fix normalization for angular dispersion

[pkienzle]

sasview 2D models have an issue with the theta parameter. If there is angular dispersion on theta, then the calculated pattern is weighted by cos(theta). This means that as the 2D pattern is rotated through 90 degrees, the overall scattering intensity decreases and increases again. Furthermore, if there is no angular dispersion, the "correction" is not applied.

It is perhaps the case that the definition of theta and phi in the models does not match theta and phi in the documentation, and that the correction should be applied on the basis of phi, or alternatively, phi and theta should be swapped and the correction continue to apply to theta.

[pkienzle]

Guinier and Fournet (1955) give the following equation 2.36 (pg 24):

<F²(h)> = ∫ ∫ F²(h,θ,ϕ)P₁(h,θ)P₂(h,phi) dθ dϕ

SasView is using:

<I(q)> = pi/2 ∫ ∫ I(q,θ,ϕ)P₁(q,θ)P₂(q,ϕ)|cos(θ)| dθ dϕ

I can't find a reference that includes the cosine term, or the pi/2

Perhaps the confusion arose from the different forms of the ellipsoid of revolution. Eq 2.32 gives the ellipsoid of revolution as:

i(h) = ∫₀^pi/2 Phi²(h a √(cos² θ + v² sin² θ) cos θ dθ

but the 1-D ellipsoid kernel employs the substitution u = sin theta so it is evaluated as

i(h) = ∫₀¹ Phi²(h a √(1 + u²(v² - 1)) du

[Note: the u variable above is incorrectly named as cos_alpha in the code.]

[pkienzle]

… comment removed …

[butler]

This should be part of sasmodels project and therefore core of 4.0 <br>
NOTE: this may be a duplicate of #492 — needs checking

Meanwhile will make this blocker

[richardh]

This possibly resolved, PAK need to test code. cos(theta) in sasview likely the same as sin(theta) in Hayter & Penfold etc. We might still consider changing all the axes?