We have looked above at the possible individual influence on the stopping
fraction determination by charged particles following muon capture, decay
electron energy spectra, and upstream-downstream acceptance variations.
Here we combine all the factors to give a general expression. When the
muons stop in ** M** different materials, located in

where

(93) |

with

In our specific case where we assume ** M = 2** and hydrogen is present only
at the upstream foil (
), while the
muons stop in gold both at the upstream and the downstream layer with the
relative fraction

where is the ratio of the effective efficiencies for gold and hydrogen both at the upstream location, and that for the upstream and the downstream gold:

The expressions above reduce to Eq. 6.11, if (

We consider the case for counters at the sides (Ege, En1) and downstream
(En2) separately. For En2, if we assume
(** k = Au, H** and

(97) | |||

(98) |

On the other hand, for Ege, En1, assuming
,
,
we have

(99) | |||

1, | (100) |

where there is some uncertainty in due to the charged particle emission effect . In addition, there is potentially a large error in the factor due to the possible non-flatness of the gold foil which could cause significant shadowing of the electrons either from gold or hydrogen

The factors
in
Eq. 6.15 were determined from detailed GEANT
simulations [218] taking into account the full geometry.
Simulation for the muon beam assuming a momentum of 27.0 MeV/c with
of 5.7% yielded the gold stopping upstream and downstream
to be
** F_{u} = 0.48** and

The correction factors, which take into account the differences in solid angle for upstream and downstream, electron energy spectra between gold and hydrogen, and effects of charged particle emission, can now be determined. Together with the assumed quantities, the values of and are given in Table 6.12. Note that assuming , is the direct ratio of the electron detection efficiencies . As mentioned before, can have relatively large uncertainties, due to possible errors in and .

The effect of these corrections on the stopping fraction is summarized in Table 6.13. The uncertainty in the estimated value of (1.04) for Ege, En1 is assumed to be 0.04, which is included quadratically in the uncertainties presented in the table. Possible errors in for Ege and En1 are not included. The error for En2 is statistical only.

The total of about **-**12% relative correction to
for Ege, En1, is dominated by the **-14**% correction due to the difference
in the relative electron detection efficiency
(which in turn is
dominated presumably by the electron energy spectrum effect), partly offset
by a +3% effect due to charged particle emission.

For En2, the correction to
is rather small as a result
of cancellation between the **-**16% relative correction, due to the
difference in the energy spectrum, and the +14% correction, due to the
difference in the upstream-downstream relative acceptance.
The latter correction in
is reasonably close to the
approximate **+10**% given on page , estimated with
simplified assumptions without detailed simulations.