Guinier Porod

Calculates the scattering for a generalized Guinier/power law object. This is an empirical model that can be used to determine the size and dimensionality of scattering objects, including asymmetric objects such as rods or platelets, and shapes intermediate between spheres and rods or between rods and platelets, and overcomes some of the deficiencies of the (Beaucage) `unified-power-rg` model (see Hammouda, 2010).

Definition

The following functional form is used

$$ I(q) = \begin{cases} \frac{G}{Q^s}\ \exp{\left[\frac{-Q^2R_g^2}{3-s} \right]} & Q \leq Q_1 \\ D / Q^m & Q \geq Q_1 \end{cases}
$$
This is based on the generalized Guinier law for such elongated objects (see the Glatter reference below). For 3D globular objects (such as spheres), $s = 0$ and one recovers the standard Guinier formula. For 2D symmetry (such as for rods) $s = 1$, and for 1D symmetry (such as for lamellae or platelets) $s = 2$. A dimensionality parameter ($3-s$) is thus defined, and is 3 for spherical objects, 2 for rods, and 1 for plates.

Enforcing the continuity of the Guinier and Porod functions and their derivatives yields

$$ Q_1 = \frac{1}{R_g} \sqrt{(m-s)(3-s)/2}
$$
and

$$ D = G \ \exp{ \left[ \frac{-Q_1^2 R_g^2}{3-s} \right]} \ Q_1^{m-s}
$$
&= \frac{G}{R_g^{m-s}} \exp \left[ -\frac{m-s}{2} \right] \left( \frac{(m-s)(3-s)}{2} \right)^{\frac{m-s}{2}}

Note that the radius of gyration for a sphere of radius $R$ is given by $R_g = R \sqrt{3/5}$. For a cylinder of radius $R$ and length $L$, $R_g^2 = \frac{L^2}{12} + \frac{R^2}{2}$ from which the cross-sectional radius of gyration for a randomly oriented thin cylinder is $R_g = R/\sqrt{2}$ and the cross-sectional radius of gyration of a randomly oriented lamella of thickness $T$ is given by $R_g = T / \sqrt{12}$.

For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the q vector is defined as

$$ q = \sqrt{q_x^2+q_y^2}
$$

Reference

B Hammouda, *A new Guinier-Porod model, J. Appl. Cryst.*, (2010), 43, 716-719
B Hammouda, *Analysis of the Beaucage model, J. Appl. Cryst.*, (2010), 43, 1474-1478

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