Guinier

Definition

This model fits the Guinier function

$$ I(q) = \text{scale} \cdot \exp{\left[ \frac{-Q^2 R_g^2 }{3} \right]} + \text{background}
$$
to the data directly without any need for linearization (*cf*. the usual plot of $\ln I(q)$ vs $q^2$). Note that you may have to restrict the data range to include small q only, where the Guinier approximation actually applies. See also the guinier_porod model.

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

$$ q = \sqrt{q_x^2 + q_y^2}
$$
In scattering, the radius of gyration $R_g$ quantifies the object's distribution of SLD (not mass density, as in mechanics) from the object's SLD centre of mass. It is defined by

$$ R_g^2 = \frac{\sum_i\rho_i\left(r_i-r_0\right)^2}{\sum_i\rho_i}
$$
where $r_0$ denotes the object's SLD centre of mass and $\rho_i$ is the SLD at a point $i$.

Notice that $R_g^2$ may be negative (since SLD can be negative), which happens when a form factor $P(Q)$ is increasing with $Q$ rather than decreasing. This can occur for core/shell particles, hollow particles, or for composite particles with domains of different SLDs in a solvent with an SLD close to the average match point. (Alternatively, this might be regarded as there being an internal inter-domain "structure factor" within a single particle which gives rise to a peak in the scattering).

To specify a negative value of $R_g^2$ in SasView, simply give $R_g$ a negative value ($R_g^2$ will be evaluated as $R_g |R_g|$). Note that the physical radius of gyration, of the exterior of the particle, will still be large and positive. It is only the apparent size from the small $Q$ data that will give a small or negative value of $R_g^2$.

References

#. A Guinier and G Fournet, *Small-Angle Scattering of X-Rays*, John Wiley & Sons, New York (1955)

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