Kevin J. Coakley, Grace L. Yang

*Statistical Engineering Division, ITL*

Bradley K. Alpert

*Mathematical and Computational Sciences Division, ITL*

M.S.Dewey,
D.Gilliam

*Ionizing Radiation Division, PL*

Researchers from
Harvard University, Los Alamos National Laboratory,
University of Washington,
University of Berlin,
and NIST plan
to produce and
confine polarized Ultra Cold Neutrons
(UCN)
in a magnetic trap.
Based on this new technology,
the neutron lifetime
will be determined
at a precision up to 100 times better than
the current value.
Along with other experimental data,
a measurement of the mean lifetime of the neutron
allows one to
test the consistency of the
standard model of electroweak interactions.
Further, the mean lifetime of the neutron is
an important parameter in astrophysical theories.
Statistical and computational work has focused on
optimal experimental design
and dynamical studies of
marginally trapped neutrons.

**Optimal Estimation. **
There will be many run cycles
of a two stage experiment.
In the first stage of each run,
neutrons from the NIST Cold Neutron Research
Facility are
guided into
a superfluid ^{4}He
bath where they
dissipate almost all their energy by
inelastic scattering.
These UCN
are confined in a magnetic trap.
After filling the trap to some level,
the neutron beam is blocked and
decay events, as well as background events,
are recorded.
Denote the duration of each stage as
*T*_{fill} and *T*_{decay}.
Two algorithms for estimating the mean lifetime are
compared.
In one method, the event time data is
summarized as a histogram.
The time endpoints
of the histogram are selected so that
the expected number of counts per bin
contributed by the decay process, is
constant.
In the second method,
the lifetime is estimated from
the complete sequence of event times.
The histogram method yields
a less variable estimate
of the mean lifetime.
The optimal strategy for
time allocation is found by
minimizing the asymptotic variance of
the lifetime (estimated from the pooled histogram data from all cycles)
as a function
*T*_{fill} and *T*_{decay},
given knowledge of the filling rate of the trap and parameters
which characterize the background process.
The validity of the asymptotic approximation
is demonstrated in Monte Carlo experiments.

**Marginally Trapped Neutrons.**
Neutrons with sufficiently high energy escape
the trap, but not immediately.
These ``marginally trapped" neutrons may decay before
escaping.
In a Monte Carlo study,
many trajectories are simulated.
For each trajectory,
we compute the escape time.
Beyond about six seconds of elapsed time,
we find that
escape times can not be predicted in
a numerically stable manner.

Figure 12: The approximate mean lifetime of the neutron
is
.
During the fill stage of
each run cycle, the expected number of confined neutrons
grows as

where
is the rate at which neutrons enter the trap,
is the mean lifetime of the neutron
and *T*_{fill} is the duration of the fill stage.
We express the
asymptotic standard error of the mean lifetime, estimated
from data pooled from all run cycles, as

where
the duration of the entire experiment is
*T*_{total}.
Above,
*log*_{10}(*T*^{*}) is
plotted as
a function of
*T*_{fill}, *T*_{decay}for the case where
and
the background
is a stationary Poisson process with
intensity rate equal to .