source: sasmodels/sasmodels/models/spherical_sld.py @ da63656

r"""
This model calculates an empirical functional form for SAS data using
SpericalSLD profile

Similarly to the OnionExpShellModel, 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 one of a number of functions including error,
power-law, and exponential functions.
This model is to calculate the scattering intensity by building a continuous
custom SLD profile against the radius of the particle.
The SLD profile is composed of a flat core, a flat solvent, a number (up to 9 )
flat shells, and the interfacial layers between the adjacent flat shells
(or core, and solvent) (see below).

.. figure:: img/spherical_sld_profile.gif

    Exemplary SLD profile

Unlike the <onion> model (using an analytical integration), the interfacial
layers here are sub-divided and numerically integrated assuming each of the
sub-layers are described by a line function.
The number of the sub-layer can be given by users by setting the integer values
of npts_inter. The form factor is normalized by the total volume of the sphere.

Definition
----------

The form factor $P(q)$ in 1D is calculated by:

.. 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 calcualted 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})
    \Big[ \frac{\sin(qr_\text{core}) - qr_\text{core} \cos(qr_\text{core})}
    {qr_\text{core}^3} \Big]

    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 } )
    \Big[ \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 }  \Big]
    -3 \rho_{ \text{flat}_i } V(r_{ \text{inter}_i })
    \Big[ \frac{\sin(qr_{\text{inter}_i}) - qr_{\text{flat}_i}
    \cos(qr_{\text{inter}_i}) } {qr_{\text{inter}_i}^3} \Big]

    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)
    \Big[ \frac{\sin(qr_N) - qr_N \cos(qr_N)} {qr_N^3} \Big]


Here we assumed that the SLDs of the core and solvent are constant against $r$.
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\Big( \frac {\pm A(r - r_{\text{flat}_i})}
    {\Delta t_{ \text{inter}_i }} \Big) +C  & \text{for} A \neq 0 \\
    B \Big( \frac {(r - r_{\text{flat}_i})}
    {\Delta t_{ \text{inter}_i }} \Big) +C  & \text{for} A = 0 \\
    \end{cases}

Power-Law

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

Erf:

.. math::
    \rho_{{inter}_i} (r) = \begin{cases}
    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 \\
    B \Big( \frac {(r - r_{\text{flat}_i} )} {\Delta t_{ \text{inter}_i }}
    \Big)  +C  & \text{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-layers. Thus B and C are determined.

Once $\rho_{\text{inter}_i}$ is found at the boundary of the sub-layer 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=0}^{npts_{\text{inter}_i} -1 }
    \int_{r_j}^{r_{j+1}} \rho_{ \text{inter}_i } (r_j)
    \frac{\sin(qr)} {qr} r^2 dr \approx

    4 \pi \sum_{j=0}^{npts_{\text{inter}_i} -1 } \Big[
    3 ( \rho_{ \text{inter}_i } ( r_{j+1} ) - \rho_{ \text{inter}_i }
    ( r_{j} ) V ( r_{ \text{sublayer}_j } )
    \Big[ \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 } \Big]

    - 3 ( \rho_{ \text{inter}_i } ( r_{j+1} ) - \rho_{ \text{inter}_i }
    ( r_{j} ) V ( r_{ \text{sublayer}_j-1 } )
    \Big[ \frac {r_{j-1}^2 \sin(\beta_\text{in})
    - (\beta_\text{in}^2-2) \cos(\beta_\text{in}) }
    {\beta_\text{in}^4 } \Big]

    + 3 \rho_{ \text{inter}_i } ( r_{j+1} )  V ( r_j )
    \Big[ \frac {\sin(\beta_\text{out}) - \cos(\beta_\text{out}) }
    {\beta_\text{out}^4 } \Big]

    - 3 \rho_{ \text{inter}_i } ( r_{j} )  V ( r_j )
    \Big[ \frac {\sin(\beta_\text{in}) - \cos(\beta_\text{in}) }
    {\beta_\text{in}^4 } \Big]
    \Big]

where

.. math::
    V(a) = \frac {4\pi}{3}a^3

    a_\text{in} ~ \frac{r_j}{r_{j+1} -r_j} \text{, } a_\text{out}
    ~ \frac{r_{j+1}}{r_{j+1} -r_j}

    \beta_\text{in} = qr_j \text{, } \beta_\text{out} = qr_{j+1}


We assume the $\rho_{\text{inter}_i} (r)$ can be approximately linear
within a sub-layer $j$

Finally form factor can be calculated by

.. math::

    P(q) = \frac{[f]^2} {V_\text{particle}} \text{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}


.. figure:: img/spherical_sld_1d.jpg

    1D plot using the default values (w/400 data point).

.. figure:: img/spherical_sld_default_profile.jpg

    SLD profile from the default values.

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

References
----------
L A Feigin and D I Svergun, Structure Analysis by Small-Angle X-Ray
and Neutron Scattering, Plenum Press, New York, (1987)

"""

from numpy import inf

name = "spherical_sld"
title = "Sperical SLD intensity calculation"
description = """
            I(q) =
               background = Incoherent background [1/cm]
        """
category = "sphere-based"

# pylint: disable=bad-whitespace, line-too-long
#            ["name", "units", default, [lower, upper], "type", "description"],
parameters = [["n_shells",          "",               1,      [0, 9],         "volume", "number of shells"],
              ["npts_inter",        "",               35,     [0, inf],        "", "number of points in each sublayer Must be odd number"],
              ["radius_core",       "Ang",            50.0,   [0, inf],       "volume", "intern layer thickness"],
              ["sld_core",          "1e-6/Ang^2",     2.07,   [-inf, inf],    "", "sld function flat"],
              ["sld_solvent",       "1e-6/Ang^2",     1.0,    [-inf, inf],    "", "sld function solvent"],
              ["func_inter0",       "",               0,      [0, 4],         "", "Erf:0, RPower:1, LPower:2, RExp:3, LExp:4"],
              ["thick_inter0",      "Ang",            50.0,   [0, inf],       "volume", "intern layer thickness for core layer"],
              ["nu_inter0",         "",               2.5,    [-inf, inf],    "", "steepness parameter for core layer"],
              ["sld_flat[n_shells]",      "1e-6/Ang^2",     4.06,   [-inf, inf],    "", "sld function flat"],
              ["thick_flat[n_shells]",    "Ang",            100.0,  [0, inf],       "volume", "flat layer_thickness"],
              ["func_inter[n_shells]",    "",               0,      [0, 4],         "", "Erf:0, RPower:1, LPower:2, RExp:3, LExp:4"],
              ["thick_inter[n_shells]",   "Ang",            50.0,   [0, inf],       "volume", "intern layer thickness"],
              ["nu_inter[n_shells]",      "",               2.5,    [-inf, inf],    "", "steepness parameter"],
              ]
# pylint: enable=bad-whitespace, line-too-long
source = ["lib/librefl.c",  "lib/sph_j1c.c", "spherical_sld.c"]

profile_axes = ['Radius (A)', 'SLD (1e-6/A^2)']
def profile(n_shells, radius_core,  sld_core,  sld_solvent, sld_flat,
            thick_flat, func_inter, thick_inter, nu_inter, npts_inter):
    """
    Returns shape profile with x=radius, y=SLD.
    """

    z = []
    beta = []
    z0 = 0
    # two sld points for core
    z.append(0)
    beta.append(sld_core)
    z.append(radius_core)
    beta.append(sld_core)
    z0 += radius_core

    for i in range(1, n_shells+2):
        dz = thick_inter[i-1]/npts_inter
        # j=0 for interface, j=1 for flat layer
        for j in range(0, 2):
            # interation for sub-layers
            for n_s in range(0, npts_inter+1):
                if j == 1:
                    if i == n_shells+1:
                        break
                    # shift half sub thickness for the first point
                    z0 -= dz#/2.0
                    z.append(z0)
                    #z0 -= dz/2.0
                    z0 += thick_flat[i]
                    sld_i = sld_flat[i]
                    beta.append(sld_flat[i])
                    dz = 0
                else:
                    nu = nu_inter[i-1]
                    # decide which sld is which, sld_r or sld_l
                    if i == 1:
                        sld_l = sld_core
                    else:
                        sld_l = sld_flat[i-1]
                    if i == n_shells+1:
                        sld_r = sld_solvent
                    else:
                        sld_r = sld_flat[i]
                    # get function type
                    func_idx = func_inter[i-1]
                    # calculate the sld
                    sld_i = intersldfunc(func_idx, npts_inter, n_s, nu,
                                            sld_l, sld_r)
                # append to the list
                z.append(z0)
                beta.append(sld_i)
                z0 += dz
                if j == 1:
                    break
    z.append(z0)
    beta.append(sld_solvent)
    z_ext = z0/5.0
    z.append(z0+z_ext)
    beta.append(sld_solvent)
    # return sld profile (r, beta)
    return np.asarray(z), np.asarray(beta)*1e-6

def ER(n_shells, radius_core, thick_inter0, thick_inter, thick_flat):
    total_thickness = thick_inter0
    total_thickness += np.sum(thick_inter[:n_shells], axis=0)
    total_thickness += np.sum(thick_flat[:n_shells], axis=0)
    return total_thickness + radius_core


demo = {
    "n_shells": 4,
    "npts_inter": 35.0,
    "radius_core": 50.0,
    "sld_core": 2.07,
    "sld_solvent": 1.0,
    "thick_inter0": 50.0,
    "func_inter0": 0,
    "nu_inter0": 2.5,
    "sld_flat":[4.0,3.5,4.0,3.5],
    "thick_flat":[100.0,100.0,100.0,100.0],
    "func_inter":[0,0,0,0],
    "thick_inter":[50.0,50.0,50.0,50.0],
    "nu_inter":[2.5,2.5,2.5,2.5],
    }

#TODO: Not working yet
tests = [
    # Accuracy tests based on content in test/utest_extra_models.py
    [{"n_shells":4,
        'npts_inter':35,
        "radius_core":50.0,
        "sld_core":2.07,
        "sld_solvent": 1.0,
        "sld_flat":[4.0,3.5,4.0,3.5],
        "thick_flat":[100.0,100.0,100.0,100.0],
        "func_inter":[0,0,0,0],
        "thick_inter":[50.0,50.0,50.0,50.0],
        "nu_inter":[2.5,2.5,2.5,2.5]
    }, 0.001, 0.001],
]