.. _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 **Authorship and Verification** * **Author:** * **Last Modified by:** * **Last Reviewed by:**