Rectangular Prism

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

This model provides the form factor, $P(q)$, for a rectangular prism.

Note that this model is almost totally equivalent to the existing `parallelepiped` model. The only difference is that the way the relevant parameters are defined here ($a$, $b/a$, $c/a$ instead of $a$, $b$, $c$) which allows use of polydispersity with this model while keeping the shape of the prism (e.g. setting $b/a = 1$ and $c/a = 1$ and applying polydispersity to *a* will generate a distribution of cubes of different sizes).

Definition

The 1D scattering intensity for this model was calculated by Mittelbach and Porod (Mittelbach, 1961), but the implementation here is closer to the equations given by Nayuk and Huber (Nayuk, 2012). Note also that the angle definitions used in the code and the present documentation correspond to those used in (Nayuk, 2012) (see Fig. 1 of that reference), with $\theta$ corresponding to $\alpha$ in that paper, and not to the usual convention used for example in the `parallelepiped` model.

In this model the scattering from a massive parallelepiped with an orientation with respect to the scattering vector given by $\theta$ and $\phi$

$$ A_P\,(q) = \frac{\sin \left( \tfrac{1}{2}qC \cos\theta \right) }{\tfrac{1}{2} qC \cos\theta} \,\times\, \frac{\sin \left( \tfrac{1}{2}qA \cos\theta \right) }{\tfrac{1}{2} qA \cos\theta} \,\times\ , \frac{\sin \left( \tfrac{1}{2}qB \cos\theta \right) }{\tfrac{1}{2} qB \cos\theta}
$$
where $A$, $B$ and $C$ are the sides of the parallelepiped and must fulfill $A \le B \le C$, $\theta$ 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 normalized form factor in 1D is obtained averaging over all possible orientations

$$ P(q) = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \, \int_0^{\frac{\pi}{2}} A_P^2(q) \, \sin\theta \, d\theta \, d\phi
$$
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 rectangular prism, $\rho_\text{p}$ is the scattering length of the parallelepiped, $\rho_\text{solvent}$ is the scattering length of the solvent, and (if the data are in absolute units) *scale* represents the volume fraction (which is unitless).

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, but the results may be not correct.

Definition of the angles for oriented core-shell parallelepipeds. 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 cylinder. The neutron or X-ray beam is along the $z$ axis.

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

Validation

Validation of the code was conducted by comparing the output of the 1D model to the output of the existing `parallelepiped` model.

References

P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211
R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854
L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659

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