helike_cs.cpp

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/* This file is part of Cloudy and is copyright (C)1978-2013 by Gary J. Ferland and
 * others.  For conditions of distribution and use see copyright notice in license.txt */
/*HeCollid evaluate collisional rates */
/*HeCSInterp interpolate on He1 collision strengths */
/*AtomCSInterp do the atom      */
/*IonCSInterp do the ions       */
/*CS_l_mixing_PS64 - find rate for l-mixing collisions by protons, for neutrals */
#include "cddefines.h"
#include "atmdat.h"
#include "conv.h"
#include "dense.h"
#include "helike.h"
#include "helike_cs.h"
#include "hydro_vs_rates.h"
#include "iso.h"
#include "lines_service.h"
#include "opacity.h"
#include "phycon.h"
#include "physconst.h"
#include "rfield.h"
#include "taulines.h"
#include "thirdparty.h"
#include "trace.h"
 
vector<double> CSTemp;
multi_arr<realnum,3> HeCS;
 
/* returns thermally-averaged Seaton 62 collision strength. */
STATIC double S62_Therm_ave_coll_str( double proj_energy_overKT, long nelem, long Collider, double deltaE, double osc_strength, double temp,
        double stat_weight, double I_energy_eV );
 
/* all of these are used in the calculation of Stark collision strengths
 * following the algorithms in Vrinceanu & Flannery (2001). */
STATIC double collision_strength_VF01( long ipISO, long nelem, long n, long l, long lp, long s, long Collider, 
        double ColliderCharge, double temp, double velOrEner, bool lgParamIsRedVel );
STATIC double L_mix_integrand_VF01( long n, long l, long lp, double bmax, double red_vel, double an, double ColliderCharge, double alpha );
STATIC double StarkCollTransProb_VF01( long int n, long int l, long int lp, double alpha, double deltaPhi);
 
 
class my_Integrand_S62
{
public:
        long nelem, Collider;
        double deltaE, osc_strength, temp, stat_weight, I_energy_eV;
 
        double operator() (double proj_energy_overKT)
        {
                double col_str = S62_Therm_ave_coll_str( proj_energy_overKT, nelem, Collider, deltaE, osc_strength,
                        temp, stat_weight, I_energy_eV );
                return col_str;
        }
};
 
class my_Integrand_VF01_E
{
public:
        long ipISO, nelem, n, l, lp, s, Collider;
        double ColliderCharge, temp, velOrEner;
        bool lgParamIsRedVel;
 
        double operator() (double EOverKT)
        {
                double col_str = collision_strength_VF01( ipISO, nelem, n, l, lp, s, Collider,
                        ColliderCharge, temp, EOverKT * temp / TE1RYD, lgParamIsRedVel );
                return exp( -1.*EOverKT ) * col_str;
        }
};
 
class my_Integrand_VF01_alpha
{
public:
        long n, l, lp;
        double bmax, red_vel, an, ColliderCharge;
 
        double operator() (double alpha)
        {
                double integrand = L_mix_integrand_VF01( n, l, lp,
                        bmax, red_vel, an, ColliderCharge, alpha );
                return integrand;
        }
};
 
 
/* These are masses relative to the proton mass of the electron, proton, he+, and alpha particle. */
static const double ColliderMass[4] = {ELECTRON_MASS/PROTON_MASS, 1.0, 4.0, 4.0};
static const double ColliderCharge[4] = {1.0, 1.0, 1.0, 2.0};
 
void HeCollidSetup( void )
{
        /* this must be longer than data path string, set in path.h/cpu.cpp */
        long i, i1, j, nelem, ipHi, ipLo;
        bool lgEOL, lgHIT;
        FILE *ioDATA;
 
#       define chLine_LENGTH 1000
        char chLine[chLine_LENGTH];
 
        DEBUG_ENTRY( "HeCollidSetup()" );
 
        /* get the collision strength data for the He 1 lines */
        if( trace.lgTrace )
                fprintf( ioQQQ," HeCollidSetup opening he1_cs.dat:");
 
        ioDATA = open_data( "he1_cs.dat", "r" );
 
        /* check that magic number is ok */
        if( read_whole_line( chLine , (int)sizeof(chLine) , ioDATA ) == NULL )
        {
                fprintf( ioQQQ, " HeCollidSetup could not read first line of he1_cs.dat.\n");
                cdEXIT(EXIT_FAILURE);
        }
        i = 1;
        i1 = (long)FFmtRead(chLine,&i,sizeof(chLine),&lgEOL);
        /*i2 = (long)FFmtRead(chLine,&i,sizeof(chLine),&lgEOL);
        i3 = (long)FFmtRead(chLine,&i,sizeof(chLine),&lgEOL);*/
 
        /* the following is to check the numbers that appear at the start of he1_cs.dat */
        if( i1 !=COLLISMAGIC )
        {
                fprintf( ioQQQ, 
                        " HeCollidSetup: the version of he1_cs.dat is not the current version.\n" );
                fprintf( ioQQQ, 
                        " HeCollidSetup: I expected to find the number %i and got %li instead.\n" ,
                        COLLISMAGIC, i1 );
                fprintf( ioQQQ, "Here is the line image:\n==%s==\n", chLine );
                cdEXIT(EXIT_FAILURE);
        }
 
        /* get the array of temperatures */
        lgHIT = false;
        while( read_whole_line( chLine , (int)sizeof(chLine) , ioDATA ) != NULL )
        {
                /* only look at lines without '#' in first col */
                if( chLine[0] == '#')
                        continue;
                
                lgHIT = true;
                lgEOL = false;
                char *chTemp = strtok(chLine," \t\n");
                while( chTemp != NULL )
                {
                        CSTemp.push_back( atof(chTemp) );
                        chTemp = strtok(NULL," \t\n");
                }
                break;
        }
        if( !lgHIT )
        {
                fprintf( ioQQQ, " HeCollidSetup could not find line in CS temperatures in he1_cs.dat.\n");
                cdEXIT(EXIT_FAILURE);
        }
        ASSERT( CSTemp.size() == 9U );
 
        /* create space for array of CS values, if not already done */
        {
                long nelem = ipHELIUM;
                long numLevs = iso_sp[ipHE_LIKE][nelem].numLevels_max - iso_sp[ipHE_LIKE][nelem].nCollapsed_max;
                HeCS.reserve( numLevs );
                for( long ipHi=1; ipHi < numLevs; ++ipHi )
                {
                        HeCS.reserve( ipHi, ipHi );
                        for( long ipLo=0; ipLo<ipHi; ++ipLo )
                                HeCS.reserve( ipHi, ipLo, CSTemp.size() );
                }
                HeCS.alloc();
        }
 
        /* now read in the CS data */
        lgHIT = false;
        nelem = ipHELIUM;
        while( read_whole_line( chLine , (int)sizeof(chLine) , ioDATA ) != NULL )
        {
                char *chTemp;
                /* only look at lines without '#' in first col */
                if( (chLine[0] == ' ') || (chLine[0]=='\n') )
                        break;
                if( chLine[0] != '#')
                {
                        lgHIT = true;
 
                        /* get lower and upper level index */
                        j = 1;
                        ipLo = (long)FFmtRead(chLine,&j,sizeof(chLine),&lgEOL);
                        ipHi = (long)FFmtRead(chLine,&j,sizeof(chLine),&lgEOL);
                        ASSERT( ipLo < ipHi );
                        if( ipHi >= iso_sp[ipHE_LIKE][nelem].numLevels_max - iso_sp[ipHE_LIKE][nelem].nCollapsed_max )
                                continue;
                        else
                        {
                                chTemp = chLine;
                                /* skip over 4 tabs to start of cs data */
                                for( long i=0; i<3; ++i )
                                {
                                        if( (chTemp = strchr_s( chTemp, '\t' )) == NULL )
                                        {
                                                fprintf( ioQQQ, " HeCollidSetup could not init cs\n" );
                                                cdEXIT(EXIT_FAILURE);
                                        }
                                        ++chTemp;
                                }
 
                                for( unsigned i=0; i< CSTemp.size(); ++i )
                                {
                                        double a;
                                        if( (chTemp = strchr_s( chTemp, '\t' )) == NULL )
                                        {
                                                fprintf( ioQQQ, " HeCollidSetup could not scan cs, current indices: %li %li\n", ipHi, ipLo );
                                                cdEXIT(EXIT_FAILURE);
                                        }
                                        ++chTemp;
                                        sscanf( chTemp , "%le" , &a );
                                        HeCS[ipHi][ipLo][i] = (realnum)a;
                                }
                        }
                }
        }
 
        /* close the data file */
        fclose( ioDATA );
 
        return;
}
 
/* Choose either AtomCSInterp or IonCSInterp */
realnum HeCSInterp(long int nelem,
                                 long int ipHi,
                                 long int ipLo,
                                 long int Collider )
{
        realnum cs, factor1;
 
        /* This variable is for diagnostic purposes:
         * a string used in the output to designate where each cs comes from.   */      
        const char *where = "      ";
 
        DEBUG_ENTRY( "HeCSInterp()" );
 
        if( !iso_ctrl.lgColl_excite[ipHE_LIKE] )
        {
                return (realnum)1E-10;
        }
 
        if( nelem == ipHELIUM )
        {
                /* do for helium */
                cs = AtomCSInterp( nelem, ipHi , ipLo, &factor1, &where, Collider );
        }
        else
        {
                /* get collision strengths for an ion */
                cs = IonCSInterp( nelem, ipHi , ipLo, &factor1, &where, Collider );
        }
 
        ASSERT( cs >= 0.f );
 
        /* in many cases the correction factor for split states has already been made,
         * if not then factor is still negative */
        /* Remove the second test here when IonCSInterp is up to par with AtomCSInterp */
        ASSERT( factor1 >= 0.f || nelem!=ipHELIUM );
        if( factor1 < 0.f )
        {
                ASSERT( iso_ctrl.lgCS_Vriens[ipHE_LIKE] );
 
                factor1 = 1.f;
        }
 
        /* take factor into account, usually 1, ratio of stat weights if within 2 3P 
         * and with collisions from collapsed to resolved levels */
        cs *= factor1;
 
        {
                /*@-redef@*/
                enum {DEBUG_LOC=false};
                /*@+redef@*/
 
                if( DEBUG_LOC && ( nelem==ipOXYGEN ) && (cs > 0.f) && (iso_sp[ipHE_LIKE][nelem].st[ipHi].n() == 2) 
                        && ( iso_sp[ipHE_LIKE][nelem].st[ipLo].n() <= 2 ) )
                        fprintf(ioQQQ,"%li\t%li\t%li\t-\t%li\t%li\t%li\t%.2e\t%s\n", 
                                iso_sp[ipHE_LIKE][nelem].st[ipLo].n(), iso_sp[ipHE_LIKE][nelem].st[ipLo].S() ,
                                iso_sp[ipHE_LIKE][nelem].st[ipLo].l(), iso_sp[ipHE_LIKE][nelem].st[ipHi].n() ,
                                iso_sp[ipHE_LIKE][nelem].st[ipHi].S(), iso_sp[ipHE_LIKE][nelem].st[ipHi].l() , cs,where);
        }
 
        return MAX2(cs, 1.e-10f);
}
 
realnum AtomCSInterp(long int nelem,
                                   long int ipHi,
                                   long int ipLo,
                                   realnum *factor1,
                                   const char **where,
                                   long int Collider )
{
        long ipArray;
        realnum cs;
 
        DEBUG_ENTRY( "AtomCSInterp()" );
 
        ASSERT( nelem == ipHELIUM );
 
        /* init values, better be positive when we exit */
        cs = -1.f; 
 
        /* this may be used for splitting up the collision strength within 2 3P when
         * the lower level is withint 2 3P, and for collisions between resolved and collapsed levels.
         * It may be set somewhere in routine, so set to negative value here as flag saying not set */
        *factor1 = -1.f;
 
        /* for most of the helium iso sequence, the order of the J levels within 2 3P 
         * in increasing energy, is 0 1 2 - the exception is atomic helium itself,
         * which is swapped, 2 1 0 */
 
        /* this branch is for upper and lower levels within 2p3P */
        if( ipLo >= ipHe2p3P0 && ipHi <= ipHe2p3P2 && Collider==ipELECTRON )
        {
                *factor1 = 1.f;
                /* atomic helium, use Berrington private comm cs */
 
                /* >>refer      he1     cs      Berrington, Keith, 2001, private communication - email follows
                > Dear Gary,
                > I could not find any literature on the He fine-structure
                > problem (but I didn't look very hard, so there may be 
                > something somewhere). However, I did a quick R-matrix run to 
                > see what the magnitude of the collision strengths are... At 
                > 1000K, I get the effective collision strength for 2^3P J=0-1, 
                >  0-2, 1-2 as 0.8,0.7,2.7; for 10000K I get 1.2, 2.1, 6.0
                */
                /* indices are the same and correct, only thing special is that energies are in inverted order...was right first time.  */
                if( ipLo == ipHe2p3P0 && ipHi == ipHe2p3P1 )
                {
                        cs = 1.2f;
                }
                else if( ipLo == ipHe2p3P0 && ipHi == ipHe2p3P2 )
                {
                        cs = 2.1f;
                }
                else if( ipLo == ipHe2p3P1 && ipHi == ipHe2p3P2 )
                {
                        cs = 6.0f;
                }
                else
                {
                        cs = 1.0f;
                        TotalInsanity();
                }
 
                *where = "Berr  ";
                /* statistical weights included */
        }
        /* >>chng 02 feb 25, Bray data should come first since it is the best we have.  */
        /* this branch is the Bray et al data, for n <= 5, where quantal calcs exist 
         * must exclude ipLo >= ipHe2p1P because they give no numbers for those */
        else if( iso_sp[ipHE_LIKE][nelem].st[ipHi].n() <= 5 && 
                ( ipHi < iso_sp[ipHE_LIKE][nelem].numLevels_max - iso_sp[ipHE_LIKE][nelem].nCollapsed_max ) &&
                nelem==ipHELIUM && HeCS[ipHi][ipLo][0] >= 0.f && Collider== ipELECTRON )
        {
                ASSERT( *factor1 == -1.f );
                ASSERT( ipLo < ipHi );
                ASSERT( ipHe2p3P0 == 3 );
 
                /* ipLo is within 2^3P  */
                if( ipLo >= ipHe2p3P0 && ipLo <= ipHe2p3P2 )
                {
                        /* *factor1 is ratio of statistical weights of level to term */
 
                        /* ipHe2p3P0, ipHe2p3P1, ipHe2p3P2 have indices 3,4,and 5, but j=0,1,and 2.     */
                        *factor1 = (2.f*((realnum)ipLo-3.f)+1.f) / 9.f;
 
                        /* ipHi must be above ipHe2p3P2 since 2p3Pj->2p3Pk taken care of above  */
                        ASSERT( ipHi > ipHe2p3P2 );
                }
                /* ipHi is within 2^3P  */
                else if( ipHi >= ipHe2p3P0 && ipHi <= ipHe2p3P2 )
                {
                        ASSERT( ipLo < ipHe2p3P0 );
 
                        *factor1 = (2.f*((realnum)ipHi-3.f)+1.f) / 9.f;
                }
                /* neither are within 2^3P...no splitting necessary     */
                else 
                {
                        *factor1 = 1.f;
                }
 
                /* SOME OF THESE ARE NOT N-CHANGING!    */
                /* Must be careful about turning each one on or off.    */
 
                /* this is the case where we have quantal calculations */
                /* >>refer      He1     cs      Bray, I., Burgess, A., Fursa, D.V., & Tully, J.A., 2000, A&AS, 146, 481-498 */
                /* check whether we are outside temperature array bounds,
                 * and use extreme value if we are */
                if( phycon.alogte <= CSTemp[0] )
                {
                        cs = HeCS[ipHi][ipLo][0];
                }
                else if( phycon.alogte >= CSTemp.back() )
                {
                        cs = HeCS[ipHi][ipLo][CSTemp.size()-1];
                }
                else
                {
                        realnum flow; 
 
                        /* find which array element within the cs vs temp array */
                        ipArray = (long)((phycon.alogte - CSTemp[0])/(CSTemp[1]-CSTemp[0]));
                        ASSERT( (unsigned)ipArray < CSTemp.size() );
                        ASSERT( ipArray >= 0 );
                        /* when taking the average, this is the fraction from the lower temperature value */
                        flow = (realnum)( (phycon.alogte - CSTemp[ipArray])/
                                (CSTemp[ipArray+1]-CSTemp[ipArray]));
                        ASSERT( (flow >= 0.f) && (flow <= 1.f) );
 
                        cs = HeCS[ipHi][ipLo][ipArray] * (1.f-flow) +
                                HeCS[ipHi][ipLo][ipArray+1] * flow;
                }
 
                *where = "Bray ";
 
                /* options to kill collisional excitation and/or l-mixing       */
                if( iso_sp[ipHE_LIKE][nelem].st[ipHi].n() == iso_sp[ipHE_LIKE][nelem].st[ipLo].n() )
                        /* iso_ctrl.lgColl_l_mixing turned off with atom he-like l-mixing collisions off command */
                        cs *= (realnum)iso_ctrl.lgColl_l_mixing[ipHE_LIKE];
                else
                {
                        /* iso_ctrl.lgColl_excite turned off with atom he-like collisional excitation off command */
                        cs *= (realnum)iso_ctrl.lgColl_excite[ipHE_LIKE];
                }
 
                ASSERT( cs >= 0.f );
                /* statistical weights included */
        }
        /* this branch, n-same, l-changing collision, and not case of He with quantal data */
        else if( (iso_sp[ipHE_LIKE][nelem].st[ipHi].n() == iso_sp[ipHE_LIKE][nelem].st[ipLo].n() ) &&
                (iso_sp[ipHE_LIKE][nelem].st[ipHi].S() == iso_sp[ipHE_LIKE][nelem].st[ipLo].S() ) )
        {
                ASSERT( *factor1 == -1.f );
                *factor1 = 1.f;
 
                /* ASSERT( iso_sp[ipHE_LIKE][nelem].st[ipHi].n() >= 3 ); */
                ASSERT( iso_sp[ipHE_LIKE][nelem].st[ipHi].n() <= iso_sp[ipHE_LIKE][nelem].n_HighestResolved_max );
 
                if( (iso_sp[ipHE_LIKE][nelem].st[ipLo].l() <=2) &&
                        abs(iso_sp[ipHE_LIKE][nelem].st[ipHi].l() - iso_sp[ipHE_LIKE][nelem].st[ipLo].l())== 1 )
                {
                        /* Use the method given in 
                         * >>refer He   CS      Seaton, M. J. 1962, Proc. Phys. Soc. 79, 1105 
                         * statistical weights included */
                        cs = (realnum)CS_l_mixing_S62(ipHE_LIKE, nelem, ipLo, ipHi, (double)phycon.te, Collider); 
                }
                else if( iso_ctrl.lgCS_Vrinceanu[ipHE_LIKE] )
                {
                        if( iso_sp[ipHE_LIKE][nelem].st[ipLo].l() >=3 &&
                                iso_sp[ipHE_LIKE][nelem].st[ipHi].l() >=3 )
                        {
                                /* Use the method given in 
                                 * >>refer He   CS      Vrinceanu, D. \& Flannery, M. R. 2001, PhysRevA 63, 032701 
                                 * statistical weights included */
                                cs = (realnum)CS_l_mixing_VF01( ipHE_LIKE,
                                        nelem,
                                        iso_sp[ipHE_LIKE][nelem].st[ipLo].n(),
                                        iso_sp[ipHE_LIKE][nelem].st[ipLo].l(),
                                        iso_sp[ipHE_LIKE][nelem].st[ipHi].l(),
                                        iso_sp[ipHE_LIKE][nelem].st[ipLo].S(),
                                        (double)phycon.te,
                                        Collider );
                        }
                        else
                        {
                                cs = 0.f;
                        }
                }
                /* this branch, l changing by one */
                else if( abs(iso_sp[ipHE_LIKE][nelem].st[ipHi].l() - iso_sp[ipHE_LIKE][nelem].st[ipLo].l())== 1)
                {
                        /* >>refer      He      cs      Pengelly, R.M., & Seaton, M.J., 1964, MNRAS, 127, 165 */
                        /* statistical weights included */
                        cs = (realnum)CS_l_mixing_PS64( 
                                nelem,
                                iso_sp[ipHE_LIKE][nelem].st[ipLo].lifetime(),
                                nelem+1.-ipHE_LIKE,
                                iso_sp[ipHE_LIKE][nelem].st[ipLo].n(),
                                iso_sp[ipHE_LIKE][nelem].st[ipLo].l(),
                                iso_sp[ipHE_LIKE][nelem].st[ipHi].g(),
                                Collider);
                }
                else
                {
                        /* l changes by more than 1, but same-n collision */
                        cs = 0.f;
                }
 
                /* ipLo is within 2^3P  */
                if( ipLo >= ipHe2p3P0 && ipLo <= ipHe2p3P2 )
                {
                        *factor1 = (2.f*((realnum)ipLo-3.f)+1.f) / 9.f;
                }
 
                /* ipHi is within 2^3P  */
                if( ipHi >= ipHe2p3P0 && ipHi <= ipHe2p3P2 )
                {
                        *factor1 = (2.f*((realnum)ipHi-3.f)+1.f) / 9.f;
                }
 
                *where = "lmix  ";
                cs *= (realnum)iso_ctrl.lgColl_l_mixing[ipHE_LIKE];
        }
 
        /* this is an atomic n-changing collision with no quantal calculations */
        else if( iso_sp[ipHE_LIKE][nelem].st[ipHi].n() != iso_sp[ipHE_LIKE][nelem].st[ipLo].n() )
        {
                ASSERT( *factor1 == -1.f );
                /* this is an atomic n-changing collision with no quantal calculations */
                /* gbar g-bar goes here */
 
                /* >>chng 02 jul 25, add option for fits to quantal cs data */
                if( iso_ctrl.lgCS_Vriens[ipHE_LIKE] )
                {                       
                        /* >>refer He CS        Vriens, L., & Smeets, A.H.M. 1980, Phys Rev A 22, 940
                         * statistical weight IS included in the routine */
                        cs = (realnum)CS_VS80(
                                                        ipHE_LIKE,
                                                        nelem,
                                                        ipHi,
                                                        ipLo,
                                                        iso_sp[ipHE_LIKE][nelem].trans(ipHi,ipLo).Emis().Aul(),
                                                        phycon.te,
                                                        Collider );
 
                        *factor1 = 1.f;
                        *where = "Vriens";
                }
                else if( iso_ctrl.lgCS_None[ipHE_LIKE] )
                {
                        cs = 0.f;
                        *factor1 = 1.f;
                        *where = "no gb";
                }
                else if( iso_ctrl.nCS_new[ipHE_LIKE] )
                {
                        *factor1 = 1.f;
                        /* Don't know if stat weights are included in this, but they're probably
                         * wrong anyway since they are based in part on the former (incorrect)
                         * implementation of Vriens and Smeets rates */
 
                        /* two different fits, allowed and forbidden */
                        if( iso_sp[ipHE_LIKE][nelem].trans(ipHi,ipLo).Emis().Aul() > 1. )
                        {
                                /* permitted lines - large A */
                                double x = 
                                        log10(MAX2(34.7,iso_sp[ipHE_LIKE][nelem].trans(ipHi,ipLo).EnergyWN()));
 
                                if( iso_ctrl.nCS_new[ipHE_LIKE] == 1 )
                                {
                                        /* this is the broken power law fit, passing through both quantal
                                         * calcs at high energy and asymptotically goes to VS at low energies */
                                        if( x < 4.5 )
                                        {
                                                /* low energy fit for permitted transitions */
                                                cs = (realnum)pow( 10. , -1.45*x + 6.75);
                                        }
                                        else
                                        {
                                                /* higher energy fit for permitted transitions */
                                                cs = (realnum)pow( 10. , -3.33*x+15.15);
                                        }
                                }
                                else if( iso_ctrl.nCS_new[ipHE_LIKE] == 2 )
                                {
                                        /* single parallel fit for permitted transitions, runs parallel to VS */
                                        cs = (realnum)pow( 10. , -2.3*x+10.3);
                                }
                                else
                                        TotalInsanity();
                        }
                        else
                        {
                                /* fit for forbidden transitions */
                                if( iso_sp[ipHE_LIKE][nelem].trans(ipHi,ipLo).EnergyWN() < 25119.f )
                                {
                                        cs = 0.631f; 
                                }
                                else
                                {
                                        cs = (realnum)pow(10., 
                                                -3.*log10(iso_sp[ipHE_LIKE][nelem].trans(ipHi,ipLo).EnergyWN())+12.8);
                                }
                        }
 
                        *where = "newgb";
                }
                else
                        TotalInsanity();
 
                /* ipLo is within 2^3P  */
                if( ipLo >= ipHe2p3P0 && ipLo <= ipHe2p3P2 )
                {
                        *factor1 = (2.f*((realnum)ipLo-3.f)+1.f) / 9.f;
                }
 
                /* ipHi is within 2^3P  */
                if( ipHi >= ipHe2p3P0 && ipHi <= ipHe2p3P2 )
                {
                        *factor1 = (2.f*((realnum)ipHi-3.f)+1.f) / 9.f;
                }
 
                /* options to turn off collisions */
                cs *= (realnum)iso_ctrl.lgColl_excite[ipHE_LIKE];
 
        }
        else
        {
                /* If spin changing collisions are prohibited in the l-mixing routine,
                 * they will fall here, and will have been assigned no collision strength.      
                 * Assign zero for now. */
                ASSERT( iso_sp[ipHE_LIKE][nelem].st[ipHi].S() != iso_sp[ipHE_LIKE][nelem].st[ipLo].S() );
                cs = 0.f;
                *factor1 = 1.f;
        }
 
        ASSERT( cs >= 0.f );
 
        return(cs);
 
}
 
/* IonCSInterp interpolate on collision strengths for element nelem */
realnum IonCSInterp( long nelem , long ipHi , long ipLo, realnum *factor1, const char **where, long Collider  )
{
        realnum cs;
 
        DEBUG_ENTRY( "IonCSInterp()" );
 
        ASSERT( nelem > ipHELIUM );
        ASSERT( nelem < LIMELM );
 
        /* init values, better be positive when we exit */
        cs = -1.f; 
 
        /* this may be used for splitting up the collision strength for collisions between
         * resolved and collapsed levels.  It may be set somewhere in routine, so set to 
         * negative value here as flag saying not set */
        *factor1 = -1.f;
 
 
        /* >>chng 02 mar 04,  the approximation here for transitions within 2p3P was not needed,
         * because the Zhang data give these transitions.  They are of the same order, but are 
         * specific to the three transitions    */
 
        /* this branch is ground to n=2 or from n=2 to n=2, for ions only       */
        /*>>refer Helike        CS      Zhang, Honglin, & Sampson, Douglas H. 1987, ApJS 63, 487        */
        if( iso_sp[ipHE_LIKE][nelem].st[ipHi].n()==2 
                && iso_sp[ipHE_LIKE][nelem].st[ipLo].n()<=2 && Collider==ipELECTRON)
        {
                *where = "Zhang";
                *factor1 = 1.;
 
                /* Collisions from gound        */
                if( ipLo == ipHe1s1S )
                {
                        switch( ipHi )
                        {
                        case 1: /* to 2tripS    */
                                cs = 0.25f/POW2(nelem+1.f);
                                break;
                        case 2: /* to 2singS    */
                                cs = 0.4f/POW2(nelem+1.f);
                                break;
                        case 3: /* to 2tripP0   */
                                cs = 0.15f/POW2(nelem+1.f);
                                break;
                        case 4: /* to 2tripP1   */
                                cs = 0.45f/POW2(nelem+1.f);
                                break;
                        case 5: /* to 2tripP2   */
                                cs = 0.75f/POW2(nelem+1.f);
                                break;
                        case 6: /* to 2singP    */
                                cs = 1.3f/POW2(nelem+1.f);
                                break;
                        default:
                                TotalInsanity();
                                break;
                        }
                        cs *= (realnum)iso_ctrl.lgColl_excite[ipHE_LIKE];
                }
                /* collisions from 2tripS to n=2        */
                else if( ipLo == ipHe2s3S )
                {
                        switch( ipHi )
                        {
                        case 2: /* to 2singS    */
                                cs = 2.75f/POW2(nelem+1.f);
                                break;
                        case 3: /* to 2tripP0   */
                                cs = 60.f/POW2(nelem+1.f);
                                break;
                        case 4: /* to 2tripP1   */
                                cs = 180.f/POW2(nelem+1.f);
                                break;
                        case 5: /* to 2tripP2   */
                                cs = 300.f/POW2(nelem+1.f);
                                break;
                        case 6: /* to 2singP    */
                                cs = 5.8f/POW2(nelem+1.f);
                                break;
                        default:
                                TotalInsanity();
                                break;
                        }
                        cs *= (realnum)iso_ctrl.lgColl_l_mixing[ipHE_LIKE];
                }
                /* collisions from 2singS to n=2        */
                else if( ipLo == ipHe2s1S )
                {
                        switch( ipHi )
                        {
                        case 3: /* to 2tripP0   */
                                cs = 0.56f/POW2(nelem+1.f);
                                break;
                        case 4: /* to 2tripP1   */
                                cs = 1.74f/POW2(nelem+1.f);
                                break;
                        case 5: /* to 2tripP2   */
                                cs = 2.81f/POW2(nelem+1.f);
                                break;
                        case 6: /* to 2singP    */
                                cs = 190.f/POW2(nelem+1.f);
                                break;
                        default:
                                TotalInsanity();
                                break;
                        }
                        cs *= (realnum)iso_ctrl.lgColl_l_mixing[ipHE_LIKE];
                }
                /* collisions from 2tripP0 to n=2       */
                else if( ipLo == ipHe2p3P0 )
                {
                        switch( ipHi )
                        {
                        case 4: /* to 2tripP1   */
                                cs = 8.1f/POW2(nelem+1.f);
                                break;
                        case 5: /* to 2tripP2   */
                                cs = 8.2f/POW2(nelem+1.f);
                                break;
                        case 6: /* to 2singP    */
                                cs = 3.9f/POW2(nelem+1.f);
                                break;
                        default:
                                TotalInsanity();
                                break;
                        }
                        cs *= (realnum)iso_ctrl.lgColl_l_mixing[ipHE_LIKE];
                }
                /* collisions from 2tripP1 to n=2       */
                else if( ipLo == ipHe2p3P1 )
                {
                        switch( ipHi )
                        {
                        case 5: /* to 2tripP2   */
                                cs = 30.f/POW2(nelem+1.f);
                                break;
                        case 6: /* to 2singP    */
                                cs = 11.7f/POW2(nelem+1.f);
                                break;
                        default:
                                TotalInsanity();
                                break;
                        }
                        cs *= (realnum)iso_ctrl.lgColl_l_mixing[ipHE_LIKE];
                }
                /* collisions from 2tripP2 to n=2       */
                else if( ipLo == ipHe2p3P2 )
                {
                        /* to 2singP    */
                        cs = 19.5f/POW2(nelem+1.f) * (realnum)iso_ctrl.lgColl_l_mixing[ipHE_LIKE];
                }
                else
                        TotalInsanity();
 
                /* statistical weights included */
        }
 
        /* this branch, n-same, l-changing collisions */
        else if( (iso_sp[ipHE_LIKE][nelem].st[ipHi].n() == iso_sp[ipHE_LIKE][nelem].st[ipLo].n() ) &&
                (iso_sp[ipHE_LIKE][nelem].st[ipHi].S() == iso_sp[ipHE_LIKE][nelem].st[ipLo].S() ) )
        {
                ASSERT( *factor1 == -1.f );
                *factor1 = 1.f;
 
                ASSERT( iso_sp[ipHE_LIKE][nelem].st[ipHi].n() <= iso_sp[ipHE_LIKE][nelem].n_HighestResolved_max );
 
                if( (iso_sp[ipHE_LIKE][nelem].st[ipLo].l() <=2) &&
                        abs(iso_sp[ipHE_LIKE][nelem].st[ipHi].l() - iso_sp[ipHE_LIKE][nelem].st[ipLo].l())== 1 )
                {
                        /* Use the method given in 
                         * >>refer He   CS      Seaton, M. J. 1962, Proc. Phys. Soc. 79, 1105 
                         * statistical weights included */
                        cs = (realnum)CS_l_mixing_S62(ipHE_LIKE, nelem, ipLo, ipHi, (double)phycon.te, Collider); 
                }
                else if( iso_ctrl.lgCS_Vrinceanu[ipHE_LIKE] )
                {
                        if( iso_sp[ipHE_LIKE][nelem].st[ipLo].l() >=3 &&
                                iso_sp[ipHE_LIKE][nelem].st[ipHi].l() >=3 )
                        {
                                /* Use the method given in 
                                 * >>refer He   CS      Vrinceanu, D. \& Flannery, M. R. 2001, PhysRevA 63, 032701 
                                 * statistical weights included */
                                cs = (realnum)CS_l_mixing_VF01( ipHE_LIKE,
                                        nelem,
                                        iso_sp[ipHE_LIKE][nelem].st[ipLo].n(),
                                        iso_sp[ipHE_LIKE][nelem].st[ipLo].l(),
                                        iso_sp[ipHE_LIKE][nelem].st[ipHi].l(),
                                        iso_sp[ipHE_LIKE][nelem].st[ipLo].S(),
                                        (double)phycon.te,
                                        Collider );
                        }
                        else
                        {
                                cs = 0.f;
                        }
                }
                /* this branch, l changing by one */
                else if( abs(iso_sp[ipHE_LIKE][nelem].st[ipHi].l() - iso_sp[ipHE_LIKE][nelem].st[ipLo].l())== 1)
                {
                        /* >>refer      He      cs      Pengelly, R.M., & Seaton, M.J., 1964, MNRAS, 127, 165 */
                        /* statistical weights included */
                        cs = (realnum)CS_l_mixing_PS64( 
                                nelem,
                                iso_sp[ipHE_LIKE][nelem].st[ipLo].lifetime(),
                                nelem+1.-ipHE_LIKE,
                                iso_sp[ipHE_LIKE][nelem].st[ipLo].n(),
                                iso_sp[ipHE_LIKE][nelem].st[ipLo].l(),
                                iso_sp[ipHE_LIKE][nelem].st[ipHi].g(),
                                Collider);
                }
                else
                {
                        /* l changes by more than 1, but same-n collision */
                        cs = 0.f;
                }
 
                /* ipHi is within 2^3P, do not need to split on ipLo.   */
                if( ipHi >= ipHe2p3P0 && ipHi <= ipHe2p3P2 )
                {
                        *factor1 = (2.f*((realnum)ipHi-3.f)+1.f) / 9.f;
                }
 
                *where = "lmix  ";
                cs *= (realnum)iso_ctrl.lgColl_l_mixing[ipHE_LIKE];
        }
 
        /* this branch, n changing collisions for ions */
        else if( iso_sp[ipHE_LIKE][nelem].st[ipHi].n() != iso_sp[ipHE_LIKE][nelem].st[ipLo].n() )
        {
                if( iso_ctrl.lgCS_Vriens[ipHE_LIKE] )
                {
                        /* this is the default branch */
                        /* >>refer He CS        Vriens, L., & Smeets, A.H.M. 1980, Phys Rev A 22, 940
                         * statistical weight is NOT included in the routine */
                        cs = (realnum)CS_VS80(
                                                        ipHE_LIKE,
                                                        nelem,
                                                        ipHi,
                                                        ipLo,
                                                        iso_sp[ipHE_LIKE][nelem].trans(ipHi,ipLo).Emis().Aul(),
                                                        phycon.te,
                                                        Collider );
 
                        *factor1 = 1.f;
                        *where = "Vriens";
                }
                else
                {
                        /* \todo 2 this branch and above now do the same thing.  Change something. */
                        /* this branch is for testing and reached with command ATOM HE-LIKE COLLISIONS VRIENS OFF */
                        fixit(); /* use Percival and Richards here */
 
                        cs = 0.f;
                        *where = "hydro";
                }
        }
        else
        {
                /* what's left are deltaN=0, spin changing collisions.
                 * These have not been accounted for.   */
                /* Make sure what comes here is what we think it is.    */
                ASSERT( iso_sp[ipHE_LIKE][nelem].st[ipHi].n() == iso_sp[ipHE_LIKE][nelem].st[ipLo].n() );
                ASSERT( iso_sp[ipHE_LIKE][nelem].st[ipHi].S() != iso_sp[ipHE_LIKE][nelem].st[ipLo].S() );
                cs = 0.f;
                *where = "spin ";
        }
 
        ASSERT( cs >= 0.f );
 
        return(cs);
}
 
 
/*CS_l_mixing_S62 - find rate for l-mixing collisions by protons, for neutrals */
/* The S62 stands for Seaton 1962 */
double CS_l_mixing_S62(
        long ipISO,
        long nelem /* the chemical element, 1 for He */,
        long ipLo /* lower level, 0 for ground */,
        long ipHi /* upper level, 0 for ground */,
        double temp,
        long Collider)
{
        /* >>refer      He      l-mixing        Seaton, M.J., 1962, Proc. Phys. Soc. */
        double coll_str;
 
        DEBUG_ENTRY( "CS_l_mixing_S62()" );
 
        if( iso_sp[ipISO][nelem].trans(ipHi,ipLo).ipCont() <= 0 )
        {
                return 0.;
        }
 
        my_Integrand_S62 func;
 
        func.temp = temp;
        func.stat_weight = iso_sp[ipISO][nelem].st[ipLo].g();
        /* >>chng 05 sep 06, RP  - update energies of excited states */
        /* func.deltaE = EVRYD*(iso_sp[ipISO][nelem].st[ipLo].xIsoLevNIonRyd -
                iso_sp[ipISO][nelem].st[ipHi].xIsoLevNIonRyd); */
        func.deltaE = iso_sp[ipISO][nelem].trans(ipHi,ipLo).EnergyErg()/EN1EV;
        func.I_energy_eV = EVRYD*iso_sp[ipISO][nelem].fb[0].xIsoLevNIonRyd;
        func.Collider = Collider;
        func.nelem = nelem;
 
        ASSERT( TRANS_PROB_CONST*POW2(func.deltaE/WAVNRYD/EVRYD) > 0. );
 
        func.osc_strength = iso_sp[ipISO][nelem].trans(ipHi,ipLo).Emis().Aul()/
                (TRANS_PROB_CONST*POW2(func.deltaE/WAVNRYD/EVRYD));
 
        Integrator<my_Integrand_S62,Gaussian32> S62;
        /* This returns a thermally averaged collision strength */
        coll_str = S62.sum( 0., 1., func );
        coll_str += S62.sum( 1., 10., func );
        ASSERT( coll_str > 0. );
 
        return coll_str;
}
 
/* The integrand for calculating the thermal average of collision strengths */
STATIC double S62_Therm_ave_coll_str( double proj_energy_overKT, long nelem, long Collider, double deltaE, double osc_strength, double temp,
        double stat_weight, double I_energy_eV )
{
 
        double integrand, cross_section, /*Rnot,*/ coll_str, zOverB2;
        double /* betanot, */ betaone, zeta_of_betaone, /* cs1, */ cs2;
        double Dubya, proj_energy;
        double reduced_mass;
 
        DEBUG_ENTRY( "S62_Therm_ave_coll_str()" );
 
        /* projectile energy in eV */
        proj_energy = proj_energy_overKT * phycon.te / EVDEGK;
 
        reduced_mass = dense.AtomicWeight[nelem]*ColliderMass[Collider]/
                (dense.AtomicWeight[nelem]+ColliderMass[Collider])*ATOMIC_MASS_UNIT;
 
        /* Rnot = 1.1229*H_BAR/sqrt(ELECTRON_MASS*deltaE*EN1EV)/Bohr_radius; in units of Bohr_radius */
 
        proj_energy *= ColliderMass[ipELECTRON]/ColliderMass[Collider];
        /* The deltaE here is to make sure that the collider has no less
         * than the energy difference between the initial and final levels. */
        proj_energy += deltaE;
        Dubya = 0.5*(2.*proj_energy-deltaE);
        ASSERT( Dubya > 0. );
 
        /* betanot = sqrt(proj_energy/I_energy_eV)*(deltaE/2./Dubya)*Rnot; */
 
        ASSERT( I_energy_eV > 0. );
        ASSERT( osc_strength > 0. );
 
        /* from equation 33 */
        zOverB2 = 0.5*POW2(Dubya/deltaE)*deltaE/I_energy_eV/osc_strength;
 
        ASSERT( zOverB2 > 0. );
 
        if( zOverB2 > 100. )
        {
                betaone = sqrt( 1./zOverB2 );
        }
        else if( zOverB2 < 0.54 )
        {
                /* Low betaone approximation */
                betaone = (1./3.)*(log(PI)-log(zOverB2)+1.28);
                if( betaone > 2.38 )
                {
                        /* average with this over approximation */
                        betaone += 0.5*(log(PI)-log(zOverB2));
                        betaone *= 0.5;
                }
        }
        else
        {
                long ip_zOverB2 = 0;
                double zetaOVERbeta2[101] = {
                        1.030E+02,9.840E+01,9.402E+01,8.983E+01,8.583E+01,8.200E+01,7.835E+01,7.485E+01,
                        7.151E+01,6.832E+01,6.527E+01,6.236E+01,5.957E+01,5.691E+01,5.436E+01,5.193E+01,
                        4.961E+01,4.738E+01,4.526E+01,4.323E+01,4.129E+01,3.943E+01,3.766E+01,3.596E+01,
                        3.434E+01,3.279E+01,3.131E+01,2.989E+01,2.854E+01,2.724E+01,2.601E+01,2.482E+01,
                        2.369E+01,2.261E+01,2.158E+01,2.059E+01,1.964E+01,1.874E+01,1.787E+01,1.705E+01,
                        1.626E+01,1.550E+01,1.478E+01,1.409E+01,1.343E+01,1.280E+01,1.219E+01,1.162E+01,
                        1.107E+01,1.054E+01,1.004E+01,9.554E+00,9.094E+00,8.655E+00,8.234E+00,7.833E+00,
                        7.449E+00,7.082E+00,6.732E+00,6.397E+00,6.078E+00,5.772E+00,5.481E+00,5.202E+00,
                        4.937E+00,4.683E+00,4.441E+00,4.210E+00,3.989E+00,3.779E+00,3.578E+00,3.387E+00,
                        3.204E+00,3.031E+00,2.865E+00,2.707E+00,2.557E+00,2.414E+00,2.277E+00,2.148E+00,
                        2.024E+00,1.907E+00,1.795E+00,1.689E+00,1.589E+00,1.493E+00,1.402E+00,1.316E+00,
                        1.235E+00,1.157E+00,1.084E+00,1.015E+00,9.491E-01,8.870E-01,8.283E-01,7.729E-01,
                        7.206E-01,6.712E-01,6.247E-01,5.808E-01,5.396E-01};
 
                ASSERT( zOverB2 >= zetaOVERbeta2[100] );
 
                /* find beta in the table */
                for( long i=0; i< 100; ++i )
                {
                        if( ( zOverB2 < zetaOVERbeta2[i] ) && ( zOverB2 >= zetaOVERbeta2[i+1] ) )
                        {
                                /* found the temperature - use it */
                                ip_zOverB2 = i;
                                break;
                        }
                }
 
                ASSERT( (ip_zOverB2 >=0) && (ip_zOverB2 < 100) );
 
                betaone = (zOverB2 - zetaOVERbeta2[ip_zOverB2]) / 
                        (zetaOVERbeta2[ip_zOverB2+1] - zetaOVERbeta2[ip_zOverB2]) *
                        (pow(10., ((double)ip_zOverB2+1.)/100. - 1.) - pow(10., ((double)ip_zOverB2)/100. - 1.)) +
                        pow(10., (double)ip_zOverB2/100. - 1.);
 
        }
 
        zeta_of_betaone = zOverB2 * POW2(betaone);
 
        /* cs1 = betanot * bessel_k0(betanot) * bessel_k1(betanot); */
        cs2 = 0.5*zeta_of_betaone + betaone * bessel_k0(betaone) * bessel_k1(betaone);
 
        /* cross_section = MIN2(cs1, cs2); */
        cross_section = cs2;
 
        /* cross section in units of PI * a_o^2 */
        cross_section *= 8. * (I_energy_eV/deltaE) * osc_strength * (I_energy_eV/proj_energy);
 
        /* convert to collision strength */
        coll_str = ConvCrossSect2CollStr( cross_section * PI*BOHR_RADIUS_CM*BOHR_RADIUS_CM, stat_weight, proj_energy/EVRYD, reduced_mass );
 
        integrand = exp( -1.*(proj_energy-deltaE)*EVDEGK/temp ) * coll_str;
        return integrand;
}
 
/*CS_l_mixing_PS64 - find rate for l-mixing collisions by protons, for neutrals */
double CS_l_mixing_PS64(
        long nelem, /* the chemical element, 1 for He */
        double tau, /* the radiative lifetime of the initial level.     */
        double target_charge,
        long n,
        long l,
        double gHi,
        long Collider)
{
        /* >>refer      H-like  l-mixing        Pengelly, R.M., & Seaton, M.J., 1964, MNRAS, 127, 165 */
        /* >>refer      He-like l-mixing        Pengelly, R.M., & Seaton, M.J., 1964, MNRAS, 127, 165 */
        double cs, Dnl,
                TwoLogDebye, TwoLogRc1, 
                factor1, factor2, 
                bestfactor,     factorpart,
                reduced_mass, reduced_mass_2_emass,
                rate;
        const double ChargIncoming = ColliderCharge[Collider];
 
        DEBUG_ENTRY( "CS_l_mixing_PS64()" );
 
        /* In this routine, two different cutoff radii are calculated, and as per PS64,
         * the least of these is selected.  We take the least positive result.  */
 
        /* This reduced mass is in grams.       */
        reduced_mass = dense.AtomicWeight[nelem]*ColliderMass[Collider]/
                (dense.AtomicWeight[nelem]+ColliderMass[Collider])*ATOMIC_MASS_UNIT;
        /* this mass always appears relative to the electron mass, so define it that way */
        reduced_mass_2_emass = reduced_mass / ELECTRON_MASS;
 
        /* equation 46 of PS64 */
        /* min on density added to prevent this from becoming large and negative
         * at very high n_e - Pengelly & Seaton did not apply this above 1e12 cm^-3 */
        /* This is 2 times the log of the Debye radius. */
        TwoLogDebye = 1.68 + log10( phycon.te / MIN2(1e11 , dense.eden ) );
        /* Brocklehurst (1971, equation 3.40) has 1.181 instead of 1.68.  This appears to be due to 
         * Pengelly and Seaton neglecting one of the two factors of PI in their Equation 33 */
        //TwoLogDebye = 1.181 + log10( phycon.te / MIN2(1e10 , dense.eden ) );
 
        /* This is 2 times the log of cutoff = 0.72v(tau), where tau is the lifetime of the level nl.   
         * This is PS64 equation 45 (same as Brocklehurst 1971 equation 3.41)  */
        TwoLogRc1 = 10.95 + log10( phycon.te * tau * tau / reduced_mass_2_emass );
 
#if 0
        /* This is 2 times the log of cutoff = 1.12 * hbar v / deltaE, where v is reduced velocity = sqrt( 2kT/mu ). */ 
        /* This is PS64 equation 29 */
        if( deltaE > 0. )
                TwoLogRc2 = 2. * log10( 1.12 * H_BAR * v / deltaE );
        else
                TwoLogRc2 = BIGDOUBLE;
#endif
 
        /* this is equation 44 of PS64 */
        Dnl = POW2( ChargIncoming / target_charge) * 6. * POW2( (double)n) *
                ( POW2((double)n) - POW2((double)l) - l - 1);
 
        ASSERT( Dnl > 0. );
        ASSERT( phycon.te  / Dnl / reduced_mass_2_emass > 0. );
 
        factorpart = (11.54 + log10( phycon.te  / Dnl / reduced_mass_2_emass ) );
 
        if( (factor1 = factorpart + TwoLogDebye) <= 0.)
                factor1 = BIGDOUBLE;
        if( (factor2 = factorpart + TwoLogRc1) <= 0.)
                factor2 = BIGDOUBLE;
 
        /* Now we find the least positive result.       */
        bestfactor = MIN2(factor1,factor2);
 
        ASSERT( bestfactor > 0. );
 
        /* If both factors are bigger than 100, toss out the result, and return SMALLFLOAT. */
        if( bestfactor > 100. )
        {
                return SMALLFLOAT;
        }
 
        /* This is the rate coefficient.   Units: cm^3 s-1      */
        rate = 9.93e-6 * sqrt( reduced_mass_2_emass  ) * Dnl / phycon.sqrte * bestfactor;
 
        /***** NB NB NB NB 
         * Brocklehurst (1971) has a factor of electron density in the rate coefficient (equation 3.38).  
         * This is NOT a proper rate, as can be seen in his equations 2.2 and 2.4.  This differs
         * from the formulism given by PS64. */
        //rate *= dense.eden;
 
        /* this is the TOTAL rate from nl to nl+/-1. So assume we can
         * divide by two to get the rate nl to either nl+1 or nl-1. */
        if( l > 0 )
                rate /=2;
 
        /* convert rate to collision strength */
        /* NB - the term in parentheses corrects for the fact that COLL_CONST is only appropriate 
         * for electron colliders and is off by reduced_mass_2_emass^-1.5 */
        cs = rate / ( COLL_CONST * pow(reduced_mass_2_emass, -1.5) ) *
                phycon.sqrte * gHi;
 
        ASSERT( cs > 0. );
 
        return cs;
}
 
/*CS_l_mixing - find collision strength for l-mixing collisions for neutrals */
/* The VF stands for Vrinceanu & Flannery 2001 */
/* >>refer      He      l-mixing        Vrinceanu, D. & Flannery, M. R. 2001, PhysRevA 63, 032701       */
/* >>refer      He      l-mixing        Hezel, T. P., Burkhardt, C. E., Ciocca, M., He, L-W., */
/* >>refercon   Leventhal, J. J. 1992, Am. J. Phys. 60, 329 */
double CS_l_mixing_VF01(long int ipISO,
                                                long int nelem,
                                                long int n,
                                                long int l,
                                                long int lp,
                                                long int s,
                                                double temp,
                                                long int Collider )
{
 
        double coll_str;
 
        DEBUG_ENTRY( "CS_l_mixing_VF01()" );
 
        my_Integrand_VF01_E func;
        func.ipISO = ipISO;
        func.nelem = nelem;
        func.n = n;
        func.l = l;
        func.lp = lp;
        func.s = s;
        func.Collider = Collider;
        func.temp = temp;
        func.ColliderCharge = ColliderCharge[Collider];
        func.lgParamIsRedVel = false;
        ASSERT( func.ColliderCharge > 0. );
 
        Integrator<my_Integrand_VF01_E, Gaussian32> VF01_E;
 
        /* no need to do this for h-like */
        if( ipISO > ipH_LIKE )
        {
                ASSERT( l != 0 );
                ASSERT( lp != 0 );
        }
 
        if( !iso_ctrl.lgCS_therm_ave[ipISO] )
        {
                /* Must do some thermal averaging for densities greater
                 * than about 10000 and less than about 1e10,
                 * because kT gives significantly different results.
                 * Still, do sparser integration than is done below */
                if( (dense.eden > 10000.) && (dense.eden < 1E10 ) )
                {
                        coll_str = VF01_E.sum( 0.0, 6.0, func );
                }
                else
                {
                        /* Do NOT average over Maxwellian */
                        coll_str = collision_strength_VF01( ipISO, nelem, n, l, lp, s, Collider,
                                ColliderCharge[Collider], temp, temp/TE1RYD, false );
                }
        }
        else
        {
                /* DO average over Maxwellian */
                coll_str = VF01_E.sum( 0.0, 1.0, func );
                coll_str += VF01_E.sum( 1.0, 10.0, func );
        }
 
        return coll_str;
}
 
STATIC double collision_strength_VF01( long ipISO, long nelem, long n, long l, long lp, long s, long Collider, double ColliderCharge, double temp, double velOrEner, bool lgParamIsRedVel )
{
        double cross_section, coll_str, RMSv, aveRadius, reduced_vel, E_Proj_Ryd;
        double ConstantFactors, reduced_mass, CSIntegral, stat_weight;
        double quantum_defect1, quantum_defect2, omega_qd1, omega_qd2, omega_qd;
        double reduced_b_max, reduced_b_min, alphamax, alphamin, step, alpha1 /*, alpha2*/;
        long ipLo, ipHi;
 
        DEBUG_ENTRY( "collision_strength_VF01()" );
 
        ASSERT( n > 0 );
 
        /* >>chng 06 may 30, move these up from below.  Also ipHi needs lp not l. */
        ipLo = iso_sp[ipISO][nelem].QuantumNumbers2Index[n][l][s];
        ipHi = iso_sp[ipISO][nelem].QuantumNumbers2Index[n][lp][s];
        stat_weight = iso_sp[ipISO][nelem].st[ipLo].g();
 
        /* no need to do this for h-like */
        if( ipISO > ipH_LIKE )
        {
                /* these shut up the lint, already done above */
                ASSERT( l > 0 );
                ASSERT( lp > 0 );
        }
 
        /* This reduced mass is in grams.       */
        reduced_mass = dense.AtomicWeight[nelem]*ColliderMass[Collider]/
                (dense.AtomicWeight[nelem]+ColliderMass[Collider])*ATOMIC_MASS_UNIT;
        ASSERT( reduced_mass > 0. );
 
        /* this is root mean squared velocity */
        /* use this as projectile velocity for thermally-averaged cross-section */
        aveRadius = (BOHR_RADIUS_CM/((double)nelem+1.-(double)ipISO))*POW2((double)n);
        ASSERT( aveRadius < 1.e-4 );
        /* >>chng 05 jul 14, as per exchange with Ryan Porter & Peter van Hoof, avoid
         * roundoff error and give ability to go beyond zinc */
        /*ASSERT( aveRadius >=  BOHR_RADIUS_CM );*/
        ASSERT( aveRadius > 3.9/LIMELM * BOHR_RADIUS_CM );
 
        /* vn = n * H_BAR/ m / r = Z * e^2 / n / H_BAR 
         * where Z is the effective charge. */
        RMSv = ((double)nelem+1.-(double)ipISO)*POW2(ELEM_CHARGE_ESU)/(double)n/H_BAR;
        ASSERT( RMSv > 0. );
 
        ASSERT( ColliderMass[Collider] > 0. );
 
        if( lgParamIsRedVel )
        {
                /* velOrEner is a reduced velocity */
                reduced_vel = velOrEner;
                /* The proton mass is needed here because the ColliderMass array is
                 * expressed in units of the proton mass, but here we need absolute mass. */
                E_Proj_Ryd = 0.5 * POW2( reduced_vel * RMSv ) * ColliderMass[Collider] *
                        PROTON_MASS / EN1RYD;
        }
        else
        {       
                /* velOrEner is a projectile energy in Rydbergs */
                E_Proj_Ryd = velOrEner;
                reduced_vel = sqrt( 2.*E_Proj_Ryd*EN1RYD/ColliderMass[Collider]/PROTON_MASS )/RMSv;
        }
 
        /* put limits on the reduced velocity.   These limits should be more than fair. */
        ASSERT( reduced_vel > 1.e-10 );
        ASSERT( reduced_vel < 1.e10 );
 
        /* Factors outside integral     */
        ConstantFactors = 4.5*PI*POW2(ColliderCharge*aveRadius/reduced_vel);
 
        /* Reduced here means in units of aveRadius: */
        reduced_b_min = 1.5 * ColliderCharge / reduced_vel;
        ASSERT( reduced_b_min > 1.e-10 );
        ASSERT( reduced_b_min < 1.e10 );
 
        if( ipISO == ipH_LIKE )
        {
                /* Debye radius: appears to be too large, results in 1/v^2 variation. */
                reduced_b_max = sqrt( BOLTZMANN*temp/ColliderCharge/dense.eden ) 
                        / (PI2*ELEM_CHARGE_ESU)/aveRadius;
        }
        else if( ipISO == ipHE_LIKE )
        {
                quantum_defect1  = (double)n- (double)nelem/sqrt(iso_sp[ipISO][nelem].fb[ipLo].xIsoLevNIonRyd);
                quantum_defect2  = (double)n- (double)nelem/sqrt(iso_sp[ipISO][nelem].fb[ipHi].xIsoLevNIonRyd);
 
                /* The magnitude of each quantum defect must be between zero and one. */
                ASSERT( fabs(quantum_defect1)  < 1.0 );
                ASSERT( fabs(quantum_defect1)  > 0.0 );
                ASSERT( fabs(quantum_defect2)  < 1.0 );
                ASSERT( fabs(quantum_defect2)  > 0.0 );
 
                /* The quantum defect precession frequencies */
                omega_qd1 = fabs( 5.* quantum_defect1 * (1.-0.6*POW2((double)l/(double)n)) / POW3( (double)n ) / (double)l );
                /* >>chng 06 may 30, this needs lp not l. */
                omega_qd2 = fabs( 5.* quantum_defect2 * (1.-0.6*POW2((double)lp/(double)n)) / POW3( (double)n ) / (double)lp );
                /* Take the average for the two levels, for reciprocity. */
                omega_qd = 0.5*( omega_qd1 + omega_qd2 );
 
                ASSERT( omega_qd > 0. );
 
                reduced_b_max = sqrt( 1.5 * ColliderCharge * n / omega_qd )/aveRadius;
        }
        else
                /* rethink this before using on other iso sequences. */
                TotalInsanity();
 
        reduced_b_max = MAX2( reduced_b_max, reduced_b_min );
        ASSERT( reduced_b_max > 0. );
 
        // set up the integrator.
        my_Integrand_VF01_alpha func;
        func.n = n;
        func.l = l;
        func.lp = lp;
        func.red_vel = reduced_vel;
        func.an = aveRadius;
        func.ColliderCharge = ColliderCharge;
        func.bmax = reduced_b_max*aveRadius;
        Integrator<my_Integrand_VF01_alpha, Gaussian32> VF01_alpha;
 
        alphamin = 1.5*ColliderCharge/(reduced_vel * reduced_b_max);
        alphamax = 1.5*ColliderCharge/(reduced_vel * reduced_b_min);
 
        ASSERT( alphamin > 0. );
        ASSERT( alphamax > 0. );
 
        alphamin = MAX2( alphamin, 1.e-30 );
        alphamax = MAX2( alphamax, 1.e-20 );
 
        CSIntegral = 0.;
 
        if( alphamax > alphamin )
        {
 
                step = (alphamax - alphamin)/5.;
                alpha1 = alphamin;
                CSIntegral += VF01_alpha.sum( alpha1, alpha1+step, func );
                CSIntegral += VF01_alpha.sum( alpha1+step, alpha1+4.*step, func );
        }
 
        /* Calculate cross section */
        cross_section = ConstantFactors * CSIntegral;
 
        /* convert to collision strength, cross section is already in cm^2 */
        coll_str = ConvCrossSect2CollStr( cross_section, stat_weight, E_Proj_Ryd, reduced_mass );
 
        coll_str = MAX2( SMALLFLOAT, coll_str);
 
        return coll_str;
}
 
STATIC double L_mix_integrand_VF01( long n, long l, long lp, double bmax, double red_vel, double an, double ColliderCharge, double alpha )
{
        double integrand, deltaPhi, b;
 
        DEBUG_ENTRY( "L_mix_integrand_VF01()" );
 
        ASSERT( alpha >= 1.e-30 );
        ASSERT( bmax > 0. );
        ASSERT( red_vel > 0. );
 
        /* >>refer He l-mixing Kazansky, A. K. & Ostrovsky, V. N. 1996, JPhysB: At. Mol. Opt. Phys. 29, 3651 */
        b = 1.5*ColliderCharge*an/(red_vel * alpha);
        /* deltaPhi is the effective angle swept out by the projector as viewed by the target.  */
        if( b < bmax )
        {
                deltaPhi = -1.*PI + 2.*asin(b/bmax);
        }
        else
        {
                deltaPhi = 0.;
        }
        integrand = 1./alpha/alpha/alpha;
        integrand *= StarkCollTransProb_VF01( n, l, lp, alpha, deltaPhi );
 
        return integrand;
}
 
STATIC double StarkCollTransProb_VF01( long n, long l, long lp, double alpha, double deltaPhi)
{
        double probability;
        double cosU1, cosU2, sinU1, sinU2, cosChiOver2, sinChiOver2, cosChi, A, B;
 
        DEBUG_ENTRY( "StarkCollTransProb_VF01()" );
 
        ASSERT( alpha > 0. );
 
        /* These are defined on page 11 of VF01 */ 
        cosU1 = 2.*POW2((double)l/(double)n) - 1.;
        cosU2 = 2.*POW2((double)lp/(double)n) - 1.;
 
        sinU1 = sqrt( 1. - cosU1*cosU1 );
        sinU2 = sqrt( 1. - cosU2*cosU2 );
 
 
        cosChiOver2 = (1. + alpha*alpha*cos( sqrt(1.+alpha*alpha) * deltaPhi ) )/(1.+alpha*alpha);
        sinChiOver2 = sqrt( 1. - cosChiOver2*cosChiOver2 );
        cosChi = 2. * POW2( cosChiOver2 ) - 1.;
 
        if( l == 0 )
        {
                if( -1.*cosU2 - cosChi < 0. )
                {
                        probability = 0.;
                }
                else
                {
                        /* Here is the initial state S case */
                        ASSERT( sinChiOver2 > 0. );
                        ASSERT( sinChiOver2*sinChiOver2 > POW2((double)lp/(double)n) );
                        /* This is equation 35 of VF01.  There is a factor of hbar missing in the denominator of the
                         * paper, but it's okay if you use atomic units (hbar = 1). */
                        probability = (double)lp/(POW2((double)n)*sinChiOver2*sqrt( POW2(sinChiOver2) - POW2((double)lp/(double)n) ) );
                }
        }
        else
        {
                double OneMinusCosChi = 1. - cosChi;
 
                if( OneMinusCosChi == 0. )
                {
                        double hold = sin( deltaPhi / 2. );
                        /* From approximation at bottom of page 10 of VF01. */
                        OneMinusCosChi = 8. * alpha * alpha * POW2( hold );
                }
 
                if( OneMinusCosChi == 0. )
                {
                        probability = 0.;
                }
                else
                {
                        /* Here is the general case */
                        A = (cosU1*cosU2 - sinU1*sinU2 - cosChi)/OneMinusCosChi;
                        B = (cosU1*cosU2 + sinU1*sinU2 - cosChi)/OneMinusCosChi;
 
                        ASSERT( B > A );
 
                        /* These are the three cases of equation 34. */
                        if( B <= 0. )
                        {
                                probability = 0.;
                        }
                        else
                        {
                                ASSERT( POW2( sinChiOver2 ) > 0. );
 
                                probability = 2.*lp/(PI* /*H_BAR* */ n*n*POW2( sinChiOver2 ));
 
                                if( A < 0. )
                                {
                                        probability *= ellpk( -A/(B-A) );
                                        probability /= sqrt( B - A );
                                }
                                else
                                {
                                        probability *= ellpk( A/B );
                                        probability /= sqrt( B );
                                }
                        }
                }
 
        }
 
        return probability;
 
}