gauss_lorentz_gel

Gauss Lorentz Gel model of scattering from a gel structure

Parameter	Description	Units	Default value
scale	Scale factor or Volume fraction	None	1
background	Source background	cm-1	0.001
gauss_scale	Gauss scale factor	None	100
cor_length_static	Static correlation length	Å	100
lorentz_scale	Lorentzian scale factor	None	50
cor_length_dynamic	Dynamic correlation length	Å	20

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

This model calculates the scattering from a gel structure, but typically a physical rather than chemical network. It is modeled as a sum of a low-q exponential decay (which happens to give a functional form similar to Guinier scattering, so interpret with care) plus a Lorentzian at higher-q values. See also the gel_fit model.

Definition

The scattering intensity \(I(q)\) is calculated as (Eqn. 5 from the reference)

\[I(q) = I_G(0) \exp(-q^2\Xi ^2/2) + I_L(0)/(1+q^2\xi^2)\]

\(\Xi\) is the length scale of the static correlations in the gel, which can be attributed to the “frozen-in” crosslinks. \(\xi\) is the dynamic correlation length, which can be attributed to the fluctuating polymer chains between crosslinks. \(I_G(0)\) and \(I_L(0)\) are the scaling factors for each of these structures. Think carefully about how these map to your particular system!

Note

The peaked structure at higher \(q\) values (Figure 2 from the reference) is not reproduced by the model. Peaks can be introduced into the model by summing this model with the gaussian_peak model.

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

\[q = \sqrt{q_x^2 + q_y^2}\]

Fig. 104 1D plot corresponding to the default parameters of the model.

Source

gauss_lorentz_gel.py

References

1. G Evmenenko, E Theunissen, K Mortensen, H Reynaers, Polymer, 42 (2001) 2907-2913

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