[8ad9619] | 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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[e42b0b9] | 15 | Unlike the <onion> model (using an analytical integration), |
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[8ad9619] | 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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[2be86e2] | 172 | parameters = [["n_shells", "", 1, [0, 9], "", "number of shells"], |
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[6f0e04f] | 173 | ["radius_core", "Ang", 50.0, [0, inf], "", "intern layer thickness"], |
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[2be86e2] | 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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[6f0e04f] | 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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[2be86e2] | 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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[6f0e04f] | 181 | ["sld_solvent", "1e-6/Ang^2", 1.0, [-inf, inf], "", "sld function solvent"], |
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[8ad9619] | 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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[669bf21] | 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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[8ad9619] | 192 | |
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| 193 | demo = dict( |
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[6f0e04f] | 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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[8ad9619] | 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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[6f0e04f] | 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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[8ad9619] | 216 | 'background': 0.0, |
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[6f0e04f] | 217 | }, 0.001, 0.001], |
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[8ad9619] | 218 | ] |
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