MODEL

Cylinder

ORIGINAL 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 monodisperse right circular
cylinder with uniform scattering length density. The form factor
is normalized by the particle volume such that P(*q*) =
scale*<*f*^{2}>/Vol + bkg, where < >
is 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 | Radius (Å) | 20.0 |

2 | Length (Å) | 400.0 |

3 | Contrast (Å^{-2}) | 3.0e-6 |

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

USAGE NOTES

The function calculated is:

and

where J_{1}(*x*) is the first order Bessel function.
Alpha is defined as the angle between the cylinder axis and the
scattering vector, **q**. The integral over alpha averages the form
factor over all possible orientations of the cylinder with respect
to **q**. The returned value is in units of [cm^{-1}], on
absolute scale.

Scattering contrast (*Dr*) = SLD (cylinder) - SLD (solvent)

Parameter[0] (scale) and Parameter[3] (contrast) are both multiplicative factors in the model and are perfectly correlated. One or both of these parameters must be held fixed during model fitting.

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.

76-point Gaussian quadrature is used to calculate the form factor. This calcualtion will be slow (approximately 10 seconds) on older machines. The smearing calculation is an additional numerical integration (20-point quadrature) and will further slow the update of the graph.

REFERENCE

Guinier, A. and G. Fournet, "Small-Angle Scattering of X-Rays", John Wiley and Sons, New York, (1955).

TEST DATASET

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