# parallelepiped

Rectangular parallelepiped with uniform scattering length density.

Parameter Description Units Default value
scale Source intensity None 1
background Source background cm-1 0.001
sld Parallelepiped scattering length density 10-6-2 4
sld_solvent Solvent scattering length density 10-6-2 1
length_a Shorter side of the parallelepiped 35
length_b Second side of the parallelepiped 75
length_c Larger side of the parallelepiped 400
theta In plane angle degree 60
phi Out of plane angle degree 60
psi Rotation angle around its own c axis against q plane degree 60

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

The form factor is normalized by the particle volume. For information about polarised and magnetic scattering, see the Polarisation/Magnetic Scattering documentation.

Definition

This model calculates the scattering from a rectangular parallelepiped
(:numref:parallelepiped-image).

Note

The edge of the solid must satisfy the condition that $$A < B < C$$. This requirement is not enforced in the model, so it is up to the user to check this during the analysis.

The 1D scattering intensity $$I(q)$$ is calculated as:

$I(q) = \frac{\text{scale}}{V} (\Delta\rho \cdot V)^2 \left< P(q, \alpha) \right> + \text{background}$

where the volume $$V = A B C$$, the contrast is defined as $$\Delta\rho = \rho_\text{p} - \rho_\text{solvent}$$, $$P(q, \alpha)$$ is the form factor corresponding to a parallelepiped oriented at an angle $$\alpha$$ (angle between the long axis C and $$\vec q$$), and the averaging $$\left<\ldots\right>$$ is applied over all orientations.

Assuming $$a = A/B < 1$$, $$b = B /B = 1$$, and $$c = C/B > 1$$, the form factor is given by (Mittelbach and Porod, 1961)

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

with

\begin{align}\begin{aligned}\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\end{aligned}\end{align}

The scattering intensity per unit volume is returned in units of cm-1.

NB: The 2nd virial coefficient of the parallelepiped is calculated based on the averaged effective radius $$(=\sqrt{A B / \pi})$$ and length $$(= C)$$ values, and used as the effective radius for $$S(q)$$ when $$P(q) \cdot S(q)$$ is applied.

To provide easy access to the orientation of the parallelepiped, we define three angles $$\theta$$, $$\phi$$ and $$\Psi$$. The definition of $$\theta$$ and $$\phi$$ is the same as for the cylinder model (see also figures below).

The angle $$\Psi$$ is the rotational angle around the $$C$$ 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 z-axis. For other $$\theta$$, $$\phi$$ values, the parallelepiped has to be first rotated $$\theta$$ degrees around $$z$$ and $$\phi$$ degrees around $$y$$, before doing a final rotation of $$\Psi$$ degrees around the resulting $$C$$ to obtain the final orientation of the parallelepiped. For example, for $$\theta = 0$$ and $$\phi = 90$$, we have that $$\Psi = 0$$ corresponds to $$A$$ along $$x$$ and $$B$$ along $$y$$, while for $$\theta = 90$$ and $$\phi = 0$$, $$\Psi = 0$$ corresponds to $$A$$ along $$z$$ and $$B$$ along $$x$$.

For a given orientation of the parallelepiped, the 2D form factor is calculated as

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

with

\begin{align}\begin{aligned}\cos\alpha &= \hat A \cdot \hat q,\\\cos\beta &= \hat B \cdot \hat q,\\\cos\gamma &= \hat C \cdot \hat q\end{aligned}\end{align}

and the scattering intensity as:

$I(q_x, q_y) = \frac{\text{scale}}{V} V^2 \Delta\rho^2 P(q_x, q_y) + \text{background}$

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.

This model is based on form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006).

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