.. _spherical-sld:

spherical_sld
=======================================================

Spherical SLD intensity calculation

=================== ========================================================== ============ =============
Parameter           Description                                                Units        Default value
=================== ========================================================== ============ =============
scale               Scale factor or Volume fraction                            None                     1
background          Source background                                          |cm^-1|              0.001
n_shells            number of shells (must be integer)                         None                     1
sld_solvent         solvent sld                                                |1e-6Ang^-2|             1
sld[n_shells]       sld of the shell                                           |1e-6Ang^-2|          4.06
thickness[n_shells] thickness shell                                            |Ang|                  100
interface[n_shells] thickness of the interface                                 |Ang|                   50
shape[n_shells]     interface shape                                            None                     0
nu[n_shells]        interface shape exponent                                   None                   2.5
n_steps             number of steps in each interface (must be an odd integer) None                    35
=================== ========================================================== ============ =============

The returned value is scaled to units of |cm^-1| |sr^-1|, absolute scale.


**Definition**

Similarly to the onion, this model provides the form factor, $P(q)$, for
a multi-shell sphere, where the interface between the each neighboring
shells can be described by the error function, power-law, or exponential
functions.  The scattering intensity is computed by building a continuous
custom SLD profile along the radius of the particle. The SLD profile is
composed of a number of uniform shells with interfacial shells between them.

.. figure:: img/spherical_sld_profile.png

    Example SLD profile

Unlike the :ref:`onion` model (using an analytical integration), the interfacial
shells here are sub-divided and numerically integrated assuming each
sub-shell is described by a line function, with *n_steps* sub-shells per
interface. The form factor is normalized by the total volume of the sphere.

.. note::

   *n_shells* must be an integer. *n_steps* must be an ODD integer.

Interface shapes are as follows:

    0: erf($\nu z$)

    1: Rpow($z^\nu$)

    2: Lpow($z^\nu$)

    3: Rexp($-\nu z$)

    4: Lexp($-\nu z$)

    5: Boucher ($(1-z^2)^(\nu/2-2)$)

The form factor $P(q)$ in 1D is calculated by [#Feigin1987]_:

.. math::

    P(q) = \frac{f^2}{V_\text{particle}} \text{ where }
    f = f_\text{core} + \sum_{\text{inter}_i=0}^N f_{\text{inter}_i} +
    \sum_{\text{flat}_i=0}^N f_{\text{flat}_i} +f_\text{solvent}

For a spherically symmetric particle with a particle density $\rho_x(r)$
the sld function can be defined as:

.. math::

    f_x = 4 \pi \int_{0}^{\infty} \rho_x(r)  \frac{\sin(qr)} {qr^2} r^2 dr


so that individual terms can be calculated as follows:

.. math::

    f_\text{core}
        &= 4 \pi \int_{0}^{r_\text{core}} \rho_\text{core}
            \frac{\sin(qr)} {qr} r^2 dr \\
        &= 3 \rho_\text{core} V(r_\text{core})
          \left[ \frac{\sin(qr_\text{core}) - qr_\text{core} \cos(qr_\text{core})}
                {qr_\text{core}^3} \right] \\
    f_{\text{inter}_i}
        &= 4 \pi \int_{\Delta t_{ \text{inter}_i } } \rho_{ \text{inter}_i }
            \frac{\sin(qr)} {qr} r^2 dr \\
    f_{\text{shell}_i}
        &= 4 \pi \int_{\Delta t_{ \text{inter}_i } } \rho_{ \text{flat}_i }
            \frac{\sin(qr)} {qr} r^2 dr \\
        &= 3 \rho_{\text{flat}_i} V (r_{\text{inter}_i}
                                       + \Delta t_{\text{inter}_i})
            \left[
                \frac{\sin(qr_{\text{inter}_i} + \Delta t_{\text{inter}_i})
                    - q (r_{\text{inter}_i} + \Delta t_{ \text{inter}_i })
                    \cos(q(r_{\text{inter}_i} + \Delta t_{\text{inter}_i}))}
                {q ( r_{\text{inter}_i} + \Delta t_{\text{inter}_i} )^3 }
            \right] \\
        &\quad {} - 3 \rho_{ \text{flat}_i } V (r_{\text{inter}_i})
            \left[
                \frac{\sin(qr_{\text{inter}_i})
                    - qr_{\text{flat}_i} \cos(qr_{\text{inter}_i})}
                {qr_{\text{inter}_i}^3}
            \right] \\
    f_\text{solvent}
        &= 4 \pi \int_{r_N}^{\infty} \rho_\text{solvent}
            \frac{\sin(qr)} {qr} r^2 dr \\
        &= 3 \rho_\text{solvent} V(r_N)
            \left[ \frac{\sin(qr_N) - qr_N \cos(qr_N)} {qr_N^3} \right]

Here we assumed that the SLDs of the core and solvent are constant in $r$.
The SLD at the interface between shells, $\rho_{\text {inter}_i}$
is calculated with a function chosen by an user, where the functions are

Exp:

.. math::

    \rho_{{inter}_i}(r) &=
    \begin{cases}
        B\, \exp\left(
            \frac{\pm A(r - r_{\text{flat}_i})}{\Delta t_{\text{inter}_i}}
        \right) + C  & \mbox{for } A \neq 0 \\
        B\, \left(
            \frac{(r - r_{\text{flat}_i})}{\Delta t_{\text{inter}_i}}
        \right) + C  & \mbox{for } A = 0 \\
    \end{cases}

Power-Law:

.. math::

    \rho_{{inter}_i}(r) &=
    \begin{cases}
        \pm B\, \left(
            \frac{(r - r_{\text{flat}_i})}{\Delta t_{ \text{inter}_i }}
            \right) ^A  + C  & \mbox{for } A \neq 0 \\
        \rho_{\text{flat}_{i+1}}  & \mbox{for } A = 0 \\
    \end{cases}

Erf:

.. math::

    \rho_{{inter}_i}(r) =
    \begin{cases}
        B\, \text{erf} \left(
            \frac{A(r - r_{\text{flat}_i})}{\sqrt{2} \Delta t_{\text{inter}_i}}
            \right) + C  & \mbox{for } A \neq 0 \\
        B\, \left(
            \frac{(r - r_{\text{flat}_i})}{\Delta t_{\text{inter}_i}}
            \right)  +C  & \mbox{for } A = 0 \\
    \end{cases}


Boucher [#Boucher1983]_:

.. math::

    \rho_{{inter}_i}(r) =
    \begin{cases}
        \pm B\, \left(1-
            (\frac{(r - r_{\text{flat}_i})}{\Delta t_{ \text{inter}_i }})^2
            \right) ^(A/2-2)  + C  & \mbox{for } A \neq 0 \\
        \rho_{\text{flat}_{i+1}}  & \mbox{for } A = 0 \\
    \end{cases}

The functions are normalized so that they vary between 0 and 1, and they are
constrained such that the SLD is continuous at the boundaries of the interface
as well as each sub-shell. Thus B and C are determined.

Once $\rho_{\text{inter}_i}$ is found at the boundary of the sub-shell of the
interface, we can find its contribution to the form factor $P(q)$

.. math::

    f_{\text{inter}_i}
        &= 4 \pi \int_{\Delta t_{\text{inter}_i} } \rho_{\text{inter}_i}
            \frac{\sin(qr)}{qr} r^2 dr \\
        &= 4 \pi \sum_{j=1}^{n_\text{steps}}
            \int_{r_j}^{r_{j+1}} \rho_{\text{inter}_i}(r_j)
                \frac{\sin(qr)}{qr} r^2 dr \\
        &\approx 4 \pi \sum_{j=1}^{n_\text{steps}}
        \Biggl[
             3 (\rho_{\text{inter}_i}(r_{j+1}) - \rho_{\text{inter}_i}(r_{j})) V (r_j)
            \left[
                \frac{r_j^2 \beta_\text{out}^2 \sin(\beta_\text{out})
                    - (\beta_\text{out}^2-2) \cos(\beta_\text{out})}
                {\beta_\text{out}^4}
            \right] \\
        &\quad {} - 3 (\rho_{\text{inter}_i}(r_{j+1}) - \rho_{\text{inter}_i}(r_{j})) V(r_{j-1})
            \left[
                \frac{r_{j-1}^2 \sin(\beta_\text{in})
                    - (\beta_\text{in}^2-2) \cos(\beta_\text{in})}
                {\beta_\text{in}^4}
            \right] \\
        &\quad {} + 3 \rho_{\text{inter}_i}(r_{j+1})  V(r_j)
            \left[
                \frac{\sin(\beta_\text{out}) - \cos(\beta_\text{out})}
                {\beta_\text{out}^4}
            \right] \\
        &\quad {} - 3 \rho_{\text{inter}_i}(r_{j})  V(r_j)
            \left[
                \frac{\sin(\beta_\text{in}) - \cos(\beta_\text{in})}
                {\beta_\text{in}^4}
            \right]
        \Biggr]

where

.. math::
    :nowrap:

    \begin{align*}
    V(a) &= \frac {4\pi}{3}a^3
        & {} & {} \\
    a_\text{in} &\sim \frac{r_j}{r_{j+1} -r_j}
        & a_\text{out} &\sim \frac{r_{j+1}}{r_{j+1} -r_j} \\
    \beta_\text{in} &= qr_j
        & \beta_\text{out} &= qr_{j+1}
    \end{align*}

We assume $\rho_{\text{inter}_j} (r)$ is approximately linear
within the sub-shell $j$.

Finally the form factor can be calculated by

.. math::

    P(q) = \frac{[f]^2} {V_\text{particle}} \mbox{ where } V_\text{particle}
    = V(r_{\text{shell}_N})

For 2D data the scattering intensity is calculated in the same way as 1D,
where the $q$ vector is defined as

.. math::

    q = \sqrt{q_x^2 + q_y^2}

.. note::

    The outer most radius is used as the effective radius for $S(Q)$
    when $P(Q) * S(Q)$ is applied.


.. figure:: img/spherical_sld_autogenfig.png

    1D plot corresponding to the default parameters of the model.


**Source**

:download:`spherical_sld.py <src/spherical_sld.py>`
$\ \star\ $ :download:`spherical_sld.c <src/spherical_sld.c>`
$\ \star\ $ :download:`sas_3j1x_x.c <src/sas_3j1x_x.c>`
$\ \star\ $ :download:`sas_erf.c <src/sas_erf.c>`
$\ \star\ $ :download:`polevl.c <src/polevl.c>`

**References**

.. [#Feigin1987] L A Feigin and D I Svergun, Structure Analysis by Small-Angle X-Ray
   and Neutron Scattering, Plenum Press, New York, (1987)
.. [#Boucher1983] B Boucher, P Chieux, P Convert, and M Tournarie,
   *Metal Physics*, 13,1339 (1983).



**Authorship and Verification**

* **Author:** Jae-Hie Cho **Date:** Nov 1, 2010
* **Last Modified by:** Paul Kienzle **Date:** Dec 20, 2016
* **Last Reviewed by:** Steve King **Date:** March 29, 2019

