- 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($
u z$)
1: Rpow($z^
u$)
2: Lpow($z^
u$)
3: Rexp($-
u z$)
4: Lexp($-
u z$)
The form factor $P(q)$ in 1D is calculated by:
.. math::
P(q) = frac{f^2}{V_ ext{particle}} ext{ where }
f = f_ ext{core} + sum_{ ext{inter}_i=0}^N f_{ ext{inter}_i} +
sum_{ ext{flat}_i=0}^N f_{ ext{flat}_i} +f_ ext{solvent}
For a spherically symmetric particle with a particle density $
ho_x(r)$
the sld function can be defined as:
.. math::
f_x = 4 pi int_{0}^{infty}
ho_x(r) frac{sin(qr)} {qr^2} r^2 dr
so that individual terms can be calculated as follows:
.. math::
f_ ext{core} &= 4 pi int_{0}^{r_ ext{core}}
ho_ ext{core}
frac{sin(qr)} {qr} r^2 dr =
3
ho_ ext{core} V(r_ ext{core})
Big[ frac{sin(qr_ ext{core}) - qr_ ext{core} cos(qr_ ext{core})}
{qr_ ext{core}^3} Big] \
f_{ ext{inter}_i} &= 4 pi int_{Delta t_{ ext{inter}_i } }
ho_{ ext{inter}_i } frac{sin(qr)} {qr} r^2 dr \
f_{ ext{shell}_i} &= 4 pi int_{Delta t_{ ext{inter}_i } }
ho_{ ext{flat}_i } frac{sin(qr)} {qr} r^2 dr =
3
ho_{ ext{flat}_i } V ( r_{ ext{inter}_i } +
Delta t_{ ext{inter}_i } )
Big[ frac{sin(qr_{ ext{inter}_i} + Delta t_{ ext{inter}_i } )
- q (r_{ ext{inter}_i} + Delta t_{ ext{inter}_i })
cos(q( r_{ ext{inter}_i} + Delta t_{ ext{inter}_i } ) ) }
{q ( r_{ ext{inter}_i} + Delta t_{ ext{inter}_i } )^3 } Big]
-3
ho_{ ext{flat}_i } V(r_{ ext{inter}_i })
Big[ frac{sin(qr_{ ext{inter}_i}) - qr_{ ext{flat}_i}
cos(qr_{ ext{inter}_i}) } {qr_{ ext{inter}_i}^3} Big] \
f_ ext{solvent} &= 4 pi int_{r_N}^{infty}
ho_ ext{solvent}
frac{sin(qr)} {qr} r^2 dr =
3
ho_ ext{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 in $r$.
The SLD at the interface between shells, $
ho_{ ext {inter}_i}$
is calculated with a function chosen by an user, where the functions are
Exp:
.. math::
ho_{{inter}_i} (r) &= egin{cases}
B expBig( frac {pm A(r - r_{ ext{flat}_i})}
{Delta t_{ ext{inter}_i }} Big) +C & mbox{for } A
eq 0 \
B Big( frac {(r - r_{ ext{flat}_i})}
{Delta t_{ ext{inter}_i }} Big) +C & mbox{for } A = 0 \
end{cases}
Power-Law:
.. math::
ho_{{inter}_i} (r) &= egin{cases}
pm B Big( frac {(r - r_{ ext{flat}_i} )} {Delta t_{ ext{inter}_i }}
Big) ^A +C & mbox{for } A
eq 0 \
ho_{ ext{flat}_{i+1}} & mbox{for } A = 0 \
end{cases}
Erf:
.. math::
ho_{{inter}_i} (r) = egin{cases}
B ext{erf} Big( frac { A(r - r_{ ext{flat}_i})}
{sqrt{2} Delta t_{ ext{inter}_i }} Big) +C & mbox{for } A
eq 0 \
B Big( frac {(r - r_{ ext{flat}_i} )} {Delta t_{ ext{inter}_i }}
Big) +C & 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 $
ho_{ ext{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_{ ext{inter}_i} &= 4 pi int_{Delta t_{ ext{inter}_i } }
ho_{ ext{inter}_i } frac{sin(qr)} {qr} r^2 dr =
4 pi sum_{j=1}^{n_ ext{steps}}
int_{r_j}^{r_{j+1}}
ho_{ ext{inter}_i } (r_j)
frac{sin(qr)} {qr} r^2 dr \
approx 4 pi sum_{j=1}^{n_ ext{steps}} Big[
3 (
ho_{ ext{inter}_i } ( r_{j+1} ) -
ho_{ ext{inter}_i }
( r_{j} ) V (r_j)
Big[ frac {r_j^2 eta_ ext{out}^2 sin(eta_ ext{out})
- (eta_ ext{out}^2-2) cos(eta_ ext{out}) }
{eta_ ext{out}^4 } Big] \
{} - 3 (
ho_{ ext{inter}_i } ( r_{j+1} ) -
ho_{ ext{inter}_i }
( r_{j} ) V ( r_{j-1} )
Big[ frac {r_{j-1}^2 sin(eta_ ext{in})
- (eta_ ext{in}^2-2) cos(eta_ ext{in}) }
{eta_ ext{in}^4 } Big] \
{} + 3
ho_{ ext{inter}_i } ( r_{j+1} ) V ( r_j )
Big[ frac {sin(eta_ ext{out}) - cos(eta_ ext{out}) }
{eta_ ext{out}^4 } Big]
- 3
ho_{ ext{inter}_i } ( r_{j} ) V ( r_j )
Big[ frac {sin(eta_ ext{in}) - cos(eta_ ext{in}) }
{eta_ ext{in}^4 } Big]
Big]
where
.. math::
:nowrap:
egin{align*}
V(a) &= frac {4pi}{3}a^3 && \
a_ ext{in} sim frac{r_j}{r_{j+1} -r_j} ext{, } & a_ ext{out}
sim frac{r_{j+1}}{r_{j+1} -r_j} \
eta_ ext{in} &= qr_j ext{, } & eta_ ext{out} &= qr_{j+1}
end{align*}
We assume $
ho_{ ext{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_ ext{particle}} mbox{ where } V_ ext{particle}
= V(r_{ ext{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
----------
.. [#] L A Feigin and D I Svergun, Structure Analysis by Small-Angle X-Ray
and Neutron Scattering, Plenum Press, New York, (1987)
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)"]
# 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
]
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.
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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