.. _capped-cylinder: capped_cylinder ======================================================= Right circular cylinder with spherical end caps and uniform SLD =========== ================================== ============ ============= 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 radius_cap Cap 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. **Definitions** Calculates the scattering from a cylinder with spherical section end-caps. Like :ref:`barbell`, this is a sphereocylinder with end caps that have a radius larger than that of the cylinder, but with the center of the end cap radius lying within the cylinder. This model simply becomes a convex lens when the length of the cylinder $L=0$. See the diagram for the details of the geometry and restrictions on parameter values. .. figure:: img/capped_cylinder_geometry.jpg Capped cylinder geometry, where $r$ is *radius*, $R$ is *radius_cap* and $L$ is *length*. Since the end cap radius $R \geq r$ and by definition for this geometry $h \le 0$, $h$ is then defined by $r$ and $R$ as $h = -\sqrt{R^2 - r^2}$ The scattered intensity $I(q)$ is calculated as .. math:: I(q) = \frac{\Delta \rho^2}{V} \left where the amplitude $A(q,\alpha)$ with the rod axis at angle $\alpha$ to $q$ is given as .. math:: A(q) =&\ \pi r^2L \frac{\sin\left(\tfrac12 qL\cos\alpha\right)} {\tfrac12 qL\cos\alpha} \frac{2 J_1(qr\sin\alpha)}{qr\sin\alpha} \\ &\ + 4 \pi R^3 \int_{-h/R}^1 dt \cos\left[ q\cos\alpha \left(Rt + h + {\tfrac12} L\right)\right] \times (1-t^2) \frac{J_1\left[qR\sin\alpha \left(1-t^2\right)^{1/2}\right]} {qR\sin\alpha \left(1-t^2\right)^{1/2}} The $\left<\ldots\right>$ brackets denote an average of the structure over all orientations. $\left< A^2(q)\right>$ is then the form factor, $P(q)$. The scale factor is equivalent to the volume fraction of cylinders, each of volume, $V$. Contrast $\Delta\rho$ is the difference of scattering length densities of the cylinder and the surrounding solvent. The volume of the capped cylinder is (with $h$ as a positive value here) .. math:: V = \pi r_c^2 L + 2\pi\left(\tfrac23R^3 + R^2h - \tfrac13h^3\right) and its radius of gyration is .. math:: R_g^2 =&\ \left[ \tfrac{12}{5}R^5 + R^4\left(6h+\tfrac32 L\right) + R^3\left(4h^2 + L^2 + 4Lh\right) + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\ &\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3 + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right] \left( 4R^3 + 6R^2h - 2h^3 + 3r^2L \right)^{-1} .. note:: The requirement that $R \geq r$ is not enforced in the model! It is up to you to restrict this during analysis. The 2D scattering intensity is calculated similar to the 2D cylinder model. .. figure:: img/cylinder_angle_definition.png Definition of the angles for oriented 2D cylinders. .. figure:: img/capped_cylinder_autogenfig.png 1D and 2D plots corresponding to the default parameters of the model. **Source** :download:`capped_cylinder.py ` $\ \star\ $ :download:`capped_cylinder.c ` $\ \star\ $ :download:`gauss76.c ` $\ \star\ $ :download:`sas_J1.c ` $\ \star\ $ :download:`polevl.c ` **References** #. H Kaya, *J. Appl. Cryst.*, 37 (2004) 223-230 #. H Kaya and N R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda and errata) #. L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659 **Authorship and Verification** * **Author:** NIST IGOR/DANSE **Date:** pre 2010 * **Last Modified by:** Paul Butler **Date:** September 30, 2016 * **Last Reviewed by:** Richard Heenan **Date:** January 4, 2017