# stacked_disks

Form factor for a stacked set of non exfoliated core/shell disks

Parameter

Description

Units

Default value

scale

Scale factor or Volume fraction

None

1

background

Source background

cm-1

0.001

thick_core

Thickness of the core disk

10

thick_layer

Thickness of layer each side of core

10

Radius of the stacked disk

15

n_stacking

Number of stacked layer/core/layer disks

None

1

sigma_d

Sigma of nearest neighbor spacing

0

sld_core

Core scattering length density

10-6-2

4

sld_layer

Layer scattering length density

10-6-2

0

sld_solvent

Solvent scattering length density

10-6-2

5

theta

Orientation of the stacked disk axis w/respect incoming beam

degree

0

phi

degree

0

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

Definition

This model provides the form factor, $$P(q)$$, for stacked discs (tactoids) with a core/layer structure which is constructed itself as $$P(q) S(Q)$$ multiplying a $$P(q)$$ for individual core/layer disks by a structure factor $$S(q)$$ proposed by Kratky and Porod in 1949[1] assuming the next neighbor distance (d-spacing) in the stack of parallel discs obeys a Gaussian distribution. As such the normalization of this “composite” form factor is relative to the individual disk volume, not the volume of the stack of disks. This model is appropriate for example for non non exfoliated clay particles such as Laponite.

The scattered intensity $$I(q)$$ is calculated as

$I(q) = N\int_{0}^{\pi /2}\left[ \Delta \rho_t \left( V_t f_t(q,\alpha) - V_c f_c(q,\alpha)\right) + \Delta \rho_c V_c f_c(q,\alpha)\right]^2 S(q,\alpha)\sin{\alpha}\ d\alpha + \text{background}$

where the contrast

$\Delta \rho_i = \rho_i - \rho_\text{solvent}$

and $$N$$ is the number of individual (single) discs per unit volume, $$\alpha$$ is the angle between the axis of the disc and $$q$$, and $$V_t$$ and $$V_c$$ are the total volume and the core volume of a single disc, respectively, and

\begin{align}\begin{aligned}f_t(q,\alpha) = \left(\frac{\sin(q(d+h)\cos{\alpha})}{q(d+h)\cos{\alpha}}\right) \left(\frac{2J_1(qR\sin{\alpha})}{qR\sin{\alpha}} \right)\\f_c(q,\alpha) = \left(\frac{\sin(qh)\cos{\alpha})}{qh\cos{\alpha}}\right) \left(\frac{2J_1(qR\sin{\alpha})}{qR\sin{\alpha}}\right)\end{aligned}\end{align}

where $$d$$ = thickness of the layer (thick_layer), $$2h$$ = core thickness (thick_core), and $$R$$ = radius of the disc (radius).

$S(q,\alpha) = 1 + \frac{1}{2}\sum_{k=1}^n(n-k)\cos{(kDq\cos{\alpha})} \exp\left[ -k(q)^2(D\cos{\alpha}~\sigma_d)^2/2\right]$

where $$n$$ is the total number of the disc stacked (n_stacking), $$D = 2(d+h)$$ is the next neighbor center-to-center distance (d-spacing), and $$\sigma_d$$ = the Gaussian standard deviation of the d-spacing (sigma_d). Note that $$D\cos(\alpha)$$ is the component of $$D$$ parallel to $$q$$ and the last term in the equation above is effectively a Debye-Waller factor term.

Note

1. Each assembly in the stack is layer/core/layer, so the spacing of the cores is core plus two layers. The 2nd virial coefficient of the cylinder is calculated based on the radius and length = n_stacking * (thick_core + 2 * thick_layer) values, and used as the effective radius for $$S(Q)$$ when $$P(Q) * S(Q)$$ is applied.

2. the resolution smearing calculation uses 76 Gaussian quadrature points to properly smear the model since the function is HIGHLY oscillatory, especially around the q-values that correspond to the repeat distance of the layers.

2d scattering from oriented stacks is calculated in the same way as for cylinders, for further details of the calculation and angular dispersions see Oriented particles.

Our model is derived from the form factor calculations implemented in a C-library provided by the NIST Center for Neutron Research[2]

Source

stacked_disks.py $$\ \star\$$ stacked_disks.c $$\ \star\$$ gauss76.c $$\ \star\$$ sas_J1.c $$\ \star\$$ polevl.c

References

See also Higgins and Benoit [3] and Guinier and Fournet [4].

Authorship and Verification

• Author: NIST IGOR/DANSE Date: pre 2010

• Last Modified by: Paul Butler and Paul Kienzle Date: November 26, 2016

• Last Reviewed by: Paul Butler and Paul Kienzle Date: November 26, 2016