Form Factor:Sphere

Jump to: navigation, search

This page provides the equations for calculating the form factor of a sphere (including derivations).


For spheres of radius R (volume ):

Form Factor Amplitude

Isotropic Form Factor Intensity

Sphere form factor.png



From NCNR SANS Models documentation:

  • Parameters:
    1.  : Intensity scaling
    2.  : sphere radius (Å)
    3.  : scattering contrast (Å−2),
    4.  : incoherent background (cm−1)


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:
    1.  : sphere radius (Å)


From IsGISAXS, Born form factors:

  • Parameters:
    1.  : sphere radius (Å)


    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


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:


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: