1 | r""" |
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2 | This model calculates an empirical functional form for SAS data using SpericalSLD profile |
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3 | |
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4 | Similarly to the OnionExpShellModel, this model provides the form factor, P(q), for a multi-shell sphere, |
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5 | where the interface between the each neighboring shells can be described by one of a number of functions |
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6 | including error, power-law, and exponential functions. This model is to calculate the scattering intensity |
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7 | by building a continuous custom SLD profile against the radius of the particle. The SLD profile is composed |
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8 | of a flat core, a flat solvent, a number (up to 9 ) flat shells, and the interfacial layers between |
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9 | the adjacent flat shells (or core, and solvent) (see below). |
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10 | |
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11 | .. figure:: img/spherical_sld_profile.gif |
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12 | |
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13 | Exemplary SLD profile |
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14 | |
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15 | Unlike the <onion> model (using an analytical integration), |
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16 | the interfacial layers here are sub-divided and numerically integrated assuming each of the sub-layers are described |
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17 | by a line function. The number of the sub-layer can be given by users by setting the integer values of npts_inter. |
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18 | The form factor is normalized by the total volume of the sphere. |
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19 | |
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20 | Definition |
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21 | ---------- |
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22 | |
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23 | The form factor $P(q)$ in 1D is calculated by: |
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24 | |
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25 | .. math:: |
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26 | |
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27 | P(q) = \frac{f^2}{V_\text{particle}} \text{ where } |
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28 | f = f_\text{core} + \sum_{\text{inter}_i=0}^N f_{\text{inter}_i} + |
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29 | \sum_{\text{flat}_i=0}^N f_{\text{flat}_i} +f_\text{solvent} |
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30 | |
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31 | For a spherically symmetric particle with a particle density $\rho_x(r)$ the sld function can be defined as: |
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32 | |
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33 | .. math:: |
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34 | |
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35 | f_x = 4 \pi \int_{0}^{\infty} \rho_x(r) \frac{\sin(qr)} {qr^2} r^2 dr |
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36 | |
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37 | |
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38 | so that individual terms can be calcualted as follows: |
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39 | |
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40 | .. math:: |
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41 | f_\text{core} = 4 \pi \int_{0}^{r_\text{core}} \rho_\text{core} \frac{\sin(qr)} {qr} r^2 dr = |
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42 | 3 \rho_\text{core} V(r_\text{core}) |
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43 | \Big[ \frac{\sin(qr_\text{core}) - qr_\text{core} \cos(qr_\text{core})} {qr_\text{core}^3} \Big] |
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44 | |
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45 | f_{\text{inter}_i} = 4 \pi \int_{\Delta t_{ \text{inter}_i } } \rho_{ \text{inter}_i } \frac{\sin(qr)} {qr} r^2 dr |
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46 | |
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47 | f_{\text{shell}_i} = 4 \pi \int_{\Delta t_{ \text{inter}_i } } \rho_{ \text{flat}_i } \frac{\sin(qr)} {qr} r^2 dr = |
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48 | 3 \rho_{ \text{flat}_i } V ( r_{ \text{inter}_i } + \Delta t_{ \text{inter}_i } ) |
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49 | \Big[ \frac{\sin(qr_{\text{inter}_i} + \Delta t_{ \text{inter}_i } ) - q (r_{\text{inter}_i} + |
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50 | \Delta t_{ \text{inter}_i }) \cos(q( r_{\text{inter}_i} + \Delta t_{ \text{inter}_i } ) ) } |
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51 | {q ( r_{\text{inter}_i} + \Delta t_{ \text{inter}_i } )^3 } \Big] |
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52 | -3 \rho_{ \text{flat}_i } V(r_{ \text{inter}_i }) |
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53 | \Big[ \frac{\sin(qr_{\text{inter}_i}) - qr_{\text{flat}_i} \cos(qr_{\text{inter}_i}) } {qr_{\text{inter}_i}^3} \Big] |
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54 | |
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55 | f_\text{solvent} = 4 \pi \int_{r_N}^{\infty} \rho_\text{solvent} \frac{\sin(qr)} {qr} r^2 dr = |
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56 | 3 \rho_\text{solvent} V(r_N) |
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57 | \Big[ \frac{\sin(qr_N) - qr_N \cos(qr_N)} {qr_N^3} \Big] |
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58 | |
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59 | |
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60 | Here we assumed that the SLDs of the core and solvent are constant against $r$. |
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61 | The SLD at the interface between shells, $\rho_{\text {inter}_i}$ |
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62 | is calculated with a function chosen by an user, where the functions are |
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63 | |
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64 | Exp: |
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65 | |
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66 | .. math:: |
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67 | \rho_{{inter}_i} (r) = \begin{cases} |
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68 | B \exp\Big( \frac {\pm A(r - r_{\text{flat}_i})} {\Delta t_{ \text{inter}_i }} \Big) +C & \text{for} A \neq 0 \\ |
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69 | B \Big( \frac {(r - r_{\text{flat}_i})} {\Delta t_{ \text{inter}_i }} \Big) +C & \text{for} A = 0 \\ |
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70 | \end{cases} |
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71 | |
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72 | Power-Law |
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73 | |
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74 | .. math:: |
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75 | \rho_{{inter}_i} (r) = \begin{cases} |
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76 | \pm B \Big( \frac {(r - r_{\text{flat}_i} )} {\Delta t_{ \text{inter}_i }} \Big) ^A +C & \text{for} A \neq 0 \\ |
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77 | \rho_{\text{flat}_{i+1}} & \text{for} A = 0 \\ |
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78 | \end{cases} |
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79 | |
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80 | Erf: |
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81 | |
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82 | .. math:: |
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83 | \rho_{{inter}_i} (r) = \begin{cases} |
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84 | B \text{erf} \Big( \frac { A(r - r_{\text{flat}_i})} {\sqrt{2} \Delta t_{ \text{inter}_i }} \Big) +C & \text{for} A \neq 0 \\ |
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85 | B \Big( \frac {(r - r_{\text{flat}_i} )} {\Delta t_{ \text{inter}_i }} \Big) +C & \text{for} A = 0 \\ |
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86 | \end{cases} |
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87 | |
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88 | The functions are normalized so that they vary between 0 and 1, and they are constrained such that the SLD |
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89 | is continuous at the boundaries of the interface as well as each sub-layers. Thus B and C are determined. |
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90 | |
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91 | Once $\rho_{\text{inter}_i}$ is found at the boundary of the sub-layer of the interface, we can find its contribution |
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92 | to the form factor $P(q)$ |
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93 | |
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94 | .. math:: |
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95 | f_{\text{inter}_i} = 4 \pi \int_{\Delta t_{ \text{inter}_i } } \rho_{ \text{inter}_i } \frac{\sin(qr)} {qr} r^2 dr = |
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96 | 4 \pi \sum_{j=0}^{npts_{\text{inter}_i} -1 } |
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97 | \int_{r_j}^{r_{j+1}} \rho_{ \text{inter}_i } (r_j) \frac{\sin(qr)} {qr} r^2 dr \approx |
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98 | |
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99 | 4 \pi \sum_{j=0}^{npts_{\text{inter}_i} -1 } \Big[ |
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100 | 3 ( \rho_{ \text{inter}_i } ( r_{j+1} ) - \rho_{ \text{inter}_i } ( r_{j} ) V ( r_{ \text{sublayer}_j } ) |
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101 | \Big[ \frac {r_j^2 \beta_\text{out}^2 \sin(\beta_\text{out}) - (\beta_\text{out}^2-2) \cos(\beta_\text{out}) } |
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102 | {\beta_\text{out}^4 } \Big] |
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103 | |
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104 | - 3 ( \rho_{ \text{inter}_i } ( r_{j+1} ) - \rho_{ \text{inter}_i } ( r_{j} ) V ( r_{ \text{sublayer}_j-1 } ) |
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105 | \Big[ \frac {r_{j-1}^2 \sin(\beta_\text{in}) - (\beta_\text{in}^2-2) \cos(\beta_\text{in}) } |
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106 | {\beta_\text{in}^4 } \Big] |
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107 | |
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108 | + 3 \rho_{ \text{inter}_i } ( r_{j+1} ) V ( r_j ) |
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109 | \Big[ \frac {\sin(\beta_\text{out}) - \cos(\beta_\text{out}) } |
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110 | {\beta_\text{out}^4 } \Big] |
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111 | |
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112 | - 3 \rho_{ \text{inter}_i } ( r_{j} ) V ( r_j ) |
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113 | \Big[ \frac {\sin(\beta_\text{in}) - \cos(\beta_\text{in}) } |
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114 | {\beta_\text{in}^4 } \Big] |
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115 | \Big] |
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116 | |
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117 | where |
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118 | |
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119 | .. math:: |
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120 | V(a) = \frac {4\pi}{3}a^3 |
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121 | |
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122 | a_\text{in} ~ \frac{r_j}{r_{j+1} -r_j} \text{, } a_\text{out} ~ \frac{r_{j+1}}{r_{j+1} -r_j} |
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123 | |
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124 | \beta_\text{in} = qr_j \text{, } \beta_\text{out} = qr_{j+1} |
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125 | |
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126 | |
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127 | We assume the $\rho_{\text{inter}_i} (r)$ can be approximately linear within a sub-layer $j$ |
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128 | |
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129 | Finally form factor can be calculated by |
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130 | |
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131 | .. math:: |
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132 | |
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133 | P(q) = \frac{[f]^2} {V_\text{particle}} \text{where} V_\text{particle} = V(r_{\text{shell}_N}) |
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134 | |
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135 | For 2D data the scattering intensity is calculated in the same way as 1D, |
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136 | where the $q$ vector is defined as |
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137 | |
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138 | .. math:: |
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139 | |
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140 | q = \sqrt{q_x^2 + q_y^2} |
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141 | |
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142 | |
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143 | .. figure:: img/spherical_sld_1d.jpg |
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144 | |
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145 | 1D plot using the default values (w/400 data point). |
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146 | |
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147 | .. figure:: img/spherical_sld_default_profile.jpg |
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148 | |
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149 | SLD profile from the default values. |
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150 | |
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151 | .. note:: |
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152 | The outer most radius is used as the effective radius for S(Q) when $P(Q) * S(Q)$ is applied. |
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153 | |
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154 | References |
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155 | ---------- |
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156 | L A Feigin and D I Svergun, Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum Press, New York, (1987) |
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157 | |
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158 | """ |
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159 | |
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160 | from numpy import inf |
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161 | |
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162 | name = "spherical_sld" |
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163 | title = "Sperical SLD intensity calculation" |
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164 | description = """ |
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165 | I(q) = |
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166 | background = Incoherent background [1/cm] |
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167 | """ |
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168 | category = "sphere-based" |
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169 | |
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170 | # pylint: disable=bad-whitespace, line-too-long |
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171 | # ["name", "units", default, [lower, upper], "type", "description"], |
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172 | parameters = [["n_shells", "", 1, [0, 9], "", "number of shells"], |
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173 | ["radius_core", "Ang", 50.0, [0, inf], "", "intern layer thickness"], |
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174 | ["sld_core", "1e-6/Ang^2", 2.07, [-inf, inf], "", "sld function flat"], |
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175 | ["sld_flat[n]", "1e-6/Ang^2", 4.06, [-inf, inf], "", "sld function flat"], |
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176 | ["thick_flat[n]", "Ang", 100.0, [0, inf], "", "flat layer_thickness"], |
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177 | ["func_inter[n]", "", 0, [0, 4], "", "Erf:0, RPower:1, LPower:2, RExp:3, LExp:4"], |
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178 | ["thick_inter[n]", "Ang", 50.0, [0, inf], "", "intern layer thickness"], |
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179 | ["inter_nu[n]", "", 2.5, [-inf, inf], "", "steepness parameter"], |
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180 | ["npts_inter", "", 35, [0, 35], "", "number of points in each sublayer Must be odd number"], |
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181 | ["sld_solvent", "1e-6/Ang^2", 1.0, [-inf, inf], "", "sld function solvent"], |
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182 | ] |
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183 | # pylint: enable=bad-whitespace, line-too-long |
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184 | #source = ["lib/librefl.c", "lib/sph_j1c.c", "spherical_sld.c"] |
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185 | |
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186 | def Iq(q, *args, **kw): |
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187 | return q |
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188 | |
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189 | def Iqxy(qx, *args, **kw): |
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190 | return qx |
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191 | |
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192 | |
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193 | demo = dict( |
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194 | n_shells=4, |
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195 | scale=1.0, |
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196 | solvent_sld=1.0, |
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197 | background=0.0, |
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198 | npts_inter=35.0, |
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199 | ) |
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200 | |
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201 | #TODO: Not working yet |
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202 | tests = [ |
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203 | # Accuracy tests based on content in test/utest_extra_models.py |
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204 | [{'npts_iter':35, |
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205 | 'sld_solv':1, |
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206 | 'radius_core':50.0, |
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207 | 'sld_core':2.07, |
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208 | 'func_inter2':0.0, |
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209 | 'thick_inter2':50, |
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210 | 'nu_inter2':2.5, |
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211 | 'sld_flat2':4, |
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212 | 'thick_flat2':100, |
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213 | 'func_inter1':0.0, |
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214 | 'thick_inter1':50, |
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215 | 'nu_inter1':2.5, |
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216 | 'background': 0.0, |
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217 | }, 0.001, 0.001], |
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218 | ] |
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