Inconsistent fuzzy or rough interfaces in fuzzy_sphere and core_shell_bicelle_elliptical_belt_rough

[richardh]

Whilst dealing with #1160 beta(Q) model changes, Paul K  noted that

Fuzzy sphere is defined as

P =  (f * exp(-½ q^2 fuzziness^2))^2

but core shell bicelle elliptical belt rough uses:

P = f^2 * exp(-½ q^2 sigma^2)

Are both of these correct?  Or should the bicelle roughness term be squared?

Do these models work correctly with structure factors?

Does the fuzziness apply as usual for the calculation for beta?

beta = < f exp(-q^2 s^2/2) >^2 / < f^2 exp(-q^2 s^2/2))^2 >

THIS REQUIRES FURTHER INPUT - HAS ANYONE ELSE SEEN PAPERS DERIVING DIFFUSE INTERFACES?

Email discussion summarised  below …

Richard wrote:  In both cases sasview follows the references given, so we either need to think harder or find some other references!

The derivation for fuzziness goes something like this - at high Q the I(Q) is essentially a 1d Fourier transform of the sld profile normal to an interface.

If the features in the sld profile were to be convoluted with a Gaussian, then the convolution theorem tells us that we multiply the I(Q) by an exponential.

Thus strictly sigma in the Gaussian has to be very small compared to the radius of the particle (which is perhaps not always so in the Stieger paper, though they don't seem to give any values for sigma, just say that Rsans = R + 2*sigma).

Should we be multiplying F(Q) or P(Q) ?    Squaring the exponential of course only rescales sigma by sqrt(2), so all depends on the precise definition of sigma.

Strey points (incorrectly) to a paper by Ruland (papers available from Richard) which does say Iobs = I. H^2 where H is the Fourier of the smoothing distribution. See also the discussion after eqn (3) in the Ciccariello paper (not that I in any way understand the details in the rest of that paper).

I could suggest that  "fuzziness" ought only to be applied after everything else, including any S(Q) and beta(Q) as it ought only to work at very high Q, but in that case it should not affect results at smaller Q anyway, so I suppose that it could go into F or F^2.

What we should really be doing, at least for spherical particles, is to use a constrained multiple linear step sld profile, which would then work properly for all particle radii. Of course for elliptical bicelles this is not possible, so we have to resort to approximations like this.

Have any of you got any further references for diffuse or fuzzy interfaces?

Yun wrote: Thanks Richard to dig out all the reference and provide detailed comments on this.

As for the calculation of the fuzzy ball, I agree with Richard's comments.

Overall, both equations are ok. But the definition of the fuzziness is slightly different.Personally, I prefer "  P =  (f * exp(-½ q^2 fuzziness^2))^2".

But since we need to have a reference for any model, I guess we can keep them as they are now since these are the "correct" equations from the references SASView provides.

For "P =  (f * exp(-½ q^2 fuzziness^2))^2" (or other equation), it is used to describe a density convolution in 3D in the real space. However, due to the isotropic properties, it reduces to the 1D problem as pointed out by Richard.

It is indeed much better defined if sigma (or fuzziness) is relatively small. I feel that when the sigma is very large, its mathematical meaning can still hold. I did not read the paper by Stieger carefully yet so not sure if they discussed this.  

As for the beta approximation, if there is  a change (polydispersity) of size or sigma, I feel that the beta factor can be still calculated in the same way. What Paul K. proposed, "beta = < f exp(-q^2 s^2/2) >^2 / < f^2 exp(-q^2 s^2/2))^2 >", seems reasonable to me.

As long as the shape fluctuation is independent of the relative locations of the center of mass of all particles, the beta approximation would be still fine. (for the full Q range? not necessarily for high Q only? ) Of course, I am not sure when the beta approximation should fail for this case. This still needs more future work.