PolyCoreShellRatio

Steve Kline 06 NOV 1998

Alan Munter 08 JULY 1999, converted to Java

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*<f²>/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.

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

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 s² 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.

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

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.