MICE Collaboration

# Multiple Coulomb Scattering of muons in Lithium Hydride

M. Bogomilov, R. Tsenov, G. Vankova-Kirilova  Department of Atomic Physics,
St. Kliment Ohridski University of Sofia, 5 James Bourchier Blvd, Sofia,
Bulgaria  Y. P. Song, J. Y. Tang  Institute of High Energy Physics, Chinese
Academy of Sciences, 19 Yuquan Rd, Shijingshan District, Beijing, China  Z. H.
Li  Sichuan University, 252 Shuncheng St, Chengdu, China  R. Bertoni, M.
Bonesini, F. Chignoli, R. Mazza  Sezione INFN Milano Bicocca and Dipartimento
di Fisica G. Occhialini, Piazza della Scienza 3, Milano, Italy  V. Palladino
Sezione INFN Napoli and Dipartimento di Fisica, Università Federico II,
Complesso Universitario di Monte S. Angelo, via Cintia, Napoli, Italy  A. de
Bari  Sezione INFN Pavia and Dipartimento di Fisica, Università di Pavia, Via
Agostino Bassi 6, Pavia, Italy  D. Orestano, L. Tortora  Sezione INFN Roma Tre
and Dipartimento di Matematica e Fisica, Università Roma Tre, Via della Vasca
Navale 84, Roma, Italy  Y. Kuno, H. Sakamoto111Current address RIKEN 2-1
Hirosawa, Wako, Saitama, Japan, A. Sato  Osaka University, Graduate School of
Science, Department of Physics, 1-1 Machikaneyamacho, Toyonaka, Osaka, Japan
S. Ishimoto  High Energy Accelerator Research Organization (KEK), Institute of
Particle and Nuclear Studies, Tsukuba, Ibaraki, Japan  M. Chung, C. K. Sung
Department of Physics, UNIST, 50 UNIST-gil, Ulsan, South Korea  F.
Filthaut222Also at Radboud University, Houtlaan 4, Nijmegen, Netherlands
Nikhef, Science Park 105, Amsterdam, Netherlands  M. Fedorov  Radboud
University, Houtlaan 4, Nijmegen, Netherlands  D. Jokovic, D. Maletic, M.
Savic  Institute of Physics, University of Belgrade, Serbia  N. Jovancevic, J.
Nikolov  Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovića
3, Novi Sad, Serbia  M. Vretenar, S. Ramberger  CERN, Esplanade des Particules
1, Geneva, Switzerland  R. Asfandiyarov, A. Blondel, F. Drielsma333Current
address SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park,
California, USA, Y. Karadzhov  DPNC, Section de Physique, Université de
Genève, 24 Quai Ernest-Ansermet, Geneva, Switzerland  G. Charnley, N. Collomb,
K. Dumbell, A. Gallagher, A. Grant, S. Griffiths, T. Hartnett, B. Martlew, A.
Moss, A. Muir, I. Mullacrane, A. Oates, P. Owens, G. Stokes, P. Warburton, C.
White  STFC Daresbury Laboratory, Keckwick Ln, Daresbury, Cheshire, UK  D.
Adams, V. Bayliss, J. Boehm, T. W. Bradshaw, C. Brown444Also at College of
Engineering, Design and Physical Sciences, Brunel University, Kingston Lane,
Uxbridge, UK, M. Courthold, J. Govans, M. Hills, J.-B. Lagrange, C. Macwaters,
A. Nichols, R. Preece, S. Ricciardi, C. Rogers, T. Stanley, J. Tarrant, M.
Tucker, S. Watson555Current address ATC, Royal Observatory Edinburgh,
Blackford Hill, Edinburgh, UK, A. Wilson  STFC Rutherford Appleton Laboratory,
Harwell Campus, Didcot, UK  R. Bayes666Current address Laurentian University,
935 Ramsey Lake Road, Sudbury, Ontario, Canada, J. C. Nugent, F. J. P. Soler
School of Physics and Astronomy, Kelvin Building, University of Glasgow,
Glasgow, UK  R. Gamet, P. Cooke  Department of Physics, University of
Liverpool, Oxford St, Liverpool, UK  V. J. Blackmore, D. Colling, A.
Dobbs777Current address OPERA Simulation Software, Network House, Langford
Locks, Kidlington, UK, P. Dornan, P. Franchini888Current address Lancaster
University, Physics avenue, Lancaster, UK, C. Hunt999Current address CERN,
Esplanade des Particules 1, Geneva, Switzerland, P. B. Jurj, A. Kurup, K.
Long, J. Martyniak, S. Middleton101010Current address School of Physics and
Astronomy, University of Manchester, Oxford Road, Manchester, UK, J.
Pasternak, M. A. Uchida111111Current address Rutherford Building, Cavendish
Laboratory, JJ Thomson Avenue, Cambridge, UK Department of Physics, Blackett
Laboratory, Imperial College London, Exhibition Road, London, UK  J. H. Cobb
Department of Physics, University of Oxford, Denys Wilkinson Building, Keble
Rd, Oxford, UK  C. N. Booth, P. Hodgson, J. Langlands, E. Overton121212Current
address Arm, City Gate, 8 St Mary’s Gate, Sheffield, UK, V. Pec, P. J. Smith,
S. Wilbur  Department of Physics and Astronomy, University of Sheffield,
Hounsfield Rd, Sheffield, UK  G. T. Chatzitheodoridis131313Also at School of
Physics and Astronomy, Kelvin Building, University of Glasgow, Glasgow,
UK,141414Also at Cockcroft Institute, Daresbury Laboratory, Sci-Tech
Daresbury, Keckwick Ln, Daresbury, Warrington, UK, A. J. Dick††footnotemark: ,
K. Ronald††footnotemark: , C. G. Whyte††footnotemark: , A. R.
Young††footnotemark:  SUPA and the Department of Physics, University of
Strathclyde, 107 Rottenrow, Glasgow, UK  S. Boyd, J. R. Greis151515Current
address TNG Technology Consulting, Beta-Strasse 13A, Unterföhring, Germany, T.
Lord, C. Pidcott161616Current address Department of Physics and Astronomy,
University of Sheffield, Hounsfield Rd, Sheffield, UK, I. Taylor171717Current
address Defence Science and Technology Laboratory, Porton Down, Salisbury, UK
Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, UK
M. Ellis181818Current address Macquarie Group, 50 Martin Place, Sydney,
Australia, R.B.S. Gardener191919Current address Inawisdom, Columba House,
Adastral park, Martlesham, Ipswich, UK, P. Kyberd, J. J.
Nebrensky202020Current address UK Atomic Energy Authority, Culham Science
Centre, Abingdon, UK College of Engineering, Design and Physical Sciences,
Brunel University, Kingston Lane, Uxbridge, UK  M. Palmer, H. Witte
Brookhaven National Laboratory, 98 Rochester St, Upton, New York, USA  D.
Adey212121Current address Institute of High Energy Physics, Chinese Academy of
Sciences, 19 Yuquan Rd, Shijingshan District, Beijing, China, A. D. Bross, D.
Bowring, P. Hanlet, A. Liu222222Current address Euclid Techlabs, 367 Remington
Blvd, Bolingbrook, Illinois, USA, D. Neuffer, M. Popovic, P. Rubinov
Fermilab, Kirk Rd and Pine St, Batavia, Illinois, USA  A. DeMello, S. Gourlay,
A. Lambert, D. Li, T. Luo, S. Prestemon, S. Virostek  Lawrence Berkeley
National Laboratory, 1 Cyclotron Rd, Berkeley, California, USA  B.
Freemire††footnotemark: , D. M. Kaplan, T. A. Mohayai232323Current address
Fermilab, Kirk Rd and Pine St, Batavia, Illinois, USA, D. Rajaram242424Current
address KLA, 2350 Green Rd, Ann Arbor, Michigan, USA, P. Snopok, Y. Torun
Illinois Institute of Technology, 10 West 35th St, Chicago, Illinois, USA  L.
M. Cremaldi, D. A. Sanders, D. J. Summers252525Deceased University of
Mississippi, University Ave, Oxford, Mississippi, USA  L. R.
Coney262626Current address European Spallation Source ERIC, Partikelgatan 2,
Lund, Sweden, G. G. Hanson, C. Heidt272727Current address Swish Analytics,
Oakland, California, USA University of California, 900 University Ave,
Riverside, California, USA

###### Abstract

Multiple Coulomb Scattering (MCS) is a well known phenomenon occurring when
charged particles traverse materials. Measurements of muons traversing low $Z$
materials made in the MuScat experiment showed that theoretical models and
simulation codes, such as GEANT4 (v7.0), over-estimated the scattering. The
Muon Ionization Cooling Experiment (MICE) measured the cooling of a muon beam
traversing a liquid hydrogen or lithium hydride (LiH) energy absorber as part
of a programme to develop muon accelerator facilities, such as a Neutrino
Factory or a Muon Collider. The energy loss and MCS that occur in the absorber
material are competing effects that alter the performance of the cooling
channel. Therefore measurements of MCS are required in order to validate the
simulations used to predict the cooling performance in future accelerator
facilities. We report measurements made in the MICE apparatus of MCS using a
LiH absorber and muons within the momentum range 160 to 245 MeV/$c$. The
measured RMS scattering width is about 9% smaller than that predicted by the
approximate formula proposed by the Particle Data Group. Data at 172, 200 and
240 MeV/$c$ are compared to the GEANT4 (v9.6) default scattering model. These
measurements show agreement with this more recent GEANT4 (v9.6) version over
the range of incident muon momenta.

††preprint: RAL-P-2022-001

##  I INTRODUCTION

Multiple Coulomb Scattering (MCS) describes the multiple interactions of
charged particles in the Coulomb field of the nuclei and electrons of a
material. Rossi and Greisen derived a simple expression for the root-mean-
square (RMS) scattering angle in the small angle approximation [1] by
integrating the Rutherford cross section [2]. The mean square scattering angle
$\left\langle\theta^{2}\right\rangle$ after multiple collisions traversing a
thickness $dz$ of material can be expressed as a function of radiation length
$X_{0}$

$\frac{d\left\langle\theta^{2}\right\rangle}{dz}=\frac{E_{s}^{2}}{p^{2}\beta^{2}}\frac{1}{X_{0}},$
(1)

where $E_{s}=21.2$ MeV/$c$, $p$ is the momentum of the charged particle and
$\beta$ its speed in units of the speed of light, $c$. The projection of the
scattering angle onto a plane containing the incident track gives the RMS
projected scattering angle
$\theta_{0}=\sqrt{\left\langle\theta^{2}/2\right\rangle}$ [3]

$\theta_{0}=\frac{14.85\textrm{ MeV/{\hbox{c}}}}{p\beta}\sqrt{\frac{\Delta
z}{X_{0}}}.$ (2)

Molière [4, 5] developed a theory of MCS based on the scattering of fast
charged particles from atomic nuclei that showed good agreement with data.
Bethe [6] improved the treatment by taking into account interactions with
electrons within the atom. The theory was subsequently improved by Fano [7] to
account for elastic and inelastic scattering.

Most of the models of MCS mentioned above reproduce data very well [8] for
small angle scatters and when the atomic number, $Z$, of the target nuclei is
large. Highland [9] compared the Molière theory with the simple formula by
Rossi and Greisen (Eq. 1), and found a distinct $Z$ dependence of the value of
$E_{s}$. As a consequence, Highland recommended that a logarithmic term be
added to the Rossi-Greisen formula to improve the agreement with Molière’s
theory, especially at low $Z$ such as for liquid hydrogen or lithium hydride.
The formula for $\theta_{0}$, the RMS width of the Gaussian approximation for
the central 98% of the projected scattering angle distribution on a plane, was
reviewed by Lynch and Dahl [10] and is now recommended by the Particle Data
Group [3] as

$\theta_{0}=\frac{13.6\textrm{ MeV/{\hbox{c}}}}{p\beta}\sqrt{\frac{\Delta
z}{X_{0}}}\left(1+0.038\ln{\frac{\Delta z}{X_{0}\beta^{2}}}\right),$ (3)

claimed to be accurate to 11% over the full range of values of $Z$.

Multiple scattering has not been well modelled for low $Z$ materials in
standard simulations. Data collected by the MuScat experiment [8] indicate
that GEANT4 v7.0 [11] and the Molière model overestimate MCS for these
materials. However, a simple Monte Carlo method, which samples the Wentzel
scattering cross section [12] to generate the MCS distributions, was shown by
Carlisle and Cobb in [13] to agree very well with muon scattering data from
the MuScat experiment. Since the time of MuScat, GEANT4 has evolved through
several versions and the comparison to data made in this analysis uses GEANT4
v9.6.

Emittance is a measure of the average spread of particle coordinates in
position and momentum phase space and has dimensions of length times angle,
e.g., mm$\cdot$radians, usually written as just mm. The Muon Ionization
Cooling Experiment (MICE) made measurements of emittance reduction in low $Z$
absorbers, i.e., those materials that can be used to reduce muon-beam
emittance via ionization cooling [14], thus providing the first observation of
the ionization cooling process [15] that can be used to cool beams of muons
for a Neutrino Factory [16] or a Muon Collider [17, 18, 19, 20]. The
normalized transverse emittance of the MICE muon beam [21] is reduced due to
energy loss and increased by the scattering in the absorber material. The rate
of change in the normalized emittance, $\epsilon_{n}$, [14] is given by

$\frac{d\epsilon_{n}}{dz}\approx-\frac{\epsilon_{n}}{p_{\mu}\beta}\Bigg{\langle}\frac{dE_{\mu}}{dz}\Bigg{\rangle}+\frac{\beta_{\perp}p_{\mu}}{2m_{\mu}}\frac{d\theta_{0}^{2}}{dz}\,,$
(4)

where $\frac{dE_{\mu}}{dz}$ is the energy loss of muons per unit distance,
$m_{\mu}$ the muon mass, $p_{\mu}$ the muon momentum and $\beta_{\perp}$ the
betatron function.

To make accurate predictions of the emittance in the absorber materials, the
model in the simulation must be validated. This is particularly important for
the prediction of the equilibrium emittance, the case when
$d\epsilon_{n}/dz=0$ and

$\epsilon_{n}=\frac{\beta_{\perp}p_{\mu}^{2}\beta}{2m_{\mu}}\frac{d\theta_{0}^{2}}{dz}\left\langle\frac{dE_{\mu}}{dz}\right\rangle^{-1}.$
(5)

This provides the minimum emittance for which cooling is effective and is
lowest for low $Z$ absorbers. There is thus great interest in performing a
detailed measurement of MCS of muons traversing low $Z$ absorbers, such as
liquid hydrogen or lithium hydride (LiH). Here, we report the first
measurement of MCS of muons in lithium hydride in the muon momentum range 160
to 245 MeV/$c$, using the MICE apparatus. Accurate MCS modelling will ensure
design studies for future facilities are as informative as possible [22]. This
paper is divided as follows: Section II outlines the MICE experiment,
describes the analysis method and defines the relevant measurement angles,
Section III describes the data collected and the event selection and Section
IV describes the data deconvolution method and the multiple scattering
results, with a final short conclusion in Section V.

##  II METHOD

The MICE configuration for the MCS measurements presented here consisted of
two scintillating fiber trackers, one upstream (US) and one downstream (DS) of
a lithium hydride absorber. Each tracker contained five stations, each
composed of three planes of scintillating fiber employing 120∘ stereo views,
immersed in helium gas [23]. Thin aluminum windows separated the helium volume
from the vacuum containing the absorber. The tracker position resolution was
determined to be 470 $\mu$m [24]. The solenoid magnets surrounding the
trackers were turned off for these measurements to allow straight-track
reconstruction of the muons before and after the absorber.

The muon beam was generated by protons with a kinetic energy of 700 MeV at the
STFC Rutherford Appleton Laboratory ISIS synchrotron facility [21, 25]
impinging on a titanium target [26, 27]. The beam line is described in [21].

A schematic diagram of the MICE cooling channel and detectors is shown in Fig.
1. A time of flight (TOF) system, consisting of three detectors (TOF0 and TOF1
upstream and TOF2 downstream of the apparatus), was used to measure the
momentum of reconstructed muons [28]. The Cherenkov detector, pre-shower
system (KL) and Electron-Muon Ranger (EMR) were used to confirm the TOF’s
particle identification performance [21, 29, 30]. The MICE coordinate system
is defined with $+z$ pointing along the beam direction towards the downstream
region, $+y$ pointing upwards and $+x$ defined to be consistent with a right-
handed coordinate system.

Figure 1: Schematic of the MICE cooling channel. The spectrometer solenoids
and focus coils were not powered during the measurements described here. A
variable thickness diffuser upstream of the trackers was fully retracted
during the measurements. Acronyms are defined in the text.

The MICE LiH absorber was a disk, 65.37$\pm$0.02 mm thick (along the $z$-axis)
and 550 mm in diameter. The absorber was coated with a thin parylene layer to
prevent the ingress of water or oxygen. The composition of the LiH disk by
weight was 81% 6Li, 4% 7Li and 14% 1H with some trace amounts of carbon,
oxygen and calcium. The density of the disk was measured to be
0.6957$\pm$0.0006 g/cm3, and the radiation length was calculated to be 70.38
g/cm2.

Multiple scattering is characterized using either the three-dimensional (3D)
angle between the initial and final momentum vectors, $\theta_{Scatt}$, or the
2D projected angle of scattering. The projected angles between the track
vectors in the $x$-$z$ ($\theta_{Y}$) and $y$-$z$ ($\theta_{X}$) planes of the
experimental coordinate system can be used, but these are only the true
projected angles if the incident muon has no component of momentum in a
direction perpendicular to these planes, i.e., the $y$ or $x$ direction
respectively. To obtain the correct projected angle, a plane of projection
must be defined for each incoming muon. The rotation calculated about an axis
in the plane defined for each incoming muon is, to a very good approximation,
the rotation around the specified axis. The precise definitions of
$\theta_{X}$ and $\theta_{Y}$ are given in the Appendix.

Table 1 shows the expected RMS projected scattering angles, $\theta_{0}$,
obtained using Eq. 3, for the LiH absorber and the material in each of the
trackers. The number of radiation lengths traversed by a muon as it passes
through the absorber was larger than that which it traversed as it passed
through the trackers hence the majority of the scattering occurs in the
absorber. Nevertheless the scattering in the trackers is significant and must
be corrected for.

Table 1: Material budget affecting particles passing through the MICE LiH absorber. The material thickness normalized by the radiation length is given with the RMS width of the scattering distribution calculated from the full PDG formula [3] in eq. 3. Note that the thickness shown for the tracker materials (He, Al windows and Scintillating Fibres) includes both trackers. |  |  |  |  $\theta_{0}$ (mrad)
---|---|---|---|---
Material |  $z$ (cm) | $z/X_{0}$ |  $\rho$ (g cm-3) | 172 MeV/$c$ | 200 MeV/$c$ | 240 MeV/$c$
Tracker He | 226 | 0.00030 | 1.663$\times$10-4 | 1.09 | 0.91 | 0.73
Al Window | 0.032 | 0.0036 | 2.699 | 4.31 | 3.58 | 2.89
Scintillating Fibres | 1.48 | 0.036 | 1.06 | 14.9 | 12.4 | 10.0
Total Tracker |  | 0.038 |  | 15.8 | 13.2 | 10.6
LiH | 6.5 | 0.0641 | 0.6957 | 21.3 | 17.7 | 14.3
Total with LiH |  | 0.1058 |  | 29.9 | 24.8 | 20.0

##  III DATA SELECTION AND RECONSTRUCTION

A coincidence of two PMTs firing in TOF1 was used to trigger readout of the
detector system including the trackers. The muon rate was such that only a
single incident particle was observed in the apparatus per readout. Data
reconstruction and simulation were carried out using MAUS (MICE Analysis and
User Software) v3.3.2 [31] (which uses GEANT4 v9.6.p02). Position and angle
reconstruction was performed using data from the MICE trackers while momentum
reconstruction was performed using data from the TOF detectors.

###  III.1 Position and Angle Reconstruction

Space points were created from the signals generated in the three
scintillating fiber planes contained in a tracker station. Multiple space
points that formed a straight line through the tracker were associated
together. Space points that did not match a possible track were rejected. A
Kalman filter [32] was used to provide an improved estimate of the track
position and angle in each tracker at the plane nearest to the absorber.

An upstream track was required for the event to be considered for analysis,
with a minimum of three space points among the five stations of the upstream
tracker. No requirement was made on the presence of a downstream track. All
scattering distributions were normalized to the number of upstream tracks
selected in the analysis. The efficiency of the trackers has been shown to be
very close to $100$% [33].

A residual misalignment between the upstream and downstream trackers was
corrected by rotating all upstream tracks by a fixed angle in the range 1–7
mrad. The final uncertainty in the rotation angles following the alignment
procedure was 0.07 mrad.

###  III.2 Momentum Reconstruction

Time of flight was used to measure the momentum of the muon at the absorber.
Two time of flight measurements were used, designated as TOF01, the time of
flight between TOF0 and TOF1, and TOF12, the time of flight between TOF1 and
TOF2. The average momentum between time of flight detectors was calculated by
evaluating

$p=\frac{m_{\mu}c}{\sqrt{\frac{t_{\mu}^{2}}{t_{e}^{2}}-1}}-\Delta
p_{BB}-p_{MC},$ (6)

which assumes the mass of the electron to be $\approx$ 0 and where $t_{\mu}$
is the time of flight of the muon and $t_{e}$ is the average time of flight of
positrons ($t_{e}=25.40$ ns for TOF01 and $27.38$ ns for TOF12). $\Delta
p_{BB}$ was an additional term which accounted for the Bethe-Bloch most
probable energy loss of the muon as it passes through matter and was chosen to
yield an optimal reconstructed momentum at the center of the absorber. When
measuring the momentum using TOF01, accounting for the material upstream of
the LiH absorber, $\Delta p_{BB}$ was of order $\sim$25 MeV/$c$ (the
correction varied as a function of muon momentum and was calculated separately
for each selected sample of muons). $p_{MC}$ accounted for the bias between
the reconstructed and true momentum observed in the Monte Carlo (MC)
simulation; this arises primarily due to the simplifying assumptions intrinsic
to Eq. 6, e.g., that the path length between the TOF detectors can be
approximated to the straight line on-axis distance between the two detectors
when in fact the particle’s trajectory may have curved through various
magnetic fields or scattered in material. $p_{MC}$ was used when calculating
the momentum with both TOF01 and TOF12 and the correction, $p_{MC}$, was
$\sim$2–6 MeV/$c$. After correction, the reconstructed data were well
described by the MC as shown in Fig. 2.

For muons reaching the end of the channel, the momentum measurement was made
using TOF1 and TOF2. In this case the absorber sits near the midpoint between
the detectors and the distance between them was larger than the distance
between TOF0 and TOF1 which results in a slightly smaller uncertainty. In the
selected samples, $\sim$ 90% of muons reach TOF2. If no hit was recorded in
TOF2, the momentum measurement was made using TOF0 and TOF1. The TOF01
distribution is shown in Fig. 3.

Characteristics of the time-of-flight samples selected using TOF01 are shown
in Table 2. The resolution of the TOF system was $\approx$70 ps which
corresponds to $\sim$4–10 MeV/$c$ depending on the momentum setting. The
agreement between the reconstructed momentum and the simulated true muon
momentum at the centre of the absorber is shown in Fig. 2 and a residual plot
($p_{\mathrm{Reconstructed}}-p_{\mathrm{Truth}}$) is shown in Fig. 2.

Table 2: Characteristics of the samples selected for model comparison; the standard deviation of the reconstructed momenta are compared with the spread of true momenta of equivalent samples selected from the simulation. Desired Momentum | Lower TOF | Upper TOF | Measured | Standard deviation | True MC Momentum
---|---|---|---|---|---
(MeV/$c$) | limit (ns) | limit (ns) |  $\langle p\rangle$ (MeV/$c$) | (MeV/$c$) | Spread (MeV/$c$)
172 | 28.60 | 28.80 | 171.55$\pm$0.06 | 4.37$\pm$0.06 | 4.82
200 | 27.89 | 28.09 | 199.93$\pm$0.07 | 5.92$\pm$0.05 | 5.97
240 | 27.16 | 27.36 | 239.76$\pm$0.13 | 8.95$\pm$0.09 | 8.21

Figure 2: Top: comparison of the reconstructed and true momentum for the MC
sample, for the bin with average momentum 200 MeV/c. Bottom: residual between
reconstructed and true momentum for the MC sample. The systematic error
associated with the momentum reconstruction is discussed in section IV.4

###  III.3 Data Collection

Six data sets were collected during the ISIS user cycle 2015/04 using muon
beams with a nominal 3 mm emittance, at three nominal momenta (172, 200 and
240 MeV/$c$). The three data sets collected with the LiH absorber in place are
referred to as ‘LiH’ data while the three data sets with no absorber in place
are referred to as ‘No Absorber’ data. The beams typically had RMS widths of
$30$–$36$ mm and divergences of $9.0$–$9.4$ mrad, after the selection
described in section III.4. The No Absorber data sets were used to determine
the scattering attributable to the tracking detectors and thus to extract the
true MCS distribution due to the LiH absorber. Two methods, described in
Section IV, were used. Positively charged muon beams were used to minimize
pion contamination, which was measured to be less than 1.4% [29]. Positron
contamination was identified and rejected using the time-of-flight system.

###  III.4 Event Selection

The data from the three nominal muon beams were merged into one sample and all
muons in the sample were treated identically. Unbiased scattering
distributions were selected from the data samples using the cuts listed in
Table 3. The fraction of events selected by each cut is also shown. Events
that produced one space point in TOF0 and one space point in TOF1 were
selected. A beam diffuser, otherwise used to increase the beam emittance, was
fully retracted for all of the runs used in this analysis. A fraction of the
muon beam traversed the diffuser ring in its retracted position, adding
additional energy loss. Any upstream tracks that traversed the outer ring of
the diffuser were removed.

Table 3: Particle selection criteria and survival rates for the muon sample with a LiH absorber. Selection |  Description | Fraction events surviving each cut
---|---|---
Upstream track selection |  Exactly one TOF0 space point, exactly one TOF1 space point and one upstream track. | 100.0%
Diffuser cut |  Upstream tracks were projected to the diffuser position. Any track outside the radius of the diffuser aperture was rejected. | 81.7%
Fiducial selection |  Upstream tracks, when projected to the far end of the downstream tracker, have a projected distance from axis less than 90 mm. | 3.7%

A fiducial selection to ensure that the unscattered downstream track was
likely to have been within the volume of the downstream tracker was also
applied. If the upstream track, when projected to the downstream end of the
downstream tracker, passed outside of the fiducial radius $r_{0}=90$ mm the
track was rejected.

Finally, particles with a time of flight between stations TOF0 and TOF1
compatible with the passage of a muon (above 26 ns) were selected. The data
were then binned in 200 ps $\Delta t_{01}$ bins (Fig. 3) to yield eleven
quasi-monochromatic samples. Most positrons, which had a TOF between 25 and 26
ns, were excluded by this binning. Three of these samples, with mean momentum
of 172, 200 and 240 MeV/$c$ and containing 0.19, 0.25 and 0.19% of the total
number of events respectively, were compared to the GEANT4 and Molière models.
The sample at 172 MeV/$c$ enabled comparison with MuScat while samples at 200
and 240 MeV/$c$ were of interest for the MICE experiment. The selected sample
sizes are shown in Table 4.

Table 4: Sample size after selection. Absorber |  $p$ (MeV/$c$) | No. of events US | No. of events DS
---|---|---|---
| 172 | 6479 | 5906
LiH | 200 | 8589 | 8112
| 240 | 5612 | 5445
| 172 | 1500 | 1469
No Absorber | 200 | 2025 | 1995
| 240 | 1394 | 1378
Figure 3: Combined time-of-flight between TOF0 and TOF1 distribution of LiH
data for all beam line settings after all selection cuts.

###  III.5 Acceptance Correction

The simulated geometric acceptance of the downstream tracker as a function of
the projected scattering angles $\theta_{X}$ and $\theta_{Y}$ is shown in Fig.
4. The acceptance depends on the scattering angle so the scattering angle
distributions must be corrected by the acceptance determined from simulation.
The acceptance data were fitted by a seventh order polynomial,

$\epsilon=a+b\theta_{i}^{2}+c\theta_{i}^{4}+d\theta_{i}^{6}+e\theta_{i}^{7},$

where $i$ is the bin number and $a,b,c$, $d$ and $e$ are fit parameters. This
smoothed fluctuations in the tails of the acceptance function.

Figure 4: The simulated fraction of events reconstructed by the trackers as a
function of scattering angle after event selection. The red curve is an
asymmetric seventh order polynomial fitted to the points and used for the
acceptance correction.

###  III.6 Comparison to simulation

The MICE MC simulation models particles arising from protons incident on the
target. G4beamline [34] was used to simulate particles from immediately after
the target to just upstream of TOF0. The remainder of MICE, including the
downstream portion of the beam line and cooling channel, was simulated using
MAUS [31]. The simulation is handled in this way to reduce the computing
resources required, as only a small subset of particles at the target is
transported to the end of the cooling channel.

A comparison between the momentum distributions for reconstructed MC and data
for the selected samples at three momenta (172, 200 and 240 MeV/$c$) is shown
in Fig. 5. The measured distributions of $x$ and $y$ positions and slopes for
the selected upstream muon samples are well described by the GEANT4 (v9.6) MC,
as illustrated in Fig. 6.

Figure 5: Comparison of reconstructed muon momentum at the center of the
absorber for the 172, 200 and 240 MeV/$c$ samples for data and simulation.

Figure 6: Comparison between Monte Carlo simulations and data for muons in the
200 MeV/$c$ sample with the LiH absorber installed. All distributions are for
the selected muons at the upstream reference plane. Top Left: $x$
distribution, top right: $y$ distribution, bottom left: $dx/dz$ distribution
and bottom right: $dy/dz$ distribution

##  IV RESULTS

###  IV.1 Raw Data MC Comparison

Table 5: Distribution widths of multiple scattering in lithium hydride and the $\chi^{2}$ comparisons between data and the GEANT4 simulation. The $\chi^{2}/N\\!D\\!F$ were calculated using the number of bins as the number of degrees of freedom. Statistical and systematic uncertainties are given for the data distributions. Only statistical uncertainties are given for the model. $p$ (MeV/$c$) | Angle |  $\theta_{\mathrm{Data}}$ (mrad) |  $\theta_{G4}$ (mrad) | $\chi^{2}/N\\!D\\!F$ | P-value
---|---|---|---|---|---
171.55 | $\theta_{X}$ | 21.16$\pm$0.28$\pm$0.48 | 21.87$\pm$0.25 | 23.67 / 31 | 0.79
171.55 | $\theta_{Y}$ | 20.97$\pm$0.27$\pm$0.48 | 21.51$\pm$0.25 | 37.86 / 31 | 0.15
199.93 | $\theta_{X}$ | 18.38$\pm$0.18$\pm$0.33 | 18.76$\pm$0.09 | 17.75 / 31 | 0.96
199.93 | $\theta_{Y}$ | 18.35$\pm$0.18$\pm$0.33 | 18.89$\pm$0.09 | 27.93 / 31 | 0.57
239.76 | $\theta_{X}$ | 15.05$\pm$0.17$\pm$0.21 | 15.69$\pm$0.06 | 8.07 / 31 | 1.00
239.76 | $\theta_{Y}$ | 15.03$\pm$0.16$\pm$0.21 | 15.55$\pm$0.06 | 8.23 / 31 | 1.00

The $\theta_{X}$ and $\theta_{Y}$ distributions from the LiH and No Absorber
data are compared to GEANT4 (v9.6) simulations in Figs. 7, 8 and 9 and the
$\theta^{2}_{Scatt}$ distribution in Fig. 10, at three momenta: 172, 200 and
240 MeV/$c$. The simulation gives an adequate description of the data; a
summary of the comparison given in Table 5. The integrals of these
distributions are between 88% and 96% demonstrating that the selection
criteria ensure high transmission for the selected sample. In this analysis
GEANT4 (v9.6) is used with the QGSP_BERT (v4.0) physics list. In this
configuration, multiple Coulomb scattering is modelled by the G4WentzelVI
model [35, 36]. The G4WentzelVI model is a mixed algorithm simulating both the
hard collisions one by one and using a multiple scattering theory to treat the
effects of the soft collisions at the end of a given step; this prevents the
number of steps in the simulation from becoming too large and also reduces the
dependence on the step length. This model is expected to provide results
similar in accuracy to single scattering but in a computationally efficient
manner. Single scattering is based on the assumption that the effect of
multiple scattering can be modelled as if the hard scatters are the sum of
many individual scatters while soft scatters are sampled from a distribution.
‘Hard’ scatters are inelastic and result in large-angle deflections and large
energy transfers. ‘Soft’ scatters are elastic and result in small-angle
deflections with small energy transfers.

Figure 7: Scattering probability functions $\theta_{X}$ and $\theta_{Y}$
reconstructed from the 172 MeV/$c$ muon beam with (top) and without (bottom)
the LiH absorber in place compared to reconstructed MC scattering
distributions. The black points are the real data and the blue open squares
are the simulated data.

Figure 8: Scattering probability functions $\theta_{X}$ and $\theta_{Y}$
reconstructed from the 200 MeV/$c$ muon beam with (top) and without (bottom)
the LiH absorber in place compared to reconstructed MC scattering
distributions. The black points are the real data and the blue open squares
are the simulated data.

Figure 9: Scattering probability functions $\theta_{X}$ and $\theta_{Y}$
reconstructed from the 240 MeV/$c$ muon beam with (top) and without (bottom)
the LiH absorber in place compared to reconstructed MC scattering
distributions. The black points are the real data and the blue open squares
are the simulated data.

Figure 10: $\theta_{Scatt}^{2}$ distributions reconstructed from the 172, 200
and 240 MeV/$c$ muon samples. The LiH absorber was in the beamline in these
samples. The black points are the real data and the blue open squares are the
reconstructed simulated data. A function $Ae^{-m\theta_{Scatt}^{2}}$ was
fitted to the data distribution and is shown by the red line with $m$
displayed on the plot.

###  IV.2 Convolution with Scattering Models

The data collected with the absorber were compared to GEANT4 and the Molière
scattering models by performing a convolution of the scattering model with No
Absorber data. The convolution,

$n_{\mathrm{conv}}(\theta)=n_{NA}(\theta)\ast n_{\mathrm{model}}(\theta),$ (7)

where $n_{conv}(\theta)$ is the forward convolved distribution,
$n_{NA}(\theta)$ is the scattering distribution measured with the No Absorber
data and $n_{model}(\theta)$ is the scattering distribution predicted by the
model, is performed by adding an angle sampled from the predicted scattering
distribution in the absorber for a given model (GEANT4 or Molière) to the
angle determined from a given trajectory selected from the No Absorber data.
This takes into account scattering in the measurement system. The trajectories
described by the sum of angles are extrapolated to the downstream tracker and
if the track would not have been contained within the downstream tracker then
it is not shown in the scattering distribution but the event is still counted
in the normalisation. The net effect is a distribution, $n_{conv}(\theta)$,
that is the convolution of the raw scattering model $n_{model}(\theta)$ with
the detector effects given by the No Absorber distribution $n_{NA}(\theta)$.
Plots of the Lithium Hydride Absorber data and the No Absorber data convolved
with either the GEANT4 simulation or the Molière model are shown in Fig. 11,
with the residuals shown in Fig. 12, and the results are summarised in Table
6.

Table 6: Distribution widths of multiple scattering in lithium hydride data compared to No Absorber data convolved with two different models of scattering (Geant4 and Molière). The $\chi^{2}/N\\!D\\!F$ were calculated using the number of bins as the number of degrees of freedom. Statistical and systematic uncertainties are given for the data distributions. $p$ (MeV/$c$) | Angle |  $\theta_{\mathrm{Data}}$ (mrad) |  $\theta_{G4}$ (mrad) | $\chi^{2}/N\\!D\\!F$ | P-value |  $\theta_{Mol}$ (mrad) | $\chi^{2}/N\\!D\\!F$ | P-value
---|---|---|---|---|---|---|---|---
171.55 | $\theta_{X}$ | 21.16$\pm$0.28$\pm$0.48 | 21.36$\pm$0.05 | 30.29 / 31 | 0.45 | 22.64$\pm$0.06 | 34.72 / 31 | 0.25
171.55 | $\theta_{Y}$ | 20.97$\pm$0.27$\pm$0.48 | 21.32$\pm$0.05 | 29.10 / 31 | 0.51 | 22.58$\pm$0.06 | 41.14 / 31 | 0.08
199.93 | $\theta_{X}$ | 18.38$\pm$0.18$\pm$0.33 | 18.09$\pm$0.03 | 21.78 / 31 | 0.86 | 19.00$\pm$0.04 | 28.04 / 31 | 0.57
199.93 | $\theta_{Y}$ | 18.35$\pm$0.18$\pm$0.33 | 18.02$\pm$0.03 | 26.98 / 31 | 0.62 | 18.98$\pm$0.04 | 35.41 / 31 | 0.23
239.76 | $\theta_{X}$ | 15.05$\pm$0.17$\pm$0.21 | 15.07$\pm$0.02 | 4.08 / 31 | 1.00 | 15.62$\pm$0.02 | 9.48 / 31 | 1.00
239.76 | $\theta_{Y}$ | 15.03$\pm$0.16$\pm$0.21 | 15.11$\pm$0.02 | 3.44 / 31 | 1.00 | 15.70$\pm$0.02 | 8.62 / 31 | 1.00

The Molière distributions for the lithium hydride absorber were calculated
using the procedure described by Gottschalk [37] for mixtures and compounds.
Pure 6LiH with a thickness of 4.498 g cm-2 was assumed. Distributions were
calculated for monoenergetic muons of 172, 200 and 240 MeV/$c$. Because the
muon energy loss is small – about 11 MeV – the muon momentum was taken to be
constant through the absorber.

Fano’s correction to the Molière distribution was used to account for the
scattering by atomic electrons. The values of the parameter $U_{\rm in}$,
which appears in the correction, were $-U_{\rm in}=3.6$ for hydrogen, as
calculated exactly by Fano, and $-U_{\rm in}=5.0$ for lithium as suggested by
Gottschalk for other materials.

A cubic spline was used to interpolate between the tabulated points of the
functions given by Molière and Bethe. Systematic errors in the calculation
arising from, for example, the description of the absorber as pure 6LiH were
estimated to be of the order of one percent.

The calculated widths, $\theta_{m}$, of the central Gaussian term of the
projected Molière distributions are given in Table 7. If scattering by
electrons is not included, i.e., Fano’s electron correction is set to zero,
the distributions are approximately twenty percent narrower. We note that
Bethe’s ansatz $Z^{2}\rightarrow Z(Z+1)$ to describe the electron contribution
is inappropriate here because the maximum kinematically allowed scattering
angle of a 200 MeV/$c$ muon by an electron is of the order of 4 milliradians,
much less than the width of the scattering distribution. The Molière
predictions shown in Table 7 differ from those shown in Table 6 as these are
the predictions solely from the Molière calculation not the Molière prediction
convolved with MICE No Absorber scattering data.

Table 7: Calculated widths, $\theta_{m}$, of the central Gaussian term of the projected Molière distribution for the lithium hydride absorber at each selected muon momentum. Momentum | $\theta_{m}$
---|---
MeV/$c$ | milliradians
172 | 20.03
200 | 16.87
240 | 13.60

Figure 11: Scattering probability functions reconstructed from the 172, 200
and 240 MeV/$c$ muon beams with the LiH absorber in place (black dots)
compared to the GEANT4 scattering model (blue dots) and the Molière model (red
dots) in LiH convolved with the No Absorber distribution.

###  IV.3 Deconvolution

To determine the underlying scattering distribution in the absorber, the
effects of scattering in non-absorber materials and the detector resolution
must be deconvolved from the measured scattering distribution. The measured
scattering distribution with the absorber in the MICE channel can be written

$s^{\prime}(i)=A(i)\sum_{k=0}^{k=31}s(k)(h(i-k)/A(i-k)),$ (8)

where $s^{\prime}(i)$ is the number of events measured in the $i$th bin with
the absorber in the channel, $s(k)$ is the scattering distribution due only to
the absorber material without the detector, $h(i-k)$ is the No Absorber
scattering distribution which includes the detector resolution and $A(i)$ is
the acceptance function at bin $(i)$. This system of linear equations can be
written in matrix form as

$\overrightarrow{s}^{\prime}=\textbf{H}\overrightarrow{s}$ (9)

where $\overrightarrow{s}^{\prime}$ is the a vector where each entry is the
number of events in a bin of the scattering distribution of all material in
the channel. Similarly for $\overrightarrow{s}$ but for a scattering
distribution of only the absorber and H is a matrix which transforms one to
the other. The unfolding step employs Gold’s deconvolution algorithm to
extract the true scattering distribution ($s$) solely due to the absorber
material, as described in [38] and implemented in the ROOT [39] TSpectrum
class. The advantages of using the Gold deconvolution algorithm are that it
does not rely on simulated data or scattering models and is a purely data-
driven technique making use of all of the data collected. The output of the
deconvolution is compared to the GEANT4 and Molière prediction in Fig. 13.

###  IV.4 Systematic uncertainties

Six contributions to the systematic uncertainty in the scattering
distributions are considered here; uncertainties in the time of flight;
measured alignment; fiducial radius; choice of plane in which to measure
scattering; effect of pion contamination; and in the deconvolution procedure.
To calculate the systematic uncertainty for the individual bins of the
scattering plots shown in Figs. 7, 8, 9, 10 and 13 the numerical derivative is
calculated with the expression

$\sigma_{sys,i}=\frac{dn_{i}}{d\alpha}\sigma_{\alpha}\approx\frac{\Delta
n_{i}}{\Delta\alpha}\sigma_{\alpha}\,,$ (10)

where $\Delta n_{i}$ is the change in the number of entries in a bin that
results from altering a parameter $\alpha$ with a known uncertainty
$\sigma_{\alpha}$ in the analysis or simulation by an amount $\Delta\alpha$.
The uncertainty in the measured width of the distribution is calculated in a
similar way using

$\sigma_{sys}\approx\frac{\Delta\theta_{0}}{\Delta\alpha}\sigma_{\alpha}\,,$
(11)

where $\Delta\theta_{0}$ is the change in the width of the scattering
distribution when measured in either the $x$ or $y$ projection. The systematic
uncertainties are reported for the RMS width of the $\theta_{X}$ distribution
($\theta_{0,X}$) and the width of the $\theta_{Y}$ distribution
($\theta_{0,Y}$) separately.

A significant systematic uncertainty is due to the TOF selection criteria
which directly impact the momentum range of the particles used in the
scattering measurement. The scale is set using the measured 70 ps uncertainty
on the time-of-flight measurement. The effect of particles incorrectly
appearing inside or outside of the 200 ps bin selection window is determined
by offsetting the No Absorber data by 200 ps and the change in the measured
scattering width is treated as the systematic uncertainty.

Uncertainties in the alignment have a direct effect on the angles measured by
the tracker. The alignment of the MICE trackers is characterized by offsets
parallel to $x$ and $y$, with an uncertainty of 0.2 mm, and angles of rotation
about the $x$ and $y$ axes, with an uncertainty of 0.07 mrad. The
uncertainties in the width of the scattering distributions were extracted from
a number of pseudo-experiments, where the alignment parameters were varied in
each iteration.

The choice of the fiducial region may systematically affect the results. A
scan over the possible values of the fiducial radius was completed and the
variation in the width of the scattering distributions for samples adjacent to
the selected value of 90 mm was used to set the uncertainty.

The definitions of the scattering angles are given in section II and the
Appendix. In the definition of the projected scattering angles, $\theta_{x}$
and $\theta_{y}$, $\hat{\textbf{v}}$ is the unit vector mutually orthogonal to
the $y$ direction and the momentum vector and $\hat{\textbf{u}}$ is the unit
vector parallel to the upstream momentum vector. They are related via the
formula

$\hat{\textbf{v}}=\hat{\textbf{s}}\times\hat{\textbf{u}},$ (12)

where $\hat{\vec{s}}$ is arbitrarily defined as $\hat{\textbf{s}}=(0,-1,0)$.
This expression defines a direction perpendicular to a plane containing the
upstream track. There are an infinite number of planes that contain this
track, so we consider the uncertainty introduced by the definition of
$\hat{\vec{s}}$ by rotating it between 0∘ and 180∘, in increments of 1∘,
around the $x$-axis, with the analysis repeated after each increment. The
resulting maximum change in measured scattering angle is included in the
systematic uncertainties in Table 8.

The MICE muon beam has pion contamination with an upper limit $f_{\pi}<1.4\%$
at 90% C.L. [29]. To measure the effect of this contamination on the
scattering measurement for muons, a Monte Carlo study was performed. The
measurement was simulated with the MICE beam, including simulated impurities,
and a pure muon sample, with the systematic error being the difference between
the two results.

The difference between the deconvolved result and the true scattering
distribution from a GEANT4 simulation was taken to be an additional source of
systematic error. This accounts for any bias introduced by the Gold
deconvolution procedure. The systematic uncertainties for the deconvolution
procedure showed significant variation from bin to bin so a parabolic
smoothing function was used to assign the systematic uncertainty to each bin.

All systematic uncertainties, and their quadratic combination, for the three
selected momenta of 172, 200 and 240 MeV/$c$ are included in Table 8. The
dominant systematic uncertainties are those in the momentum scale of the TOF
system and the deconvolution procedure.

Table 8: Systematic uncertainties associated with the width of the scattering distributions of $\theta_{0,X}$ and $\theta_{0,Y}$ in three representative momentum bins.  $p$ (MeV/$c$) | Type | $\Delta\theta_{0,X}$ | $\Delta\theta_{0,Y}$
---|---|---|---
|  | (mrad) | (mrad)
| TOF selection | 0.64 | 0.64
| Alignment | $<0.01$ | 0.01
171.55 | Fiducial radius | $<0.01$ | $<0.01$
|  $\theta$ angle definition | $<0.01$ | $<0.01$
|  $\pi$ contamination | $<0.01$ | $<0.01$
| Deconvolution | 1.25 | 1.19
| Total sys. | 1.39 | 1.35
| TOF selection | 0.29 | 0.29
| Alignment | 0.02 | $<0.01$
199.93 | Fiducial radius | 0.01 | 0.01
|  $\theta$ definition | $<0.01$ | $<0.01$
|  $\pi$ contamination | $<0.01$ | $<0.01$
| Deconvolution | 0.70 | 0.47
| Total sys. | 0.73 | 0.54
| TOF selection | 0.27 | 0.27
| Alignment | $<0.01$ | $<0.01$
239.76 | Fiducial radius | 0.01 | 0.01
|  $\theta$ definition | $<0.01$ | $<0.01$
|  $\pi$ contamination | 0.01 | 0.01
| Deconvolution | 0.27 | 0.41
| Total sys. | 0.36 | 0.49

Figure 12: Scattering residuals between data with the LiH absorber and No
Absorber data convolved with either GEANT4 or the Molière scattering models in
LiH for the 172, 200 and 240 MeV/$c$ samples. The residuals are normalized to
the estimated uncertainty in the data in each bin. The agreement improves at
higher momentum where the scattering distributions are narrower.

Figure 13: Projected $\theta_{X}$ and $\theta_{Y}$ multiple scattering
probability functions at 172, 200 and 240 MeV/$c$ after deconvolution. The
GEANT4 and Molière scattering distributions in LiH are provided for
comparison.

###  IV.5 Model comparisons

The residual between the scattering distribution in data and that predicted by
the models is used to quantify the level of agreement between data and
simulation. The normalised residual is defined as

$\text{residual}=\frac{p_{\mathrm{data}}(\theta_{i})-p_{\mathrm{simulation}}(\theta_{i})}{\sqrt{\sigma^{2}_{\mathrm{stat}}+\sum\sigma^{2}_{\mathrm{sys},i}}}$
(13)

where $p_{\mathrm{data}}(\theta_{i})$ is the probability of scattering at
angle $\theta_{i}$ measured with the MICE data and
$p_{\mathrm{simulation}}(\theta_{i})$ is the probability of scattering
predicted by the corresponding model. The systematic uncertainties
$\sigma^{2}_{\mathrm{sys},i}$, discussed in Sec. IV.4, are calculated and
summed in quadrature on a bin by bin level. The $\chi^{2}$ derived from these
residuals appears in Table 5. The $\chi^{2}$ between the scattering
distribution from the data and that predicted by the model is calculated using

$\chi^{2}=\sum_{i=0}^{N}\frac{\left(p_{\mathrm{data}}(\theta_{i})-p_{\mathrm{simulation}}(\theta_{i})\right)^{2}}{{\sigma_{\mathrm{stat}}^{2}+\sum_{\mathrm{sys}}\sigma^{2}_{\mathrm{sys},i}}}$
(14)

where $N$ is the number of bins and $\mathrm{sys}$ is the number of systematic
errors. The $\chi^{2}$ was calculated using 31 data points and demonstrates
good agreement between data and MC. The $\chi^{2}$ calculation in Eqn. 14 was
repeated for both the forward convolution comparison to real data and for the
comparison between the deconvolved data and the GEANT4 and Molière models. The
systematic uncertainties are added on a bin by bin basis in the calculation of
the $\chi^{2}$ in Eq. 14.

There is very little difference between the GEANT4 simulation, the Molière
calculations and the deconvolved data. The deconvolved $\theta_{X}$ and
$\theta_{Y}$ multiple scattering distributions on lithium hydride for the 172,
200 and 240 MeV/$c$ muon samples are shown in Fig. 13, and these are compared
with a GEANT4 LiH simulation and the Molière calculation.

The distributions of the projections in $\theta_{X}$ and $\theta_{Y}$ were
characterized using a Gaussian fit within a $\pm 45$ mrad range, with the
results shown in Table 9 for deconvolved data using the Gold deconvolution
algorithm and the true distributions extracted from the GEANT4 simulation and
the Molière model calculation. The table shows that the deconvolved
$\theta_{X}$ and $\theta_{Y}$ projections of the scattering distributions are
approximately consistent with the GEANT4 and Molière distributions, but the
Molière distribution is systematically wider than the rest and significantly
wider than that given by GEANT4.

Table 9: Widths of best fit Gaussian fitted to central $\pm 45$ mrad of scattering distributions after deconvolution compared to GEANT4 and Molière models. Statistical and systematic uncertainties are given for the data distributions. Only statistical uncertainties are given for the GEANT4 model. $p$ (MeV/$c$) | Angle |  $\theta^{\mathrm{meas}}_{\mathrm{Gold}}$ (mrad) | $\theta^{\mathrm{true}}_{G4}$ | $\theta^{\mathrm{true}}_{\mathrm{Moli\grave{e}re}}$
---|---|---|---|---
|  |  | (mrad) | (mrad)
171.55 | $\theta_{X}$ | 19.03$\pm$0.26$\pm$1.39 | 18.62$\pm$0.13 | 20.03
171.55 | $\theta_{Y}$ | 18.95$\pm$0.24$\pm$1.35 | 18.59$\pm$0.12 | 20.03
199.93 | $\theta_{X}$ | 16.59$\pm$0.17$\pm$0.73 | 15.82$\pm$0.05 | 16.87
199.93 | $\theta_{Y}$ | 16.36$\pm$0.17$\pm$0.55 | 15.82$\pm$0.05 | 16.87
239.76 | $\theta_{X}$ | 13.29$\pm$0.17$\pm$0.37 | 13.16$\pm$0.04 | 13.60
239.76 | $\theta_{Y}$ | 13.21$\pm$0.16$\pm$0.49 | 13.10$\pm$0.04 | 13.60

###  IV.6 Momentum-dependent measurements

The selected samples are plotted as a function of mean momentum for each
sample, to confirm the dependence of the widths of the scattering
distributions on momentum. The number of events contained in each TOF bin is
between 3500 and 9000 events. The deconvolved scattering widths as a function
of momentum are shown in Fig. 14. The widths, $\theta_{0}$, are fitted to

$\theta_{0}=\frac{13.6\,[\mathrm{MeV}/c]a}{p\beta},$ (15)

where $a$ is a fit coefficient, motivated by Eq. 3, where the $\beta$
dependence of the log term is negligible, changing the calculated value by
less than 1%.

Figure 14: The results of the scattering analysis using data in a number of
momentum bins. Scattering widths are reported after application of the Gold
deconvolution.

The coefficient, $a$, is compared with the prediction from the PDG formula in
Eq. 3. The values of the coefficients, $a$, determined from the fits to the
$\theta_{0,X}$ and $\theta_{0,Y}$ distributions are shown in Table 10. The
numerical derivative of the momentum with respect to TOF of the sample was
calculated and used to assess the systematic uncertainty associated with the
measurement.

Measurements using the projected angles are systematically smaller than the
PDG prediction. The average of the two fits to the $\theta_{0,X}$ and
$\theta_{0,Y}$ muon scattering widths as a function of momentum yields
$a=208.1\pm 1.5$ mrad, which is 9% smaller than the value proposed by the PDG
formula, $a=226.7$ mrad, but still within the uncertainties of that
approximate formula, Eq. 3, which is quoted as accurate to 11%.

Table 10: Results of the fit to the scattering widths as a function of momentum, given by Eq. 15. The value predicted by the PDG is also shown. Angle |  $a$ (mrad)
---|---
$\theta_{0,X}$ | 206.6$\pm$2.1
$\theta_{0,Y}$ | 210.2$\pm$2.1
PDG | 226.7

##  V CONCLUSIONS

Presented here is an analysis of the LiH multiple Coulomb scattering data
taken during ISIS user run 2015/04 using MICE. These data were compared to the
GEANT4 (v9.6) default scattering model [11] and the full Molière calculation
[4, 5]. A $\chi^{2}$ statistic was used to make quantitative statements about
the validity of the proposed models. Three approaches are taken; the measured
LiH and No Absorber scattering distributions were compared to GEANT4, the
forward convolution using the No Absorber data was compared to both GEANT4 and
the Molière model and the deconvolution of the LiH scattering data using the
No Absorber data was compared to both GEANT4 and the Molière model. In all
cases the GEANT4 scattering widths agreed with the measured data at each of
the nominal momenta, but the Molière model produces systematically wider
distributions.

The momentum dependence of scattering was examined by selecting 200 ps time of
flight samples from the muon beam data. The momentum dependence from 160 to
245 MeV/$c$ was compared to the dependence in Eq. 3, from the PDG [3], and it
was found that the measured RMS scattering width is about 9% smaller than the
approximate PDG estimation, but within the latter’s stated uncertainty.

*

##  Appendix A Definition of Scattering Angles

The projections of the scattering angle onto the $y$-$z$ or $x$-$z$ plane,
angles $\theta_{X}$ and $\theta_{Y}$, are defined by considering the inner
product of the downstream momentum $\textbf{p}_{DS}$ with the component of the
upstream momentum vector $\textbf{p}_{US}$, perpendicular to the projection
plane. The scattering projection into the plane defined by the momentum vector
and the $y$-axis is

$\theta_{Y}=\arctan\left(\frac{\textbf{p}_{DS}\cdot\hat{\textbf{v}}}{\textbf{p}_{DS}\cdot\hat{\textbf{u}}}\right)=\arctan\left(\frac{\textbf{p}_{DS}\cdot(\hat{\textbf{y}}\times\textbf{p}_{US})|\textbf{p}_{US}|}{(\textbf{p}_{DS}\cdot\textbf{p}_{US})|\hat{\textbf{y}}\times\textbf{p}_{US}|}\right),$
(16)

where $\hat{\textbf{y}}$ is the unit vector in the $y$ direction,
$\hat{\textbf{v}}=\hat{\textbf{y}}\times\textbf{p}_{US}/|\hat{\textbf{y}}\times\textbf{p}_{US}|$
is the unit vector mutually orthogonal to the $y$ direction and the momentum
vector and $\hat{\textbf{u}}=\textbf{p}_{US}/|\textbf{p}_{US}|$ is the unit
vector parallel to the upstream momentum vector. A scattering angle in the
perpendicular plane must then be defined as

$\theta_{X}=\arctan\left(|\textbf{p}_{US}|\frac{\textbf{p}_{DS}\cdot(\textbf{p}_{US}\times(\hat{\textbf{y}}\times\textbf{p}_{US}))}{|\textbf{p}_{US}\times(\hat{\textbf{y}}\times\textbf{p}_{US})|\textbf{p}_{DS}\cdot\textbf{p}_{US}}\right),$
(17)

where the downstream vector is now projected onto the unit vector
$\hat{\textbf{v}}=\textbf{p}_{US}\times(\hat{\textbf{y}}\times\textbf{p}_{US})/|\textbf{p}_{US}\times(\hat{\textbf{y}}\times\textbf{p}_{US})|$.
These two expressions can be expressed in terms of the gradients of the muon
tracks before and after the scatters,

$\displaystyle\theta_{Y}$ $\displaystyle=$
$\displaystyle\arctan\left\\{\frac{\sqrt{1+\left(\frac{dx}{dz}\right)_{US}^{2}+\left(\frac{dy}{dz}\right)_{US}^{2}}}{\sqrt{1+\left(\frac{dx}{dz}\right)_{US}^{2}}}\left(\frac{\left(\frac{dx}{dz}\right)_{DS}-\left(\frac{dx}{dz}\right)_{US}}{1+\left(\frac{dx}{dz}\right)_{US}\left(\frac{dx}{dz}\right)_{DS}+\left(\frac{dy}{dz}\right)_{US}\left(\frac{dy}{dz}\right)_{DS}}\right)\right\\},$
(18)
$\theta_{X}=\arctan\left\\{\sqrt{\frac{1+\left(\frac{dx}{dz}\right)_{US}^{2}+\left(\frac{dy}{dz}\right)_{US}^{2}}{\left(1+\left(\frac{dx}{dz}\right)_{US}^{2}+\left(\frac{dy}{dz}\right)_{US}^{2}\right)\left(1+\left(\frac{dx}{dz}\right)^{2}_{US}\right)}}\right.\times\left.\left(\frac{\left(\frac{dy}{dz}\right)_{DS}\left(1+\left(\frac{dx}{dz}\right)_{US}^{2}\right)+\left(\left(\frac{dx}{dz}\right)_{DS}\left(\frac{dx}{dz}\right)_{US}-1\right)\left(\frac{dy}{dz}\right)_{US}}{1+\left(\frac{dx}{dz}\right)_{US}\left(\frac{dx}{dz}\right)_{DS}+\left(\frac{dy}{dz}\right)_{US}\left(\frac{dy}{dz}\right)_{DS}}\right)\right\\}.$
(19)

In the approximation of small angles (i.e.
$\frac{dx}{dz}\approx\frac{dy}{dz}\ll 1$) these produce the more familiar
forms

$\theta_{X}=\left(\frac{dy}{dz}\right)_{DS}-\left(\frac{dy}{dz}\right)_{US}$
(20)

for scattering about the $x$-axis or

$\theta_{Y}=\left(\frac{dx}{dz}\right)_{DS}-\left(\frac{dx}{dz}\right)_{US}$
(21)

for scattering about the $y$-axis. The more exact expressions, equations 18
and 19, are used throughout for this analysis.

###### Acknowledgements.

The work described here was made possible by grants from the Science and
Technology Facilities Council (UK), the Department of Energy and the National
Science Foundation (USA), the Istituto Nazionale di Fisica Nucleare (Italy),
the European Union under the European Union’s Framework Programme 7 (AIDA
project, grant agreement number 262025; TIARA project, grant agreement number
261905; and EuCARD), the Japan Society for the Promotion of Science, the
National Research Foundation of Korea (number NRF-2016R1A5A1013277), the
Ministry of Education, Science and Technological Development of the Republic
of Serbia, the Institute of High Energy Physics/Chinese Academy of Sciences
fund for collaboration between the People’s Republic of China and the USA, and
the Swiss National Science Foundation in the framework of the SCOPES
programme. We gratefully acknowledge all sources of support. We are grateful
for the support given to us by the staff of the STFC Rutherford Appleton and
Daresbury laboratories. We acknowledge the use of Grid computing resources
deployed and operated by GridPP in the UK, http://www.gridpp.ac.uk/.

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