With these Monte Carlo parameters fixed, we can now compare the time of
flight spectrum of the emitted muonic tritium with Monte Carlo simulations
to test theoretical cross sections based on few-body theory. We shall focus
on the measurement of the position of the Ramsauer-Townsend minimum ** E_{R}**,
since this is important both as a source of the
beam and as a test
of the quantum three-body calculations.

The Nuclear Atlas cross sections [16,17] for the muonic
processes were used as a nominal input to the Monte Carlo. Molecular and
condensed matter effects [167,170,216] are expected to
play negligible roles in the transport properties of muonic atoms at the
energies above a few eV. Iterative calculations were performed by
multiplying the energy scale of the
elastic scattering cross
section by a constant factor ,
thus shifting the RT minimum as
.
The resulting simulated time spectrum is fitted
to the experimental data with one free parameter (relative normalization),
and
was calculated for each value of .
For both the
data and the MC, a longitudinal spatial cut of
** z=[-10,10]** mm was applied
to select the vacuum region.

Figure 7.4 shows an example of such a fit (top) and its
residuals (bottom). Plotted with error bars is the
time spectrum
from the standard emission target^{} (1000
T
H** _{2}** with 0.1% T

Illustrated in Fig. 7.5 is the global trend of total
versus the energy scaling factor ,
while
Fig. 7.6 shows the details near the minimum. The horizontal
axis is plotted against the inverse of square root of
to reflect
our sensitivity to the time of flight, rather than the
energy^{}.

Dependences of the various parameters were investigated in detail for the potential systematic effects. Figs. 7.5 and 7.6 show some examples of such investigations including:

- 1.
- Muon transfer rate from protons to tritons,
,
as input
to the simulation. Our standard input uses an experimental rate measured
by our collaboration,
s
[83] for this process, but fits using the theoretical energy-dependent value of from the Nuclear Atlas cross section [17] were also tried.^{-1} - 2.
- Time interval of the fit region. Region A:
s, Region B:*t*=[0,6]s (see Fig. 7.4). If our fit model is correct, the fit results should be independent of fit region except to add a constant value in total , which was confirmed in Figs. 7.5, 7.6.*t*=[0.1,3] - 3.
- MWPC resolution parameter . Our nominal value 1.3 mm was varied from 1.0 mm to 1.5 mm (only some selections are shown in Fig. 7.6.)
- 4.
- Beam width parameters characterizing a flat top Gaussian distribution
in the
plane. 1010 mm (nominal) and 1515 mm were plotted.*XY* - 5.
- Muon beam stopping distribution in the
direction of the emission layer.*z*

: Gaussian with the peak at the surface of the layer with the standard deviation width of half the layer thickness.

: Gaussian with the peak at the centre of the layer with the standard deviation width of 0.4 times layer thickness.

No symbol: uniformstopping distribution.*Z*

For the nominal values of parameters, we obtained the best fit with
(
in the figures). The value of shifted between
1.05 to 1.15 depending on the parameters (see Fig 7.6).
In addition, we estimate that the uncertainty in the drift distance scale
can give rise to a shift in
of order .
With the two
major systematic errors (due to the parameters (1)-(5) and the distance
scale) added in quadrature, and the statistical errors much smaller, as can
be observed in Fig. 7.6, our measurement indicates a scaling
factor of
,
*i.e.*, the Ramsauer-Townsend minimum
energy of
eV (*c.f.* the theortecial minimum,
** E_{R}^{th} = 12.4** eV). The results of a similar analysis for a measurement
using an emission target with a tritium concentration of
0.3%

In conclusion, we have reported in this chapter, (1) the first observation of in vacuum, (2) the first quantitative spectroscopic evidence of the Ramsauer-Townsend effect in an exotic system, confirming the theoretical RT minimum energy in the elastic scattering at the 10% level, an accuracy sufficent for our goal of molecular formation rate measurements.