Changeset adc753d in sasmodels


Ignore:
Timestamp:
Oct 6, 2016 6:12:24 PM (5 years ago)
Author:
butler
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:

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  • sasmodels/models/core_shell_bicelle.py

    ra0fee3b radc753d  
    22Definition 
    33---------- 
     4 
    45This model provides the form factor for a circular cylinder with a 
    56core-shell scattering length density profile. Thus this is a variation 
     
    89factor is normalized by the particle volume. 
    910 
    10 .. _core-shell-bicelle-geometry: 
    1111 
    1212.. figure:: img/core_shell_bicelle_geometry.png 
     
    2121   and core regions in order to estimate appropriate starting parameters. 
    2222 
     23Given the scattering length densities (sld) $\rho_c$, the core sld, $\rho_f$, 
     24the face sld, $\rho_r$, the rim sld and $\rho_s$ the solvent sld, the 
     25scattering length density variation along the cylinder axis is: 
     26 
     27.. math:: 
     28 
     29    \rho(r) =  
     30      \begin{cases}  
     31      &\rho_c \text{ for } 0 \lt r \lt R; -L \lt z\lt L \\[1.5ex] 
     32      &\rho_f \text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; 
     33      L \lt z\lt (L+2t) \\[1.5ex] 
     34      &\rho_r\text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; L \lt z\lt (L+2t) 
     35      \end{cases} 
     36 
     37The form factor for the bicelle is calculated in cylindrical coordinates, where 
     38$\alpha$ is the angle between the $Q$ vector and the cylinder axis, to give: 
     39 
     40.. math:: 
     41 
     42    I(Q,\alpha) = \frac{\text{scale}}{V} \cdot 
     43        F(Q,\alpha)^2 + \text{background} 
     44where 
     45 
     46.. math:: 
     47 
     48    \begin{align}     
     49    F(Q,\alpha) = &\frac{1}{V_t} \bigg[  
     50    (\rho_c - \rho_f) V_c \frac{J_1(QRsin \alpha)}{QRsin\alpha}\frac{2 \cdot QLcos\alpha}{QLcos\alpha} \\ 
     51    &+(\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} \\ 
     52    &+(\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} 
     53    \bigg] 
     54    \end{align}  
     55 
     56where $V_t$ is the total volume of the bicelle, $V_c$ the volume of the core, 
     57$V_{c+f}$ the volume of the core plus the volume of the faces, $R$ is the radius 
     58of the core, $L$ the length of the core, $t_f$ the thickness of the face, $t_r$ 
     59the thickness of the rim and $J_1$ the usual first order bessel function. 
     60 
    2361The output of the 1D scattering intensity function for randomly oriented 
    24 cylinders is then given by the equation above. 
     62cylinders is then given by integrating over all possible $\theta$ and $\phi$. 
    2563 
    2664The *theta* and *phi* parameters are not used for the 1D output. 
     
    4078---------- 
    4179 
    42 .. [#Matusmori] `N Matsumori and M Murata <http://dx.doi.org/10.1039/C0NP0000 
    43    2G>`_, *Nat. Prod. Rep.* 27 (2010) 1480-1492 
    44 .. [#] L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and 
    45    Neutron Scattering,* Plenum Press, New York, (1987) 
     80.. [#] D Singh (2009). *Small angle scattering studies of self assembly in 
     81   lipid mixtures*, John's Hopkins University Thesis (2009) 223-225. `Available 
     82   from Proquest <http://search.proquest.com/docview/304915826?accountid 
     83   =26379>`_ 
    4684 
    4785Authorship and Verification 
     
    4987 
    5088* **Author:** NIST IGOR/DANSE **Date:** pre 2010 
    51 * **Last Modified by:** Paul Butler **Date:** Septmber 30, 2016 
    52 * **Last Reviewed by:** Under Review **Date:** October 5, 2016 
    53  
     89* **Last Modified by:** Paul Butler **Date:** September 30, 2016 
     90* **Last Reviewed by:** Richard Heenan **Date:** October 5, 2016 
    5491""" 
    5592 
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