Methods for optimisation for c and t value pairs

[subindev-d]

• Decide the range of values for c and t
• Optimistaion for the values: Do we need to run on all the values or methods like gradient descent works for this
• Implement optimisation

[MilanSCD]

From the data for Palladium Isotopes in Megumi's paper I think we can constrain c and t in fermi charge distribution model such that:

• C is close to the nuclear radius $r = 1.25 \times A^{1/3} fm$
• C and t combined are close to the nuclear radius eg. $r^2 -a < c^2 + t^2 < r^2 + b$ or a constraint with a similar relationship.

So far for palladium I get optimal values for mean square error when $c \approx r$ and $t < 0.1$

I don't know where the default value $t=2.3fm$ comes from, but if anyone knows what appropriate values are that would be appreciated.

scipy minimisers in python work well and we could do a similar thing in c++ using the dlib library.

[MilanSCD]

from the development guide c and t are linked to the uniformed charged sphere by equation:
$c^2 + at^2 = R^2$
where $a = \frac{7 \cdot\pi^2}{48 \cdot \ln(3)^2}$
and R is the radius of the uniform sphere.
We can therefore restrict c and t with:
$|c^2 + a \cdot t^2 - R^2| \leq \epsilon$

So far this is consistent with the results for palladium.

[MilanSCD]

we can parameterise this ellipse equation into polar co ordinates to make a rectangular domain for minimisation.

$r(\theta) = \frac{R}{\sqrt{\cos^2\theta+a \cdot \sin^2\theta}}, \quad 0 \leq \theta \leq \frac{\pi}{2}, \quad R_0 - \epsilon \leq R \leq R_0 + \epsilon$

we can minimise the least squares for the expermental values and mudirac using $\theta$ and $R$

when the minimum is found we can convert back to $c$ and $t$ with:

$c = r(\theta, R)\cos\theta$ and $t = r(\theta, R) \sin\theta$

[MichaelHeines]

@MilanSCD: On your question "I don't know where the default value t=2.3 fm comes from, but if anyone knows what appropriate values are that would be appreciated."

The typical value of t=2.3 fm (equivalently a or z = 0.523) is from general trends in electron scattering. Muonic atoms are generally not that sensitive to the shape of the distribution, but electron scattering is very good at this (see the de Vries et al. compilation from 1987 for more info). This is why the Barrett radius + V2 correction in combined analysis of muons + e scattering gives the best values, you combine radius sensitivity from muons with shape sensitivity from scattering.

Likely, this is the cause of the large relative uncertainties in the paper from the Japanese group. Maybe their initial point of the fit was closer so that they were left in a different local minimum than you were or a small extra QED correction ever so slightly altered the chi2 surface. I suspect if you would scan an area around the minimum you found, you will see that by tweaking c slightly, we can get large variations of optimal values in t that are equally/similarly good.

Hope this helps