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catima/calculations.cpp
hrocho a2041ab72d X0 Material
2020-12-13 02:41:51 +01:00

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#include <math.h>
#include <algorithm>
#include <cassert>
#include "catima/calculations.h"
#include "catima/build_config.h"
#include "catima/constants.h"
#include "catima/data_ionisation_potential.h"
#include "catima/data_atima.h"
#include "catima/generated_LS_coeff.h"
#include "catima/nucdata.h"
#include "catima/storage.h"
#include "catima/srim.h"
#ifdef GLOBAL
extern "C"
{
#include "global/globallib.h"
}
#endif
namespace catima{
double reduced_energy_loss_unit(const Projectile &p, const Target &t){
double zpowers = pow(p.Z,0.23)+pow(t.Z,0.23);
double asum = p.A + t.A;
return 32.53*t.A*1000*p.T*p.A/(p.Z*t.Z*asum*zpowers); //projectile energy is converted from MeV/u to keV
}
double dedx_n(const Projectile &p, const Material &mat){
double w;
double sum=0.0;
for(int i=0;i<mat.ncomponents();i++){
auto t = mat.get_element(i);
w = mat.weight_fraction(i);
sum += w*dedx_n(p,t);
}
return sum;
}
double dedx_n(const Projectile &p, const Target &t){
double zpowers = pow(p.Z,0.23)+pow(t.Z,0.23);
double asum = p.A + t.A;
double epsilon = 32.53*t.A*1000*p.T*p.A/(p.Z*t.Z*asum*zpowers); //projectile energy is converted from MeV/u to keV
double sn=0;
if(epsilon<=0){
return 0.0;
}
else if(epsilon<=30){
assert(p.A>0);
assert(epsilon>0);
sn = log1p(1.1383*epsilon)/ (2*(epsilon + 0.01321*pow(epsilon,0.21226) + 0.19593*pow(epsilon,0.5)));
}
else{
assert(p.A>0);
assert(epsilon>0);
sn = log(epsilon)/(2*epsilon);
}
sn = 100*8.4621*p.Z*t.Z*p.A*sn*Avogadro/(asum*zpowers*t.A);
return sn;
}
double bethek_dedx_e(const Projectile &p,const Material &mat, const Config &c){
double w;
double sum=0.0;
for(int i=0;i<mat.ncomponents();i++){
auto t = mat.get_element(i);
w = mat.weight_fraction(i);
sum += w*bethek_dedx_e(p,t,c,mat.I());
}
return sum;
}
double bethek_dedx_e(const Projectile &p, const Target &t, const Config &c, double I){
assert(t.Z>0 && p.Z>0);
assert(t.A>0 && p.A>0);
assert(p.T>0.0);
if(p.T==0)return 0.0;
double gamma=1.0 + p.T/atomic_mass_unit;
double beta2=1.0-1.0/(gamma*gamma);
double beta = sqrt(beta2);
double zp_eff = z_effective(p,t,c);
assert(zp_eff>=0);
double Ipot = (I>0.0)?I:ipot(t.Z);
assert(Ipot>0);
double f1 = dedx_constant*pow(zp_eff,2.0)*t.Z/(beta2*t.A);
double f2 = log(2.0*electron_mass*1000000*beta2/Ipot);
double eta = beta*gamma;
if(!(c.corrections&corrections::no_shell_correction) && eta>=0.13){ //shell corrections
double cor = (+0.422377*pow(eta,-2)
+0.0304043*pow(eta,-4)
-0.00038106*pow(eta,-6))*1e-6*pow(Ipot,2)
+(+3.858019*pow(eta,-2)
-0.1667989*(pow(eta,-4))
+0.00157955*(pow(eta,-6)))*1.0e-9*pow(Ipot,3);
f2 = f2 -cor/t.Z;
}
f2+=2*log(gamma) -beta2;
double barkas=1.0;
if(!(c.corrections&corrections::no_barkas)){
barkas = bethek_barkas(zp_eff,eta,t.Z);
}
double delta = bethek_density_effect(beta, t.Z);
double LS = 0.0;
if(!(c.corrections&corrections::no_lindhard)){
//double LS = bethek_lindhard(p);
LS = precalculated_lindhard(p);
}
double result = (f2)*barkas + LS - delta/2.;
result *=f1;
if( (p.T>50000.0) && !(c.corrections&corrections::no_highenergy)){
result += pair_production(p,t);
result += bremsstrahlung(p,t);
}
return result;
}
double bethek_barkas(double zp_eff,double eta, double zt){
double V2FVA[4]={0.33,0.30,0.26,0.23};
double VA[4]={1.,2.,3.,4.};
double v1 = eta/(fine_structure*sqrt(zt));
double v2fv;
if(v1 >= 4){
v2fv = 0.45 / sqrt(v1);
}
else if((v1 >= 1) && v1 < 4){//VALUES FROM THE JACKSON MC CARTHY FUNCTION //PHYS. REV. B 6 4131 P4136
int i;
for(i=1; i<4; i++){
if( VA[i] >= v1){
v2fv = V2FVA[i-1]+(v1-VA[i-1])*(V2FVA[i]-V2FVA[i-1])/(VA[i]-VA[i-1]);
break;
}
}
}
else{
v2fv=0;
}
return 1.0+2.0 * zp_eff * v2fv /(v1*v1*sqrt(zt));
}
double bethek_density_effect(double beta, int zt){
double gamma = 1/sqrt(1-(beta*beta));
double x = log(beta * gamma) / 2.3025851;
int i;
double del = 0;
if(zt>97){ // check if data are available, if not take highest z data
zt=97;
}
i = zt-1;
if (x < density_effect::x0[i] ){
if(density_effect::del_0[i] > 0.)del = density_effect::del_0[i] * pow(10.0,(2.*(x-density_effect::x0[i])));
}
else {
del = 4.6052 * x - density_effect::c[i];
if ( density_effect::x0[i]<= x && x <= density_effect::x1[i] ) del += density_effect::a[i] * pow((density_effect::x1[i] - x),density_effect::m[i]);
}
return del;
}
double bethek_lindhard(const Projectile &p){
const double compton=3.05573356675e-3; // 1.18 fm / Compton wavelength
double rho = exp(log(p.A)/3.0)*compton;
double gamma=1.0 + p.T/atomic_mass_unit;
double beta2=1.0-1.0/(gamma*gamma);
double beta = sqrt(beta2);
double eta = p.Z*fine_structure/beta;
double beta_gamma_R = beta*gamma*rho;
double sum = 0;
int n=1;
if(gamma < 10.0/rho){
double dk[3];
double dmk = 0;
double dkm1 = 0;
while(n<5000){
double k0 = n;
int max = (n==1)?3:2;
for(int i=0;i<max;i++){
double k = k0;
if(i==1)k=-k0 - 1.0;
if(i==2)k=-k0;
double l = (k>0)?k:-k-1.0;
double signk = (k>0)?1:((k<0)?-1:0);
double sk = sqrt(k*k-fine_structure*fine_structure*p.Z*p.Z);
std::complex<double> cexir_n (k,-eta/gamma);
std::complex<double> cexir_den (sk,-eta);
std::complex<double> cexir = std::sqrt(cexir_n/cexir_den);
std::complex<double> csketa (sk + 1.0, eta);
std::complex<double> cpiske(0.0,(PI*(l-sk)/2.0) - lngamma(csketa).imag());
std::complex<double> cedr = cexir*std::exp(cpiske);
double H=0;
//std::complex<double> ceds(0.0,0.0);
// finite struct part
std::complex<double> cmsketa (-sk + 1.0, eta);
std::complex<double> cexis_den (-sk,-eta);
std::complex<double> cexis = std::sqrt(cexir_n/cexis_den);
std::complex<double> cpimske(0.0,(PI*(l+sk)/2.0) - lngamma(cmsketa).imag());
std::complex<double> ceds = cexis*std::exp(cpimske);
std::complex<double> cmbeta_gamma_R(0,-beta_gamma_R);
std::complex<double> c2beta_gamma_R(0,2.0*beta_gamma_R);
std::complex<double> c2sk_1 (2.0*sk+1,0);
std::complex<double> cm2sk_1 (-2.0*sk+1,0);
std::complex<double> clambda_r = cexir*std::exp(cmbeta_gamma_R)*hyperg(csketa,c2sk_1,c2beta_gamma_R);
std::complex<double> clambda_s = cexis*std::exp(cmbeta_gamma_R)*hyperg(cmsketa,cm2sk_1,c2beta_gamma_R);
std::complex<double> cGrGs = lngamma(cm2sk_1);
double GrGs = clambda_r.imag()/clambda_s.imag();
GrGs *= exp( lngamma(csketa).real()
- lngamma(cmsketa).real()
- lngamma(c2sk_1).real()
+ cGrGs.real()
+ 2.0*sk*log(2.0*beta_gamma_R));
if(cos(cGrGs.imag()) < 1.0)GrGs*=-1;
if(fabs(GrGs)>1.0e-9){
double FrGr = sqrt((gamma-1)/(gamma+1)) * clambda_r.real()/clambda_r.imag();
double FsGs = sqrt((gamma-1)/(gamma+1)) * clambda_s.real()/clambda_s.imag();
double gz = -1.0*signk*(rho*gamma + 1.5*p.Z*fine_structure);
double z1 = -1.0*signk*p.Z;
double b0 = 1.0;
double a0 = (1.0 + 2.0*fabs(k))*b0/(rho-gz);
double a1 = 0.5*(gz+rho)*b0;
double an = a1;
double anm1 = a0;
double bnm1 = b0;
double asum = a0;
double bsum = b0;
double nn = 1.0;
while(fabs(anm1/asum)>1e-6 && fabs(anm1/asum)>1e-6){
double bn = ((rho-gz)*an + fine_structure*z1*anm1/2.0)/(2.0*nn+2.0*fabs(k)+1.0);
double anp1 = ((gz+rho)*bn - fine_structure*z1*bnm1/2.0)/(2.0*nn + 2.0);
asum += an;
bsum += bn;
nn += 1.0;
anm1 = an;
an = anp1;
bnm1 = bn;
}
double figi= (k>0) ? asum/bsum : bsum/asum;
H = (FrGr - figi)/(figi-FsGs)* GrGs;
}
else
H = 0;
dk[i] = std::arg(cedr + H*ceds);
}
if(n>1)dk[2] = dmk;
double sdm2 = sin(dk[2]-dk[1]);
double term1 = k0*(k0+1.0)*sdm2*sdm2/(eta*eta*(2.0*k0 + 1.0));
if(n>1){
double sd2 = sin(dk[0]-dkm1);
term1 += k0*(k0-1.0)*sd2*sd2/(eta*eta*(2.0*k0 - 1.0));
}
double sdd = sin(dk[0]-dk[2]);
double term2 = k0*sdd*sdd/(eta*eta*(4.0*k0*k0 - 1.0));
double term3 = term1 - 1.0/k0;
sum += term2 + term3;
n += 1;
dmk = dk[1];
dkm1 = dk[0];
}// end of while n<100
}
else{ // ultrarelativistic limit
sum = -log(beta_gamma_R) - 0.2;
}
return sum + (0.5*beta2);
}
double bethek_lindhard_X(const Projectile &p){
const double compton=3.05573356675e-3; // 1.18 fm / Compton wavelength
double rho = exp(log(p.A)/3.0)*compton;
double gamma=1.0 + p.T/atomic_mass_unit;
double beta2=1.0-1.0/(gamma*gamma);
double beta = sqrt(beta2);
double eta = p.Z*fine_structure/beta;
double beta_gamma_R = beta*gamma*rho;
double sum = 0;
int n=1;
if(1){
double dk[4];
double dmk = 0;
double dmkp1 = 0;
double dkm1 = 0;
double dkm2 = 0;
while(n<1000){
double k0 = n;
//int max = (n==1)?4:2;
int max = 4;
for(int i=0;i<max;i++){
double k=k0;
if(i==1)k=-k0 - 1.0;
if(i==2 && n==1)k=-k0;
if(i==3)k=-k0 - 2.0;
double l = (k>0)?k:-k-1.0;
double signk = (k>0)?1:((k<0)?-1:0);
double sk = sqrt(k*k-fine_structure*fine_structure*p.Z*p.Z);
std::complex<double> cexir_n (k,-eta/gamma);
std::complex<double> cexir_den (sk,-eta);
std::complex<double> cexir = std::sqrt(cexir_n/cexir_den);
std::complex<double> csketa (sk + 1.0, eta);
std::complex<double> cpiske(0.0,(PI*(l-sk)/2.0) - lngamma(csketa).imag());
std::complex<double> cedr = cexir*std::exp(cpiske);
double H=0;
//std::complex<double> ceds(0.0,0.0);
// finite struct part
std::complex<double> cmsketa (-sk + 1.0, eta);
std::complex<double> cexis_den (-sk,-eta);
std::complex<double> cexis = std::sqrt(cexir_n/cexis_den);
std::complex<double> cpimske(0.0,(PI*(l+sk)/2.0) - lngamma(cmsketa).imag());
std::complex<double> ceds = cexis*std::exp(cpimske);
std::complex<double> cmbeta_gamma_R(0,-beta_gamma_R);
std::complex<double> c2beta_gamma_R(0,2.0*beta_gamma_R);
std::complex<double> c2sk_1 (2.0*sk+1,0);
std::complex<double> cm2sk_1 (-2.0*sk+1,0);
std::complex<double> clambda_r = cexir*std::exp(cmbeta_gamma_R)*hyperg(csketa,c2sk_1,c2beta_gamma_R);
std::complex<double> clambda_s = cexis*std::exp(cmbeta_gamma_R)*hyperg(cmsketa,cm2sk_1,c2beta_gamma_R);
std::complex<double> cGrGs = lngamma(cm2sk_1);
double GrGs = clambda_r.imag()/clambda_s.imag();
GrGs *= exp( lngamma(csketa).real()
- lngamma(cmsketa).real()
- lngamma(c2sk_1).real()
+ cGrGs.real()
+ 2.0*sk*log(2.0*beta_gamma_R));
if(cos(cGrGs.imag()) < 1.0)GrGs*=-1;
if(fabs(GrGs)>1.0e-9){
double FrGr = sqrt((gamma-1)/(gamma+1)) * clambda_r.real()/clambda_r.imag();
double FsGs = sqrt((gamma-1)/(gamma+1)) * clambda_s.real()/clambda_s.imag();
double gz = -1.0*signk*(rho*gamma + 1.5*p.Z*fine_structure);
double z1 = -1.0*signk*p.Z;
double b0 = 1.0;
double a0 = (1.0 + 2.0*fabs(k))*b0/(rho-gz);
double a1 = 0.5*(gz+rho)*b0;
double an = a1;
double anm1 = a0;
double bnm1 = b0;
double asum = a0;
double bsum = b0;
double nn = 1.0;
while(fabs(anm1/asum)>1e-6 ){
double bn = ((rho-gz)*an + fine_structure*z1*anm1/2.0)/(2.0*nn+2.0*fabs(k)+1.0);
double anp1 = ((gz+rho)*bn - fine_structure*z1*bnm1/2.0)/(2.0*nn + 2.0);
asum += an;
bsum += bn;
nn += 1.0;
anm1 = an;
an = anp1;
bnm1 = bn;
}
double figi= (k>0) ? asum/bsum : bsum/asum;
H = (FrGr - figi)/(figi-FsGs)* GrGs;
}
else
H = 0;
dk[i] = std::arg(cedr + H*ceds);
}
if(n>1)dk[2] = dmk;
double strterm1p = 0;
double strterm1n = 0;
double strterm2 = 0;
double strterm3 = 0;
double eta2 = eta*eta;
double sdm2 = sin(dk[0]-dkm2);
if(n>2){
strterm1p = sdm2*sdm2*(k0-1)*(k0-2)/((2.0*k0 - 1.0)*(2.0*k0-3.0));
}
sdm2 = sin(dk[2]-dk[3]);
strterm1n = sdm2*sdm2*(-k0-1)*(-k0-2)/((-2.0*k0 - 1.0)*(-2.0*k0-3.0));
if(n>1){
double sd2 = sin(dk[0]-dmkp1);
strterm2 += (k0-1.0)*sd2*sd2/((2.0*k0 - 3.0)*(4.0*k0*k0 - 1.0));
}
double sdd = sin(dk[0]-dk[1]);
strterm3 = sdd*sdd*(k0+1.0)*((1/(4.0*k0*k0 -1.0))+(1/(4*(k0+1.0)*(k0+1.0) - 1.0)))/(2.0*k0 + 1.0);
//sum += k0*(strterm1p + strterm1n + (strterm2*2) + strterm3)/eta2;
sum += k0*(strterm1p + strterm1n + (strterm2*2) + strterm3)/eta2;
sum += - (2.0/k0);
//std::cout<<n<<" "<<strterm1p<<" "<<strterm1n<<" "<<strterm2<<" "<<strterm3<<" "<<sum<<std::endl;
n += 1;
dmk = dk[1];
dkm2 = dkm1;
dkm1 = dk[0];
dmkp1 = dk[2];
}// end of while n<100
}
else{ // ultrarelativistic limit
}
double res = 2*bethek_lindhard(p) - sum - beta2;
return (res>=0)?res:0.0;
}
double pair_production(const Projectile &p, const Target &t){
double gamma=1.0 + p.T/atomic_mass_unit;
double d = 1.0/sqrt(gamma);
double logd2 = log(d*d);
double l0 = log(2.0*gamma);
double Zt13 = 183.0*power(t.Z,-1.0/3.0);
double L0screen = (19.0/9.0)*log(Zt13/(1.0 + 25.018803808*Zt13/gamma));
double L1 = (4178.0/(81.0*PI*PI))
- (21.0/27.0)
- (248.0*l0/(27.0*PI*PI))
+ ( ((28.0*l0/(9.0)) - 446.0/27.0)*logd2/(PI*PI))
+ (14.0*logd2*logd2/(9.0*PI*PI));
L1 *= d;
double s = 0.25*dedx_constant*fine_structure*fine_structure*p.Z*p.Z*t.Z*t.Z*gamma*(1.0+(1.0/t.Z))*(L0screen + L1)/t.A;
return (s<0.0)?0.0:s;
};
double bremsstrahlung(const Projectile &p, const Target &t){
double gamma=1.0 + p.T/atomic_mass_unit;
double R = 1.18*(catima::power(p.A, 1.0/3.0) + catima::power(t.A, 1.0/3.0));
double Lbs = log1p(2.0*gamma*0.1*hbar*c_light/atomic_mass_unit/R/p.A);
double C = dedx_constant*fine_structure*(electron_mass/atomic_mass_unit);
return 16.0*C*gamma*p.Z*p.Z*p.Z*p.Z*t.Z*t.Z*Lbs/(t.A*p.A*3.0*4.0*PI);
};
double sezi_dedx_e(const Projectile &p, const Material &mat, const Config &c){
double w;
double sum=0.0;
bool use95 = c.low_energy == low_energy_types::srim_95;
double T = p.T;
for(int i=0;i<mat.ncomponents();i++){
auto t = mat.get_element(i);
w = mat.weight_fraction(i);
sum += w*srim_dedx_e(p.Z,t.Z,T, use95)/t.A;
}
return 100*sum*Avogadro; // returning MeV/g/cm2
}
double gamma_from_T(double T){
return 1.0 + T/atomic_mass_unit;
};
double beta_from_T(double T){
double gamma = gamma_from_T(T);
return sqrt(1.0-1.0/(gamma*gamma));
}
double p_from_T(double T, double M){
return sqrt(T*(T+2*atomic_mass_unit))*M;
}
double energy_straggling_firsov(double z1,double energy, double z2, double m2){
double gamma = gamma_from_T(energy);
double beta2=1.0-1.0/(gamma*gamma);
double factor=4.8184E-3*pow(z1+z2,8.0/3.0)/m2;
return factor*beta2/fine_structure/fine_structure;
}
double angular_scattering_variance(const Projectile &p, const Target &t){
if(p.T<=0)return 0.0;
double e=p.T;
double _p = p_from_T(e,p.A);
double beta = _p/((e+atomic_mass_unit)*p.A);
double lr = radiation_length(t.Z,t.A);
return 198.81 * pow(p.Z,2)/(lr*pow(_p*beta,2));
}
double angular_scattering_variance(const Projectile &p, const Material &mat){
if(p.T<=0)return 0.0;
double e=p.T;
double _p = p_from_T(e,p.A);
double beta = _p/((e+atomic_mass_unit)*p.A);
double X0 = radiation_length(mat);
return 198.81 * pow(p.Z,2)/(X0*pow(_p*beta,2));
}
/// radiation lengths are taken from Particle Data Group 2014
double radiation_length(int z, double m){
double lr = 0;
if(z==1){return 63.04;}
if(z==2){return 94.32;}
if(z==3){return 82.78;}
if(z==4){return 65.19;}
if(z==6){return 42.7;}
if(z==7){return 37.99;}
if(z==8){return 34.24;}
if(z==9){return 32.93;}
if(z==10){return 28.94;}
if(z==13){return 24.01;}
if(z==14){return 21.82;}
if(z==17){return 19.28;}
if(z==18){return 19.55;}
if(z==22){return 16.16;}
if(z==26){return 13.84;}
if(z==29){return 12.86;}
if(z==32){return 12.25;}
if(z==50){return 8.82;}
if(z==54){return 8.48;}
if(z==74){return 6.76;}
if(z==78){return 6.54;}
if(z==79){return 6.46;}
if(z==82){return 6.37;}
if(z==92){return 6.00;}
double z2 = z*z;
double z_13 = 1.0/pow(z,1./3.);
double z_23 = z_13*z_13;
double a2 = fine_structure*fine_structure*z2;
double a4 = a2*a2;
double a6 = a4*a2;
lr= 716.405*m/(z2* (log(184.15*z_13) + log(1194.0*z_23)/z - -1.202*a2 + 1.0369*a4 - 1.008*a6/(1+a2) ) );
return lr;
}
double radiation_length(const Material &material){
double sum = 0;
for(int i=0;i<material.ncomponents();i++){
auto t = material.get_element(i);
double w = material.weight_fraction(i);
sum += w/radiation_length(t.Z,t.A);
}
return 1/sum;
}
double precalculated_lindhard(const Projectile &p){
double T = p.T;
int z = (int)p.Z ;
if(z>LS_MAX_Z)z=LS_MAX_Z;
if(p.T<ls_coefficients::ls_energy_table(0))T=ls_coefficients::ls_energy_table(0);
double da = (p.A - element_atomic_weight(z))/element_atomic_weight(z);
z = z-1;
double v1 = EnergyTable_interpolate(ls_coefficients::ls_energy_table,T,ls_coefficients::ls_coefficients_a[z]);
double v2 = EnergyTable_interpolate(ls_coefficients::ls_energy_table,T,ls_coefficients::ls_coefficients_ahi[z]);
//double dif = ls_ahi(T) - ls_a(T);
//return ls_a(T)+(dif*da/ls_coefficients::a_rel_increase);
double dif = v2 - v1;
return v1+(dif*da/ls_coefficients::a_rel_increase);
}
double precalculated_lindhard_X(const Projectile &p){
double T = p.T;
int z = (int)p.Z ;
if(z>LS_MAX_Z)z=LS_MAX_Z;
//if(p.T<ls_coefficients::ls_energy_table(0))T=ls_coefficients::ls_energy_table(0);
if(p.T<ls_coefficients::ls_energy_table(0))return 1.0;
double da = (p.A - element_atomic_weight(z))/element_atomic_weight(z);
z = z-1;
double v1 = EnergyTable_interpolate(ls_coefficients::ls_energy_table,T,ls_coefficients::ls_X_coefficients_a[z]);
double v2 = EnergyTable_interpolate(ls_coefficients::ls_energy_table,T,ls_coefficients::ls_X_coefficients_ahi[z]);
//double dif = ls_X_ahi(T) - ls_X_a(T);
//return ls_X_a(T)+(dif*da/ls_coefficients::a_rel_increase);
double dif = v2 - v1;
return v1+(dif*da/ls_coefficients::a_rel_increase);
}
double dedx_variance(const Projectile &p, const Target &t, const Config &c){
double gamma = gamma_from_T(p.T);
double cor=0;
double beta = beta_from_T(p.T);
double beta2 = beta*beta;
double zp_eff = z_effective(p,t,c);
double f = domega2dx_constant*pow(zp_eff,2)*t.Z/t.A;
if( (c.calculation == omega_types::atima) ){
cor = 24.89 * pow(t.Z,1.2324)/(electron_mass*1e6 * beta2)*
log( 2.0*electron_mass*1e6*beta2/(33.05*pow(t.Z,1.6364)));
cor = std::max(cor, 0.0 );
}
//double X = bethek_lindhard_X(p);
double X = precalculated_lindhard_X(p);
X *= gamma*gamma;
if(p.T<30.0)
return std::min(f*(X+cor), energy_straggling_firsov(p.Z, p.T, t.Z,t.A));
else
return f*(X+cor);
}
double z_effective(const Projectile &p,const Target &t, const Config &c){
if(c.z_effective == z_eff_type::none){
return p.Q;
}
double gamma=1.0 + p.T/atomic_mass_unit;
double beta = sqrt(1.0-1.0/(gamma*gamma));
if(c.z_effective == z_eff_type::pierce_blann){
return z_eff_Pierce_Blann(p.Z, beta);
}
else if(c.z_effective == z_eff_type::anthony_landorf){
return z_eff_Anthony_Landford(p.Z, beta, t.Z);
}
else if(c.z_effective == z_eff_type::hubert){
return z_eff_Hubert(p.Z, p.T, t.Z);
}
else if(c.z_effective == z_eff_type::winger){
return z_eff_Winger(p.Z, beta, t.Z);
}
else if(c.z_effective == z_eff_type::global){
return z_eff_global(p.Z, p.T, t.Z);
}
else if(c.z_effective == z_eff_type::atima14){
return z_eff_atima14(p.Z, p.T, t.Z);
}
else if(c.z_effective == z_eff_type::schiwietz){
return z_eff_Schiwietz(p.Z, beta, t.Z);
}
else{
return 0.0;
}
}
double z_eff_Pierce_Blann(double z, double beta){
return z*(1.0-exp(-0.95*fine_structure_inverted*beta/pow(z,2.0/3.0)));
}
double z_eff_Anthony_Landford(double pz, double beta, double tz){
double B = 1.18-tz*(7.5e-03 - 4.53e-05*tz);
double A = 1.16-tz*(1.91e-03 - 1.26e-05*tz);
return pz*(1.0-(A*exp(-fine_structure_inverted*B*beta/pow(pz,2.0/3.0))));
}
double z_eff_Hubert(double pz, double E, double tz){
double lntz = log(tz);
double x1,x2,x3,x4;
if(E<2.5)
return 0.0;
if(tz == 4){
x1 = 2.045 + 2.0*exp(-0.04369*pz);
x2 = 7.0;
x3 = 0.2643;
x4 = 0.4171;
}
else if(tz==6){
x1 = 2.584 + 1.91*exp(-0.03958*pz);
x2 = 6.933;
x3 = 0.2433;
x4 = 0.3969;
}
else{
x1 = (1.164 + 0.2319*exp(-0.004302*tz)) + 1.658*exp(-0.0517*pz);
x2 = 8.144 + 0.09876*lntz;
x3 = 0.314 + 0.01072*lntz;
x4 = 0.5218 + 0.02521*lntz;
}
return pz*(1-x1*exp(-x2*catima::power(E,x3)*catima::power(pz,-x4)));
}
double z_eff_Winger(double pz, double beta, double tz){
double c0,c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11,c12,c13;
double x, lnz, lnzt, a0,a1,a2,a3,a4;
c0 = 0.4662;
c1 = 0.5491;
c2 = 0.7028;
c3 = 0.1089;
c4 = 0.001644;
c5 = 0.5155;
c6 = 0.05633;
c7 = 0.005447;
c8 = 0.8795;
c9 = 1.091;
c10= 0.0008261;
c11= 2.848;
c12= 0.2442;
c13= 0.00009293;
// the following is from Helmut to correct winger formula for H and He target
if(tz==1){
tz = 2.6;
}
if(tz==2){
tz = 2.8;
}
x = beta /0.012 /pow(pz,0.45);
lnz =log(pz);
lnzt=log(tz);
a0 = -c0;
a1 = -c1 * exp( c2 *lnz - c3 *lnz*lnz +c4*lnz*lnz*lnz -c5*lnzt + c6 *lnzt*lnzt);
a2 = c7 * exp( c8 *lnz - c9 *lnzt);
a3 = -c10 * exp( c11*lnz - c12*lnz*lnz*lnz);
a4 = -c13;
return pz * (1. - exp(a0 +a1*x +a2*x*x +a3*x*x*x +a4*x*x*x*x));
}
double z_eff_global(double pz, double E, double tz){
if(E>2000)
return pz;
else
#ifdef GLOBAL
return global_qmean(pz, tz, E);
#else
assert(false);
return -1;
#endif
}
double z_eff_Schiwietz(double pz, double beta, double tz){
double scaled_velocity;
double c1, c2;
double x;
scaled_velocity = catima::power(pz,-0.543)*beta/bohr_velocity;
c1 = 1-0.26*exp(-tz/11.0)*exp(-(tz-pz)*(tz-pz)/9);
c2 = 1+0.030*scaled_velocity*log(tz);
x = c1*catima::power(scaled_velocity/c2/1.54,1+(1.83/pz));
return pz*((8.29*x) + (x*x*x*x))/((0.06/x) + 4 + (7.4*x) + (x*x*x*x) );
}
double z_eff_atima14(double pz, double T, double tz){
double qmean = 0.0;
#ifdef GLOBAL
double qpb;
double qhigh,qwinger,qglobal=0.0;
double beta = beta_from_T(T);
double gamma = gamma_from_T(T);
double emax, emin;
qpb = z_eff_Pierce_Blann(pz,beta);
if(T>30.0 && T<1500.0 && pz>28){
double c1 = 1.4;
double c2 = 0.28;
double c3 = 0.04;
qglobal = z_eff_global(pz,T,tz);
qglobal = (qglobal - qpb)*c1/catima::power(tz+1,c2)*(1-exp(-c3*T)) + qpb;
}
emax = 1500.;
emin = 1000.;
if(T>emax){
qhigh = pz * (1.0-exp(-180.0*beta*catima::power(gamma,0.18)*catima::power(pz,-0.82)*catima::power(tz,0.1)));
qmean = qhigh;
}
else if(T>=emin && T<=emax){
qhigh = pz * (1.0-exp(-180.0*beta*catima::power(gamma,0.18)*catima::power(pz,-0.82)*catima::power(tz,0.1)));
if(pz<=28){
qwinger = z_eff_Winger(pz,beta,tz);
qmean = ((emax-T)*qwinger + (T-emin)*qhigh)/(emax-emin);
}
else{
qmean = ((emax-T)*qglobal + (T-emin)*qhigh)/(emax-emin);
}
}
else{
emax = 70.0;
emin = 30.0;
if(pz<=28){
qwinger = z_eff_Winger(pz,beta,tz);
qmean = qwinger;
}
else{
if(T>=emax){
qmean = qglobal;
}
else if(T<emin){
qwinger = z_eff_Winger(pz,beta,tz);
qmean = qwinger;
}
else
{
qwinger = z_eff_Winger(pz,beta,tz);
qmean = ((emax-T)*qwinger + (T-emin)*qglobal)/(emax-emin);
}
}
}
#else
assert(false);
#endif
return qmean;
}
std::complex<double> hyperg(const std::complex<double> &a,
const std::complex<double> &b,
const std::complex<double> &z){
double dm = 0.0;
std::complex<double> term(1.0, 0.0);
std::complex<double> sumterm(1.0, 0.0);
std::complex<double> previousterm;
do {
previousterm = term;
dm += 1.0;
std::complex<double> Cm(dm-1.0, 0.0);
term = previousterm * ((a + Cm)/(b + Cm)) * (z/dm);
sumterm += term;
} while( std::abs(term) > 1.0e-6 && std::abs(previousterm) > 1.0e-6 );
return(sumterm);
}
std::complex<double> lngamma( const std::complex<double> &z )
{
const static double coeff[6] = {76.18009172947146,
-86.50532032941677,
24.01409824083091,
-1.231739572450155,
0.1208650973866179e-2,
-0.5395239384953e-5};
double x, y;
if(z.real() > 0) {
x=z.real()-1.0;
y=z.imag();
} else {
x=-z.real();
y=-z.imag();
}
double r = sqrt((x+5.5)*(x+5.5)+y*y);
double aterm1=y*log(r);
double aterm2=(x+0.5)*atan2(y,(x+5.5))-y;
double lterm1=(x+0.5)*log(r);
double lterm2=-y*atan2(y,(x+5.5)) - (x+5.5) + 0.5*log(2.0*PI);
double num=0.0;
double denom=1.000000000190015;
for(int j=1;j<7;j++){
double fj=(double)j;
double cterm=coeff[j-1]/((x+fj)*(x+fj)+y*y);
num+=cterm;
denom+=(x+fj)*cterm;
}
num*=-y;
double aterm3=atan2(num,denom);
double lterm3 = 0.5*log(num*num + denom*denom);
std::complex<double> result(lterm1+lterm2+lterm3,aterm1+aterm2+aterm3);
if(z.real() < 0){
std::complex<double> lpi(log(PI), 0.0);
result = lpi - (result + std::log(std::sin(PI*z)));
}
return(result);
}
}
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