Form Factor:Cylindrical symmetry

Derivation

Form Factor

Assuming a particle is cylindrically-symmetric:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho(\mathbf{r}) = \rho(r,\phi,z) = \rho(r)}

We of course use cylindrical coordinates:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}=(x,y,z)=(r \cos\phi, r \sin\phi, z) }

We can take advantage of the cylindrical symmetry by rotating any candidate q-vector into the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_x,q_z} plane, eliminating the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_y} component:

$\mathbf {q} =(q_{x},0,q_{z})$

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{alignedat}{2}\mathbf {q} \cdot \mathbf {r} &=q_{x}x+q_{y}y+q_{z}z\\&=q_{x}r\cos \phi +q_{z}z\\\end{alignedat}}}

The form factor is (note that the integration limits in z define the particle size in that direction):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} F(\mathbf{q}) & = \int \rho(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V \\ & = \int\limits_{z=0}^{L}\int\limits_{\phi=0}^{2 \pi}\int\limits_{r=0}^{\infty} \rho(r) e^{i \mathbf{q} \cdot \mathbf{r} } r \mathrm{d}r \mathrm{d}\phi \mathrm{d}z \\ & = \int\limits_{0}^{L}\int\limits_{0}^{2 \pi}\int\limits_{0}^{\infty} \rho(r) e^{i (q_x r \cos \phi + q_z z) } r \mathrm{d}r \mathrm{d}\phi \mathrm{d}z \\ & = \left( \int\limits_{0}^{L} e^{i q_z z } \mathrm{d}z \right) \int\limits_{0}^{\infty} r \rho(r) \left ( \int\limits_{0}^{2 \pi} e^{i q_x r \cos \phi } \mathrm{d}\phi \right) \mathrm{d}r \\ & = \left( \left[ \frac{1}{i q_z}e^{i q_z z} \right]_{z=0}^{L} \right) \int\limits_{0}^{\infty} r \rho(r) \left ( \int\limits_{0}^{2 \pi} e^{i q_x r \cos \phi } \mathrm{d}\phi \right) \mathrm{d}r \\ & = \left( \frac{e^{i q_z L} - 1}{i q_z} \right) \int\limits_{0}^{\infty} r \rho(r) \left ( \int\limits_{0}^{2 \pi} e^{i q_x r \cos \phi } \mathrm{d}\phi \right) \mathrm{d}r \\ \end{alignat} }