reparameterize Teubner-Strey

[pkienzle]

The current implementation of Teubner-Strey use parameters scale, c1, c2 and background. Since scale and background are automatic parameters in sasmodels, this was changed to a2, c1 and c2.

Propose changing the parameters to the more physically meaningful repeat length d, correlation length xi and a scale factor. This will give more useful parameter uncertainties.

A little bit of algebra gives a2/c2 and c1/c2 in terms of xi and d:

a2/c2 = (xi^-2 + (2 pi/d)²)²
c1/c2 = (xi^-2 - (2 pi/d)²)²

The scale factor comes from the normalized form (Schubert, 1994):

I(q) = (8 pi c2 eta² / xi) / (a2 + c1 q² + c2 q⁴)

where

eta² = phi(1-phi)(delta rho)²

for volume fraction phi and scattering length density difference delta rho.

The following parameters should be sufficient:

sld, sld_solvent, phi in (0,1), xi > 0, and d > 0

using

I(q) = (8 pi phi(1-phi)(delta rho)²) / ((1+(2 pi xi/d)²)² + (1-(2 pi xi/d)²)² q² + q⁴)

with two of phi, sld and sld_solvent fixed.

[Paul Kienzle <pkienzle@…>]

In [bc87305316c560a93eeaf77db05eb6721733d636]:

Fix slit height resolution problem. Still issues with slit width. Refs #472

[butler]

This is the same request as ticket #530 by Wim Bouwman. There he is concerned about the instabilities with this parametrization. Gives reference and attachment for how to change — should compare against Paul Kienzle's approach above.

[butler]

with the 4.0 release being way late, we clearly don't have the bandwidth to deal with such a change now. Moving to +1

[butler]

Moving back to 4.0 as part of ticket #643 admonition to finalize parameter and model names so as to remain compatible in the future.

[ajj]

Correct reparameterisation is as per (Schubert, 1994) and (Endo, 2001)

a2/c2 = (xi^-2 + (2 pi/d)²)²
c1/c2 = 2(xi^-2 - (2 pi/d)²) ←- difference is here compared to original ticket text

The scale factor comes from the normalized form (Schubert, 1994):

I(q) = (8 pi c2 eta² / xi) / (a2 + c1 q² + c2 q⁴)

where

eta² = phi(1-phi)(delta rho)²

for volume fraction phi and scattering length density difference delta rho.

The following parameters should be sufficient:

sld, sld_solvent, phi in (0,1), xi > 0, and d > 0

using

I(q) = (8 pi phi(1-phi)(delta rho)²) / ((xi^-2+(2 pi/d)²)² + 2(xi^-2 - (2 pi/d)²) q² + q⁴)

with two of phi, sld and sld_solvent fixed.

[ajj]

So it seems that we have had various implementations over time:

1) Igor + libigor
2) sansview + libigor
3) a python version
4) a c++ version replacing the python version (ILL 2012)
5) version in sasmodels

in leaving (1) it seems that correct normalisation of parameters was not transferred and so 2→5 are all "incorrectly" normalised. They all have the correct functional form, but the physical meaning of the parameters and how one extracts the physical meaning varies.

The update with this ticket will aim to match (1), since the citation is from a paper with Steve Kline as an author (along with R Strey) and he wrote that model, so we will take it as definitive.

[ajj]

In caddb142198aa5e7856a0850214d72611c3b6ba8/sasmodels:

Updated and re-parameterised Teubner-Strey model.

Closes #472
Closes #530