OrientedMagneticChains

This plug-in model calculates oriented core-shell chains, with the option of
adding a magnetic SLD to each layer. The chain scattering is the incoherent
sum of a user-defined combination of singletons, dimers, trimers,
quadramers, and pentamers. Note that no matter the numerical values
selected for the amount of each chain type, the fraction of each will be
normalized such that the sum of the chain type fractions is unity. A
normalization radius parameter is also included that so that the scale will
be equivalent to the volume fraction corresponding to that size of the
nanoparticle (e.g. if the amount of iron oxide nanoparticle material in a
solution is known, but the amount of surfactant is not, a normalization
radius corresponding to the iron oxide radius will return a scale equal to
the volume fraction of iron oxide in solution).

The chains are preferentially oriented about the x-direction, with a Gaussian
FWHM in degrees set by user-input sigma. From here, the user can choose
the viewing angle w.r.t. the x-axis (0 degrees and 90 degrees would be
common slices). Alternatively, the user can select the 2D View before
plotting the model to see a 2D plot; plotting 2D after starting a 1D plot may
result in having the 1D plot determined by the viewing angle be symmetrically
rotated about 2D, which would is not physically correct.

The magnetism of the chains comes in three varieties, selected using the
parameter magnetic_orientation: 1 = core-shells with random magnetic
alignment from particle-to-particle (but with a shared direction per core and
shell), 2 = magnetic moments aligned along the chain axis, and 3 = magnetic
moments aligned along the x-axis (regardless of chain orientation).
The scattering amplitude form factor is calculated in same way as the core-shell
sphere model (Guinier, 1955), and it is then multiplied by a complex structure
factor that depends on chain length:
$$ P(q) = \frac{\text{scale}}{V} F^2(q)*(\text{real \ phase}^2 + \text{img \ phase}^2) + \text{background} $$
where
$$ \text{real \ phase} = 1.0 + \sum_{k=0}^{4} \sum_{n=0}^{k} cos(k*length*(Q_X*cos(\theta) + Q_Y*sin(\theta)*cos(\phi)) $$
and
$$ \text{img \ phase} = 0.0 + \sum_{k=0}^{4} \sum_{n=0}^{k} sin(k*length*(Q_X*cos(\theta) + Q_Y*sin(\theta)*cos(\phi)) $$
and
$$ F(q) = \frac{3}{V_s}\left[
V_c(\rho_c-\rho_s)\frac{\sin(qr_c)-qr_c\cos(qr_c)}{(qr_c)^3} +
V_s(\rho_s-\rho_\text{solv})\frac{\sin(qr_s)-qr_s\cos(qr_s)}{(qr_s)^3}
\right] $$.

Here $V_s$ is the volume of the whole particle, $V_c$ is the volume of the
core, $r_s$ = $radius$ + $thickness$ is the radius of the particle, $r_c$
is the radius of the core, $\rho_c$ is the scattering length density of the
core, $\rho_s$ is the scattering length density of the shell,
$\rho_\text{solv}$, is the scattering length density of the solvent.

Theta is the angle between Qx-Qy and the x-axis, and it is sampled in 2 degree
increments in 45 steps. Phi is the rotation of the vector set by theta in a
half-cone about the x-axis in 4 steps of 45 degrees (finer steps are possible, but
would slow the code and don't noticeably affect the model).

Please note, this model will soon be updated to include polarization analysis
(currently unpolarized) and will include ability to change the orientation angle
(now set to x). It will also include a fully randomly oriented option.