1 | # Note: model title and parameter table are inserted automatically |
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2 | r""" |
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3 | This calculates the interparticle structure factor for a square well fluid |
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4 | spherical particles. The mean spherical approximation (MSA) closure was |
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5 | used for this calculation, and is not the most appropriate closure for |
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6 | an attractive interparticle potential. This solution has been compared |
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7 | to Monte Carlo simulations for a square well fluid, showing this calculation |
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8 | to be limited in applicability to well depths $\epsilon < 1.5$ kT and |
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9 | volume fractions $\phi < 0.08$. |
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10 | |
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11 | Positive well depths correspond to an attractive potential well. Negative |
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12 | well depths correspond to a potential "shoulder", which may or may not be |
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13 | physically reasonable. The stickyhardsphere model may be a better choice in |
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14 | some circumstances. Computed values may behave badly at extremely small $qR$. |
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15 | |
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16 | The well width $(\lambda)$ is defined as multiples of the particle diameter |
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17 | $(2 R)$. |
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18 | |
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19 | The interaction potential is: |
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20 | |
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21 | .. image:: img/squarewell.png |
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22 | |
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23 | .. math:: |
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24 | |
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25 | U(r) = \begin{cases} |
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26 | \infty & r < 2R \\ |
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27 | -\epsilon & 2R \leq r < 2R\lambda \\ |
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28 | 0 & r \geq 2R\lambda |
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29 | \end{cases} |
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30 | |
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31 | where $r$ is the distance from the center of the sphere of a radius $R$. |
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32 | |
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33 | In sasview the effective radius may be calculated from the parameters |
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34 | used in the form factor $P(q)$ that this $S(q)$ is combined with. |
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35 | |
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36 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, |
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37 | where the $q$ vector is defined as |
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38 | |
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39 | .. math:: |
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40 | |
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41 | q = \sqrt{q_x^2 + q_y^2} |
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42 | |
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43 | References |
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44 | ---------- |
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45 | |
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46 | .. [#] R V Sharma, K C Sharma, *Physica*, 89A (1977) 213 |
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47 | |
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48 | Source |
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49 | ------ |
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50 | |
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51 | `squarewell.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/squarewell.py>`_ |
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52 | |
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53 | Authorship and Verification |
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54 | ---------------------------- |
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55 | |
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56 | * **Author:** |
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57 | * **Last Modified by:** |
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58 | * **Last Reviewed by:** |
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59 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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60 | """ |
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61 | |
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62 | import numpy as np |
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63 | from numpy import inf |
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64 | |
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65 | name = "squarewell" |
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66 | title = "Square well structure factor, with MSA closure" |
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67 | description = """\ |
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68 | [Square well structure factor, with MSA closure] |
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69 | Interparticle structure factor S(Q)for a hard sphere fluid with |
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70 | a narrow attractive well. Fits are prone to deliver non-physical |
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71 | parameters, use with care and read the references in the full manual. |
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72 | In sasview the effective radius will be calculated from the |
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73 | parameters used in P(Q). |
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74 | """ |
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75 | category = "structure-factor" |
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76 | structure_factor = True |
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77 | single = False |
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78 | |
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79 | #single = False |
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80 | # ["name", "units", default, [lower, upper], "type","description"], |
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81 | parameters = [ |
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82 | # [ "name", "units", default, [lower, upper], "type", |
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83 | # "description" ], |
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84 | ["radius_effective", "Ang", 50.0, [0, inf], "volume", |
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85 | "effective radius of hard sphere"], |
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86 | ["volfraction", "", 0.04, [0, 0.08], "", |
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87 | "volume fraction of spheres"], |
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88 | ["welldepth", "kT", 1.5, [0.0, 1.5], "", |
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89 | "depth of well, epsilon"], |
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90 | ["wellwidth", "diameters", 1.2, [1.0, inf], "", |
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91 | "width of well in diameters (=2R) units, must be > 1"], |
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92 | ] |
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93 | |
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94 | # No volume normalization despite having a volume parameter |
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95 | # This should perhaps be volume normalized? |
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96 | form_volume = """ |
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97 | return 1.0; |
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98 | """ |
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99 | |
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100 | Iq = """ |
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101 | // single precision is very poor at extreme small Q, would need a Taylor series |
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102 | double req,phis,edibkb,lambda,struc; |
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103 | double sigma,eta,eta2,eta3,eta4,etam1,etam14,alpha,beta,gamm; |
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104 | double x,sk,sk2,sk3,sk4,t1,t2,t3,t4,ck; |
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105 | double S,C,SL,CL; |
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106 | x= q; |
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107 | |
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108 | req = radius_effective; |
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109 | phis = volfraction; |
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110 | edibkb = welldepth; |
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111 | lambda = wellwidth; |
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112 | |
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113 | sigma = req*2.; |
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114 | eta = phis; |
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115 | eta2 = eta*eta; |
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116 | eta3 = eta*eta2; |
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117 | eta4 = eta*eta3; |
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118 | etam1 = 1. - eta; |
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119 | etam14 = etam1*etam1*etam1*etam1; |
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120 | // temp borrow sk for an intermediate calc |
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121 | sk = 1.0 +2.0*eta; |
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122 | sk *= sk; |
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123 | alpha = ( sk + eta3*( eta-4.0 ) )/etam14; |
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124 | beta = -(eta/3.0) * ( 18. + 20.*eta - 12.*eta2 + eta4 )/etam14; |
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125 | gamm = 0.5*eta*( sk + eta3*(eta-4.) )/etam14; |
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126 | |
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127 | // calculate the structure factor |
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128 | |
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129 | sk = x*sigma; |
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130 | sk2 = sk*sk; |
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131 | sk3 = sk*sk2; |
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132 | sk4 = sk3*sk; |
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133 | SINCOS(sk,S,C); |
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134 | SINCOS(lambda*sk,SL,CL); |
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135 | t1 = alpha * sk3 * ( S - sk * C ); |
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136 | t2 = beta * sk2 * 2.0*( sk*S - (0.5*sk2 - 1.)*C - 1.0 ); |
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137 | t3 = gamm*( ( 4.0*sk3 - 24.*sk ) * S - ( sk4 - 12.0*sk2 + 24.0 )*C + 24.0 ); |
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138 | t4 = -edibkb*sk3*(SL +sk*(C - lambda*CL) - S ); |
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139 | ck = -24.0*eta*( t1 + t2 + t3 + t4 )/sk3/sk3; |
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140 | struc = 1./(1.-ck); |
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141 | |
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142 | return(struc); |
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143 | """ |
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144 | |
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145 | def random(): |
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146 | """Return a random parameter set for the model.""" |
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147 | pars = dict( |
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148 | scale=1, background=0, |
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149 | radius_effective=10**np.random.uniform(1, 4.7), |
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150 | volfraction=np.random.uniform(0.00001, 0.08), |
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151 | welldepth=np.random.uniform(0, 1.5), |
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152 | wellwidth=np.random.uniform(1, 1.2), |
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153 | ) |
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154 | return pars |
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155 | |
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156 | demo = dict(radius_effective=50, volfraction=0.04, welldepth=1.5, |
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157 | wellwidth=1.2, radius_effective_pd=0, radius_effective_pd_n=0) |
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158 | # |
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159 | tests = [ |
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160 | [{'scale': 1.0, 'background': 0.0, 'radius_effective': 50.0, |
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161 | 'volfraction': 0.04, 'welldepth': 1.5, 'wellwidth': 1.2, |
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162 | 'radius_effective_pd': 0}, [0.001], [0.97665742]], |
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163 | ] |
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164 | # ADDED by: converting from sasview RKH ON: 16Mar2016 |
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