5) * pow((1+sqrt(y/y_a[nelem-1])),-P[nelem-1]);
                pcs = sigma[nelem-1]*F;
        }
        else if( nelem>=ipLITHIUM && nelem<=ipCALCIUM && n<11 && OP_Helike_NumPts[nelem][n][l][s]>0 )
        {
                // use TOPbase cross-sections
                long numDataPoints = OP_Helike_NumPts[nelem][n][l][s];
                ASSERT( numDataPoints > 0 );
                ASSERT( OP_Helike_Xsectn[nelem][n][l][s] != NULL );

                if( EgammaRyd < OP_Helike_Energy[nelem][n][l][s][numDataPoints-1] )
                {
                        pcs = linint( OP_Helike_Energy[nelem][n][l][s], OP_Helike_Xsectn[nelem][n][l][s], numDataPoints, EgammaRyd );
                }
                else
                {
                        // use a E^-3 tail 
                        pcs = OP_Helike_Xsectn[nelem][n][l][s][numDataPoints-1] * POW3( OP_Helike_Energy[nelem][n][l][s][numDataPoints-1]/EgammaRyd );
                }
        }
        else 
        {
                /* To everything else we apply a hydrogenic routine.    */
                pcs = (1.e18)*H_photo_cs(rel_photon_energy , n, l, nelem);
        }

        ASSERT( pcs > 0. && pcs < 1.E10 );

        return pcs;
}

STATIC double GetHS98CrossSection( long n, long l, long s, double EgammaRyd )
{
        double pcs;
        ASSERT( n<=25 );
        ASSERT( l<=4 );
        ASSERT( s==0 || s==1 );

        // use Hummer & Storey 1998 cross-sections
        if( EgammaRyd < HS_He1_Energy[n][l][s][NUM_HS98_DATA_POINTS-1] )
        {
                pcs = linint( HS_He1_Energy[n][l][s], HS_He1_Xsectn[n][l][s], NUM_HS98_DATA_POINTS, EgammaRyd );
        }
        else
        {
                // use a E^-3 tail 
                pcs = HS_He1_Xsectn[n][l][s][NUM_HS98_DATA_POINTS-1] * POW3( HS_He1_Energy[n][l][s][NUM_HS98_DATA_POINTS-1]/EgammaRyd );
        }

        return pcs;
}

#if 1
/* >>refer      He-like RR      Seaton, M.J. 1959, MNRAS 119, 81S */
double Recomb_Seaton59( long nelem, double temp, long n)
{
        double lambda = TE1RYD * nelem * nelem / temp;
        /* smallest x ever used here should be lowest Z, highest T, highest n...
         * using helium, logt = 10., and n = 1000, we get xmin = 1.5789E-11.    */
        double x = lambda / n / n;
        double AlphaN;
        double SzeroOfX = 0.;
        double SoneOfX = 0.;
        double StwoOfX = 0.;
        double SnOfLambda = 0.;
        double lowerlimit, upperlimit, step;

        fixit("the variant below should be faster and more accurate, but needs more testing");
        Xn_S59 = x;

        /* Equation 12  */
        lowerlimit = x;
        step = 3. * x;
        upperlimit = lowerlimit + step;
        SzeroOfX = qg32( lowerlimit, upperlimit, ExponentialInt);

        do
        {
                lowerlimit = upperlimit;
                step *= 2;
                upperlimit = lowerlimit + step;
                SzeroOfX += qg32( lowerlimit, upperlimit, ExponentialInt);
        } while ( upperlimit < 20. );

        /* This must be placed inside integral...too big to be 
         * handled separately.  
        SzeroOfX *= exp( x );   */

        /* Equations 13 and 14 */
        lowerlimit = 0.;
        step = 0.5;
        upperlimit = lowerlimit + step;
        SoneOfX = qg32(lowerlimit, upperlimit, X1Int);
        StwoOfX = qg32(lowerlimit, upperlimit, X2Int);

        do
        {
                lowerlimit = upperlimit;
                step *= 2;
                upperlimit = lowerlimit + step;
                SoneOfX += qg32( lowerlimit, upperlimit, X1Int);
                StwoOfX += qg32( lowerlimit, upperlimit, X2Int);
        } while ( upperlimit < 200. );

        SoneOfX *= 0.1728 * powpq( x, 1, 3 );
        StwoOfX *= -0.0496 * powpq( x, 2, 3 );

        /* Equation 11  */
        SnOfLambda = SzeroOfX + powpq(1./lambda, 1, 3)*SoneOfX + powpq(1./lambda, 2, 3)*StwoOfX;

        if( false )
        {
                double SSzeroOfX = e1_scaled(x);
                double gamma13 = tgamma(1./3.);
                double gamma23 = PI2/(sqrt(3.)*gamma13);
                double x13 = cbrt(x);
                double x23 = pow2(x13);
                double SSoneOfX = 0.1728*((3.*x+1)*igamc_scaled(1./3.,x)*gamma13 - 3.*x13);
                double SStwoOfX = -0.0496*(((1.5*x+2.)*x+1.)*igamc_scaled(2./3.,x)*gamma23 - 1.5*x23*(x+1.));
                double SSnOfLambda = SSzeroOfX + powpq(1./lambda, 1, 3)*SSoneOfX + powpq(1./lambda, 2, 3)*SStwoOfX;

                dprintf( ioQQQ, "%e %e %e %e %e old %e new %e\n", x, SzeroOfX/SSzeroOfX-1., SoneOfX/SSoneOfX-1.,
                         StwoOfX/SStwoOfX-1., SnOfLambda/SSnOfLambda-1., SnOfLambda, SSnOfLambda );
        }

        AlphaN = 5.197E-14 * nelem * powpq(x, 3, 2) * SnOfLambda;

        return AlphaN;
}

STATIC double ExponentialInt( double v )
{
        double Integrand;

        Integrand = exp( -v + Xn_S59) / v;

        return Integrand;
}

STATIC double X1Int( double u )
{
        double Integrand;

        Integrand = powpq(1./(u + 1.), 5, 3) * (u - 1.) * exp( -Xn_S59 * u );

        return Integrand;
}

STATIC double X2Int( double u )
{
        double Integrand;

        Integrand = powpq(1./(u + 1.), 7, 3) * (u*u + 4./3.*u + 1.) * exp( -Xn_S59 * u );

        return Integrand;
}
#else
/* >>refer      He-like RR      Seaton, M.J. 1959, MNRAS 119, 81S */
double Recomb_Seaton59(long nelem, double temp, long n)
{
        double lambda = TE1RYD * nelem * nelem / temp;
        /* smallest x ever used here should be lowest Z, highest T, highest n...
         * using helium, logt = 10., and n = 1000, we get xmin = 1.5789E-11.    */
        double x = lambda / n / n;

        /* Equation 12  */
        double SzeroOfX = e1_scaled(x);

        /* Equations 13 and 14 */
        double gamma13 = tgamma(1./3.);
        double gamma23 = PI2/(sqrt(3.)*gamma13);
        double x13 = cbrt(x);
        double x23 = pow2(x13);
        double SoneOfX = 0.172826*((3.*x+1)*igamc_scaled(1./3.,x)*gamma13 - 3.*x13);
        double StwoOfX = -0.0495957*(((1.5*x+2.)*x+1.)*igamc_scaled(2./3.,x)*gamma23 - 1.5*x23*(x+1.));

        /* Equation 11  */
        double z13 = cbrt(1./lambda);
        double SnOfLambda = (StwoOfX*z13 + SoneOfX)*z13 + SzeroOfX;

        return 5.197e-14 * nelem * powpq(x,3,2) * SnOfLambda;
}
#endif