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

Parameter Description Units Default value
scale Source intensity 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 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 Orientation of the stacked disk in the plane of the detector degree 0

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


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.


Fig. 31 Geometry of a single core/layer disk

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.


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.

To provide easy access to the orientation of the stacked disks, we define the axis of the cylinder using two angles \(\theta\) and \(\varphi\).


Fig. 32 Examples of the angles against the detector plane.

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


Fig. 33 1D and 2D plots corresponding to the default parameters of the model.


[1]O Kratky and G Porod, J. Colloid Science, 4, (1949) 35
[2]S R Kline, J Appl. Cryst., 39 (2006) 895
[3]J S Higgins and H C Benoit, Polymers and Neutron Scattering, Clarendon, Oxford, 1994
[4]A Guinier and G Fournet, Small-Angle Scattering of X-Rays, John Wiley and Sons, New York, 1955

Authorship and Verification

  • Author: NIST IGOR/DANSE Date: pre 2010
  • Last Modified by: Paul Butler and Paul Kienzle on: November 26, 2016
  • Last Reviewed by: Paul Butler and Paul Kienzle on: November 26, 2016