Form Factor:Sphere
This page provides the equations for calculating the form factor of a sphere (including derivations).
Contents
Equations
For spheres of radius R (volume ):
Form Factor Amplitude
Isotropic Form Factor Intensity
Sources
NCNR
From NCNR SANS Models documentation:
- Parameters:
- : Intensity scaling
- : sphere radius (Å)
- : scattering contrast (Å^{−2}),
- : incoherent background (cm^{−1})
Pedersen
From Pedersen review, Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting Jan Skov Pedersen, Advances in Colloid and Interface Science 1997, 70, 171. doi: 10.1016/S0001-8686(97)00312-6
- Parameters:
- : sphere radius (Å)
IsGISAXS
From IsGISAXS, Born form factors:
- Parameters:
- : sphere radius (Å)
Code
def sphere(self, q, r, scale=1.0, contrast=0.1, background=0.0): V = (4/3)*numpy.pi*(r**3) return (scale/V)*(( 3*V*contrast*(sin(q*r)-q*r*cos(q*r) )/( (q*r)**3 ) )**2) + background
Derivations
Form Factor
For a sphere of radius R, the volume is:
We can use a spherical coordinates, where denotes the angle with respect to the axis, and is the in-plane angle (i.e. with respect to the axis):
Where the form factor is:
We take advantage of spherical symmetry. E.g. we can rotate any q onto a particular axis, such as . So that:
And so:
A simple variable substitution:
Yields:
Using the fact that:
We integrate:
Form Factor at q=0
At very small q:
Isotropic Form Factor Intensity
To average over all possible orientations, we use:
For a sphere:
Note that the spherical symmetry guarantees that the integrand does not depend on or :
Isotropic Form Factor Intensity at q=0
At q=0, we expect:
Isotropic Form Factor Intensity at large q
Note that:
For large q, the term dominates the numerator:
The oscillation of the numerator is overwhelmed by the decay of the denominator: