MODEL

PolyCoreShellRatio

AUTHOR/MODIFICATION

Steve Kline 06 NOV 1998

Alan Munter 08 JULY 1999, converted to Java

APPROVED FOR DISTRIBUTION

DESCRIPTION

Calculates the form factor for a polydisperse spherical particle
with a core-shell structure. In this model, the ratio
R(core)/R(core+shell) is held constant. The form factor is normalized
by the mean particle volume such that P(*q*) =
scale*<*f*^{2}>/Vol + bkg, where *f* is the
single particle scattering amplitude and the average < > is
over a 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 | Average Shell Thickness (Å) | 10.0 |

3 | Overall Polydispersity (0-1) | 0.2 |

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 it is
quietly changed to 1 for the calculation. Polydispersity, *p* =
*s*/R_{t}, where *s*^{2} is the variance of
the distribution and R_{t} is the total particle radius,
R_{t} = R_{c} +*t*. 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 here *r* denotes the total particle radius, R_{c} +
*t*, and *z* is the width parameter of the Schulz
distribution, .

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*). However, no
interparticle interference effects are included in this calculation.

REFERENCE

J. B. Hayter in "Physics of Amphiphiles--Micelles, Vesicles, and
Microemulsions" Eds. V. DeGiorgio; M. Corti, pp. 59-93,**1983**.
Eqns: 32-37

TEST DATASET

This example dataset is produced by calculating the PolyCoreShellRatio
using 128 data points, *q*_{min} = 0.001 Å^{-1},
*q*_{max} = 0.7 Å^{-1} and the above default
parameter values.