# core_shell_parallelepiped

Rectangular solid with a core-shell structure.

Parameter |
Description |
Units |
Default value |
---|---|---|---|

scale |
Scale factor or Volume fraction |
None |
1 |

background |
Source background |
cm |
0.001 |

sld_core |
Parallelepiped core scattering length density |
10 |
1 |

sld_a |
Parallelepiped A rim scattering length density |
10 |
2 |

sld_b |
Parallelepiped B rim scattering length density |
10 |
4 |

sld_c |
Parallelepiped C rim scattering length density |
10 |
2 |

sld_solvent |
Solvent scattering length density |
10 |
6 |

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 |

thick_rim_a |
Thickness of A rim |
Å |
10 |

thick_rim_b |
Thickness of B rim |
Å |
10 |

thick_rim_c |
Thickness of C rim |
Å |
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 three dimensions of the core of the
parallelepiped (strictly here a cuboid) may be given in *any* size order as
long as the particles are randomly oriented (i.e. take on all possible
orientations see notes on 2D below). 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 form factor is normalized by the particle volume \(V\) such that

where \(\langle \ldots \rangle\) is an average over all possible orientations of the rectangular solid, and the usual \(\Delta \rho^2 \ V^2\) term cannot be pulled out of the form factor term due to the multiple slds in the model.

The core of the solid is defined by the dimensions \(A\), \(B\), \(C\) here shown such that \(A < B < C\).

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

The total volume of the solid is thus given as

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, and \(\beta\) is the angle between the projection of the particle in the \(xy\) detector plane and the \(y\) axis.

and

with

and

where \(\rho_\text{core}\), \(\rho_\text{A}\), \(\rho_\text{B}\) and \(\rho_\text{C}\) are the scattering lengths 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.

Note

the code actually implements two substitutions: \(d(cos\alpha)\) is substituted for -\(sin\alpha \ d\alpha\) (note that in the parallelepiped code this is explicitly implemented with \(\sigma = cos\alpha\)), and \(\beta\) is set to \(\beta = u \pi/2\) so that \(du = \pi/2 \ d\beta\). Thus both integrals go from 0 to 1 rather than 0 to \(\pi/2\).

**FITTING NOTES**

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 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, where \(\theta\) and \(\phi\) define the orientation of the director in the laboratory reference frame of the beam direction (\(z\)) and detector plane (\(x-y\) plane), while the angle \(\Psi\) is effectively the rotational angle around the particle \(C\) 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 order of rotations matters. In particular, the parallelepiped must first be rotated \(\theta\) degrees in the \(x-z\) plane before rotating \(\phi\) degrees around the \(z\) axis (in the \(x-y\) plane). Applying orientational distribution to the particle orientation (i.e jitter to one or more of these angles) can get more confusing as jitter is defined

**NOT**with respect to the laboratory frame but the particle reference frame. It is thus highly recommended to read Oriented particles for further details of the calculation and angular dispersions.

Note

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

**Validation**

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.

**Source**

`core_shell_parallelepiped.py`

\(\ \star\ \) `core_shell_parallelepiped.c`

\(\ \star\ \) `gauss76.c`

**References**

P Mittelbach and G Porod,

*Acta Physica Austriaca*, 14 (1961) 185-211 Equations (1), (13-14). (in German)D Singh (2009).

*Small angle scattering studies of self assembly in lipid mixtures*, Johns Hopkins University Thesis (2009) 223-225. Available from ProquestOnsager,

*Ann. New York Acad. Sci.*, 51 (1949) 627-659

**Authorship and Verification**

**Author:**NIST IGOR/DANSE**Date:**pre 2010**Converted to sasmodels by:**Miguel Gonzalez**Date:**February 26, 2016**Last Modified by:**Paul Kienzle**Date:**October 17, 2017**Last Reviewed by:**Paul Butler**Date:**May 24, 2018 - documentation updated