.. _vesicle:

vesicle
=======================================================

Vesicle model representing a hollow sphere

=========== ======================================= ============ =============
Parameter   Description                             Units        Default value
=========== ======================================= ============ =============
scale       Scale factor or Volume fraction         None                     1
background  Source background                       |cm^-1|              0.001
sld         vesicle shell scattering length density |1e-6Ang^-2|           0.5
sld_solvent solvent scattering length density       |1e-6Ang^-2|          6.36
volfraction volume fraction of shell                None                  0.05
radius      vesicle core radius                     |Ang|                  100
thickness   vesicle shell thickness                 |Ang|                   30
=========== ======================================= ============ =============

The returned value is scaled to units of |cm^-1| |sr^-1|, absolute scale.


**Definition**

This model provides the form factor, *P(q)*, for an unilamellar vesicle and is
effectively identical to the hollow sphere reparameterized to be
more intuitive for a vesicle and normalizing the form factor by the volume of
the shell. The 1D scattering intensity is calculated in the following way
[#Guinier1955]_:

.. math::

    P(q) = \frac{\phi}{V_\text{shell}} \left[
           \frac{3V_{\text{core}}({\rho_{\text{solvent}}
           - \rho_{\text{shell}})j_1(qR_{\text{core}})}}{qR_{\text{core}}}
           + \frac{3V_{\text{tot}}(\rho_{\text{shell}}
           - \rho_{\text{solvent}}) j_1(qR_{\text{tot}})}{qR_{\text{tot}}}
           \right]^2 + \text{background}


where $\phi$ is the volume fraction of shell material, $V_{shell}$ is the volume
of the shell, $V_{\text{cor}}$ is the volume of the core, $V_{\text{tot}}$ is
the total volume, $R_{\text{core}}$ is the radius of the core, $R_{\text{tot}}$
is the outer radius of the shell, $\rho_{\text{solvent}}$ is the scattering
length density of the solvent (which is the same as for the core in this case),
$\rho_{\text{scale}}$ is the scattering length density of the shell, background
is a flat background level (due for example to incoherent scattering in the
case of neutrons), and $j_1$ is the spherical Bessel function
$j_1 = (\sin(x) - x \cos(x))/ x^2$.

The functional form is identical to a "typical" core-shell structure, except
that the scattering is normalized by the volume that is contributing to the
scattering, namely the volume of the shell alone, the scattering length density
of the core is fixed the same as that of the solvent, the scale factor when the
data are on an absolute scale is equivalent to the volume fraction of material
in the shell rather than the entire core+shell sphere, and the parameterization
is done in terms of the core radius = $R_{\text{core}}$ and the shell
thickness = $R_{\text{tot}} - R_{\text{core}}$.

.. figure:: img/vesicle_geometry.jpg

    Vesicle geometry.

The 2D scattering intensity is the same as *P(q)* above, regardless of the
orientation of the *q* vector which is defined as

.. math::

    q = \sqrt{q_x^2 + q_y^2}


NB: The outer most radius (= *radius* + *thickness*) is used as the effective
radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied.



.. figure:: img/vesicle_autogenfig.png

    1D plot corresponding to the default parameters of the model.


**Source**

:download:`vesicle.py <src/vesicle.py>`
$\ \star\ $ :download:`vesicle.c <src/vesicle.c>`
$\ \star\ $ :download:`sas_3j1x_x.c <src/sas_3j1x_x.c>`

**References**

.. [#Guinier1955] A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*,
   John Wiley and Sons, New York, (1955)

**Authorship and Verification**

* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Paul Butler **Date:** March 20, 2016
* **Last Reviewed by:** Paul Butler **Date:** September 7, 2018

