4.7 Momentum resolution
4.7.1 Simulations
A muon beam of central momentum 200 MeV/cis generated 1 m upstream of the centre of the EMR. The particles are fired in the direction of the detector with a momentum spread varied in the range 0–50 MeV/c. The angular and spacial spreads are fixed to match a beam with a 6 mm transverse emittance.
The Monte Carlo hits associated with a single muon stopping in the scintillating volume are represented as red crosses in figure4.29, in thexz andyz projections. The hits are digitized and reconstructed following the same procedure used to process data to produce track points. The linear track points are in agreement with the true trajectory of the particle but do not resolve the small scatterings about the central trend. A higher order polynomial would only increase the uncertainty on the fitting parameters.
z [mm]
−450−400−350−300−250−200−150−100−50 0
x [mm]
30 35 40 45 50 55
Truth
Reconstructed [simulation]
MICE
MAUS v3.0.0
z [mm]
−450−400−350−300−250−200−150−100−50 0
y [mm]
−6
−4
−2 0 2 4 6 8
Figure 4.29:Interactions of a single muon with the scintillating volume of the EMR (red crosses) and reconstructed track points (black squares).
At each step of the propagation, the simulation records the amount of polystyrene traversed by the particle before interacting. The increments are added together to form the true MC range,R. The reconstructed range, ˆR, is produced with equation4.37. The residuals are represented in figure4.30 for the monochromatic 200 MeV/c muon beam. The distribution peaks very close to zero and exhibits a slightly larger tail on the left-hand side.
This is a consequence of the insufficient resolution to reconstruct the small-scale scattering presented previously. The ∼4 mm RMS is dominated by the intrinsic resolution of the detector on the end point of the track. The uncertainty on the fitting parameters, the straggling about the linear fit and the variable number of holes encountered also contribute slightly.
-R [mm]
R
−050 −40 −30 −20 −10 0 10 20 30 40 50
500 1000 1500 2000 2500 3000 3500
Entries 9998
Mean −0.2041 ± 0.04133 RMS 4.121 ± 0.02923 [simulation]
MICE
MAUS v3.0.0
Figure 4.30:Distribution of residuals between the reconstructed range, ˆR, and the true range,R, of a 200 MeV/cmonochromatic muon beam in the EMR.
The impinging total momentum, ˆp, is estimated by inverting the relation in equation4.38. The truth,p, is provided at generation by the simulation framework. A muon beam of large momentum spread is used to scan over the range of momenta encountered in MICE. The relative RMS resolution is represented as a function of momentum in figure4.31. Below 80 MeV/c, the range is less than 3 cm, i.e. the thickness of two bars, and the resolution diverges. Above 300 MeV/c, the muons do not stop in the scintillating volume.
The EMR achieves resolutions as low as∼1.5 % on the total momentum for momenta above 150 MeV/c. For a 200 MeV/c muon, this corresponds to an uncertainty of ∼3 MeV/c, better than the longitudinal momentum resolution of the MICE trackers. The reconstructed momentum is consistent with the true momentum throughout the range of interest.
p [MeV/c]
100 150 200 250 300
-p)/p [%]p(
−3
−2
−1 0 1 2 3
[simulation]
MICE
MAUS v3.0.0
p [MeV/c]
100 150 200 250 300
RMS [%]
1.5 2 2.5 3 3.5
Figure 4.31: Relative residuals between the reconstructed momentum, ˆp, and the true momentum,p, as a function of the true momentum. The squares represent the central values in each bin. The dark blue error bars show the uncertainty on the mean and the light blue error bars the bin RMS, graphed in the lower pad.
4.7.2 Measurement
The entire data set recorded during the October 2013 run is used in this anal-ysis. The TOF momentum is reconstructed from the time-of-flight between TOF1 and TOF2, as explained in section4.5.2. The muons are selected and the range is reconstructed using equation 4.37. The range distribution is represented as a function of the TOF momentum in figure4.32. The plateau observed around 80 cm corresponds to the depth of the EMR detector.
1 10 102
[MeV/c]
pd
150 200 250 300 350 400 450 500
R [mm]
0 100 200 300 400 500 600 700 800 900 1000
MICE
ISIS Cycle 2013/03 Run 5384-5598 MAUS v1.2.0
Figure 4.32:Reconstructed muon range as a function of the longitudinal momentum, pz, reconstructed from the time-of-flight between TOF1 and TOF2,t12.
In the CSDA, the range is a monotonic function of the impinging momen-tum,p0, as expressed by the integral in equation4.38. To make a comparison with data, the momentum inferred from the time-of-flight is corrected for the energy loss in TOF2 and the KL following the prescriptions in section4.5.3 to yield p0. The theoretical function is obtained at a given momentum by numerically integrating the full Bethe’s stopping-power formula. The comparison between data and theory is represented in figure4.33. The the-oretical prediction closely follows the data. The CSDA is a good working approximation to reconstruct the momentum from a range measurement.
[MeV/c]
Figure 4.33:Average range measured in the EMR as a function of the momentum corrected for energy loss in TOF2 and the KL,p0 (blue points) compared to the range predicted in the Continuously Slowing Down Approximation (CSDA).
The momentum of the muon impinging the EMR,p0, can be estimated by inverting the relation in equation4.38. In this analysis, there are three main sources of uncertainty on the momentum measurement. The spread in the time-of-flight measurement,σt'150 ps, and the distance between the stations, σs ' 0.5 mm, results in an uncertainty on the downstream momentum,pd, expressed as
σTOF(pd) =
(pd/mc)2+ 1, the Lorentz gamma factor. This uncertainty grows rapidly as the velocity of the muon approaches the speed of light, i.e. as γ→+∞. The energy loss straggling in TOF2 and the KL,σdE/dx, estimated from a Geant4 simulation in figure 4.18 as a function of p0, is another significant source of uncertainty at low momentum. Given that the uncertainties are not correlated, the resolution of the EMR is deconvolved
from the other sources of error through σEMR =q
σ2−σTOF2 −σdE/dx2 . (4.40) The uncertainty that originates from the TOF measurement is estimated by binningpd as a functionp0 to yield a probability density function in each bin,fi(pd). The uncertainty in bini is computed as
σT OF =Z +∞
−∞
fi(pd)σTOF(pd)dpd. (4.41) Figure 4.34 shows a compilation of the uncertainties originating from the different components of this analysis. For small values of the range, the EMR has a poor resolution and does not produce reliable momentum reconstruction. Above 150 MeV/c, the total spread fluctuates around its nominal value of 10–15 MeV/cuntil the last bin where the value is biased by the muons exiting the detector. The EMR itself, subtracting the errors from KL and TOF, achieves resolutions down to∼5 MeV/cfor larger range, consistent with the simulation. The simulation seems to underestimate the energy straggling that stems from the energy loss in TOF2 and the KL.
80 100 120 140 160 180 200 220 240 260 280
[MeV/c]0p
100 150 200 250 300
MICE
ISIS Cycle 2013/03 Run 5384-5598 MAUS v1.2.0
[MeV/c]
p0
80 100 120 140 160 180 200 220 240 260 280
RMS [MeV/c] 5 10 15 20
25 σ
σTOF σEMR dE/dx σ
Figure 4.34:Reconstructed muon range as a function of the longitudinal momentum, pz, reconstructed from the time-of-flight between TOF1 and TOF2,t12.