Muon decay electron time spectrum fit

Fits of muon decay electron time spectra to the exponential functions were performed taking into account the finite size of the time bins.

For the muon stopping in ** M** different elements, it obviously holds in
the continuous approximation that

where are the numbers of muons in the element

where

On the other hand, the decay electron detection rate, which we measure in
our experiments, is

where is the detector efficiency for decay electrons from the element

where is the muon decay rate in free space, and, for the element

We used . Note that because of muon loss channels it is , rather than , which is in the normalization factor in the final line of Eq. C.5.

The actual experimental time spectra are histogrammed in a finite size of
time bin ,
hence are discrete function of ** t_{i}** (
).
To reflect this, Eq. C.5 is integrated
over ,

Evaluating Eq. C.6, we have

This can be rearranged as

which can be expanded in to ,

Eq. C.9 is an explicit expression for the intuitively obvious conclusion that, in the limit of small , counts in a histogram bin are propotional to its bin size. It also shows that the use of the integrated fitting fucntion Eq. C.7 is necessary unless ,

Thus, our fit function
is, with a background term
*bkgd*^{}