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Pyramid6

A truncated pyramid (frustum) based on a regular hexagon.

Constructor

Pyramid6(R, H, alpha)

Parameters:

  • R, edge of the regular hexagonal base
  • H, height
  • alpha, dihedral angle $\alpha$ between the base and a side face. For $\alpha>90^\circ$, the upper face is larger than the base.

The angle must satisfy

$$0 < \alpha < 180^\circ.$$

For $\alpha<90^\circ$, the upper edge must not become negative, so the height must satisfy

$$H \le \dfrac{\sqrt3\tan\alpha}{2}R.$$

For $\alpha\ge90^\circ$, there is no additional height constraint.

Note that the orthographic projection does not show $\alpha$ but the angle $\beta$ between base and a side edge. They are related through $\sqrt3\tan\alpha = 2\tan\beta$.

Usage

As for any other Form factor.

Implementation

Class Pyramid6 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 $$ V=\dfrac{3}{4}R^3\tan\alpha\Big[1-\Big(1-\dfrac{2H}{\sqrt3R\tan\alpha}\Big)^3\Big]. $$

More special:

  • Prism6, if $\alpha=90^{\circ}$.

Example

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

Generated by Examples/sas/sas-ff.py (particle_geometry=Pyramid6).

History

Was named “Cone6” until BornAgain 1.19.

Corresponds to “Cone6” in IsGISAXS Lazzari, IsGISAXS manual v2.6 (2006), Eq. 2.31 and “Cone with six fold symmetry” Renaud et al, Surf Sci Rep 64, 255 (2009), Eq. 222. In BornAgain 1.6, redefined to let the $x$ axis lie in a mirror plane (rotated by $30^{\circ}$ with respect to the previous version).

Up to BornAgain 1.5 by numerical integration, as in IsGISAXS. Since BornAgain 1.6 higher speed and better accuracy are achieved by using the generic form factor of a polyhedron Wuttke, J Appl Cryst 54, 580 (2021) with series expansions near singularities.