two_lorentzian

This model calculates an empirical functional form for SAS data characterized by two Lorentzian-type functions.

Parameter	Description	Units	Default value
scale	Scale factor or Volume fraction	None	1
background	Source background	cm-1	0.001
lorentz_scale_1	First power law scale factor	None	10
lorentz_length_1	First Lorentzian screening length	Å	100
lorentz_exp_1	First exponent of power law	None	3
lorentz_scale_2	Second scale factor for broad Lorentzian peak	None	1
lorentz_length_2	Second Lorentzian screening length	Å	10
lorentz_exp_2	Second exponent of power law	None	2

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

Definition

The scattering intensity \(I(q)\) is calculated as

\[I(q) = \frac{A}{1 +(Q\xi_1)^n} + \frac{C}{1 +(Q\xi_2)^m} + \text{B}\]

where \(A\) = Lorentzian scale factor #1, \(C\) = Lorentzian scale #2, \(\xi_1\) and \(\xi_2\) are the corresponding correlation lengths, and \(n\) and \(m\) are the respective power law exponents (set \(n = m = 2\) for Ornstein-Zernicke behaviour).

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. 124 1D plot corresponding to the default parameters of the model.

Source

two_lorentzian.py

References

None.

Authorship and Verification

• Author: NIST IGOR/DANSE Date: pre 2010

• Last Modified by: Piotr rozyczko Date: January 29, 2016

• Last Reviewed by: Paul Butler Date: March 21, 2016