Displaced core-shell ellipsoid model

Displaced core-shell ellipsoid model - absolute scale version.

**Description**

This model is identical in geometry and physics to ``core_shell_displ_core``,
but works in **absolute scale** (cm\ :sup:`-1`). Instead of a dimensionless
contrast ratio, the user supplies the scattering length densities (SLDs) of
the core, shell, and solvent directly, together with the particle volume
fraction. The model then computes all contrasts internally and returns the
absolute intensity without any additional ``scale`` factor.

Because the absolute intensity is computed internally, the SASView global
``scale`` parameter should be left at **1** and ``background`` set to the
measured incoherent flat background.

**Model Geometry**

Same as ``core_shell_displ_core``:

- Core ellipsoid: equatorial semi-axis *R*, aspect ratio *eps* = c/a.
- Coaxial shell: equatorial outer radius *R* + *d_shell*, where
*d_shell* = |thickness_shell| + 2 |sigma_outer|.
- Core centre displaced by ``displacement`` perpendicular to the symmetry
axis.

**Form Factor Decomposition**

The scattering amplitude at a given size and orientation is

.. math::

F(q) = \Delta\rho_\text{shell}\, V_\text{out}\,\phi(q r_\text{out})
+ (\Delta\rho_\text{core} - \Delta\rho_\text{shell})\,
V_\text{core}\,\phi(q r_\text{core})\,e^{i q\delta\sin\theta}

where :math:`\Delta\rho_X = \text{sld}_X - \text{sld\_solvent}` and
:math:`\phi(x) = 3j_1(x)/x`.

**Absolute Intensity**

.. math::

I(q) = 10^{-4}\,\frac{\varphi}{\langle V\rangle}
\bigl[\langle F^2(q)\rangle + \langle F(q)\rangle^2
(S(q)-1)\bigr]\,B(q) + I_\text{poly}(q)

where the factor :math:`10^{-4}` converts
:math:`(10^{-6}\,\text{Ang}^{-2})^2 \cdot \text{Ang}^3 \cdot \text{Ang}^{-3}`
= :math:`10^{-12}\,\text{Ang}^{-1}` to cm\ :sup:`-1`
(:math:`1\,\text{Ang}^{-1} = 10^8\,\text{cm}^{-1}`).

:math:`\langle\cdot\rangle` denotes an average over the Gaussian size
distribution and all particle orientations (each 50-point Gauss quadrature).
:math:`\langle V\rangle` is the Gaussian-weighted mean particle volume used
as the number-density denominator.

The SASView framework then applies:

.. math::

I_\text{total}(q) = \text{scale}\cdot I(q) + \text{background}

Keep ``scale = 1`` for true absolute scale.

**Parameter Definitions**

``scale``
Overall scale factor — leave at **1** for absolute scale. Only change
if a systematic calibration offset is needed.

``background`` (cm\ :sup:`-1`)
Flat incoherent background. Set to the measured empty-cell or
incoherent level.

``radius`` (*R*, Ang)
Mean equatorial semi-axis of the core ellipsoid.

``aspect_ratio`` (*eps* = c/a)
Core polar-to-equatorial semi-axis ratio.
eps = 1 -> sphere; eps > 1 -> prolate; eps < 1 -> oblate.

``volfraction`` (*phi*)
Volume fraction of particles. Together with the mean particle volume
:math:`\langle V\rangle`, it sets the absolute scattering level:
:math:`n = \varphi/\langle V\rangle`.

``sld_core`` (10\ :sup:`-6` Ang\ :sup:`-2`)
Scattering length density of the core material.

``sld_shell`` (10\ :sup:`-6` Ang\ :sup:`-2`)
Scattering length density of the shell material.

``sld_solvent`` (10\ :sup:`-6` Ang\ :sup:`-2`)
Scattering length density of the solvent (D\ :sub:`2`\ O ~ 6.35,
H\ :sub:`2`\ O ~ -0.56).

``thickness_shell`` (*d*, Ang)
Equatorial shell thickness. Effective shell extent used in the
geometry: *d_shell* = |d| + 2 |sigma_outer|.

``volfraction_hs`` (*η*)
Hard-sphere volume fraction for the Percus-Yevick S(q).
Set to 0 to disable inter-particle interactions.

``radius_hs`` (*R*\ :sub:`HS`, Ang)
Hard-sphere interaction radius. Not necessarily equal to the geometric
particle radius; active only when ``volfraction_hs`` > 0.

``sigma_rel`` (*sigma_R/R*)
Relative Gaussian polydispersity of the core radius.
0 = monodisperse (a minimum of 10\ :sup:`-4` is enforced internally).

``power_law`` (*A*\ :sub:`10`)
Amplitude of the empirical low-*q* correction
:math:`B(q) = 1 + A_{10}(0.001/q)^m`. Set to 0 to disable.

``exponent_2`` (*A*\ :sub:`11`)
Sets the power-law exponent: *m* = |A\ :sub:`11`| + 2.

``sigma_outer`` (*sigma*\ :sub:`out`, Ang)
Debye–Waller roughness of the outer shell surface.

``rg_polymer`` (*R*\ :sub:`g`, Ang)
Radius of gyration of free polymer chains (Debye background).

``scale_polymer`` (cm\ :sup:`-1`)
Zero-*q* intensity of the Debye polymer contribution.
This is an *absolute* amplitude (cm\ :sup:`-1`); set to 0 to disable.

``displacement`` (*delta*, Ang)
Core-centre offset perpendicular to the symmetry axis.
Clamped to [0, *d_shell*] internally. 0 = concentric particle.

``sigma_core`` (*sigma*\ :sub:`core`, Ang)
Debye–Waller roughness of the core–shell interface.

**References**

- Spinozzi et al., *Biophys. J.* (2002)
- Orthaber et al., *J. Appl. Cryst.* 33, 218–225 (2000)

Example Data: