# cylinder¶

Right circular cylinder with uniform scattering length density.

Parameter

Description

Units

Default value

scale

Scale factor or Volume fraction

None

1

background

Source background

cm-1

0.001

sld

Cylinder scattering length density

10-6-2

4

sld_solvent

Solvent scattering length density

10-6-2

1

20

length

Cylinder length

400

theta

cylinder axis to beam angle

degree

60

phi

degree

60

The returned value is scaled to units of cm-1 sr-1, absolute scale.

For information about polarised and magnetic scattering, see the Polarisation/Magnetic Scattering documentation.

Definition

The output of the 2D scattering intensity function for oriented cylinders is given by (Guinier, 1955)

$I(q,\alpha) = \frac{\text{scale}}{V} F^2(q,\alpha) + \text{background}$

where

$F(q,\alpha) = 2 (\Delta \rho) V \frac{\sin \left(\tfrac12 qL\cos\alpha \right)} {\tfrac12 qL \cos \alpha} \frac{J_1 \left(q R \sin \alpha\right)}{q R \sin \alpha}$

and $$\alpha$$ is the angle between the axis of the cylinder and $$\vec q$$, $$V =\pi R^2L$$ is the volume of the cylinder, $$L$$ is the length of the cylinder, $$R$$ is the radius of the cylinder, and $$\Delta\rho$$ (contrast) is the scattering length density difference between the scatterer and the solvent. $$J_1$$ is the first order Bessel function.

For randomly oriented particles:

$P(q)=F^2(q)=\int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)d\alpha}$

The output of the 1D scattering intensity function for randomly oriented cylinders is thus given by

$I(q) = \frac{\text{scale}}{V} \int_0^{\pi/2} F^2(q,\alpha) \sin \alpha\ d\alpha + \text{background}$

NB: The 2nd virial coefficient of the cylinder is calculated based on the radius and length values, and used as the effective radius for $$S(q)$$ when $$P(q) \cdot S(q)$$ is applied.

For 2d scattering from oriented cylinders, we define the direction of the axis of the cylinder using two angles $$\theta$$ (note this is not the same as the scattering angle used in q) and $$\phi$$. Those angles are defined in Fig. 21 , for further details see Oriented Particles.

The $$\theta$$ and $$\phi$$ parameters to orient the cylinder only appear in the model when fitting 2d data.

Validation

Validation of the code was done by comparing the output of the 1D model to the output of the software provided by the NIST (Kline, 2006). The implementation of the intensity for fully oriented cylinders was done by averaging over a uniform distribution of orientations using

$P(q) = \int_0^{\pi/2} d\phi \int_0^\pi p(\theta) P_0(q,\theta) \sin \theta\ d\theta$

where $$p(\theta,\phi) = 1$$ is the probability distribution for the orientation and $$P_0(q,\theta)$$ is the scattering intensity for the fully oriented system, and then comparing to the 1D result.

Source

cylinder.py $$\ \star\$$ cylinder.c $$\ \star\$$ gauss76.c $$\ \star\$$ sas_J1.c $$\ \star\$$ polevl.c

References

1. Pedersen, Adv. Colloid Interface Sci., 70 (1997) 171-210

1. Fournet, Bull. Soc. Fr. Mineral. Cristallogr., 74 (1951) 39-113

1. Onsager, Ann. New York Acad. Sci., 51 (1949) 627-659

Authorship and Verification

• Author: