# correlation_length

Calculates an empirical functional form for SAS data characterized by a low-Q signal and a high-Q signal.

Parameter Description Units Default value
scale Source intensity None 1
background Source background cm-1 0.001
lorentz_scale Lorentzian Scaling Factor None 10
porod_scale Porod Scaling Factor None 1e-06
cor_length Correlation length, xi, in Lorentzian 50
porod_exp Porod Exponent, n, in q^-n None 3
lorentz_exp Lorentzian Exponent, m, in 1/( 1 + (q.xi)^m) -2 2

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

Definition

The scattering intensity I(q) is calculated as

$I(Q) = \frac{A}{Q^n} + \frac{C}{1 + (Q\xi)^m} + \text{background}$

The first term describes Porod scattering from clusters (exponent = $$n$$) and the second term is a Lorentzian function describing scattering from polymer chains (exponent = $$m$$). This second term characterizes the polymer/solvent interactions and therefore the thermodynamics. The two multiplicative factors $$A$$ and $$C$$, and the two exponents $$n$$ and $$m$$ are used as fitting parameters. (Respectively porod_scale, lorentz_scale, porod_exp and lorentz_exp in the parameter list.) The remaining parameter $$\xi$$ (cor_length in the parameter list) is a correlation length for the polymer chains. Note that when $$m=2$$ this functional form becomes the familiar Lorentzian function. Some interpretation of the values of $$A$$ and $$C$$ may be possible depending on the values of $$m$$ and $$n$$.

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

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

References

B Hammouda, D L Ho and S R Kline, Insight into Clustering in Poly(ethylene oxide) Solutions, Macromolecules, 37 (2004) 6932-6937