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Bipyramid4

A compound of two truncated pyramids with a common square base and opposite orientations.

Constructor

Bipyramid4(L, H, r_H, alpha)

Parameters:

  • L, edge length of the shared quadratic base
  • H, height of the bottom pyramid
  • r_H, nonnegative height ratio $r_{H}$ between the top and bottom pyramid
  • alpha, dihedral angle $\alpha$ between the base and a side face.

Constraints:

$$r_H \geq 0, \qquad 0 < \alpha \leq 90^\circ,$$

$$H \leq \dfrac{\tan\alpha}{2} L, \qquad r_{H} H \leq \dfrac{\tan\alpha}{2} L.$$

Usage

As for any other Form factor.

Implementation

Class Bipyramid4 inherits from the interface class Formfactor.

Form factor computation is based on the generic form factor of a polyhedron provided by libformfactor .

Volume has been validated against $$V=\dfrac{L^3\tan\alpha}{6}\Big[2-\Big(1-\dfrac{2H}{L\tan\alpha}\Big)^3-\Big(1-\dfrac{2r_{H}H}{L\tan\alpha}\Big)^3\Big].$$

More special:

Example

Scattering by uncorrelated, oriented bipyramids for horizontal incidence. Rotation around $z$ axis:

Generated by Examples/formfactor/sas-ff.py (particle_geometry=Bipyramid4).

History

Was incorrectly named “Cuboctahedron” up to BornAgain 1.19.

Agrees with “Cubooctaedron” form factor of IsGISAXS Lazzari, IsGISAXS manual v2.6 (2006), Eq. 2.33; Renaud et al, Surf Sci Rep 64, 255 (2009), Eq. 218, except for different parametrization $L = 2R_{IsGISXAXS}$. Since BornAgain-1.6 implemented using the generic polyhedron form factor Wuttke, J Appl Cryst 54, 580 (2021).