MODEL

HollowCylinder

AUTHOR/MODIFICATION

Steve Kline 20 JAN 1999

Alan Munter 08 JUL 1999, converted to Java

Alan Munter 20 OCT 2000, corrected a calculation error.

APPROVED FOR DISTRIBUTION

DESCRIPTION

Calculates the form factor for a monodisperse, hollow, right
circular cylinder (or a tube). The inside and outside of the tube
have the same scattering length density and the tube itself is of
uniform SLD. The form factor is normalized by the tube material
volume ONLY such that P(*q*) = scale*<*f***f*>/Vol
+ bkg, where *f* is the scattering amplitude and the < >
denote an average over all possible orientations of the cylinder.

Resolution smeared version is also provided.

VARIABLES

Input Variables (default values):

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

0 | Scale | 1.0 |

1 | Core Radius (Å) | 20.0 |

2 | Shell Radius (Å) | 30.0 |

3 | Length (Å) | 400.0 |

4 | Contrast (Å^{-2}) | 3.0e-6 |

5 | Incoherent Background (cm^{-1}) | 0.0 |

**USAGE** **NOTES**

The function calculated is:

where J_{1} is the first order Bessel function. The
integral over *x* is the orientaional average and the returned
form factor is scaled to units of [cm^{-1}].

contrast = SLD (shell) - SLD (solvent)

The shell thickness is uniform over the cylinder radius. There is
no material covering the ends of the tube. This mean that for the
default case above, the total *diameter* of the cylinder is
2*30 = 60 Å and that the total *length* is 400 Å.
The 40 Å diameter core is assumed to be the same scattering
length density as the solvent, giving no contribution to the scattered
intensity.

The form factor is normalized to the volume of material comprising the shell such that . Note that this is different than the total excluded volume of the hollow cylinder, which is .

Parameter[0] (scale) and Parameter[4] (contrast) are multiplicative factors in the model and are perfectly correlated. Only one of these parameters should be left free during model fitting.

The user should ensure that the shell radius is always larger than the core radius.

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

Feigin, L. A, and D. I. Svergun, "Structure Analysis by Small-Angle X-Ray and Neutron Scattering", Plenum Press, New York, (1987).

TEST DATASET

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