prism

prism of different cross-sections

Parameter

Description

Units

Default value

scale

Scale factor or Volume fraction

None

1

background

Source background

cm-1

0.001

sld

Prism scattering length density

10-6Å-2

126

sld_solvent

Solvent scattering length density

10-6Å-2

9.4

n_sides

Number of sides

None

5

radius_average

Average radius

Å

500

length

Length

Å

5000

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

This model provides the form factor for a right prism which cross-section is a regular polygon. 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

We consider particles having the shape of a right prism of length L and a cross section made of a regular polygon with n sides as illustrated in the figure below.

../../_images/nanoprisms_geometry.jpg

The size of a regular polygon can be characterized by its edge length E or by R, the circumradius of the polygon shown in black in the figure, with :

\[R = \frac{E}{2\sin(\pi/n)}\]

The apothem is the radius of the dashed circle, it is defined as:

\[R\cos(\pi/n)\]

For comparison purposes, it is convenient to introduce an average radius R_ave (shown in red in the figure). The area of the n-sided regular polygon is given by :

\[A = \pi R^{2} \, \mathrm{sinc}\left(\frac{2\pi}{n}\right) = \pi R_{\mathrm{ave}}^{2}\]

and the volume of the nanoprism is therefore given by:

\[V = L \times {A}\]

where R_ave is the radius of the equivalent disc having the same area as the n-sided polygon. It is also the squared average of the distance from the center of the polygon to any point of its perimeter. It is related to the circumradius R by :

\[R_{\mathrm{ave}}^{2} = R^{2} \, \mathrm{sinc}\left(\frac{2\pi}{n}\right)\]

Form factor for a prism: Following Wuttke’s expression, the form factor \(F(\mathbf{q})\) for any right prism can be decomposed into the product of two factors. One factor corresponds to the component \(\mathbf{q}_{\perp}\) of the scattering vector that is perpendicular to the cross section and depends only on the length \(L\). The other factor is coplanar with the cross section and involves the component \(\mathbf{q}_{\|}\); it depends on the number of sides \(n\) and the edge length \(E\) of the polygon. The perpendicular factor is:

\[f_{\perp}(\mathbf{q}_{\perp}, L) = L \mathrm{sinc}( \frac{(\mathbf{q}_{\perp} \cdot \hat{\mathbf{n}})\, L}{2} )\]

where \(\hat{\mathbf{n}}\) is the direction normal to the cross section. The length \(L\) gives rise to a standard sinc function for the form factor.

On the other hand, the parallel factor for a regular \(n\)-sided polygon of circumradius \(R\) can be expressed as

\[f_{\|}(\mathbf{q}_{\|}, n, R) = \frac{2}{i q_{\|}^{2}} \sum_{j=1}^{n} \hat{\mathbf{n}} \cdot \left( \mathbf{q}_{\|} \times \mathbf{E}_{j} \right) \, \mathrm{sinc}\left( \mathbf{q}_{\|} \cdot \mathbf{E}_{j} \right) \exp( i \mathbf{q}_{\|} \cdot \mathbf{M}_{j})\]

In the sum over all edges, \(\mathbf{M}_{j}\) is the vector joining the center of the polygon to the middle of the jth edge and \(\mathbf{E}_{j}\) is the half-edge vector.

The scattered intensity for one prism is given by:

\[I(\mathbf{q}, n, R, L) = \left| F(\mathbf{q}, n, R, L) \right|^{2} = \left| f_{\perp}(\mathbf{q}_{\perp}, L) f_{\|}(\mathbf{q}_{\|}, n, R) \right|^{2}\]

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

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

Fig. 76 Fibonacci sphere using N=5810 points.

Validation

The model has been tested against experimental data obtained on gold nanoprisms with pentagonal cross-section (see J. Marcone et al. JAC 2025). Moreover, comparisons with Debye formula calculations were made using DebyeCalculator library (https://github.com/FrederikLizakJohansen/DebyeCalculator). Good agreement was found at q < 0.1 1/Angstrom.

../../_images/prism_autogenfig.png

Fig. 77 1D plot corresponding to the default parameters of the model.

Source

prism.py

References

  1. Jules Marcone et al. “Form factor of prismatic particles for small-angle scattering analysis”. In: J. Appl. Cryst. (2025) 58, 543-552 DOI: https://doi.org/10.1107/S1600576725000676

  2. Wei-Ren Chen et al. “Scattering functions of Platonic solids”. In: Journal of Applied Crystallography - J. Appl. Cryst. (June 2011). DOI: https://doi.org/10.1107/S0021889811011691

  3. Croset, Bernard, “Form factor of any polyhedron: a general compact formula and its singularities” In: J. Appl. Cryst. (2017). 50, 1245–1255 DOI: https://doi.org/10.1107/S1600576717010147

  4. Wuttke, J. Numerically stable form factor of any polygon and polyhedron J Appl Cryst 54, 580-587 (2021) DOI: https://doi.org/10.1107/S1600576721001710

Authorship and Verification