core_shell_parallelepiped

Rectangular solid with a core-shell structure.

Parameter Description Units Default value
scale Source intensity None 1
background Source background cm-1 0.001
sld_core Parallelepiped core scattering length density 10-6-2 1
sld_a Parallelepiped A rim scattering length density 10-6-2 2
sld_b Parallelepiped B rim scattering length density 10-6-2 4
sld_c Parallelepiped C rim scattering length density 10-6-2 2
sld_solvent Solvent scattering length density 10-6-2 6
length_a Shorter side of the parallelepiped 35
length_b Second side of the parallelepiped 75
length_c Larger side of the parallelepiped 400
thick_rim_a Thickness of A rim 10
thick_rim_b Thickness of B rim 10
thick_rim_c Thickness of C rim 10
theta c axis to beam angle degree 0
phi rotation about beam degree 0
psi rotation about c axis degree 0

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

Definition

Calculates the form factor for a rectangular solid with a core-shell structure. The thickness and the scattering length density of the shell or “rim” can be different on each (pair) of faces.

The form factor is normalized by the particle volume \(V\) such that

\[I(q) = \text{scale}\frac{\langle f^2 \rangle}{V} + \text{background}\]

where \(\langle \ldots \rangle\) is an average over all possible orientations of the rectangular solid.

The function calculated is the form factor of the rectangular solid below. The core of the solid is defined by the dimensions \(A\), \(B\), \(C\) such that \(A < B < C\).

../../_images/core_shell_parallelepiped_geometry.jpg

There are rectangular “slabs” of thickness \(t_A\) that add to the \(A\) dimension (on the \(BC\) faces). There are similar slabs on the \(AC\) \((=t_B)\) and \(AB\) \((=t_C)\) faces. The projection in the \(AB\) plane is then

../../_images/core_shell_parallelepiped_projection.jpg

The volume of the solid is

\[V = ABC + 2t_ABC + 2t_BAC + 2t_CAB\]

meaning that there are “gaps” at the corners of the solid.

The intensity calculated follows the parallelepiped model, with the core-shell intensity being calculated as the square of the sum of the amplitudes of the core and the slabs on the edges.

the scattering amplitude is computed for a particular orientation of the core-shell parallelepiped with respect to the scattering vector and then averaged over all possible orientations, where \(\alpha\) is the angle between the \(z\) axis and the \(C\) axis of the parallelepiped, \(\beta\) is the angle between projection of the particle in the \(xy\) detector plane and the \(y\) axis.

\[\begin{split}F(Q) &= (\rho_\text{core}-\rho_\text{solvent}) S(Q_A, A) S(Q_B, B) S(Q_C, C) \\ &+ (\rho_\text{A}-\rho_\text{solvent}) \left[S(Q_A, A+2t_A) - S(Q_A, Q)\right] S(Q_B, B) S(Q_C, C) \\ &+ (\rho_\text{B}-\rho_\text{solvent}) S(Q_A, A) \left[S(Q_B, B+2t_B) - S(Q_B, B)\right] S(Q_C, C) \\ &+ (\rho_\text{C}-\rho_\text{solvent}) S(Q_A, A) S(Q_B, B) \left[S(Q_C, C+2t_C) - S(Q_C, C)\right]\end{split}\]

with

\[S(Q, L) = L \frac{\sin \tfrac{1}{2} Q L}{\tfrac{1}{2} Q L}\]

and

\[\begin{split}Q_A &= \sin\alpha \sin\beta \\ Q_B &= \sin\alpha \cos\beta \\ Q_C &= \cos\alpha\end{split}\]

where \(\rho_\text{core}\), \(\rho_\text{A}\), \(\rho_\text{B}\) and \(\rho_\text{C}\) are the scattering length of the parallelepiped core, and the rectangular slabs of thickness \(t_A\), \(t_B\) and \(t_C\), respectively. \(\rho_\text{solvent}\) is the scattering length of the solvent.

FITTING NOTES

If the scale is set equal to the particle volume fraction, \(\phi\), the returned value is the scattered intensity per unit volume, \(I(q) = \phi P(q)\). However, no interparticle interference effects are included in this calculation.

There are many parameters in this model. Hold as many fixed as possible with known values, or you will certainly end up at a solution that is unphysical.

The returned value is in units of cm-1, on absolute scale.

NB: The 2nd virial coefficient of the core_shell_parallelepiped is calculated based on the the averaged effective radius \((=\sqrt{(A+2t_A)(B+2t_B)/\pi})\) and length \((C+2t_C)\) values, after appropriately sorting the three dimensions to give an oblate or prolate particle, to give an effective radius, for \(S(Q)\) when \(P(Q) * S(Q)\) 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 long_c axis. For example, \(\Psi = 0\) when the short_b axis is parallel to the x-axis of the detector.

For 2d, constraints must be applied during fitting to ensure that the inequality \(A < B < C\) is not violated, and hence the correct definition of angles is preserved. The calculation will not report an error, but the results may be not correct.

../../_images/parallelepiped_angle_definition.png

Fig. 59 Definition of the angles for oriented core-shell parallelepipeds. Note that rotation \(\theta\), initially in the \(xz\) plane, is carried out first, then rotation \(\phi\) about the \(z\) axis, finally rotation \(\Psi\) is now around the axis of the cylinder. The neutron or X-ray beam is along the \(z\) axis.

../../_images/parallelepiped_angle_projection.png

Fig. 60 Examples of the angles for oriented core-shell parallelepipeds against the detector plane.

../../_images/core_shell_parallelepiped_autogenfig.png

Fig. 61 1D and 2D plots corresponding to the default parameters of the model.

References

[1]P Mittelbach and G Porod, Acta Physica Austriaca, 14 (1961) 185-211 Equations (1), (13-14). (in German)
[2]D Singh (2009). Small angle scattering studies of self assembly in lipid mixtures, Johns Hopkins University Thesis (2009) 223-225. Available from Proquest

Authorship and Verification

  • Author: NIST IGOR/DANSE Date: pre 2010
  • Converted to sasmodels by: Miguel Gonzales Date: February 26, 2016
  • Last Modified by: Paul Kienzle Date: October 17, 2017
  • Cross-checked against hollow rectangular prism and rectangular prism for equal thickness overlapping sides, and by Monte Carlo sampling of points within the shape for non-uniform, non-overlapping sides.