Capped Cylinder - capped_cylinder.c
#define INVALID(v) (v.radius_cap < v.radius)
// Integral over a convex lens kernel for t in [h/R,1]. See the docs for
// the definition of the function being integrated.
// q is the magnitude of the q vector.
// h is the length of the lens "inside" the cylinder. This negative wrt the
// definition of h in the docs.
// radius_cap is the radius of the lens
// length is the cylinder length, or the separation between the lens halves
// theta is the angle of the cylinder wrt q.
static double
_cap_kernel(double qab, double qc, double h, double radius_cap,
double half_length)
{
// translate a point in [-1,1] to a point in [lower,upper]
const double upper = 1.0;
const double lower = h/radius_cap; // integral lower bound
const double zm = 0.5*(upper-lower);
const double zb = 0.5*(upper+lower);
// cos term in integral is:
// cos (q (R t - h + L/2) cos(theta))
// so turn it into:
// cos (m t + b)
// where:
// m = q R cos(theta)
// b = q(L/2-h) cos(theta)
const double m = radius_cap*qc; // cos argument slope
const double b = (half_length-h)*qc; // cos argument intercept
const double qab_r = radius_cap*qab; // Q*R*sin(theta)
double total = 0.0;
for (int i=0; i<GAUSS_N; i++) {
const double t = GAUSS_Z[i]*zm + zb;
const double radical = 1.0 - t*t;
const double bj = sas_2J1x_x(qab_r*sqrt(radical));
const double Fq = cos(m*t + b) * radical * bj;
total += GAUSS_W[i] * Fq;
}
// translate dx in [-1,1] to dx in [lower,upper]
const double integral = total*zm;
const double cap_Fq = 2.0*M_PI*cube(radius_cap)*integral;
return cap_Fq;
}
static double
_fq(double qab, double qc, double h, double radius_cap, double radius, double half_length)
{
const double cap_Fq = _cap_kernel(qab, qc, h, radius_cap, half_length);
const double bj = sas_2J1x_x(radius*qab);
const double si = sas_sinx_x(half_length*qc);
const double cyl_Fq = 2.0*M_PI*radius*radius*half_length*bj*si;
const double Aq = cap_Fq + cyl_Fq;
return Aq;
}
static double
form_volume(double radius, double radius_cap, double length)
{
// cap radius should never be less than radius when this is called
// Note: volume V = 2*V_cap + V_cyl
//
// V_cyl = pi r_cyl^2 L
// V_cap = 1/6 pi h_c (3 r_cyl^2 + h_c^2) = 1/3 pi h_c^2 (3 r_cap - h_c)
//
// The docs for capped cylinder give the volume as:
// V = pi r^2 L + 2/3 pi (R-h)^2 (2R + h)
// where r_cap=R and h = R - h_c.
//
// The first part is clearly V_cyl. The second part requires some work:
// (R-h)^2 => h_c^2
// (2R+h) => 2R+ h_c-h_c + h => 2R + (R-h)-h_c + h => 3R-h_c
// And so:
// 2/3 pi (R-h)^2 (2R + h) => 2/3 pi h_c^2 (3 r_cap - h_c)
// which is 2 V_cap, using the second form above.
//
// In this function we are going to use the first form of V_cap
//
// V = V_cyl + 2 V_cap
// = pi r^2 L + pi hc (r^2 + hc^2/3)
// = pi (r^2 (L+hc) + hc^3/3)
const double hc = radius_cap - sqrt(radius_cap*radius_cap - radius*radius);
return M_PI*(radius*radius*(length+hc) + hc*hc*hc/3.0);
}
static double
radius_from_excluded_volume(double radius, double radius_cap, double length)
{
const double hc = radius_cap - sqrt(radius_cap*radius_cap - radius*radius);
const double length_tot = length + 2.0*hc;
return 0.5*cbrt(0.75*radius*(2.0*radius*length_tot + (radius + length_tot)*(M_PI*radius + length_tot)));
}
static double
radius_from_volume(double radius, double radius_cap, double length)
{
const double vol_cappedcyl = form_volume(radius,radius_cap,length);
return cbrt(vol_cappedcyl/M_4PI_3);
}
static double
radius_from_totallength(double radius, double radius_cap, double length)
{
const double hc = radius_cap - sqrt(radius_cap*radius_cap - radius*radius);
return 0.5*length + hc;
}
static double
radius_effective(int mode, double radius, double radius_cap, double length)
{
switch (mode) {
default:
case 1: // equivalent cylinder excluded volume
return radius_from_excluded_volume(radius, radius_cap, length);
case 2: // equivalent volume sphere
return radius_from_volume(radius, radius_cap, length);
case 3: // radius
return radius;
case 4: // half length
return 0.5*length;
case 5: // half total length
return radius_from_totallength(radius, radius_cap,length);
}
}
static void
Fq(double q,double *F1, double *F2, double sld, double solvent_sld,
double radius, double radius_cap, double length)
{
const double h = sqrt(radius_cap*radius_cap - radius*radius);
const double half_length = 0.5*length;
// translate a point in [-1,1] to a point in [0, pi/2]
const double zm = M_PI_4;
const double zb = M_PI_4;
double total_F1 = 0.0;
double total_F2 = 0.0;
for (int i=0; i<GAUSS_N ;i++) {
const double theta = GAUSS_Z[i]*zm + zb;
double sin_theta, cos_theta; // slots to hold sincos function output
SINCOS(theta, sin_theta, cos_theta);
const double qab = q*sin_theta;
const double qc = q*cos_theta;
const double Aq = _fq(qab, qc, h, radius_cap, radius, half_length);
// scale by sin_theta for spherical coord integration
total_F1 += GAUSS_W[i] * Aq * sin_theta;
total_F2 += GAUSS_W[i] * Aq * Aq * sin_theta;
}
// translate dx in [-1,1] to dx in [lower,upper]
const double form_avg = total_F1 * zm;
const double form_squared_avg = total_F2 * zm;
// Contrast
const double s = (sld - solvent_sld);
*F1 = 1.0e-2 * s * form_avg;
*F2 = 1.0e-4 * s * s * form_squared_avg;
}
static double
Iqac(double qab, double qc,
double sld, double solvent_sld, double radius,
double radius_cap, double length)
{
const double h = sqrt(radius_cap*radius_cap - radius*radius);
const double Aq = _fq(qab, qc, h, radius_cap, radius, 0.5*length);
// Multiply by contrast^2 and convert to cm-1
const double s = (sld - solvent_sld);
return 1.0e-4 * square(s * Aq);
}
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