[3330bb4] | 1 | r""" |
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[8400bc5] | 2 | Definition |
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| 3 | ---------- |
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| 4 | |
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[3330bb4] | 5 | Similarly to the onion, this model provides the form factor, $P(q)$, for |
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| 6 | a multi-shell sphere, where the interface between the each neighboring |
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| 7 | shells can be described by the error function, power-law, or exponential |
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| 8 | functions. The scattering intensity is computed by building a continuous |
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| 9 | custom SLD profile along the radius of the particle. The SLD profile is |
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| 10 | composed of a number of uniform shells with interfacial shells between them. |
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| 11 | |
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| 12 | .. figure:: img/spherical_sld_profile.png |
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| 13 | |
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| 14 | Example SLD profile |
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| 15 | |
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[0507e09] | 16 | Unlike the :ref:`onion` model (using an analytical integration), the interfacial |
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[3330bb4] | 17 | shells here are sub-divided and numerically integrated assuming each |
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| 18 | sub-shell is described by a line function, with *n_steps* sub-shells per |
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| 19 | interface. The form factor is normalized by the total volume of the sphere. |
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| 20 | |
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[627b68b] | 21 | .. note:: |
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| 22 | |
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| 23 | *n_shells* must be an integer. *n_steps* must be an ODD integer. |
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| 24 | |
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[8400bc5] | 25 | Interface shapes are as follows: |
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[3330bb4] | 26 | |
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[2ad5d30] | 27 | 0: erf($\nu z$) |
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[ee60aa7] | 28 | |
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[2ad5d30] | 29 | 1: Rpow($z^\nu$) |
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[ee60aa7] | 30 | |
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[2ad5d30] | 31 | 2: Lpow($z^\nu$) |
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[ee60aa7] | 32 | |
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[2ad5d30] | 33 | 3: Rexp($-\nu z$) |
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[ee60aa7] | 34 | |
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[2ad5d30] | 35 | 4: Lexp($-\nu z$) |
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[3330bb4] | 36 | |
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| 37 | The form factor $P(q)$ in 1D is calculated by: |
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| 38 | |
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| 39 | .. math:: |
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| 40 | |
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| 41 | P(q) = \frac{f^2}{V_\text{particle}} \text{ where } |
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| 42 | f = f_\text{core} + \sum_{\text{inter}_i=0}^N f_{\text{inter}_i} + |
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| 43 | \sum_{\text{flat}_i=0}^N f_{\text{flat}_i} +f_\text{solvent} |
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| 44 | |
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| 45 | For a spherically symmetric particle with a particle density $\rho_x(r)$ |
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| 46 | the sld function can be defined as: |
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| 47 | |
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| 48 | .. math:: |
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| 49 | |
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| 50 | f_x = 4 \pi \int_{0}^{\infty} \rho_x(r) \frac{\sin(qr)} {qr^2} r^2 dr |
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| 51 | |
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| 52 | |
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| 53 | so that individual terms can be calculated as follows: |
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| 54 | |
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| 55 | .. math:: |
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| 56 | |
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| 57 | f_\text{core} &= 4 \pi \int_{0}^{r_\text{core}} \rho_\text{core} |
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| 58 | \frac{\sin(qr)} {qr} r^2 dr = |
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| 59 | 3 \rho_\text{core} V(r_\text{core}) |
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| 60 | \Big[ \frac{\sin(qr_\text{core}) - qr_\text{core} \cos(qr_\text{core})} |
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[ca04add] | 61 | {qr_\text{core}^3} \Big] \\ |
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[3330bb4] | 62 | f_{\text{inter}_i} &= 4 \pi \int_{\Delta t_{ \text{inter}_i } } |
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[ca04add] | 63 | \rho_{ \text{inter}_i } \frac{\sin(qr)} {qr} r^2 dr \\ |
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[3330bb4] | 64 | f_{\text{shell}_i} &= 4 \pi \int_{\Delta t_{ \text{inter}_i } } |
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| 65 | \rho_{ \text{flat}_i } \frac{\sin(qr)} {qr} r^2 dr = |
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| 66 | 3 \rho_{ \text{flat}_i } V ( r_{ \text{inter}_i } + |
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| 67 | \Delta t_{ \text{inter}_i } ) |
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| 68 | \Big[ \frac{\sin(qr_{\text{inter}_i} + \Delta t_{ \text{inter}_i } ) |
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| 69 | - q (r_{\text{inter}_i} + \Delta t_{ \text{inter}_i }) |
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| 70 | \cos(q( r_{\text{inter}_i} + \Delta t_{ \text{inter}_i } ) ) } |
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| 71 | {q ( r_{\text{inter}_i} + \Delta t_{ \text{inter}_i } )^3 } \Big] |
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| 72 | -3 \rho_{ \text{flat}_i } V(r_{ \text{inter}_i }) |
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| 73 | \Big[ \frac{\sin(qr_{\text{inter}_i}) - qr_{\text{flat}_i} |
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[ca04add] | 74 | \cos(qr_{\text{inter}_i}) } {qr_{\text{inter}_i}^3} \Big] \\ |
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[3330bb4] | 75 | f_\text{solvent} &= 4 \pi \int_{r_N}^{\infty} \rho_\text{solvent} |
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| 76 | \frac{\sin(qr)} {qr} r^2 dr = |
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| 77 | 3 \rho_\text{solvent} V(r_N) |
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| 78 | \Big[ \frac{\sin(qr_N) - qr_N \cos(qr_N)} {qr_N^3} \Big] |
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| 79 | |
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| 80 | Here we assumed that the SLDs of the core and solvent are constant in $r$. |
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| 81 | The SLD at the interface between shells, $\rho_{\text {inter}_i}$ |
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| 82 | is calculated with a function chosen by an user, where the functions are |
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| 83 | |
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| 84 | Exp: |
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| 85 | |
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| 86 | .. math:: |
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| 87 | |
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| 88 | \rho_{{inter}_i} (r) &= \begin{cases} |
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| 89 | B \exp\Big( \frac {\pm A(r - r_{\text{flat}_i})} |
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| 90 | {\Delta t_{ \text{inter}_i }} \Big) +C & \mbox{for } A \neq 0 \\ |
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| 91 | B \Big( \frac {(r - r_{\text{flat}_i})} |
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| 92 | {\Delta t_{ \text{inter}_i }} \Big) +C & \mbox{for } A = 0 \\ |
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| 93 | \end{cases} |
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| 94 | |
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[0507e09] | 95 | Power-Law: |
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[3330bb4] | 96 | |
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| 97 | .. math:: |
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| 98 | |
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| 99 | \rho_{{inter}_i} (r) &= \begin{cases} |
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| 100 | \pm B \Big( \frac {(r - r_{\text{flat}_i} )} {\Delta t_{ \text{inter}_i }} |
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| 101 | \Big) ^A +C & \mbox{for } A \neq 0 \\ |
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| 102 | \rho_{\text{flat}_{i+1}} & \mbox{for } A = 0 \\ |
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| 103 | \end{cases} |
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| 104 | |
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| 105 | Erf: |
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| 106 | |
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| 107 | .. math:: |
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[0507e09] | 108 | |
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[3330bb4] | 109 | \rho_{{inter}_i} (r) = \begin{cases} |
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| 110 | B \text{erf} \Big( \frac { A(r - r_{\text{flat}_i})} |
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| 111 | {\sqrt{2} \Delta t_{ \text{inter}_i }} \Big) +C & \mbox{for } A \neq 0 \\ |
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| 112 | B \Big( \frac {(r - r_{\text{flat}_i} )} {\Delta t_{ \text{inter}_i }} |
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| 113 | \Big) +C & \mbox{for } A = 0 \\ |
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| 114 | \end{cases} |
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| 115 | |
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| 116 | The functions are normalized so that they vary between 0 and 1, and they are |
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| 117 | constrained such that the SLD is continuous at the boundaries of the interface |
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| 118 | as well as each sub-shell. Thus B and C are determined. |
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| 119 | |
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| 120 | Once $\rho_{\text{inter}_i}$ is found at the boundary of the sub-shell of the |
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| 121 | interface, we can find its contribution to the form factor $P(q)$ |
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| 122 | |
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| 123 | .. math:: |
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| 124 | |
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| 125 | f_{\text{inter}_i} &= 4 \pi \int_{\Delta t_{ \text{inter}_i } } |
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| 126 | \rho_{ \text{inter}_i } \frac{\sin(qr)} {qr} r^2 dr = |
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| 127 | 4 \pi \sum_{j=1}^{n_\text{steps}} |
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| 128 | \int_{r_j}^{r_{j+1}} \rho_{ \text{inter}_i } (r_j) |
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[ca04add] | 129 | \frac{\sin(qr)} {qr} r^2 dr \\ |
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| 130 | \approx 4 \pi \sum_{j=1}^{n_\text{steps}} \Big[ |
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[3330bb4] | 131 | 3 ( \rho_{ \text{inter}_i } ( r_{j+1} ) - \rho_{ \text{inter}_i } |
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| 132 | ( r_{j} ) V (r_j) |
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| 133 | \Big[ \frac {r_j^2 \beta_\text{out}^2 \sin(\beta_\text{out}) |
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| 134 | - (\beta_\text{out}^2-2) \cos(\beta_\text{out}) } |
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[ca04add] | 135 | {\beta_\text{out}^4 } \Big] \\ |
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| 136 | {} - 3 ( \rho_{ \text{inter}_i } ( r_{j+1} ) - \rho_{ \text{inter}_i } |
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[3330bb4] | 137 | ( r_{j} ) V ( r_{j-1} ) |
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| 138 | \Big[ \frac {r_{j-1}^2 \sin(\beta_\text{in}) |
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| 139 | - (\beta_\text{in}^2-2) \cos(\beta_\text{in}) } |
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[ca04add] | 140 | {\beta_\text{in}^4 } \Big] \\ |
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| 141 | {} + 3 \rho_{ \text{inter}_i } ( r_{j+1} ) V ( r_j ) |
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[3330bb4] | 142 | \Big[ \frac {\sin(\beta_\text{out}) - \cos(\beta_\text{out}) } |
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| 143 | {\beta_\text{out}^4 } \Big] |
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| 144 | - 3 \rho_{ \text{inter}_i } ( r_{j} ) V ( r_j ) |
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| 145 | \Big[ \frac {\sin(\beta_\text{in}) - \cos(\beta_\text{in}) } |
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| 146 | {\beta_\text{in}^4 } \Big] |
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| 147 | \Big] |
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| 148 | |
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| 149 | where |
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| 150 | |
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| 151 | .. math:: |
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| 152 | :nowrap: |
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| 153 | |
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| 154 | \begin{align*} |
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| 155 | V(a) &= \frac {4\pi}{3}a^3 && \\ |
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[ca04add] | 156 | a_\text{in} \sim \frac{r_j}{r_{j+1} -r_j} \text{, } & a_\text{out} |
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| 157 | \sim \frac{r_{j+1}}{r_{j+1} -r_j} \\ |
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| 158 | \beta_\text{in} &= qr_j \text{, } & \beta_\text{out} &= qr_{j+1} |
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[3330bb4] | 159 | \end{align*} |
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| 160 | |
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| 161 | We assume $\rho_{\text{inter}_j} (r)$ is approximately linear |
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| 162 | within the sub-shell $j$. |
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| 163 | |
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| 164 | Finally the form factor can be calculated by |
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| 165 | |
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| 166 | .. math:: |
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| 167 | |
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| 168 | P(q) = \frac{[f]^2} {V_\text{particle}} \mbox{ where } V_\text{particle} |
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| 169 | = V(r_{\text{shell}_N}) |
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| 170 | |
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| 171 | For 2D data the scattering intensity is calculated in the same way as 1D, |
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| 172 | where the $q$ vector is defined as |
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| 173 | |
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| 174 | .. math:: |
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| 175 | |
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| 176 | q = \sqrt{q_x^2 + q_y^2} |
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| 177 | |
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| 178 | .. note:: |
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| 179 | |
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| 180 | The outer most radius is used as the effective radius for $S(Q)$ |
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| 181 | when $P(Q) * S(Q)$ is applied. |
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| 182 | |
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| 183 | References |
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| 184 | ---------- |
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[8400bc5] | 185 | |
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| 186 | .. [#] L A Feigin and D I Svergun, Structure Analysis by Small-Angle X-Ray |
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| 187 | and Neutron Scattering, Plenum Press, New York, (1987) |
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| 188 | |
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[0507e09] | 189 | Source |
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| 190 | ------ |
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| 191 | |
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| 192 | `spherical_sld.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/spherical_sld.py>`_ |
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| 193 | |
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| 194 | `spherical_sld.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/spherical_sld.c>`_ |
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[8400bc5] | 195 | |
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| 196 | Authorship and Verification |
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[abe4255] | 197 | --------------------------- |
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[8400bc5] | 198 | |
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| 199 | * **Author:** Jae-Hie Cho **Date:** Nov 1, 2010 |
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| 200 | * **Last Modified by:** Paul Kienzle **Date:** Dec 20, 2016 |
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[627b68b] | 201 | * **Last Reviewed by:** Steve King **Date:** March 29, 2019 |
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[0507e09] | 202 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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[3330bb4] | 203 | """ |
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| 204 | |
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| 205 | import numpy as np |
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| 206 | from numpy import inf, expm1, sqrt |
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| 207 | from scipy.special import erf |
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| 208 | |
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| 209 | name = "spherical_sld" |
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[627b68b] | 210 | title = "Spherical SLD intensity calculation" |
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[3330bb4] | 211 | description = """ |
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| 212 | I(q) = |
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| 213 | background = Incoherent background [1/cm] |
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| 214 | """ |
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| 215 | category = "shape:sphere" |
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| 216 | |
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[63a7fe8] | 217 | SHAPES = ["erf(|nu|*z)", "Rpow(z^|nu|)", "Lpow(z^|nu|)", |
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| 218 | "Rexp(-|nu|z)", "Lexp(-|nu|z)"] |
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[3330bb4] | 219 | |
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| 220 | # pylint: disable=bad-whitespace, line-too-long |
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| 221 | # ["name", "units", default, [lower, upper], "type", "description"], |
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[627b68b] | 222 | parameters = [["n_shells", "", 1, [1, 10], "volume", "number of shells (must be integer)"], |
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[3330bb4] | 223 | ["sld_solvent", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "solvent sld"], |
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| 224 | ["sld[n_shells]", "1e-6/Ang^2", 4.06, [-inf, inf], "sld", "sld of the shell"], |
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| 225 | ["thickness[n_shells]", "Ang", 100.0, [0, inf], "volume", "thickness shell"], |
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| 226 | ["interface[n_shells]", "Ang", 50.0, [0, inf], "volume", "thickness of the interface"], |
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[63a7fe8] | 227 | ["shape[n_shells]", "", 0, [SHAPES], "", "interface shape"], |
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[e8eff7b] | 228 | ["nu[n_shells]", "", 2.5, [1, inf], "", "interface shape exponent"], |
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[3330bb4] | 229 | ["n_steps", "", 35, [0, inf], "", "number of steps in each interface (must be an odd integer)"], |
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| 230 | ] |
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| 231 | # pylint: enable=bad-whitespace, line-too-long |
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| 232 | source = ["lib/polevl.c", "lib/sas_erf.c", "lib/sas_3j1x_x.c", "spherical_sld.c"] |
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| 233 | single = False # TODO: fix low q behaviour |
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[71b751d] | 234 | have_Fq = True |
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[a34b811] | 235 | radius_effective_modes = ["outer radius"] |
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[3330bb4] | 236 | |
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| 237 | profile_axes = ['Radius (A)', 'SLD (1e-6/A^2)'] |
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| 238 | |
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| 239 | SHAPE_FUNCTIONS = [ |
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| 240 | lambda z, nu: erf(nu/sqrt(2)*(2*z-1))/(2*erf(nu/sqrt(2))) + 0.5, # erf |
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| 241 | lambda z, nu: z**nu, # Rpow |
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| 242 | lambda z, nu: 1 - (1-z)**nu, # Lpow |
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| 243 | lambda z, nu: expm1(-nu*z)/expm1(-nu), # Rexp |
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| 244 | lambda z, nu: expm1(nu*z)/expm1(nu), # Lexp |
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| 245 | ] |
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| 246 | |
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| 247 | def profile(n_shells, sld_solvent, sld, thickness, |
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| 248 | interface, shape, nu, n_steps): |
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| 249 | """ |
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| 250 | Returns shape profile with x=radius, y=SLD. |
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| 251 | """ |
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| 252 | |
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| 253 | n_shells = int(n_shells + 0.5) |
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| 254 | n_steps = int(n_steps + 0.5) |
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| 255 | z = [] |
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| 256 | rho = [] |
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| 257 | z_next = 0 |
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| 258 | # two sld points for core |
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| 259 | z.append(z_next) |
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| 260 | rho.append(sld[0]) |
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| 261 | |
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| 262 | for i in range(0, n_shells): |
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| 263 | z_next += thickness[i] |
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| 264 | z.append(z_next) |
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| 265 | rho.append(sld[i]) |
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| 266 | dz = interface[i]/n_steps |
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| 267 | sld_l = sld[i] |
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| 268 | sld_r = sld[i+1] if i < n_shells-1 else sld_solvent |
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| 269 | fun = SHAPE_FUNCTIONS[int(np.clip(shape[i], 0, len(SHAPE_FUNCTIONS)-1))] |
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| 270 | for step in range(1, n_steps+1): |
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| 271 | portion = fun(float(step)/n_steps, max(abs(nu[i]), 1e-14)) |
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| 272 | z_next += dz |
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| 273 | z.append(z_next) |
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| 274 | rho.append((sld_r - sld_l)*portion + sld_l) |
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| 275 | z.append(z_next*1.2) |
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| 276 | rho.append(sld_solvent) |
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| 277 | # return sld profile (r, beta) |
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| 278 | return np.asarray(z), np.asarray(rho) |
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| 279 | |
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[b297ba9] | 280 | # TODO: no random parameter generator for spherical SLD. |
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[3330bb4] | 281 | |
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| 282 | demo = { |
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| 283 | "n_shells": 5, |
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| 284 | "n_steps": 35.0, |
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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": [50.0]*5, |
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| 289 | "shape": [0]*5, |
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| 290 | "nu": [2.5]*5, |
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| 291 | } |
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| 292 | |
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| 293 | tests = [ |
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| 294 | # Results checked against sasview 3.1 |
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| 295 | [{"n_shells": 5, |
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| 296 | "n_steps": 35, |
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| 297 | "sld_solvent": 1.0, |
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| 298 | "sld": [2.07, 4.0, 3.5, 4.0, 3.5], |
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| 299 | "thickness": [50.0, 100.0, 100.0, 100.0, 100.0], |
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| 300 | "interface": [50]*5, |
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| 301 | "shape": [0]*5, |
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| 302 | "nu": [2.5]*5, |
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| 303 | }, 0.001, 750697.238], |
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| 304 | ] |
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