1 | double _pearl_necklace_kernel(double q, double radius, double edge_separation, |
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2 | double thick_string, double num_pearls, double sld_pearl, |
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3 | double sld_string, double sld_solv); |
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4 | double form_volume(double radius, double edge_separation, |
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5 | double string_thickness, double number_of_pearls); |
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6 | double sinc(double x); |
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7 | |
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8 | double Iq(double q, double radius, double edge_separation, |
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9 | double string_thickness, double number_of_pearls, double sld, |
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10 | double string_sld, double solvent_sld); |
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11 | double Iqxy(double qx, double qy, double radius, double edge_separation, |
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12 | double string_thickness, double number_of_pearls, double sld, |
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13 | double string_sld, double solvent_sld); |
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14 | |
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15 | double ER(double radius, double edge_separation, |
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16 | double string_thickness, double number_of_pearls); |
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17 | double VR(double radius, double edge_separation, |
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18 | double string_thickness, double number_of_pearls); |
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19 | |
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20 | // From Igor library |
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21 | double _pearl_necklace_kernel(double q, double radius, double edge_separation, double thick_string, |
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22 | double num_pearls, double sld_pearl, double sld_string, double sld_solv) |
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23 | { |
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24 | double contrast_pearl = sld_pearl - sld_solv; |
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25 | double contrast_string = sld_string - sld_solv; |
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26 | |
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27 | //total volume |
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28 | double pi = 4.0*atan(1.0); |
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29 | double tot_vol = form_volume(radius, edge_separation, thick_string, num_pearls); |
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30 | double string_vol = edge_separation * pi * pow((thick_string / 2.0), 2); |
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31 | double pearl_vol = 4.0 /3.0 * pi * pow(radius, 3); |
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32 | double num_strings = num_pearls - 1; |
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33 | //each mass |
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34 | double m_r= contrast_string * string_vol; |
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35 | double m_s= contrast_pearl * pearl_vol; |
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36 | double psi, gamma, beta; |
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37 | //form factors |
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38 | double sss; //pearls |
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39 | double srr; //strings |
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40 | double srs; //cross |
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41 | double A_s, srr_1, srr_2, srr_3; |
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42 | double form_factor; |
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43 | |
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44 | //sine functions of a pearl |
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45 | psi = sin(q*radius); |
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46 | psi -= q * radius * cos(q * radius); |
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47 | psi /= pow((q * radius), 3); |
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48 | psi *= 3.0; |
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49 | |
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50 | // Note take only 20 terms in Si series: 10 terms may be enough though. |
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51 | gamma = Si(q* edge_separation); |
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52 | gamma /= (q* edge_separation); |
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53 | beta = Si(q * (edge_separation + radius)); |
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54 | beta -= Si(q * radius); |
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55 | beta /= (q* edge_separation); |
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56 | |
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57 | // center to center distance between the neighboring pearls |
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58 | A_s = edge_separation + 2.0 * radius; |
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59 | |
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60 | // form factor for num_pearls |
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61 | sss = 1.0 - pow(sinc(q*A_s), num_pearls ); |
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62 | sss /= pow((1.0-sinc(q*A_s)), 2); |
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63 | sss *= -sinc(q*A_s); |
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64 | sss -= num_pearls/2.0; |
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65 | sss += num_pearls/(1.0-sinc(q*A_s)); |
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66 | sss *= 2.0 * pow((m_s*psi), 2); |
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67 | |
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68 | // form factor for num_strings (like thin rods) |
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69 | srr_1 = -pow(sinc(q*edge_separation/2.0), 2); |
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70 | |
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71 | srr_1 += 2.0 * gamma; |
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72 | srr_1 *= num_strings; |
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73 | srr_2 = 2.0/(1.0-sinc(q*A_s)); |
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74 | srr_2 *= num_strings * pow(beta, 2); |
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75 | srr_3 = 1.0 - pow(sinc(q*A_s), num_strings); |
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76 | srr_3 /= pow((1.0-sinc(q*A_s)), 2); |
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77 | srr_3 *= -2.0 * pow(beta, 2); |
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78 | |
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79 | // total srr |
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80 | srr = srr_1 + srr_2 + srr_3; |
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81 | srr *= pow(m_r, 2); |
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82 | |
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83 | // form factor for correlations |
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84 | srs = 1.0; |
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85 | srs -= pow(sinc(q*A_s), num_strings); |
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86 | srs /= pow((1.0-sinc(q*A_s)), 2); |
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87 | srs *= -sinc(q*A_s); |
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88 | srs += (num_strings/(1.0-sinc(q*A_s))); |
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89 | srs *= 4.0 * (m_r * m_s * beta * psi); |
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90 | |
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91 | form_factor = sss + srr + srs; |
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92 | form_factor /= (tot_vol * 1.0e4); // norm by volume and A^-1 to cm^-1 |
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93 | |
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94 | return (form_factor); |
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95 | } |
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96 | |
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97 | double form_volume(double radius, double edge_separation, |
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98 | double string_thickness, double number_of_pearls) |
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99 | { |
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100 | double total_vol; |
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101 | |
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102 | double pi = 4.0*atan(1.0); |
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103 | double number_of_strings = number_of_pearls - 1.0; |
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104 | |
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105 | double string_vol = edge_separation * pi * pow((string_thickness / 2.0), 2); |
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106 | double pearl_vol = 4.0 /3.0 * pi * pow(radius, 3); |
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107 | |
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108 | total_vol = number_of_strings * string_vol; |
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109 | total_vol += number_of_pearls * pearl_vol; |
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110 | |
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111 | return(total_vol); |
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112 | } |
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113 | |
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114 | double sinc(double x) |
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115 | { |
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116 | double num = sin(x); |
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117 | double denom = x; |
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118 | return num/denom; |
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119 | } |
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120 | |
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121 | |
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122 | double Iq(double q, double radius, double edge_separation, |
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123 | double string_thickness, double number_of_pearls, double sld, |
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124 | double string_sld, double solvent_sld) |
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125 | { |
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126 | double value = 0.0; |
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127 | double tot_vol = 0.0; |
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128 | |
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129 | value = _pearl_necklace_kernel(q, radius, edge_separation, string_thickness, |
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130 | number_of_pearls, sld, string_sld, solvent_sld); |
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131 | tot_vol = form_volume(radius, edge_separation, string_thickness, number_of_pearls); |
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132 | |
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133 | return value*tot_vol; |
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134 | } |
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135 | |
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136 | double Iqxy(double qx, double qy, double radius, double edge_separation, |
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137 | double string_thickness, double number_of_pearls, double sld, |
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138 | double string_sld, double solvent_sld) |
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139 | { |
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140 | double q = sqrt(qx*qx + qy*qy); |
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141 | return(Iq(q, radius, edge_separation, string_thickness, number_of_pearls, |
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142 | sld, string_sld, solvent_sld)); |
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143 | } |
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144 | |
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145 | |
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146 | double ER(double radius, double edge_separation, |
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147 | double string_thickness, double number_of_pearls) |
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148 | { |
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149 | double tot_vol = form_volume(radius, edge_separation, string_thickness, number_of_pearls); |
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150 | double pi = 4.0*atan(1.0); |
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151 | |
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152 | double rad_out = pow((3.0*tot_vol/4.0/pi), 0.33333); |
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153 | |
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154 | return rad_out; |
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155 | } |
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156 | double VR(double radius, double edge_separation, |
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157 | double string_thickness, double number_of_pearls) |
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158 | { |
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159 | return 1.0; |
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160 | } |
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