number of steps in each interface (must be an odd integer)
None
35
The returned value is scaled to units of cm-1 sr-1, absolute scale.
Definition
Similarly to the onion, this model provides the form factor, \(P(q)\), for
a multi-shell sphere, where the interface between the each neighboring
shells can be described by the error function, power-law, or exponential
functions. The scattering intensity is computed by building a continuous
custom SLD profile along the radius of the particle. The SLD profile is
composed of a number of uniform shells with interfacial shells between them.
Unlike the onion model (using an analytical integration), the interfacial
shells here are sub-divided and numerically integrated assuming each
sub-shell is described by a line function, with n_steps sub-shells per
interface. The form factor is normalized by the total volume of the sphere.
Note
n_shells must be an integer. n_steps must be an ODD integer.
Interface shapes are as follows:
0: erf(\(\nu z\))
1: Rpow(\(z^\nu\))
2: Lpow(\(z^\nu\))
3: Rexp(\(-\nu z\))
4: Lexp(\(-\nu z\))
5: Boucher (\((1-z^2)^(\nu/2-2)\))
The form factor \(P(q)\) in 1D is calculated by [1]:
\[P(q) = \frac{f^2}{V_\text{particle}} \text{ where }
f = f_\text{core} + \sum_{\text{inter}_i=0}^N f_{\text{inter}_i} +
\sum_{\text{flat}_i=0}^N f_{\text{flat}_i} +f_\text{solvent}\]
For a spherically symmetric particle with a particle density \(\rho_x(r)\)
the sld function can be defined as:
Here we assumed that the SLDs of the core and solvent are constant in \(r\).
The SLD at the interface between shells, \(\rho_{\text {inter}_i}\)
is calculated with a function chosen by an user, where the functions are
Exp:
\[\begin{split}\rho_{{inter}_i}(r) &=
\begin{cases}
B\, \exp\left(
\frac{\pm A(r - r_{\text{flat}_i})}{\Delta t_{\text{inter}_i}}
\right) + C & \mbox{for } A \neq 0 \\
B\, \left(
\frac{(r - r_{\text{flat}_i})}{\Delta t_{\text{inter}_i}}
\right) + C & \mbox{for } A = 0 \\
\end{cases}\end{split}\]
The functions are normalized so that they vary between 0 and 1, and they are
constrained such that the SLD is continuous at the boundaries of the interface
as well as each sub-shell. Thus B and C are determined.
Once \(\rho_{\text{inter}_i}\) is found at the boundary of the sub-shell of the
interface, we can find its contribution to the form factor \(P(q)\)
