source: sasmodels/sasmodels/models/lib/wrc_cyl.c @ e5a8f33

/*
    Functions for WRC implementation of flexible cylinders. See
    W R Chen, P D Butler and L J Magid,
    Incorporating Intermicellar Interactions in the Fitting of
    SANS Data from Cationic Wormlike Micelles.
    Langmuir, 22(15) 2006 6539-6548
*/

static double
Rgsquare(double L, double b)
{
    const double x = L/b;
    // Use Horner's method to evaluate Pedersen eq 15:
    //     alpha^2 = [1.0 + (x/3.12)^2 + (x/8.67)^3] ^ (0.176/3)
    const double alphasq =
        pow(1.0 + x*x*(1.027284681130835e-01 + 1.534414548417740e-03*x),
            5.866666666666667e-02);
    return alphasq*L*b/6.0;
}

static double
Rgsquareshort(double L, double b)
{
    const double r = b/L;  // = 1/n_b in Pedersen ref.
    return Rgsquare(L, b) * (1.0 + r*(-1.5 + r*(1.5 + r*0.75*expm1(-2.0/r))));
}

static double
w_WR(double x)
{
    // Pedersen eq. 16:
    //    w = [1 + tanh((x-C4)/C5)]/2
    const double C4 = 1.523;
    const double C5 = 0.1477;
    return 0.5 + 0.5*tanh((x - C4)/C5);
}

static double
Sdebye(double qsq)
{
#if FLOAT_SIZE>4
#  define DEBYE_CUTOFF 0.25  // 1e-15 error
#else
#  define DEBYE_CUTOFF 1.0  // 4e-7 error
#endif

    if (qsq < DEBYE_CUTOFF) {
        const double x = qsq;
        // mathematica: PadeApproximant[2*Exp[-x^2] + x^2-1)/x^4, {x, 0, 8}]
        const double A1=1./15., A2=1./60, A3=0., A4=1./75600.;
        const double B1=2./5., B2=1./15., B3=1./180., B4=1./5040.;
        return ((((A4*x + A3)*x + A2)*x + A1)*x + 1.)
             / ((((B4*x + B3)*x + B2)*x + B1)*x + 1.);
    } else {
        return 2.*(expm1(-qsq) + qsq)/(qsq*qsq);
    }
}

static double
a_long(double q, double L, double b/*, double p1, double p2, double q0*/)
{
    const double p1 = 4.12;
    const double p2 = 4.42;
    const double q0 = 3.1;

    // Constants C1, ..., C5 derived from least squares fit (Pedersen, eq 13,16)
    const double C1 = 1.22;
    const double C2 = 0.4288;
    const double C3 = -1.651;
    const double C4 = 1.523;
    const double C5 = 0.1477;
    const double miu = 0.585;

    const double C = (L/b>10.0 ? 3.06*pow(L/b, -0.44) : 1.0);
    //printf("branch B-%d q=%g L=%g b=%g\n", C==1.0, q, L, b);
    const double r2 = Rgsquare(L,b);
    const double r = sqrt(r2);
    const double qr_b = q0*r/b;
    const double qr_b_sq = qr_b*qr_b;
    const double qr_b_4 = qr_b_sq*qr_b_sq;
    const double qr_b_miu = pow(qr_b, -1.0/miu);
    const double em1_qr_b_sq = expm1(-qr_b_sq);
    const double sech2 = 1.0/square(cosh((qr_b-C4)/C5));
    const double w = w_WR(qr_b);

    const double t1 = pow(q0, 1.0 + p1 + p2)/(b*(p1-p2));
    const double t2 = C/(15.0*L) * (
        + 14.0*b*b*em1_qr_b_sq/(q0*qr_b_sq)
        + 2.0*q0*r2*exp(-qr_b_sq)*(11.0 + 7.0/qr_b_sq));
    const double t11 = ((C3*qr_b_miu + C2)*qr_b_miu + C1)*qr_b_miu;
    const double t3 = r*sech2/(2.*C5)*t11;
    const double t4 = r*(em1_qr_b_sq + qr_b_sq)*sech2 / (C5*qr_b_4);
    const double t5 = -4.0*r*qr_b*em1_qr_b_sq/qr_b_4 * (1.0 - w);
    const double t10 = 2.0*(em1_qr_b_sq + qr_b_sq)/qr_b_4 * (1.0 - w); //=Sdebye*(1-w)
    const double t6 = 4.0*b/q0 * t10;
    const double t7 = r*((-3.0*C3*qr_b_miu -2.0*C2)*qr_b_miu -1.0*C1)*qr_b_miu/(miu*qr_b);
    const double t9 = C*b/L * (4.0 - exp(-qr_b_sq) * (11.0 + 7.0/qr_b_sq) + 7.0/qr_b_sq)/15.0;
    const double t12 = b*b*M_PI/(L*q0*q0) + t2 + t3 - t4 + t5 - t6 + t7*w;
    const double t13 = -b*M_PI/(L*q0) + t9 + t10 + t11*w;

    const double a1 = pow(q0,p1)*t13 - t1*pow(q0,-p2)*(t12 + b*p1/q0*t13);
    const double a2 = t1*pow(q0,-p1)*(t12 + b*p1/q0*t13);

    const double ans = a1*pow(q*b, -p1) + a2*pow(q*b, -p2) + M_PI/(q*L);
    return ans;
}

static double
_short(double r2, double exp_qr_b, double L, double b, double p1short,
        double p2short, double q0)
{
    const double qr2 = q0*q0 * r2;
    const double b3 = b*b*b;
    const double q0p = pow(q0, -4.0 + p1short);

    double yy = 1.0/(L*r2*r2) * b/exp_qr_b*q0p
        * (8.0*b3*L
           - 8.0*b3*exp_qr_b*L
           + 2.0*b3*exp_qr_b*L*p2short
           - 2.0*b*exp_qr_b*L*p2short*qr2
           + 4.0*b*exp_qr_b*L*qr2
           - 2.0*b3*L*p2short
           + 4.0*b*L*qr2
           - M_PI*exp_qr_b*qr2*q0*r2
           + M_PI*exp_qr_b*p2short*qr2*q0*r2);

    return yy;
}
static double
a_short(double qp, double L, double b
        /*double p1short, double p2short*/, double q0)
{
    const double p1short = 5.36;
    const double p2short = 5.62;

    const double r2 = Rgsquareshort(L,b);
    const double exp_qr_b = exp(r2*square(q0/b));
    const double pdiff = p1short - p2short;
    const double a1 = _short(r2,exp_qr_b,L,b,p1short,p2short,q0)/pdiff;
    const double a2= -_short(r2,exp_qr_b,L,b,p2short,p1short,q0)/pdiff;
    const double ans = a1*pow(qp*b, -p1short) + a2*pow(qp*b, -p2short) + M_PI/(qp*L);
    return ans;
}


static double
Sexv(double q, double L, double b)
{
    // Pedersen eq 13, corrected by Chen eq A.5, swapping w and 1-w
    const double C1=1.22;
    const double C2=0.4288;
    const double C3=-1.651;
    const double miu = 0.585;
    const double qr = q*sqrt(Rgsquare(L,b));
    const double qr_miu = pow(qr, -1.0/miu);
    const double w = w_WR(qr);
    const double t10 = Sdebye(qr*qr)*(1.0 - w);
    const double t11 = ((C3*qr_miu + C2)*qr_miu + C1)*qr_miu;

    return t10 + w*t11;
}

// Modified by Yun on Oct. 15,
static double
Sexv_new(double q, double L, double b)
{
    const double qr = q*sqrt(Rgsquare(L,b));
    const double qr2 = qr*qr;
    const double C = (L/b > 10.0) ? 3.06*pow(L/b, -0.44) : 1.0;
    const double t9 = C*b/L * (4.0 - exp(-qr2) * (11.0 + 7.0/qr2) + 7.0/qr2)/15.0;

    const double Sexv_orig = Sexv(q, L, b);

    // calculating the derivative to decide on the correction (cutoff) term?
    // Note: this is modified from WRs original code
    const double del=1.05;
    const double qdel = (Sexv(q*del,L,b) - Sexv_orig)/(q*(del - 1.0));

    if (qdel < 0) {
        //printf("branch A1-%d q=%g L=%g b=%g\n", C==1.0, q, L, b);
        return t9 + Sexv_orig;
    } else {
        //printf("branch A2-%d q=%g L=%g b=%g\n", C==1.0, q, L, b);
        const double w = w_WR(qr);
        const double t10 = Sdebye(qr*qr)*(1.0 - w);
        return t9 + t10;
    }
}


static double
Sk_WR(double q, double L, double b)
{
    const double Rg_short = sqrt(Rgsquareshort(L, b));
    double q0short = fmax(1.9/Rg_short, 3.0);
    double ans;


    if( L > 4*b ) { // L > 4*b : Longer Chains
        if (q*b <= 3.1) {
            ans = Sexv_new(q, L, b);
        } else { //q(i)*b > 3.1
            ans = a_long(q, L, b /*, p1, p2, q0*/);
        }
    } else { // L <= 4*b : Shorter Chains
        if (q*b <= q0short) { // q*b <= fmax(1.9/Rg_short, 3)
            //printf("branch C-%d q=%g L=%g b=%g\n", square(q*Rg_short)<DEBYE_CUTOFF, q, L, b);
            // Note that q0short is usually 3, but it will be greater than 3
            // small enough b, depending on the L/b ratio:
            //     L/b == 1 => b < 2.37
            //     L/b == 2 => b < 1.36
            //     L/b == 3 => b < 1.00
            //     L/b == 4 => b < 0.816
            // 2017-10-01 pkienzle: moved low q approximation into Sdebye()
            ans = Sdebye(square(q*Rg_short));
        } else {  // q*b > max(1.9/Rg_short, 3)
            //printf("branch D q=%g L=%g b=%g\n", q, L, b);
            ans = a_short(q, L, b /*, p1short, p2short*/, q0short);
        }
    }

    return ans;
}