The previous sections demonstrate that the level of transmission achieved in the MICE cooling channel in its Step IV configuration does not straight-forwardly allow for the measurement of an unbiased emittance change. The loss of large amplitude particles prevents the treatment of the beam as an ensemble. Single-particle transverse amplitude offers a route to observing an increase of phase space density at the core of the beam.
6.6.1 Generalization
Transverse amplitude,A⊥, as computed in equation3.14, assumes a single underlying covariance matrix to describe the full ensemble. Unless the transmission losses are isotropic in amplitude, the shape of the covariance matrix is not preserved and neither is the amplitude measurement. In addition, the drift space between the focus coil and the downstream section of the magnetic channel may introduce nonlinear effects, i.e. that do not conserve emittance, further tampering with the amplitude reconstruction.
One must generalise the concept of amplitude to preserve unbiased values at the core, regardless of tail losses or beam filamentation. The approach to recovering the true particle amplitude is iteratively to remove the highest amplitude from the sample. At each iteration, the covariance matrix and the beam centroid are updated through
hxαin−1= 1
n−1(nhxαin−xnα), (6.17) Σn−1αβ = n−1
n−2Σnαβ− n
n−1(xnα− hxαin) xnβ− hxβin
, (6.18)
withα, β=x, y, px, py. The amplitudes of the remaining particles are then updated with the new values and the process is repeated.
This approach is tested on a Gaussian beam to determine if it reconstructs the expected amplitude profile. The left panel of figure6.23shows a scatter plot of 104particles sampled from a⊥= 6 mm beam. In four dimensions, the amplitude follows a chi-squared distribution with four degrees of freedom, i.e. A⊥ ∼⊥χ24. The amplitude distribution is represented for traditional and corrected definitions of amplitude in the right panel. The two histograms are consistent; the Kolmogorov-Smirnov test returns a probability of∼1.
Both distributions follow the expected underlyingχ24trend.
The beam is passed through a nonlinear lens of aberration coefficient Cα = −10−4mm−2 and an aperture removes any particle of radiusR >
150 mm. The scatter plot of the beam particles in the (x, px) plane after transport are represented on the left panel of figure 6.24. Despite large tails pulling the global covariance matrix, the corrected amplitude algorithm
1
Figure 6.23: (Left) Corrected amplitude scatter plot of a Gaussian beam of⊥= 6 mm. (Right) Regular and corrected amplitude distribution of the Gaussian beam.
recovers the true shape of the beam at the core. The right panel shows the amplitude distribution before and after correction. The generalised method preserves the true underlying density function at the core of the beam.
1
Figure 6.24: (Left) Corrected amplitude scatter plot of a non-Gaussian beam.
(Right) Regular and corrected amplitude distribution of the Gaussian beam.
6.6.2 Poincaré sections
A Poincaré section is a lower-dimensional transverse section of phase space.
It allows for the reduction of a high-dimensional system in a series of two-dimensional sections representing the elements of the covariance matrix. In the four dimensional transverse phase space, a set ofd(d−1)/2 = 6 sections fully represents the off-diagonal elements of the covariance matrix.
Figure 6.25represents the six Poincaré sections of the transverse phase space in the 6 mm beam, before and after going through the lithium hydride absorber. Each point represents the projection of a particle onto the section.
Its colour depicts the corrected transverse amplitude of the particle. Particles of amplitude larger than 100 mm are not represented. The (x, px) and (y, py) sections relate to theα⊥ transverse Twiss parameter, the (x, y) section to β⊥ and the (px, py) section toγ⊥.
−1
Figure 6.25: Poincaré sections of the transverse phase space of the 6 mm muon beam upstream (top) and downstream (bottom) of the absorber in the 2017/02-7 magnetic channel setting.
6.6.3 Correction factors
The trackers provide a measurement of phase space upstream and down-stream the absorber which folds in the detector resolution and inefficiencies.
A scheme is devised to correct the amplitude distribution for the effect of the reconstruction. A correction must be applied to each reconstructed bin content, ˜N⊥i, to yield the expected true bin content, N⊥i. The correction is driven for each setting by a large simulated sample of order 106muons in the upstream trackers transported to the downstream section.
The first biasing effect that is taken into account is the detector resolution.
The Monte Carlo simulation is used to compare the reconstructed amplitudes, including digitization effect, to the true underlying amplitudes of the particles that have been reconstructed. A migration matrix is produced for each setting. Its elements,Mij, represent the probability of a particle in theith true amplitude bin, ¯Bi⊥, to fall into the jth reconstructed bin, ˜B⊥j. The content of each bin is then corrected through
N¯⊥i =
n
X
j=1
MijN˜⊥j. (6.19)
with ¯N⊥i the number of particles that fall in bin ¯B⊥i and ˜N⊥i the number of particles that fall in bin ˜Bj⊥. The migration matrices are graphically represented in figure 6.26 for the 2017/02-7 magnetic channel. The large majority of particles fall in the expected on-diagonal bins.
The second biasing factor is the detector inefficiency. Efficiency is defined with respect to the sample selected in the upstream tracker. There is no inefficiency in the upstream tracker as the sample being analysed is the one selected, regardless of what was selected out. A muon that makes it to the downstream tracker, is contained within its fiducial volume but is not reconstructed introduces an inefficiency in TKD. A particle that is not a muon but is included in the final sample constitutes an impurity.
Figure6.27represents the number of selected true tracks in theithbin, N¯⊥i, over the total number of true Monte Carlo tracks that fall in theith amplitude bin,N⊥i, in the downstream tracker, for the 2017/02-7 magnetic channel. The efficiency factorei= ¯N⊥i/N⊥i is used to correct the true number of entries in binithrough
N⊥i =e−1i N¯⊥i =e−1i
n
X
j=1
MijN˜⊥j. (6.20) The corrected content of each amplitude bin is presented in this chapter.
Equation6.20 effectively introduces a scaling factor on each bin, ci = N⊥i/N˜⊥i. An uncertainty on this factor introduces an uncertainty on the amplitude bin content. This is discussed in the next subsection.
0
[mm]Reconstructed A
10 20 30 40 50 60 70 80 90
90 MICE [simulation]
ISIS Cycle 2017/03 Run setting 7 MAUS v3.2.0
[mm]
True A
[mm]Reconstructed A
10 20 30 40 50 60 70 80 90
[mm]Reconstructed A
10 20 30 40 50 60 70 80 90
[mm]Reconstructed A
10 20 30 40 50 60 70 80 90
[mm]Reconstructed A
10 20 30 40 50 60 70 80 90
[mm]Reconstructed A
10 20 30 40 50 60 70 80 90
Figure 6.26:Amplitude reconstruction migration matrices in the upstream and the downstream samples for the 2017/02-7 magnetic channel setting.
[mm]
1.8 MICE [simulation]
ISIS Cycle 2017/03
Figure 6.27:Amplitude reconstruction efficiency factors in the downstream tracker for the 2017/02-7 cooling channel.
6.6.4 Uncertainties
The first source of uncertainty on the measurement is statistical in nature.
In a counting measurement such as this one, each bin is associated with a standard Poisson error. For sample sizes of order 105 muons in total, the statistical uncertainties in individual bins are expected to be of order 1 %.
The correction scheme assumes perfect knowledge of the hardware and the magnetic field. An uncertainty on these assumptions introduces a systematic uncertainty on the measurement. The main sources of uncertainty on the
measurement are expected to be the misalignment of the trackers with the magnetic field, the field strength and the tracker material density. For both trackers, a large simulation is produced in each of the following scenarios:
• Tracker displaced by 1 mm;
• Tracker rotated by 1 mrad;
• Field strength in the centre coil varied by 1 %;
• Field strength in each end coil varied by 5 %;
• Tracker material density varied by 25 %.
In each scenario, the simulation is generated with a sample of identical size to the baseline and the correction factors,ci, are calculated. Figure6.28 represents the uncertainty introduced on each bin due to the six afore-mentioned uncertainties in the upstream and downstream trackers, for the 2017/02-7 cooling channel. The total systematic uncertainty is defined as the quadratic sum of all the sources of uncertainty. The addition of the different sources usually amounts to an uncertainty on the bin content of order 1 % or less. No specific source of uncertainty stands out from the others as they all produce errors of the same magnitude. The bins corresponding to amplitudes larger than 50 mm are not represented as they lie beyond the expected region of the cooling signal.
[mm]
Figure 6.28:Systematic uncertainties on the amplitude distributions in the upstream and the downstream tracker samples for the 2017/02-7 cooling channel.
6.6.5 Distributions
The techniques developed in the previous sections are used to evaluate the amplitude distribution of the muon beam particles at the upstream and downstream reference planes. The top panel of figure6.29shows the corrected amplitude distributions before and after going through the absorber in the 2017/02-7 cooling channel1. Each bin is associated with a total uncertainty defined as the quadratic sum of its statistical and systematic components.
The 3 mm beam is dominated by nonlinear heating as the beam is heavily mismatched in the upstream section. In the presence of lithium hydride, both the 6 and 10 mm beams exhibit a significant increase in low amplitude bins, as expected, but show an exaggerated reduction at higher amplitudes due to scraping. The number of low amplitude particles is roughly preserved for settings where the absorber is absent.
The distributions are used to evaluate the Cumulative Density Function (CDF) ratio. The CDF is defined, at an amplitudez, as
F(z) =p(A⊥< z), (6.21) i.e. the probability ofA⊥ falling belowz. The ratio of the downstream CDF and the upstream one thus reads
R(z) = Fd(z)
Fu(z) =2− z/d⊥+ 2
e−z/(2d⊥)
2−(z/u⊥+ 2)e−z/(2u⊥). (6.22) Theoretically, the CDF ratio equates (u⊥/d⊥)2 atz= 0. It then follows an exponential decay that converges to 1 at infinity. For each amplitude bin, the ratio is computed for particles that fall below the upper edge of the bin.
The bottom panel of figure6.29shows the measured CDF ratios, in the data and the simulation. In the presence of the lithium hydride absorber, a very strong and significant increase in the number of low amplitude muons is observed in the 6 and 10 mm beams. This indicates that particles migrated to regions of higher density, effectively cooling the beam.