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