2.1.1.4. Cylinder¶
Right circular cylinder with uniform scattering length density.
| Parameter | Description | Units | Default value |
|---|---|---|---|
| scale | Source intensity | None | 1 |
| background | Source background | cm-1 | 0 |
| sld | Cylinder scattering length density | 10-6Å-2 | 4 |
| solvent_sld | Solvent scattering length density | 10-6Å-2 | 1 |
| radius | Cylinder radius | Å | 20 |
| length | Cylinder length | Å | 400 |
| theta | In plane angle | degree | 60 |
| phi | Out of plane angle | degree | 60 |
The returned value is scaled to units of cm-1.
The form factor is normalized by the particle volume.
For information about polarised and magnetic scattering, click here_.
Definition¶
The output of the 2D scattering intensity function for oriented cylinders is given by (Guinier, 1955)
where
and \(\alpha\) is the angle between the axis of the cylinder and \(\vec q\), \(V\) 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.
To provide easy access to the orientation of the cylinder, we define the axis of the cylinder using two angles \(\theta\) and \(\phi\). Those angles are defined in figure 1.
Figure 1: Definition of the angles for oriented cylinders.
Figure 2: Examples of the angles for oriented cylinders against the detector plane.
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.
The output of the 1D scattering intensity function for randomly oriented cylinders is then given by
The theta and phi parameters are not used for the 1D output. Our implementation of the scattering kernel and the 1D scattering intensity use the c-library from NIST.
Validation¶
Validation of our code was done by comparing the output of the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure 3 shows a comparison of the 1D output of our model and the output of the NIST software.
Figure 3: Comparison of the SasView scattering intensity for a cylinder with the output of the NIST SANS analysis software. The parameters were set to: scale = 1.0, radius = 20 Å, length = 400 Å, contrast = 3e-6 Å-2, and background = 0.01 cm-1.
In general, averaging over a distribution of orientations is done by evaluating the following
where \(p(\theta,\phi)\) is the probability distribution for the orientation and \(P_0(Q,\alpha)\) is the scattering intensity for the fully oriented system. Since we have no other software to compare the implementation of the intensity for fully oriented cylinders, we can compare the result of averaging our 2D output using a uniform distribution \(p(\theta, \phi) = 1.0\). Figure 4 shows the result of such a cross-check.
Figure 4: Comparison of the intensity for uniformly distributed cylinders calculated from our 2D model and the intensity from the NIST SANS analysis software. The parameters used were: scale = 1.0, radius = 20 Å, length = 400 Å, contrast = 3e-6 Å-2, and background = 0.0 cm-1.