1 | r""" |
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2 | This model provides the form factor, $P(q)$, for a multi-shell sphere where |
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3 | the scattering length density (SLD) of each shell is described by an |
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4 | exponential, linear, or constant function. The form factor is normalized by |
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5 | the volume of the sphere where the SLD is not identical to the SLD of the |
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6 | solvent. We currently provide up to 9 shells with this model. |
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7 | |
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8 | NB: *radius* represents the core radius $r_0$ and |
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9 | *thickness[k]* represents the thickness of the shell, $r_{k+1} - r_k$. |
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10 | |
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11 | Definition |
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12 | ---------- |
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13 | |
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14 | The 1D scattering intensity is calculated in the following way |
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15 | |
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16 | .. math:: |
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17 | |
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18 | P(q) = [f]^2 / V_\text{particle} |
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19 | |
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20 | where |
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21 | |
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22 | .. math:: |
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23 | :nowrap: |
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24 | |
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25 | \begin{align*} |
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26 | f &= f_\text{core} |
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27 | + \left(\sum_{\text{shell}=1}^N f_\text{shell}\right) |
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28 | + f_\text{solvent} |
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29 | \end{align*} |
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30 | |
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31 | The shells are spherically symmetric with particle density $\rho(r)$ and |
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32 | constant SLD within the core and solvent, so |
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33 | |
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34 | .. math:: |
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35 | :nowrap: |
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36 | |
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37 | \begin{align*} |
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38 | f_\text{core} |
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39 | &= 4\pi\int_0^{r_\text{core}} \rho_\text{core} |
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40 | \frac{\sin(qr)}{qr}\, r^2\,\mathrm{d}r |
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41 | &= 3\rho_\text{core} V(r_\text{core}) |
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42 | \frac{j_1(qr_\text{core})}{qr_\text{core}} \\ |
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43 | f_\text{shell} |
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44 | &= 4\pi\int_{r_{\text{shell}-1}}^{r_\text{shell}} |
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45 | \rho_\text{shell}(r)\frac{\sin(qr)}{qr}\,r^2\,\mathrm{d}r \\ |
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46 | f_\text{solvent} |
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47 | &= 4\pi\int_{r_N}^\infty |
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48 | \rho_\text{solvent}\frac{\sin(qr)}{qr}\,r^2\,\mathrm{d}r |
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49 | &= -3\rho_\text{solvent}V(r_N)\frac{j_1(q r_N)}{q r_N} |
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50 | \end{align*} |
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51 | |
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52 | where the spherical bessel function $j_1$ is |
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53 | |
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54 | .. math:: |
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55 | |
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56 | j_1(x) = \frac{\sin(x)}{x^2} - \frac{\cos(x)}{x} |
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57 | |
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58 | and the volume is $V(r) = \frac{4\pi}{3}r^3$. The volume of the particle |
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59 | is determined by the radius of the outer shell, so $V_\text{particle} = V(r_N)$. |
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60 | |
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61 | Now lets consider the SLD of a shell defined by |
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62 | |
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63 | .. math:: |
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64 | |
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65 | \rho_\text{shell}(r) = \begin{cases} |
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66 | B\exp\left(A(r-r_{\text{shell}-1})/\Delta t_\text{shell}\right) |
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67 | + C & \mbox{for } A \neq 0 \\ |
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68 | \rho_\text{in} = \text{constant} & \mbox{for } A = 0 |
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69 | \end{cases} |
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70 | |
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71 | An example of a possible SLD profile is shown below where |
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72 | $\rho_\text{in}$ and $\Delta t_\text{shell}$ stand for the |
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73 | SLD of the inner side of the $k^\text{th}$ shell and the |
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74 | thickness of the $k^\text{th}$ shell in the equation above, respectively. |
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75 | |
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76 | For $A > 0$, |
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77 | |
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78 | .. math:: |
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79 | |
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80 | f_\text{shell} &= 4 \pi \int_{r_{\text{shell}-1}}^{r_\text{shell}} |
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81 | \left[ B\exp |
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82 | \left(A (r - r_{\text{shell}-1}) / \Delta t_\text{shell} \right) + C |
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83 | \right] \frac{\sin(qr)}{qr}\,r^2\,\mathrm{d}r \\ |
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84 | &= 3BV(r_\text{shell}) e^A h(\alpha_\text{out},\beta_\text{out}) |
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85 | - 3BV(r_{\text{shell}-1}) h(\alpha_\text{in},\beta_\text{in}) |
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86 | + 3CV(r_{\text{shell}}) \frac{j_1(\beta_\text{out})}{\beta_\text{out}} |
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87 | - 3CV(r_{\text{shell}-1}) \frac{j_1(\beta_\text{in})}{\beta_\text{in}} |
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88 | |
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89 | for |
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90 | |
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91 | .. math:: |
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92 | :nowrap: |
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93 | |
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94 | \begin{align*} |
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95 | B&=\frac{\rho_\text{out} - \rho_\text{in}}{e^A-1} |
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96 | & C &= \frac{\rho_\text{in}e^A - \rho_\text{out}}{e^A-1} \\ |
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97 | \alpha_\text{in} &= A\frac{r_{\text{shell}-1}}{\Delta t_\text{shell}} |
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98 | & \alpha_\text{out} &= A\frac{r_\text{shell}}{\Delta t_\text{shell}} \\ |
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99 | \beta_\text{in} &= qr_{\text{shell}-1} |
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100 | & \beta_\text{out} &= qr_\text{shell} \\ |
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101 | \end{align*} |
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102 | |
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103 | where $h$ is |
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104 | |
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105 | .. math:: |
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106 | |
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107 | h(x,y) = \frac{x \sin(y) - y\cos(y)}{(x^2+y^2)y} |
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108 | - \frac{(x^2-y^2)\sin(y) - 2xy\cos(y)}{(x^2+y^2)^2y} |
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109 | |
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110 | |
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111 | For $A \sim 0$, e.g., $A = -0.0001$, this function converges to that of the |
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112 | linear SLD profile with |
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113 | $\rho_\text{shell}(r) \approx A(r-r_{\text{shell}-1})/\Delta t_\text{shell})+B$, |
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114 | so this case is equivalent to |
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115 | |
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116 | .. math:: |
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117 | :nowrap: |
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118 | |
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119 | \begin{align*} |
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120 | f_\text{shell} |
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121 | &= |
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122 | 3 V(r_\text{shell}) \frac{\Delta\rho_\text{shell}}{\Delta t_\text{shell}} |
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123 | \left[\frac{ |
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124 | 2 \cos(qr_\text{out}) |
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125 | + qr_\text{out} \sin(qr_\text{out}) |
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126 | }{ |
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127 | (qr_\text{out})^4 |
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128 | }\right] \\ |
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129 | &{} |
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130 | -3 V(r_\text{shell}) \frac{\Delta\rho_\text{shell}}{\Delta t_\text{shell}} |
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131 | \left[\frac{ |
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132 | 2\cos(qr_\text{in}) |
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133 | +qr_\text{in}\sin(qr_\text{in}) |
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134 | }{ |
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135 | (qr_\text{in})^4 |
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136 | }\right] \\ |
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137 | &{} |
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138 | +3\rho_\text{out}V(r_\text{shell}) \frac{j_1(qr_\text{out})}{qr_\text{out}} |
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139 | -3\rho_\text{in}V(r_{\text{shell}-1}) \frac{j_1(qr_\text{in})}{qr_\text{in}} |
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140 | \end{align*} |
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141 | |
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142 | For $A = 0$, the exponential function has no dependence on the radius (so that |
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143 | $\rho_\text{out}$ is ignored in this case) and becomes flat. We set the constant |
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144 | to $\rho_\text{in}$ for convenience, and thus the form factor contributed by |
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145 | the shells is |
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146 | |
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147 | .. math:: |
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148 | |
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149 | f_\text{shell} = |
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150 | 3\rho_\text{in}V(r_\text{shell}) |
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151 | \frac{j_1(qr_\text{out})}{qr_\text{out}} |
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152 | - 3\rho_\text{in}V(r_{\text{shell}-1}) |
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153 | \frac{j_1(qr_\text{in})}{qr_\text{in}} |
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154 | |
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155 | .. figure:: img/onion_geometry.png |
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156 | |
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157 | Example of an onion model profile. |
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158 | |
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159 | The 2D scattering intensity is the same as $P(q)$ above, regardless of the |
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160 | orientation of the $q$ vector which is defined as |
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161 | |
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162 | .. math:: |
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163 | |
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164 | q = \sqrt{q_x^2 + q_y^2} |
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165 | |
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166 | NB: The outer most radius is used as the effective radius for $S(q)$ |
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167 | when $P(q) S(q)$ is applied. |
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168 | |
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169 | References |
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170 | ---------- |
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171 | |
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172 | L A Feigin and D I Svergun, |
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173 | *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, |
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174 | Plenum Press, New York, 1987. |
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175 | """ |
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176 | |
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177 | # |
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178 | # Give a polynomial $\rho(r) = Ar^3 + Br^2 + Cr + D$ for density, |
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179 | # |
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180 | # .. math:: |
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181 | # |
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182 | # f = 4 \pi \int_a^b \rho(r) \sin(qr)/(qr) \mathrm{d}r = h(b) - h(a) |
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183 | # |
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184 | # where |
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185 | # |
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186 | # .. math:: |
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187 | # |
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188 | # h(r) = \frac{4 \pi}{q^6}\left[ |
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189 | # (q^3(4Ar^3 + 3Br^2 + 2Cr + D) - q(24Ar + 6B)) \sin(qr) |
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190 | # - (q^4(Ar^4 + Br^3 + Cr^2 + Dr) - q^2(12Ar^2 + 6Br + 2C) + 24A) \cos(qr) |
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191 | # \right] |
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192 | # |
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193 | # Use the monotonic spline to get the polynomial coefficients for each shell. |
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194 | # |
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195 | # Order 0 |
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196 | # |
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197 | # .. math:: |
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198 | # |
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199 | # h(r) = \frac{4 \pi}{q^3} \left[ |
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200 | # - \cos(qr) (Ar) q |
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201 | # + \sin(qr) (A) |
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202 | # \right] |
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203 | # |
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204 | # Order 1 |
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205 | # |
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206 | # .. math:: |
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207 | # |
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208 | # h(r) = \frac{4 \pi}{q^4} \left[ |
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209 | # - \cos(qr) ( Ar^2 + Br) q^2 |
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210 | # + \sin(qr) ( Ar + B ) q |
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211 | # + \cos(qr) (2A ) |
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212 | # \right] |
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213 | # |
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214 | # Order 2 |
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215 | # |
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216 | # .. math:: |
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217 | # h(r) = \frac{4 \pi}{q^5} \left[ |
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218 | # - \cos(qr) ( Ar^3 + Br^2 + Cr) q^3 |
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219 | # + \sin(qr) (3Ar^2 + 2Br + C ) q^2 |
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220 | # + \cos(qr) (6Ar + 2B ) q |
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221 | # - \sin(qr) (6A ) |
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222 | # |
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223 | # Order 3 |
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224 | # |
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225 | # h(r) = \frac{4 \pi}{q^6}\left[ |
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226 | # - \cos(qr) ( Ar^4 + Br^3 + Cr^2 + Dr) q^4 |
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227 | # + \sin(qr) ( 4Ar^3 + 3Br^2 + 2Cr + D ) q^3 |
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228 | # + \cos(qr) (12Ar^2 + 6Br + 2C ) q^2 |
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229 | # - \sin(qr) (24Ar + 6B ) q |
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230 | # - \cos(qr) (24A ) |
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231 | # \right] |
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232 | # |
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233 | # Order p |
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234 | # |
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235 | # h(r) = \frac{4 \pi}{q^{2}} |
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236 | # \sum_{k=0}^p -\frac{d^k\cos(qr)}{dr^k} \frac{d^k r\rho(r)}{dr^k} (qr)^{-k} |
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237 | # |
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238 | # Given the equation |
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239 | # |
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240 | # f = sum_(k=0)^(n-1) h_k(r_(k+1)) - h_k(r_k) |
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241 | # |
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242 | # we can rearrange the terms so that |
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243 | # |
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244 | # f = sum_0^(n-1) h_k(r_(k+1)) - sum_0^(n-1) h_k(r_k) |
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245 | # = sum_1^n h_(k-1)(r_k) - sum_0^(n-1) h_k(r_k) |
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246 | # = h_(n-1)(r_n) - h_0(r_0) + sum_1^(n-1) [h_(k-1)(r_k) - h_k(r_k)] |
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247 | # = h_(n-1)(r_n) - h_0(r_0) - sum_1^(n-1) h_(Delta k)(r_k) |
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248 | # |
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249 | # where |
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250 | # |
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251 | # h_(Delta k)(r) = h(Delta rho_k, r) |
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252 | # |
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253 | # for |
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254 | # |
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255 | # Delta rho_k = (A_k-A_(k-1)) r^p + (B_k-B_(k-1)) r^(p-1) + ... |
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256 | # |
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257 | # Using l'H\^opital's Rule 6 times on the order 3 polynomial, |
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258 | # |
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259 | # lim_(q->0) h(r) = (140D r^3 + 180C r^4 + 144B r^5 + 120A r^6)/720 |
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260 | # |
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261 | |
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262 | from __future__ import division |
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263 | |
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264 | from math import fabs, exp, expm1 |
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265 | |
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266 | import numpy as np |
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267 | from numpy import inf, nan |
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268 | |
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269 | name = "onion" |
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270 | title = "Onion shell model with constant, linear or exponential density" |
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271 | |
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272 | description = """\ |
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273 | Form factor of mutishells normalized by the volume. Here each shell is |
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274 | described by an exponential function; |
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275 | |
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276 | I) For A_shell != 0, |
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277 | f(r) = B*exp(A_shell*(r-r_in)/thick_shell)+C |
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278 | where |
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279 | B=(sld_out-sld_in)/(exp(A_shell)-1) |
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280 | C=sld_in-B. |
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281 | Note that in the above case, the function becomes a linear function |
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282 | as A_shell --> 0+ or 0-. |
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283 | |
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284 | II) For the exact point of A_shell == 0, |
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285 | f(r) = sld_in ,i.e., it crosses over flat function |
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286 | Note that the 'sld_out' becaomes NULL in this case. |
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287 | |
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288 | background:background, |
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289 | rad_core0: radius of sphere(core) |
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290 | thick_shell#:the thickness of the shell# |
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291 | sld_core0: the SLD of the sphere |
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292 | sld_solv: the SLD of the solvent |
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293 | sld_shell: the SLD of the shell# |
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294 | A_shell#: the coefficient in the exponential function |
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295 | """ |
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296 | |
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297 | category = "shape:sphere" |
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298 | |
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299 | # TODO: n is a volume parameter that is not polydisperse |
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300 | |
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301 | # pylint: disable=bad-whitespace, line-too-long |
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302 | # ["name", "units", default, [lower, upper], "type","description"], |
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303 | parameters = [ |
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304 | ["sld_core", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "Core scattering length density"], |
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305 | ["radius_core", "Ang", 200., [0, inf], "volume", "Radius of the core"], |
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306 | ["sld_solvent", "1e-6/Ang^2", 6.4, [-inf, inf], "sld", "Solvent scattering length density"], |
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307 | ["n_shells", "", 1, [0, 10], "volume", "number of shells"], |
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308 | ["sld_in[n_shells]", "1e-6/Ang^2", 1.7, [-inf, inf], "sld", "scattering length density at the inner radius of shell k"], |
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309 | ["sld_out[n_shells]", "1e-6/Ang^2", 2.0, [-inf, inf], "sld", "scattering length density at the outer radius of shell k"], |
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310 | ["thickness[n_shells]", "Ang", 40., [0, inf], "volume", "Thickness of shell k"], |
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311 | ["A[n_shells]", "", 1.0, [-inf, inf], "", "Decay rate of shell k"], |
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312 | ] |
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313 | # pylint: enable=bad-whitespace, line-too-long |
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314 | |
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315 | source = ["lib/sas_3j1x_x.c", "onion.c"] |
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316 | single = False |
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317 | have_Fq = True |
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318 | |
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319 | profile_axes = ['Radius (A)', 'SLD (1e-6/A^2)'] |
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320 | def profile(sld_core, radius_core, sld_solvent, n_shells, |
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321 | sld_in, sld_out, thickness, A): |
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322 | """ |
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323 | Returns shape profile with x=radius, y=SLD. |
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324 | """ |
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325 | n_shells = int(n_shells+0.5) |
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326 | total_radius = 1.25*(sum(thickness[:n_shells]) + radius_core + 1) |
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327 | dz = total_radius/400 # 400 points for a smooth plot |
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328 | |
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329 | z = [] |
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330 | rho = [] |
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331 | |
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332 | # add in the core |
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333 | z.append(0) |
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334 | rho.append(sld_core) |
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335 | z.append(radius_core) |
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336 | rho.append(sld_core) |
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337 | |
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338 | # add in the shells |
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339 | for k in range(int(n_shells)): |
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340 | # Left side of each shells |
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341 | z_current = z[-1] |
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342 | z.append(z_current) |
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343 | rho.append(sld_in[k]) |
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344 | |
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345 | if fabs(A[k]) < 1.0e-16: |
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346 | # flat shell |
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347 | z.append(z_current + thickness[k]) |
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348 | rho.append(sld_in[k]) |
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349 | else: |
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350 | # exponential shell |
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351 | # num_steps must be at least 1, so use floor()+1 rather than ceil |
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352 | # to protect against a thickness0. |
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353 | num_steps = np.floor(thickness[k]/dz) + 1 |
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354 | slope = (sld_out[k] - sld_in[k]) / expm1(A[k]) |
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355 | const = (sld_in[k] - slope) |
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356 | for z_shell in np.linspace(0, thickness[k], num_steps+1): |
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357 | z.append(z_current+z_shell) |
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358 | rho.append(slope*exp(A[k]*z_shell/thickness[k]) + const) |
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359 | |
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360 | # add in the solvent |
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361 | z.append(z[-1]) |
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362 | rho.append(sld_solvent) |
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363 | z.append(total_radius) |
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364 | rho.append(sld_solvent) |
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365 | |
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366 | return np.asarray(z), np.asarray(rho) |
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367 | |
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368 | def ER(radius_core, n_shells, thickness): |
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369 | """Effective radius""" |
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370 | n = int(n_shells[0]+0.5) |
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371 | return np.sum(thickness[:n], axis=0) + radius_core |
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372 | |
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373 | demo = { |
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374 | "sld_solvent": 2.2, |
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375 | "sld_core": 1.0, |
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376 | "radius_core": 100, |
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377 | "n_shells": 4, |
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378 | "sld_in": [0.5, 1.5, 0.9, 2.0], |
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379 | "sld_out": [nan, 0.9, 1.2, 1.6], |
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380 | "thickness": [50, 75, 150, 75], |
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381 | "A": [0, -1, 1e-4, 1], |
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382 | # Could also specify them individually as |
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383 | # "A1": 0, "A2": -1, "A3": 1e-4, "A4": 1, |
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384 | #"radius_core_pd_n": 10, |
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385 | #"radius_core_pd": 0.4, |
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386 | #"thickness4_pd_n": 10, |
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387 | #"thickness4_pd": 0.4, |
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388 | } |
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