1 | # Note: model title and parameter table are inserted automatically |
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2 | r"""Calculate the interparticle structure factor for monodisperse |
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3 | spherical particles interacting through hard sphere (excluded volume) |
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4 | interactions. |
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5 | May be a reasonable approximation for other shapes of particles that |
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6 | freely rotate, and for moderately polydisperse systems. Though strictly |
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7 | the maths needs to be modified (no \Beta(Q) correction yet in sasview). |
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8 | |
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9 | radius_effective is the effective hard sphere radius. |
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10 | volfraction is the volume fraction occupied by the spheres. |
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11 | |
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12 | In sasview the effective radius may be calculated from the parameters |
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13 | used in the form factor $P(q)$ that this $S(q)$ is combined with. |
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14 | |
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15 | For numerical stability the computation uses a Taylor series expansion |
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16 | at very small $qR$, there may be a very minor glitch at the transition point |
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17 | in some circumstances. |
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18 | |
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19 | The S(Q) uses the Percus-Yevick closure where the interparticle |
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20 | potential is |
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21 | |
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22 | .. math:: |
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23 | |
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24 | U(r) = \begin{cases} |
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25 | \infty & r < 2R \\ |
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26 | 0 & r \geq 2R |
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27 | \end{cases} |
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28 | |
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29 | where $r$ is the distance from the center of the sphere of a radius $R$. |
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30 | |
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31 | For a 2D plot, the wave transfer is defined as |
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32 | |
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33 | .. math:: |
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34 | |
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35 | q = \sqrt{q_x^2 + q_y^2} |
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36 | |
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37 | |
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38 | References |
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39 | ---------- |
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40 | |
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41 | J K Percus, J Yevick, *J. Phys. Rev.*, 110, (1958) 1 |
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42 | """ |
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43 | |
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44 | import numpy as np |
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45 | from numpy import inf |
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46 | |
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47 | name = "hardsphere" |
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48 | title = "Hard sphere structure factor, with Percus-Yevick closure" |
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49 | description = """\ |
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50 | [Hard sphere structure factor, with Percus-Yevick closure] |
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51 | Interparticle S(Q) for random, non-interacting spheres. |
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52 | May be a reasonable approximation for other shapes of |
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53 | particles that freely rotate, and for moderately polydisperse |
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54 | systems. Though strictly the maths needs to be modified - |
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55 | which sasview does not do yet. |
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56 | radius_effective is the hard sphere radius |
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57 | volfraction is the volume fraction occupied by the spheres. |
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58 | """ |
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59 | category = "structure-factor" |
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60 | structure_factor = True |
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61 | single = False # TODO: check |
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62 | |
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63 | # ["name", "units", default, [lower, upper], "type","description"], |
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64 | parameters = [["radius_effective", "Ang", 50.0, [0, inf], "", |
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65 | "effective radius of hard sphere"], |
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66 | ["volfraction", "", 0.2, [0, 0.74], "", |
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67 | "volume fraction of hard spheres"], |
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68 | ] |
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69 | |
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70 | Iq = r""" |
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71 | double D,A,B,G,X,X2,X4,S,C,FF,HARDSPH; |
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72 | // these are c compiler instructions, can also put normal code inside the "if else" structure |
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73 | #if FLOAT_SIZE > 4 |
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74 | // double precision |
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75 | // orig had 0.2, don't call the variable cutoff as PAK already has one called that! |
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76 | // Must use UPPERCASE name please. |
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77 | // 0.05 better, 0.1 OK |
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78 | #define CUTOFFHS 0.05 |
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79 | #else |
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80 | // 0.1 bad, 0.2 OK, 0.3 good, 0.4 better, 0.8 no good |
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81 | #define CUTOFFHS 0.4 |
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82 | #endif |
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83 | |
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84 | if(fabs(radius_effective) < 1.E-12) { |
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85 | HARDSPH=1.0; |
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86 | //printf("HS1 %g: %g\n",q,HARDSPH); |
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87 | return(HARDSPH); |
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88 | } |
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89 | // removing use of pow(xxx,2) and rearranging the calcs |
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90 | // of A, B & G cut ~40% off execution time ( 0.5 to 0.3 msec) |
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91 | X = 1.0/( 1.0 -volfraction); |
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92 | D= X*X; |
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93 | A= (1.+2.*volfraction)*D; |
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94 | A *=A; |
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95 | X=fabs(q*radius_effective*2.0); |
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96 | |
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97 | if(X < 5.E-06) { |
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98 | HARDSPH=1./A; |
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99 | //printf("HS2 %g: %g\n",q,HARDSPH); |
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100 | return(HARDSPH); |
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101 | } |
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102 | X2 =X*X; |
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103 | B = (1.0 +0.5*volfraction)*D; |
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104 | B *= B; |
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105 | B *= -6.*volfraction; |
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106 | G=0.5*volfraction*A; |
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107 | |
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108 | if(X < CUTOFFHS) { |
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109 | // RKH Feb 2016, use Taylor series expansion for small X |
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110 | // else no obvious way to rearrange the equations to avoid |
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111 | // needing a very high number of significant figures. |
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112 | // Series expansion found using Mathematica software. Numerical test |
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113 | // in .xls showed terms to X^2 are sufficient |
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114 | // for 5 or 6 significant figures, but I put the X^4 one in anyway |
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115 | //FF = 8*A +6*B + 4*G - (0.8*A +2.0*B/3.0 +0.5*G)*X2 +(A/35. +B/40. +G/50.)*X4; |
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116 | // refactoring the polynomial makes it very slightly faster (0.5 not 0.6 msec) |
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117 | //FF = 8*A +6*B + 4*G + ( -0.8*A -2.0*B/3.0 -0.5*G +(A/35. +B/40. +G/50.)*X2)*X2; |
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118 | |
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119 | FF = 8.0*A +6.0*B + 4.0*G + ( -0.8*A -B/1.5 -0.5*G +(A/35. +0.0125*B +0.02*G)*X2)*X2; |
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120 | |
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121 | // combining the terms makes things worse at smallest Q in single precision |
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122 | //FF = (8-0.8*X2)*A +(3.0-X2/3.)*2*B + (4+0.5*X2)*G +(A/35. +B/40. +G/50.)*X4; |
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123 | // note that G = -volfraction*A/2, combining this makes no further difference at smallest Q |
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124 | //FF = (8 +2.*volfraction + ( volfraction/4. -0.8 +(volfraction/100. -1./35.)*X2 )*X2 )*A + (3.0 -X2/3. +X4/40.)*2.*B; |
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125 | HARDSPH= 1./(1. + volfraction*FF ); |
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126 | //printf("HS3 %g: %g\n",q,HARDSPH); |
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127 | return(HARDSPH); |
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128 | } |
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129 | X4=X2*X2; |
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130 | SINCOS(X,S,C); |
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131 | |
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132 | // RKH Feb 2016, use version FISH code as is better than original sasview one |
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133 | // at small Q in single precision, and more than twice as fast in double. |
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134 | //FF=A*(S-X*C)/X + B*(2.*X*S -(X2-2.)*C -2.)/X2 + G*( (4.*X2*X -24.*X)*S -(X4 -12.*X2 +24.)*C +24. )/X4; |
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135 | // refactoring the polynomial here & above makes it slightly faster |
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136 | |
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137 | FF= (( G*( (4.*X2 -24.)*X*S -(X4 -12.*X2 +24.)*C +24. )/X2 + B*(2.*X*S -(X2-2.)*C -2.) )/X + A*(S-X*C))/X ; |
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138 | HARDSPH= 1./(1. + 24.*volfraction*FF/X2 ); |
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139 | |
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140 | // changing /X and /X2 to *MX1 and *MX2, no significantg difference? |
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141 | //MX=1.0/X; |
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142 | //MX2=MX*MX; |
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143 | //FF= (( G*( (4.*X2 -24.)*X*S -(X4 -12.*X2 +24.)*C +24. )*MX2 + B*(2.*X*S -(X2-2.)*C -2.) )*MX + A*(S-X*C)) ; |
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144 | //HARDSPH= 1./(1. + 24.*volfraction*FF*MX2*MX ); |
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145 | |
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146 | // grouping the terms, was about same as sasmodels for single precision issues |
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147 | // FF=A*(S/X-C) + B*(2.*S/X - C +2.0*(C-1.0)/X2) + G*( (4./X -24./X3)*S -(1.0 -12./X2 +24./X4)*C +24./X4 ); |
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148 | // HARDSPH= 1./(1. + 24.*volfraction*FF/X2 ); |
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149 | // remove 1/X2 from final line, take more powers of X inside the brackets, stil bad |
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150 | // FF=A*(S/X3-C/X2) + B*(2.*S/X3 - C/X2 +2.0*(C-1.0)/X4) + G*( (4./X -24./X3)*S -(1.0 -12./X2 +24./X4)*C +24./X4 )/X2; |
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151 | // HARDSPH= 1./(1. + 24.*volfraction*FF ); |
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152 | //printf("HS4 %g: %g\n",q,HARDSPH); |
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153 | return(HARDSPH); |
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154 | """ |
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155 | |
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156 | def random(): |
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157 | pars = dict( |
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158 | scale=1, background=0, |
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159 | radius_effective=10**np.random.uniform(1, 4), |
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160 | volfraction=10**np.random.uniform(-2, 0), # high volume fraction |
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161 | ) |
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162 | return pars |
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163 | |
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164 | # ER defaults to 0.0 |
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165 | # VR defaults to 1.0 |
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166 | |
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167 | demo = dict(radius_effective=200, volfraction=0.2, |
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168 | radius_effective_pd=0.1, radius_effective_pd_n=40) |
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169 | # Q=0.001 is in the Taylor series, low Q part, so add Q=0.1, |
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170 | # assuming double precision sasview is correct |
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171 | tests = [ |
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172 | [{'scale': 1.0, 'background' : 0.0, 'radius_effective' : 50.0, |
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173 | 'volfraction' : 0.2, 'radius_effective_pd' : 0}, |
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174 | [0.001, 0.1], [0.209128, 0.930587]], |
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175 | ] |
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176 | # ADDED by: RKH ON: 16Mar2016 using equations from FISH as better than |
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177 | # orig sasview, see notes above. Added Taylor expansions at small Q. |
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