Form and Structure Factors - Molass Library Essence

General Note¶

References:

Simultaneous estimation of the form factor and structure factor for globular particles in small-angle scattering
Computational Reverse-Engineering Analysis for Scattering Experiments for Form Factor and Structure Factor Determination (“P(q) and S(q) CREASE”)

Definition for SEC-SAXS Solution Context¶

For the discussion in the following chapters, the definitions of form factor and structure factor are presented here. According to Gommes et al. (2021), these two factors are pedagogically distinguished as follows:

form factor — scattering by individual particles
structure factor — scattering by collections of particles

In simple terms, we can understand them pragmatically as:

$I(q) \sim P(q)$ when interpaticle effects are ignored (i.e., $S(q)=1$ in this case)
$I(q) \sim P(q)S(q)$ when interpaticle effects are considered.

Form Factor¶

For an individual particle, the form factor is defined as

I(q) = \rho^2V^2P(q) \qquad \qquad (1)

(1)

where

$I(q)$ : scattered intensity
$\rho$ : electron density
$V$ : particle volume
$P(q)$ : form factor (normalized so that P(q) = 1 for small q).

For a sample, ignoring interparticle effects, it can be written as

I(q) = c\rho^2V^2P(q) \qquad \qquad (2)

(2)

where

$c$ : particle concentration.

Structure Factor¶

When you consider the interparticle effects, they are expressed as

I(q) = c\rho^2V^2P(q)S(q) \qquad \qquad (3)

(3)

where

$S(q)$ : structure factor.

In other words, we can just think that the structure factor is a reprentaion of interparticle effects.

Form Factor Models¶

Following Pedersen (1997), we will give some known examples of form factor models.

Homogeneous Sphere¶

from molass import get_version
assert get_version() >= '0.5.0', "This tutorial requires molass version 0.5.0 or higher."
import numpy as np
import matplotlib.pyplot as plt
from molass.Shapes import Sphere
from molass.DensitySpace import VoxelSpace
from molass.SAXS.Models.Formfactors import homogeneous_sphere

q = np.linspace(0.005, 0.7, 100)
R = 30

I = homogeneous_sphere(q, R)

sphere = Sphere(radius=10)
space = VoxelSpace(64, sphere)

fig = plt.figure(figsize=(12, 5))
fig.suptitle('Form factor of a Homogeneous Sphere')
ax1 = fig.add_subplot(121, projection='3d')
ax1.set_title('Electron density')
space.plot_as_dots(ax1)

ax2 = fig.add_subplot(122)
ax2.set_title('SAXS intensity')
ax2.set_yscale('log')
ax2.plot(q, I, label='Analitical formula')
ax2.legend()
ax2.set_xlabel('q (1/Å)')

Tri-axial Ellipsoid¶

from molass.Shapes import Ellipsoid
from molass.DensitySpace import VoxelSpace
from molass.SAXS.Models.Formfactors import tri_axial_ellipsoid
I = tri_axial_ellipsoid(q, 30, 20, 10)

ellipsoid = Ellipsoid(30, 20, 10)
space = VoxelSpace(64, ellipsoid)

fig = plt.figure(figsize=(12, 5))
fig.suptitle('Form factor of a Tri-axial Ellipsoid')
ax1 = fig.add_subplot(121, projection='3d')
ax1.set_title('Electron density')
space.plot_as_dots(ax1)

ax2 = fig.add_subplot(122)
ax2.set_title('SAXS intensity')
ax2.set_yscale('log')
ax2.plot(q, I, label='Analitical formula')
ax2.legend()
ax2.set_xlabel('q (1/Å)')

Structure Factor Models¶

Following Kinning & Thomas (1984), we will give some known examples of structure factor models.

from molass import get_version
assert get_version() >= '0.5.0', "This tutorial requires molass version 0.5.0 or higher."
from molass.SAXS.Theory.DjKinning1984 import demo1
demo1()

References¶

Heil, C. M., Ma, Y., Bharti, B., & Jayaraman, A. (2023). Computational Reverse-Engineering Analysis for Scattering Experiments for Form Factor and Structure Factor Determination (“P(q) and S(q) CREASE”). JACS Au, 3(3), 889–904. 10.1021/jacsau.2c00697
Gommes, C. J., Jaksch, S., & Frielinghaus, H. (2021). Small-angle scattering for beginners. Journal of Applied Crystallography, 54(6), 1832–1843. 10.1107/S1600576721010293
Pedersen, J. S. (1997). Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting. Advances in Colloid and Interface Science, 70, 171–210. https://doi.org/10.1016/S0001-8686(97)00312-6
Kinning, D. J., & Thomas, E. L. (1984). Hard-sphere interactions between spherical domains in diblock copolymers. Macromolecules, 17(9), 1712–1718.