hollow_rectangular_prism_thin_walls

Hollow rectangular parallelepiped with thin walls.

Parameter Description Units Default value
scale Scale factor or Volume fraction None 1
background Source background cm-1 0.001
sld Parallelepiped scattering length density 10-6Å-2 6.3
sld_solvent Solvent scattering length density 10-6Å-2 1
length_a Shorter side of the parallelepiped Å 35
b2a_ratio Ratio sides b/a Å 1
c2a_ratio Ratio sides c/a Å 1

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

Definition This model provides the form factor, \(P(q)\), for a hollow rectangular prism with infinitely thin walls. It computes only the 1D scattering, not the 2D. The 1D scattering intensity for this model is calculated according to the equations given by Nayuk and Huber[1].

Assuming a hollow parallelepiped with infinitely thin walls, edge lengths \(A \le B \le C\) and presenting an orientation with respect to the scattering vector given by \(\theta\) and \(\phi\), where \(\theta\) is the angle between the \(z\) axis and the longest axis of the parallelepiped \(C\), and \(\phi\) is the angle between the scattering vector (lying in the \(xy\) plane) and the \(y\) axis, the form factor is given by

\[P(q) = \frac{1}{V^2} \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}} [A_L(q)+A_T(q)]^2 \sin\theta\,d\theta\,d\phi\]

where

\[\begin{split}V &= 2AB + 2AC + 2BC \\ A_L(q) &= 8 \times \frac{ \sin \left( \tfrac{1}{2} q A \sin\phi \sin\theta \right) \sin \left( \tfrac{1}{2} q B \cos\phi \sin\theta \right) \cos \left( \tfrac{1}{2} q C \cos\theta \right) }{q^2 \, \sin^2\theta \, \sin\phi \cos\phi} \\ A_T(q) &= A_F(q) \times \frac{2\,\sin \left( \tfrac{1}{2} q C \cos\theta \right)}{q\,\cos\theta}\end{split}\]

and

\[A_F(q) = 4 \frac{ \cos \left( \tfrac{1}{2} q A \sin\phi \sin\theta \right) \sin \left( \tfrac{1}{2} q B \cos\phi \sin\theta \right) } {q \, \cos\phi \, \sin\theta} + 4 \frac{ \sin \left( \tfrac{1}{2} q A \sin\phi \sin\theta \right) \cos \left( \tfrac{1}{2} q B \cos\phi \sin\theta \right) } {q \, \sin\phi \, \sin\theta}\]

The 1D scattering intensity is then calculated as

\[I(q) = \text{scale} \times V \times (\rho_\text{p} - \rho_\text{solvent})^2 \times P(q)\]

where \(V\) is the surface area of the rectangular prism, \(\rho_\text{p}\) is the scattering length density of the parallelepiped, \(\rho_\text{solvent}\) is the scattering length density of the solvent, and (if the data are in absolute units) scale is related to the total surface area.

The 2D scattering intensity is not computed by this model.

Validation

Validation of the code was conducted by qualitatively comparing the output of the 1D model to the curves shown in (Nayuk, 2012[1]).

../../_images/hollow_rectangular_prism_thin_walls_autogenfig.png

Fig. 67 1D plot corresponding to the default parameters of the model.

Source

hollow_rectangular_prism_thin_walls.py \(\ \star\ \) hollow_rectangular_prism_thin_walls.c \(\ \star\ \) gauss76.c

References

See also Onsager [2].

[1](1, 2) R Nayuk and K Huber, Z. Phys. Chem., 226 (2012) 837-854
[2]
  1. Onsager, Ann. New York Acad. Sci., 51 (1949) 627-659

Authorship and Verification

  • Author: Miguel Gonzales Date: February 26, 2016
  • Last Modified by: Paul Kienzle Date: October 15, 2016
  • Last Reviewed by: Paul Butler Date: September 07, 2018