Factor 4 missing in Touschek scattering B2 expression

[oscarxblanco]

Dear all,

there is a missing factor 4 in function _get_vals of at/pyat/at/acceptance/touschek.py lines

at/pyat/at/acceptance/touschek.py

Lines 127 to 129 in b016b77

	B2sq = bg2i*bg2i*(numpy.diff(bs, axis=1).T**2 +
	sigh2*sigh2*numpy.prod(bxy2*dt2, axis=1) /
	numpy.prod(sigb2*sigb2, axis=1))

    B2sq = bg2i*bg2i*(numpy.diff(bs, axis=1).T**2 +
                      sigh2*sigh2*numpy.prod(bxy2*dt2, axis=1) /
                      numpy.prod(sigb2*sigb2, axis=1))

bg2i is calculated above as $1/(2\beta^2\gamma^2)$.
bg2i is wrongly put as multiplicative factor of the whole expression.

Those lines should calculate the $B^2_2$ factor in Eq. (34) on Piwinski's article titled 'The Touschek effect in Strong Focusing Storage Rings'. https://arxiv.org/abs/physics/9903034

where the second term on right hand side is not divided by 4.

[swhite2401]

Hi @oscarxblanco, good catch!
Does this correction have a big impact on the lifetime calculation?
I copied this from the matlab implement, is it there as well did you see the same issue there or is it something I introduced when translating it?

[oscarxblanco]

The second term in $B_2^2$ is a crossed-term (x-y) that depends on the product of energy spread, the dispersions (x and y), and the inverse of the transverse emittances.

It has an impact in lattices with vertical dispersion ($\eta_y\neq0,\eta'_y\neq0$) either due to lattice design, limits on the optics correction or transverse coupling.

However, not many lattices have vertical dispersion.

I have checked the matlab implementation and it does not have this issue. It also has the answer to an approximation that I would suggest to document also on pyat. In matlab the function TLT_IntPiw_k uses the first term in the the modified Bessel function approximation for large numbers: $I_0 \approx \frac{\exp{x}}{\sqrt{2\pi x}}(1+\frac{1}{8x}(1+ ...))$
it is explicitly written in the comments. In pyat, the same logic is used in the function int_piwinski but there is no explanation on why is done like that.

[swhite2401]

Ok. Thanks for this analysis. I will correct this as suggested then.

[swhite2401]

Correction proposed in #765