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 10-6-2 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 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.
../_images/parallelepiped_geometry.jpg

Fig. 1.66 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:

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

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

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

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.

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 Oriented particles .

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 x-axis. For other \(\theta\), \(\phi\) values, the parallelepiped has to be first rotated \(\theta\) degrees in the \(z-x\) plane and then \(\phi\) degrees around the \(z\) axis, before doing a final rotation of \(\Psi\) degrees around the resulting \(C\) axis of the particle to obtain the final orientation of the parallelepiped.

../_images/parallelepiped_angle_definition.png

Fig. 1.67 Figure 2: Definition of the angles for oriented parallelepiped, shown with \(A<B<C\).

../_images/parallelepiped_angle_projection.png

Fig. 1.68 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 as discussed in Oriented particles .

For a given orientation of the parallelepiped, the 2D 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

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

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.

../_images/parallelepiped_autogenfig.png

Fig. 1.69 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