truncated_octahedron

Truncated Octahedron.

Parameter

Description

Units

Default value

scale

Scale factor or Volume fraction

None

1

background

Source background

cm-1

0.001

sld

Octahedron scattering length density

10-6Å-2

126

sld_solvent

Solvent scattering length density

10-6Å-2

9.4

radius_a

half height along a axis

Å

400

b2a_ratio

Ratio b/a

None

1

c2a_ratio

Ratio c/a

None

1

truncation

truncation ratio, 0 for octahedron and 0.5 for cuboctahedron

None

0

theta

c axis to beam angle

degree

0

phi

rotation about beam

degree

0

psi

rotation about c axis

degree

0

The returned value is scaled to units of cm-1 sr-1, absolute scale.

This model provides the form factor P(q) for a general octahedron. It can be a regular octahedron, with all edges of equal length. Or a general shape with different elongations along the three perpendicular two-fold axes. It includes the possibility to add an adjustable square truncation at each of the six vertices. This model includes the general cuboctahedron shape for the maximum value of truncation. The form factor expression is obtained by analytical integration over the volume of the shape. This model is constructed in a similar way as the rectangular prism model. It contains both the form factor for a reference orientation and the 1D form factor after orientation average (Gauss-Legendre).

Definition

This model computes the form factor of a general octahedron by defining its size through its circumradius \(radius_a\) (parameter called radius_a in the model), the elongations through the ratios \(\frac{b}{a}\) and \(\frac{c}{a}\) (parameters called b2a_ratio and c2a_ratio in the model) and the truncation level through the truncation ratio t (parameter called truncation in the model).

Indeed, the general octahedron is defined by its dimensions along its three perpendicular two-fold axes along x, y and z directions. \(radius_a\) (parameter called radius_a in the model), \(radius_b\) and \(radius_c\) are the distances from the center of the general octahedron to its 6 vertices, which are equivalent to the circumradiuses of the general octahedron along the three directions.

Coordinates of the six vertices are:

\[\begin{split}(radius_a,\ 0,\ 0) \\ (-radius_a,\ 0,\ 0) \\ (0,\ radius_b,\ 0) \\ (0,\ -radius_b,\ 0) \\ (0,\ 0,\ radius_c) \\ (0,\ 0,\ -radius_c)\end{split}\]

Truncation adds a square facet for each vertex that is perpendicular to a 2-fold axis. The resulting shape consists of six squares and eight hexagons, which may be irregular depending on the three dimensions. The truncation ratio t (parameter called truncation in the model) is defined as: 0 ≤ t ≤ 0.5, 0 corresponding to no truncation (full octahedron) and 0.5 corresponding to the maximum truncation (cuboctahedron). For the following formulas, we will use the notation \(t_inv = 1 - t\). Indeed, a square facet crosses the x, y, z directions at distances equal to \(t_{\mathrm{inv}} \, radius_a\), \(t_{\mathrm{inv}} \, radius_b\) and \(t_{\mathrm{inv}} \, radius_c\).

A regular octahedron corresponds to:

\[radius_a = radius_b = radius_c, \quad t = 0\]

A regular cuboctahedron shape with 6 squares and 8 triangles corresponds to:

\[radius_a = radius_b = radius_c, \quad t = \frac{1}{2}\]

The volume of the general shape including truncation is given by:

\[V = \frac{4}{3}\, radius_{\text{a}}^{3}\, \frac{b}{a}\, \frac{c}{a}\,\bigl(1 - 3t^{3}\bigr)\]

The general octahedron is made of eight triangular faces. The three edge lengths are:

\[A_{\text{edge}}^{2} = radius_{\text{a}}^{2} + radius_{\text{b}}^{2},\qquad B_{\text{edge}}^{2} = radius_{\text{a}}^{2} + radius_{\text{c}}^{2},\qquad C_{\text{edge}}^{2} = radius_{\text{b}}^{2} + radius_{\text{c}}^{2}\]

For a regular shape (no elongation):

\[\frac{b}{a} = \frac{c}{a} = 1\]
\[A_{\text{edge}} = B_{\text{edge}} = C_{\text{edge}} = radius_{\text{a}} \sqrt{2}\]
\[radius_{\text{a}} = radius_{\text{b}} = radius_{\text{c}} = A_{\text{edge}} / \sqrt{2}\]
\[V = \frac{4}{3} \, radius_{\text{a}}^{3} \, \bigl(1 - 3 t^3 \bigr)\]

The reference orientation of the shape is: a along x, b along y and c along z. Amplitude of the form factor AP for the reference orientation of the shape reads

\[AP(q,\theta,\phi) = \frac{6}{1 - 3t^3}\,(AA + BB + CC)\]
\[\begin{split}AA = \frac{1}{2\,(q_y^2 - q_z^2)\,(q_y^2 - q_x^2)}\Big[(q_y - q_x)\sin\big(q_y t - q_x t_{\text{inv}}\big) + (q_y + q_x)\sin\big(q_y t + q_x t_{\text{inv}}\big)\Big] \\ + \frac{1}{2\,(q_z^2 - q_x^2)\,(q_z^2 - q_y^2)}\Big[(q_z - q_x)\sin\big(q_z t - q_x t_{\text{inv}}\big) + (q_z + q_x)\sin\big(q_z t + q_x t_{\text{inv}}\big)\Big]\end{split}\]
\[\begin{split}BB = \frac{1}{2\,(q_z^2 - q_x^2)\,(q_z^2 - q_y^2)}\Big[(q_z - q_y)\sin\big(q_z t - q_y t_{\text{inv}}\big) + (q_z + q_y)\sin\big(q_z t + q_y t_{\text{inv}}\big)\Big] \\ + \frac{1}{2\,(q_x^2 - q_y^2)\,(q_x^2 - q_z^2)}\Big[(q_x - q_y)\sin\big(q_x t - q_y t_{\text{inv}}\big) + (q_x + q_y)\sin\big(q_x t + q_y t_{\text{inv}}\big)\Big]\end{split}\]
\[\begin{split}CC = \frac{1}{2\,(q_x^2 - q_y^2)\,(q_x^2 - q_z^2)}\Big[(q_x - q_z)\sin\big(q_x t - q_z t_{\text{inv}}\big) + (q_x + q_z)\sin\big(q_x t + q_z t_{\text{inv}}\big)\Big] \\ + \frac{1}{2\,(q_y^2 - q_z^2)\,(q_y^2 - q_x^2)}\Big[(q_y - q_z)\sin\big(q_y t - q_z t_{\text{inv}}\big) + (q_y + q_z)\sin\big(q_y t + q_z t_{\text{inv}}\big)\Big]\end{split}\]

Capital Qx Qy Qz are the three components in [A-1] of the scattering vector. qx qy qz are rescaled components (no unit) for computing AA, BB and CC terms.

\[Q_x = q\,\sin\theta\,\cos\phi, \qquad Q_y = q\,\sin\theta\,\sin\phi, \qquad Q_z = q\,\cos\theta\]
\[q_x = Q_x \, radius_{\text{a}},\qquad q_y = Q_y \, radius_{\text{b}},\qquad q_z = Q_z \, radius_{\text{c}}\]

θ is the angle between the scattering vector and the z axis. ϕ is the rotation angle in the xy plane.

The octahedron is in its reference orientation, with the c-axis aligned along z and the a-axis aligned along x.

The 1D form factor P(q) corresponds to the orientation average with all the possible orientations having the same probability. Instead of rotating the shape through all the possible orientations in the integral, it is equivalent to integrate the 3D scattering vector over a sphere of radius q with the shape in its reference orientation.

\[P(q) = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \, \int_0^{\frac{\pi}{2}} A_P^2(q) \, \sin\theta \, d\theta \, d\phi\]

And the 1D scattering intensity is calculated as

\[I(q) = \text{scale} \times V \times (\rho_\text{p} - \rho_\text{solvent})^2 \times P(q)\]

where V is the volume of the truncated octahedron, ρ is the scattering length inside the volume, ρ solvent is the scattering length of the solvent, and (if the data are in absolute units) scale represents the volume fraction (which is unitless).

../_images/octa-truncated.png

Fig. 81 Figure 1: Truncated octahedron shape for different truncation.

../_images/octahedrons_intensity_plot.png

Fig. 82 Figure 2: Scattering intensity of a cuboctahedron (t=0.5) and a regular octahedron (t=0) of radius_a = 400 Å.

Validation

Validation of the code is made using numerical checks. Comparisons with Debye formula calculations were made using DebyeCalculator library (https://github.com/FrederikLizakJohansen/DebyeCalculator). Good agreement was found at q < 0.1 1/Å.

../_images/truncated_octahedron_autogenfig.png

Fig. 83 Figure 3: 1D and 2D plots corresponding to the default parameters of the model.

Source

truncated_octahedron.py \(\ \star\ \) truncated_octahedron.c \(\ \star\ \) adaptive.c

References

  1. Li, X., Shew, C., He, L., Meilleur, F., Myles, D. A. A., Liu, E., Zhang, Y., Smith, G. S., Herwig, K. W., Pynn, R., & Chen, W. (2011). Scattering functions of Platonic solids. Journal Of Applied Crystallography, 44(3), 545‑557. https://doi.org/10.1107/s0021889811011691

  2. Croset, B. (2017). Form factor of any polyhedron : a general compact formula and its singularities. Journal Of Applied Crystallography, 50(5), 1245‑1255. https://doi.org/10.1107/s1600576717010147

  3. Wuttke, J. (2021). Numerically stable form factor of any polygon and polyhedron. Journal Of Applied Crystallography, 54(2), 580‑587. https://doi.org/10.1107/s1600576721001710

Authorship and Verification