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static double
shell_volume(double length_a, double b2a_ratio, double c2a_ratio, double thickness)
{
const double length_b = length_a * b2a_ratio;
const double length_c = length_a * c2a_ratio;
const double form_volume = length_a * length_b * length_c;
const double a_core = length_a - 2.0*thickness;
const double b_core = length_b - 2.0*thickness;
const double c_core = length_c - 2.0*thickness;
const double core_volume = a_core * b_core * c_core;
return form_volume - core_volume;
}
static double
form_volume(double length_a, double b2a_ratio, double c2a_ratio, double thickness)
{
const double length_b = length_a * b2a_ratio;
const double length_c = length_a * c2a_ratio;
const double form_volume = length_a * length_b * length_c;
return form_volume;
}
static double
radius_from_excluded_volume(double length_a, double b2a_ratio, double c2a_ratio)
{
const double r_equiv = sqrt(length_a*length_a*b2a_ratio/M_PI);
const double length_c = length_a*c2a_ratio;
return 0.5*cbrt(0.75*r_equiv*(2.0*r_equiv*length_c + (r_equiv + length_c)*(M_PI*r_equiv + length_c)));
}
static double
radius_effective(int mode, double length_a, double b2a_ratio, double c2a_ratio, double thickness)
// NOTE length_a is external dimension!
{
switch (mode) {
default:
case 1: // equivalent cylinder excluded volume
return radius_from_excluded_volume(length_a, b2a_ratio, c2a_ratio);
case 2: // equivalent outer volume sphere
return cbrt(cube(length_a)*b2a_ratio*c2a_ratio/M_4PI_3);
case 3: // half length_a
return 0.5 * length_a;
case 4: // half length_b
return 0.5 * length_a*b2a_ratio;
case 5: // half length_c
return 0.5 * length_a*c2a_ratio;
case 6: // equivalent outer circular cross-section
return length_a*sqrt(b2a_ratio/M_PI);
case 7: // half ab diagonal
return 0.5*sqrt(square(length_a) * (1.0 + square(b2a_ratio)));
case 8: // half diagonal
return 0.5*sqrt(square(length_a) * (1.0 + square(b2a_ratio) + square(c2a_ratio)));
}
}
static void
Fq(double q,
double *F1,
double *F2,
double sld,
double solvent_sld,
double length_a,
double b2a_ratio,
double c2a_ratio,
double thickness)
{
const double length_b = length_a * b2a_ratio;
const double length_c = length_a * c2a_ratio;
const double a_half = 0.5 * length_a;
const double b_half = 0.5 * length_b;
const double c_half = 0.5 * length_c;
const double vol_total = length_a * length_b * length_c;
const double vol_core = 8.0 * (a_half-thickness) * (b_half-thickness) * (c_half-thickness);
//Integration limits to use in Gaussian quadrature
const double v1a = 0.0;
const double v1b = M_PI_2; //theta integration limits
const double v2a = 0.0;
const double v2b = M_PI_2; //phi integration limits
double outer_sum_F1 = 0.0;
double outer_sum_F2 = 0.0;
for(int i=0; i<GAUSS_N; i++) {
const double theta = 0.5 * ( GAUSS_Z[i]*(v1b-v1a) + v1a + v1b );
double sin_theta, cos_theta;
SINCOS(theta, sin_theta, cos_theta);
const double termC1 = sas_sinx_x(q * c_half * cos(theta));
const double termC2 = sas_sinx_x(q * (c_half-thickness)*cos(theta));
double inner_sum_F1 = 0.0;
double inner_sum_F2 = 0.0;
for(int j=0; j<GAUSS_N; j++) {
const double phi = 0.5 * ( GAUSS_Z[j]*(v2b-v2a) + v2a + v2b );
double sin_phi, cos_phi;
SINCOS(phi, sin_phi, cos_phi);
// Amplitude AP from eqn. (13), rewritten to avoid round-off effects when arg=0
const double termA1 = sas_sinx_x(q * a_half * sin_theta * sin_phi);
const double termA2 = sas_sinx_x(q * (a_half-thickness) * sin_theta * sin_phi);
const double termB1 = sas_sinx_x(q * b_half * sin_theta * cos_phi);
const double termB2 = sas_sinx_x(q * (b_half-thickness) * sin_theta * cos_phi);
const double AP1 = vol_total * termA1 * termB1 * termC1;
const double AP2 = vol_core * termA2 * termB2 * termC2;
inner_sum_F1 += GAUSS_W[j] * (AP1-AP2);
inner_sum_F2 += GAUSS_W[j] * square(AP1-AP2);
}
inner_sum_F1 *= 0.5 * (v2b-v2a);
inner_sum_F2 *= 0.5 * (v2b-v2a);
outer_sum_F1 += GAUSS_W[i] * inner_sum_F1 * sin(theta);
outer_sum_F2 += GAUSS_W[i] * inner_sum_F2 * sin(theta);
}
outer_sum_F1 *= 0.5*(v1b-v1a);
outer_sum_F2 *= 0.5*(v1b-v1a);
// Normalize as in Eqn. (15) without the volume factor (as cancels with (V*DelRho)^2 normalization)
// The factor 2 is due to the different theta integration limit (pi/2 instead of pi)
const double form_avg = outer_sum_F1/M_PI_2;
const double form_squared_avg = outer_sum_F2/M_PI_2;
// Multiply by contrast^2. Factor corresponding to volume^2 cancels with previous normalization.
const double contrast = sld - solvent_sld;
// Convert from [1e-12 A-1] to [cm-1]
*F1 = 1.0e-2 * contrast * form_avg;
*F2 = 1.0e-4 * contrast * contrast * form_squared_avg;
}
static double
Iqabc(double qa, double qb, double qc,
double sld,
double solvent_sld,
double length_a,
double b2a_ratio,
double c2a_ratio,
double thickness)
{
const double length_b = length_a * b2a_ratio;
const double length_c = length_a * c2a_ratio;
const double a_half = 0.5 * length_a;
const double b_half = 0.5 * length_b;
const double c_half = 0.5 * length_c;
const double vol_total = length_a * length_b * length_c;
const double vol_core = 8.0 * (a_half-thickness) * (b_half-thickness) * (c_half-thickness);
// Amplitude AP from eqn. (13)
const double termA1 = sas_sinx_x(qa * a_half);
const double termA2 = sas_sinx_x(qa * (a_half-thickness));
const double termB1 = sas_sinx_x(qb * b_half);
const double termB2 = sas_sinx_x(qb * (b_half-thickness));
const double termC1 = sas_sinx_x(qc * c_half);
const double termC2 = sas_sinx_x(qc * (c_half-thickness));
const double AP1 = vol_total * termA1 * termB1 * termC1;
const double AP2 = vol_core * termA2 * termB2 * termC2;
// Multiply by contrast^2. Factor corresponding to volume^2 cancels with previous normalization.
const double delrho = sld - solvent_sld;
// Convert from [1e-12 A-1] to [cm-1]
return 1.0e-4 * square(delrho * (AP1-AP2));
}
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