source: sasmodels/sasmodels/models/pearl_necklace.c @ 9272cbd

double _pearl_necklace_kernel(double q, double radius, double edge_sep,
        double thick_string, double num_pearls, double sld_pearl,
        double sld_string, double sld_solv);
double form_volume(double radius, double edge_sep,
        double thick_string, double num_pearls);

double Iq(double q, double radius, double edge_sep,
        double thick_string, double num_pearls, double sld, 
        double string_sld, double solvent_sld);

#define INVALID(v) (v.thick_string >= v.radius || v.num_pearls <= 0)

// From Igor library
double _pearl_necklace_kernel(double q, double radius, double edge_sep, double thick_string,
        double num_pearls, double sld_pearl, double sld_string, double sld_solv)
{
        //relative slds
        double contrast_pearl = sld_pearl - sld_solv;
        double contrast_string = sld_string - sld_solv;
        
        // number of string segments
        num_pearls = floor(num_pearls + 0.5); //Force integer number of pearls
        double num_strings = num_pearls - 1.0;
        
        //Pi
        double pi = 4.0*atan(1.0);
        
        // center to center distance between the neighboring pearls
        double A_s = edge_sep + 2.0 * radius;
        
        // Repeated Calculations
        double sincasq = sinc(q*A_s);
        double oneminussinc = 1 - sincasq;
        double q_r = q * radius;
        double q_edge = q * edge_sep;
        
        // each volume
        double string_vol = edge_sep * pi * thick_string * thick_string / 4.0;
        double pearl_vol = 4.0 / 3.0 * pi * radius * radius * radius;

        //total volume
        double tot_vol;
        //each masses
        double m_r= contrast_string * string_vol;
        double m_s= contrast_pearl * pearl_vol;
        double psi, gamma, beta;
        //form factors
        double sss, srr, srs; //cross
        double srr_1, srr_2, srr_3;
        double form_factor;
        tot_vol = num_strings * string_vol;
        tot_vol += num_pearls * pearl_vol;

        //sine functions of a pearl
        psi = sin(q_r);
        psi -= q_r * cos(q_r);
        psi *= 3.0;
        psi /= q_r * q_r * q_r;

        // Note take only 20 terms in Si series: 10 terms may be enough though.
        gamma = Si(q_edge);
        gamma /= (q_edge);
        beta = Si(q * (A_s - radius));
        beta -= Si(q_r);
        beta /= q_edge;

        // form factor for num_pearls
        sss = 1.0 - pow(sincasq, num_pearls);
        sss /= oneminussinc * oneminussinc;
        sss *= -sincasq;
        sss -= num_pearls / 2.0;
        sss += num_pearls / oneminussinc;
        sss *= 2.0 * m_s * psi * m_s * psi;

        // form factor for num_strings (like thin rods)
        srr_1 = -sinc(q_edge/2.0) * sinc(q_edge/2.0);

        srr_1 += 2.0 * gamma;
        srr_1 *= num_strings;
        srr_2 = 2.0/oneminussinc;
        srr_2 *= num_strings;
        srr_2 *= beta * beta;
        srr_3 = 1.0 - pow(sincasq, num_strings);
        srr_3 /= oneminussinc * oneminussinc;
        srr_3 *= beta * beta;
        srr_3 *= -2.0;

        // total srr
        srr = srr_1 + srr_2 + srr_3;
        srr *= m_r * m_r;

        // form factor for correlations
        srs = 1.0;
        srs -= pow(sincasq, num_strings);
        srs /= oneminussinc * oneminussinc;
        srs *= -sincasq;
        srs += num_strings/oneminussinc;
        srs *= 4.0;
        srs *= (m_r * m_s * beta * psi);

        form_factor = sss + srr + srs;
        form_factor /= (tot_vol * 1.0e4); // norm by volume and A^-1 to cm^-1

        return (form_factor);
}

double form_volume(double radius, double edge_sep,
        double thick_string, double num_pearls)
{
        double total_vol;

        double pi = 4.0*atan(1.0);
        double number_of_strings = num_pearls - 1.0;
        
        double string_vol = edge_sep * pi * thick_string * thick_string / 4.0;
        double pearl_vol = 4.0 / 3.0 * pi * radius * radius * radius;

        total_vol = number_of_strings * string_vol;
        total_vol += num_pearls * pearl_vol;

        return(total_vol);
}

double Iq(double q, double radius, double edge_sep,
        double thick_string, double num_pearls, double sld, 
        double string_sld, double solvent_sld)
{
        double value, tot_vol;
        
        value = _pearl_necklace_kernel(q, radius, edge_sep, thick_string,
                num_pearls, sld, string_sld, solvent_sld);
        tot_vol = form_volume(radius, edge_sep, thick_string, num_pearls);

        return value*tot_vol;
}