PolyCoreShell

Steve Kline 06 NOV 1998

Alan Munter 08 JULY 1999, converted to Java

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.

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

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 s² 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.

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

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.