Pringle-Schmidt Helices

This is the Pringle-Schmidt equation for fitting the helical form factor of an infinitely long helix formed by two helical tapes wrapped around each other at the angle $\phi$.

$$I(q) = \frac{\pi}{q L} \sum^{\inf}_{n = 0} \epsilon_{n} \cos^2 \left( \frac{n \phi}{2} \right) \frac{\sin^{2} \left( n \omega / 2 \right)}{\left( n \omega / 2 \right)^2} \left[ g_{n} \left( q, R, a \right) \right]^2$$

where

$$g_{n} \left( q, R, a \right) = 2 R^{-2} \left(1 - a^{2} \right) \times \int^{R}_{aR} r dr J_{n} \left[ q r \left( 1 - q^{2}_{n}) \right)^{1/2} \right]$$

and

$$q_{n} = \frac{2 \pi n}{P q} .$$

References

1) O. A. Pringle and P. W. Schmidt, Journal of Applied Crystallography, 1971, 4, 290-293, DOI: 10.1107/S002188987100699X
2) C. V. Teixeira, H. Amenitsch, T. Fukushima et al., Journal of Applied Crystallography, 2010, 43, 850-857, DOI: 10.1107/S0021889810015736
The fitting equation can be found in the latter paper as equations 15 & 16