[2222134] | 1 | double form_volume(double radius, double radius_cap, double length); |
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[994d77f] | 2 | double Iq(double q, double sld, double solvent_sld, |
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[2222134] | 3 | double radius, double radius_cap, double length); |
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[994d77f] | 4 | double Iqxy(double qx, double qy, double sld, double solvent_sld, |
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[2222134] | 5 | double radius, double radius_cap, double length, double theta, double phi); |
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[5d4777d] | 6 | |
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[2222134] | 7 | #define INVALID(v) (v.radius_cap < v.radius) |
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[2f5c6d4] | 8 | |
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[5d4777d] | 9 | // Integral over a convex lens kernel for t in [h/R,1]. See the docs for |
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| 10 | // the definition of the function being integrated. |
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| 11 | // q is the magnitude of the q vector. |
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| 12 | // h is the length of the lens "inside" the cylinder. This negative wrt the |
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| 13 | // definition of h in the docs. |
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[2222134] | 14 | // radius_cap is the radius of the lens |
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[5d4777d] | 15 | // length is the cylinder length, or the separation between the lens halves |
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[2a0b2b1] | 16 | // theta is the angle of the cylinder wrt q. |
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[50e1e40] | 17 | static double |
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[2a0b2b1] | 18 | _cap_kernel(double qab, double qc, double h, double radius_cap, |
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| 19 | double half_length) |
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[5d4777d] | 20 | { |
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[50e1e40] | 21 | // translate a point in [-1,1] to a point in [lower,upper] |
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[994d77f] | 22 | const double upper = 1.0; |
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[2222134] | 23 | const double lower = h/radius_cap; // integral lower bound |
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[139c528] | 24 | const double zm = 0.5*(upper-lower); |
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| 25 | const double zb = 0.5*(upper+lower); |
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[50e1e40] | 26 | |
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[f4cf580] | 27 | // cos term in integral is: |
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[2a0b2b1] | 28 | // cos (q (R t - h + L/2) cos(theta)) |
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[f4cf580] | 29 | // so turn it into: |
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| 30 | // cos (m t + b) |
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| 31 | // where: |
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[2a0b2b1] | 32 | // m = q R cos(theta) |
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| 33 | // b = q(L/2-h) cos(theta) |
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| 34 | const double m = radius_cap*qc; // cos argument slope |
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| 35 | const double b = (half_length-h)*qc; // cos argument intercept |
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| 36 | const double qab_r = radius_cap*qab; // Q*R*sin(theta) |
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[994d77f] | 37 | double total = 0.0; |
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[5d4777d] | 38 | for (int i=0; i<76 ;i++) { |
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[139c528] | 39 | const double t = Gauss76Z[i]*zm + zb; |
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[994d77f] | 40 | const double radical = 1.0 - t*t; |
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[2a0b2b1] | 41 | const double bj = sas_2J1x_x(qab_r*sqrt(radical)); |
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[50e1e40] | 42 | const double Fq = cos(m*t + b) * radical * bj; |
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[5d4777d] | 43 | total += Gauss76Wt[i] * Fq; |
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| 44 | } |
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| 45 | // translate dx in [-1,1] to dx in [lower,upper] |
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[50e1e40] | 46 | const double integral = total*zm; |
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[3a48772] | 47 | const double cap_Fq = 2.0*M_PI*cube(radius_cap)*integral; |
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[50e1e40] | 48 | return cap_Fq; |
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[5d4777d] | 49 | } |
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| 50 | |
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[5bddd89] | 51 | static double |
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[2a0b2b1] | 52 | _fq(double qab, double qc, double h, double radius_cap, double radius, double half_length) |
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[5bddd89] | 53 | { |
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[2a0b2b1] | 54 | const double cap_Fq = _cap_kernel(qab, qc, h, radius_cap, half_length); |
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| 55 | const double bj = sas_2J1x_x(radius*qab); |
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| 56 | const double si = sas_sinx_x(half_length*qc); |
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[3a48772] | 57 | const double cyl_Fq = 2.0*M_PI*radius*radius*half_length*bj*si; |
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[5bddd89] | 58 | const double Aq = cap_Fq + cyl_Fq; |
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| 59 | return Aq; |
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| 60 | } |
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| 61 | |
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[2222134] | 62 | double form_volume(double radius, double radius_cap, double length) |
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[5d4777d] | 63 | { |
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| 64 | // cap radius should never be less than radius when this is called |
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[34756fd] | 65 | |
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| 66 | // Note: volume V = 2*V_cap + V_cyl |
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| 67 | // |
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| 68 | // V_cyl = pi r_cyl^2 L |
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| 69 | // 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) |
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| 70 | // |
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| 71 | // The docs for capped cylinder give the volume as: |
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| 72 | // V = pi r^2 L + 2/3 pi (R-h)^2 (2R + h) |
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| 73 | // where r_cap=R and h = R - h_c. |
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| 74 | // |
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| 75 | // The first part is clearly V_cyl. The second part requires some work: |
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| 76 | // (R-h)^2 => h_c^2 |
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[50e1e40] | 77 | // (2R+h) => 2R+ h_c-h_c + h => 2R + (R-h)-h_c + h => 3R-h_c |
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[34756fd] | 78 | // And so: |
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| 79 | // 2/3 pi (R-h)^2 (2R + h) => 2/3 pi h_c^2 (3 r_cap - h_c) |
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| 80 | // which is 2 V_cap, using the second form above. |
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| 81 | // |
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| 82 | // In this function we are going to use the first form of V_cap |
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| 83 | // |
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| 84 | // V = V_cyl + 2 V_cap |
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[5d4777d] | 85 | // = pi r^2 L + pi hc (r^2 + hc^2/3) |
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[34756fd] | 86 | // = pi (r^2 (L+hc) + hc^3/3) |
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[2222134] | 87 | const double hc = radius_cap - sqrt(radius_cap*radius_cap - radius*radius); |
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[50e1e40] | 88 | return M_PI*(radius*radius*(length+hc) + hc*hc*hc/3.0); |
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[5d4777d] | 89 | } |
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| 90 | |
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[50e1e40] | 91 | double Iq(double q, double sld, double solvent_sld, |
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[2222134] | 92 | double radius, double radius_cap, double length) |
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[5d4777d] | 93 | { |
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[2222134] | 94 | const double h = sqrt(radius_cap*radius_cap - radius*radius); |
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[50e1e40] | 95 | const double half_length = 0.5*length; |
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[5d4777d] | 96 | |
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[50e1e40] | 97 | // translate a point in [-1,1] to a point in [0, pi/2] |
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| 98 | const double zm = M_PI_4; |
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| 99 | const double zb = M_PI_4; |
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[994d77f] | 100 | double total = 0.0; |
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[5d4777d] | 101 | for (int i=0; i<76 ;i++) { |
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[2a0b2b1] | 102 | const double theta = Gauss76Z[i]*zm + zb; |
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| 103 | double sin_theta, cos_theta; // slots to hold sincos function output |
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| 104 | SINCOS(theta, sin_theta, cos_theta); |
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| 105 | const double qab = q*sin_theta; |
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| 106 | const double qc = q*cos_theta; |
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| 107 | const double Aq = _fq(qab, qc, h, radius_cap, radius, half_length); |
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| 108 | // scale by sin_theta for spherical coord integration |
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| 109 | total += Gauss76Wt[i] * Aq * Aq * sin_theta; |
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[5d4777d] | 110 | } |
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| 111 | // translate dx in [-1,1] to dx in [lower,upper] |
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[50e1e40] | 112 | const double form = total * zm; |
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[5d4777d] | 113 | |
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[50e1e40] | 114 | // Contrast |
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[994d77f] | 115 | const double s = (sld - solvent_sld); |
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[50e1e40] | 116 | return 1.0e-4 * s * s * form; |
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[5d4777d] | 117 | } |
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| 118 | |
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| 119 | |
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[994d77f] | 120 | double Iqxy(double qx, double qy, |
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[50e1e40] | 121 | double sld, double solvent_sld, double radius, |
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[2222134] | 122 | double radius_cap, double length, |
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[50e1e40] | 123 | double theta, double phi) |
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[5d4777d] | 124 | { |
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[5bddd89] | 125 | double q, sin_alpha, cos_alpha; |
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| 126 | ORIENT_SYMMETRIC(qx, qy, theta, phi, q, sin_alpha, cos_alpha); |
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[2a0b2b1] | 127 | const double qab = q*sin_alpha; |
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| 128 | const double qc = q*cos_alpha; |
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[5d4777d] | 129 | |
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[2222134] | 130 | const double h = sqrt(radius_cap*radius_cap - radius*radius); |
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[2a0b2b1] | 131 | const double Aq = _fq(qab, qc, h, radius_cap, radius, 0.5*length); |
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[50e1e40] | 132 | |
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| 133 | // Multiply by contrast^2 and convert to cm-1 |
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[994d77f] | 134 | const double s = (sld - solvent_sld); |
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[50e1e40] | 135 | return 1.0e-4 * square(s * Aq); |
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[5d4777d] | 136 | } |
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