1 | static double |
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2 | sc_Zq(double qa, double qb, double qc, double dnn, double d_factor) |
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3 | { |
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4 | // Rewriting equations for efficiency, accuracy and readability, and so |
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5 | // code is reusable between 1D and 2D models. |
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6 | const double a1 = qa; |
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7 | const double a2 = qb; |
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8 | const double a3 = qc; |
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9 | |
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10 | const double arg = -0.5*square(dnn*d_factor)*(a1*a1 + a2*a2 + a3*a3); |
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11 | |
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12 | // Numerator: (1 - exp(a)^2)^3 |
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13 | // => (-(exp(2a) - 1))^3 |
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14 | // => -expm1(2a)^3 |
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15 | // Denominator: prod(1 - 2 cos(xk) exp(a) + exp(a)^2) |
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16 | // => exp(a)^2 - 2 cos(xk) exp(a) + 1 |
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17 | // => (exp(a) - 2 cos(xk)) * exp(a) + 1 |
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18 | const double exp_arg = exp(arg); |
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19 | const double Zq = -cube(expm1(2.0*arg)) |
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20 | / ( ((exp_arg - 2.0*cos(dnn*a1))*exp_arg + 1.0) |
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21 | * ((exp_arg - 2.0*cos(dnn*a2))*exp_arg + 1.0) |
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22 | * ((exp_arg - 2.0*cos(dnn*a3))*exp_arg + 1.0)); |
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23 | |
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24 | return Zq; |
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25 | } |
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26 | |
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27 | // occupied volume fraction calculated from lattice symmetry and sphere radius |
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28 | static double |
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29 | sc_volume_fraction(double radius, double dnn) |
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30 | { |
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31 | return sphere_volume(radius/dnn); |
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32 | } |
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33 | |
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34 | static double |
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35 | form_volume(double radius) |
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36 | { |
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37 | return sphere_volume(radius); |
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38 | } |
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39 | |
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40 | |
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41 | static double Iq(double q, double dnn, |
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42 | double d_factor, double radius, |
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43 | double sld, double solvent_sld, |
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44 | double n, double sym) |
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45 | { |
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46 | double phi_m, phi_b, theta_m, theta_b; |
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47 | if (sym>0.) { |
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48 | // translate a point in [-1,1] to a point in [0, 2 pi] |
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49 | phi_m = M_PI_4; |
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50 | phi_b = M_PI_4; |
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51 | // translate a point in [-1,1] to a point in [0, pi] |
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52 | theta_m = M_PI_4; |
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53 | theta_b = M_PI_4; |
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54 | } else { |
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55 | // translate a point in [-1,1] to a point in [0, 2 pi] |
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56 | phi_m = M_PI; |
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57 | phi_b = M_PI; |
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58 | // translate a point in [-1,1] to a point in [0, pi] |
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59 | theta_m = M_PI_2; |
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60 | theta_b = M_PI_2; |
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61 | } |
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62 | |
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63 | #if 0 |
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64 | double outer_sum = 0.0; |
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65 | for(int i=0; i<150; i++) { |
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66 | double inner_sum = 0.0; |
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67 | const double theta = Gauss150Z[i]*theta_m + theta_b; |
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68 | double sin_theta, cos_theta; |
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69 | SINCOS(theta, sin_theta, cos_theta); |
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70 | const double qc = q*cos_theta; |
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71 | const double qab = q*sin_theta; |
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72 | for(int j=0;j<150;j++) { |
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73 | const double phi = Gauss150Z[j]*phi_m + phi_b; |
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74 | double sin_phi, cos_phi; |
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75 | SINCOS(phi, sin_phi, cos_phi); |
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76 | const double qa = qab*cos_phi; |
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77 | const double qb = qab*sin_phi; |
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78 | const double fq = _sq_sc(qa, qb, qc, dnn, d_factor); |
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79 | inner_sum += Gauss150Wt[j] * fq; |
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80 | } |
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81 | inner_sum *= phi_m; // sum(f(x)dx) = sum(f(x)) dx |
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82 | outer_sum += Gauss150Wt[i] * inner_sum * sin_theta; |
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83 | } |
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84 | outer_sum *= theta_m; |
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85 | #else |
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86 | double outer_sum = 0.0; |
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87 | for(int i=0; i<(int)n; i++) { |
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88 | double inner_sum = 0.0; |
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89 | const double theta = (i*2./n-1.)*theta_m + theta_b; |
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90 | double sin_theta, cos_theta; |
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91 | SINCOS(theta, sin_theta, cos_theta); |
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92 | const double qc = q*cos_theta; |
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93 | const double qab = q*sin_theta; |
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94 | for(int j=0;j<(int)n;j++) { |
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95 | const double phi = (j*2./n-1.)*phi_m + phi_b; |
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96 | double sin_phi, cos_phi; |
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97 | SINCOS(phi, sin_phi, cos_phi); |
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98 | const double qa = qab*cos_phi; |
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99 | const double qb = qab*sin_phi; |
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100 | const double form = sc_Zq(qa, qb, qc, dnn, d_factor); |
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101 | inner_sum += form; |
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102 | } |
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103 | inner_sum *= phi_m; // sum(f(x)dx) = sum(f(x)) dx |
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104 | outer_sum += inner_sum * sin_theta; |
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105 | } |
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106 | outer_sum *= theta_m/(n*n); |
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107 | #endif |
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108 | double Zq; |
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109 | if (sym > 0.) { |
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110 | Zq = outer_sum/M_PI_2; |
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111 | } else { |
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112 | Zq = outer_sum/(4.0*M_PI); |
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113 | } |
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114 | |
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115 | return Zq; |
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116 | const double Pq = sphere_form(q, radius, sld, solvent_sld); |
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117 | return sc_volume_fraction(radius, dnn) * Pq * Zq; |
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118 | } |
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119 | |
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120 | |
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121 | static double Iqxy(double qx, double qy, |
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122 | double dnn, double d_factor, double radius, |
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123 | double sld, double solvent_sld, |
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124 | double n, double sym, |
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125 | double theta, double phi, double psi) |
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126 | { |
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127 | double q, zhat, yhat, xhat; |
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128 | ORIENT_ASYMMETRIC(qx, qy, theta, phi, psi, q, xhat, yhat, zhat); |
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129 | const double qa = q*xhat; |
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130 | const double qb = q*yhat; |
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131 | const double qc = q*zhat; |
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132 | |
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133 | q = sqrt(qa*qa + qb*qb + qc*qc); |
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134 | const double Pq = sphere_form(q, radius, sld, solvent_sld); |
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135 | const double Zq = sc_Zq(qa, qb, qc, dnn, d_factor); |
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136 | return sc_volume_fraction(radius, dnn) * Pq * Zq; |
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137 | } |
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