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SphericalSegment

A spherical segment, obtained from a spherical ball by two parallel cuts.

Constructor

SphericalSegment(R, t, b)

Parameters:

  • $ R $, radius
  • $ t $, cut from top
  • $ b $, cut from bottom

Constraint:

$ t+b \le 2R $

Usage

As for any other Form factor.

Implementation

Class SphericalSegment inherits from the interface class Formfactor.

Computation involves numerical integration in vertical direction,

$$ F(\mathbf{q})=2\pi \exp[iq_z(R-b)] \int_{-R+b}^{R-t} \text{d}z \space r_z^2 \frac{J_1(q_{||}r_z)}{q_{||}r_z} \exp(iq_z z), $$

with the notation

$$ q_{||} := \sqrt{q_x^2 + q_y^2}, \quad r_z:=\sqrt{R^2-z^2} $$

Volume has been validated against $$ V=\dfrac{\pi}{3} [ 4R^3 - 3R(t^2+b^2) + (t^3 + b^3) ]. $$

More general:

More special:

Example

Scattering by uncorrelated, oriented truncated spheres for horizontal incidence. Rotation around $y$ axis:

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

History

Agrees with the IsGISAXS form factor “Sphere” Lazzari, IsGISAXS manual v2.6 (2006), Eq. 2.32 and “Truncated sphere” Renaud et al, Surf Sci Rep 64, 255 (2009), Eq. 228. It is not “Truncated sphere” of FitGISAXS, which is without top removal Babonneau, FitGISAXS v130531 (2013).