# stacked_disks

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 | Å | 10 |

thick_layer | Thickness of layer each side of core | Å | 10 |

radius | Radius of the stacked disk | Å | 15 |

n_stacking | Number of stacked layer/core/layer disks | None | 1 |

sigma_d | Sigma of nearest neighbor spacing | Å | 0 |

sld_core | Core scattering length density | 10^{-6}Å^{-2} |
4 |

sld_layer | Layer scattering length density | 10^{-6}Å^{-2} |
0 |

sld_solvent | Solvent scattering length density | 10^{-6}Å^{-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.

**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

where the contrast

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

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

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.

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

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

**References**

[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