Parallelepiped

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

This model calculates the scattering from a rectangular solid (`parallelepiped-image`). If you need to apply polydispersity, see also `rectangular-prism`. For information about polarised and magnetic scattering, see the `magnetism` documentation.

.. _parallelepiped-image:

Parallelepiped with the corresponding definition of sides.

The three dimensions of the parallelepiped (strictly here a cuboid) may be given in *any* size order as long as the particles are randomly oriented (i.e. take on all possible orientations see notes on 2D below). To avoid multiple fit solutions, especially with Monte-Carlo fit methods, it may be advisable to restrict their ranges. There may be a number of closely similar "best fits", so some trial and error, or fixing of some dimensions at expected values, may help.

The form factor is normalized by the particle volume and the 1D scattering intensity $I(q)$ is then calculated as:

.. Comment by Miguel Gonzalez: I am modifying the original text because I find the notation a little bit confusing. I think that in most textbooks/papers, the notation P(Q) is used for the form factor (adim, P(Q=0)=1), although F(q) seems also to be used. But here (as for many other models), P(q) is used to represent the scattering intensity (in cm-1 normally). It would be good to agree on a common notation.

$$ I(q) = \frac{\text{scale}}{V} (\Delta\rho \cdot V)^2 \left< P(q, \alpha, \beta) \right> + \text{background}
$$
where the volume $V = A B C$, the contrast is defined as $\Delta\rho = \rho_\text{p} - \rho_\text{solvent}$, $P(q, \alpha, \beta)$ is the form factor corresponding to a parallelepiped oriented at an angle $\alpha$ (angle between the long axis C and $\vec q$), and $\beta$ (the angle between the projection of the particle in the $xy$ detector plane and the $y$ axis) and the averaging $\left<\ldots\right>$ is applied over all orientations.

form factor is given by (Mittelbach and Porod, 19_)
Assuming $a = A/B < 1$, $b = B /B = 1$, and $c = C/B > 1$, the

$$ P(q, \alpha) = \int_0^1 \phi_Q\left(\mu \sqrt{1-\sigma^2},a\right) \left[S(\mu c \sigma/2)\right]^2 d\sigma
$$
with

$$ \phi_Q(\mu,a) = \int_0^1 \left\{S\left[\frac{\mu}{2}\cos\left(\frac{\pi}{2}u\right)\right] S\left[\frac{\mu a}{2}\sin\left(\frac{\pi}{2}u\right)\right] \right\}^2 du \\ S(x) = \frac{\sin x}{x} \\ \mu = qB
$$
where substitution of $\sigma = cos\alpha$ and $\beta = \pi/2 u$ have been applied.

For **oriented** particles, the 2D scattering intensity, $I(q_x, q_y)$, is given as:

$$ I(q_x, q_y) = \frac{\text{scale}}{V} (\Delta\rho \cdot V)^2 P(q_x, q_y) + \text{background}
$$
.. Comment by Miguel Gonzalez: This reflects the logic of the code, as in parallelepiped.c the call to _pkernel returns $P(q_x, q_y)$ and then this is multiplied by $V^2 * (\Delta \rho)^2$. And finally outside parallelepiped.c it will be multiplied by scale, normalized by $V$ and the background added. But mathematically it makes more sense to write $I(q_x, q_y) = \text{scale} V \Delta\rho^2 P(q_x, q_y) + \text{background}$, with scale being the volume fraction.

Where $P(q_x, q_y)$ for a given orientation of the form factor is calculated as

$$ P(q_x, q_y) = \left[\frac{\sin(\tfrac{1}{2}qA\cos\alpha)}{(\tfrac{1} {2}qA\cos\alpha)}\right]^2 \left[\frac{\sin(\tfrac{1}{2}qB\cos\beta)}{(\tfrac{1} {2}qB\cos\beta)}\right]^2 \left[\frac{\sin(\tfrac{1}{2}qC\cos\gamma)}{(\tfrac{1} {2}qC\cos\gamma)}\right]^2
$$
with

$$ \cos\alpha = \hat A \cdot \hat q, \\ \cos\beta = \hat B \cdot \hat q, \\ \cos\gamma = \hat C \cdot \hat q
$$

FITTING NOTES ~~~~~~~~~~~~~

#. The 2nd virial coefficient of the parallelepiped is calculated based on the averaged effective radius, after appropriately sorting the three dimensions, to give an oblate or prolate particle, $(=\sqrt{AB/\pi})$ and length $(= C)$ values, and used as the effective radius for $S(q)$ when $P(q) \cdot S(q)$ is applied.

#. For 2d data the orientation of the particle is required, described using angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, where $\theta$ and $\phi$ define the orientation of the director in the laboratry reference frame of the beam direction ($z$) and detector plane ($x-y$ plane), while the angle $\Psi$ is effectively the rotational angle around the particle $C$ axis. For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the $B$ axis oriented parallel to the y-axis of the detector with $A$ along the x-axis. For other $\theta$, $\phi$ values, the order of rotations matters. In particular, the parallelepiped must first be rotated $\theta$ degrees in the $x-z$ plane before rotating $\phi$ degrees around the $z$ axis (in the $x-y$ plane). Applying orientational distribution to the particle orientation (i.e `jitter` to one or more of these angles) can get more confusing as `jitter` is defined **NOT** with respect to the laboratory frame but the particle reference frame. It is thus highly recmmended to read `orientation` for further details of the calculation and angular dispersions.

.. note:: For 2d, constraints must be applied during fitting to ensure that the order of sides chosen is not altered, and hence that the correct definition of angles is preserved. For the default choice shown here, that means ensuring that the inequality $A < B < C$ is not violated, The calculation will not report an error, but the results may be not correct.

.. _parallelepiped-orientation:

Definition of the angles for oriented parallelepiped, shown with $A<B<C$.

Examples of the angles for an oriented parallelepiped against the detector plane.

.. Comment by Paul Butler I am commenting this section out as we are trying to minimize the amount of oritentational detail here and encourage the user to go to the full orientation documentation so that changes can be made in just one place. below is the commented paragrah: On introducing "Orientational Distribution" in the angles, "distribution of theta" and "distribution of phi" parameters will appear. These are actually rotations about axes $\delta_1$ and $\delta_2$ of the parallelepiped, perpendicular to the $a$ x $c$ and $b$ x $c$ faces. (When $\theta = \phi = 0$ these are parallel to the $Y$ and $X$ axes of the instrument.) The third orientation distribution, in $\psi$, is about the $c$ axis of the particle, perpendicular to the $a$ x $b$ face. Some experimentation may be required to understand the 2d patterns fully as discussed in `orientation` .

Validation

Validation of the code was done by comparing the output of the 1D calculation to the angular average of the output of a 2D calculation over all possible angles.

References

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

Authorship and Verification

**Author:** NIST IGOR/DANSE **Date:** pre 2010
**Last Modified by:** Paul Kienzle **Date:** April 05, 2017
**Last Reviewed by:** Miguel Gonzales and Paul Butler **Date:** May 24,
2018 - documentation updated