1 | r""" |
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2 | Similarly to the onion, this model provides the form factor, $P(q)$, for |
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3 | a multi-shell sphere, where the interface between the each neighboring |
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4 | shells can be described by the error function, power-law, or exponential |
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5 | functions. The scattering intensity is computed by building a continuous |
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6 | custom SLD profile along the radius of the particle. The SLD profile is |
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7 | composed of a number of uniform shells with interfacial shells between them. |
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8 | |
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9 | .. figure:: img/spherical_sld_profile.png |
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10 | |
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11 | Example SLD profile |
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12 | |
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13 | Unlike the <onion> model (using an analytical integration), the interfacial |
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14 | shells here are sub-divided and numerically integrated assuming each |
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15 | sub-shell is described by a line function, with *n_steps* sub-shells per |
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16 | interface. The form factor is normalized by the total volume of the sphere. |
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17 | |
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18 | Interface shapes are as follows:: |
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19 | |
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20 | 0: erf(|nu|*z) |
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21 | 1: Rpow(z^|nu|) |
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22 | 2: Lpow(z^|nu|) |
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23 | 3: Rexp(-|nu|z) |
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24 | 4: Lexp(-|nu|z) |
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25 | |
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26 | Definition |
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27 | ---------- |
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28 | |
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29 | The form factor $P(q)$ in 1D is calculated by: |
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30 | |
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31 | .. math:: |
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32 | |
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33 | P(q) = \frac{f^2}{V_\text{particle}} \text{ where } |
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34 | f = f_\text{core} + \sum_{\text{inter}_i=0}^N f_{\text{inter}_i} + |
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35 | \sum_{\text{flat}_i=0}^N f_{\text{flat}_i} +f_\text{solvent} |
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36 | |
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37 | For a spherically symmetric particle with a particle density $\rho_x(r)$ |
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38 | the sld function can be defined as: |
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39 | |
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40 | .. math:: |
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41 | |
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42 | f_x = 4 \pi \int_{0}^{\infty} \rho_x(r) \frac{\sin(qr)} {qr^2} r^2 dr |
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43 | |
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44 | |
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45 | so that individual terms can be calculated as follows: |
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46 | |
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47 | .. math:: |
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48 | |
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49 | f_\text{core} &= 4 \pi \int_{0}^{r_\text{core}} \rho_\text{core} |
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50 | \frac{\sin(qr)} {qr} r^2 dr = |
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51 | 3 \rho_\text{core} V(r_\text{core}) |
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52 | \Big[ \frac{\sin(qr_\text{core}) - qr_\text{core} \cos(qr_\text{core})} |
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53 | {qr_\text{core}^3} \Big] |
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54 | |
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55 | f_{\text{inter}_i} &= 4 \pi \int_{\Delta t_{ \text{inter}_i } } |
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56 | \rho_{ \text{inter}_i } \frac{\sin(qr)} {qr} r^2 dr |
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57 | |
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58 | f_{\text{shell}_i} &= 4 \pi \int_{\Delta t_{ \text{inter}_i } } |
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59 | \rho_{ \text{flat}_i } \frac{\sin(qr)} {qr} r^2 dr = |
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60 | 3 \rho_{ \text{flat}_i } V ( r_{ \text{inter}_i } + |
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61 | \Delta t_{ \text{inter}_i } ) |
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62 | \Big[ \frac{\sin(qr_{\text{inter}_i} + \Delta t_{ \text{inter}_i } ) |
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63 | - q (r_{\text{inter}_i} + \Delta t_{ \text{inter}_i }) |
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64 | \cos(q( r_{\text{inter}_i} + \Delta t_{ \text{inter}_i } ) ) } |
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65 | {q ( r_{\text{inter}_i} + \Delta t_{ \text{inter}_i } )^3 } \Big] |
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66 | -3 \rho_{ \text{flat}_i } V(r_{ \text{inter}_i }) |
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67 | \Big[ \frac{\sin(qr_{\text{inter}_i}) - qr_{\text{flat}_i} |
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68 | \cos(qr_{\text{inter}_i}) } {qr_{\text{inter}_i}^3} \Big] |
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69 | |
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70 | f_\text{solvent} &= 4 \pi \int_{r_N}^{\infty} \rho_\text{solvent} |
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71 | \frac{\sin(qr)} {qr} r^2 dr = |
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72 | 3 \rho_\text{solvent} V(r_N) |
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73 | \Big[ \frac{\sin(qr_N) - qr_N \cos(qr_N)} {qr_N^3} \Big] |
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74 | |
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75 | |
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76 | Here we assumed that the SLDs of the core and solvent are constant in $r$. |
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77 | The SLD at the interface between shells, $\rho_{\text {inter}_i}$ |
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78 | is calculated with a function chosen by an user, where the functions are |
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79 | |
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80 | Exp: |
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81 | |
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82 | .. math:: |
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83 | |
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84 | \rho_{{inter}_i} (r) &= \begin{cases} |
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85 | B \exp\Big( \frac {\pm A(r - r_{\text{flat}_i})} |
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86 | {\Delta t_{ \text{inter}_i }} \Big) +C & \mbox{for } A \neq 0 \\ |
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87 | B \Big( \frac {(r - r_{\text{flat}_i})} |
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88 | {\Delta t_{ \text{inter}_i }} \Big) +C & \mbox{for } A = 0 \\ |
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89 | \end{cases} |
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90 | |
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91 | Power-Law |
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92 | |
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93 | .. math:: |
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94 | |
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95 | \rho_{{inter}_i} (r) &= \begin{cases} |
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96 | \pm B \Big( \frac {(r - r_{\text{flat}_i} )} {\Delta t_{ \text{inter}_i }} |
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97 | \Big) ^A +C & \mbox{for } A \neq 0 \\ |
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98 | \rho_{\text{flat}_{i+1}} & \mbox{for } A = 0 \\ |
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99 | \end{cases} |
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100 | |
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101 | Erf: |
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102 | |
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103 | .. math:: |
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104 | \rho_{{inter}_i} (r) = \begin{cases} |
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105 | B \text{erf} \Big( \frac { A(r - r_{\text{flat}_i})} |
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106 | {\sqrt{2} \Delta t_{ \text{inter}_i }} \Big) +C & \mbox{for } A \neq 0 \\ |
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107 | B \Big( \frac {(r - r_{\text{flat}_i} )} {\Delta t_{ \text{inter}_i }} |
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108 | \Big) +C & \mbox{for } A = 0 \\ |
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109 | \end{cases} |
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110 | |
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111 | The functions are normalized so that they vary between 0 and 1, and they are |
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112 | constrained such that the SLD is continuous at the boundaries of the interface |
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113 | as well as each sub-shell. Thus B and C are determined. |
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114 | |
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115 | Once $\rho_{\text{inter}_i}$ is found at the boundary of the sub-shell of the |
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116 | interface, we can find its contribution to the form factor $P(q)$ |
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117 | |
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118 | .. math:: |
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119 | |
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120 | f_{\text{inter}_i} &= 4 \pi \int_{\Delta t_{ \text{inter}_i } } |
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121 | \rho_{ \text{inter}_i } \frac{\sin(qr)} {qr} r^2 dr = |
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122 | 4 \pi \sum_{j=1}^{n_\text{steps}} |
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123 | \int_{r_j}^{r_{j+1}} \rho_{ \text{inter}_i } (r_j) |
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124 | \frac{\sin(qr)} {qr} r^2 dr |
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125 | |
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126 | &\approx 4 \pi \sum_{j=1}^{n_\text{steps}} \Big[ |
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127 | 3 ( \rho_{ \text{inter}_i } ( r_{j+1} ) - \rho_{ \text{inter}_i } |
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128 | ( r_{j} ) V (r_j) |
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129 | \Big[ \frac {r_j^2 \beta_\text{out}^2 \sin(\beta_\text{out}) |
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130 | - (\beta_\text{out}^2-2) \cos(\beta_\text{out}) } |
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131 | {\beta_\text{out}^4 } \Big] |
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132 | |
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133 | &{} - 3 ( \rho_{ \text{inter}_i } ( r_{j+1} ) - \rho_{ \text{inter}_i } |
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134 | ( r_{j} ) V ( r_{j-1} ) |
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135 | \Big[ \frac {r_{j-1}^2 \sin(\beta_\text{in}) |
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136 | - (\beta_\text{in}^2-2) \cos(\beta_\text{in}) } |
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137 | {\beta_\text{in}^4 } \Big] |
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138 | |
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139 | &{} + 3 \rho_{ \text{inter}_i } ( r_{j+1} ) V ( r_j ) |
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140 | \Big[ \frac {\sin(\beta_\text{out}) - \cos(\beta_\text{out}) } |
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141 | {\beta_\text{out}^4 } \Big] |
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142 | - 3 \rho_{ \text{inter}_i } ( r_{j} ) V ( r_j ) |
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143 | \Big[ \frac {\sin(\beta_\text{in}) - \cos(\beta_\text{in}) } |
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144 | {\beta_\text{in}^4 } \Big] |
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145 | \Big] |
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146 | |
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147 | where |
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148 | |
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149 | .. math:: |
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150 | :nowrap: |
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151 | |
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152 | \begin{align*} |
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153 | V(a) &= \frac {4\pi}{3}a^3 && \\ |
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154 | a_\text{in} &\sim \frac{r_j}{r_{j+1} -r_j} \text{, } &a_\text{out} |
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155 | &\sim \frac{r_{j+1}}{r_{j+1} -r_j} \\ |
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156 | \beta_\text{in} &= qr_j \text{, } &\beta_\text{out} &= qr_{j+1} |
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157 | \end{align*} |
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158 | |
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159 | |
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160 | We assume $\rho_{\text{inter}_j} (r)$ is approximately linear |
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161 | within the sub-shell $j$. |
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162 | |
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163 | Finally the form factor can be calculated by |
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164 | |
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165 | .. math:: |
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166 | |
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167 | P(q) = \frac{[f]^2} {V_\text{particle}} \mbox{ where } V_\text{particle} |
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168 | = V(r_{\text{shell}_N}) |
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169 | |
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170 | For 2D data the scattering intensity is calculated in the same way as 1D, |
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171 | where the $q$ vector is defined as |
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172 | |
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173 | .. math:: |
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174 | |
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175 | q = \sqrt{q_x^2 + q_y^2} |
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176 | |
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177 | .. note:: |
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178 | |
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179 | The outer most radius is used as the effective radius for $S(Q)$ |
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180 | when $P(Q) * S(Q)$ is applied. |
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181 | |
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182 | References |
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183 | ---------- |
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184 | L A Feigin and D I Svergun, Structure Analysis by Small-Angle X-Ray |
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185 | and Neutron Scattering, Plenum Press, New York, (1987) |
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186 | |
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187 | """ |
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188 | |
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189 | import numpy as np |
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190 | from numpy import inf, expm1, sqrt |
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191 | from scipy.special import erf |
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192 | |
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193 | name = "spherical_sld" |
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194 | title = "Sperical SLD intensity calculation" |
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195 | description = """ |
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196 | I(q) = |
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197 | background = Incoherent background [1/cm] |
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198 | """ |
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199 | category = "shape:sphere" |
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200 | |
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201 | SHAPES = [["erf(|nu|*z)", "Rpow(z^|nu|)", "Lpow(z^|nu|)", |
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202 | "Rexp(-|nu|z)", "Lexp(-|nu|z)"]] |
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203 | |
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204 | # pylint: disable=bad-whitespace, line-too-long |
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205 | # ["name", "units", default, [lower, upper], "type", "description"], |
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206 | parameters = [["n_shells", "", 1, [1, 10], "volume", "number of shells"], |
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207 | ["sld_solvent", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "solvent sld"], |
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208 | ["sld[n_shells]", "1e-6/Ang^2", 4.06, [-inf, inf], "sld", "sld of the shell"], |
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209 | ["thickness[n_shells]", "Ang", 100.0, [0, inf], "volume", "thickness shell"], |
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210 | ["interface[n_shells]", "Ang", 50.0, [0, inf], "volume", "thickness of the interface"], |
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211 | ["shape[n_shells]", "", 0, SHAPES, "", "interface shape"], |
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212 | ["nu[n_shells]", "", 2.5, [0, inf], "", "interface shape exponent"], |
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213 | ["n_steps", "", 35, [0, inf], "", "number of steps in each interface (must be an odd integer)"], |
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214 | ] |
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215 | # pylint: enable=bad-whitespace, line-too-long |
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216 | source = ["lib/polevl.c", "lib/sas_erf.c", "lib/sph_j1c.c", "spherical_sld.c"] |
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217 | single = False # TODO: fix low q behaviour |
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218 | |
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219 | profile_axes = ['Radius (A)', 'SLD (1e-6/A^2)'] |
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220 | |
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221 | SHAPE_FUNCTIONS = [ |
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222 | lambda z, nu: erf(nu/sqrt(2)*(2*z-1))/(2*erf(nu/sqrt(2))) + 0.5, # erf |
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223 | lambda z, nu: z**nu, # Rpow |
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224 | lambda z, nu: 1 - (1-z)**nu, # Lpow |
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225 | lambda z, nu: expm1(-nu*z)/expm1(-nu), # Rexp |
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226 | lambda z, nu: expm1(nu*z)/expm1(nu), # Lexp |
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227 | ] |
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228 | |
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229 | def profile(n_shells, sld_solvent, sld, thickness, |
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230 | interface, shape, nu, n_steps): |
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231 | """ |
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232 | Returns shape profile with x=radius, y=SLD. |
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233 | """ |
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234 | |
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235 | z = [] |
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236 | rho = [] |
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237 | z0 = 0 |
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238 | # two sld points for core |
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239 | z.append(0) |
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240 | rho.append(sld[0]) |
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241 | |
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242 | for i in range(0, int(n_shells)): |
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243 | z0 += thickness[i] |
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244 | z.append(z0) |
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245 | rho.append(sld[i]) |
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246 | dz = interface[i]/n_steps |
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247 | sld_l = sld[i] |
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248 | sld_r = sld[i+1] if i < n_shells-1 else sld_solvent |
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249 | fn = SHAPE_FUNCTIONS[int(np.clip(shape[i], 0, len(SHAPE_FUNCTIONS)-1))] |
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250 | for step in range(1, n_steps+1): |
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251 | portion = fn(float(step)/n_steps, max(abs(nu[i]), 1e-14)) |
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252 | z0 += dz |
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253 | z.append(z0) |
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254 | rho.append((sld_r - sld_l)*portion + sld_l) |
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255 | z.append(z0*1.2) |
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256 | rho.append(sld_solvent) |
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257 | # return sld profile (r, beta) |
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258 | return np.asarray(z), np.asarray(rho) |
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259 | |
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260 | |
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261 | def ER(n_shells, thickness, interface): |
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262 | n_shells = int(n_shells) |
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263 | total = (np.sum(thickness[:n_shells], axis=1) |
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264 | + np.sum(interface[:n_shells], axis=1)) |
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265 | return total |
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266 | |
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267 | |
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268 | demo = { |
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269 | "n_shells": 5, |
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270 | "n_steps": 35.0, |
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271 | "sld_solvent": 1.0, |
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272 | "sld":[2.07,4.0,3.5,4.0,3.5], |
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273 | "thickness":[50.0,100.0,100.0,100.0,100.0], |
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274 | "interface":[50.0,50.0,50.0,50.0], |
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275 | "shape": [0,0,0,0,0], |
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276 | "nu":[2.5,2.5,2.5,2.5,2.5], |
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277 | } |
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278 | |
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279 | #TODO: Not working yet |
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280 | """ |
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281 | tests = [ |
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282 | # Accuracy tests based on content in test/utest_extra_models.py |
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283 | [{"n_shells": 5, |
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284 | "n_steps": 35, |
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285 | "sld_solvent": 1.0, |
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286 | "sld": [2.07, 4.0, 3.5, 4.0, 3.5], |
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287 | "thickness": [50.0, 100.0, 100.0, 100.0, 100.0], |
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288 | "interface": [50]*5, |
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289 | "shape": [0]*5, |
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290 | "nu": [2.5]*5, |
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291 | }, 0.001, 0.001], |
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292 | ] |
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293 | """ |
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