1 | #pragma rtGlobals=1 // Use modern global access method. |
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2 | |
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3 | //////////////////////////////////////////////// |
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4 | // this model calculation is for the scattered intensity from a dispersion of polydisperse spheres |
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5 | // hard sphere interactions are NOT included |
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6 | // the polydispersity in radius is a Schulz distribution |
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7 | // |
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8 | // TWO polulations of spheres are considered |
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9 | // |
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10 | // 31 DEC 03 SRK |
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11 | //////////////////////////////////////////////// |
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12 | #include "SchulzSpheres" |
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13 | |
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14 | Proc PlotBimodalSchulzSpheres(num,qmin,qmax) |
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15 | Variable num=128,qmin=0.001,qmax=0.7 |
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16 | Prompt num "Enter number of data points for model: " |
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17 | Prompt qmin "Enter minimum q-value (A^-1) for model: " |
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18 | Prompt qmax "Enter maximum q-value (A^-1) for model: " |
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19 | |
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20 | Make/O/D/n=(num) xwave_bss,ywave_bss |
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21 | xwave_bss = alog(log(qmin) + x*((log(qmax)-log(qmin))/num)) |
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22 | Make/O/D coef_bss = {0.01,200,0.2,1e-6,0.05,25,0.2,1e-6,6.4e-6,0.001} |
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23 | make/o/t/N=10 parameters_bss |
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24 | parameters_bss[0,3] = {"volume fraction(1)","Radius (1) (A)","polydispersity(1)","SLD(1) (A^-2)"} |
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25 | parameters_bss[4,9] = {"volume fraction(2)","Radius (2)","polydispersity(2)","SLD(2)","SLD (solvent)","background (cm-1 sr-1)"} |
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26 | Edit parameters_bss,coef_bss |
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27 | ywave_bss := BimodalSchulzSpheres(coef_bss,xwave_bss) |
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28 | Display ywave_bss vs xwave_bss |
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29 | ModifyGraph log=1,marker=29,msize=2,mode=4 |
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30 | Label bottom "q (A\\S-1\\M)" |
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31 | Label left "Intensity (cm\\S-1\\M)" |
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32 | AutoPositionWindow/M=1/R=$(WinName(0,1)) $WinName(0,2) |
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33 | End |
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34 | |
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35 | |
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36 | /////////////////////////////////////////////////////////// |
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37 | |
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38 | Proc PlotSmearedBimodalSchulzSpheres() |
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39 | //no input parameters necessary, it MUST use the experimental q-values |
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40 | // from the experimental data read in from an AVE/QSIG data file |
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41 | |
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42 | // if no gQvals wave, data must not have been loaded => abort |
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43 | if(ResolutionWavesMissing()) |
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44 | Abort |
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45 | endif |
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46 | |
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47 | // Setup parameter table for model function |
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48 | Make/O/D smear_coef_bss = {0.01,200,0.2,1e-6,0.05,25,0.2,1e-6,6.4e-6,0.001} |
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49 | make/o/t/N=10 smear_parameters_bss |
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50 | smear_parameters_bss[0,3] = {"volume fraction(1)","Radius (1) (A)","polydispersity(1)","SLD(1) (A^-2)"} |
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51 | smear_parameters_bss[4,9] = {"volume fraction(2)","Radius (2)","polydispersity(2)","SLD(2)","SLD (solvent)","background (cm-1 sr-1)"} |
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52 | Edit smear_parameters_bss,smear_coef_bss |
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53 | |
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54 | // output smeared intensity wave, dimensions are identical to experimental QSIG values |
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55 | // make extra copy of experimental q-values for easy plotting |
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56 | Duplicate/O $gQvals smeared_bss,smeared_qvals |
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57 | SetScale d,0,0,"1/cm",smeared_bss |
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58 | |
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59 | smeared_bss := SmearedBimodalSchulzSpheres(smear_coef_bss,$gQvals) |
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60 | Display smeared_bss vs smeared_qvals |
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61 | ModifyGraph log=1,marker=29,msize=2,mode=4 |
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62 | Label bottom "q (A\\S-1\\M)" |
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63 | Label left "Intensity (cm\\S-1\\M)" |
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64 | AutoPositionWindow/M=1/R=$(WinName(0,1)) $WinName(0,2) |
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65 | End |
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66 | |
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67 | // Calculates some characteristic parameters for bimodal Shulz distribution |
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68 | Macro NumberDensity_Bimodal() |
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69 | |
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70 | Variable nden1,nden2,phi1,phi2,R1,R2,Ravg,p1,p2,Rg1,Rg2,I1_0,I2_0,I0,Sv1,Sv2,Sv,vpoly1,vpoly2 |
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71 | Variable z1,z2,v2poly1,v2poly2 |
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72 | if(WaveExists(coef_bss)==0) |
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73 | abort "You need to plot the model first to create the coefficient table" |
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74 | Endif |
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75 | |
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76 | phi1 = coef_bss[0] // volume fraction, mode 1 |
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77 | phi2 = coef_bss[4] // volume fraction, mode 1 |
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78 | R1 = coef_bss[1] // mean radius, mode 1(A) |
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79 | R2 = coef_bss[5] // mean radius, mode 1(A) |
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80 | p1 = coef_bss[2] // polydispersity, mode 1 |
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81 | p2 = coef_bss[6] // polydispersity, mode 1 |
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82 | |
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83 | z1 = (1/p1)^2-1 |
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84 | z2 = (1/p2)^2-1 |
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85 | // average particle volume |
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86 | vpoly1 = 4*Pi/3*(z1+3)*(z1+2)/(z1+1)/(z1+1)*r1^3 |
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87 | vpoly2 = 4*Pi/3*(z2+3)*(z2+2)/(z2+1)/(z2+1)*r2^3 |
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88 | //average particle volume^2 |
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89 | v2poly1 = (4*Pi/3)^2*(z1+6)*(z1+5)*(z1+4)*(z1+3)*(z1+2)/((z1+1)^5)*r1^6 |
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90 | v2poly2 = (4*Pi/3)^2*(z2+6)*(z2+5)*(z2+4)*(z2+3)*(z2+2)/((z2+1)^5)*r2^6 |
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91 | nden1 = phi1/vpoly1 //nden in 1/A^3 |
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92 | nden2 = phi2/vpoly2 //nden in 1/A^3 |
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93 | |
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94 | rg1 = r1*((3*(z1+8)*(z1+7))/5/(z1+1)/(z1+1))^0.5 // in A |
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95 | rg2 = r2*((3*(z2+8)*(z2+7))/5/(z2+1)/(z2+1))^0.5 // in A |
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96 | sv1 = 1.0e8*3*phi1*(z1+1)/R1/(z1+3) // in 1/cm |
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97 | sv2 = 1.0e8*3*phi2*(z2+1)/R2/(z2+3) // in 1/cm |
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98 | I1_0 = 1.0e8*nden1*v2poly1*(coef_bss[3]-coef_bss[8])^2 // 1/cm/sr |
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99 | I2_0 = 1.0e8*nden2*v2poly2*(coef_bss[7]-coef_bss[8])^2 // 1/cm/sr |
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100 | |
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101 | Print "mode 1 number density (A^-3) = ",nden1 |
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102 | Print "mode 2 number density (A^-3) = ",nden2 |
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103 | |
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104 | Ravg = (nden1*R1+nden2*R2)/(nden1+nden2) |
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105 | |
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106 | Print "mean radius, mode 1 (A) = ",R1 |
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107 | Print "mean radius, mode 2 (A) = ",R2 |
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108 | Print "mean radius, total (A) = ",Ravg |
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109 | Print "polydispersity, mode 1 (sig/avg) = ",p1 |
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110 | Print "polydispersity, mode 2 (sig/avg) = ",p2 |
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111 | Print "volume fraction, mode 1 = ",phi1 |
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112 | Print "volume fraction, mode 2 = ",phi2 |
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113 | |
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114 | Print "Guinier Radius, mode 1 (A) = ",Rg1 |
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115 | Print "Guinier Radius, mode 2 (A) = ",Rg2 |
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116 | I0 = I1_0+I2_0 |
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117 | Print "Forward scattering cross-section, mode 1 (cm-1 sr-1) I(0)= ",I1_0 |
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118 | Print "Forward scattering cross-section, mode 2 (cm-1 sr-1) I(0)= ",I2_0 |
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119 | Print "Forward scattering cross-section, total (cm-1 sr-1) I(0)= ",I0 |
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120 | Sv = Sv1+Sv2 |
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121 | Print "Interfacial surface area per unit sample volume, mode 1 (cm-1) Sv= ",Sv1 |
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122 | Print "Interfacial surface area per unit sample volume, mode 2 (cm-1) Sv= ",Sv2 |
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123 | Print "Interfacial surface area per unit sample volume, total (cm-1) Sv= ",Sv |
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124 | End |
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125 | |
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126 | |
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127 | // Plots bimodal size distribution |
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128 | Macro Plot_Bimodal_Distribution() |
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129 | |
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130 | variable p1,p2,r1,r2,z1,z2,phi1,phi2,f1,f2,nden1,nden2,vpoly1,vpoly2,maxr |
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131 | |
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132 | if(WaveExists(coef_bss)==0) |
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133 | abort "You need to plot the model first to create the coefficient table" |
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134 | Endif |
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135 | phi1 = coef_bss[0] // volume fraction, mode 1 |
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136 | phi2 = coef_bss[4] // volume fraction, mode 1 |
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137 | R1 = coef_bss[1] // mean radius, mode 1(A) |
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138 | R2 = coef_bss[5] // mean radius, mode 1(A) |
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139 | p1 = coef_bss[2] // polydispersity, mode 1 |
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140 | p2 = coef_bss[6] // polydispersity, mode 1 |
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141 | |
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142 | z1 = (1/p1)^2-1 |
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143 | z2 = (1/p2)^2-1 |
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144 | // average particle volume |
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145 | vpoly1 = 4*Pi/3*(z1+3)*(z1+2)/(z1+1)/(z1+1)*r1^3 |
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146 | vpoly2 = 4*Pi/3*(z2+3)*(z2+2)/(z2+1)/(z2+1)*r2^3 |
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147 | |
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148 | nden1 = phi1/vpoly1 //nden in 1/A^3 |
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149 | nden2 = phi2/vpoly2 //nden in 1/A^3 |
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150 | f1 = nden1/(nden1+nden2) |
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151 | f2 = nden2/(nden1+nden2) |
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152 | |
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153 | Make/O/D/N=1000 Bimodal_Schulz_distribution |
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154 | if (r1>r2) then |
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155 | maxr = r1*(1+6*p1) |
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156 | else |
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157 | maxr = r2*(1+6*p2) |
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158 | endif |
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159 | |
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160 | SetScale/I x, 0, maxr, Bimodal_Schulz_distribution |
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161 | Bimodal_Schulz_distribution = f1*Schulz_Point_bss(x,r1,z1)+f2*Schulz_Point_bss(x,r2,z2) |
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162 | Display Bimodal_Schulz_distribution |
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163 | Label left "f(R) (normalized)" |
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164 | Label bottom "R (A)" |
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165 | legend |
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166 | End |
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167 | |
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168 | /////////////////////////////////////////////////////////////// |
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169 | // unsmeared model calculation |
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170 | /////////////////////////// |
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171 | Function BimodalSchulzSpheres(w,k) : FitFunc |
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172 | Wave w // the coefficient wave |
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173 | Variable k // the x values, as a variable |
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174 | |
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175 | Variable ans=0 |
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176 | Make/O/D/N=6 temp_coef_1,temp_coef_2 //coefficient waves for each population |
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177 | temp_coef_1[0] = w[0] |
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178 | temp_coef_1[1] = w[1] |
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179 | temp_coef_1[2] = w[2] |
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180 | temp_coef_1[3] = w[3] |
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181 | temp_coef_1[4] = w[8] |
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182 | temp_coef_1[5] = 0 |
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183 | |
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184 | //second population |
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185 | temp_coef_2[0] = w[4] |
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186 | temp_coef_2[1] = w[5] |
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187 | temp_coef_2[2] = w[6] |
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188 | temp_coef_2[3] = w[7] |
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189 | temp_coef_2[4] = w[8] |
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190 | temp_coef_2[5] = 0 //always zero - background is added in the final step |
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191 | |
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192 | //calculate both models and sum (add background here) |
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193 | ans = SchulzSpheres(temp_coef_1,k) |
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194 | ans += SchulzSpheres(temp_coef_2,k) |
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195 | ans += w[9] //background |
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196 | |
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197 | return(ans) |
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198 | End |
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199 | |
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200 | Function Schulz_Point_bss(x,avg,zz) |
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201 | Variable x,avg,zz |
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202 | |
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203 | Variable dr |
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204 | |
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205 | dr = zz*ln(x) - gammln(zz+1)+(zz+1)*ln((zz+1)/avg)-(x/avg*(zz+1)) |
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206 | return (exp(dr)) |
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207 | End |
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208 | |
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209 | // this is all there is to the smeared calculation! |
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210 | Function SmearedBimodalSchulzSpheres(w,x) :FitFunc |
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211 | Wave w |
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212 | Variable x |
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213 | |
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214 | Variable ans |
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215 | SVAR sq = gSig_Q |
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216 | SVAR qb = gQ_bar |
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217 | SVAR sh = gShadow |
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218 | SVAR gQ = gQVals |
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219 | |
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220 | //the name of your unsmeared model is the first argument |
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221 | ans = Smear_Model_20(BimodalSchulzSpheres,$sq,$qb,$sh,$gQ,w,x) |
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222 | |
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223 | return(ans) |
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224 | End |
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225 | |
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