spherical_sld¶
Spherical SLD intensity calculation
Parameter |
Description |
Units |
Default value |
---|---|---|---|
scale |
Scale factor or Volume fraction |
None |
1 |
background |
Source background |
cm-1 |
0.001 |
n_shells |
number of shells (must be integer) |
None |
1 |
sld_solvent |
solvent sld |
10-6Å-2 |
1 |
sld[n_shells] |
sld of the shell |
10-6Å-2 |
4.06 |
thickness[n_shells] |
thickness shell |
Å |
100 |
interface[n_shells] |
thickness of the interface |
Å |
50 |
shape[n_shells] |
interface shape |
None |
0 |
nu[n_shells] |
interface shape exponent |
None |
2.5 |
n_steps |
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.

Fig. 90 Figure 1: Example SLD profile¶
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(\(-\nu z\))
The form factor \(P(q)\) in 1D is calculated by:
For a spherically symmetric particle with a particle density \(\rho_x(r)\) the sld function can be defined as:
so that individual terms can be calculated as follows:
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:
Power-Law:
Erf:
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 \(P(q)\)
where
We assume \(\rho_{\text{inter}_j} (r)\) is approximately linear within the sub-shell \(j\).
Finally the form factor can be calculated by
For 2D data the scattering intensity is calculated in the same way as 1D, where the \(q\) vector is defined as
Note
The outer most radius is used as the effective radius for \(S(Q)\) when \(P(Q) * S(Q)\) is applied.

Fig. 91 Figure 2: 1D plot corresponding to the default parameters of the model.¶
Source
spherical_sld.py
\(\ \star\ \) spherical_sld.c
\(\ \star\ \) lib/sas_3j1x_x.c
\(\ \star\ \) lib/sas_erf.c
\(\ \star\ \) lib/polevl.c
References
L A Feigin and D I Svergun, Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum Press, New York, (1987)
Authorship and Verification
Author: Jae-Hie Cho Date: Nov 1, 2010
Last Modified by: Paul Kienzle Date: Dec 20, 2016
Last Reviewed by: Steve King Date: March 29, 2019