Hollow Rectangular Prism Thin Walls

# rectangular_prism model # Note: model title and parameter table are inserted automatically Definition

This model provides the form factor, $P(q)$, for a hollow rectangular prism with infinitely thin walls. It computes only the 1D scattering, not the 2D. The 1D scattering intensity for this model is calculated according to the equations given by Nayuk and Huber[#Nayuk2012]_.

Assuming a hollow parallelepiped with infinitely thin walls, edge lengths $A \le B \le C$ and presenting an orientation with respect to the scattering vector given by $\theta$ and $\phi$, where $\theta$ is the angle between the $z$ axis and the longest axis of the parallelepiped $C$, and $\phi$ is the angle between the scattering vector (lying in the $xy$ plane) and the $y$ axis, the form factor is given by

$$ P(q) = \frac{1}{V^2} \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}} [A_L(q)+A_T(q)]^2 \sin\theta\,d\theta\,d\phi
$$
where

$$ V = 2AB + 2AC + 2BC \\ A_L(q) = 8 \times \frac{ \sin \left( \tfrac{1}{2} q A \sin\phi \sin\theta \right) \sin \left( \tfrac{1}{2} q B \cos\phi \sin\theta \right) \cos \left( \tfrac{1}{2} q C \cos\theta \right) }{q^2 \, \sin^2\theta \, \sin\phi \cos\phi} \\ A_T(q) = A_F(q) \times \frac{2\,\sin \left( \tfrac{1}{2} q C \cos\theta \right)}{q\,\cos\theta}
$$
and

$$ A_F(q) = 4 \frac{ \cos \left( \tfrac{1}{2} q A \sin\phi \sin\theta \right) \sin \left( \tfrac{1}{2} q B \cos\phi \sin\theta \right) } {q \, \cos\phi \, \sin\theta} + 4 \frac{ \sin \left( \tfrac{1}{2} q A \sin\phi \sin\theta \right) \cos \left( \tfrac{1}{2} q B \cos\phi \sin\theta \right) } {q \, \sin\phi \, \sin\theta}
$$
The 1D scattering intensity is then calculated as

$$ I(q) = \text{scale} \times V \times (\rho_\text{p} - \rho_\text{solvent})^2 \times P(q)
$$
where $V$ is the surface area of the rectangular prism, $\rho_\text{p}$ is the scattering length density of the parallelepiped, $\rho_\text{solvent}$ is the scattering length density of the solvent, and (if the data are in absolute units) *scale* is related to the total surface area.

**The 2D scattering intensity is not computed by this model.**

Validation

Validation of the code was conducted by qualitatively comparing the output of the 1D model to the curves shown in (Nayuk, 2012[#Nayuk2012]_).

References

L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659
.. [#Nayuk2012] R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854

Authorship and Verification

**Author:** Miguel Gonzales **Date:** February 26, 2016
**Last Modified by:** Paul Kienzle **Date:** October 15, 2016
**Last Reviewed by:** Paul Butler **Date:** September 07, 2018