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Particle form factor

Theory

A particle shape is described through the density function n(r) in real space, or the form factor F(q)=∫d3reiqrn(r) in reciprocal space.

For a “hard” particle, n(r) is an indicator function with value 1 inside, 0 outside. For a “soft” particle, n(r) has a smooth transition from the inside limit of 1 to the outside limit of 0.

At large q, the exponential factor in the integral defining F(q) oscillates rapidly. This leads to substantial errors if q is of the same order or larger than the inverse detector bin length. In such cases, we recommend Monte-Carlo integration over detector bins.

By construction, F(q) is a holomorphic function. Therefore expressions previously derived for real wave vectors q also hold for complex q.

Implementation

The interface class IFormFactor cannot be instantiated directly. Instead, one has to choose one of the following child classes:

  • hard particles:
    • spheroids:
    • cylinders and cones:
    • polygonal prisms:
    • pyramids (frusta):
      • rectangular
      • trigonal
      • quadratic: Pyramid4, Bipyramid4
      • hexagonal
    • other polyhedra:
  • soft particles:
    • spherical:

Using Python scripting, it is also possible to define a custom form factor.

Usage

Form factors are used to define the shape of particles:

ff = ba.Cylinder(5*nm, 5*nm)
particle = ba.Particle(material_Particle, ff)

Literature

Form factor catalogues in earlier software:

Stable algorithm for computing form factors of polyhedra: Wuttke 2021.