Ignore:
Timestamp:
Oct 6, 2016 6:12:24 PM (5 years ago)
Branches:
master, core_shell_microgels, costrafo411, magnetic_model, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
Children:
b3f2a24
Parents:
a86b9f4
Message:

bicelle model documentation standardized, ref to original derivation and
equations added to rst documentation. re #696 and #646

File:
1 edited

### Legend:

Unmodified
 ra0fee3b Definition ---------- This model provides the form factor for a circular cylinder with a core-shell scattering length density profile. Thus this is a variation factor is normalized by the particle volume. .. _core-shell-bicelle-geometry: .. figure:: img/core_shell_bicelle_geometry.png and core regions in order to estimate appropriate starting parameters. Given the scattering length densities (sld) $\rho_c$, the core sld, $\rho_f$, the face sld, $\rho_r$, the rim sld and $\rho_s$ the solvent sld, the scattering length density variation along the cylinder axis is: .. math:: \rho(r) = \begin{cases} &\rho_c \text{ for } 0 \lt r \lt R; -L \lt z\lt L \\[1.5ex] &\rho_f \text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; L \lt z\lt (L+2t) \\[1.5ex] &\rho_r\text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; L \lt z\lt (L+2t) \end{cases} The form factor for the bicelle is calculated in cylindrical coordinates, where $\alpha$ is the angle between the $Q$ vector and the cylinder axis, to give: .. math:: I(Q,\alpha) = \frac{\text{scale}}{V} \cdot F(Q,\alpha)^2 + \text{background} where .. math:: \begin{align} F(Q,\alpha) = &\frac{1}{V_t} \bigg[ (\rho_c - \rho_f) V_c \frac{J_1(QRsin \alpha)}{QRsin\alpha}\frac{2 \cdot QLcos\alpha}{QLcos\alpha} \\ &+(\rho_f - \rho_r) V_{c+f} \frac{J_1(QRsin\alpha)}{QRsin\alpha}\frac{2 \cdot Q(L+t_f)cos\alpha}{Q(L+t_f)cos\alpha} \\ &+(\rho_r - \rho_s) V_t \frac{J_1(Q(R+t_r)sin\alpha)}{Q(R+t_r)sin\alpha}\frac{2 \cdot Q(L+t_f)cos\alpha}{Q(L+t_f)cos\alpha} \bigg] \end{align} where $V_t$ is the total volume of the bicelle, $V_c$ the volume of the core, $V_{c+f}$ the volume of the core plus the volume of the faces, $R$ is the radius of the core, $L$ the length of the core, $t_f$ the thickness of the face, $t_r$ the thickness of the rim and $J_1$ the usual first order bessel function. The output of the 1D scattering intensity function for randomly oriented cylinders is then given by the equation above. cylinders is then given by integrating over all possible $\theta$ and $\phi$. The *theta* and *phi* parameters are not used for the 1D output. ---------- .. [#Matusmori] N Matsumori and M Murata _, *Nat. Prod. Rep.* 27 (2010) 1480-1492 .. [#] L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering,* Plenum Press, New York, (1987) .. [#] D Singh (2009). *Small angle scattering studies of self assembly in lipid mixtures*, John's Hopkins University Thesis (2009) 223-225. Available from Proquest _ Authorship and Verification * **Author:** NIST IGOR/DANSE **Date:** pre 2010 * **Last Modified by:** Paul Butler **Date:** Septmber 30, 2016 * **Last Reviewed by:** Under Review **Date:** October 5, 2016 * **Last Modified by:** Paul Butler **Date:** September 30, 2016 * **Last Reviewed by:** Richard Heenan **Date:** October 5, 2016 """