1.5.4. 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 | 10-6Å-2 | 4 |
sld_solvent | Solvent scattering length density | 10-6Å-2 | 1 |
length_a | Shorter side of the parallelepiped | Å | 35 |
length_b | Second side of the parallelepiped | Å | 75 |
length_c | Larger side of the parallelepiped | Å | 400 |
theta | c axis to beam angle | degree | 60 |
phi | rotation about beam | degree | 60 |
psi | rotation about c axis | 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). If you need to apply polydispersity, see also rectangular_prism.

Fig. 1.60 Figure 1: Parallelepiped with the corresponding definition of sides.
The three dimensions of the parallelepiped (strictly here a cuboid) may be given in any size order. 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 1D scattering intensity \(I(q)\) is calculated as:
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)
with
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, 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.
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. 1.61 Figure 2: Definition of the angles for oriented parallelepiped, shown with \(A<B<C\).

Fig. 1.62 Figure 3: Examples of the angles for an oriented parallelepiped against the detector plane.
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. (Earlier implementations had numerical integration issues in some circumstances when orientation distributions passed through 90 degrees, such situations, with very broad distributions, should still be approached with care.)
For a given orientation of the parallelepiped, the 2D form factor is calculated as
with
and the scattering intensity as:
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.

Fig. 1.63 Figure 4: 1D and 2D plots corresponding to the default parameters of the 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
Authorship and Verification
- Author: This model is based on form factor calculations implemented
- in a c-library provided by the NIST Center for Neutron Research (Kline, 2006).
- Last Modified by: Paul Kienzle Date: April 05, 2017
- Last Reviewed by: Richard Heenan Date: April 06, 2017