How to simulate an isotope shift measurement

[Malamala]

Hello! My questions are based on this paper talking about King plot non-linearities. Assuming I have 3 isotopes and 2 transitions, I would like to know how well I should measure the transitions (i.e. what uncertainty on the transition value) in order to reach a given sensitivity for the new physics parameter. What I am thinking of doing is to generate data using equation 5 (i.e. without any new physics, assuming I know the masses and changes in charge radii), which of course if I plug in equation 9 will give me $\alpha_{NP} = 0$. However, even if $\alpha_{NP} = 0$, I can still use the error propagation mentioned below equation 9 to get the error on $\alpha_{NP}$. So basically I will get $\alpha_{NP} = 0 \pm d\alpha_{NP}$ and from here I can set an upper bound on $\alpha_{NP} < d\alpha_{NP}$ at 1 sigma level. And based on the value of $d\alpha_{NP}$ I am aiming for, I can get the needed uncertainty on the transitions frequencies. However, in practice, $\alpha_{NP}$ won't be zero. It will be smaller than $d\alpha_{NP}$, but not zero and the upper limit will be $\alpha_{NP} + d\alpha_{NP}$, which in principle can be up to 2 times bigger than $d\alpha_{NP}$ alone. Given that I know what upper bound I aim for, how can I get the needed uncertainty on the transitions in this general case? Thank you!

 

[Twigg]

It's not clear to me why you will need a simulation to do this calculation. If you assume a non-zero value of $\alpha_{NP}$, don't you just need to calculate the derivatives of $\frac{\partial \alpha_{NP}}{\partial \nu_i}$ for $i = 1,2$ being the two transition frequencies, and perform standard propagation of error?

Alternatively, for more accuracy, just run a simple monte carlo. Some algebra will let you derive an expression like $\nu_2 = f(\nu_1,\alpha_{NP})$ by solving the equation for $\alpha_{NP}$ for $\nu_2$. Then introduce some measurement noise by adding uncorrelated random numbers to both $\nu_1$ and $\nu_2$, with equal variance (assuming you measure both transitions with equal uncertainty). Then just find the scatter on your observed values of $\alpha_{NP}$. Was that clear?