text:
# Characterisation of the muon beams for the Muon Ionisation Cooling
Experiment

The MICE Collaboration  D. Adams  STFC Rutherford Appleton Laboratory, Harwell
Oxford, Didcot, UK  D. Adey  Department of Physics, University of Warwick,
Coventry, UK  Now at Fermilab, Batavia, IL, USA  A. Alekou  Department of
Physics, Blackett Laboratory, Imperial College London, London, UK  Now at
CERN, Geneva, Switzerland  M. Apollonio  Department of Physics, Blackett
Laboratory, Imperial College London, London, UK  Now at Diamond Light Source,
Harwell Science and Innovation Campus, Didcot, Oxfordshire, UK  R.
Asfandiyarov  DPNC, Section de Physique, Université de Genève, Geneva,
Switzerland  J. Back  Department of Physics, University of Warwick, Coventry,
UK  G. Barber  Department of Physics, Blackett Laboratory, Imperial College
London, London, UK  P. Barclay  STFC Rutherford Appleton Laboratory, Harwell
Oxford, Didcot, UK  A. de Bari  Sezione INFN Pavia and Dipartimento di Fisica
Nucleare e Teorica, Pavia, Italy  R. Bayes  School of Physics and Astronomy,
Kelvin Building, The University of Glasgow, Glasgow, UK  V. Bayliss  STFC
Rutherford Appleton Laboratory, Harwell Oxford, Didcot, UK  R. Bertoni
Sezione INFN Milano Bicocca, Dipartimento di Fisica G. Occhialini, Milano,
Italy  V. J. Blackmore  Department of Physics, University of Oxford, Denys
Wilkinson Building, Oxford, UK  email<EMAIL_ADDRESS>A.
Blondel  DPNC, Section de Physique, Université de Genève, Geneva, Switzerland
S. Blot  Enrico Fermi Institute, University of Chicago, Chicago, IL, USA  M.
Bogomilov  Department of Atomic Physics, St. Kliment Ohridski University of
Sofia, Sofia, Bulgaria  M. Bonesini  Sezione INFN Milano Bicocca, Dipartimento
di Fisica G. Occhialini, Milano, Italy  C. N. Booth  Department of Physics and
Astronomy, University of Sheffield, Sheffield, UK  D. Bowring  Lawrence
Berkeley National Laboratory, Berkeley, CA, USA  S. Boyd  Department of
Physics, University of Warwick, Coventry, UK  T. W. Bradshaw  STFC Rutherford
Appleton Laboratory, Harwell Oxford, Didcot, UK  U. Bravar  University of New
Hampshire, Durham, NH, USA  A. D. Bross  Fermilab, Batavia, IL, USA  M.
Capponi  Sezione INFN Roma Tre e Dipartimento di Fisica, Roma, Italy  T.
Carlisle  Department of Physics, University of Oxford, Denys Wilkinson
Building, Oxford, UK  G. Cecchet  Sezione INFN Pavia and Dipartimento di
Fisica Nucleare e Teorica, Pavia, Italy  G. Charnley  The Cockcroft Institute,
Daresbury Science and Innovation Centre,Daresbury, Cheshire, UK  J. H. Cobb
Department of Physics, University of Oxford, Denys Wilkinson Building, Oxford,
UK  D. Colling  Department of Physics, Blackett Laboratory, Imperial College
London, London, UK  N. Collomb  STFC Daresbury Laboratory, Daresbury,
Cheshire, UK  L. Coney  University of California, Riverside, CA, USA  P. Cooke
Department of Physics, University of Liverpool, Liverpool, UK  M. Courthold
STFC Rutherford Appleton Laboratory, Harwell Oxford, Didcot, UK  L. M.
Cremaldi  University of Mississippi, Oxford, MS, USA  A. DeMello  Lawrence
Berkeley National Laboratory, Berkeley, CA, USA  A. Dick  Department of
Physics, University of Strathclyde, Glasgow, UK  A. Dobbs  Department of
Physics, Blackett Laboratory, Imperial College London, London, UK  P. Dornan
Department of Physics, Blackett Laboratory, Imperial College London, London,
UK  S. Fayer  Department of Physics, Blackett Laboratory, Imperial College
London, London, UK  F. Filthaut  NIKHEF, Amsterdam, The Netherlands  Also at
Radboud University Nijmegen, Nijmegen, The Netherlands  A. Fish  Department of
Physics, Blackett Laboratory, Imperial College London, London, UK  T.
Fitzpatrick  Fermilab, Batavia, IL, USA  R. Fletcher  University of
California, Riverside, CA, USA  D. Forrest  School of Physics and Astronomy,
Kelvin Building, The University of Glasgow, Glasgow, UK  V. Francis  STFC
Rutherford Appleton Laboratory, Harwell Oxford, Didcot, UK  B. Freemire
Illinois Institute of Technology, Chicago, IL, USA  L. Fry  STFC Rutherford
Appleton Laboratory, Harwell Oxford, Didcot, UK  A. Gallagher  STFC Daresbury
Laboratory, Daresbury, Cheshire, UK  R. Gamet  Department of Physics,
University of Liverpool, Liverpool, UK  S. Gourlay  Lawrence Berkeley National
Laboratory, Berkeley, CA, USA  A. Grant  STFC Daresbury Laboratory, Daresbury,
Cheshire, UK  J. S. Graulich  DPNC, Section de Physique, Université de Genève,
Geneva, Switzerland  S. Griffiths  The Cockcroft Institute, Daresbury Science
and Innovation Centre,Daresbury, Cheshire, UK  P. Hanlet  Illinois Institute
of Technology, Chicago, IL, USA  O. M. Hansen  CERN, Geneva, Switzerland  Also
at University of Oslo, Norway  G. G. Hanson  University of California,
Riverside, CA, USA  P. Harrison  Department of Physics, University of Warwick,
Coventry, UK  T. L. Hart  University of Mississippi, Oxford, MS, USA  T.
Hartnett  STFC Daresbury Laboratory, Daresbury, Cheshire, UK  T. Hayler  STFC
Rutherford Appleton Laboratory, Harwell Oxford, Didcot, UK  C. Heidt
University of California, Riverside, CA, USA  M. Hills  STFC Rutherford
Appleton Laboratory, Harwell Oxford, Didcot, UK  P. Hodgson  Department of
Physics and Astronomy, University of Sheffield, Sheffield, UK  A. Iaciofano
Sezione INFN Roma Tre e Dipartimento di Fisica, Roma, Italy  S. Ishimoto  High
Energy Accelerator Research Organization (KEK), Institute of Particle and
Nuclear Studies, Tsukuba, Ibaraki, Japan  G. Kafka  Illinois Institute of
Technology, Chicago, IL, USA  D. M. Kaplan  Illinois Institute of Technology,
Chicago, IL, USA  Y. Karadzhov  DPNC, Section de Physique, Université de
Genève, Geneva, Switzerland  Y. K. Kim  Enrico Fermi Institute, University of
Chicago, Chicago, IL, USA  D. Kolev  Department of Atomic Physics, St. Kliment
Ohridski University of Sofia, Sofia, Bulgaria  Y. Kuno  Osaka University,
Graduate School of Science, Department of Physics, Toyonaka, Osaka, Japan  P.
Kyberd  Brunel University, Uxbridge, UK  W. Lau  Department of Physics,
University of Oxford, Denys Wilkinson Building, Oxford, UK  J. Leaver
Department of Physics, Blackett Laboratory, Imperial College London, London,
UK  M. Leonova  Fermilab, Batavia, IL, USA  D. Li  Lawrence Berkeley National
Laboratory, Berkeley, CA, USA  A. Lintern  STFC Rutherford Appleton
Laboratory, Harwell Oxford, Didcot, UK  M. Littlefield  Brunel University,
Uxbridge, UK  K. Long  Department of Physics, Blackett Laboratory, Imperial
College London, London, UK  G. Lucchini  Sezione INFN Milano Bicocca,
Dipartimento di Fisica G. Occhialini, Milano, Italy  T. Luo  University of
Mississippi, Oxford, MS, USA  C. Macwaters  STFC Rutherford Appleton
Laboratory, Harwell Oxford, Didcot, UK  B. Martlew  The Cockcroft Institute,
Daresbury Science and Innovation Centre,Daresbury, Cheshire, UK  J. Martyniak
Department of Physics, Blackett Laboratory, Imperial College London, London,
UK  A. Moretti  Fermilab, Batavia, IL, USA  A. Moss  The Cockcroft Institute,
Daresbury Science and Innovation Centre,Daresbury, Cheshire, UK  A. Muir  The
Cockcroft Institute, Daresbury Science and Innovation Centre,Daresbury,
Cheshire, UK  I. Mullacrane  The Cockcroft Institute, Daresbury Science and
Innovation Centre,Daresbury, Cheshire, UK  J. J. Nebrensky  Brunel University,
Uxbridge, UK  D. Neuffer  Fermilab, Batavia, IL, USA  A. Nichols  STFC
Rutherford Appleton Laboratory, Harwell Oxford, Didcot, UK  R. Nicholson
Department of Physics and Astronomy, University of Sheffield, Sheffield, UK
J. C. Nugent  School of Physics and Astronomy, Kelvin Building, The University
of Glasgow, Glasgow, UK  Y. Onel  Department of Physics and Astronomy,
University of Iowa, Iowa City, IA, USA  D. Orestano  Sezione INFN Roma Tre e
Dipartimento di Fisica, Roma, Italy  E. Overton  Department of Physics and
Astronomy, University of Sheffield, Sheffield, UK  P. Owens  The Cockcroft
Institute, Daresbury Science and Innovation Centre,Daresbury, Cheshire, UK  V.
Palladino  Sezione INFN Napoli and Dipartimento di Fisica, Università Federico
II,Complesso Universitario di Monte S. Angelo, Napoli, Italy  J. Pasternak
Department of Physics, Blackett Laboratory, Imperial College London, London,
UK  F. Pastore  Sezione INFN Roma Tre e Dipartimento di Fisica, Roma, Italy
C. Pidcott  Department of Physics, University of Warwick, Coventry, UK  M.
Popovic  Fermilab, Batavia, IL, USA  R. Preece  STFC Rutherford Appleton
Laboratory, Harwell Oxford, Didcot, UK  S. Prestemon  Lawrence Berkeley
National Laboratory, Berkeley, CA, USA  D. Rajaram  Illinois Institute of
Technology, Chicago, IL, USA  S. Ramberger  CERN, Geneva, Switzerland  M. A.
Rayner  Department of Physics, University of Oxford, Denys Wilkinson Building,
Oxford, UK  Now at DPNC, Université de Genève, Geneva, Switzerland  S.
Ricciardi  STFC Rutherford Appleton Laboratory, Harwell Oxford, Didcot, UK  A.
Richards  Department of Physics, Blackett Laboratory, Imperial College London,
London, UK  T. J. Roberts  Muons, Inc., Batavia, IL, USA  M. Robinson
Department of Physics and Astronomy, University of Sheffield, Sheffield, UK
C. Rogers  STFC Rutherford Appleton Laboratory, Harwell Oxford, Didcot, UK  K.
Ronald  Department of Physics, University of Strathclyde, Glasgow, UK  P.
Rubinov  Fermilab, Batavia, IL, USA  R. Rucinski  Fermilab, Batavia, IL, USA
I. Rusinov  Department of Atomic Physics, St. Kliment Ohridski University of
Sofia, Sofia, Bulgaria  H. Sakamoto  Osaka University, Graduate School of
Science, Department of Physics, Toyonaka, Osaka, Japan  D. A. Sanders
University of Mississippi, Oxford, MS, USA  E. Santos  Department of Physics,
Blackett Laboratory, Imperial College London, London, UK  T. Savidge
Department of Physics, Blackett Laboratory, Imperial College London, London,
UK  P. J. Smith  Department of Physics and Astronomy, University of Sheffield,
Sheffield, UK  P. Snopok  Illinois Institute of Technology, Chicago, IL, USA
F. J. P. Soler  School of Physics and Astronomy, Kelvin Building, The
University of Glasgow, Glasgow, UK  T. Stanley  STFC Rutherford Appleton
Laboratory, Harwell Oxford, Didcot, UK  D. J. Summers  University of
Mississippi, Oxford, MS, USA  M. Takahashi  Department of Physics, Blackett
Laboratory, Imperial College London, London, UK  J. Tarrant  STFC Rutherford
Appleton Laboratory, Harwell Oxford, Didcot, UK  I. Taylor  Department of
Physics, University of Warwick, Coventry, UK  L. Tortora  Sezione INFN Roma
Tre e Dipartimento di Fisica, Roma, Italy  Y. Torun  Illinois Institute of
Technology, Chicago, IL, USA  R. Tsenov  Department of Atomic Physics, St.
Kliment Ohridski University of Sofia, Sofia, Bulgaria  C. D. Tunnell
Department of Physics, University of Oxford, Denys Wilkinson Building, Oxford,
UK  G.Vankova  Department of Atomic Physics, St. Kliment Ohridski University
of Sofia, Sofia, Bulgaria  V. Verguilov  DPNC, Section de Physique, Université
de Genève, Geneva, Switzerland  S. Virostek  Lawrence Berkeley National
Laboratory, Berkeley, CA, USA  M. Vretenar  CERN, Geneva, Switzerland  K.
Walaron  School of Physics and Astronomy, Kelvin Building, The University of
Glasgow, Glasgow, UK  S. Watson  STFC Rutherford Appleton Laboratory, Harwell
Oxford, Didcot, UK  C. White  The Cockcroft Institute, Daresbury Science and
Innovation Centre,Daresbury, Cheshire, UK  C. G. Whyte  Department of Physics,
University of Strathclyde, Glasgow, UK  A. Wilson  STFC Rutherford Appleton
Laboratory, Harwell Oxford, Didcot, UK  H. Wisting  DPNC, Section de Physique,
Université de Genève, Geneva, Switzerland  M. Zisman  Lawrence Berkeley
National Laboratory, Berkeley, CA, USA

(Received: date / Accepted: date – Entered later)

###### Abstract

A novel single-particle technique to measure emittance has been developed and
used to characterise seventeen different muon beams for the Muon Ionisation
Cooling Experiment (MICE). The muon beams, whose mean momenta vary from 171 to
281 MeV/$c$, have emittances of approximately 1.2–2.3 $\pi$ mm-rad
horizontally and 0.6–1.0 $\pi$ mm-rad vertically, a horizontal dispersion of
90–190 mm and momentum spreads of about 25 MeV/$c$. There is reasonable
agreement between the measured parameters of the beams and the results of
simulations. The beams are found to meet the requirements of MICE.

##  1 Introduction

A future high-energy Neutrino Factory or Muon Collider will require an intense
source of muons. The large volume of phase space occupied by muons at
production must be reduced before they are accelerated and stored. The short
muon lifetime prohibits the use of conventional cooling techniques; another
technique must be developed to maximise the muon flux delivered to a storage
ring.

Ionisation cooling is the only practical approach. A muon passing through a
low-$Z$ material loses energy by ionisation, reducing its transverse and
longitudinal momenta. The longitudinal momentum is restored by accelerating
cavities, with the net effect of reducing the divergence of the beam and thus
the transverse phase space the beam occupies.

The muon beams at the front-end of a Neutrino Factory or Muon Collider will be
similar. They are expected to have a very large transverse normalised
emittance of $\varepsilon_{N}\approx$ 12–20 $\pi$ mm-rad and momentum spreads
of 20 MeV/$c$ or more about a central momentum of 200 MeV/$c$. The transverse
emittance must be reduced to 2- 5 $\pi$ mm-rad (depending on the subsequent
acceleration scheme) for a Neutrino Factory [1, 2, 3, 4]. Further transverse
and longitudinal cooling is required for a Muon Collider. Emittances of
$0.4\,\pi$ mm-rad and $1\,\pi$ mm-rad are desired in the transverse and
longitudinal planes respectively, where the latter is achieved by emittance
exchange [5].

The Muon Ionisation Cooling Experiment (MICE) will be the first experiment to
demonstrate the practicality of muon ionisation cooling. This paper describes
measurements of the muon beams that will be used by MICE.

##  2 The MICE Experiment

Figure 1: MICE upstream beam line.

MICE will measure the ionisation cooling efficiency of one “Super Focus-Focus”
(SFOFO) lattice cell [6] based on the cooling-channel design of Neutrino
Factory Feasibility Study 2 [1]. A detailed description of the cooling cell is
contained in [7]. Since ionisation cooling depends on momentum (via the
dependence of energy loss and multiple scattering in materials), the MICE
experiment has been designed to measure the performance of the cell for beams
of 140 to 240 MeV/$c$ with large momentum spreads; liquid hydrogen and other
low-$Z$ absorber materials will be studied.

The expected reduction in emittance ($\approx$10% using liquid hydrogen) is
too small to be measured conventionally, where methods typically attain 10%
precision. MICE will therefore make single-particle measurements using
scintillating fibre trackers [8] inside superconducting solenoids (the
“spectrometer solenoids”) at each end of the cooling cell. Cherenkov detectors
and time-of-flight (TOF) detectors provide upstream particle identification;
the TOFs will also allow the muons to be timed with respect to the RF phase. A
pre-shower detector and a muon ranger will provide particle identification
downstream of the cooling section.

###  2.1 MICE beam requirements

For a realistic demonstration of cooling the beams used should closely
resemble those expected at the front end of a Neutrino Factory, _i.e._ , they
should have a large momentum spread and a large normalised emittance. The
emittance must be variable to allow the equilibrium emittance—which depends on
the absorber material and the optics of the channel—to be measured.

The MICE beam line has been designed to produce beams of three different
emittances at each of three central momenta. These beams are named by the
convention “$\left(\varepsilon_{N},p_{z}\right)$” according to their
normalised transverse emittance at the entrance to the cooling section and
longitudinal momentum at the centre of the first absorber. The nominal (RMS)
input emittances are $\varepsilon_{N}=3$, 6 and $10\,\pi$ mm-rad; the central
momenta are 140, 200 and 240 MeV/$c$. The baseline beam configuration is
$\left(\varepsilon_{N},p_{z}\right)$ = ($6\,\pi$ mm-rad, 200 MeV/$c$).

The beams of different emittances will be generated by means of a “diffuser”,
which allows a variable thickness of high-$Z$ material to be inserted into the
beam at the entrance to the upstream spectrometer solenoid. Scattering
increases the emittance of the beam to the desired values and, as a
consequence of the energy lost in the material, beams with a higher emittance
downstream of the diffuser must have a higher momentum upstream. An important
requirement is that the muon beam downstream of the diffuser is correctly
matched in the spectrometer solenoid.

###  2.2 MICE muon beam line

The new muon beam line for MICE (at the ISIS proton synchrotron, Rutherford
Appleton Laboratory) is shown in Figure 1 and described at length in [9]. A
titanium target [10] samples the proton beam, creating pions that are captured
by the upstream quadrupoles (Q1–3) and momentum-selected by the first dipole
(D1). The beam is transported to the Decay Solenoid, which focuses the pions
and captures the decay muons.

The second dipole (D2) can be tuned to select muons emitted backwards in the
pion rest frame to obtain a high purity muon beam. The final transport is
through two large-aperture quadrupole triplets, Q4–6 and Q7–9, that focus the
beam onto the diffuser. Each of the three quadrupole triplets is arranged to
focus-defocus-focus in the horizontal plane; the beam line can be operated in
either polarity. The optics of this section are determined by the desired
emittance of the beam in the cooling channel and the requirement of matching
into the spectrometer solenoid.

A time-of-flight station (TOF0) and two aerogel Cherenkov detectors are
located just after the Q4–6 triplet; a second time-of-flight station (TOF1) is
located after the final triplet (Q7–9). A low-mass scintillating-fibre beam-
position monitor (BPM) is located close to Q7. For the $\mu^{+}$ beams, a
variable thickness polyethylene absorber is introduced upstream of D2 to
reduce the flux of protons incident on TOF0.

The TOF detectors are described in [9, 11]. Each station consists of two
perpendicular ($x,y$) planes of 25.4 mm thick scintillator slabs. Each end of
each slab is coupled to a fast photomultiplier and subsequent electronics
[12]. The measured timing resolutions are $\sigma_{t}=55$ ps and
$\sigma_{t}=53$ ps at TOF0 and TOF1 respectively [13]. The differences in the
arrival times of light at each end of the slabs are used to obtain transverse
position measurements with resolutions of $\sigma_{x}=9.8$ mm at TOF0 and
$\sigma_{x}=11.4$ mm at TOF1 [14].

###  2.3 Beam line design

The initial design of the baseline $\left(\varepsilon_{N},p_{z}\right)$ =
($6\,\pi$ mm-rad, 200 MeV/$c$) $\mu^{+}$ beam was made using the TURTLE beam
transport code [15] assuming a 1 cm thick lead diffuser. The design was then
optimised with G4beamline [16], with matching conditions in the upstream
spectrometer solenoid of $\alpha_{x}=\alpha_{y}=0$ and
$\beta_{x}=\beta_{y}=333$ mm [9]. The baseline beam design does not compensate
for horizontal dispersion introduced at D2. The remaining
$\left(\varepsilon_{N},p_{z}\right)$ beam settings were obtained by scaling
the magnet currents of the baseline case according to the the local muon
momentum, accounting for the energy loss of muons in the beam line material,
_i.e._ scaled by a factor $p_{\mathrm{new}}/p_{\mathrm{base}}$. Hence, the
beam line will transport 18 different beams to the cooling channel with
$\varepsilon_{N}=3,6,10\,\pi$ mm-rad and $p_{z}=140,200,240$ MeV/$c$, after
the diffuser, in two beam polarities.

The “re-scaled” beam line settings will transport muons to the cooling channel
with the desired momenta but are not necessarily matched in the first
spectrometer as scattering in the diffuser changes the optical parameters.
Because the diffuser is thin, the beta function will decrease by the same
ratio as the emittance is increased and therefore the final optics and
diffuser thicknesses cannot be determined until the inherent emittances of the
input beams are known.

The beam line was commissioned in MICE “Step I” in 2010–2011. Data were taken
for eight positive and nine negative beam settings to verify the beam line
design and determine the characteristics of the beams, in particular their
momentum distributions, emittances and dispersions. The result of the
commissioning is presented below.

##  3 Characterising the MICE beams

Emittance is the area occupied by a charged particle beam, in two, four, or
six-dimensional trace-space, given by $\varepsilon=\sqrt{\det\Sigma}$ where
$\Sigma$ is the covariance matrix. In two dimensions,

$\Sigma=\left(\begin{array}[]{cc}\sigma_{xx}&\sigma_{xx^{\prime}}\\\
\sigma_{x^{\prime}x}&\sigma_{x^{\prime}x^{\prime}}\end{array}\right)\equiv\left(\begin{array}[]{cc}\varepsilon\beta&-\varepsilon\alpha\\\
-\varepsilon\alpha&\varepsilon\gamma\end{array}\right),$

where, for example, $\sigma_{xx}=\overline{x}\,\overline{x}-\overline{xx}$ and
$\overline{x}$ denotes the mean. The covariance matrix can also be expressed
in terms of the Twiss parameters $\alpha,\beta,\gamma$, and $\varepsilon$
giving a full parameterisation of the beam.

Several different methods exist for measuring the emittance of beams [17].
Commonly, beam profile monitors are used to measure the RMS beam size,
$\sigma_{x}$, at several positions. At least three profile monitors are
required to determine the three elements of the covariance matrix and hence
the emittance of the beam; the transfer matrices between the profile monitors
must be known. These methods do not require individual particles to be tracked
but are ultimately limited by the spatial resolution of the detectors and the
intensity of the beam.

By contrast, the MICE muon beam is large in spatial extent and its intensity
is low compared to conventional primary beams. The emittance and optical
parameters of such a beam can be measured if either the trace space co-
ordinates, $(x,x^{\prime}),(y,y^{\prime})$ of individual particles can be
measured at a single plane or, as in the new method described here, the
spatial co-ordinates of individual particles are measured at two detectors and
the transfer matrix between the detectors is known.

In the later Steps of MICE the beam emittance will be measured by a
scintillating fibre tracker inside a 4T solenoid. This detector was not
present during Step I and the new method was developed to characterise the
beam using only the two TOF detectors. The relative times and $(x,y)$
positions of single particles are measured in the two TOF stations and muons
are selected by broad time-of-flight cuts. Each muon is tracked through the
Q7–9 quadrupole triplet, determining the trace space angles $x^{\prime}$ and
$y^{\prime}$ at each plane. Simultaneously, the muon momenta is measured by
time-of-flight, which is important as the beam has a large momentum spread and
the transfer matrix between the two detectors depends strongly on momentum.
The covariance matrix of the beam is then obtained from a large sample of
muons so measured. The method is described briefly below; its detailed
development is given in [14].

Figure 2: The time-of-flight system and beam line section used to characterise
the beam.

Figure 2 shows the section of beam line used for the measurements. The pole
tip radius of the quadrupoles is 176 mm. The two TOF detectors have active
areas of 400 mm $\times$ 400 mm and 420 mm $\times$ 420 mm, respectively, and
were separated by 7.705 m during the 2010 commissioning; their combined time
resolution of 76 ps allows the momenta of muons to be determined with a
resolution of $\sigma_{p}=3.7$ MeV/$c$ for $p_{z}=230$ MeV/$c$.

###  3.1 Measurement technique

The measurement algorithm proceeds iteratively. An initial estimate of $p_{z}$
is made by assuming that a muon travels along the $z$-axis between the two TOF
counters. This estimate is used to determine the $x$ and $y$ transfer
matrices, $M_{x}(p_{z})$ and $M_{y}(p_{z})$, between TOF0 and TOF1. Once the
transfer matrices are known, the trace-space vectors $(x_{0},x_{0}^{\prime})$
and $(y_{0},y_{0}^{\prime})$, and $(x_{1},x_{1}^{\prime})$ and
$(y_{1},y_{1}^{\prime})$, at TOF0 and TOF1 respectively, are obtained from the
position measurements $(x_{0},y_{0})$ and $(x_{1},y_{1})$ by a rearrangement
of the transport equations:

$\left(\begin{array}[]{c}x_{1}\\\
x_{1}^{\prime}\end{array}\right)=M_{x}\left(\begin{array}[]{c}x_{0}\\\
x_{0}^{\prime}\end{array}\right),$ $\left(\begin{array}[]{c}y_{1}\\\
y_{1}^{\prime}\end{array}\right)=M_{y}\left(\begin{array}[]{c}y_{0}\\\
y_{0}^{\prime}\end{array}\right)\,.$ (1)

Explicitly

$\left(\begin{array}[]{c}x_{0}^{\prime}\\\
x_{1}^{\prime}\end{array}\right)=\frac{1}{M_{12}}\left(\begin{array}[]{cc}-M_{11}&1\\\
-1&M_{22}\end{array}\right)\left(\begin{array}[]{c}x_{0}\\\
x_{1}\end{array}\right),$ (2)

where $M_{ij}$ are the (momentum dependent) elements of $M_{x}$, and similarly
for $(y_{0}^{\prime},y_{1}^{\prime})$. The estimates of
$(x_{0},x_{0}^{\prime})$, $(y_{0},y_{0}^{\prime})$, and $p_{z}$ are used to
track the muon between TOF0 and TOF1 and obtain an improved estimate of the
trajectory and a correction, $\Delta s$, to the path length. To ensure
convergence to a stable solution, only half the predicted $\Delta s$ was
applied before recalculating the momentum from the time-of-flight; the
procedure was repeated ten times for each muon although a convergent solution
was found after typically five iterations. Finally, a small correction
($\approx$ 1.5 MeV/$c$) is applied to account for energy loss in the material,
including air, between the TOF counters.

Figure 3:  Quadrupole gradients for the ($6\,\pi$ mm-rad, 200 MeV/$c$)
baseline muon beam (colour online).

In order to obtain the transfer matrices, the focusing gradients of
quadrupoles Q7–9 were determined by fitting the results of an OPERA [18] field
model of the quadrupole with two hyperbolic tangent functions [14]. Figure 3
shows the focusing gradients of the Q7–9 triplet. The quadrupoles are thick
and their fields overlap substantially. A more computationally efficient, and
sufficiently accurate, “top-hat” model of the magnets was used to obtain
$\Delta s$ [14].

Equation 2 for $x_{1}^{\prime}$, which is used to determine the horizontal
emittance at TOF1, can be expressed as

$x_{1}^{\prime}=A(p_{z})x_{0}+B(p_{z})x_{1}$

and mutatis mutandis for $y^{\prime}$. The coefficients $A(p_{z})$ and
$B(p_{z})$ for the baseline (6, 200) beam, with mean $p_{z}\approx 230$
MeV/$c$, are shown in Figure 4. Both $A$ and $B$ are strongly momentum
dependent below 200 MeV/$c$.

Figure 4:  The reconstruction coefficients $A(p_{z})$ (top) and $B(p_{z})$
(bottom) for the (6, 200) baseline muon beam. The solid (blue) lines are for
$x^{\prime}$ (horizontal); the dashed (red) lines are for $y^{\prime}$
(vertical).

The procedure described above enabled the reconstruction of the trace space
vectors at both TOF counters as well as the momenta of single muons. The path
length correction, which could be as much as 15–20 mm, was necessary to avoid
a systematic underestimate of $p_{z}$ of about 4 MeV/$c$.

The momentum distributions and the $(x_{1},x_{1}^{\prime})$ and
$(y_{1},y_{1}^{\prime})$ covariance matrices, $\Sigma_{x,y}$, at the upstream
side of TOF1 for each measured beam were obtained from all the muons recorded
for that beam. The effective optical parameters and the emittances of each
beam were deduced from $\Sigma_{x,y}$ as described in Section 4.3. The
systematic uncertainty on the measurements is discussed in Section 4.4

###  3.2 Monte Carlo simulations of the MICE beam line

Monte Carlo simulations were made for six of the 18 possible beam settings to
check the beam line design software. The ($6\,\pi$ mm-rad, $p_{z}$ = 140, 200,
240 MeV/$c$) $\mu^{+}$ and $\mu^{-}$ beams were simulated in two steps.
G4beamline was used to track particles from the target as far as TOF0; the
G4MICE Monte Carlo [19] was then used to track muons between TOF0 and TOF1.
Both simulations contained descriptions of the material in, and surrounding,
the beam line and magnet models, including the apertures of the quadrupoles
Q4–9, using the optics designed for the corresponding beams. The simulations
suggest that the final emittance of the beams before the diffuser is $\approx
1\,\pi$ mm-rad, partly due to scattering in the material in the beam line but
limited by the aperture of the quadrupoles. Dispersion in the horizontal plane
due to D2 is expected.

###  3.3 Performance of the reconstruction algorithm

Figure 5:  Difference between reconstructed and true momenta for a simulated
200 MeV/$c$ muon beam. Figure 6: Reconstructed trace space angles versus true
simulated values.

The performance of the reconstruction algorithm was determined by smearing the
true simulated coordinates of the muons at each TOF plane with the measured
time and position resolutions of the TOFs. The trace-space vectors were
reconstructed by the method described in Section 3.1 to obtain a set of
simulated reconstructed muons.

Figure 5 shows, for a simulated $(6,200)\,\mu^{-}$ beam, the difference
between reconstructed and true momenta. The RMS width of the distribution of
3.7 MeV/$c$ confirms that the momentum resolution is dominated by the timing
resolution of the TOF system. It is sufficiently small to measure the large
expected widths of approximately 20 MeV/$c$ of the momentum distributions.

Figure 6 shows the agreement between the true and reconstructed angles,
$x_{1}^{\prime}$ and $y_{1}^{\prime}$ for the simulated $(6,200)\,\mu^{-}$
beam. The average angular resolutions, $\sigma_{x_{1}^{\prime}}$ and
$\sigma_{y_{1}^{\prime}}$, for this beam are approximately 29 and 8 mrad
respectively. They are determined by the position resolution of the TOF
counters and multiple scattering, and depend on momentum as $x_{1}^{\prime}$
and $y_{1}^{\prime}$ are obtained from position measurements using the
momentum-dependent elements of the transfer matrix. The angular resolutions
are small but not negligible compared with the expected widths of the
$x_{1}^{\prime}$ and $y_{1}^{\prime}$ distributions.

##  4 Results of the measurements and comparison with simulations

Data were taken for eight positive and nine negative re-scaled beams that,
when used in conjunction with the diffuser, will generate the full range of
desired emittances (see Section 2.3); the polarity of the decay solenoid was
kept the same for both positive and negative beams. Muons in the data were
selected by broad time-of-flight cuts chosen for each nominal beam momentum.

The simulations used in this analysis suggest that the pion contamination at
TOF0 of the $\mu^{-}$ data is about one percent and less than five percent for
the $\mu^{+}$ sample [14]. Recent measurements [20] indicate a somewhat
smaller pion contamination.

Figure 7: Reconstructed longitudinal momentum distributions at TOF1 for six
MICE beams compared with simulations. The dotted (red), dash-dotted (blue) and
shaded distributions are simulation, reconstructed simulation and data
respectively. Distributions are normalised to contain equal numbers of events.

###  4.1 Longitudinal momentum

Table 1: Mean and RMS widths of the longitudinal momentum distributions for six beams compared with the corresponding simulations.  |  | Data | Simulation
---|---|---|---
Beam | Mean $p_{z}$ | RMS $p_{z}$ | Mean $p_{z}$ | RMS $p_{z}$
|  | MeV/$c$ | MeV/$c$ | MeV/$c$ | MeV/$c$
| $\left(6,140\right)$ | 176.4$\pm$2.3 | 22.8$\pm$0.3 | 173.7$\pm$2.1 | 19.5$\pm$0.2
$\mu^{-}$ | $\left(6,200\right)$ | 232.2$\pm$2.5 | 23.6$\pm$0.3 | 229.3$\pm$0.8 | 21.0$\pm$0.1
| $\left(6,240\right)$ | 271.0$\pm$3.7 | 24.5$\pm$0.3 | 270.5$\pm$0.9 | 22.2$\pm$0.1
| $\left(6,140\right)$ | 176.5$\pm$2.0 | 24.4$\pm$0.3 | 176.6$\pm$3.7 | 25.5$\pm$0.5
$\mu^{+}$ | $\left(6,200\right)$ | 229.2$\pm$2.4 | 25.9$\pm$0.3 | 230.8$\pm$1.4 | 28.9$\pm$0.2
| $\left(6,240\right)$ | 267.7$\pm$2.9 | 25.8$\pm$0.3 | 269.2$\pm$4.2 | 31.3$\pm$0.5

Figure 7 shows the distributions of $p_{z}$ at TOF1 for six beams compared
with the results of the simulations. Overall the measured and simulated
distributions agree well in shape and width. The $\mu^{+}$ beams have a
slightly greater momentum spread than the $\mu^{-}$ beams, due to energy loss
fluctuations in the proton absorber. The agreement between the measured and
simulated momentum distributions is better for the $\mu^{-}$ beams than it is
for the $\mu^{+}$ beams. Since the mean momentum is dictated by D2, the
agreement between the measured and simulated mean momenta at TOF1 confirms the
beam line design. The mean momenta and the RMS widths of the measured beams
are given in Table 1. The systematic error on $p_{z}$ is mainly due to the
$\pm 35$ ps calibration uncertainty on the absolute time of flight value [14]
and is estimated to be less than 3 MeV/$c$ for all momenta below 300 MeV/$c$.

###  4.2 Transverse spatial distributions

Figure 8:  Spatial distributions in the transverse plane at TOF1 for
simulation (left), reconstructed simulation (centre), and data (right) for a
(6, 200) $\mu^{-}$ beam, normalised to the same total contents. Simulated
muons in the shaded area cross uncalibrated regions of TOF1 and are excluded
from further analysis.
---
Figure 9:  Root mean square beam widths, $\sigma_{x}$, $\sigma_{y}$, at TOF1
versus $p_{z}$. Solid black circles: $\mu^{-}$ data, open black circles:
$\mu^{+}$ data, solid red triangles: $\mu^{-}$ simulation, open red triangles:
$\mu^{+}$ simulation. The nominal “$p_{z}=140$” MeV/$c$ beams correspond to
momenta in the range 170–190 MeV/$c$, “$p_{z}=200$” to 220–250 MeV/$c$, and
“$p_{z}=240$” to 250–290 MeV/$c$.

Figure 8 shows a comparison of the spatial distributions in the transverse
plane at TOF1 for a simulated (6, 200) $\mu^{-}$ beam before and after
reconstruction, and data for the same beam. The effect of smearing by the
reconstruction procedure is small. Muons crossing the shaded area are excluded
from the simulation (and hence the reconstruction) as they pass through
uncalibrated regions of TOF1. Since muons must cross both a horizontal and
vertical slab of the TOF to be counted, these regions are excluded from the
data. Figure 9 shows the RMS horizontal and vertical beam sizes, $\sigma_{x}$
and $\sigma_{y}$, versus mean $p_{z}$ for all the measured beams and the six
simulated beams. The sizes of the positive and negative muon beams are very
similar both vertically and horizontally. The measured vertical beam size is
about 10–20% smaller than suggested by the simulations. The horizontal beam
size is about 10% smaller than the $\mu^{-}$ simulations but wider than the
$\mu^{+}$ simulations.

Figure 10: Horizontal ($x$) and vertical ($y$) trace space distributions at
TOF1 for simulation (top), reconstructed simulation (centre) and data (bottom)
for a (6, 200) $\mu^{-}$ beam. The ellipses correspond to $\chi^{2}=6$ (see
text).  Figure 11:  Distributions of $\chi^{2}$ for data (solid, shaded,
black) and reconstructed simulation (dash-dot, blue) for the (6, 200)
$\mu^{-}$ beam. Left: horizontal $\chi^{2}$; right: vertical $\chi^{2}$ at
TOF1.

Figure 10 shows the horizontal ($x,x^{\prime}$) and vertical ($y,y^{\prime}$)
trace-space distributions at TOF1 for the (6, 200) $\mu^{-}$ beam and the same
distributions from the simulations with and without smearing due to the
reconstruction. There is very good qualitative agreement between data and
reconstructed simulations in both the horizontal and vertical trace spaces.
The smearing due to the reconstruction is apparent. The distributions have a
dense core and diffuse halo. The boundaries of the distributions reflect the
apertures of the quadrupoles, principally Q9, transported to the TOF1
measurement plane downstream, and the size of TOF1. The vertical divergence of
the beam is approximately three times smaller than the horizontal divergence.

Figure 11 shows the $x$ and $y$ amplitude distributions of muons in the (6,
200) $\mu^{-}$ beam at TOF1 in terms of $\chi^{2}_{x,y}$ where

$\chi^{2}_{x}=[(x-\bar{x}),(x^{\prime}-\bar{x}^{\prime})]\Sigma_{x}^{-1}[(x-\bar{x}),(x^{\prime}-\bar{x}^{\prime})]^{T}={A_{x}}/{{\varepsilon_{x}}}\,,$

$A_{x}$ is the amplitude of a muon in trace-spaceaaaThis is sometimes referred
to as ‘single particle emittance’ [21]. and
${\varepsilon_{x}}=\sqrt{\det\Sigma_{x}}$ is the emittance of the ensemble.
The distributions of $\chi^{2}$ for the reconstructed simulation are shown for
comparison. The initial exponential behaviour of the distribution suggests
that the beam has a quasi-Gaussian core up to $\chi^{2}\approx 6$ and a non-
Gaussian tail. The high amplitude tails of the distributions are slightly
larger for the data than for the simulation.

###  4.3 Determination of emittances and effective optical parameters

The optical parameters and emittances of each beam were determined from the
covariance matrices [22] as

$\displaystyle\varepsilon_{x}$ $\displaystyle=$
$\displaystyle\sqrt{\det\Sigma_{x}},$ $\displaystyle\beta_{x}$
$\displaystyle=$ $\displaystyle\frac{\Sigma_{11}}{\varepsilon_{x}},$
$\displaystyle\alpha_{x}$ $\displaystyle=$
$\displaystyle-\frac{\Sigma_{12}}{\varepsilon_{x}},$

and similarly for $y$. Each of the beams, however, has a large momentum spread
and $\alpha$ and $\beta$ vary with momentum. The parameters determined from
the measurements are therefore effective parameters which describe the
distributions in trace-space.

The reconstructed covariance matrices at TOF1 will differ from the true
covariance matrices because of the finite spatial and angular resolution of
the reconstruction, and multiple scattering in the air between the TOFs (which
cannot be included in the simple transfer matrices used). The finite
resolution leads to a small increase in the apparent emittance of the beams;
scattering will lead to an underestimate of the emittances.

A small correction was made for the effects of resolution and scattering by
subtracting a “resolution” matrix from each measured covariance matrix. The
resolution matrices were estimated from the simulations by taking the
difference between the covariance matrices of the reconstructed and true
simulated beams. These resolution matrices were subtracted from the measured
covariance matrices to obtain corrected, measured covariance matrices, _i.e._
,

$\displaystyle\Sigma_{\rm Corrected}$ $\displaystyle=$
$\displaystyle\Sigma_{\rm Measured}-\Sigma_{\rm Resolution}$ $\displaystyle=$
$\displaystyle\Sigma_{\rm Measured}-(\Sigma_{\rm Reco-sim}-\Sigma_{\rm True-
sim})\,.$

Since simulations were made for only the six (6 $\pi$ mm-rad, $p_{z}$) beams,
the resolution matrices estimated for these beams have been used to correct
the measured covariance matrices for other beams at the same nominal momentum.
As variances are very sensitive to outliers, muons in the high amplitude tails
of the $(x,x^{\prime})$ and $(y,y^{\prime})$ distributions were excluded by
requiring $\chi^{2}_{x,y}<6$ before the corrected covariance matrices were
calculated. The ellipses on Figure 10 show the areas of the distributions
included by this cut.

Figure 12: RMS emittance ellipses in ($x,x^{\prime}$) trace-space for data
without correction for the measurement resolution (black dotted line),
corrected data (black solid line) and true simulation (red dashed line).

Figure 12 shows, for the six beams for which simulations were made, the
horizontal ($x,x^{\prime}$) RMS emittance ellipses for the uncorrected data,
the data after correction for resolution and the true simulation. The effect
of the resolution correction is to reduce the apparent emittance of the beams
and to rotate the ellipses into better alignment with the true simulation.

Figure 13: Horizontal emittance after correction for measurement resolution
and multiple scattering versus mean $p_{z}$ of the seventeen measured beams.
Solid black circles: $\mu^{-}$ data, open black circles: $\mu^{+}$ data, solid
red triangles: $\mu^{-}$ simulation, open red triangles: $\mu^{+}$ simulation.
The nominal “$p_{z}=140$” MeV/$c$ beams correspond to momenta in the range
170–190 MeV/$c$, “$p_{z}=200$” to 220–250 MeV/$c$, and “$p_{z}=240$” to
250–290 MeV/$c$. Figure 14: Vertical emittance after correction for
measurement resolution and multiple scattering versus mean $p_{z}$ of the
seventeen measured beams. Solid black circles: $\mu^{-}$ data, open black
circles: $\mu^{+}$ data, solid red triangles: $\mu^{-}$ simulation, open red
triangles: $\mu^{+}$ simulation. The nominal “$p_{z}=140$” MeV/$c$ beams
correspond to momenta in the range 170–190 MeV/$c$, “$p_{z}=200$” to 220–250
MeV/$c$, and “$p_{z}=240$” to 250–290 MeV/$c$.

The measured emittances discussed below have not been corrected upward for the
$\chi^{2}_{x,y}<6$ requirement, which has also been applied to the simulated
data, because the long non-Gaussian tail of the amplitude distribution (see
Figure 11) is not well-described by the simulations. For a pure Gaussian
distribution 5% of the muons would have $\chi^{2}>6$ and the correction would
increase the measured values of emittance by approximately 20%.

Figure 13 shows the measured horizontal emittances, after resolution
correction, of all the seventeen beams versus the mean $p_{z}$ of the beam and
the true emittances of the six simulated beams. The correction reduces the
measured emittances by $0.6\,\pi$ mm-rad on average; the largest correction is
$-0.7\,\pi$ mm-rad for the (10,140) $\mu^{+}$ beam. Figure 14 shows the
measured vertical emittances of all the seventeen beams versus the mean
$p_{z}$ of the beam and the emittances of the six simulated beams. The
correction increases the measured vertical emittances by about 10%. Clipping
occurs in the vertical plane as Q4 and Q7 are vertically defocusing. This
collimates the beam, resulting in more uniform emittances compared to the
horizontal plane.

###  4.4 Systematic uncertainties

The error bars shown on Figures 13 and 14 include both statistical and
systematic errors. Sources of systematic error fall into three broad
categories; those that affect the transverse position measurement, momentum
reconstruction, and path length corrections. The largest contribution to the
uncertainty on the emittance measurement derives from the effective speed of
light in the TOF slabs, which directly determines the measured RMS width of
the spatial and angular distributions. The various sources, summarised in
Table 2, were determined by examining the change in the reconstructed
emittance and optical parameters when the positions of the TOF detectors and
magnet currents were varied in simulation.

The TOF offsets arise from the uncertainty on their surveyed positions in the
beam line. In each instance, a simulation was produced with one TOF offset by
up to 1 cm in $x,y$ or $z$ and the muon positions recorded. These positions
were input into the reconstruction procedure, which assumes the beam line
elements are located as given by survey. The largest uncertainty occurs when
the TOFs are offset in the longitudinal ($z$) direction, which directly
affects the momentum measurement by altering the distance $\Delta L$.

The uncertainty on the quadrupole triplet position in survey was investigated
in the same manner as for the TOFs. However, since this does not affect the
distance, $\Delta L$, it has a negligible effect on the momentum calculation
and a plays a minor role in the path length correction assigned to a muon. The
currents in the quadrupoles are known to better than 1%, and the effect of
changing these currents was determined. A change in the quadrupole current by
1% has a small effect on the reconstructed path length of a muon, when
compared to the nominal currents, and is a minor source of uncertainty on the
emittance measurement. The uncertainty on $p_{z}$ has a much larger effect on
the transfer matrix used than any scaling due to an uncertainty on the
quadrupole currents (_cf._ Figure 4).

###  4.5 Results

Table 2: Contributions to the errors on the emittance measurements as percentage relative error. Source | $\delta\epsilon_{x}$ | $\delta\alpha_{x}$ | $\delta\beta_{x}$ | $\delta\eta_{x}$ | $\delta\eta_{x}^{\prime}$ | $\delta\epsilon_{y}$ | $\delta\alpha_{y}$ | $\delta\beta_{y}$ |  $\delta p_{z}$
---|---|---|---|---|---|---|---|---|---
TOF1 offsets | $x$ | 0.47 | 0.74 | 0.47 | 1.39 | 0.69 | 0.014 | 0.05 | $\approx 0$ |  $\approx 0$
$y$ | $\approx 0$ | 0.01 | $\approx 0$ | 0.29 | 0.17 | 0.02 | 0.06 | $\approx 0$ | 0.71
TOF0 offsets | $x$ | 0.04 | 0.07 | 0.03 | 0.13 | 0.22 | $\approx 0$ | 0.08 | 0.01 | $\approx 0$
$y$ | $\approx 0$ | $\approx 0$ | $\approx 0$ | 0.02 | $\approx 0$ | $\approx 0$ | $\approx 0$ | $\approx 0$ | $\approx 0$
$\Delta L$ | 2.10 | 0.32 | 2.11 | 1.69 | 3.30 | 2.74 | 30.17 | 2.78 | 0.71
Q789 currents | 0.051 | $\approx 0$ | 0.03 | 0.008 | 0.02 | 0.04 | 0.036 | 0.035 | 0.002
Q789 offsets | $x$ | 0.08 | 0.13 | 0.08 | 0.17 | 0.99 | $\approx 0$ | 0.08 | 0.01 | $\approx 0$
$y$ | $\approx 0$ | $\approx 0$ | $\approx 0$ | 0.01 | $\approx 0$ | $\approx 0$ | 0.01 | $\approx 0$ | $\approx 0$
Effective $c$ in scintillator | 4.87 | 0.05 | 5.23 | 2.22 | 1.59 | 4.05 | 41.27 | 4.09 | 0.11
Total (%) | 5.32 | 0.82 | 5.66 | 3.14 | 3.87 | 4.89 | 51.12 | 4.94 | 1.02

Table 3: The characterised Step I beams. | Beam |  $\langle p_{z}\rangle$ (MeV/$c$) |  $\sigma_{pz}$ (MeV/$c$) |  $\varepsilon_{x}$ ($\pi$ mm-rad) | $\alpha_{x}$ | $\beta_{x}(m)$ |  $\varepsilon_{y}$ ($\pi$ mm-rad) | $\alpha_{y}$ |  $\beta_{y}$ (m)
---|---|---|---|---|---|---|---|---|---
| $\varepsilon_{N}$ | $p_{z}$ |  |  |  |  |  |  |  |
$\mu^{-}$ | 3 | 140 | 171.58$\pm$2.39 | 22.81$\pm$ 0.32 | 2.28$\pm$0.12 | 0.50$\pm$0.01 | 1.49$\pm$0.09 | 0.95$\pm$0.05 | -0.55$\pm$0.28 | 3.62$\pm$0.18
200 | 223.24$\pm$2.72 | 24.02$\pm$ 0.29 | 1.74$\pm$0.09 | 0.49$\pm$0.01 | 1.69$\pm$0.10 | 0.78$\pm$0.04 | -0.50$\pm$0.25 | 3.71$\pm$0.19
240 | 260.55$\pm$3.24 | 24.49$\pm$ 0.30 | 1.49$\pm$0.08 | 0.49$\pm$0.01 | 1.80$\pm$0.10 | 0.75$\pm$0.04 | -0.41$\pm$0.21 | 3.65$\pm$0.18
6 | 140 | 176.43$\pm$2.27 | 22.83$\pm$ 0.29 | 2.17$\pm$0.12 | 0.52$\pm$0.01 | 1.57$\pm$0.09 | 0.96$\pm$0.05 | -0.54$\pm$0.28 | 3.64$\pm$0.18
200 | 232.22$\pm$2.51 | 23.62$\pm$ 0.26 | 1.53$\pm$0.08 | 0.55$\pm$0.01 | 1.85$\pm$0.10 | 0.78$\pm$0.04 | -0.51$\pm$0.26 | 3.80$\pm$0.19
240 | 270.96$\pm$3.65 | 24.53$\pm$ 0.33 | 1.51$\pm$0.08 | 0.48$\pm$0.01 | 1.80$\pm$0.10 | 0.73$\pm$0.04 | -0.39$\pm$0.20 | 3.51$\pm$0.18
10 | 140 | 183.46$\pm$2.35 | 22.75$\pm$ 0.29 | 2.01$\pm$0.11 | 0.53$\pm$0.01 | 1.62$\pm$0.09 | 0.92$\pm$0.05 | -0.56$\pm$-0.29 | 3.68$\pm$0.18
200 | 247.23$\pm$3.56 | 24.20$\pm$ 0.35 | 1.23$\pm$0.07 | 0.59$\pm$0.01 | 2.22$\pm$0.13 | 0.75$\pm$0.04 | -0.52$\pm$-0.27 | 3.81$\pm$0.19
240 | 281.89$\pm$3.65 | 25.28$\pm$ 0.33 | 1.65$\pm$0.09 | 0.56$\pm$0.01 | 1.82$\pm$0.10 | 0.64$\pm$0.03 | -0.39$\pm$0.20 | 3.40$\pm$0.17
$\mu^{+}$ | 3 | 200 | 222.69$\pm$2.40 | 26.49$\pm$ 0.29 | 1.98$\pm$0.11 | 0.49$\pm$0.01 | 1.58$\pm$0.09 | 0.83$\pm$0.04 | -0.40$\pm$0.20 | 3.44$\pm$0.17
240 | 257.97$\pm$2.83 | 26.37$\pm$ 0.29 | 1.59$\pm$0.08 | 0.57$\pm$0.01 | 1.87$\pm$0.11 | 0.76$\pm$0.04 | -0.31$\pm$0.16 | 3.40$\pm$0.17
6 | 140 | 176.45$\pm$1.98 | 24.36$\pm$ 0.27 | 2.32$\pm$0.12 | 0.45$\pm$0.01 | 1.50$\pm$0.09 | 0.95$\pm$0.05 | -0.48$\pm$0.25 | 3.59$\pm$0.18
200 | 229.16$\pm$2.36 | 25.87$\pm$ 0.27 | 1.91$\pm$0.10 | 0.50$\pm$0.01 | 1.61$\pm$0.09 | 0.81$\pm$0.04 | -0.38$\pm$0.19 | 3.42$\pm$0.17
240 | 267.65$\pm$2.85 | 25.79$\pm$ 0.28 | 1.69$\pm$0.09 | 0.54$\pm$0.01 | 1.76$\pm$0.10 | 0.76$\pm$0.04 | 0.26$\pm$0.14 | 3.23$\pm$0.16
10 | 140 | 182.42$\pm$2.05 | 23.87$\pm$ 0.27 | 2.16$\pm$0.12 | 0.47$\pm$0.01 | 1.56$\pm$0.09 | 0.92$\pm$0.05 | -0.48$\pm$0.24 | 3.59$\pm$0.18
200 | 243.39$\pm$2.65 | 26.77$\pm$ 0.29 | 1.66$\pm$0.09 | 0.51$\pm$0.01 | 1.78$\pm$0.10 | 0.78$\pm$0.04 | -0.38$\pm$0.19 | 3.37$\pm$0.17
240 | 274.77$\pm$2.94 | 24.79$\pm$ 0.27 | 1.78$\pm$0.09 | 0.51$\pm$0.01 | 1.65$\pm$0.09 | 0.76$\pm$0.04 | -0.22$\pm$0.11 | 3.07$\pm$0.15

The measured emittances and optical parameters are given in Table 3. The
horizontal and vertical beta functions lie in the ranges 1.49
m$<\beta_{x}<2.22$ m and 3.07 m$<\beta_{y}<3.81$ m. The values of the
horizontal and vertical $\alpha$ parameters, $0.45<\alpha_{x}<0.59$ and
$-0.56<\alpha_{y}<-0.22$, show that the beams converge to a horizontal focus
roughly 700 mm downstream of TOF1 but diverge vertically. The emittances will
be increased by scattering in TOF1.

The measured horizontal emittances and simulations agree to within 10%. Some
of the emittance of the beams can be attributed to multiple scattering in
TOF0. The emittance growth in $x$ ($y$) is expected to be
$\Delta\varepsilon_{x,y}^{2}=\sigma_{x,y}^{2}\theta_{\rm ms}^{2}$ where

$\theta_{\rm ms}^{2}=(13.6{\rm\,MeV/}c)^{2}/(p^{2}\beta^{2})\Delta X/X_{0}$

is the mean square scattering angle in the $\Delta X=0.125X_{0}$ of material
in TOF0. For 200 MeV/$c$ muons and $\sigma_{x}=70$ mm,
$\Delta\varepsilon=1.9\,\pi$ mm-rad for a beam of zero divergence, although
the effective emittance at TOF1 is limited by the aperture of the Q7–9
triplet. The fall in measured emittance with increasing $p_{z}$ seen in
Figures 13 and 14 can be attributed to scattering via the dependence of
$\theta_{\rm ms}$ on $p_{z}$.

There is some emittance growth in the $\approx 8$ m of air between TOF0 and
TOF1. Since the Q7–9 triplet focusses horizontally but is weakly defocusing
vertically, this emittance growth is less in the horizontal than the vertical
plane. For an on-axis beam, $\delta\varepsilon_{y}$ is estimated to be less
than $0.4\,\pi$ mm-rad. The resolution correction described previously
includes a small upwards correction for this emittance growth, and has the
largest effect on the measured vertical emittance. The remaining disagreement
between the measured and simulated vertical emittances can be attributed to
the difference in RMS vertical beam sizebbbThe RMS beam size in Figure 9 is
calculated and shown without the $\chi^{2}<6$ cut to demonstrate the physical
size of the beam, whereas the emittance calculation includes it. shown in
Figure 9.

Figure 15: The horizontal dispersion coefficient, $\eta$, versus mean $p_{z}$
for the seventeen beams. Solid black circles: $\mu^{-}$ data, open black
circles: $\mu^{+}$ data, Solid red triangles: $\mu^{-}$ simulation smeared
with measurement resolution. The nominal “$p_{z}=140$” MeV/$c$ beams
correspond to momenta in the range 170–190 MeV/$c$, “$p_{z}=200$” to 220–250
MeV/$c$, and “$p_{z}=240$” to 250–290 MeV/$c$.

The measured horizontal emittances shown in Figure 13 include (for both data
and simulation) the effect of dispersion. The dispersion in $x$ at the exit of
the D2 bending magnet is transformed by the optics of the beam transport into
dispersion in $x$ and $x^{\prime}$ at the TOF1 measurement plane. The
intrinsic horizontal emittances of the beams have been obtained from the
covariance matrices by subtracting the dispersion characterised by $\eta$ and
$\eta^{\prime}$ [23]:

$\displaystyle\Sigma_{11}$
$\displaystyle\rightarrow\Sigma_{11}-\eta^{2}\delta^{2}$
$\displaystyle\Sigma_{12}$
$\displaystyle\rightarrow\Sigma_{12}-\eta\eta^{\prime}\delta^{2}$
$\displaystyle\Sigma_{11}$
$\displaystyle\rightarrow\Sigma_{11}-\eta^{{}^{\prime}2}\delta^{2}$

where $\eta=\langle x\delta\rangle/\langle\delta^{2}\rangle$,
$\eta^{\prime}=\langle x^{\prime}\delta\rangle/\langle\delta^{2}\rangle$ and
$\delta=(p_{z}-\bar{p}_{z})/\bar{p}_{z}$. Figure 15 shows $\eta$ versus
$\langle p_{z}\rangle$ for all the beams and the simulations for the three
negative beams. The dispersions are similar for the $\mu^{+}$ and $\mu^{-}$
beams and are reproduced by the simulations for the negative beams. The
positive beam simulations are not shown as they did not reproduce the data
well. The reason for this is under investigation. The dispersion-corrected
intrinsic horizontal emittances and $\eta$ and $\eta^{\prime}$ are given in
Table 4. The intrinsic horizontal emittances are, on average, $0.25\,\pi$ mm-
rad smaller than the effective horizontal emittances.

Table 4: Horizontal dispersion and the intrinsic emittances of the Step I beams. | Beam |  $\eta_{x}$ (mm) |  $\eta^{\prime}_{x}$ (rad) |  $\varepsilon_{x}$ ($\pi$ mm-rad) | $\alpha_{x}$ |  $\beta_{x}$ (m)
---|---|---|---|---|---|---
| $\varepsilon_{N}$ | $p_{z}$ |  |  |  |  |
$\mu^{-}$ | 3 | 140 | 90.28 | 0.07 | 2.08$\pm$0.11 | 0.60$\pm$0.01 | 1.56$\pm$0.09
200 | 123.78 | 0.09 | 1.53$\pm$0.08 | 0.65$\pm$0.01 | 1.82$\pm$0.10
240 | 137.58 | 0.11 | 1.26$\pm$0.07 | 0.68$\pm$0.01 | 1.99$\pm$0.11
6 | 140 | 89.37 | 0.08 | 1.97$\pm$0.11 | 0.64$\pm$0.01 | 1.66$\pm$0.09
200 | 106.27 | 0.10 | 1.31$\pm$0.07 | 0.72$\pm$0.01 | 2.06$\pm$0.12
240 | 157.91 | 0.11 | 1.26$\pm$0.07 | 0.68$\pm$0.01 | 1.98$\pm$0.11
10 | 140 | 96.03 | 0.07 | 1.83$\pm$0.10 | 0.64$\pm$0.01 | 1.71$\pm$0.10
200 | 132.78 | 0.08 | 1.04$\pm$0.06 | 0.79$\pm$0.01 | 2.47$\pm$0.14
240 | 145.71 | 0.11 | 1.40$\pm$0.08 | 0.75$\pm$0.01 | 2.02$\pm$0.12
$\mu^{+}$ | 3 | 200 | 122.96 | 0.03 | 1.85$\pm$0.10 | 0.56$\pm$0.00 | 1.58$\pm$0.09
240 | 156.47 | 0.03 | 1.45$\pm$0.08 | 0.66$\pm$0.01 | 1.87$\pm$0.11
6 | 140 | 95.91 | 0.04 | 2.18$\pm$0.12 | 0.52$\pm$0.00 | 1.51$\pm$0.09
200 | 131.16 | 0.04 | 1.76$\pm$0.09 | 0.58$\pm$0.00 | 1.62$\pm$0.09
240 | 172.97 | 0.04 | 1.54$\pm$0.08 | 0.64$\pm$0.01 | 1.76$\pm$0.10
10 | 140 | 103.27 | 0.04 | 2.03$\pm$0.11 | 0.54$\pm$0.01 | 1.57$\pm$0.09
200 | 138.50 | 0.03 | 1.53$\pm$0.08 | 0.59$\pm$0.01 | 1.78$\pm$0.10
240 | 189.64 | 0.04 | 1.61$\pm$0.09 | 0.61$\pm$0.01 | 1.64$\pm$0.09

##  5 Summary

A single-particle method for measuring the properties of the muon beams to be
used by MICE has been developed. Timing measurements using two time-of-flight
counters allow the momentum of single muons to be measured with a resolution
of better than 4 MeV/$c$ and a systematic error of $<3$ MeV/$c$. The ability
to measure $p_{z}$ to this precision will complement the momentum measurements
of the solenoidal spectrometers. For low transverse amplitude particles, the
measurement of $p_{z}$ in the TOF counters is expected to have better
resolution than that of the spectrometers, which are primarily designed for
measuring $p_{t}$.

The same method allows the trace-space distributions at the entrance to MICE
to be measured to $\approx 5\%$ and hence the emittances and dispersions of
the beams. The emittances are found to be approximately 1.2–2.3 $\pi$ mm-rad
horizontally and 0.6–1.0 $\pi$ mm-rad vertically; the average horizontal
dispersion, $\eta$, is measured to be 129 mm, although it depends on the
nominal $(\varepsilon_{N},p_{z})$ beam setting. The positive and negative muon
beams are found to have very similar properties.

As a final check on the suitability of the beams for use by MICE, a set of
measured muons for the (6, 200) baseline beam was propagated from TOF1 to the
diffuser and through a simulation of the experiment. Even without further
software selection (for example, on the rather asymmetric momentum
distribution) the beam was found to be relatively well matched [24]. In
practice, some further fine-tuning of the magnet currents and diffuser
thickness should be sufficient to generate a well-matched beam suitable for
the demonstration of ionisation cooling by MICE.

### Acknowledgements

The work described here was made possible by grants from Department of Energy
and National Science Foundation (USA), the Instituto Nazionale di Fisica
Nucleare (Italy), the Science and Technology Facilities Council (UK), the
European Community under the European Commission Framework Programme 7, the
Japan Society for the Promotion of Science and the Swiss National Science
Foundation, in the framework of the SCOPES programme, whose support we
gratefully acknowledge. We are also grateful to the staff of ISIS for the
reliable operation of ISIS.

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license: Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/

authors:
The MICE Collaboration: D. Adams (16), D. Adey (23, b), A. Alekou (19,
  c), M. Apollonio (19, d), R. Asfandiyarov (12), J. Back (23), G. Barber (19),
  P. Barclay (16), A. de Bari (5), R. Bayes (17), V. Bayliss (16), R. Bertoni
  (3), V. J. Blackmore (20, a), A. Blondel (12), S. Blot (29), M. Bogomilov
  (1), M. Bonesini (3), C. N. Booth (21), D. Bowring (27), S. Boyd (23), T. W.
  Bradshaw (16), U. Bravar (30), A. D. Bross (25), M. Capponi (6), T. Carlisle
  (20), G. Cecchet (5), G. Charnley (14), J. H. Cobb (20), D. Colling (19), N.
  Collomb (15), L. Coney (34), P. Cooke (18), M. Courthold (16), L. M. Cremaldi
  (33), A. DeMello (27), A. Dick (22), A. Dobbs (19), P. Dornan (19), S. Fayer
  (19), F. Filthaut (10, f), A. Fish (19), T. Fitzpatrick (25), R. Fletcher
  (34), D. Forrest (17), V. Francis (16), B. Freemire (28), L. Fry (16), A.
  Gallagher (15), R. Gamet (18), S. Gourlay (27), A. Grant (15), J. S. Graulich
  (12), S. Griffiths (14), P. Hanlet (28), O. M. Hansen (11, h), G. G. Hanson
  (34), P. Harrison (23), T. L. Hart (33), T. Hartnett (15), T. Hayler (16), C.
  Heidt (34), M. Hills (16), P. Hodgson (21), A. Iaciofano (6), S. Ishimoto
  (9), G. Kafka (28), D. M. Kaplan (28), Y. Karadzhov (12), Y. K. Kim (29), D.
  Kolev (1), Y. Kuno (8), P. Kyberd (24), W. Lau (20), J. Leaver (19), M.
  Leonova (25), D. Li (27), A. Lintern (16), M. Littlefield (24), K. Long (19),
  G. Lucchini (3), T. Luo (33), C. Macwaters (16), B. Martlew (14), J.
  Martyniak (19), A. Moretti (25), A. Moss (14), A. Muir (14), I. Mullacrane
  (14), J. J. Nebrensky (24), D. Neuffer (25), A. Nichols (16), R. Nicholson
  (21), J. C. Nugent (17), Y. Onel (31), D. Orestano (6), E. Overton (21), P.
  Owens (14), V. Palladino (4), J. Pasternak (19), F. Pastore (6), C. Pidcott
  (23), M. Popovic (25), R. Preece (16), S. Prestemon (27), D. Rajaram (28), S.
  Ramberger (11), M. A. Rayner (20, j), S. Ricciardi (16), A. Richards (19), T.
  J. Roberts (26), M. Robinson (21), C. Rogers (16), K. Ronald (22), P. Rubinov
  (25), R. Rucinski (25), I. Rusinov (1), H. Sakamoto (8), D. A. Sanders (33),
  E. Santos (19), T. Savidge (19), P. J. Smith (21), P. Snopok (28), F. J. P.
  Soler (17), T. Stanley (16), D. J. Summers (33), M. Takahashi (19), J.
  Tarrant (16), I. Taylor (23), L. Tortora (6), Y. Torun (28), R. Tsenov (1),
  C. D. Tunnell (20), G.Vankova (1), V. Verguilov (12), S. Virostek (27), M.
  Vretenar (11), K. Walaron (17), S. Watson (16), C. White (14), C. G. Whyte
  (22), A. Wilson (16), H. Wisting (12), M. Zisman (27) ((1) Department of
  Atomic Physics, St. Kliment Ohridski University of Sofia, Sofia, Bulgaria (2)
  Institute for Cryogenic and Superconductivity Technology, Harbin Institute of
  Technology, Harbin, PR China (3) Sezione INFN Milano Bicocca, Dipartimento di
  Fisica G. Occhialini, Milano, Italy (4) Sezione INFN Napoli and Dipartimento
  di Fisica, Universit\`a Federico II, Complesso Universitario di Monte S.
  Angelo, Napoli, Italy (5) Sezione INFN Pavia and Dipartimento di Fisica
  Nucleare e Teorica, Pavia, Italy (6) Sezione INFN Roma Tre e Dipartimento di
  Fisica, Roma, Italy (7) Kyoto University Research Reactor Institute, Osaka,
  Japan (8) Osaka University, Graduate School of Science, Department of
  Physics, Toyonaka, Osaka, Japan (9) High Energy Accelerator Research
  Organization (KEK), Institute of Particle and Nuclear Studies, Tsukuba,
  Ibaraki, Japan (10) NIKHEF, Amsterdam, The Netherlands (11) CERN, Geneva,
  Switzerland (12) DPNC, Section de Physique, Universit\'e de Gen\`eve, Geneva,
  Switzerland (13) Paul Scherrer Institut, Villigen, Switzerland (14) The
  Cockcroft Institute, Daresbury Science and Innovation Centre, Daresbury,
  Cheshire, UK (15) STFC Daresbury Laboratory, Daresbury, Cheshire, UK (16)
  STFC Rutherford Appleton Laboratory, Harwell Oxford, Didcot, UK (17) School
  of Physics and Astronomy, Kelvin Building, The University of Glasgow,
  Glasgow, UK (18) Department of Physics, University of Liverpool, Liverpool,
  UK (19) Department of Physics, Blackett Laboratory, Imperial College London,
  London, UK (20) Department of Physics, University of Oxford, Denys Wilkinson
  Building, Oxford, UK (21) Department of Physics and Astronomy, University of
  Sheffield, Sheffield, UK (22) Department of Physics, University of
  Strathclyde, Glasgow, UK (23) Department of Physics, University of Warwick,
  Coventry, UK (24) Brunel University, Uxbridge, UK (25) Fermilab, Batavia, IL,
  USA (26) Muons, Inc., Batavia, IL, USA (27) Lawrence Berkeley National
  Laboratory, Berkeley, CA, USA (28) Illinois Institute of Technology, Chicago,
  IL, USA (29) Enrico Fermi Institute, University of Chicago, Chicago, IL, USA
  (30) University of New Hampshire, Durham, NH, USA (31) Department of Physics
  and Astronomy, University of Iowa, Iowa City, IA, USA (32) Jefferson Lab,
  Newport News, VA, USA (33) University of Mississippi, Oxford, MS, USA (34)
  University of California, Riverside, CA, USA (35) Brookhaven National
  Laboratory, Upton, NY, USA (a) email: v.blackmore1@physics.ox.ac.uk (b) Now
  at Fermilab, Batavia, IL, USA (c) Now at CERN, Geneva, Switzerland (d) Now at
  Diamond Light Source, Harwell Science and Innovation Campus, Didcot,
  Oxfordshire, UK (f) Also at Radboud University Nijmegen, Nijmegen, The
  Netherlands (g) Permanent address Institute of Physics, Universit\'e
  Catholique de Louvain, Louvain-la-Neuve, Belgium (h) Also at University of
  Oslo, Norway (j) Now at DPNC, Universit\'e de Gen\`eve, Geneva, Switzerland
  (k) Now at University of Huddersfield, UK)

submitter: Victoria Blackmore