Displaced core-shell ellipsoids

This model calculates the scattering intensity from core-shell ellipsoids where the core centre is displaced relative to the shell centre. The model accounts for:
- Size polydispersity (Gaussian distribution of particle radius, 50-point quadrature)
- Orientation averaging (50-point angular quadrature)
- Hard-sphere structure factor (Percus-Yevick closure)
- Debye (Gaussian chain) background contribution from free polymer

**Model Geometry**

The core is an ellipsoid with equatorial semi-axis *R* and aspect ratio *eps* (semi-axes *R*, *R*, *R eps*). The shell is a coaxial ellipsoid whose equatorial outer radius is *R + d_shell*, where
$$
d_\text{shell} = |d| + 2\,|\sigma_\text{out}|
$$
with *d* = ``thickness_shell`` and $`\sigma_\text{out}$ = ``sigma_outer``. The core centre is offset by ``displacement`` in the direction perpendicular to the symmetry axis.

**Mathematical Basis**

The scattering intensity returned by the kernel (before SASView applies the global ``scale`` and ``background``) is:
$$
I(q) = \bigl[ P(q) + \langle F(q)\rangle^2\,(S(q)-1) \bigr]\,B(q)
+ I_\text{poly}(q)
$$
where
$$
P(q) = \frac{\langle F^2(q)\rangle}{F_0^2}, \qquad
B(q) = 1 + A_{10}\!\left(\frac{q_0}{q}\right)^{\!m},
\qquad m = |A_{11}|+2, \quad q_0 = 0.001\,\text{Ang}^{-1}
$$
and

- $P(q)$ — size- and orientation-averaged normalised form factor
- $\langle F(q)\rangle/F_{\theta}$ — mean reduced amplitude (used with S(q))
- $S(q)$ — Percus-Yevick hard-sphere structure factor
- $B(q)$ — empirical power-law low-*q* correction
- $I_\text{poly}(q)$ — Debye scattering from Gaussian polymer chains

The global intensity is then:
$$
I_\text{total}(q) = \text{scale}\cdot I(q) + \text{background}
$$
**Parameter Definitions**

``scale`` and ``background``
Added automatically by SASView. ``scale`` is an overall multiplicative factor (units of the desired absolute intensity scale). ``background`` is a flat incoherent baseline in cm\:sup:`-1`.

``radius`` (*R*, Ang)
Mean equatorial semi-axis of the core ellipsoid, i.e. the half-length along the two axes perpendicular to the symmetry axis. This is the
centre of the Gaussian size distribution. Typical values: 5–200 Ang.

``aspect_ratio`` (*eps* = c/a, dimensionless)
Ratio of the polar semi-axis to the equatorial semi-axis of the core.

- eps = 1 -> sphere
- eps > 1 -> prolate (cigar-shaped)
- eps < 1 -> oblate (disc-shaped)

The outer shell is constructed so that it has the same equatorial thickness *d_shell* everywhere: the shell's own aspect ratio is *(R·eps + d_shell) / (R + d_shell)*, which differs from eps unless d_shell = 0.

``volfraction_hs`` (*eta*, dimensionless)
Volume fraction of particles used in the Percus-Yevick hard-sphere structure factor *S(q)*. Set to 0 to suppress inter-particle interactions (i.e. *S(q)* = 1). Physical upper limit for random close packing is ~ 0.64.

``radius_hs`` (*R*\ :sub:`HS`, Ang)
Effective radius of the hard-sphere potential in the Percus-Yevick calculation. This is the centre-to-centre distance at contact and need not equal the geometric particle radius; it is often larger to account for a hydration layer, electrostatic repulsion, or a polymer brush. Only meaningful when ``volfraction_hs`` > 0.

``thickness_shell`` (*d*, Ang)
Equatorial thickness of the shell layer. The quantity actually used in the geometry is the effective shell extent
$$
d_\text{shell} = |d| + 2\,|\sigma_\text{out}|
$$
so the outer radius of the ellipsoid is *R + d_shell*. Note that ``sigma_outer`` also contributes to the shell extent (see below).

``contrast_shell`` (*A*\ :sub:`8`, dimensionless)
Ratio of the shell SLD contrast to the core SLD contrast, $\Delta\rho_\text{shell}/\Delta\rho_\text{core}$, where each $`\Delta\rho$ is measured relative to the solvent. The core form factor amplitude is weighted by (1 − contrast_shell):
$$
F(q) = V_\text{out}\,\phi(q\,r_\text{out})
- V_\text{core}\,(1 - A_8)\,\phi(q\,r_\text{core})
$$
Limiting cases:

- A_8 = 0 — the shell is invisible ($\delta\rho_shell = 0$); the particle appears as a hollow ellipsoid.
- A_8 = 1 — core and shell have equal contrast; the particle looks uniform (solid ellipsoid, no internal interface visible).
- A_8 < 0 — shell and core scatter with opposite sign (e.g.a lipid membrane in D\ :sub:`2`\ O where the hydrocarbon core has negative contrast and the head-group shell has positive contrast).

``sigma_rel`` (*sigma*\ :sub:`R`/R, dimensionless)
Relative standard deviation of the Gaussian size distribution of the core radius: :math:`\sigma_R = \sigma_\text{rel}\cdot R`. The distribution is sampled symmetrically over 3 sigma\ :sub:`R` using 50 Gauss points. Example: ``sigma_rel`` = 0.05 means 5 percent polydispersity. Set to 0 for a monodisperse sample (the kernel uses a minimum width of 10\ :sup:`-4`*R to avoid division by zero).

``power_law`` (*A*\ :sub:`10`, dimensionless)
Amplitude of the empirical low-*q* power-law correction
$$
B(q) = 1 + A_{10}\!\left(\frac{q_0}{q}\right)^{\!m}
$$
with *q*\ :sub:`0` = 0.001 Ang\ :sup:`-1` (fixed reference wavevector) and *m* = |``exponent_2``| + 2. This term accounts for scattering from large-scale structures (e.g. fractal aggregates or density fluctuations) that dominate at very low *q*. Set to 0 to disable.

``exponent_2`` (*A*\ :sub:`11`, dimensionless)
Controls the exponent of the power-law correction through *m* = |A\ :sub:`11`| + 2. The absolute value ensures *m* >= 2, which enforces a Porod-like decay at high *q* and prevents the correction from diverging. Larger values of |A\ :sub:`11`| confine the correction to a narrower low-*q* range.

``sigma_outer`` (*sigma*\ :sub:`out`, Ang)
Debye–Waller (Gaussian) roughness of the outer shell surface. The outer form factor amplitude is multiplied by $\exp(-\tfrac{1}{2}(q\,\sigma_\text{out})^2)$, smearing the outer interface. *sigma*\ :sub:`out` also adds 2 sigma\ :sub:`out` to the effective shell thickness *d_shell*. Set to 0 for a perfectly sharp outer surface.

``rg_polymer`` (*R*\ :sub:`g`, Ang)
Radius of gyration of free Gaussian polymer chains in solution. Enters the Debye scattering function
$$
P_\text{Debye}(u) = \frac{2(\mathrm{e}^{-u}-1+u)}{u^2},
\quad u = (q\,R_g)^2
$$
Only used when ``scale_polymer`` nonequal to 0.

``scale_polymer`` (*A*\ :sub:`14`, cm\ :sup:`-1`)
Amplitude (zero-*q* intensity) of the Debye polymer contribution: $I_\text{poly}(q) = A_{14}\,P_\text{Debye}(q)$. Set to 0 to disable. Note that this term is *not* multiplied by the global ``scale``, so it represents an absolute polymer contribution independent of the particle scattering amplitude.

``displacement`` (*delta*, Ang)
Distance by which the core centre is displaced from the shell centre, in the direction *perpendicular* to the ellipsoid symmetry axis.The displacement introduces an asymmetric contrast profile, generating a phase factor :math:`\cos(q\,\delta\,\sin\theta)` in the form factor (where *theta* is the angle between *q* and the symmetry axis). This modulates the form factor oscillations and can model, for example, an off-centre protein domain inside a lipid vesicle. If *delta* exceeds *d_shell* the value is reflected back: effectively *delta* is clamped to [0, *d_shell*]. Set to 0 for a concentric core-shell particle.

``sigma_core`` (*sigma*\ :sub:`core`, Ang)
Debye–Waller (Gaussian) roughness of the core–shell interface. The core amplitude is multiplied by $exp(-\tfrac{1}{2}(q\,\sigma_\text{core})^2)$, smearing the inner interface. Analogous to ``sigma_outer`` but applied at the inner surface. Set to 0 for a sharp core–shell boundary.

**Notes on the Fortran Reference**

* The Fortran source had a bug in the amplitude accumulator: ``FFF = SUM1*STEP`` was used instead of ``SUM1X*STEP``, which caused the structure-factor correction term $\langle F\rangle^2(S-1)$ to vanish silently. This has been corrected.
* The Fortran normalisation denominator used the angle-dependent ``RAO`` (last integration point) instead of the angle-independent ``RO``; this has been corrected.

**References**

- Correa et al., International Journal of Biological Macromolecules, 152016 (2026)
- Spinozzi et al., *Biophys. J.* (2002)
- Orthaber et al., *J. Appl. Cryst.* 33, 218–225 (2000)