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
1Average Core Radius (Å)60.0
2Average Shell Thickness (Å)10.0
3Overall Polydispersity (0-1)0.2
4Core SLD (Å-2)1.0e-6
5Shell SLD (Å-2)2.0e-6
6Solvent SLD (Å-2)3.0e-6
7incoherent Background(cm-1)0.000


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

returned value

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/Rt, where s2 is the variance of the distribution and Rt is the total particle radius, Rt = Rc +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>:

Equation 1 where Equation 2 and here r denotes the total particle radius, Rc + t, and z is the width parameter of the Schulz distribution, Equation 3.

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, qmin = 0.001 Å-1, qmax = 0.7 Å-1 and the above default parameter values.

Poly Core Shell Ratio Example Dataset