# 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-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

degree

0

psi

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

$I(q) = \frac{\text{scale}}{V} \langle P(q,\alpha,\beta) \rangle + \text{background}$

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

$V = ABC + 2t_ABC + 2t_BAC + 2t_CAB$

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.

$P(q)=\frac {\int_{0}^{\pi/2}\int_{0}^{\pi/2}F^2(q,\alpha,\beta) \ sin\alpha \ d\alpha \ d\beta} {\int_{0}^{\pi/2} \ sin\alpha \ d\alpha \ d\beta}$

and

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

with

$S(Q_X, L) = L \frac{\sin (\tfrac{1}{2} Q_X L)}{\tfrac{1}{2} Q_X L}$

and

$\begin{split}Q_A &= q \sin\alpha \sin\beta \\ Q_B &= q \sin\alpha \cos\beta \\ Q_C &= q \cos\alpha\end{split}$

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

1. 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.

2. 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.

3. 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

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

1. Onsager, 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