Hollow Rectangular Prism

Description:

# 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 parallelepiped with a wall of thickness $\Delta$. The 1D scattering intensity for this model is calculated by forming the difference of the amplitudes of two massive parallelepipeds differing in their outermost dimensions in each direction by the same length increment $2\Delta$ ([#Nayuk2012]_ Nayuk, 2012).

As in the case of the massive parallelepiped model (`rectangular-prism`), the scattering amplitude is computed for a particular orientation of the parallelepiped with respect to the scattering vector and then averaged over all possible orientations, giving

$$P(q) = \frac{1}{V^2} \frac{2}{\pi} \times \, \int_0^{\frac{\pi}{2}} \, \int_0^{\frac{\pi}{2}} A_{P\Delta}^2(q) \, \sin\theta \, d\theta \, d\phi$$
where $\theta$ is the angle between the $z$ axis and the longest axis of the parallelepiped, $\phi$ is the angle between the scattering vector (lying in the $xy$ plane) and the $y$ axis, and

\begin{align*} A_{P\Delta}(q) & = A B C \left[\frac{\sin \bigl( q \frac{C}{2} \cos\theta \bigr)} {\left( q \frac{C}{2} \cos\theta \right)} \right] \left[\frac{\sin \bigl( q \frac{A}{2} \sin\theta \sin\phi \bigr)} {\left( q \frac{A}{2} \sin\theta \sin\phi \right)}\right] \left[\frac{\sin \bigl( q \frac{B}{2} \sin\theta \cos\phi \bigr)} {\left( q \frac{B}{2} \sin\theta \cos\phi \right)}\right] \\ & - 8 \left(\frac{A}{2}-\Delta\right) \left(\frac{B}{2}-\Delta\right) \left(\frac{C}{2}-\Delta\right) \left[ \frac{\sin \bigl[ q \bigl(\frac{C}{2}-\Delta\bigr) \cos\theta \bigr]} {q \bigl(\frac{C}{2}-\Delta\bigr) \cos\theta} \right] \left[ \frac{\sin \bigl[ q \bigl(\frac{A}{2}-\Delta\bigr) \sin\theta \sin\phi \bigr]} {q \bigl(\frac{A}{2}-\Delta\bigr) \sin\theta \sin\phi} \right] \left[ \frac{\sin \bigl[ q \bigl(\frac{B}{2}-\Delta\bigr) \sin\theta \cos\phi \bigr]} {q \bigl(\frac{B}{2}-\Delta\bigr) \sin\theta \cos\phi} \right] \end{align*}
where $A$, $B$ and $C$ are the external sides of the parallelepiped fulfilling $A \le B \le C$, and the volume $V$ of the parallelepiped is

$$V = A B C \, - \, (A - 2\Delta) (B - 2\Delta) (C - 2\Delta)$$
The 1D scattering intensity is then calculated as

$$I(q) = \text{scale} \times V \times (\rho_\text{p} - \rho_\text{solvent})^2 \times P(q) + \text{background}$$
where $\rho_\text{p}$ 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* represents the volume fraction (which is unitless) of the rectangular shell of material (i.e. not including the volume of the solvent filled core).

For 2d data the orientation of the particle is required, described using angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, for further details of the calculation and angular dispersions see `orientation` . The angle $\Psi$ is the rotational angle around the long *C* axis. For example, $\Psi = 0$ when the *B* axis is parallel to the *x*-axis of the detector.

For 2d, constraints must be applied during fitting to ensure that the inequality $A < B < C$ is not violated, and hence the correct definition of angles is preserved. The calculation will not report an error if the inequality is *not* preserved, but the results may be not correct.

Definition of the angles for oriented hollow rectangular prism. Note that rotation $\theta$, initially in the $xz$ plane, is carried out first, then rotation $\phi$ about the $z$ axis, finally rotation $\Psi$ is now around the axis of the prism. The neutron or X-ray beam is along the $z$ axis.

Examples of the angles for oriented hollow rectangular prisms against the detector plane.

Validation

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

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