1 | double form_volume(double radius, double thickness, double alpha, double beta); |
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2 | |
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3 | double Iq(double q, |
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4 | double radius, |
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5 | double thickness, |
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6 | double alpha, |
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7 | double beta, |
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8 | double sld, |
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9 | double sld_solvent); |
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10 | |
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11 | |
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12 | static |
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13 | void _integrate_bessel( |
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14 | double radius, |
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15 | double alpha, |
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16 | double beta, |
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17 | double q_sin_psi, |
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18 | double q_cos_psi, |
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19 | double n, |
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20 | double *Sn, |
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21 | double *Cn) |
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22 | { |
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23 | // translate gauss point z in [-1,1] to a point in [0, radius] |
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24 | const double zm = 0.5*radius; |
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25 | const double zb = 0.5*radius; |
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26 | |
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27 | // evaluate at Gauss points |
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28 | double sumS = 0.0; // initialize integral |
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29 | double sumC = 0.0; // initialize integral |
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30 | double r; |
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31 | for (int i=0; i < GAUSS_N; i++) { |
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32 | r = GAUSS_Z[i]*zm + zb; |
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33 | |
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34 | const double qrs = r*q_sin_psi; |
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35 | const double qrrc = r*r*q_cos_psi; |
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36 | |
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37 | double y = GAUSS_W[i] * r * sas_JN(n, beta*qrrc) * sas_JN(2*n, qrs); |
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38 | double S, C; |
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39 | SINCOS(alpha*qrrc, S, C); |
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40 | sumS += y*S; |
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41 | sumC += y*C; |
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42 | } |
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43 | |
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44 | *Sn = zm*sumS / (radius*radius); |
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45 | *Cn = zm*sumC / (radius*radius); |
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46 | } |
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47 | |
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48 | static |
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49 | double _sum_bessel_orders( |
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50 | double radius, |
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51 | double alpha, |
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52 | double beta, |
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53 | double q_sin_psi, |
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54 | double q_cos_psi) |
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55 | { |
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56 | //calculate sum term from n = -3 to 3 |
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57 | //Note 1: |
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58 | // S_n(-x) = (-1)^S_n(x) |
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59 | // => S_n^2(-x) = S_n^2(x), |
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60 | // => sum_-k^k Sk = S_0^2 + 2*sum_1^kSk^2 |
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61 | //Note 2: |
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62 | // better precision to sum terms from smaller to larger |
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63 | // though it doesn't seem to make a difference in this case. |
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64 | double Sn, Cn, sum; |
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65 | sum = 0.0; |
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66 | for (int n=3; n>0; n--) { |
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67 | _integrate_bessel(radius, alpha, beta, q_sin_psi, q_cos_psi, n, &Sn, &Cn); |
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68 | sum += 2.0*(Sn*Sn + Cn*Cn); |
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69 | } |
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70 | _integrate_bessel(radius, alpha, beta, q_sin_psi, q_cos_psi, 0, &Sn, &Cn); |
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71 | sum += Sn*Sn+ Cn*Cn; |
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72 | return sum; |
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73 | } |
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74 | |
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75 | static |
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76 | double _integrate_psi( |
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77 | double q, |
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78 | double radius, |
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79 | double thickness, |
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80 | double alpha, |
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81 | double beta) |
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82 | { |
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83 | // translate gauss point z in [-1,1] to a point in [0, pi/2] |
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84 | const double zm = M_PI_4; |
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85 | const double zb = M_PI_4; |
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86 | |
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87 | double sum = 0.0; |
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88 | for (int i = 0; i < GAUSS_N; i++) { |
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89 | double psi = GAUSS_Z[i]*zm + zb; |
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90 | double sin_psi, cos_psi; |
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91 | SINCOS(psi, sin_psi, cos_psi); |
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92 | double bessel_term = _sum_bessel_orders(radius, alpha, beta, q*sin_psi, q*cos_psi); |
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93 | double sinc_term = square(sas_sinx_x(q * thickness * cos_psi / 2.0)); |
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94 | double pringle_kernel = 4.0 * sin_psi * bessel_term * sinc_term; |
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95 | sum += GAUSS_W[i] * pringle_kernel; |
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96 | } |
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97 | |
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98 | return zm * sum; |
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99 | } |
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100 | |
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101 | double form_volume(double radius, double thickness, double alpha, double beta) |
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102 | { |
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103 | return M_PI*radius*radius*thickness; |
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104 | } |
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105 | |
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106 | static double |
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107 | radius_from_excluded_volume(double radius, double thickness) |
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108 | { |
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109 | return 0.5*cbrt(0.75*radius*(2.0*radius*thickness + (radius + thickness)*(M_PI*radius + thickness))); |
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110 | } |
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111 | |
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112 | static double |
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113 | effective_radius(int mode, double radius, double thickness, double alpha, double beta) |
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114 | { |
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115 | switch (mode) { |
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116 | default: |
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117 | case 1: // equivalent cylinder excluded volume |
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118 | return radius_from_excluded_volume(radius, thickness); |
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119 | case 2: // equivalent volume sphere |
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120 | return cbrt(M_PI*radius*radius*thickness/M_4PI_3); |
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121 | case 3: // radius |
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122 | return radius; |
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123 | } |
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124 | } |
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125 | |
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126 | double Iq( |
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127 | double q, |
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128 | double radius, |
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129 | double thickness, |
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130 | double alpha, |
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131 | double beta, |
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132 | double sld, |
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133 | double sld_solvent) |
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134 | { |
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135 | double form = _integrate_psi(q, radius, thickness, alpha, beta); |
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136 | double contrast = sld - sld_solvent; |
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137 | double volume = M_PI*radius*radius*thickness; |
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138 | return 1.0e-4*form * square(contrast * volume); |
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139 | } |
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