.. _cylinder:
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 |1e-6Ang^-2| 4
sld_solvent Solvent scattering length density |1e-6Ang^-2| 1
radius Cylinder radius |Ang| 20
length Cylinder length |Ang| 400
theta cylinder axis to beam angle degree 60
phi rotation about beam 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 :ref:`magnetism` documentation.
**Definition**
The output of the 2D scattering intensity function for oriented cylinders is
given by (Guinier, 1955)
.. math::
P(q,\alpha) = \frac{\text{scale}}{V} F^2(q,\alpha).sin(\alpha) + \text{background}
where
.. math::
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:
.. math::
F^2(q)=\int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)d\alpha}=\int_{0}^{1}{F^2(q,u)du}
Numerical integration is simplified by a change of variable to $u = cos(\alpha)$
with $sin(\alpha)=\sqrt{1-u^2}$.
The output of the 1D scattering intensity function for randomly oriented
cylinders is thus given by
.. math::
P(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
:numref:`cylinder-angle-definition` , for further details see
:ref:`orientation`.
.. _cylinder-angle-definition:
.. figure:: img/cylinder_angle_definition.png
Angles $\theta$ and $\phi$ orient the cylinder relative to the beam line
coordinates, where the beam is along the $z$ axis. Rotation $\theta$,
initially in the $xz$ plane, is carried out first, then rotation $\phi$
about the $z$ axis. Orientation distributions are described as rotations
about two perpendicular axes $\delta_1$ and $\delta_2$ in the frame of
the cylinder itself, which when $\theta = \phi = 0$ are parallel to the
$Y$ and $X$ axes.
.. figure:: img/cylinder_angle_projection.png
Examples for oriented cylinders.
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
.. math::
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.
.. figure:: img/cylinder_autogenfig.png
1D and 2D plots corresponding to the default parameters of the model.
**Source**
:download:`cylinder.py `
$\ \star\ $ :download:`cylinder.c `
$\ \star\ $ :download:`gauss76.c `
$\ \star\ $ :download:`sas_J1.c `
$\ \star\ $ :download:`polevl.c `
**References**
#. J. Pedersen, *Adv. Colloid Interface Sci.*, 70 (1997) 171-210
#. G. Fournet, *Bull. Soc. Fr. Mineral. Cristallogr.*, 74 (1951) 39-113
#. L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659
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