- Categories
- Shape-Independent
- Guinier
- guinier.py
Guinier - guinier.py
r"""
Definition
----------
This model fits the Guinier function
.. math::
I(q) = ext{scale} cdot exp{left[ frac{-Q^2 R_g^2 }{3}
ight]}
+ ext{background}
to the data directly without any need for linearisation (*cf*. the usual
plot of $ln I(q)$ vs $q^2$ ). Note that you may have to restrict the data
range to include small q only, where the Guinier approximation actually
applies. See also the guinier_porod model.
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}
In scattering, the radius of gyration $R_g$ quantifies the objects's
distribution of SLD (not mass density, as in mechanics) from the objects's
SLD centre of mass. It is defined by
.. math:: R_g^2 = frac{sum_i
ho_ileft(r_i-r_0
ight)^2}{sum_i
ho_i}
where $r_0$ denotes the object's SLD centre of mass and $
ho_i$ is the SLD at
a point $i$.
Notice that $R_g^2$ may be negative (since SLD can be negative), which happens
when a form factor $P(Q)$ is increasing with $Q$ rather than decreasing. This
can occur for core/shell particles, hollow particles, or for composite
particles with domains of different SLDs in a solvent with an SLD close to the
average match point. (Alternatively, this might be regarded as there being an
internal inter-domain "structure factor" within a single particle which gives
rise to a peak in the scattering).
To specify a negative value of $R_g^2$ in SasView, simply give $R_g$ a negative
value ($R_g^2$ will be evaluated as $R_g |R_g|$). Note that the physical radius
of gyration, of the exterior of the particle, will still be large and positive.
It is only the apparent size from the small $Q$ data that will give a small or
negative value of $R_g^2$.
References
----------
.. [#] A Guinier and G Fournet, *Small-Angle Scattering of X-Rays*, John Wiley & Sons, New York (1955)
Authorship and Verification
----------------------------
* **Author:**
* **Last Modified by:**
* **Last Reviewed by:**
"""
import numpy as np
from numpy import inf
name = "guinier"
title = ""
description = """
I(q) = scale.exp ( - rg^2 q^2 / 3.0 )
List of default parameters:
scale = scale
rg = Radius of gyration
"""
category = "shape-independent"
# ["name", "units", default, [lower, upper], "type","description"],
parameters = [["rg", "Ang", 60.0, [-inf, inf], "", "Radius of Gyration"]]
Iq = """
double exponent = fabs(rg)*rg*q*q/3.0;
double value = exp(-exponent);
return value;
"""
def random():
"""Return a random parameter set for the model."""
scale = 10**np.random.uniform(1, 4)
# Note: compare.py has Rg cutoff of 1e-30 at q=1 for guinier, so use that
# log_10 Ae^(-(q Rg)^2/3) = log_10(A) - (q Rg)^2/ (3 ln 10) > -30
# => log_10(A) > Rg^2/(3 ln 10) - 30
q_max = 1.0
rg_max = np.sqrt(90*np.log(10) + 3*np.log(scale))/q_max
rg = 10**np.random.uniform(0, np.log10(rg_max))
pars = dict(
#background=0,
scale=scale,
rg=rg,
)
return pars
# parameters for demo
demo = dict(scale=1.0, background=0.001, rg=60.0)
# parameters for unit tests
tests = [[{'rg' : 31.5}, 0.005, 0.992756]]
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