truncated_tetrahedron

Truncated tetrahedron

Parameter

Description

Units

Default value

scale

Scale factor or Volume fraction

None

1

background

Source background

cm-1

0.001

sld

tetrahedron scattering length density

10-6Å-2

126

sld_solvent

Solvent scattering length density

10-6Å-2

9.4

radius

Circumradius of the full tetrahedron

Å

100

truncation

truncation, 0 for full tetrahedron, 0.5 for octahedron

None

0

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

This model provides the form factor for truncated tetrahedrons. Orientation averaging is done by using the Fibonacci quadrature. This quadrature provides a quasi-uniform distribution of points on the unit sphere using the golden ratio. The number of points to generate on the unit sphere is set to 500 points, it usually provides a good balance between accuracy and computational efficiency.

Definition

This model allows calculating the form factor of a truncated tetrahedron by defining its initial size through its circumradius \(R\) (parameter called radius in the model) and its truncation rate \(t\) (parameter called truncation in the model) . To define the truncated tetrahedron form factor, the regular tetrahedron form factor has to be defined first. The truncated tetrahedron form factor will then be obtained by subtracting the four smaller tetrahedrons at its vertices. So first, let’s consider a regular tetrahedron. The size of the tetrahedron is described by its circumradius \(R\) , which is the radius of the circumscribed sphere. The relationship between the circumradius and the edge length is also implemented. The edge length \(L\) and volume \(V_T\) are given by:

\[L = \frac{4}{\sqrt{6}} \, R\]
\[V_{T} = \frac{\sqrt{2}}{12} \, L^3\]
../../_images/tetra_truncation.png

Fig. 84 Regular tetrahedron with circumradius \(R\) and edge length \(L\). After truncation, the true edge length is \((1-2t)L\).

The four vertices of the tetrahedron, with \(\mathbf{v}_0\) at the origin, are defined as:

\[\mathbf{v}_0 = (0,\ 0,\ 0)\]
\[\mathbf{v}_1 = (\frac{L}{\sqrt{2}},\ \frac{L}{\sqrt{2}},\ 0)\]
\[\mathbf{v}_2 = (0,\ \frac{L}{\sqrt{2}},\frac{L}{\sqrt{2}})\]
\[\mathbf{v}_3 = (\frac{L}{\sqrt{2}},\ 0,\frac{L}{\sqrt{2}})\]

The form factor amplitude of a regular tetrahedron is derived using the projection method described by Yang and al. [4]. The tetrahedron is defined by four vertices \((\mathbf{v}_0, \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3)\), where \(\mathbf{v}_0\) is placed at the origin and serves as the projection origin. The three remaining vertices are specified such that the normal vector of each face points away from the tetrahedron. The form factor amplitude is defined as the Fourier transform of the particle shape:

\[F(\mathbf{q}) = \int_V \exp(i\mathbf{q} \cdot \mathbf{r}) \, d\mathbf{r}\]

The transformation matrix \(\mathbf{T}\) is built from the three edge vectors originating at \(\mathbf{v}_0\), with each vector forming one column:

\[\mathbf{T} = \bigl[\, \mathbf{v}_1 \mid \mathbf{v}_2 \mid \mathbf{v}_3 \,\bigr]\]

where \((x_i, y_i, z_i)\) are the coordinates of vertex \(\mathbf{v}_i\). Introducing the shorthand notation \(Q_i = \mathbf{q} \cdot \mathbf{v}_i\), the general tetrahedral form factor is given by:

\[\begin{split}F_\mathrm{T}(\mathbf{q}) = |\det(\mathbf{T})| \left\{ \frac{i}{Q_1 (Q_1 - Q_2)(Q_1 - Q_3)} \exp(i Q_1) + \frac{i}{Q_2 (Q_2 - Q_1)(Q_2 - Q_3)} \exp(i Q_2) \\ + \frac{i}{Q_3 (Q_3 - Q_2)(Q_3 - Q_1)} \exp(i Q_3) - \frac{i}{Q_1 Q_2 Q_3} \right\}\end{split}\]

The truncated tetrahedron is a polyhedron obtained by truncating the vertices of a regular tetrahedron.

../../_images/truncated_tetrahedron.png

Fig. 85 Tetrahedron with different truncatures.

The volume of the truncated tetrahedron is given by:

\[V_{T_{truncated}} = V_{T} (1 - 4t^3)\]

\(t\) represents the truncation level. At the maximum value \(t=1/2\), the corresponding shape is a regular octahedron. At the minimum value \(t=0\), the shape is a regular tetrahedron. At \(t=1/3\), the corresponding shape is the Friauf polyhedron, in which all edges are equal. The truncated tetrahedron form factor is also obtained by subtracting the four smaller tetrahedrons at its vertices. It is expressed as:

\[F_{T_{\text{truncated}}} (\mathbf{q},t, \mathbf{v_1},\mathbf{v_2},\mathbf{v_3}) = F_{T} (\mathbf{q},\mathbf{v_1},\mathbf{v_2},\mathbf{v_3}) - F_{T} (\mathbf{q},t\mathbf{v_1},t\mathbf{v_2},t\mathbf{v_3}) \left(1 + e^{i(1-t)\mathbf{q} \cdot \mathbf{v_1}} + e^{i(1-t)\mathbf{q} \cdot \mathbf{v_2}} + e^{i(1-t)\mathbf{q} \cdot \mathbf{v_3}}\right)\]

where the phase terms correspond to four translations inside the shape.

Singularities: the expression of the regular tetreahedron form factor presents numerical singularities in four distinct cases, each handled analytically.

Case 1 — \(\mathbf{q}\) perpendicular to vertex \(\mathbf{v}_i\) (\(Q_i = 0\), with \(Q_j, Q_k \neq 0\) and \(Q_j \neq Q_k\)):

\[F_P(\mathbf{q}) = |\det(\mathbf{T})| \left\{ \frac{i}{Q_j^2 (Q_j - Q_k)} \exp(i Q_j) + \frac{i}{Q_k^2 (Q_k - Q_j)} \exp(i Q_k) + \frac{i(Q_j + Q_k + i Q_j Q_k)}{Q_j^2 \, Q_k^2} \right\}\]

Case 2 — \(\mathbf{q}\) perpendicular to edge \(\mathbf{v}_i \mathbf{v}_j\) (\(Q_i = Q_j\), with \(Q_k \neq Q_i\)):

\[F_E(\mathbf{q}) = |\det(\mathbf{T})| \left\{ \frac{i}{Q_k (Q_k - Q_i)^2} \exp(i Q_k) + \frac{i Q_k - 2i Q_i - Q_i^2 + Q_i Q_k}{Q_i^2 (Q_k - Q_i)^2} \exp(i Q_i) - \frac{i}{Q_i^2 \, Q_k} \right\}\]

Case 3 — \(\mathbf{q}\) perpendicular to a face (\(Q_i = Q_j = Q_k \neq 0\)):

\[F_F(\mathbf{q}) = |\det(\mathbf{T})| \left\{ \frac{-i}{Q_i^3} + \frac{2i + 2Q_i - i Q_i^2}{2 Q_i^3} \exp(i Q_i) \right\}\]

The formula can be expressed by three alternatives (i.e. \(Q_i = q · v_1\) or \(Q_i = q · v_2\) or \(Q_i = q · v_3\) ).

Case 4 — \(\mathbf{q} \to 0\) (first-order Taylor expansion):

\[F_V(\mathbf{q}) = V_\mathrm{T} + |\det(\mathbf{T})| \, \frac{i(Q_1 + Q_2 + Q_3)}{24}\]

where the tetrahedral volume is \(V_\mathrm{T} = |\det(\mathbf{T})| / 6\).

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

The sphere is sampled using Fibonacci quadrature to provide a quasi-uniform distribution of points on the unit sphere. The distribution of the N points is computed using the golden ratio (see fibonacci.py). Each point of the quadrature on the unit sphere corresponds to a vector \(\mathbf{u}_{j}\). In the sum, all weights \(w_j\) are taken identical and equal to \(\frac{1}{N}\).

\[P(q) = \sum_{j=1}^{N} w_j I(q\mathbf{u}_{j}, R, t)\]
../../_images/fibonacci_sphere.png

Fig. 86 Fibonacci sphere using N=5810 points.

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 tetrahedron, 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).

Validation

The model has been tested against experimental data obtained on gold tetrahedrons at synchrotron Soleil. Moreover, comparisons with Debye formula calculations were made using DebyeCalculator library (https://github.com/FrederikLizakJohansen/DebyeCalculator). Good agreement was found at q < 1 1/Å.

../../_images/truncated_tetrahedron_autogenfig.png

Fig. 87 1D plot corresponding to the default parameters of the model.

Source

truncated_tetrahedron.py

References

  1. Patterson, A. L. (1939). The Diffraction of X-Rays by Small Crystalline Particles. Physical Review, 56(10), 972‑977. https://doi.org/10.1103/physrev.56.972

  2. 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

  3. 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

  4. Yang, T., Chen, X., Zhang, J., Ma, J., & Liu, S. (2023). Form factor of any polyhedron and its singularities derived from a projection method. Journal of Applied Crystallography, 56(1), 167–177. https://doi.org/10.1107/s160057672201130x

Authorship and Verification