MODEL

PolyCoreShell

AUTHOR/MODIFICATION

Steve Kline 06 NOV 1998

Alan Munter 08 JULY 1999, converted to Java

APPROVED FOR DISTRIBUTION

DESCRIPTION

Calculates the form factor for polydispersed spherical particles with a core-shell structure. The spherical particles have a polydisperse core with a constant shell thickness. The form factor is normalized by the average particle volume such that

P(*q*) = scale*<*f***f*>/Vol + bkg, where
*f* is the single particle scattering amplitude, appropriately
averaged over the Schulz distribution of radii.

Resolution smeared version is also provided.

VARIABLES

Input Variables (default values):

Parameter | Variable | Value |
---|---|---|

0 | Scale | 1.0 |

1 | Average Core Radius (Å) | 60.0 |

2 | Core Polydispersity (0-1) | 0.2 |

3 | Shell Thickness (Å) | 10.0 |

4 | Core SLD (Å^{-2}) | 1.0e-6 |

5 | Shell SLD (Å^{-2}) | 2.0e-6 |

6 | Solvent SLD (Å^{-2}) | 3.0e-6 |

7 | incoherent Background(cm^{-1}) | 0.000 |

**USAGE** **NOTES**

The returned value is scaled to units of [cm^{-1}], on
absolute scale.

Parameter[0] (scale) is correlated with the SLD's that describe the particle. No more than one of these parameters can be free during model fitting.

Parameter[2] (polydispersity) is constrained to keep
it within its physical limits of (0,1). If a value larger than 1 is
entered that value is silently converted to 1 for the calculation.
Polydispersity, p = s/R_{c},
where s^{2} is the variance of the distribution and
R_{c} is the mean core
radius, Parameter[1]. For a more complete description of the Schulz
distribution, see: J. Hayter in "Physics of Amphiphiles - Micelles,
Vesicles and Microemulsions" V. DeGiorgio and M. Corti, Eds. (1983)
p. 69.

The returned form factor is normalized by the average particle volume <V>:

where

and *z* is the width parameter of the Schulz distribution,
.

Setting the SLD of the core and shell equal to simulate a uniform
sphere results in a mathematical (divide by zero) error, so if this
occurs the difference is silently made equal to 10^{-10}.
To simulate a uniform SLD sphere, set the shell thickness to zero
(the SLD of the shell will now make no contribution to the scattering).

The average particle diameter is 2*(Radius + shell thickness)

If the scale factor Parameter[0] is set equal to the particle
volume fraction, *phi*, the returned value is the scattered
intensity per unit volume, I(*q*) = *phi**P(*q*).

No interparticle interference effects are included in this calculation.

This function is useful for vesicles or liposomes by setting the SLD (core) = SLD (solvent), thus calculating scattering from a spherical shell with constant thickness, but polydisperse inner radius.

REFERENCE

Bartlett, P.; Ottewill, R. H. *J. Chem. Phys.*, **1992**,
*96*, 3306.

TEST DATASET

This example dataset is produced by calculating the PolyCoreShell
using 256 data points, *q*_{min} = 0.001 Å^{-1},
*q*_{max} = 0.7 Å^{-1} and the above default
parameter values.