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/* $Id$ */
/*
* scatter.c
*
* written by Sebastian Voecking <seb.voeck@uni-muenster.de>
*
* See scatter.h for details
*
* Included in this file are function from Ferenc Glueck for calculation of
* cross sections.
*/
//#include "scatter.h"
#include <math.h>
#include <stdio.h>
#include <time.h>
#include "constants.h"
#include "random.h"
double sigmael(double E);
double sigmaexc(double E);
double sigmaion(double E);
void randomel(double E, double *Eloss, double *theta);
void randomexc(double E, double *Eloss, double *theta);
void randomion(double E, double *Eloss, double *theta);
static double sigmaBC(double E);
static double sigmadiss10(double E);
static double sigmadiss15(double E);
static double Del(double E, double c);
static double Dexc(double E, double c);
static double sumexc(double K);
static double Dinel(double E, double c);
static void gensecelen(double E, double *W);
static void subrn(double* u, int len);
static double lagrange(int n, double *xn, double *fn, double x);
static const double emass = 18780; // Electron mass in atomic units
static const double a02 = 28e-22; // Bohr radius squared
static const double clight = 137; // velocity of light in atomic units
static const double R = 13.6; // Ryberg energy in eV
static double density; /* Density of the residual gas */
static double event_x; /* Distance to the interaction point */
static double x; /* Already moved distance */
static double totalx;
static double energy_loss;
static double scatter_angle;
static FILE* output = NULL;
double sigmael(double E) {
// This function computes the total elastic cross section of
// electron scatt. on molecular hydrogen.
// See: Liu, Phys. Rev. A35 (1987) 591,
// Trajmar, Phys Reports 97 (1983) 221.
// E: incident electron energy in eV
// sigmael: cross section in m^2
static double e[14] = {0., 1.5, 5., 7., 10., 15., 20., 30., 60., 100., 150., 200., 300., 400.};
static double s[14] = {9.6, 13., 15., 12., 10., 7., 5.6, 3.3, 1.1, 0.9, 0.5, 0.36, 0.23, 0.15};
double gam, sigma = 0., T;
int i;
T = E / 27.2;
if (E >= 400.) {
gam = (emass + T) / emass;
sigma = gam * gam * M_PI / (2. * T)*(4.2106 - 1. / T) * a02;
} else
for (i = 0; i <= 12; i++)
if (E >= e[i] && E < e[i + 1])
sigma = 1.e-20 * (s[i]+(s[i + 1] - s[i])*(E - e[i]) / (e[i + 1] - e[i]));
return sigma;
}
double sigmaexc(double E) {
// This function computes the electronic excitation cross section of
// electron scatt. on molecular hydrogen.
// E: incident electron energy in eV,
// sigmaexc: cross section in m^2
double sigma;
if (E < 9.8)
sigma = 1.e-40;
else if (E >= 9.8 && E <= 250.)
sigma = sigmaBC(E) + sigmadiss10(E) + sigmadiss15(E);
else
sigma = 4. * M_PI * a02 * R / E * (0.80 * log(E / R) + 0.28);
// sigma=sigmainel(E)-sigmaion(E);
return sigma;
}
double sigmaion(double E) {
// This function computes the total ionization cross section of
// electron scatt. on molecular hydrogen.
// E: incident electron energy in eV,
// sigmaion: total ionization cross section of
// e+H2 --> e+e+H2^+ or e+e+H^+ +H
// process in m^2.
//
// E<250 eV: Eq. 5 of J. Chem. Phys. 104 (1996) 2956
// E>250: sigma_i formula on page 107 in
// Phys. Rev. A7 (1973) 103.
// Good agreement with measured results of
// PR A 54 (1996) 2146, and
// Physica 31 (1965) 94.
//
static double B = 15.43, U = 15.98;
double sigma, t, u, S, r, lnt;
if (E < 16.)
sigma = 1.e-40;
else if (E >= 16. && E <= 250.) {
t = E / B;
u = U / B;
r = R / B;
S = 4. * M_PI * a02 * 2. * r*r;
lnt = log(t);
sigma = S / (t + u + 1.)*(lnt / 2. * (1. - 1. / (t * t)) + 1. - 1. / t - lnt / (t + 1.));
} else
sigma = 4. * M_PI * a02 * R / E * (0.82 * log(E / R) + 1.3);
return sigma;
}
double sigmaBC(double E) {
// This function computes the sigmaexc electronic excitation
// cross section to the B and C states, with energy loss
// about 12.5 eV.
// E is incident electron energy in eV,
// sigmaexc in m^2
static double aB[9] = {-4.2935194e2, 5.1122109e2, -2.8481279e2,
8.8310338e1, -1.6659591e1, 1.9579609,
-1.4012824e-1, 5.5911348e-3, -9.5370103e-5};
static double aC[9] = {-8.1942684e2, 9.8705099e2, -5.3095543e2,
1.5917023e2, -2.9121036e1, 3.3321027,
-2.3305961e-1, 9.1191781e-3, -1.5298950e-4};
double lnsigma, lnE, lnEn, sigmaB, Emin, sigma, sigmaC;
int n;
sigma = 0.;
Emin = 12.5;
lnE = log(E);
lnEn = 1.;
lnsigma = 0.;
if (E < Emin)
sigmaB = 0.;
else {
for (n = 0; n <= 8; n++) {
lnsigma += aB[n] * lnEn;
lnEn = lnEn*lnE;
}
sigmaB = exp(lnsigma);
}
sigma += sigmaB;
// sigma=0.;
// C state:
Emin = 15.8;
lnE = log(E);
lnEn = 1.;
lnsigma = 0.;
if (E < Emin)
sigmaC = 0.;
else {
for (n = 0; n <= 8; n++) {
lnsigma += aC[n] * lnEn;
lnEn = lnEn*lnE;
}
sigmaC = exp(lnsigma);
}
sigma += sigmaC;
return sigma * 1.e-4;
}
//////////////////////////////////////////////////////////////////
double sigmadiss10(double E) {
// This function computes the sigmadiss10 electronic
// dissociative excitation
// cross section, with energy loss
// about 10 eV.
// E is incident electron energy in eV,
// sigmadiss10 in m^2
static double a[9] = {-2.297914361e5, 5.303988579e5, -5.316636672e5,
3.022690779e5, -1.066224144e5, 2.389841369e4,
-3.324526406e3, 2.624761592e2, -9.006246604};
double lnsigma, lnE, lnEn, Emin, sigma;
int n;
// E is in eV
sigma = 0.;
Emin = 9.8;
lnE = log(E);
lnEn = 1.;
lnsigma = 0.;
if (E < Emin)
sigma = 0.;
else {
for (n = 0; n <= 8; n++) {
lnsigma += a[n] * lnEn;
lnEn = lnEn*lnE;
}
sigma = exp(lnsigma);
}
return sigma * 1.e-4;
}
//////////////////////////////////////////////////////////////////
double sigmadiss15(double E) {
// This function computes the sigmadiss15 electronic
// dissociative excitation
// cross section, with energy loss
// about 15 eV.
// E is incident electron energy in eV,
// sigmadiss15 in m^2
static double a[9] = {-1.157041752e3, 1.501936271e3, -8.6119387e2,
2.754926257e2, -5.380465012e1, 6.573972423,
-4.912318139e-1, 2.054926773e-2, -3.689035889e-4};
double lnsigma, lnE, lnEn, Emin, sigma;
int n;
// E is in eV
sigma = 0.;
Emin = 16.5;
lnE = log(E);
lnEn = 1.;
lnsigma = 0.;
if (E < Emin)
sigma = 0.;
else {
for (n = 0; n <= 8; n++) {
lnsigma += a[n] * lnEn;
lnEn = lnEn*lnE;
}
sigma = exp(lnsigma);
}
return sigma * 1.e-4;
}
void randomel(double E, double *Eloss, double *theta) {
// This subroutine generates energy loss and polar scatt. angle according to
// electron elastic scattering in molecular hydrogen.
// Input:
// E: incident electron energy in eV.
// Output:
// *Eloss: energy loss in eV
// *theta: change of polar angle in degrees
static double H2molmass = 69.e6;
double T, c, b, u[3], G, a, gam, K2, Gmax;
int i;
if (E >= 250.)
Gmax = 1.e-19;
else if (E < 250. && E >= 150.)
Gmax = 2.5e-19;
else
Gmax = 1.e-18;
T = E / 27.2;
gam = 1. + T / (clight * clight); // relativistic correction factor
b = 2. / (1. + gam) / T;
for (i = 1; i < 5000; i++) {
subrn(u, 2);
c = 1. + b - b * (2. + b) / (b + 2. * u[1]);
K2 = 2. * T * (1. + gam) * fabs(1. - c); // momentum transfer squared
a = (4. + K2)*(4. + K2) / (gam * gam);
G = a * Del(E, c);
if (G > Gmax * u[2]) break;
}
*theta = acos(c)*180. / M_PI;
*Eloss = 2. * emass / H2molmass * (1. - c) * E;
return;
}
void randomexc(double E, double *Eloss, double *theta) {
// This subroutine generates energy loss and polar scatt. angle according to
// electron excitation scattering in molecular hydrogen.
// Input:
// E: incident electron energy in eV.
// Output:
// *Eloss: energy loss in eV
// *theta: change of polar angle in degrees
static int iff = 0;
static double sum[1001], fmax, Ecen = 12.6 / 27.21;
double T, c = 0., u[3], K, xmin, ymin, ymax, x, y, fy, dy, pmax;
double D, Dmax;
int i, j, n = 0, N, v = 0;
// Energy values of the excited electronic states:
// (from Mol. Phys. 41 (1980) 1501, in Hartree atomic units)
static double En[7] = {12.73 / 27.2, 13.2 / 27.2, 14.77 / 27.2, 15.3 / 27.2,
14.93 / 27.2, 15.4 / 27.2, 13.06 / 27.2};
// Probability numbers of the electronic states:
// (from testelectron7.c calculation )
static double p[7] = {35.86, 40.05, 6.58, 2.26, 9.61, 4.08, 1.54};
// Energy values of the B vibrational states:
// (from: Phys. Rev. A51 (1995) 3745 , in Hartree atomic units)
static double EB[28] = {0.411, 0.417, 0.423, 0.428, 0.434, 0.439, 0.444, 0.449,
0.454, 0.459, 0.464, 0.468, 0.473, 0.477, 0.481, 0.485,
0.489, 0.493, 0.496, 0.500, 0.503, 0.507, 0.510, 0.513,
0.516, 0.519, 0.521, 0.524};
// Energy values of the C vibrational states:
// (from: Phys. Rev. A51 (1995) 3745 , in Hartree atomic units)
static double EC[14] = {0.452, 0.462, 0.472, 0.481, 0.490, 0.498, 0.506, 0.513,
0.519, 0.525, 0.530, 0.534, 0.537, 0.539};
// Franck-Condon factors of the B vibrational states:
// (from: Phys. Rev. A51 (1995) 3745 )
static double pB[28] = {4.2e-3, 1.5e-2, 3.0e-2, 4.7e-2, 6.3e-2, 7.3e-2, 7.9e-2,
8.0e-2, 7.8e-2, 7.3e-2, 6.6e-2, 5.8e-2, 5.1e-2, 4.4e-2,
3.7e-2, 3.1e-2, 2.6e-2, 2.2e-2, 1.8e-2, 1.5e-2, 1.3e-2,
1.1e-2, 8.9e-3, 7.4e-3, 6.2e-3, 5.2e-3, 4.3e-3, 3.6e-3};
// Franck-Condon factors of the C vibrational states:
// (from: Phys. Rev. A51 (1995) 3745 )
static double pC[14] = {1.2e-1, 1.9e-1, 1.9e-1, 1.5e-1, 1.1e-1, 7.5e-2, 5.0e-2,
3.3e-2, 2.2e-2, 1.4e-2, 9.3e-3, 6.0e-3, 3.7e-3, 1.8e-3};
T = 20000. / 27.2;
//
xmin = Ecen * Ecen / (2. * T);
ymin = log(xmin);
ymax = log(8. * T + xmin);
dy = (ymax - ymin) / 1000.;
// Initialization of the sum[] vector, and fmax calculation:
if (iff == 0) {
fmax = 0;
for (i = 0; i <= 1000; i++) {
y = ymin + dy*i;
K = exp(y / 2.);
sum[i] = sumexc(K);
if (sum[i] > fmax) fmax = sum[i];
}
fmax = 1.05 * fmax;
iff = 1;
}
//
// Scattering angle *theta generation:
//
T = E / 27.2;
if (E >= 100.) {
xmin = Ecen * Ecen / (2. * T);
ymin = log(xmin);
ymax = log(8. * T + xmin);
dy = (ymax - ymin) / 1000.;
// Generation of y values with the Neumann acceptance-rejection method:
for (j = 1; j < 5000; j++) {
subrn(u, 2);
y = ymin + (ymax - ymin) * u[1];
K = exp(y / 2.);
fy = sumexc(K);
if (fmax * u[2] < fy) break;
}
// Calculation of c=cos(theta) and theta:
x = exp(y);
c = 1. - (x - xmin) / (4. * T);
*theta = acos(c)*180. / M_PI;
} else {
if (E <= 25.)
Dmax = 60.;
else if (E > 25. && E <= 35.)
Dmax = 95.;
else if (E > 35. && E <= 50.)
Dmax = 150.;
else
Dmax = 400.;
for (j = 1; j < 5000; j++) {
subrn(u, 2);
c = -1. + 2. * u[1];
D = Dexc(E, c)*1.e22;
if (Dmax * u[2] < D) break;
}
*theta = acos(c)*180. / M_PI;
}
// Energy loss *Eloss generation:
// First we generate the electronic state, using the Neumann
// acceptance-rejection method for discrete distribution:
N = 7; // the number of electronic states in our calculation
pmax = p[1]; // the maximum of the p[] values
for (j = 1; j < 5000; j++) {
subrn(u, 2);
n = (int) (N * u[1]);
if (u[2] * pmax < p[n]) break;
}
if (n < 0) n = 0;
if (n > 6) n = 6;
if (n > 1) // Bp, Bpp, D, Dp, EF states
{
*Eloss = En[n]*27.2;
return;
}
if (n == 0) // B state; we generate now a vibrational state,
// using the Frank-Condon factors
{
N = 28; // the number of B vibrational states in our calculation
pmax = pB[7]; // maximum of the pB[] values
for (j = 1; j < 5000; j++) {
subrn(u, 2);
v = (int) (N * u[1]);
if (u[2] * pmax < pB[v]) break;
}
if (v < 0) v = 0;
if (v > 27) v = 27;
*Eloss = EB[v]*27.2;
}
if (n == 1) // C state; we generate now a vibrational state,
// using the Franck-Condon factors
{
N = 14; // the number of C vibrational states in our calculation
pmax = pC[1]; // maximum of the pC[] values
for (j = 1; j < 5000; j++) {
subrn(u, 2);
v = (int) (N * u[1]);
if (u[2] * pmax < pC[v]) break;
}
if (v < 0) v = 0;
if (v > 13) v = 13;
*Eloss = EC[v]*27.2;
}
return;
}
void randomion(double E, double *Eloss, double *theta) {
// This subroutine generates energy loss and polar scatt. angle according to
// electron ionization scattering in molecular hydrogen.
// Input:
// E: incident electron energy in eV.
// Output:
// *Eloss: energy loss in eV
// *theta: change of polar angle in degrees
// The kinetic energy of the secondary electron is: Eloss-15.4 eV
//
static double Ei = 15.45 / 27.21;
double c, b, u[3], K, xmin, ymin, ymax, x, y, T, G, W, Gmax;
double q, h, F, Fmin, Fmax, Gp, Elmin, Elmax, qmin, qmax, El, wmax;
double WcE, Jstarq, WcstarE, w, D2ion;
int j;
double K2, KK, fE, kej, ki, kf, Rex, arg, arctg;
int i;
double st1, st2;
//
// I. Generation of theta
// -----------------------
Gmax = 1.e-20;
if (E < 200.)
Gmax = 2.e-20;
T = E / 27.2;
xmin = Ei * Ei / (2. * T);
b = xmin / (4. * T);
ymin = log(xmin);
ymax = log(8. * T + xmin);
// Generation of y values with the Neumann acceptance-rejection method:
for (j = 1; j < 5000; j++) {
subrn(u, 2);
y = ymin + (ymax - ymin) * u[1];
K = exp(y / 2.);
c = 1. + b - K * K / (4. * T);
G = K * K * (Dinel(E, c) - Dexc(E, c));
if (Gmax * u[2] < G) break;
}
// y --> x --> c --> theta
x = exp(y);
c = 1. - (x - xmin) / (4. * T);
*theta = acos(c)*180. / M_PI;
//
// II. Generation of Eloss, for fixed theta
// ----------------------------------------
//
// For E<=100 eV we use subr. gensecelen
// (in this case no correlation between theta and Eloss)
if (E <= 100.) {
gensecelen(E, &W);
*Eloss = 15.45 + W;
return;
}
// For theta>=20 the free electron model is used
// (with full correlation between theta and Eloss)
if (*theta >= 20.) {
*Eloss = E * (1. - c * c);
return;
}
// For E>100 eV and theta<20: analytical first Born approximation
// formula of Bethe for H atom (with modification for H2)
//
// Calc. of wmax:
if (*theta >= 0.7)
wmax = 1.1;
else if (*theta <= 0.7 && *theta > 0.2)
wmax = 2.;
else if (*theta <= 0.2 && *theta > 0.05)
wmax = 4.;
else
wmax = 8.;
// We generate the q value according to the Jstarq pdf. We have to
// define the qmin and qmax limits for this generation:
K = sqrt(4. * T * (1. - Ei / (2. * T) - sqrt(1. - Ei / T) * c));
Elmin = Ei;
Elmax = (E + 15.45) / 2. / 27.2;
qmin = Elmin / K - K / 2.;
qmax = Elmax / K - K / 2.;
//
q = qmax;
Fmax = 1. / 2. + 1. / M_PI * (q / (1. + q * q) + atan(q));
q = qmin;
Fmin = 1. / 2. + 1. / M_PI * (q / (1. + q * q) + atan(q));
h = Fmax - Fmin;
// Generation of Eloss with the Neumann acceptance-rejection method:
for (j = 1; j < 5000; j++) {
// Generation of q with inverse transform method
// (we use the Newton-Raphson method in order to solve the nonlinear eq.
// for the inversion) :
subrn(u, 2);
F = Fmin + h * u[1];
y = 0.;
for (i = 1; i <= 30; i++) {
G = 1. / 2. + (y + sin(2. * y) / 2.) / M_PI;
Gp = (1. + cos(2. * y)) / M_PI;
y = y - (G - F) / Gp;
if (fabs(G - F) < 1.e-8) break;
}
q = tan(y);
// We have the q value, so we can define El, and calculate the weight:
El = q * K + K * K / 2.;
// First Born approximation formula of Bethe for e-H ionization:
KK = K;
ki = sqrt(2. * T);
kf = sqrt(2. * (T - El));
K2 = 4. * T * (1. - El / (2. * T) - sqrt(1. - El / T) * c);
if (K2 < 1.e-9) K2 = 1.e-9;
K = sqrt(K2); // momentum transfer
Rex = 1. - K * K / (kf * kf) + K2 * K2 / (kf * kf * kf * kf);
kej = sqrt(2. * fabs(El - Ei) + 1.e-8);
st1 = K2 - 2. * El + 2.;
if (fabs(st1) < 1.e-9) st1 = 1.e-9;
arg = 2. * kej / st1;
if (arg >= 0.)
arctg = atan(arg);
else
arctg = atan(arg) + M_PI;
st1 = (K + kej)*(K + kej) + 1.;
st2 = (K - kej)*(K - kej) + 1.;
fE = 1024. * El * (K2 + 2. / 3. * El) / (st1 * st1 * st1 * st2 * st2 * st2) *
exp(-2. / kej * arctg) / (1. - exp(-2. * M_PI / kej));
D2ion = 2. * kf / ki * Rex / (El * K2) * fE;
K = KK;
//
WcE = D2ion;
Jstarq = 16. / (3. * M_PI * (1. + q * q)*(1. + q * q));
WcstarE = 4. / (K * K * K * K * K) * Jstarq;
w = WcE / WcstarE;
if (wmax * u[2] < w) break;
}
//
*Eloss = El * 27.2;
//
return;
}
double Del(double E, double c) {
// This subroutine computes the differential cross section
// Del= d sigma/d Omega of elastic electron scattering
// on molecular hydrogen.
// See: Nishimura et al., J. Phys. Soc. Jpn. 54 (1985) 1757.
// Input: E= electron kinetic energy in eV
// c= cos(theta), where theta is the polar scatt. angle
// Del: in m^2/steradian
static double Cel[50] = {
-0.512, -0.512, -0.509, -0.505, -0.499,
-0.491, -0.476, -0.473, -0.462, -0.452,
-0.438, -0.422, -0.406, -0.388, -0.370,
-0.352, -0.333, -0.314, -0.296, -0.277,
-0.258, -0.239, -0.221, -0.202, -0.185,
-0.167, -0.151, -0.135, -0.120, -0.105,
-0.092, -0.070, -0.053, -0.039, -0.030,
-0.024, -0.019, -0.016, -0.014, -0.013,
-0.012, -0.009, -0.008, -0.006, -0.005,
-0.004, -0.003, -0.002, -0.002, -0.001
};
static double e[10] = {0., 3., 6., 12., 20., 32., 55., 85., 150., 250.};
static double t[10] = {0., 10., 20., 30., 40., 60., 80., 100., 140., 180.};
static double D[9][10] = {
{2.9, 2.7, 2.5, 2.1, 1.8, 1.2, 0.9, 1., 1.6, 1.9},
{4.2, 3.6, 3.1, 2.5, 1.9, 1.1, 0.8, 0.9, 1.3, 1.4},
{6., 4.4, 3.2, 2.3, 1.8, 1.1, 0.7, 0.54, 0.5, 0.6},
{6., 4.1, 2.8, 1.9, 1.3, 0.6, 0.3, 0.17, 0.16, 0.23},
{4.9, 3.2, 2., 1.2, 0.8, 0.3, 0.15, 0.09, 0.05, 0.05},
{5.2, 2.5, 1.2, 0.64, 0.36, 0.13, 0.05, 0.03, 0.016, 0.02},
{4., 1.7, 0.7, 0.3, 0.16, 0.05, 0.02, 0.013, 0.01, 0.01},
{2.8, 1.1, 0.4, 0.15, 0.07, 0.02, 0.01, 0.007, 0.004, 0.003},
{1.2, 0.53, 0.2, 0.08, 0.03, 0.0074, 0.003, 0.0016, 0.001, 0.0008}
};
double T, K2, K, d, st1, st2, DH, gam, Delreturn = 0., CelK, Ki, theta;
int i, j;
T = E / 27.2;
if (E >= 250.) {
gam = 1. + T / (clight * clight); // relativistic correction factor
K2 = 2. * T * (1. + gam)*(1. - c);
if (K2 < 0.) K2 = 1.e-30;
K = sqrt(K2);
if (K < 1.e-9) K = 1.e-9; // momentum transfer
d = 1.4009; // distance of protons in H2
st1 = 8. + K2;
st2 = 4. + K2;
// DH is the diff. cross section for elastic electron scatt.
// on atomic hydrogen within the first Born approximation :
DH = 4. * st1 * st1 / (st2 * st2 * st2 * st2) * a02;
// CelK calculation with linear interpolation.
// CelK is the correction of the elastic electron
// scatt. on molecular hydrogen compared to the independent atom
// model.
if (K < 3.) {
i = (int) (K / 0.1);
Ki = i * 0.1;
CelK = Cel[i]+(K - Ki) / 0.1 * (Cel[i + 1] - Cel[i]);
} else if (K >= 3. && K < 5.) {
i = (int) (30 + (K - 3.) / 0.2);
Ki = 3. + (i - 30)*0.2;
CelK = Cel[i]+(K - Ki) / 0.2 * (Cel[i + 1] - Cel[i]);
} else if (K >= 5. && K < 9.49) {
i = (int) (40 + (K - 5.) / 0.5);
Ki = 5. + (i - 40)*0.5;
CelK = Cel[i]+(K - Ki) / 0.5 * (Cel[i + 1] - Cel[i]);
} else
CelK = 0.;
Delreturn = 2. * gam * gam * DH * (1. + sin(K * d) / (K * d))*(1. + CelK);
} else {
theta = acos(c)*180. / M_PI;
for (i = 0; i <= 8; i++)
if (E >= e[i] && E < e[i + 1])
for (j = 0; j <= 8; j++)
if (theta >= t[j] && theta < t[j + 1])
Delreturn = 1.e-20 * (D[i][j]+(D[i][j + 1] - D[i][j])*
(theta - t[j]) / (t[j + 1] - t[j]));
}
return Delreturn;
}
////////////////////////////////////////////////////////////////
double Dexc(double E, double c) {
// This subroutine computes the differential cross section
// Del= d sigma/d Omega of excitation electron scattering
// on molecular hydrogen.
// Input: E= electron kinetic energy in eV
// c= cos(theta), where theta is the polar scatt. angle
// Dexc: in m^2/steradian
double K2, K, sigma = 0., T, theta;
static double EE = 12.6 / 27.2;
static double e[5] = {0., 25., 35., 50., 100.};
static double t[9] = {0., 10., 20., 30., 40., 60., 80., 100., 180.};
static double D[4][9] = {
{60., 43., 27., 18., 13., 8., 6., 6., 6.},
{95., 70., 21., 9., 6., 3., 2., 2., 2.,},
{150., 120., 32., 8., 3.7, 1.9, 1.2, 0.8, 0.8},
{400., 200., 12., 2., 1.4, 0.7, 0.3, 0.2, 0.2}
};
int i, j;
//
T = E / 27.2;
if (E >= 100.) {
K2 = 4. * T * (1. - EE / (2. * T) - sqrt(1. - EE / T) * c);
if (K2 < 1.e-9) K2 = 1.e-9;
K = sqrt(K2); // momentum transfer
sigma = 2. / K2 * sumexc(K) * a02;
} else if (E <= 10.)
sigma = 0.;
else {
theta = acos(c)*180. / M_PI;
for (i = 0; i <= 3; i++)
if (E >= e[i] && E < e[i + 1])
for (j = 0; j <= 7; j++)
if (theta >= t[j] && theta < t[j + 1])
sigma = 1.e-22 * (D[i][j]+(D[i][j + 1] - D[i][j])*
(theta - t[j]) / (t[j + 1] - t[j]));
}
return sigma;
}
double sumexc(double K) {
static double Kvec[15] = {0., 0.1, 0.2, 0.4, 0.6, 0.8, 1., 1.2, 1.5, 1.8, 2., 2.5, 3., 4., 5.};
static double fvec[7][15] ={
{2.907e-1, 2.845e-1, 2.665e-1, 2.072e-1, 1.389e-1, // B
8.238e-2, 4.454e-2, 2.269e-2, 7.789e-3, 2.619e-3,
1.273e-3, 2.218e-4, 4.372e-5, 2.889e-6, 4.247e-7},
{3.492e-1, 3.367e-1, 3.124e-1, 2.351e-1, 1.507e-1, // C
8.406e-2, 4.214e-2, 1.966e-2, 5.799e-3, 1.632e-3,
6.929e-4, 8.082e-5, 9.574e-6, 1.526e-7, 7.058e-9},
{6.112e-2, 5.945e-2, 5.830e-2, 5.072e-2, 3.821e-2, // Bp
2.579e-2, 1.567e-2, 8.737e-3, 3.305e-3, 1.191e-3,
6.011e-4, 1.132e-4, 2.362e-5, 1.603e-6, 2.215e-7},
{2.066e-2, 2.127e-2, 2.137e-2, 1.928e-2, 1.552e-2, // Bpp
1.108e-2, 7.058e-3, 4.069e-3, 1.590e-3, 5.900e-4,
3.046e-4, 6.142e-5, 1.369e-5, 9.650e-7, 1.244e-7},
{9.405e-2, 9.049e-2, 8.613e-2, 7.301e-2, 5.144e-2, // D
3.201e-2, 1.775e-2, 8.952e-3, 2.855e-3, 8.429e-4,
3.655e-4, 4.389e-5, 5.252e-6, 9.010e-8, 7.130e-9},
{4.273e-2, 3.862e-2, 3.985e-2, 3.362e-2, 2.486e-2, // Dp
1.612e-2, 9.309e-3, 4.856e-3, 1.602e-3, 4.811e-4,
2.096e-4, 2.498e-5, 2.905e-6, 5.077e-8, 6.583e-9},
{0.000e-3, 2.042e-3, 7.439e-3, 2.200e-2, 3.164e-2, // EF
3.161e-2, 2.486e-2, 1.664e-2, 7.562e-3, 3.044e-3,
1.608e-3, 3.225e-4, 7.120e-5, 6.290e-6, 1.066e-6}};
static double EeV[7] = {12.73, 13.20, 14.77, 15.3, 14.93, 15.4, 13.06};
int n, j, jmin = 0, nmax;
double En, f[7], x4[4], f4[4], sum;
//
sum = 0.;
nmax = 6;
for (n = 0; n <= nmax; n++) {
En = EeV[n] / 27.21; // En is the excitation energy in Hartree atomic units
if (K >= 5.)
f[n] = 0.;
else if (K >= 3. && K <= 4.)
f[n] = fvec[n][12]+(K - 3.)*(fvec[n][13] - fvec[n][12]);
else if (K >= 4. && K <= 5.)
f[n] = fvec[n][13]+(K - 4.)*(fvec[n][14] - fvec[n][13]);
else {
for (j = 0; j < 14; j++) {
if (K >= Kvec[j] && K <= Kvec[j + 1]) jmin = j - 1;
}
if (jmin < 0) jmin = 0;
if (jmin > 11) jmin = 11;
for (j = 0; j <= 3; j++) {
x4[j] = Kvec[jmin + j];
f4[j] = fvec[n][jmin + j];
}
f[n] = lagrange(4, x4, f4, K);
}
sum += f[n] / En;
}
return sum;
}
///////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////
double Dinel(double E, double c) {
// This subroutine computes the differential cross section
// Dinel= d sigma/d Omega of inelastic electron scattering
// on molecular hydrogen, within the first Born approximation.
// Input: E= electron kinetic energy in eV
// c= cos(theta), where theta is the polar scatt. angle
// Dinel: in m2/steradian
static double Cinel[50] = {
-0.246, -0.244, -0.239, -0.234, -0.227,
-0.219, -0.211, -0.201, -0.190, -0.179,
-0.167, -0.155, -0.142, -0.130, -0.118,
-0.107, -0.096, -0.085, -0.076, -0.067,
-0.059, -0.051, -0.045, -0.039, -0.034,
-0.029, -0.025, -0.022, -0.019, -0.016,
-0.014, -0.010, -0.008, -0.006, -0.004,
-0.003, -0.003, -0.002, -0.002, -0.001,
-0.001, -0.001, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000
};
static double Ei = 0.568;
double T, K2, K, st1, F, DH, Dinelreturn, CinelK, Ki;
int i;
if (E < 16.)
return Dexc(E, c);
T = E / 27.2;
K2 = 4. * T * (1. - Ei / (2. * T) - sqrt(1. - Ei / T) * c);
if (K2 < 1.e-9) K2 = 1.e-9;
K = sqrt(K2); // momentum transfer
st1 = 1. + K2 / 4.;
F = 1. / (st1 * st1); // scatt. formfactor of hydrogen atom
// DH is the diff. cross section for inelastic electron scatt.
// on atomic hydrogen within the first Born approximation :
DH = 4. / (K2 * K2)*(1. - F * F) * a02;
// CinelK calculation with linear interpolation.
// CinelK is the correction of the inelastic electron
// scatt. on molecular hydrogen compared to the independent atom
// model.
if (K < 3.) {
i = (int) (K / 0.1);
Ki = i * 0.1;
CinelK = Cinel[i]+(K - Ki) / 0.1 * (Cinel[i + 1] - Cinel[i]);
} else if (K >= 3. && K < 5.) {
i = (int) (30 + (K - 3.) / 0.2);
Ki = 3. + (i - 30)*0.2;
CinelK = Cinel[i]+(K - Ki) / 0.2 * (Cinel[i + 1] - Cinel[i]);
} else if (K >= 5. && K < 9.49) {
i = (int) (40 + (K - 5.) / 0.5);
Ki = 5. + (i - 40)*0.5;
CinelK = Cinel[i]+(K - Ki) / 0.5 * (Cinel[i + 1] - Cinel[i]);
} else
CinelK = 0.;
Dinelreturn = 2. * DH * (1. + CinelK);
return Dinelreturn;
}
////////////////////////////////////////////////////////////////////
void gensecelen(double E, double *W) {
// This subroutine generates secondary electron energy W
// from ionization of incident electron energy E, by using
// the Lorentzian of Aseev et al. (Eq. 8).
// E and W in eV.
static double Ei = 15.45, eps2 = 14.3, b = 6.25;
static double B;
double D, A, eps, a, u, epsmax;
static int iff = 0;
if (iff == 0) {
B = atan((Ei - eps2) / b);
iff = 1;
}
epsmax = (E + Ei) / 2.;
A = atan((epsmax - eps2) / b);
D = b / (A - B);
u = random_get();
a = b / D * (u + D / b * B);
eps = eps2 + b * tan(a);
*W = eps - Ei;
return;
}
void subrn(double *u, int len) {
// This subroutine computes random numbers u[1],...,u[len]
// in the (0,1) interval. It uses the 0<IJKLRANDOM<900000000
// integer as initialization seed.
// In the calling program the dimension
// of the u[] vector should be larger than len (the u[0] value is
// not used).
// For each IJKLRANDOM
// numbers the program computes completely independent random number
// sequences (see: F. James, Comp. Phys. Comm. 60 (1990) 329, sec. 3.3).
//
// remark by T. Thuemmler:
// same random numbers appear each time one restarts the whole program
//
static long IJKLRANDOM = 100;
static int iff = 0;
static long ijkl, ij, kl, i, j, k, l, ii, jj, m, i97, j97, ivec;
static float s, t, uu[98], c, cd, cm, uni;
if (iff == 0) {
ijkl = IJKLRANDOM;
if (ijkl < 1 || ijkl >= 900000000) ijkl = 1;
ij = ijkl / 30082;
kl = ijkl - 30082 * ij;
i = ((ij / 177) % 177) + 2;
j = (ij % 177) + 2;
k = ((kl / 169) % 178) + 1;
l = kl % 169;
for (ii = 1; ii <= 97; ii++) {
s = 0;
t = 0.5;
for (jj = 1; jj <= 24; jj++) {
m = (((i * j) % 179) * k) % 179;
i = j;
j = k;
k = m;
l = (53 * l + 1) % 169;
if ((l * m) % 64 >= 32) s = s + t;
t = 0.5 * t;
}
uu[ii] = s;
}
c = 362436. / 16777216.;
cd = 7654321. / 16777216.;
cm = 16777213. / 16777216.;
i97 = 97;
j97 = 33;
iff = 1;
}
for (ivec = 1; ivec <= len; ivec++) {
uni = uu[i97] - uu[j97];
if (uni < 0.) uni = uni + 1.;
uu[i97] = uni;
i97 = i97 - 1;
if (i97 == 0) i97 = 97;
j97 = j97 - 1;
if (j97 == 0) j97 = 97;
c = c - cd;
if (c < 0.) c = c + cm;
uni = uni - c;
if (uni < 0.) uni = uni + 1.;
if (uni == 0.) {
uni = uu[j97]*0.59604644775391e-07;
if (uni == 0.) uni = 0.35527136788005e-14;
}
u[ivec] = uni;
}
// cout << endl<< "random: " << u[1] << endl << flush;
return;
}
double lagrange(int n, double *xn, double *fn, double x) {
int i, j;
double f, a[100], b[100], aa, bb;
f = 0.;
for (j = 0; j < n; j++) {
for (i = 0; i < n; i++) {
a[i] = x - xn[i];
b[i] = xn[j] - xn[i];
}
a[j] = b[j] = aa = bb = 1.;
for (i = 0; i < n; i++) {
aa = aa * a[i];
bb = bb * b[i];
}
f += fn[j] * aa / bb;
}
return f;
}
/* The following functions are used to calculate the new impulse after an
* interaction.
* In these functions a vector is a double[3] array and a matrix a double[9]
* array.*/
/* Sets the matrix to a rotation matrix which performs a rotation around the
* y axis of the angle theta and then around the z axis of phi. theta and
* phi are given in radians */
static void matrix_rot(double theta, double phi, double* matrix) {
double cos_theta = cos(theta);
double sin_theta = sin(theta);
double cos_phi = cos(phi);
double sin_phi = sin(phi);
matrix[0] = cos_theta*cos_phi;
matrix[1] = -sin_phi;
matrix[2] = sin_theta*cos_phi;
matrix[3] = cos_theta*sin_phi;
matrix[4] = cos_phi;
matrix[5] = sin_theta*sin_phi;
matrix[6] = -sin_theta;
matrix[7] = 0;
matrix[8] = cos_theta;
}
/* Multiplicates the matrix a with the vector x and returns the result as
* vector y*/
static void matrix_vec(double* a, double* x, double* y) {
int i, j;
for (i = 0; i < 3; i++) {
y[i] = 0;
for (j = 0; j < 3; j++) {
y[i] += a[3 * i + j] * x[j];
}
}
}
/* Returns the length of the vector x */
static double vector_abs(double* x) {
double x2 = 0;
int i;
for (i = 0; i < 3; i++)
x2 += x[i] * x[i];
return sqrt(x2);
}
/* Returns the polar angle phi of the vector x in radians */
static double vector_phi(double* x) {
return atan2(x[1], x[0]);
}
/* Returns the azimutal angle theta of the vector x in radians */
static double vector_theta(double* x) {
return acos(x[2] / vector_abs(x));
}
/* Creates a vector from spheric coordinates and returns it as x */
static void vector_spheric(double r, double theta, double phi, double* x) {
double sin_theta = sin(theta);
double cos_theta = cos(theta);
double sin_phi = sin(phi);
double cos_phi = cos(phi);
x[0] = r * sin_theta * cos_phi;
x[1] = r * sin_theta * sin_phi;
x[2] = r * cos_theta;
}
static double sigma_total(double E) {
return 1e4 * (sigmaion(E) // sigma for ionisation of H2
+ sigmaexc(E) // sigma for excitation of H2
+ sigmael(E)); // sigma for elastic scattering
}
/*
void scatter_init(double pressure, double temperature)
{
random_set_method(RANDOM_CW);
//random_seed(time(0));
//output = fopen("scatter.dat", "a");
density = pressure / BOLTZMANN / temperature;
energy_loss = 0;
scatter_angle = 0;
scatter_set_output_file(NULL);
scatter_reset();
}
*/
void scatter_reset() {
x = 0;
totalx = 0;
event_x = -log(random_get());
}
void scatter_move(double dx, double E) {
double sigma = sigma_total(E);
x += dx * density * sigma;
totalx += dx;
//fprintf(stderr, "%e\n", x);
}
int scatter_has_event() {
return (x >= event_x);
}
/*
Interaction scatter_event(double *P, double E)
{
double el = sigmael(E);
double exc = sigmaexc(E);
double ion = sigmaion(E);
double random = (el + exc + ion) * random_get();
double phi_old = vector_phi(P);
double theta_old = vector_theta(P);
double eloss, angle, theta, phi;
double matrix[9];
double y[3];
double p2;
double Enew;
Interaction interaction;
if(random < el) {
randomel(E, &eloss, &angle);
interaction = ELASTIC;
}
else if(random < exc) {
randomexc(E, &eloss, &angle);
interaction = EXCITATION;
}
else {
randomion(E, &eloss, &angle);
interaction = IONIZATION;
}
Enew = (E - eloss) * CHARGEUNIT;
p2 = Enew*Enew/C/C + 2*Enew*ELECTRON_MASS;
theta = M_PI / 180 * angle;
phi = 2 * M_PI * random_get();
vector_spheric(sqrt(p2), theta, phi, y);
matrix_rot(theta_old, phi_old, matrix);
matrix_vec(matrix, y, P);
energy_loss = eloss;
scatter_angle = angle;
scatter_reset(E);
if(output)
fprintf(output, "%f\t%d\t%f\t%f\n", E, interaction, eloss, angle);
return interaction;
}
*/
double scatter_get_energy_loss() {
return energy_loss;
}
double scatter_get_scatter_angle() {
return scatter_angle;
}
double scatter_free_path(double E) {
return 1 / density / sigma_total(E);
}
void scatter_set_output_file(const char* filename) {
if (output) {
fclose(output);
output = NULL;
}
if (filename) {
output = fopen(filename, "w");
if (output)
fprintf(output, "#Energy\tEvent\tEloss\tAngle\n");
else
fprintf(stderr,
"Warning: Cannot write to file '%s'. Debugging output will be"
"omitted.\n", filename);
}
}
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