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.


This model calculates the scattering from a rectangular parallelepiped
If you need to apply polydispersity, see also rectangular_prism.

Fig. 57 Parallelepiped with the corresponding definition of sides.


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)

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


\[ \begin{align}\begin{aligned}\phi_Q(\mu,a) &= \int_0^1 \left\{S\left[\frac{\mu}{2}\cos\left(\frac{\pi}{2}u\right)\right] S\left[\frac{\mu a}{2}\sin\left(\frac{\pi}{2}u\right)\right] \right\}^2 du\\S(x) &= \frac{\sin x}{x}\\\mu &= qB\end{aligned}\end{align} \]

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 \(x\).


Fig. 58 Definition of the angles for oriented parallelepipeds.


Fig. 59 Examples of the angles for oriented parallelepipeds against the detector plane.

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

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


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

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 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).


Fig. 60 1D and 2D plots corresponding to the default parameters of the model.


P Mittelbach and G Porod, Acta Physica Austriaca, 14 (1961) 185-211

R Nayuk and K Huber, Z. Phys. Chem., 226 (2012) 837-854