Limiting cases of cylinder model

[dirk]

The cylinder model should give the formfactor of a long, infinitely thin rod or thin disk if diameter or lengh is set to zero, respectively.

[pkienzle]

More practically, the cylinder model does not support carbon nanotubes, with length 10 um and diameter 1 nm.

Calculations for different numbers of integration steps attached. Simulations were run with explore/symint.py in the ticket-776-orientation branch, along with the I(q, theta) function that is being integrated for a few different q points.

The current code supports lengths up to 0.1 um. Switching to a 150 point gaussian will allow up to 1-2 um. A 1000 point gaussian will brings the error down to 1.5% for the 10 um length.

[pkienzle]

​Inada 2005 (supplemental) gives the following for an infinite core-shell cylinder with core radius=0:

    I(q) = pi L/q [drho V 2 J1(q r)/(q r)]^2 

This equation gives the same curve shape as the finite cylinder, but the scale is off. Instead use:

    I(q) = pi/L V/q [drho 2 J1(q r)/(q r)]^2

This matches the peaks well, but underestimates the dips. With minimal polydispersity or resolution the precise value of the dips won't matter and this approximation will be good enough. It breaks down at low q for shorter cylinder lengths.

Note: sasmodels kernels use V rather than V² because the polydispersity loop does volume weighted normalization, which still doesn't explain the 1/L vs. L difference.

[pkienzle]

Infinitely thin rod:

2Si(qL)/(qL) - [sin(qL/2)/(qL/2)]^2

Infinitely thin disk:

2/(qR)^2 [ 1 - 2 J1(2qR)/2qR ]

Pedersen, J.S., 1997. Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting. Advances in Colloid and Interface Science 70, 171–210. doi:10.1016/S0001-8686(97)00312-6