guinier

Parameter Description Units Default value
scale Scale factor or Volume fraction None 1
background Source background cm-1 0.001
rg Radius of Gyration Å 60

The returned value is scaled to units of cm-1 sr-1, absolute scale.

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 linearisation (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 objects’s distribution of SLD (not mass density, as in mechanics) from the objects’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 \(R_g^2\).

../../_images/guinier_autogenfig.png

Fig. 103 1D plot corresponding to the default parameters of the model.

Source

guinier.py

References

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

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