- Categories
- Sphere
- Spherical Sld
- spherical_sld.py
Spherical Sld - spherical_sld.py
r"""
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.
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
"""
import numpy as np
from numpy import inf, expm1, sqrt
from scipy.special import erf
name = "spherical_sld"
title = "Spherical SLD intensity calculation"
description = """
I(q) =
background = Incoherent background [1/cm]
"""
category = "shape:sphere"
SHAPES = ["erf(|nu|*z)", "Rpow(z^|nu|)", "Lpow(z^|nu|)",
"Rexp(-|nu|z)", "Lexp(-|nu|z)", "Boucher((1-z^2)^(1/2*nu-2))",]
# pylint: disable=bad-whitespace, line-too-long
# ["name", "units", default, [lower, upper], "type", "description"],
parameters = [["n_shells", "", 1, [1, 10], "volume", "number of shells (must be integer)"],
["sld_solvent", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "solvent sld"],
["sld[n_shells]", "1e-6/Ang^2", 4.06, [-inf, inf], "sld", "sld of the shell"],
["thickness[n_shells]", "Ang", 100.0, [0, inf], "volume", "thickness shell"],
["interface[n_shells]", "Ang", 50.0, [0, inf], "volume", "thickness of the interface"],
["shape[n_shells]", "", 0, [SHAPES], "", "interface shape"],
["nu[n_shells]", "", 2.5, [1, inf], "", "interface shape exponent"],
["n_steps", "", 35, [0, inf], "", "number of steps in each interface (must be an odd integer)"],
]
# pylint: enable=bad-whitespace, line-too-long
source = ["lib/polevl.c", "lib/sas_erf.c", "lib/sas_3j1x_x.c", "spherical_sld.c"]
single = False # TODO: fix low q behaviour
have_Fq = True
radius_effective_modes = ["outer radius"]
profile_axes = ['Radius (A)', 'SLD (1e-6/A^2)']
SHAPE_FUNCTIONS = [
lambda z, nu: erf(nu/sqrt(2)*(2*z-1))/(2*erf(nu/sqrt(2))) + 0.5, # erf
lambda z, nu: z**nu, # Rpow
lambda z, nu: 1 - (1-z)**nu, # Lpow
lambda z, nu: expm1(-nu*z)/expm1(-nu), # Rexp
lambda z, nu: expm1(nu*z)/expm1(nu), # Lexp
lambda z, nu: 1 - (1 - z**2)**(0.5*nu-2.0), # Boucher
]
def profile(n_shells, sld_solvent, sld, thickness,
interface, shape, nu, n_steps):
"""
Returns shape profile with x=radius, y=SLD.
"""
n_shells = int(n_shells + 0.5)
n_steps = int(n_steps + 0.5)
z = []
rho = []
z_next = 0
# two sld points for core
z.append(z_next)
rho.append(sld[0])
for i in range(0, n_shells):
z_next += thickness[i]
z.append(z_next)
rho.append(sld[i])
dz = interface[i]/n_steps
sld_l = sld[i]
sld_r = sld[i+1] if i < n_shells-1 else sld_solvent
fun = SHAPE_FUNCTIONS[int(np.clip(shape[i], 0, len(SHAPE_FUNCTIONS)-1))]
for step in range(1, n_steps+1):
portion = fun(float(step)/n_steps, max(abs(nu[i]), 1e-14))
z_next += dz
z.append(z_next)
rho.append((sld_r - sld_l)*portion + sld_l)
z.append(z_next*1.2)
rho.append(sld_solvent)
# return sld profile (r, beta)
return np.asarray(z), np.asarray(rho)
# TODO: no random parameter generator for spherical SLD.
# Another interesting demo case, again because the default function is boring.
demo = {
"n_shells": 5,
"n_steps": 35.0,
"sld_solvent": 1.0,
"sld": [2.07, 4.0, 3.5, 4.0, 3.5],
"thickness": [50.0, 100.0, 100.0, 100.0, 100.0],
"interface": [50.0]*5,
"shape": [0]*5,
"nu": [2.5]*5,
}
tests = [
# Results checked against sasview 3.1
[{"n_shells": 5,
"n_steps": 35,
"sld_solvent": 1.0,
"sld": [2.07, 4.0, 3.5, 4.0, 3.5],
"thickness": [50.0, 100.0, 100.0, 100.0, 100.0],
"interface": [50]*5,
"shape": [0]*5,
"nu": [2.5]*5,
}, 0.001, 750697.238],
]
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