Muon Capture Schemes for the Neutrino Factory

Stephen Brooks
Trinity College, Oxford

Thesis submitted in fulfilment of the requirements for the degree ofDoctor of Philosophy at the University of Oxford
Hilary Term, 2010

Abstract
The proposed neutrino factory, a facility for precision measurements of neutrino oscillations, requires directional neutrino beams to reach the required sensitivities. Among the
few sources of such beams is the decay of muons travelling at relativistic speeds, therefore
an intense source of muons and subsequent rapid acceleration must be designed so they
can reach the required energy before decaying.
This thesis considers several stages in this process: pions are produced from a proton
beam hitting a target and pion yield optima are determined as a function of target design
parameters and the proton energy. Issues related to producing the original proton beam
are also discussed. The pions decay to a beam of muons, confined by a channel of solenoids
and other components known as the muon front end. A design for this is found that meetsthe requirement of 1021 muons per operational year [1].
The computer code MARS15 [2] is used to simulate the target, with benchmarks
against GEANT4 [3] and initial results from the HARP experiment [4]. The author’s
code Muon1 [5] is used for muon tracking, with its techniques also explained in the thesis.
To find the highest-yielding arrangement of magnets and accelerating components
from the target onwards, Muon1 incorporates an optimisation feature where almost allparameters of the beamline can be varied. This produces a high-dimensionality search
space where the best muon yield is sought using a genetic algorithm. As each individual
evaluation of a design is itself a time-consuming simulation with tens of thousands ofparticles, the code has been deployed as a distributed computing project that is able toperform millions of simulations per optimisation.

Contents
1 Introduction
1
1.1 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Neutrino Beam Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 The Neutrino Factory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Muon Front End Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Guide to This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Particle Tracking Code (Muon1)
2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Runge–Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Single Particle Error Study . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Multi-Particle, Full Beamline Error Study . . . . . . . . . . . . . .
2.3 Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Off-Axis Series Solution . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Axial Field of a Sheet Solenoid . . . . . . . . . . . . . . . . . . . .
2.3.3 Field Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Particle Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Statistical Weighting and Multiple Decays . . . . . . . . . . . . . .
2.4.2 Two-body Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Many-body Decay Algorithm . . . . . . . . . . . . . . . . . . . . .
2.4.4 The Probability Density (spin averaged) . . . . . . . . . . . . . . .
2.4.5 The Primal Distribution . . . . . . . . . . . . . . . . . . . . . . . .
2.4.6 Results and Efficiency . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.7 Dalitz Plot Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Collision Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Proximity and Bounding Boxes . . . . . . . . . . . . . . . . . . . .
2.5.2 Collision Detection Regions for Components . . . . . . . . . . . . .
2.5.3 High-accuracy Isosurface Method . . . . . . . . . . . . . . . . . . .
2.6 RF Components and Phasing . . . . . . . . . . . . . . . . . . . . . . . . .

13
13
16
16
18
21
24
24
26
28
30
31
31
32
33
35
36
37
39
39
40
41
43

3 Target Design Considerations
3.1 Proton Driver Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Bunch Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Thermal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44
46
49
50

i

3.4
3.5

Geometry of the Solid Target Mechanism . . . . . . . . . . . . . . . . . . . 52
Beam Dumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Pion Production Simulations
4.1 Modelling Particle Production in MARS15 . . . . . . . . . . . . . . . . . .
4.1.1 Pion Capture Probability Grid . . . . . . . . . . . . . . . . . . . . .
4.1.2 Difference in Yield Between Probability Grids . . . . . . . . . . . .
4.2 Proton Energy and Rod Material . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Variation of Heat Deposition with Energy . . . . . . . . . . . . . .
4.3 Rod Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Variation of Heat Deposition with Length . . . . . . . . . . . . . .
4.4 Rod Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Rod with a Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Proton Beam Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Proton Beam Radius . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Circular Uniform Beam vs. Circular Parabolic . . . . . . . . . . . .
4.6 Downstream Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Pion Production from Kaons . . . . . . . . . . . . . . . . . . . . . .
4.6.2 Redistribution of Losses . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Summary of Optimal Target Parameters . . . . . . . . . . . . . . . . . . .

58
58
60
64
66
70
73
74
76
78
79
79
81
81
81
82
84

5 Benchmarking Pion Production
5.1 Comparison with GEANT4 . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Energy Cutoffs in Hadronic Models . . . . . . . . . . . . . . . . . .
5.2 Comparison with Early HARP Data . . . . . . . . . . . . . . . . . . . . .

85
85
86
89

6 Muon Energy Loss and Scattering Algorithm
97
6.1 Principle of Muon Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 Cooling Rates from Linearised Optics . . . . . . . . . . . . . . . . . . . . . 100
6.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3 Using the ELMS Scattering Model for Liquid Hydrogen . . . . . . . . . . . 104
6.4 Accelerated Scattering Tracking Algorithm . . . . . . . . . . . . . . . . . . 106
6.4.1 Comparison with ICOOL . . . . . . . . . . . . . . . . . . . . . . . . 108
6.5 Integration with Muon1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7 Design and Optimisation System111
7.1 Muon1 Lattice Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.1.1 A Simple Beamline of Components . . . . . . . . . . . . . . . . . . 111
7.1.2 Cell Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.1.3 Optimisable Variables . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.1.4 Example with Scripting Language . . . . . . . . . . . . . . . . . . . 114
7.2 Figures of Merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.3 Muon1 Result (and Queue) Files . . . . . . . . . . . . . . . . . . . . . . . 118
7.4 Genetic/Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . 120
7.4.1 Sine Wave Interpolation Test . . . . . . . . . . . . . . . . . . . . . 122
ii

7.5

7.4.2 Sequential Parameters Test . . . . . . . . . . . . . . . . . . . . . . 125
7.4.3 Example Lattice Optimisation . . . . . . . . . . . . . . . . . . . . . 128
Distributed Computing Project . . . . . . . . . . . . . . . . . . . . . . . . 133

8 Results from Optimisation
139
8.1 Decay Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.2 Chicane and 400 MeV Muon Linac . . . . . . . . . . . . . . . . . . . . . . 148
8.3 Phase Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8.3.1 Alternative Design for Two Signs . . . . . . . . . . . . . . . . . . . 157
8.4 Phase Rotation and 400 MeV Linac . . . . . . . . . . . . . . . . . . . . . . 160
8.5 Direct 900 MeV Linacs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
9 Conclusion

166

Bibliography

169

A MARS.INP Template File175

B Modification to User Subroutine BEG1 in m1504.f

177

C The Linearised Beam Optics Model
178
C.1 Equations of Motion in z . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
C.2 Transfer Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
C.3 Beam Shape and Covariance Matrices . . . . . . . . . . . . . . . . . . . . . 181
C.4 Un-normalised and Normalised Emittances . . . . . . . . . . . . . . . . . . 182
C.5 Twiss Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
D ICOOL Input File186

E Listings of the Optimised Designs

187

iii

Chapter 1
Introduction
The following sections describe the problem of detecting the very small mass of neutrinos
and experimental methods for doing this, particularly the neutrino factory that is studied
here. Attention then focusses on the muon front end section, as this is critical to theefficiency of such a facility. Lastly, the methodology used for the optimisation of frontend designs in the rest of this thesis is outlined.

1.1

Neutrino Oscillations

The original theory of neutrinos [6] by Pauli in 1930 formulated them as massless particles
with their momentum and energy alone sufficient to explain their interactions. Originally,
these interactions were inferred from the continuous energy spectrum of electrons frombeta decay but in 1956 a neutrino from a nuclear reactor was observed being absorbed by
an atomic nucleus [7], confirming its status as a particle.
The first observations that did not agree with the massless neutrino model were early
results from the Homestake solar neutrino detector [8], which from 1968 onwards showeda factor of ∼3 deficit in the number of electron neutrinos (νe ) received on Earth compared with astrophysical models of solar neutrino production. At this stage it was unclear
whether the deficit, referred to as the ‘solar neutrino problem’, was caused by errors in
the astrophysical model, oscillation between the neutrino flavours (already proposed theoretically [9]), or something else. The Super-Kamiokande experiment [10], operating from1996, was able to resolve this ambiguity by measuring oscillations in neutrinos produced
from the decay of muons in the Earth’s atmosphere. As the production sites of these neutrinos form a spherical shell around the Earth, the zenith angle of the incoming neutrinos
at the detector determines how far away they were produced (up to ∼13000 km away) andthis range of baseline distances is suitable for observing νµ oscillations. Further review ofthe evidence for neutrino oscillations is available in [11]; this section will now explain thephysics behind the process.
For a stationary particle of rest mass m and wavefunction |ψi, the mass–(total) energy
equivalence Et = mc2 becomes the quantum equation of motion Êt |ψi = mc2 |ψi, assuming that ‘a particle’ in this sense is always a pure mass eigenstate. The quantum energy
1Figure 1.1: Decay of a positive muon via the weak nuclear force (left) as occurs in theneutrino factory decay ring. The neutrino and antineutrino produced may undergo further
weak interactions in a distant neutrino detector (right), possibly after oscillating to adifferent flavour.
˙ where dot denotes ∂ , so the time solution to theoperator is given by Êt |ψi := i~|ψi,∂tequation is |ψ(t)i = exp(mc2 t/i~) |ψ(0)i with the exponential of a pure imaginary number
producing a constant rate of phase precession with angular frequency ωm := −mc2 /~.
Assuming that there are three neutrinos |νi i with masses mi for i ∈ {1, 2, 3}, consider
the possible production mechanisms. A weak reaction such as the muon decay µ+ →ν̄µ e+ νe is mediated by the W + boson as shown on the left in figure 1.1. The importantfeature is that a weak vertex for a lepton of flavour α ∈ {e, µ, τ }Pwill produce a flavoured∗|νi i, where U ∗ isneutrino that is a combination of the mass eigenstates |να i := 3i=1 Uαithe complex conjugate of the 3×3 Maki–Nakagawa–Sakata (MNS) matrix. The equivalent
formula with antineutrinos on both sides uses U insteadof U ∗ . U isPunitary (U −1 = U T ∗ ),P −1∗
so the inverse relation can be simplified to |νi i = α Uiα |να i = α Uαi |να i.
We can now model the production and detection of a neutrino in an oscillation experiment that produces να and detects them using lepton flavour β. Transporting |να i asdefined above through proper times τi in the rest frames of the mass eigenstates |νi i will
apply independent phase advances to the components as follows:|να (τ )i =

3
X

∗
Uαi
exp(iωmi τi ) |νi i .

i=1

The detection probability of a νβ flavoured neutrino is Pαβ = |hνβ |να (τ )i|2 . All observable
operators in quantum theory are Hermitian and hence have orthonormal eigenvectors,
meaning that hνi |νj i = δij and we can expand the probability formula taking advantage
of some of these being zero:

Pαβ =

3
X
i=1

Uβi hνi |

3
X

2
∗
Uαj
exp(iωmj τj ) |νj i

j=1

2

3
X

=

2
∗
Uβi Uαi
exp(iωmi τi )

i=1
3
X

3
X

i=1

j=1

∗
Uβi exp(iωmi τi )
Uαi

=

3
X

=

∗
|Uαi
Uβi |2 +i=1

X∗exp(−iωmj τj )Uαj Uβj∗∗UαiUβi Uαj Uβjexp(i(ωmi τi − ωmj τj )).

i6=jAt this point, note that
3
X

2
∗
Uαi
Uβi

i=1

=

3
X

∗
Uβi |2 +
|Uαi

i=1

X

∗
∗
Uβi Uαj Uβj
Uαi

i6=j

but because of the unitarity of UP, the squared item on the left-hand side is the ‘dot∗
product’ of two orthogonal rows: 3i=1 Uαi
Uβi = δαβ . Using this to replace the first term
gives the resultX∗∗Pαβ = δαβ +Uαi
Uβi Uαj Uβjexp(i(ωmi τi − ωmj τj )) − 1 .
i6=j

The three mass eigenstates |νi i are treated as remaining in superposition even though
they propagate at slightly different speeds. By the uncertainty principle, the neutrino
νi will not be perfectly confined in its rest-frame coordinates (cτi , ξi ): there will be asmall range ∆ξi around zero containing some amplitude for the neutrino, which all has
the same phase ωmi τi . In most situations this is larger than the very small classicaldistance expected between the neutrinos. If the neutrinos are emitted at the common
origin of the coordinate systems and received at (ct, L) in the laboratory frame (L being
the baseline length), the Lorentz transform gives τi = γi t − βi γi L/c for the neutrino νithat has relativistic factors βi and γi in the laboratory frame.
If Et is the total energy the neutrino was produced with, then⇒ωmi τi = −mi c2 /~ (γi t − βi γi L/c)= − mi c2 γi t − mi cβi γi L /~= − (Et t − pi L) /~ωmi τi − ωmj τj = ((pi − pj )L) /~.

p
The truncation of the power series pi = Et2 /c2 − m2i c2 ' Et /c−m2i c3 /2Et , valid in theseultra-relativistic cases may now be used to get1ωmi τi − ωmj τj =~



m2j − m2i 3
cL
2Et



and
Pαβ = δαβ +

X


∗
∗
Uαi
Uβi Uαj Uβjexp(i(c3 /~)∆m2ji (L/2Et )) − 1 ,

i6=j

3

where ∆m2ij := m2i − m2j . This formula reveals the oscillatory nature of these detection
probabilities as a function of L: each term is either a constant or a multiple of a linearlyadvancing phase factor (the sum is real because the imaginary parts cancel on swapping
i and j). If one is interested in a particular mass splitting, aiming for a phase advance of
π for maximum signal in its corresponding term suggests the following baseline:(c3 /~)∆m2ji (L/2Et ) = π
2π~ Et⇒ L =,c3 ∆m2Et [GeV].
or L [km] ' 1.240∆m2 [(eV/c2 )2 ]For example, neutrinos from nuclear reactions, either on Earth or in the Sun, have anenergy spectrum of the order of a few MeV. For 3 MeV neutrinos in particular, L would be∼1.4 km if ∆m223 were used, or ∼47 km in the case of oscillations caused by the ν1 –ν2 massdifference. The ∼1.4 km baseline forms the basis of nuclear reactor neutrino oscillation
experiments [12, 13]. For experiments through the Earth with much longer baselines, theinteraction of neutrinos with electrons in matter must be included [14], which changes theoscillation wavelengths from those given in vacuum by the formula above.
Oscillation experiments measure oscillation probabilities, which depend not just on∆m2ij but also the elements of the MNS matrix U . As U is unitary, the complex equivalentof a rotation matrix, it can be parameterised in terms of angles: the standard way of doing
this is given on page 12 of [15]. There are three mixing angles θ12 , θ13 and θ23 , which
determine the extent of neutrino mixing, plus one CP-violating phase δ [16]. It is thegoal of neutrino oscillation physics to measure all these parameters as well as the ∆m2ij ,though individual experiments may be sensitive to different subsets of the parameters.

1.2

Neutrino Beam Facilities

Natural sources of neutrinos create an evenly-distributed but fairly weak flux over theentire Earth’s surface, whereas nuclear reactors produce a concentrated flux from what isroughly a point source but emit neutrinos evenly in all directions. The best configuration
for an oscillation experiment would be to have as many of the neutrinos as possible going
through the detector, which requires a neutrino beam that stays concentrated over a long
distance from the source, requiring anisotropic emission of neutrinos.
The only known way of practically achieving this is to take an interaction, such as a
decay, which emits neutrinos isotropically in the rest frame of the parent particle, thenpre-accelerate the parent particle to relativistic speeds so the Lorentz boost transforms
the distribution of neutrino emission directions into a forward-directed beam in the experimental frame. For relativistic γ  1 this produces a beam opening angle proportional
to 1/γ and a flux enhancement factor of γ 2 relative to an isotropic source.
The source of beamed neutrinos taken together with the detector(s) at the other endof the baseline(s) is referred to as a neutrino beam facility.
4

Three main methods of producing boosted neutrino emission have been proposed.
• Conventional neutrino beams. Any experiment in which a beam of GeV-orderenergy hits a fixed target will generate large numbers of pions (π + and π − ), which if
left alone will decay via π + → µ+ νµ , or the charge conjugate π − → µ− ν̄µ . The pions
are mostly forward-directed because they retain some of the original beam’s momentum and, as they are charged, may be focussed using elements such as solenoids ora magnetic horn. Examples include MINOS [17] and K2K [18].
– Superbeams. An obvious way to increase the flux of the conventional beam
is to increase the mean power of the primary beam. The term superbeam,for ‘super-conventional beam’, refers to experiments such as [19] that use
megawatt-order (or higher) beams on the target.
• Beta beams. Beta decay and positron emission decay are naturally-occurring process that occur in certain radioisotopes. At the level of nucleons, these processes
are n → p e− ν̄e and p → n e+ νe respectively, producing electron-flavoured neutrinos
complementary to the muon-flavoured ones from superbeams. A beta beam experiment [20] consists of a source of radioactive ions, which are then accelerated to aring where they decay, emitting neutrino beams in their direction of travel. The
decay ring incorporates straight sections pointing at the desired detector sites.
• Neutrino factories. Although muons are discarded in conventional neutrino beams,
they also undergo a neutrino-producing decay µ+ → ν̄µ e+ νe (and its charge conjugate), which is used in the proposed neutrino factory [21]. This produces neutrinosof two flavours with equal fluxes, which is a useful calibration. The muons survive
for longer than pions, which provides (just) enough time for them to be accelerated,
resulting in a more collimated beam from a decay ring similar to beta beams.
– Low-energy neutrino factories. Removing some of the muon acceleration
system from the conventional neutrino factory gives a smaller, cheaper facility
that produces lower-energy neutrinos [22]. Detector technology can be optimised to perform better for these low energy interactions.
Figure 1.2 compares the sensitivities of these different facilities to neutrino oscillation
parameters that have not currently been measured to high precision. The angle θ13 andCP-violating δ could still be zero consistent with current data and although there aresome limits on |∆m231 |, it is not known whether the difference between the masses ispositive or negative. A full discussion of the physics reach of the neutrino factory andother experiments is available in the ISS physics report [15]. The neutrino factory provides
the best performance but requires the largest and most complex facility.

5Figure 1.2: Sensitivity of neutrino beam facilities to δ 6= 0 (top left), the sign of ∆m231(bottom left) and θ13 6= 0 (bottom right). The measurements become more difficult as θ13
approaches zero and for certain values of δ. Values to the right and below each coloured
band can definitely be measured at that facility and within the band it depends on thespecific variant of the design used. The low-energy neutrino factory (not shown) performs
comparably to the beta beam, as shown in figure 21 of [23].

6Figure 1.3: Possible neutrino factory layouts, drawn by C.R. Prior based on the conclusions of the ISS [24].

1.3

The Neutrino Factory

The neutrino factory is an accelerator-based neutrino source [21] intended to measure themass splitting ∆m231 and entries of the MNS matrix including Ue3 = e−iδ sin θ13 , whereθ13 is the ν1 –ν3 mixing angle. As presented on pages 21–22 of [15], the best fit valueof ∆m231 using data available at the time of publication was 2.6 × 10−3 (eV/c2 )2 . Using
this in the baseline formula derived in section 1.1 (and assuming matter effects are small)gives a length of 477 km per GeV of neutrino energy, suggesting that neutrinos of up toroughly 27 GeV can be used with baselines that form a chord across the Earth’s interior.
The high energy allows a directed neutrino beam to be created by the Lorentz-boosted
decay of µ+ → ν̄µ e+ νe (and its charge conjugate), where the decay products acquire
105 MeV of kinetic energy in the muon rest frame. We therefore expect a neutrino beamof divergence 2–5 mrad when a beam of muons decays in-flight at these energies. This
angle is proportional to the relativistic factor 1/γ, so the flux at the detector is a factorof γ 2 ∼ 105 above isotropic emission from sources such as nuclear reactors. A fast muon
beam produces neutrinos with energies almost uniformly distributed between zero andthe beam energy, so muons of energy 20–50 GeV are required.
The neutrino factory aims to produce 1021 muon decays per year [1], making it sufficiently sensitive to the mixing angle θ13 , which is currently constrained only by reactor
7experiments. A schematic of a neutrino factory is shown in figure 1.3. So far, the mostcomplete design study of a neutrino factory has been the Muon Collider Collaboration’s
Feasibility Study II [25] in 2001, which was updated with international consensus in 2007
by the International Scoping Study (ISS) [24].
It is difficult to produce an intense beam of muons at 20–50 GeV: muons have a mean
life of only τµ = 2.2 µs and must be produced as secondary particles from another accelerator. Multi-GeV protons are directed onto a high-Z target to maximise the number ofstrong interactions that produce pions. Each pion then decays to a muon (π + → µ+ νµ )in an average time of 26 ns. The mean beam power of the proton accelerator must be
at least 4 MW for sufficient flux and additionally it must be pulsed to provide extracted
bunches of a few ns duration for the time-dependent muon capture systems downstream.
An accelerator with these properties is referred to as a proton driver [26].
The rate of neutrinos of each flavour produced is equal to the rate of muons decaying
in the storage ring, given byRµ = Rp eπ/p emachine e−τ /τµ(achieving 1021 per operational year requires Rµ = 1014 s−1 ). Here Rp is the rate ofprotons hitting the target, eπ/p is the number of pions produced per incident proton andτ is the proper time experienced by a muon while going through the acceleration chain.
The remaining term can be expressed as the product of the muon capture, accelerationand storage efficiencies:
emachine = efront eacc estore .
The work in this report mainly focusses on improving the efficiency of the target eπ/p andthe muon front end efront . However, most terms in the equation cannot be considered inisolation: at constant beam power the proton energy will affect Rp and eπ/p jointly. Also,
the muon decay in the simulations described in this thesis is not handled in continuum assuggested by the equation. Instead, each muon decays individually, so the factor e−τfront /τµwill automatically be included in results from front end simulations.
Particular engineering challenges are involved in:• The muon front end. The target creates a large, high divergence beam andpion decays produce additional momentum variation. Typically only a few percent
can be captured both transversely and longitudinally. Superconducting solenoids ofseveral Tesla are required for transverse capture, while RF phase rotation systems
or magnetic chicanes are options for longitudinal matching.
• Muon acceleration. The lifetime of the muon is extended to γτµ in the laboratory
frame, so time spent at low energies should be minimised and acceleration must be
rapid. Conventional synchrotrons typically take at least 10 ms for the magnetic field
to ramp from low to high, so are far too slow. Therefore fixed-field, multiple-passmachines such as RLA (Recirculating Linac) and FFAG (Fixed-Field Alternating
Gradient) accelerators are under consideration. However, the most rapid acceleration possible is always achieved by a straight linac, which is used as the first stage.

8

Figure 1.4: Parts of the muon front end as investigated in this thesis, in simulation (top)and schematic form (bottom). The phase rotator may work in either the bunching or
antibunching direction depending on whether a bending chicane or RF cavities are used.

1.4

Muon Front End Structure

The muon front end is normally defined to be the piece of neutrino factory beamline
extending from the target (which the protons hit) up to the start of muon acceleration.
Most designs start with a solenoidal focussing channel (the decay channel ) to capture
pions from the target as they decay to muons, including a powerful solenoid of up to
20 Tesla (20 T) enclosing the target itself. Longitudinal (energy vs. time of flight) phasespace is then manipulated, normally after pion decay, to have a beam suitable for themuon accelerator. This second process is known as phase rotation. A typical front end
studied in this thesis is shown in figure 1.4.
Each stage of the front end improves one aspect of the beam distribution, which is
initially a multidirectional spray of various particles including pions from the target (seesections 3.3 and 4.1.1). The steps listed below prepare a beam of muons that can be
accelerated to the storage ring with low losses apart from the unavoidable muon decays.
9• The decay channel. A strong magnetic field causes charged particles to spiral around the field lines instead of continuing in a straight line. This principle
is used to deal with the particles emitted from the target, which can have hundreds of MeV/c of transverse momentum, making angles of tens of degrees withthe forward direction. A field of 20 T along the accelerator axis can keep a large
fraction of the particles within 10 cm of the axis, although it will not reverse thedirection of particles emitted with a backward longitudinal momentum component.
In a constant field the particles would continue on perfect helices with their largetransverse momentum being conserved in magnitude. However, if the field decreases
along the axis, the zero divergence property of magnetic fields implies there must
be an outward-pointing component of the field vector. This produces a force on
charged particles in the direction opposing their natural direction of spiralling inthe solenoid field, decreasing the transverse part of their momentum. Since particle
energy is conserved in magnetic fields, the decrease in transverse momentum mustbe accompanied by an increase in longitudinal (forward) momentum. All this meansthat if the magnetic field decreases significantly following the target, the transverse
momentum and angle of travel relative to the accelerator axis will both be reduced.
A magnetic field of some magnitude is still required to confine the particles in later
sections, with decay channel fields in this thesis typically reducing to 3–4 T. Superconducting solenoids can produce the above sequence of fields.
Out of the charged particles from the target, pions are the primary decay parentsof muons, with kaons contributing a few percent when present. The pions havethe longer mean lifetime of the two of roughly 26 ns [27] in their rest frame andthe forward pions have relativistic factors of γ ∼ 3, making the mean flight distance∼20 metres before decay. This means that most decay channels are 30 metres long or
more, though variation of this was allowed in the design process. The decay impartsa non-negligible momentum kick (tens of MeV/c) to the daughter muon, causing
the beam to diverge while decays are still in progress. The end product of the decay
channel is a beam of predominantly muons, with little decay activity, transversely
confined within a channel of solenoids. Longitudinally the beam has ‘debunched’
since being produced at the target, i.e. the particles continue at different forward
speeds, spreading out as time progresses.
• Phase rotation. Periodic RF systems such as those in the muon linac require thebeam to occupy a certain stable region of the energy vs. arrival time phase space
known as an RF bucket. Particles within this region will oscillate around the centre of
the bucket, which follows the acceleration waveform, whereas those outside will losesynchronisation and not be fully accelerated. The bucket has a time span inversely
proportional to the RF frequency and an energy range proportional to the applied
voltage, but is always ‘upright’ in the sense that correlation is not desired between
time and energy. The beam from the decay channel has a very strong correlation
since the highest energy muons will arrive first: as such, it occupies greater time andenergy ranges than its phase space area requires and attempting direct accelerationof it would result in RF voltages higher (and more expensive) than necessary.
The phase rotation section removes this correlation by systematically offsetting theenergy or time of arrival of the particles. Energy offset is achieved by another RF
system before the linac that is phased to decelerate the leading muons and accelerate10the stragglers in the bunch (schemes exist with one or many bunches), leading to asmaller energy range at the output. Time-of-arrival offset, known as reverse phaserotation, uses magnetic bends to put the higher energy particles on longer paths,
reducing the time spread at the output. Variants of both schemes are studied inthis thesis: a reverse phase rotation scheme in section 8.2 and RF phase rotators in
section 8.3 onwards.
The phase rotation channel typically continues the transverse beam confinement
using solenoids as in the decay channel, with the RF cavities placed between them.
The resulting beam has the stable transverse optics inherited from the decay channelin addition to a decorrelated longitudinal phase space.
• Ionisation cooling. For some proposed muon accelerators, such as FFAGs, thebeam from the decay channel is still too large to fit in the dynamic aperture of theaccelerator’s magnets, within which the transverse dynamics are stable. Scatteringof muons through materials placed in the beamline is used to drain transverse momentum, with RF acceleration replenishing forward momentum, leading to a smaller
beam. The interaction is predominantly by knocking electrons from the atoms ofthe material, hence the name muon ionisation cooling. This technique is not usedin the front ends studied herein, though chapter 6 explains the principle further andconstructs a computer model of this beam scattering. The resulting transversely
smaller beam would be compatible with smaller aperture muon linacs.
The UK neutrino factory design differs slightly from the ISS baseline [24] because
the UK team have been investigating the possibility of a solid metal target instead ofa mercury jet [28, 29]. The performance in terms of pion yield is quite similar between
the two approaches, metals such as tungsten and tantalum being close to mercury in
the periodic table, though some adaptations of the decay channel are necessary such as
locating the residual proton beam dump further downstream (see section 3.5). Design
issues related to this change are discussed throughout chapter 3.
The front ends in this thesis also differ from other baselines because they do notassume a muon ionisation cooling channel is used to reduce the beam size before entering
the acceleration systems. A buncher and cooling system may be added to make the beamsmall enough to go through the optics of FFAGs but this is not attempted. Instead, it
is suggested that a linear muon accelerator with a larger acceptance may be a practical
option all the way to the decay ring energy (25 GeV in the ISS). The reasoning is that theneutrino factory shown in figure 1.3 requires building an estimated 2.04 km of beamline
from the target to the storage rings, excluding transfer lines (0.87 km of this is in the eight
RLA teardrop arcs). If the average gradient of the first ISS RLA’s accelerating section(8.5 MV/m) was used to cover the range 0.9–25 GeV, that would require 2.84 km of linac
to be added to a front end of 0.34 km given in section 8.5, for a total of 3.18 km and nointernal transfer lines. The relativistic shrinkage of the beam in the high-energy half of thelinac may allow this part to be shortened further, either with compact transverse optics or
through high-gradient cavities (see end of section 8.1). Thus the linac option is not greatly
longer than the multi-technique system, while the use of fewer different technologies could
potentially lead to higher reliability or cost savings during manufacturing.
11

1.5

Guide to This Thesis

An automated approach is used to search for good front end designs.
The basis for this is a particle tracking code called Muon1 [5], whichcalculates the number of muons that survive a given configuration ofmagnets and RF cavities.

Chapter 2

Tracking requires an initial distribution of pions from the target, which
can be calculated with hadronic interaction codes such as MARS15 [2]used here. The target itself can be varied with a view to improving
yields, so after a discussion of the design issues,

Chapter 3

the target parameters are optimised

Chapter 4

and the resulting pion production checked against some experimental
data and another code.

Chapter 5

A module for simulating muon cooling in Muon1 is also developed.

Chapter 6

With the tracking code ready to use, an optimiser is constructed thatloops Muon1 to search a specified design space for good muon yields
using a genetic algorithm. This is deployed as a distributed computing
project.

Chapter 7

Finally, different approaches for the front end and early muon linac
are run as different design spaces in the distributed optimiser and theresulting best lattices discussed.

Chapter 8

12Chapter 2
Particle Tracking Code (Muon1)
This chapter explains the principles of particle tracking as used in the Muon1 code [5],including some analysis of error sources. Models for magnetic fields are derived, together
with algorithms for generating particle decays and locating collisions. The more specificaspects of specifying the accelerator as input to the code are discussed in section 7.1.
Muon1 was written beginning in 2001, at a time when two other codes for muon front
end simulations, ICOOL [30] and G4MICE [31] were in development. ICOOL had manyof the required features but tracked particles with beamline distance as the independent
variable. It was observed that some pions emitted from the target went backwards (seesection 4.1.1), so a code that tracked in time was preferred to investigate their behaviour
including magnetic bottle or mirror effects around the target solenoids. G4MICE, a combination of GEANT4 [3] tracking and materials with magnetic field models for solenoids,
did track in time but many of the GEANT4 material models proved not to be as accurate as those in other codes (see sections 5.1 and 6.3). It also had many external librarydependencies and did not lend itself to a self-contained small distribution that could be
downloaded and installed on Windows, as was used for the distributed computing projectin chapter 7. Finally, neither of the other two codes had genetic algorithm optimisation
capabilities, although these could conceivably have been wrapped around them in someway. Muon1 provided a more integrated package.

2.1

Equations of Motion

The force on a particle of charge q travelling through electric and magnetic fields E(x, t)
and B(x, t) is given by the Lorentz law:dp
= F = q(E + v × B),
dt
where p is the particle momentum. The time integration of this equation is complicatedsomewhat by the relativistic definition of momentum:
p = mγv
γ = γ(v) = p
13

1
1 − (v/c)2,

where v = |v| and the particle has rest mass m. Particles must be represented by a phasespace vector that encodes information about both their position and velocity (or momentum). There is some choice of how to parameterise the phase space, though it always has
3 + 3 = 6 variables. The Muon1 tracking code represents position by a cartesian vector
x relative to a fixed origin, the other options being some sort of polar coordinate system
and/or changing coordinate system to align with the axis of the accelerator components.
This is in the laboratory rest frame, another choice would have been to move the origin
with some “central particle” in the beam. Finally, some codes when in the acceleratoraligned frame use “z” as the independent variable instead of time and store the arrival
times of particles at a particular plane. These alternative techniques can provide fastertracking codes in some cases (for instance when tracking in z only requires a single component to be considered at a time) with variables that are more physically interpretable.
Because the simulation of the muon front end starts from a pion production target, withparticles going in many directions including backwards, a 3D approach that makes noassumptions is favoured for its generality. The large velocity spreads, even when going
forwards, limit the usefulness of taking a frame co-moving with the beam.
For the three other components of the phase space vector, γv is chosen. Early versionsof the code used v but this started to lose accuracy for higher-energy particles. Since
Et = mγc2 , inverting the expression for γ(v) gives the velocity as a function of theparticle’s total energy:ppv = c 1 − γ −2 = c 1 − (Et /mc2 )−2 .
The −2 index on the ratio of the total energy to the rest energy means the amount
subtracted from 1 becomes small at a quadratic rate, after which the square root moves it
even closer to unity (and v to c). This closeness results in poor energy resolution when v isnumerically quantised to triplets of floating point numbers. The vector γv (actually equal
to p/m, so proportional to momentum) does not suffer from this problem and tends toinfinity for larger energies, maintaining resolution in energy as seen in the ultra-relativistic
limit |γv| ' γc = Et /mc where the quantities become proportional. For low speeds γ ' 1and γv ' v approaches the classical velocity.
It will be useful to have expressions for the derivative of the phase space vector (x, γv)
with respect to laboratory time t (the independent variable). By definition,
γv
dx
=v=
dt
γ
but the γ in the denominator must be calculated from the speed information alreadystored in the phase space vector (i.e. |γv| = γv) this time, not just from v as before.
Although this can be done purely by algebraic manipulation of the expression for γ(v),which produces quite messy formulae that later cancel to something simple, a physics
argument can give the result directly. Start with the (classical, relativistic) total energy
equation for a moving particle
Et2 = (pc)2 + (mc2 )2 ,where p = |p| and note that p = mγv and Et = mγc2 . Substituting and then cancellingmc from within the brackets gives(mγc2 )2 = (mγvc)2 + (mc2 )214⇒(γc)2 = (γv)2 + c2p(γv)2 + c2
⇒ γ =
,
c

which can be used as the expression for γ in
γv
γvc
dx
=
=p
.
dt
γ
|γv|2 + c2
The rate of change of the other component comes from the force law as opposed tokinematics:d(p/m)
F
q
d(γv)
=
=
= (E + v × B).
dt
dt
m
m
There is again the problem of having v instead of γv. However, the program can bearranged to calculate dx
first and use the stored result in the expression
dt


d(γv)
dx
q
E+
=
×B .
dt
m
dt
The complete time derivative of the phase space vector for the tracking problem can now
be written out explicitly:√ γvc2 2|γv| +c
d
x
 
 
= q
,
γvc
γv
dt
E(x, t) + √ 2 2 × B(x, t)m|γv| +c

where the functional dependencies have become unsuppressed to demonstrate how eachelement of the derivative depends on elements of the phase space vector and only one
other variable, t. This last external dependency can be eliminated by making the current
laboratory time part of the particle’s vector, though in a real program this would not be
stored repeatedly for each particle! As a result, the derivative becomes a proper functionof the vector:√ γvc2 2|γv| +c
x
x
 
 
d 


γv  = f  γv  =  q E(x, t) + √ γvc
.
×
B(x,
t)
m
2
2
dt


|γv| +c
t
t
1

15

2.2

Runge–Kutta Methods

The previous section produced a “time integration” problem of finding the solution s(t) of
a differential equation with generic formṡ = f (s),) and the initial condition s(t0 ) is specified towhere ṡ denotes the time derivative ( dsdtmake the solution unique. In cases where there is not an exact analytic solution for the
trajectory, as is usually the case for realistic magnetic fields, the calculation can proceed ina series of small timesteps, each of which calculates a numerical approximation to s(t + δt)from s(t).
Note that incremental time-stepping is generally the slowest class of method for trajectory calculation. In lower dimensionality cases (e.g. the two-dimensional phase spaceof longitudinal dynamics) the mapping s(t0 ) → s(t1 ) can be approximated by evaluatingcomparatively few tracks and then interpolating, so tracking can be avoided for the majority of the particles. Additionally, for beams that travel very close to the machine axis,
power expansions in deviations from a central trajectory can give good results even for
the full phase space (see for instance [32]). However, the pion beam to be captured in aneutrino factory front end has a very high emittance that is intrinsic to the way it is generated from nuclear interactions in the target, so particles may move into regions far fromthe central trajectory where a truncated power series is no longer a good approximation.

2.2.1

Principle

Considering a small but finite timestep of length δt, the definition of the derivative as alimit immediately suggests the following approximation:s(t + δt) − s(t)' ṡ(t)δt⇒ s(t + δt) ' s(t) + δt ṡ(t)= s(t) + δt f (s(t)).
This is known as Euler’s forward difference method and is the simplest numerical integration method available for these problems. To obtain information about the error
behaviour of this method, Taylor expand s(t + δt) about t:
Error = s(t) + δt f (s(t)) − s(t + δt)= s(t) + δt ṡ(t) − (s(t) + δt ṡ(t) + 21 δt2 s̈(t) + O(δt3 ))= − 12 δt2 s̈(t) + O(δt3 ),which shows that the error in a single timestep is proportional to δt2 (plus terms thatare O(δt3 ), i.e. order δt3 and higher). Tracking through the entire time interval requires(t1 − t0 )/δt of these steps, giving a final error proportional to δt (t1 − t0 ) provided theerrors are additive. Because the rate of error accumulation is generally proportional toδt1 , Euler integration is known as a first-order method. This means that to increase its
16

tracking accuracy by an order of magnitude, an order of magnitude more computing timeis required, which leads to very large compute times to produce accurate results.
Fortunately there are higher-order methods available that cancel more of the Taylor
expansion of s(t + δt), leaving only higher powers of δt remaining. A simple example of
how this can work is the method that uses ṡ from the central location t + 21 δt insteadof from t to determine the direction of the step, which gives it second-order (step error∝ δt3 ) accuracy as the s̈ term is now taken account of. The central location itself has tobe worked out from an Euler step as shown below.
s(t + δt) = s(t) + δt ṡ(t + 12 δt) + O(δt3 )1= s(t) + δt f s(t + 2 δt) + O(δt3 )= s(t) + δt f s(t) + 21 δt ṡ(t) + O(δt2 ) + O(δt3 )1= s(t) + δt f s(t) + 2 δt f (s(t)) + O(δt3 )Notice how the approximation is expressed in a form that depends only on s evaluated at
the current time (and not any other time, or its derivatives) and evaluations of the known
function f . This ensures that it is computable and leads to the general form of an N -step
Runge–Kutta method:s0 = s(t);sn = s(t) + δt f

n−1
X

!
αni sifor 1 ≤ n ≤ N ;i=0s(t + δt) 'NXβi s i ,i=0where the coefficients αni and βi can be carefully chosen to cancel lower powers of δt in
the error expression, leaving a method with a high order of accuracy. This technique wasoriginally developed at the end of the 19th century for solving differential equations byhand [33] but for a more modern introduction with applications to computer programming,see [34].
The Muon1 code uses the fourth-order Runge–Kutta method defined as follows.
s0
s1
s2
s3
s4
s(t + δt)

=
=
=
=
=
'

s(t)
s(t) + δt f (s0 )s(t) + δt f ( 21 s0 + 12 s1 )s(t) + δt f ( 21 s0 + 12 s2 )s(t) + δt f (s3 )1
s + 13 s2 + 31 s3 + 16 s4
6 1

This order is chosen because of the relatively rapid convergence with decreasing step size
and the fact that increasing to fifth order requires an additional two calculation steps, as
opposed to orders up to 4 where only one more step is required for each successive order.
17

Figure 2.1: Path of the muon in the solenoid test problem.

2.2.2

Single Particle Error Study

The two main deviations from the ideal higher-order error behaviour in accelerator applications are that:
1. The errors are not always additive. When tracked paths diverge from each other,errors can be amplified on a timescale determined by the field gradients in themachine. A simple case of this is in a defocusing quadrupole magnet where off-axis
distance and hence distances between particles increase exponentially.
2. In a tracking code, f (s) is not always analytic. Often the input for a code will specify
B(x) as a ‘field map’ with values defined at the points of a grid. The interpolation
used to generate field values between these points is generally not completely smooth,so steps that straddle the border will acquire an error of a lower power of δt thanintended.
The first of these in particular makes creating a useful error analysis applicable to thegeneral case impossible. Instead, some empirical tracking must be done when fixing atimestep, to ensure it is appropriate for the problem at hand. Below, this is done for twocases: one tracking through a solenoid field and the other tracking through a complex field
map extracted from an OPERA [35] model of a muon cooling ring. In both of these, thefinal particle position and velocity obtained from timesteps ranging from 10−12 to 10−8
seconds is compared to an ‘accurate’ result using 10−13 seconds.
Test 1: Solenoid with Fringe Field
In this test an 180 MeV muon enters a B∞ =20 T solenoid that is 40 cm long and 10 cm
in radius. The 20 T value is only an asymptotic central field for an infinitely long sheet
coil solenoid with the same current density. The actual
starts with the field calculation

a
b
1
√
√
analytic expression for the on-axis field, Bz = 2 B∞ a2 +1 + b2 +1 where a = z/Rsol
18

Figure 2.2: Dependence of errors on timestep size for the solenoid test problem.

and b = LRsolsol−z . This is extrapolated off the central axis by a cubic power expansion in r
derived from imposing ∇ · B = ∇ × B = 0 on a cylindrically symmetric problem (this is
fully described in section 2.3.1). The muon is tracked for 5 ns, starting (3 cm,2 cm) off-axis
at the entrance plane of the solenoid and moving at an angle of 48.5◦ to the central axis,
so that it spirals around and then leaves via the other end (see figure 2.1).
Figure 2.2 shows the errors in the final phase space coordinates for the range oftimesteps investigated. The right-hand side of the figure is blank because the muon
hit the solenoid and was lost for those timesteps. The larger timesteps do not work due tothe confined nature of this problem: the muon would travel from one side of the solenoid
bore to the other in about 1 ns, so the timestep must be much smaller than that. Additionally, it must be much smaller than the Larmor period (2πm0 γ/qB = 0.9985 ns here)of its helical path within the solenoid, which is also short due to the large 20 T field.
Most striking is the very simple dependence of the error on the timestep below some
initial irregularities. This comes purely from the δt4 leading order error term and can beseen from the graph to decrease exactly four orders of magnitude for each order of magnitude in timestep. This dependency is so clean owing to the completely analytic natureof the B field that was used, for just one solenoid with no cut-offs or other irregularities.
19

Figure 2.3: Muon path through the cooling ring field maps.

Figure 2.4: Dependence of errors on timestep size in the cooling ring field map test
problem.
20

Test 2: Field Maps from OPERA
Here another 180 MeV muon is tracked for 50 ns through roughly one third of a turn in
the Rees cooling ring† [36]. It starts (3 cm,2 cm) off-axis, but parallel, to test tracking of
the transverse oscillations in the orbit visible in figure 2.3. The maps were generated in
the magnetostatic code OPERA-3d and sampled at cubic grids of 5 cm spacing. Trilinear
interpolation was used between the eight grid points at the corners of each cube to obtainthe field values in the continuum. This gives a continuous B field but it is not differentiable
at the cube boundaries.
The scale of the problem is larger here (metres) so many of the longer timesteps dowork to some extent and now appear on the graph (figure 2.4). An initial steep decrease(possibly ∝ δt4 ) gives way to a slower decline as the error becomes dominated by lower
terms. A δt2 term is expected from the fixed number of crossings of field map cube
boundaries, each of length δt and making a mis-estimate of the field proportional to that
length, giving δt2 overall. However, the final falloff looks even slower than that. An errorof order δt will come from any field discontinuities, such as edges of the field maps where
the field is nonzero. There, the mis-estimate is of fixed magnitude so the error decreases
only as δt, which dominates as δt tends to zero.
The timestep of 10−11 seconds (0.01 ns) has been chosen as the default for Muon1
as this gives position errors of 25 µm in the first test and below 10 µm in the second,which are comparable to magnet alignment tolerances. The velocity errors decrease instep with the positional ones, demonstrating good convergence in all parameters. Energy
conservation in magnetic fields is not enforced other than by the structure of the Lorentz
force law, so small errors in particle energy occur with Runge-Kutta tracking but decreasein the same manner as the velocity errors.

2.2.3

Multi-Particle, Full Beamline Error Study

The previous section studied the error behaviour of a single particle trajectory, whereas
there must be thousands of particles in a simulation if yield figures are to be statisticallymeaningful. The problem with running a numerical algorithm on a large number of inputs(initial particle positions here) is that it drastically increases the probability that one ofthose inputs encounters a pathological case, perhaps near a boundary, discontinuity or
singularity in some function. Therefore we cannot believe the convergence behaviour is‘good’ without also checking it for an entire particle beam.
A particular decay channel setup (roughly 30 m of beamline) was run for a pion input
file consisting of 20936 particles; on one simulation 8963 particles got to the end of thechannel. This number may include multiple decay products from a single particle, as
discussed in section 2.4.1 and will vary slightly due to the random nature of the decays
unless a fixed seed is used. Simulations were run for 120 ns, which is enough time for the
majority of particles of interest to reach the end of the channel and then the distributions
compared for timesteps of 5, 10, 15 and 20 ps (recalling that 10 ps was recommended from†

Muon1 tracking of this ring showed the optics to be unstable, which is why a full turn is not usedand no cooling ring appears in this thesis.

21the single particle work), giving the results shown in table 2.1.
Table 2.1: Distance errors between particles in pairs of beam distributions produced withdifferent choices of Runge–Kutta timestep in the decay channel simulation.
Pair of timesteps compared10 ps, 5 ps
15 ps, 5 ps
15 ps, 10 ps
20 ps, 5 ps
20 ps, 10 ps
20 ps, 15 ps

Maximum error (m)0.002339
0.017983
0.015655
0.075524
0.073302
0.058233

RMS error (m)0.000452
0.003466
0.003015
0.014397
0.013971
0.011084

Mean error (m)0.000266
0.002050
0.001784
0.008522
0.008268
0.006550

The error measure between two entire distributions can be computed in a number ofways. Here, the positions of the particles in one distribution were compared with theircounterparts in another, yielding a list of distances. The maximum, root mean square(RMS) and mean of these distances were calculated for lists coming from each possible
pair of distributions. As the method being used is fourth order, in an ideal case eachmeasure of error would be expected to follow the formula
Error = K|δt41 − δt42 | = Kδt4eff .
Here the ‘effective timestep’ has been defined as1

δteff = |δt41 − δt42 | 4 ,which is the timestep expected to have the same error from the true solution (δt → 0) as
the two compared finite timesteps have from each other, assuming ideal, linearly additive
fourth order error scaling. δteff may be plotted against the three error measures and powerlaw curves fitted as shown in figure 2.5.
These errors would be large for some accelerator applications but the comparison point
for the muon front end is that the solenoids have an inner radius typically of the orderof 30 cm. Thus the sub-millimetre RMS error at δt=10 ps and maximum error of 2.3 mm
are good enough to evaluate the transmission. Note that the maximum error is always
much larger than the averaged errors because it picks the ‘worst’ particle, with significant
chances of initial trajectory errors being amplified in (for example) the rapidly-changing
fringe fields of the solenoids.
The fact that the exponents in the power laws fitted to the graph are much closer to
5 than to 4 is a puzzle. It is possible that errors are not accumulating as δt5 (T /δt) ∝ δt4
but instead cancelling for most of the channel as the particle oscillates from one side to
the other, leaving only a fixed number of significant erroneous timesteps (δt5 N ).

22

Figure 2.5: Power laws fitted to errors between tracking results for an entire beam under
different timesteps.

23

2.3

Magnetic Fields

The tracking algorithm described in sections 2.1 and 2.2 requires the magnetic field B(x)to be evaluated at essentially arbitrary points: the Runge–Kutta evaluation nodes neareach particle on any given timestep. There are several ways of providing this information
on a computer, the following sections describe an algorithm for computing the field of
solenoids and a field map representation for general magnets, requiring external datafrom a magnet design program.

2.3.1

Off-Axis Series Solution

A useful technique for calculating cylindrically-symmetric magnetic fields (expressed asfunctions B z (z, r) and B r (z, r)) is that if the field on the axis B z (z, 0) is known, Maxwell’sequations in free space can be used to solve for a power expansion of the field in r up to
the first conductor or object:z

B (z, r) =∞X

Bnz (z)rnr

B (z, r) =

n=0

∞
X

Bnr (z)rn .

n=0

The functions Bnz (z) and Bnr (z) are to be determined for n ≥ 0. Putting r = 0 in theseequations makes rn = 0 for n ≥ 1, leaving only the r0 (n = 0) terms:
B z (z, 0) = B0z (z)B r (z, 0) = B0r (z).

The B r component is the magnitude of the field pointing radially outwards but no suchdirection exists at r = 0, forcing B r (z, 0) = 0 to ensure continuity. Both Bnz (z) and Bnr (z)are now determined for n = 0.
For a static magnetic field in free space, Maxwell’s equations simplify to ∇ · B = 0,
∇ × B = 0. Let ez and er be the unit vectors in the z and r directions at any point inspace. Taking first the scalar equation,
0 =
=
=
=

∇·B
∇ · (B z ez + B r er )∇B z · ez + B z ∇ · ez + ∇B r · er + B r ∇ · er∂z B z + B z ∇ · ez + ∂r B r + B r ∇ · er .

Note that ∇ · ez = 0 and ∇ · er = 1/r. The series expansion may be substituted in to get
0 = ∂z B z + ∂r B r + 1r B r
∞
∞
∞
X
X
X
z0
n
r
n−1
=
Bn (z)r +Bn (z)nr+
Bnr (z) 1r rn
=

n=0

n=0

∞
X

∞
X

n=0

Bnz 0 (z)rn +n=0

Bnr (z)(n + 1)rn−1 ,n=0where 0 is used to represent the derivative for a function of a single variable.
24

As the previous equation is true for all r, coefficients in rn may be equated:r
0 = Bnz 0 (z) + Bn+1(z)(n + 2)
−1r
⇒ Bn+1(z) =Bnz 0 (z)n+2

for n ≥ 0.

This is one half of a recurrence relation to obtain the Bnz,r functions for n > 0. The otherhalf comes from the magnetic vector equation:
0 = ∇×B= ∇ × (B z ez + B r er )= ∇B z × ez + B z ∇ × ez + ∇B r × er + B r ∇ × er .
Here, ∇ × ez = ∇ × er = 0. The cross product will select only the perpendicular partsof the gradient, so ∇f (z, r) × er = ∂z f (z, r)eθ and ∇f (z, r) × ez = −∂r f (z, r)eθ . Thus
0 = ∇B z × ez + ∇B r × er
= −∂r B z eθ + ∂z B r eθ
!
∞
∞
X
X
Bnr 0 (z)rn eθ .
Bnz (z)nrn−1 +
=
−
n=0

n=0

This is now in a form to equate coefficients in rn eθ , givingz
0 = −Bn+1(z)(n + 1) + Bnr 0 (z)1z⇒ Bn+1(z) =Bnr 0 (z)n+1

for n ≥ 0,

which is the other half of the recurrence relation. The relations zig-zag between the r andz functions as n increases.
Using these rules repeatedly,= B0r (z) = B1z (z) = B2r (z) = B3z (z) = · · ·= − 12 B0z 0 (z)= − 21 21 B0z 00 (z)= 14 12 21 B0z 000 (z)···(−1)n z (2n)zB(z)
B2n (z) =22n n!2 0(−1)n+1
r
B2n+1
(z) =
B z (2n+1) (z).
22n+1 n!(n + 1)! 0
0r
B1 (z)
B2z (z)B3r (z)Thus the full field can be calculated up to the radius of convergence of the series, provided
that the nth derivatives of the on-axis field function B0z (z) are also computable.

25

Figure 2.6: Geometry of the current loop and sheet solenoid field evalulation.

2.3.2

Axial Field of a Sheet Solenoid

The Biot-Savart Law gives the magnetic field generated by a loop carrying current I = |I|
as
I
I×r
µ0
ds,
B(x) =
4π|r|3where r is the vector from the point on the loop to x, I is the vector current (i.e. withdirection parallel to the curve) and ds is the element of arc length.
Suppose there is a circular loop of radius R in the x–y plane, centred on the origin.
The field value at the location (0, 0, z) is of interest, because the on-axis field of the ‘sheet
solenoid’ (of zero thickness but finite length) can be constructed by integrating the fields
of many such loops. Parameterising s(θ) = −R(cos θ, sin θ, 0) = −R er for θ ∈ [0, 2π]
gives r(θ) = R er + z ez and ds = R dθ. The vector current is I = −I eθ . ThereforeZµ0 2π −I eθ × (R er + z ez )B(0, 0, z) =
R dθ
4π 0
(R2 + z 2 )3/2
Z 2π
µ0 IR
−eθ × (R er + z ez ) dθ=4π(R2 + z 2 )3/2 0
Z 2π
µ0 IR
=
R ez − z er dθ4π(R2 + z 2 )3/2 0µ0 IR=
2πR ez4π(R2 + z 2 )3/2
µ0 IR2=ez .
2(R2 + z 2 )3/2
For a sheet solenoid of the same radius occupying the interval [a, b] in the z-axis, theon-axis field is the integral of the above expression:Z z−aµ0 I` R2z
B0 (z) =dx
2
2 3/2
z−b 2(R + x )Zµ0 I` R2 z−a
1
dx,
=
2
2 3/2
2
z−b (R + x )26

where x is the offset of the field evaluation point from each infinitesimal current loop andI` is the current per unit length in the sheet. Integrating gives the final formula
B0z (z)z−aµ0 I` R2x√=
2
R2 R2 + x2 x=z−b
µ0 I`
=
2

z−a

z−b

p
−p
R2 + (z − a)2R2 + (z − b)2!
.

Expression for the nth Derivative
The axial field and its derivatives may be expressed more succinctly as followsµ0 I`(f (z − a) − f (z − b))2µ0 I` (n)f (z − a) − f (n) (z − b) ,B0z (n) (z) =2√using the function f (x) = x/ R2 + x2 . To find f (n) (x) it simplifies matters to realisethatpn (x)√
f (n) (x) =2(R + x2 )n R2 + x2
B0z (z) =with pn (x) a polynomial and p0 (x) = x for consistency in the n = 0 case. The recurrence
relation for these polynomials is extracted in the following way.
d (n)
f (x)
dx 

d
pn (x)√=
dx (R2 + x2 )n R2 + x2−(n + 12 )pn 0 (x)√√=+ pn (x)2x(R2 + x2 )n R2 + x2
(R2 + x2 )n+1 R2 + x2
(2n + 1)xpn (x)pn 0 (x)√√−.
=(R2 + x2 )n R2 + x2 (R2 + x2 )n+1 R2 + x2

f (n+1) (x) =pn+1 (x)√R 2 + x2

(R2 + x2 )n+1√
Multiplying both sides by (R2 + x2 )n+1 R2 + x2 gives
pn+1 (x) = (R2 + x2 )pn 0 (x) − (2n + 1)xpn (x).
A computer would work out the coefficients of these polynomials once per solenoid byfilling in a two-dimensional array, then calculate f (n) and hence B0z (n) (z) to be used in theoff-axis series found in section 2.3.1.
For additional speed, Muon1 has a version of this technique up to and including n = 3(cubic terms) hard-coded into its solenoid field evaluation function. The off-axis series
will not converge beyond the radius of the solenoid and high order terms beyond cubic
extend the accurate field region by successively smaller amounts.
27

2.3.3

Field Maps

Sometimes a magnetic (or electric) field cannot be represented analytically, or perhaps
the analytic series defining it is prohibitively time-consuming to evaluate. In these cases
it is effective to precalculate a table of values of the field at certain points. More advanced
techniques such as spline interpolation are also possible, improving accuracy at the expenseof computation time.
Currently Muon1 includes the method of ‘trilinear interpolation’ on a regular cuboidal
grid. Let the field map have grid spacings of δx, δy and δz in the three axes and containvalues of B = (B x , B y , B z ) for locations from (0, 0, 0) to (P δx, Qδy, Rδz) inclusive. Thisxtorequires 3(P + 1)(Q + 1)(R + 1) numeric values in total, which shall be labelled B000x
BP QR for the x components and similarly for y and z.
Let the field value be required at (x, y, z) in coordinates relative to the field map,
which may be a rotated or displaced component in the accelerator frame. This point is in
the cuboid between BX0 Y0 Z0 and BX1 Y1 Z1 where X0 = bx/δxc, X1 = dx/δxe and similarly
for Y and Z. Trilinear interpolation derives the field at (x, y, z) from the field values atthe eight corners of this cuboid, but is best explained by starting with a simpler case.
Simple (uni-)linear interpolation approximates the field value along one of the edgesof our cuboid in the following way:B(x, Y δy, Zδz) ' (1 − p)BX0 Y Z + pBX1 Y Z .
The fractional value p ∈ [0, 1] is defined as (x − X0 δx)/δx so that it equals zero at BX0 Y Z
and tends to one when approaching BX1 Y Z (we only allow it to equal one when at the topedge of the last cell in the field map). So the field is approximated as a linear function inx that varies smoothly between the nodes. If we consider this equation for Y = Y0 and Y1
in turn, then linearly interpolate on the y-axis between those points, we get the bilinear
approximation
B(x, y, Zδz) ' (1 − q)((1 − p)BX0 Y0 Z + pBX1 Y0 Z ) + q((1 − p)BX0 Y1 Z + pBX1 Y1 Z )on the x–y planes of the grid, where q = (y − Y0 δy)/δy. The final step should now be
obvious, we interpolate in z between any point on two adjacent x–y planes, setting Z toZ0 and Z1 :B(x, y, z) ' (1 − r)((1 − q)((1 − p)BX0 Y0 Z0 + pBX1 Y0 Z0 ) + q((1 − p)BX0 Y1 Z0 + pBX1 Y1 Z0 ))+ r((1 − q)((1 − p)BX0 Y0 Z1 + pBX1 Y0 Z1 ) + q((1 − p)BX0 Y1 Z1 + pBX1 Y1 Z1 )),with r = (z − Z0 δz)/δz. This is the formula for trilinear interpolation as used in Muon1.
It may not be obvious from this expression that the order in which the interpolations weretaken (here x then y, then z) does not affect the final result. The symmetry is revealed
by multiplying out into individual terms
B(x, y, z) ' p0 q0 r0 BX0 Y0 Z0 + p0 q0 r1 BX0 Y0 Z1 + p0 q1 r0 BX0 Y1 Z0 + p0 q1 r1 BX0 Y1 Z1+ p1 q0 r0 BX1 Y0 Z0 + p1 q0 r1 BX1 Y0 Z1 + p1 q1 r0 BX1 Y1 Z0 + p1 q1 r1 BX1 Y1 Z1 ,where p0 = 1 − p and p1 = p, with analogous definitions for q and r. All eight possible binary combinations of subscripts (corresponding to the cuboid vertices) are used togenerate all the terms, without preference for a particular axis.
28

Figure 2.7: (Left) Plot of the on-plane magnetic fields in a coarse (10 cm grid spacing)
field map, where the field vectors are perpendicular to the plane of the diagram and thesigned magnitude is shown (scales are in Tesla). (Right) Plot of the error bound derived
from second derivatives for each square in this plane. The desired path of the particles isfrom the bottom-left of the view to the top-right, through the regions of low error.
If x < 0, x > P δx, or a similar situation on the y- or z-axes, then the point is outside
the range of the field map and B = 0 is returned. This means one must be careful
not to introduce discontinuities by having significant field magnitudes at the edges whereparticles may enter or leave the field map, because these can degrade the convergence ofthe tracking method (as discussed in section 2.2.2). However, Muon1 does permit multiple
field maps (or other magnetic elements) to overlap with each other, in which case the fieldvalues from the different magnets are added together to give the total field. This field will
still be physically valid because Maxwell’s equations in free space are linear in B (this is
known as the ‘superposition principle’ for magnetic fields).
The accuracy of linear interpolation depends on the second and higher derivatives of
the B field and the size of the mesh cells. It is always true that if there are small featuresin the field that fall in between grid sampling points, they will not be approximated well,so we assume that the grid is fine enough for the error to be dominated by the leading termin the Taylor series that is different between the approximation and the actual function.
Note that linear interpolation is exact for linear functions, so the terms of interest will bequadratic. Furthermore, for trilinear interpolation, even the quadratic mixed terms xy,
xz and yz are approximated exactly because they are linear in x, y and z individually ateach stage of the multidimensional interpolation procedure given above. So the quadratic
terms of interest are in general:Q(x, y, z) = qx x2 + qy y 2 + qz z 2 ,where the q are arbitrary vector coefficients. The three terms above will be dealt withseparately by the algorithm as it interpolates along each axis in turn. The error for x2 will
be a quadratic with the same leading coefficient but the linear interpolation subtracted
so that it equals zero at the neighbouring points X0 δx and X1 δx. There is only one such29

function, (x − X0 δx)(x − X1 δx), so the leading term error, Error(x, y, z), isqx (x − X0 δx)(x − X1 δx) + qy (y − Y0 δy)(y − Y1 δy) + qz (z − Z0 δz)(z − Z1 δz).
This becomes more informative when we note that, considering the local Taylor series of2
B truncated to quadratic order, qx ' 21 ∂∂xB2 (x, y, z). Also we are only interested in valuesof x ∈ [X0 δx, X1 δx], an interval of length δx with a root of (x − X0 δx)(x − X1 δx) at
either end. The extremum of this function is at the centre of this interval, 21 δx distantfrom either root and the function has value − 14 δx2 there. Thus if cubic and higher terms
can be neglected, we may bound the error anywhere in the cuboid:


∂ 2B
∂ 2B
1 ∂ 2B222|Error(x, y, z)| ≤
δx +
δy +
δz .
8
∂x2
∂y 2
∂z 2
The results of applying this formula to a field map as used in the Rees bending chicane [37] simulation is shown in figure 2.7. Large errors exist around where the field was
sampled inside iron pole pieces of the magnet, but the errors are smaller (1% level) along
the path that the particles are intended to traverse. The manifestation of field errors
as particle position errors is dependent on the optics of the beamline in question, there
is no good general-purpose
R error bound. More favourably, total bending is proportional
to the integrated field B d` along the particle trajectory, so there is some scope forerror cancellation because negative and positive convexity regions, leading to under- andover-estimation in the field components, tend to alternate along the path.

2.4

Particle Decays

The muon front end is unusual in that the particles being transmitted (initially π ± ) will
decay (mainly into µ± ) on a timescale comparable to how long it takes them to traverse
parts of the accelerator, specifically the decay channel section. The mean lifetime of astationary charged pion is τπ± = 26.033 ns [27], giving a typical 250 MeV pion a mean
path length of vγτ = 20.338 metres.
In the rest frame of the particle, decay is quantum mechanically a perfect Poissonprocess. Therefore instances of actual decay times, measured as proper times since theparticle’s creation, can be generated using a probability distribution such that P (t) ∝e−t/τ . The cumulative distribution function for this is
F (t) = 1 − e−t/τ ,(t ≥ 0)which can be inverted to giveF −1 (x) = −τ ln(1 − x).

(0 ≤ x < 1)

The reason F −1 is useful is that taking uniformly distributed numbers x ∈ [0, 1) from thecomputer’s (pseudo-)random number generator will ensure that t = −τ ln(1 − x) has thedesired distribution for the decay times [38]. These times are generated when the particle
starts and compared to the particle’s internal proper time clock, which is incremented byδt/γ each timestep of the simulation, to determine when a decay is triggered.
30

2.4.1

Statistical Weighting and Multiple Decays

Chapter 4 describes how Muon1 takes output from a hadronic particle showering code
as initial data for its tracking. The (macro-)particles in this sort of output are often‘weighted’ according to the relative probability of each being produced, so these weights
must be retained during tracking and used correctly in evaluating the muon transmission.
For example, if each particle i has weight wi ,X
wiTransmitted
Tµ
Xi
.
= 100
% Muon transmission = 100 muons
Wµwi
Muons iIf the indices i are permuted in a random order, then the transmission of muon i is arandom event with probability p ' Tµ /Wµ (strictly this ratio is only an estimate of thetrue value of p). Thus the weight transmitted by this muon is wi with probability p and
0 with probability 1 − p, giving a mean of pwi and variance pwi2 − (pwi )2 = p(1 − p)wi2 .
Since variances add linearly, the variance of the transmitted weight isVar(Tµ ) =Xp(1 − p)wi2 'Muons iTµ (Wµ − Tµ ) X 2
wi ,Wµ2
Muons iwhich gives the standard deviation of the muon yield from tracking (ignoring pion production uncertainties on Wµ ) asps X
Tµ (Wµ − Tµ )σ%muons ' 100
wi2 .
Wµ2
Muons iFor fixed transmission and total muon weight, the final term is made smaller by having
more macro-particles with smaller weights. Many pions (roughly three quarters) are lostin the very early stages of the simulation before they have time to decay, leaving relatively
few pions later on. If each pion with weight w is instead treated as N pions with weight
w/N , this will produce a greater number of decay products (muons) with which to evaluate
the transmission. However, as the original pion will remain on the same trajectory unless
it decays, there is no reason to actually track N copies of it, so Muon1 treats it as apion with weight w decreasing to NN−1 w, NN−2 w, etc. each time it decays. This does notincrease the effective number of particles in the simulation by a full factor of N because
the number of original pions is still fixed, but it does help, especially when working underan optimisation in which small differences in yield can give clues as to the direction of
better designs.

2.4.2

Two-body Algorithm

The original implementation of particle decays in the Muon1 tracking code only admitted
two-body processes. These can be dealt with by assuming the rest frame of the parent
particle and producing two decay products with equal and opposite isotropically-selected31momenta and magnitude such as to conserve energy. This has now been replaced by themore general many-body decay algorithm described in section 2.4.3.
Even the simpler algorithm, however, is not free of implementation details. To transform the decay products’ momenta back to the laboratory frame would be easy in one
dimension, by Lorentz boosting the energy-momentum vector by the velocity v = βc:

 

 

Et
γ βγ
Et
γEt + βγpc
Λ(β)
=
=
.
pc
βγ γ
pc
γpc + βγEt
Unfortunately, the decaying particle will not be travelling along a single axis and theboost formula for a general vector velocity v = βc is required. To avoid dealing withmultiple rotation matrices, the momentum can be decomposed into a part perpendicularto β, which remains unchanged, and a parallel part that undergoes the transformation:p = pk + p⊥

where

pk =

p·β
β
β2

and

p⊥ = p −

p·β
β.
β2

Here, β = |β| and γ is defined from this scalar beta. Letting only pk be transformed, theone dimensional equivalent of p is (p · β)/β, which satisfies pk = p(β/β). The result of
the general Lorentz transformation is then γEt + γ(p · β)c
Et
=
,
Λ(β)
pc
γpk c + βγEt + p⊥ c
where terms like βγEt from the one-dimensional formula have been multiplied by β/β togive them the correct direction.
All that remains is to calculate the momentum of the decay products in the rest frameof the parent particle. Suppose they have masses m1 and m2 and will acquire equal andopposite momenta of magnitude p on decay. As momentum is conserved by construction,
the one condition is energy conservation:pp
mc2 = (m1 c2 )2 + (pc)2 + (m2 c2 )2 + (pc)2 .
√√This may be solved for p using the “Q = R + S” algorithm described in section 2.4.4.
A randomised vector direction p̂ is needed to generate the momenta p1 = pp̂ andp2 = −pp̂ in the parent particle’s rest frame. In Muon1, p̂ is generated uniformly on thesurface of a unit sphere by taking random points x within the cube [−1, 1]3 until one falls
within the unit sphere (i.e. |x| ≤ 1) and then returning p̂ = x/|x|.

2.4.3

Many-body Decay Algorithm

The problem of an n-body decay imposes four constraints on the 3n degrees of freedom
found in the momentum of the decay products: three for overall conservation of momentum and one for energy. The final states can therefore be found on a (3n − 4)-dimensional
manifold in the multi-particle phase space. This is two-dimensional for two-body decays(and can be made into a sphere by the method and symmetry argument above), but even32for three-body decays it is already five-dimensional and rotational symmetry can remove
at most three of these.
Faced with this problem of a complicated surface to generate the distribution on, aMonte Carlo routine that covers the right surface with points (at some known density)
is obtained first and then the density is corrected afterwards by rejecting a calculated
proportion of them and re-running the original routine as necessary (this is known asrejection sampling). Formally, the first routine generates the primal probability density
F (pi ) on the feasible manifold, then any other desired distribution f (pi ) can be derived
from this by only accepting each primal point with probability ∝ f (pi )/F (pi ). Notethat this should not be larger than 1, so the ratio must be bounded above and scaled
accordingly.

2.4.4

The Probability Density (spin averaged)

From the 2004 Particle Physics Booklet, section 38.4, the probability of a particle with 4momentum P and randomly-selected spin decaying into n particles of 4-momenta p1 , p2 . . . pn
is governed by4P ∝ δ (P −= δ(E −

n
X

pi )

i=1
n
X

n
Y
d3 pi
i=1

Ei

Ei )δ 3 (P −

i=1

n
X
i=1

pi )

n
Y
d3 pi
i=1

Ei

.

To simplify the derivation, in this and the next subsection only, the E and Ei will always
be total energies (Et elsewhere) and units will be used such that c = 1.
The delta functions can be multiplied into the main coefficient using the following rule,
where f is a function with one root f (a) = 0:
δ(f (x))g(x) =g(a)δ(x − a).
|f 0 (a)|Applying this three times, considering the probability expression as a function of pxn , pyn
and pzn in turn, reduces it to:
"
#
!
n
n
n−1
X
Y
X
d3n P
1
∝ δ(E −Ei )
δ 3 pn − (P −pi ) ,
d3 p1 d3 p2 . . . d3 pn
E
i
i=1
i=1
i=1
which when integrated over pn givesnnXYd3n−3 P1
∝ δ(E −
,
Ei )
3
3
3
d p1 d p2 . . . d pn−1
Ei
i=1
i=1

after which the substitution pn = P −momentum and En .

Pn−1

i=1 pi33

should be used to determine the lastThis needs to be integrated one more time to remove
the last delta function, so aPn−2
degree of freedom will be taken from pn−1 . Let p∗ = P − i=1 pi so that pn−1 + pn = p∗
to conserve 3-momentum. Now perform a Lorentz transformation into a primed frame
where p0∗ = 0, so that p0n−1 = −p0n = pp̂. Integrating over p ≥ 0 will remove the delta
function, leaving the two degrees of freedom in p̂ parameterising the remaining freedomof these last two momenta.
Starting with the expression written in the primed frame:nnXYd3n−3 P100∝ δ(E −
Ei )
0
0 3 0
3
3
d p1 d p2 . . . d pn−1
Ei0
i=1
i=1

...change p01 through p0n−2 back to the original frame using d3 p0i = (Ei0 /Ei )d3 pi :
n−2
n
Y 1
X
d3n−3 P
1
0
0
∝
δ(E
−
E
)
i
0
En0 i=1 Ei
d3 p1 d3 p2 . . . d3 pn−2 d3 p0n−1
En−1
i=1

...and then use the substitution, which implies d3 p0n−1 = p2 dpd2 p̂, to get:nn−2Xp2 dp Y 1
d3n−4 P00
∝ δ(E −Ei ) 0.
d3 p1 d3 p2 . . . d3 pn−2 d2 p̂
En−1 En0 i=1 Ei
i=1Now that varying p is the concern, a “constant” part may be removed from the argument of the delta function
0

δ(E −

n
X

0
Ei0 ) = δ(E∗0 − (En−1+ En0 )),

i=1

where E∗0 = E 0 −Pn−20i=1 Ei . The modulus of the derivative of this argument:

d
d
0
(En−1
+ En0 ) =
dp
dp


q
p
p2 + m2n−1 + p2 + m2n

p
p
+p
= p
2
2
2
p + mn−1
p + m2n


0
En−1
+ En0
1
1
= p
=
p
,
+
0
0
En−1
En0
En−1
En0
...must be divided out when the argument of the delta function is linearised:n−2
Y 1
d3n−4 P
pdp
∝
δ(p
−
p
)
0
0
d3 p1 d3 p2 . . . d3 pn−2 d2 p̂
En−1 + En0 i=1 Ei

=

n−2
Y 1
p
0
En−1 + En0 i=1 Ei

with p0 being the value of p ≥ 0 for which energy is conserved.
34

,
p=p0

p0 is the root of the delta function, so
q
q
0
0
0
2
2
E∗ = En−1 + En = p0 + mn−1 + p20 + m2n ,which is of the form
⇒
⇒
⇒
⇒

√
√
Q= R+ S√√Q− R= S√
Q2 − 2Q R + R = S√
Q2 + R − S = 2Q R
(Q2 + R − S)2 = 4Q2 R.

(∗)
(∗)

This may be used on the expression now, but note that in the squaring operations marked(*) the implication goes only one way, so any value of p0 derived using the following must
be checked back to make sure the squaring operation did not equate things that originally
had opposite signs! The squared-up expression is2

2(E∗0 + m2n−1 − m2n )2 = 4E∗0 (p20 + m2n−1 ),giving the root explicitly as
s
p0 =

2.4.5

(E∗0 2 + m2n−1 − m2n )2− m2n−1 .
2
0
4E∗

The Primal Distribution

The previous section gave the probability distribution with respect to an element of the
momenta p1 . . . pn−2 and the unit direction p̂ governing the last two particles. Thus if a
uniform distribution is used for the n−2 momenta and an isotropic one for p̂ in the primal
distribution, the (3n − 4)-dimensional probability density will be exactly proportional to
the selection probability.
Generating an isotropic p̂ is easy; what is not so obvious is how to choose a ‘domain’in which the pi are uniformly spread. As |pi | ≤ Ei ≤ E = m0 (i.e. the mass of theparent particle), there is at least a crude limit on how large the momenta can be. Herenote that many of these choices for p1 . . . pn−2 miss the manifold entirely, a fact whichbecomes apparent when attempting to enforce conservation of energy leads to particles
with negative energy, or taking square-roots of negative quantities. In these cases, thesample should be rejected, so in effect there are two levels of rejection sampling, one toget onto the manifold and a second to adjust the density. The first level does not affectthe relative proportions between any of the probabilities within the manifold, so has noadverse consequences other than wasted CPU time.
The final ingredient for a workable implementation of rejection sampling is an upperbound on the probability density derived in the previous section, so that it may be divided
out to give selection probabilities of no more than 1. By noting that Ei ≥ mi ⇒ 1/Ei ≤1/mi , one obtains
n−2
n−2
Y 1
Y 1
p
p
≤
,
0
0
En−1
+ En0 i=1 Ei
En−1+ En0 i=1 mi35

leaving only the front term to deal with. Remembering that |p0n−1 | = |p0n | = p, it follows
0that En−1≥ p and En0 ≥ p so that the denominator is at least 2p, resulting in
n−2
n−2
Y 1
p
1Y 1
≤
.
0
En−1
2 i=1 mi+ En0 i=1 Ei

The probability to use is therefore
n−2
Y mi
2p
P1 = 0.
En−1 + En0 i=1 EiThis is workable for several cases, but when there are particles of small mass (such as
electrons or neutrinos) in the decay products, the ratio mi /Ei = 1/γi becomes very small
when these acquire a significant fraction of the mass–energy of the heavier parent particle.
The formula does not select any samples at all when photons are produced! The problem
arises from the choice of a uniform momentum distribution for each of the first n − 2particles not being a very good ‘fit’ to the 1/Ei in the probability density, particularly
when the energy is dominated by momentum rather than rest mass.
pWith this in mind, the bound can be modified to use the fact that Ei = p2i + m2i ≥max{|pi |, mi } so 1/Ei ≤ min{1/|pi |, 1/mi }. However, this is no longer a constant withrespect to pi , so to have it divided out, a primal distribution must be generated withF (pi ) ∝ min{1/|pi |, 1/mi }. This is uniform for |pi | ≤ mi , so retaining the previous cutoff
of |pi | ≤ m0 , the ratio of the uniform part to the outer part isZ m0
d|pi |
m2 m2 − m2i
4
3 1:
4π|pi |2πmi
= i : 0
.
3
mi
|pi |
3
2
mi
Since 2|pi |d|pi | = d|pi |2 , the outer part corresponds to choosing |pi |2 uniformly on[m2i , m20 ]. Generating events from these two parts in the ratio above, for each i up ton − 2, gives the required primal distribution.
Now dividing out min{1/|pi |, 1/mi } instead of just 1/mi gives the new selection probability asn−2
Y max{|pi |, mi }2p
P2 = 0,En−1 + En0 i=1
Ei√which is clearly better conditioned, as terms in the product part never fall below 1/ 2.

2.4.6

Results and Efficiency

The two versions of the algorithm described above were implemented and tested on avariety of decay modes for the kaon: the results are shown in table 2.2. Note that for
3-body and higher decays, there is sometimes a choice of which particles to associate the‘special’ momenta pn−1 and pn with. This affects the speed of the algorithms, particularly
the first version, so these possibilities are broken down individually in the cases where theparticles have different mass.
36

Table 2.2: Mean number of primal samples required to generate one valid decay. Figures
are an average over 10000 successful decays except where another number of decays is
indicated in parentheses. ‘inf’ denotes that it was infeasible to run the algorithm, as no
valid decays were generated after 107 samples.
Decay (special pair)
K + → e+ νe
K + → µ+ νµ
K + → π 0 e+ νe(π 0 , e+ )(π 0 , νe )(e+ , νe )K + → π+π0π0(π + , π 0 )(π 0 , π 0 )+K → π+π+π−
K + → π + π − e+ νe(π + , π − )(π, e+ )(π, νe )(e+ , νe )Using P1
1
1.0483Using P2
1
1.0485inf
6.74994123.71 (2425) 6.5441
15.74196.4251151.285
146.47184.18160.5826
61.5364
75.1907inf
inf
8 × 105 (25)3746.72 (2669)

666.969652.936654.098
636.432

The second version of the algorithm using P2 displays a much more consistent speed
across decays with the same number of daughter particles, with no decays in the test set
it could not solve. Time increases rapidly with number of daughter particles but since nomore than four are required for the particles in Muon1 and each calculation is fairly fast,the ∼650 iterations needed for 4-body decays are not a major problem.

2.4.7

Dalitz Plot Test

A standard plot that can be used to test the correctness of the routine is the Dalitz plot ofm212 against m223 , where m2ij = (mi +mj )2 −|pi +pj |2 . The correct phase-space distribution
ought to be uniformly spread on the allowed region when projected into this plane. Sucha plot is shown in figure 2.8 generated from many runs of the P2 -based version of thealgorithm.
The uniformity of the enclosed area is visually apparent, as is the sharpness of theedge in phase space of feasible decays. This provides additional confidence that no errors
have been made in the coding of this mathematically complicated routine.

37

Figure 2.8: Dalitz plot for the K + → π + π + π − decay, showing uniform coverage of theallowed area.

38

2.5

Collision Detection

When a particle goes outside the accelerator aperture, for instance into the coil of asolenoid, it is removed from the simulation and counted as a loss† . The code should be
able to tell, similarly, when a particle has entered a region of scattering material or passed
through a surface at which the beam is being logged or an aperture restriction imposed.
Proximity detection is also needed to determine which components contribute significantly
to the magnetic field at a given point in space. More complex collisional algorithms areneeded for the scattering calculation itself, but this is presented in sections 6.4 and 6.5.

2.5.1

Proximity and Bounding Boxes

For tracking codes that step in z, it is often unambiguous which component a particle isinside, so the field and aperture restriction can be applied directly. For codes that stepin t there will be particles at a range of z that may be in different components. The
simplest method would be to check against every component to determine the field andaperture set (the set containing all allowed locations for particles):# Components

B(x) =X# Components

BComponent n (x)

n=1

A=

\

AComponent n .

n=1

A direct implementation of this on a computer unfortunately adds a calculation of length(# Particles)(# Components) to each timestep. This is bad scaling behaviour in the
majority of situations where only a few components have any significant effect for eachparticle. Note that the superposition principle used for the magnetic fields here means thatcomponents containing saturating iron or other nonlinear materials must not be placed
too far into the fringe fields of others; a solution is to treat ‘interacting’ objects as a single
contiguous component using a field map. Note also that a collision is defined as a particle
moving outside the allowed aperture A and hence into its complement Ā = R3 \A, whichmeans that if the effects of each component are to be localised, the ‘solid’ collision zoneĀ must be bounded, as opposed to some codes (particularly in z) in which the disallowed
region may extend to infinity.
The efficiency problem mainly hinges on the detection of proximity to components.
Once the set of nearby components has been determined, only the B fields from theseneed to be summed and only collisions with these components need to be checked for. If
this detection is performed by a scan of all components on every timestep, it also risks bad
scaling behaviour. However, the calculation that is being repeated is now just proximity
detection rather than the full collision and field calculation; it is therefore desirable to
make this as fast to compute as possible. The approach taken is of bounding cuboids C
constructed such that
{x : |B(x)| ≤ B } ∪ Ā ⊆ C = [x1 , x2 ] × [y1 , y2 ] × [z1 , z2 ],†In reality, some particles (particularly muons) are not absorbed and can scatter back into the aperture
but Muon1 does not include this possibility. This makes a slightly pessimistic error in yield, but theprobability of a muon that scatters out of the walls rejoining the beam for an extended period is minimal.

39

so only 6 comparisons (and possibly fewer) are required to determine whether a point(x, y, z) is within C, which contains all of the collision zone and significant field region.
Speed-of-Light Optimisation
An additional optimisation exists so that the scan for nearby components is only necessary
on every N th timestep, although there is some tradeoff. If a list of nearby components is
to be valid for N timesteps (a time N δt) from the time it is constructed, it must refer toall components the particle could possibly enter into the bounding cuboid of during thattime. The particle’s trajectory is not known in advance but we would hope that it willremain within the light cone given by ∆x ≤ c∆t ≤ cN δt. The set of all points that canreach C as defined above within time N δt is
ExpandcN δt (C) = {x : ∃y ∈ C, |y − x| ≤ cN δt}⊆ [x1 − cN δt, x2 + cN δt] × [y1 − cN δt, y2 + cN δt] × [z1 − cN δt, z2 + cN δt],so we may just use a larger cuboid, C extended by cN δt outwards in every direction, as avalid bounding box for use on N successive steps. The tradeoff in this optimisation is that
if N is made too large, the cuboids will also become large and time will be spent adding
the fields of (and checking for collisions with) components that are far away from theparticle but could have been reached by it in the time N δt since the nearby component
list was generated.
Proximity Grid Optimisation
The scan can be made even faster by using the principle of a hash table, one of the fastest
known ways of speeding up search procedures, at the expense of some extra memory
usage. A three dimensional cuboidal grid is constructed, in Muon1’s case up to 50 cells
on a side, covering the bounding cuboid of the entire accelerator. In each cell is put alist of the components that have bounding boxes somewhere in the volume of that cell.
Thus when it is time to recalculate the nearby components for a particle, the cell in thegrid containing that particle is looked up and the subsequent search for bounding boxes
restricted to just those components listed in that cell.
The upshot of these optimisations, applied in Muon1 with N = 10 for the ‘speed-oflight’ optimisation, is that the time taken by proximity detection is very small comparedto the time used for Runge-Kutta stepping.

2.5.2

Collision Detection Regions for Components

Muon1 defines collision regions automatically for certain components. The collision zone
used for a sheet solenoid of length L and radius R is by defaultĀsol = {x : 0 ≤ z ≤ L, R ≤ r ≤ 5R},
where (z, r) are x expressed in cylindrical coordinates coaxial with the solenoid, withorigin at the centre of one end. The factor 5 can be changed by the user by specifying the40

OuterRadius attribute of the Solenoid component type to be some length other than
5R, see section 7.1 for how components are input to Muon1 using lattice files.
The other object that is specifically solid in the tracking code is the rod-shaped target,
modelled as a solid cylinder that absorbs particles reentering it:Ārod = {x : 0.005L ≤ z ≤ 0.995L, r ≤ 0.99R}.
The coordinates are as before, but the peculiar 1% shrinkage of the zone is to makesure that particles starting exactly on the surface, which have some rounding in theircoordinates as output from the particle production code, are never erroneously destroyed
at their start location.
In addition to these two rules, Muon1 has a more general system that allows purecollision regions (or regions of material) to be specified separately from particular accelerator components. These are known as Blockage components. As the lattice format
supports arbitrary displacements and rotations of objects from any point in the beamline
and the beamline location may be left constant by not specifying a Length attribute forthese components, complex shapes may be built up from the primitive blockages. The
primitives supported as of Muon1 version 4.44 are cuboids, cylinders, spheroids (including
spheres), annuli and toroids. Cones may have some use in the future.
As well as solid objects it is often desirable to impose a restriction on particles going
through a plane in the accelerator, usually a plane perpendicular to the accelerator axis.
Types of Aperture component were introduced for this purpose, which detect when a particle has crossed the plane, calculate the relative coordinates on the plane of the crossing
point and check against the aperture restriction. Currently only circular and rectangularapertures are supported, although additionally the MinEnergy and MaxEnergy attributes
may be set to stop particles outside a given energy range. Apertures are also used as thefinishing component, which stops all particles on the plane and flags them as ‘finished’(unless they were outside the aperture), which is used in figure-of-merit calculation.

2.5.3

High-accuracy Isosurface Method

Detecting exactly where a particle crosses a surface (an aperture plane or the boundaryof a collision zone) is a common operation in all of the above mechanisms. Initial implementations used a simple method that checked whether the particle was inside a collisionzone of any nearby component on each timestep and stopped it at its current location if
it was. Apertures were dealt with by waiting for the current and old (previous timestep)
positions of a particle to be on opposite sides of the plane and then taking a straight line
between them to find the intersection with the aperture itself.
However these simple algorithms suffer from some problems. The point of using
Runge–Kutta integration was to allow accurate tracking with a medium-sized timestep:for the chosen δt = 10−11 s, cδt ' 3 mm is large enough to jump over the corner of, forinstance, a solenoid’s annular region in a single step (see figure 2.9). This will change
the transmission figure, which is constrained ultimately by the bore sizes, because thebore radius (which has been as small as 15 cm in simulations) is now up to 2% larger
and 4% bigger in area. This becomes worse (up to 8% effect on transmission) when it
41

Figure 2.9: Diagram of how a (slightly exaggerated!) timestep may jump a particle overthe corner of the collision region (red particle) or miss the collision region (green particle).
The dotted line is the axis of symmetry of the solenoid being considered.
is considered that the transverse bores are really clipping the amplitude in 4-dimensionaltransverse phase space. The procedure of taking a straight line to detect the collision with
an aperture plane is also inaccurate, though this is a second order effect in step length sosmaller.
The solution is to detect when collisions with surfaces may have occurred and solve
iteratively for the collision point. To detect collisions reliably for a solid object requires
more than just checking if the particle is in Ā at each timestep. Instead, the solid object’s
surface is defined by smooth isosurfaces that extend outside the object from the corners
if necessary: these are the thin lines shown in figure 2.9. An isosurface is the solution setof an equation f (x) = 0 where f is some smooth function. If the particle has crossed anyof these surfaces during a timestep, which can be easily seen by one of the f s changing
sign, the intersection points are solved for and checked to see if they fall on the surfaceof the collision region or on some part of the isosurface extending into empty space. Thismethod is still not perfect because it will not detect skimming collisions when a particle
curves into and then out of an isosurface during a single timestep, but it improves things
greatly, making missed collisions a second order (curvature) effect.
Solving for the collision point can be done by a number of standard methods. Repeated42bisection would work even for non-isosurfaces but as f is provided, a faster root finding
technique like the secant method can be applied to f (x(t)) = 0. In fact, Muon1 uses a
combination of these two techniques, reverting to bisection if the secant method becomes
divergent. Note that calculating x(t) for a general time in the timestep requires theRunge–Kutta step to be called as a subroutine of the collision point finder. Adding
material scattering complicates the process further and the full stack of stepping routines
is given in section 6.5.

2.6

RF Components and Phasing

Muon1 uses a simplified model for an RF cavity where the acceleration or deceleration
takes place instantaneously as the particle crosses a plane placed where the midplane of
the real cavity would be, usually perpendicular to the beamline. Cavities are defined by
the peak voltage V0 a particle can experience, their angular frequency ω = 2πf and an
absolute phase φ. Note that the V0 specified to Muon1 should already be reduced toaccount for the transit time factor of the cavity, since it is the maximum accelerating
voltage given to a particle rather a definition of the electric field. Muon1 actually takes
input in the form of the frequency f and relative phase φr and derives ω and φ (see below).
If the RF cavity has forward-pointing unit vector n̂ and a particle of charge q andvelocity v passes through the cavity plane at simulation time t with offset ∆x from thecavity centre, the following change in momentum is applied:

ω
qV0 
cos(ωt + φ) n̂ + 2 sin(ωt + φ) v × (∆x × n̂) .
∆p =v · n̂
2c
This is derived by assuming that in a cavity of notional length ` the particle experiences
average electric and magnetic fields ofV0cos(ωt + φ) n̂
`
V0 ω
hBi =
sin(ωt + φ) ∆x × n̂
2`c2
hEi =

and

during a total transit time of `/v · n̂ and using dp/dt = q(E + v × B). The form of hBi is
the solution of Maxwell’s wave equations that has B = 0 on the n̂ axis and is consistent
with the expression for hEi.
The absolute phase φ is calculated from the relative phase φr specified in the Muon1
input files by one of two possible methods. If a Synchronous-particle component has
been defined and inserted at the beginning of the beamline, it is tracked through thesimulation before all other particles, with the time that it crosses each RF plane defining
the relative phase of zero. Absolute phases are then calculated fromωt + φ = ω(t − tcross ) + φr⇒

φ = φr − ωtcross .

If the synchronous particle is lost in the simulation, Muon1 calculates the crossing times
from its initial velocity and the arc lengths of the beamline reference axis. If a synchronous
particle is not specified, absolute phases are set equal to the phases in the input file.
43

Chapter 3
Target Design Considerations
The reference proton beam power used in the ISS [24] and most other neutrino factorystudies is 4 MW. This is quoted as the mean beam power during nominal operation of the
facility, including all inter-pulse gaps (the pulsed power is at the terawatt level). Althoughpowers greater than 4 MW are possible, they would represent an increasingly large leapin proton accelerator technology over that currently demonstrated. Lower powers would
produce so few muons that it would become hard to achieve the 1021 decays per 107 s ofrunning time required for physics results in a reasonable timescale. The figure has perhaps
a factor of two of elasticity in either direction but fixing it is essential to define the timeand cost scale of the baseline design.
Several constraints mean that the concentration of power in the target becomes aconsiderable difficulty at the 4 MW level. Most of the heat deposition in the target
will occur in the first two hadronic interaction lengths (around 20 cm for heavy metals);
however, the radius should be much smaller than an interaction length (a few centimetres)
to prevent pions being reabsorbed as they exit the target. This leaves a very small volume
to dissipate the considerable heat load and this cannot be done with a static target as
for radiation cooling the powers are a factor of ≥ 12 too high, as shown in section 3.3.
Water cooling comes closer to feasibility but falls short due to lack of surface area in the

Table 3.1: Features of a solid pion production target compared with a liquid mercury jet.
Advantages of solid target
Does not depend on hydrodynamics
No liquid cavitation
Mercury vapour damages vacuum,
require beam window
No chance of contamination fromspillage
Targets may be changed sideways atlower velocity than longitudinally

Disadvantages of solid target
Requires mechanical system forchanging targets
Mechanical joints may produce dust
Proton pulse intensity must not
shatter target
Requires the first solenoid coil to be
split

44

Figure 3.1: Main components of the UK solid target station.
target. Therefore the target material must be replaced for each pulse, either by moving
the target material or steering the beam onto different targets, the former currently being
considered more practical.
The UK neutrino factory design proposes a solid pion-production target as opposed to
the liquid mercury jet suggested in the US study [25]. The solid ‘target’ would actually
be a series of solid targets in a loop that take turns to be hit by the proton beam pulse,
whereas the liquid jet vaporises after producing the pions and is flowing at such a ratethat new liquid will be in place by the next pulse (which arrives at a 50 Hz rate in theliquid target baseline). This changes the set of limitations and challenges in the target as
set out in table 3.1.
Proof-of-principle experiments in high power targets for the neutrino factory (and otherhigh-power facilities) are being conducted at the time of writing, most notably MERIT [39]for the liquid mercury jet and wire shock test studies [29] for the solid target. Interested
readers are encouraged to follow the references as the detail of these experiments is outside
the scope of this thesis.

45

3.1

Proton Driver Parameters

With the beam power P fixed, the main remaining parameters are the proton beam kinetic
energy (Ek ) and the pulse repetition frequency f . Given these, the number of particles per
pulse (N ) can be derived from the formula for the mean power P = f N Ek . The energy
per pulse will increase as f is decreased, so for a solid target it is unfavourable to use low
repetition rates. Increasing the repetition rate eventually causes the power consumptionof the pulsed RF systems in the muon accelerators to become unacceptable, with the ISS
recommending a compromise value of f = 50 Hz, which will also be used here. Given this,the number of protons per pulse is given byN=

4 × 106
4 × 106
4.9932 × 1014
P
=
=
=
.
f Ek
50Ek
50(109 e)Ek [GeV]
Ek [GeV]

There is a threshold for pion production that puts a lower bound on Ek . Pions areproduced at low energies by nucleon-on-nucleon collisions of the form N +N → N +N +π
like p + n → p + p + π − , with other production channels becoming available at higher
energies, see section 4.6.1 for an example. This means the centre of mass energy must
exceed the masses of two nucleons and a pion (a charged pion for our uses), which is
2.016 GeV for p + p + π − . The centre of mass energy satisfies
2= (E1 + E2 )2 − (p1 + p2 )2 ,Ecmwhere Ei are the total energies of the two colliding particles and everything is measured
in energy equivalents: GeV/c for momenta and GeV/c2 for masses. The second particle
is a nucleus at rest, so p2 = 0 and p
E2 = m2 . The incoming particle has total energy
E1 = m1 + Ek and momentum p1 = E12 − m21 . Some rearrangement gives the required
incoming proton energy as
E 2 − (m1 + m2 )2
Ek = cm.
2m2
Thus Ek must be at least 286.5 MeV for the reaction p + n → p + p + π − and at least
292.3 MeV for the reaction p + p → p + n + π + . Near to this threshold the cross section
will still be very low and the outgoing pions may have such low energy that they becomereabsorbed in the target. Previously [25] beam energies of tens of GeV have been foundto achieve sufficient pion yield; a detailed investigation of where the optimum energy is
can be found in section 4.2.
Dependent on the optimal energy is the choice of proton accelerator technology. Table
3.2 lists examples of machines that have particular energies, together with their current
status and beam power.
At low energies, acceleration can be done with a single-pass linear accelerator, the mostambitious proposal in this class being the 8 GeV “Project X” concept from Fermilab [44].
The CERN SPL (Superconducting Proton Linac) design is still ambitious at 3.5 GeV in its
second design report [42] (with the option to upgrade as far as 5 GeV if necessary) but has
been designed to a much higher level of detail. The main technical disadvantage of using
a linac as the neutrino factory proton driver is that the neutrino factory Scoping Study
report [24] recommends a 1–3 ns RMS bunch length whereas a linac generally produces a46

Table 3.2: Planned and existing high-power pulsed proton accelerators. Beam energy
figures in brackets are not values on the primary sampling of the energy range in thesimulations described in this chapter, which is given by the remaining figures. Repetition
frequencies are 50 Hz unless otherwise stated.
Proton driver
CERN SPL I [40]ISIS upgrade RCS 1 [41]

Ek [GeV] Power/Schedule
2.2
Study, 4 MW, 75 Hz, superseded
3
Study, 1 MW
Proposal (first 160 MeV approved),
CERN SPL II [42](3.5)
4–5 MW ultimate
4
Green-field RAL synchrotron [43]5
Study, 4 MW
ISIS upgrade RCS 2 [41]6
Study, 5 MW
FNAL Project X linac [44]8Study, 2 MW, 10 Hz
Scoping Study FFAG option [24]10
Study, 4 MW
ISR tunnel RAL synchrotron [43]15
Study, 4 MW, 25 Hz
FNAL RCS proton driver [45](16)
Study, 1.2 MW, 15 Hz
20
BNL/AGS upgrade [25](24)
Study, 1 MW, 2.5 Hz
CERN slow-cycling synchrotron [46] 30
Study, 4 MW, 8.33 Hz
40
JPARC main ring [47]50
Beam in 2009, 0.75 MW, 0.3 Hz
75
100Running at 0.4 MW, upgradable to
FNAL injector/NuMI [48]120
1.2 MW by 2013, 0.75 Hz

Table 3.3: Proton driver specification from the International Design Study for a Neutrino
Factory (IDS-NF) baseline, version 1 [49] in 2008. Note: that study uses a liquid target.
Proton Driver
Proton power
Proton kinetic energy
Pulses per second
Bunches per pulse
Minimum time between bunches
Maximum time for all bunches
RMS proton bunch length

47

4 MW
5–15 GeV
50
3
17 µs
40 µs
1–3 ns‘pulse’ of order 1 ms in length on each repetition. This means that the pulse must first be
accumulated in a ring at the end of the linac to reduce its length to the circumference ofthe ring, corresponding to a time closer to 1 µs. Once the beam is in a ring, irrespective ofwhether that ring is an accumulator for a linac or the acceleration ring of a synchrotron or
FFAG, it must be further compressed to the final nanosecond timescale. This is achieved
via careful use of RF systems in the ring to shorten the bunch to less than the natural
bunch length during acceleration or accumulation. The main problem in this phase at low
energies is the longitudinal ‘space charge’ repulsion of particles from each other, whichis enhanced by shortening the bunch and thus increasing its charge density. The spacecharge is also reduced steeply by increasing the beam energy (relativistic time dilationand space contraction allow more charge in the same volume of the laboratory frame)
and the decrease of beam current that comes with keeping the power constant at higher
energies. Thus the SPL proposal has the most difficult bunch compression system intable 3.2, with a high voltage of RF in the compression ring. The advantage of linacs isthat their overall intensity can be made higher than an accelerator chain containing rings
and the associated injection and extraction, since there is no timing constraint on theacceleration: linacs may even be operated in continuous or “CW” mode if the components
can cope with the heat loading. This is the reason that the SPL and Project X proposals
are actually multi-use machines where proton intensity over and above that required forthe neutrino factory is used for other applications such as nuclear physics experiments or
spallation neutron sources.
At higher energies, circular accelerators dominate, as the length (and thus cost) of alinac become prohibitively large. There is also a trend for the repetition rate of theseaccelerators to decrease as energy increases, which is for two reasons: the amount of RF
voltage around the ring limits the rate at which protons can attain high energies and theconstant power requirement means less average current of protons is needed, so bunches
can be accelerated less often. With the highest energy rings, cost is again driven up due
to an increase in length, but this time because the available dipole magnetic fields are nolonger able to bend the high-energy protons around on a smaller radius of curvature. At
low energies, the main challenge for circular accelerators is the increased beam current andsharply-increased space charge. Thus there could be said to be an optimum towards the
mid-range of options shown in the table. Finally, note that the circular accelerators comein two types: synchrotrons in which the magnetic fields are varied during the acceleration
cycle to scale with the beam momentum and FFAGs in which the variation of magnetic
field strength is provided spatially across the accelerating aperture that the beam orbit
moves over with energy. FFAGs are still an active area of research and there is not yet aconclusive cost/benefit comparison of these two ring types. As of 2009, no FFAG has been
demonstrated to work in the same power or space charge regime as existing high-power
proton synchrotrons.

48

Figure 3.2: Bunch timing structure of a neutrino factory with a solid target, f = 50 Hz
and h = 5 bunches per pulse.

3.2

Bunch Spacing

The RF system of the final proton acceleration (or storage) ring provides an additional
control over the bunch structure of the beam hitting the target. To make sure the longitudinal dynamics are consistent from turn to turn, the RF system must return to thesame phase each time the centre of a bunch comes back to the cavity location, excluding
slow adiabatic adjustments to the phase used in the acceleration programme. This requirement can be satisfied by making the RF frequency equal to the revolution frequencyof the ring, but it can also be an integer multiple of this, with the multiplier known as h,
the harmonic number. For example in the main ISIS ring [50] h = 2 and in the upgrade
ring for a neutrino factory proton driver, h = 6 [41]. A consequence of having h > 1 isthat another bunch at an angle of 2π/h behind the first in the ring experiences exactly
the same RF programme and the same transverse dynamics too. It is therefore efficient
to use all the stable phase space by filling all these RF ‘buckets’ in a ring if it is possible.
The consequence for the downstream systems is that if the final proton ring is designedwith a harmonic number h > 1, then each machine pulse will produce up to h bunches
that will hit the target with some delay from each other. If the entire turn is extracted
at once then the spacing between bunches will be the same it was in the ring, usually lessthan 1 µs. As the rise-time of the extraction kicker must be less than the time between
bunches for successful extraction (unless there is an empty bucket used for this purpose),
it is usually possible to have the kicker activate for only one of the bunches, which issent to the target, then wait for a period of time before kicking out another and so on.
Thus the delay between the bunches may be increased from the natural spacing, though
in a synchrotron it cannot be increased so far as to make regular bunches at frequency
f h because at some point the main magnet field must return to a low value in orderpick up low-energy bunches to be accelerated in the next pulse. A storage ring would
make regular spacing possible but at considerable cost; it must also be stressed that withregular extraction the frequency f h essentially becomes the repetition rate, so f h ' 50 Hz
would then be desirable. Usually the bunches are extracted close enough together thatthe resulting muon trains could all go through the muon acceleration system during asingle pulse of its RF.
49

The target will react to this pulse substructure in a different way depending on whether
it is solid or liquid. For a liquid jet, it is preferred that the bunches arrive close together(within 40 µs in table 3.3) before disruption or boiling of the mercury moves it out of thebeam path. For a solid target, if the bunch spacing is sufficiently large to approach theoscillation time of acoustic (or shock) waves in the target rod, then several smaller shocks
can be initiated at different phases, reducing or even cancelling the overall shock effect.
This is shown in [51] where the oscillation time in a simulated 2 cm diameter tantalum
rod is just under 10 µs and the overall stress amplitude is reduced by a factor of ∼5 byspreading the input shocks over a period of 10 µs rather than triggering them all at once.
This would give the overall timing structure shown in figure 3.2.
One final aspect of the proton ring RF system that affects the bunches on the target
is the bunch compression scheme that manipulates longitudinal phase space to reducethe duration of each bunch to ∼1 ns RMS. This is either done with programming of themain RF or an additional higher-frequency system and will act on all the bunches in
the ring. It is required because introducing too much time spread into the distributionof muons from the target will increase their longitudinal emittance. It also means each
proton bunch is virtually instantaneous from the point of view of the target material,
meaning that thermal shock occurs, because the temperature has increased (from energy
deposition) with no commensurate expansion of the atomic lattice. The shocks in fact
propagate inwards from the target surface, which is able to expand into free space first.

3.3

Thermal Effects

The management of heating in the target caused by energy deposition from the proton
beam and its secondary particles is one of the primary considerations in the target design.
As mentioned earlier, the neutrino factory target cannot be made arbitrarily large in orderto spread the heat load, so techniques for dealing with high energy densities are necessary.
Additionally, surrounding the target in a coolant would lead to the reabsorption of pions
and additional heat deposition, so radiative cooling schemes are considered.
The inner pie chart of figure 3.3 shows that the neutrino factory target is quite anefficient target for particle production, with most of the proton beam energy going into
secondary particles rather than heat. The protons emerging at angles within 5 degreesof the rod’s forward axis are singled out because they produce a ‘hot spot’ if they hit
the target chamber wall at too close a range, with consequences discussed in section 3.5.
These protons mostly consist of those that have been multiply scattered through the highZ target material without experiencing a nuclear interaction. As such, they still have mostof their original energy, ∼9.5 GeV for the 10 GeV proton beam example.
There is a terminology issue as to whether the neutrino factory rod is a ‘stopping target’or not. The answer is that most of the primary protons undergo a nuclear interaction
even in the 20 cm long case, with the proportion increasing as the rod becomes longer,
though not attaining 100% because a few percent of the protons will always scatter out of
the sides of the rod. However, ‘stopping’ in this sense does not imply that all of the powerin the original beam is deposited as heat, as it is in neutron spallation targets, becausesuch a large fraction of energetic secondaries escape permanently from the rod.
50

Figure 3.3: Where the energy from a 10 GeV proton beam goes when it hits a 20 cm long,
2 cm diameter tantalum rod target (MARS15 [2] calculation). For the variation of thesefigures with proton energy and length of rod, see sections 4.2.1 and 4.3.1 respectively.
The outer pie chart of figure 3.3 splits up the kinetic energy of the secondary particles
by species. Positive and negative pions are the desired particles and constitute fairly large
segments of the pie, carrying 24% of the original 4 MW power. Kaons also appear thatcan decay to usable pions later on. A final histogram by angle of emitted radiation (figure3.4) shows that the target chamber will have to cope both with a diffuse component of
the radiation and with the forward-directed ‘hot spot’ mostly coming from the protons.
Apart from the target itself, the radiation will be most intense (> 100 W/cm2 ) on the
inner bore of the radiation shielding just inside the initial 20 T solenoid, meaning thisshielding must contain extensive water cooling channels.
In the solid target design, the cooling of the target itself is achieved by radiation fromthe surface. The formula for thermally radiated power from an object of surface area A
and temperature T is
P = σAT 4 ,where  is the emissivity, which may also depend on temperature, and σ is the StefanBoltzmann constant. In the approximation of continuous, volumetrically uniform irradiation and perfect thermal conductivity of the bar (i.e. an optimistic scenario), P can be set
equal to the mean input power of heat deposited in the target. With the parameters used
for figure 3.3 and a 4 MW proton beam, P = 514 kW and A = 0.0132 m2 . The emissivityof tantalum or tungsten is ∼0.32 at high temperatures [52] and decreases as T reduces.
By rearrangement of the previous equation, the equilibrium temperature at which powerradiated equals power input isr
P
4= 6808 K.
T =
σA
51

Figure 3.4: Graph showing the directionality of emitted power for each particle species,
predicted by MARS15 [2] for the same setup as figure 3.3.
The melting point of tantalum is 3290 K and that of tungsten is 3695 K, so even under
optimistic assumptions, a single bar of this size will not be able to radiatively dissipate
the required amount of heat without melting.
It is now clear why the UK neutrino factory solid target design involves multiple bars
that are replaced for each pulse. For N bars the above calculation can be redone with
P = 514/N kW, showing that N ≥ 12 is required for tungsten not to melt and N ≥ 19 fortantalum. These figures do not include margins for error and so on, with more realistic
scenarios having N ≥ 200 bars [53]. Tungsten demonstrates greater strength at hightemperatures so is currently the preferred material option, though its pion production
behaviour is very similar to the tantalum investigated in chapter 4 due to their proximity
in the periodic table.

3.4

Geometry of the Solid Target Mechanism

Figure 3.5 shows a solid target chain concept in cross section, showing how the solenoid
coils would have to be arranged such as to let the chain of rods pass through. Simulationsin this thesis using Muon1 do not model the coils to this level of complexity (as the target
design is still under development) but instead assume the magnetic field from a single
20 T initial coil.
To see how this affects the magnetic field profile, figure 3.6 shows how with coils set
52

Figure 3.5: Schematic diagram of the solenoid arrangement surrounding the chain of solid
targets, from [53].
at 50 A/mm2 , peak fields of 20 T can be created. One layout does this with a solid coiland the other splits the coil with enough room for the target and chain to pass through.
The dip in on-axis field between the two coils in the second case goes no lower than 17 T.
A few pions are tracked in the pictures, with one in the second case becoming trappedin a magnetic bottle in the field dip. However, that pion has a very small longitudinal
momentum as evidenced by the steep pitch angle of its spiralling motion, so falls outside
the region of ‘interesting’ pions defined in section 4.1.1.
Although the target station for the solid target mechanism is not yet engineered, we
may usefully inherit features from the US Feasibility Study II [25] target cross section,
reproduced in figure 3.7, which incorporates the correct amounts of shielding for theradiation fluxes produced. Any mechanism for removing the targets laterally will require
the normal conducting initial coils as well as the superconducting coil and its cryostat to
be split and the vacuum area extended to contain the chain or wheel of targets.
There are other options besides a chain for moving the solid target through the interaction area. Figure 3.8 shows the concept of a spokeless wheel or rim, which intersects
through holes in the first solenoid and rotates continuously with targets attached so they
are lined up with the proton beam for every 50 Hz pulse. Another option that is being
considered is to have a powdered solid moving through the target station fast enough thateach proton pulse hits a fresh volume of material. Figure 3.9 shows a tungsten powder
jet formed by a proof-of-principle experiment at RAL [54, 55]. The powder may haveadvantages because it cannot shatter or fail mechanically in the same ways as a solid rod.

53

Figure 3.6: Cross sections of solid (left) and split (right) coil layouts around the target.
The axial field profile Bz (0, 0, z) is superimposed for a coil current of 50 A/mm2 . The
peak field in each case is 20 T. The red target rectangle is the 20 cm length case with the
30 cm, 3 cm diameter option shown by a fainter line.

Figure 3.7: A cross section of the US Feasibility Study II [25] target station. Normalconducting solenoids are shown in green and superconducting ones in blue. Water-cooledtungsten carbide shielding is yellow. The UK targets from figure 3.6 are shown to scalein the bottom-left corner.
54

Figure 3.8: Cross-section of the rotating spokeless target wheel. The shaded annulus (left)
is the coil of the first solenoid and target rods are attached to the wheel rim (right) withlong axes pointing into the page. Adapted from [56].

Figure 3.9: Photograph of an unconfined tungsten powder jet produced experimentally[54, 55]. The jet is flowing leftwards from the nozzle (right) at 3.7 m/s and the exposed
portion of the jet is ∼20 cm long as measured by the 1/5/10 mm graph paper behind.

55

Figure 3.10: The angular distribution of remnant proton beam power within 5◦ of theincoming proton beam axis, from the same situation as in figure 3.3 (MARS15 simulation).

3.5

Beam Dumping

The UK target station beam dump differs from Study II, which uses the collection poolof the mercury jet to act as the beam dump for the 974 kW of forward directed protons(as defined in figure 3.3). The UK design uses a smaller angle between the incoming
protons and the magnetic field axis, meaning that the protons will hit the side further
down the muon decay channel. There a conventional water-cooled beam dump will absorb
the energy over a larger area, as the protons impinge at a shallower angle. The large angle
is needed in the liquid mercury design to keep the entire mercury flow loop within theinitial cryostat, whereas solid target designs do not have this limitation.
Calculating the peak areal power density requires knowing something about the angular distribution of the forward-directed protons. In a given solid angle Ω (measured in
steradians), the heating power H is the sum of the kinetic energies of protons arriving perunit time whose velocities lie within that angle. The rate of heat deposition will be directly
proportional to P , the incoming proton beam power, so H/P is a constant independentof P . In the limit of small solid angles the dimensionless quantity (1/P )dH/dΩ evaluatedat any given angle describes the concentration of secondary protons there. Figure 3.10 isa plot of (1/P )dH/dΩ against deflection angle and the peak of 239 (W/W)sr−1 is used infurther calculations.
If the beam dump is situated a distance L from the target and the remnant proton56

beam hits it with angle of incidence θ, the areal power density on the dump is
dPdump1 dH sin θ
=P
.
dA
P dΩ
L2
Values calculated using this formula are shown in table 3.4. This is not entirely accurate
as the Study II figure is calculated using the tantalum rod target rather than the mercury
jet, but shows the improvement gained by a larger distance to the dump and shallower
incidence angle. Note that these power deposition figures are also strongly influenced by
the target length (which reabsorbs protons), detailed in section 4.3.1.
Table 3.4: Comparison of the locations of the beam dump for the forward-directed proton‘hot spot’ in Study II and UKNF designs. Peak areal powers are calculated using theformula given in the main text for a 4 MW incoming proton beam.
Parameter
Angle proton beam to magnetic axis
Distance beam dump downstream
Distance beam dump off axis
Peak areal power deposition

57

Study II
66.7
3.75
0.25
453

UKNF
14
25
0.35
2.14

mrad
m
m
W/cm2

Chapter 4
Pion Production Simulations
Muon front end simulations require an initial set of pion coordinates from the target. This
chapter describes hadronic simulations that obtain pion distributions and allow these to
be compared across different target parameters to find the best solution (sections 4.1–4.5).
The target optimisation is decoupled from the muon front end in terms of yield, thoughsome choices in the target have indirect effects on the later accelerator (section 4.6).

4.1

Modelling Particle Production in MARS15

The newest available version (15.04) of the MARS particle production code [2, 57] wasused for the simulations. Later versions are likely to have updated models based on datafrom the HARP experiment [4]. As MARS15.04 does not, a benchmark of its results
against some experimental data from HARP is given in section 5.2 to help gauge thelevel of accuracy of the current MARS models in the region of interest. The series oftarget simulations in the following sections uses the reference simulation in table 4.1 as
the starting point from which to vary parameters of interest.
Table 4.1: Parameters of the target reference simulation. This configuration emerged earlyon as being somewhere near the optimum for pion production, so served as a starting point
for investigations of variations in the parameters.
Code Used
Target Material
Target Shape
Target Dimensions
Protons Per Monte Carlo Run
Proton Beam Profile
Proton Direction
Magnetic Field

MARS15
Tantalum
Cylindrical
20 cm length × 1 cm radius
105
Parallel beam, radius same as rod, parabolically distributed on circular target endface
Parallel to target cylinder axis
None used (simulation only within target)

58

Figure 4.1: Schematic of the target reference simulation.

In the simulation, the proton beam hits the target end-on as shown in figure 4.1.
MARS was configured to record all π + and π − leaving through any of the target cylinder’ssurfaces. See appendix A for the input file used for these simulations. The target also
produces copious quantities of neutrons; indeed, a stopping tantalum target is sometimes
used in dedicated neutron production machines.
A radial parabolic distribution ρ(r) ∝ 1 − (r/R)2 , 0 ≤ r ≤ R, was used for the
protons, with R the radius of the rod (see appendix B for generating routine). This ispreferable to the rectangular and Gaussian beams that are defaults in MARS, as it is thesame shape as the rod and does not lead to a proportion of the beam missing the target
entirely. Another option frequently used in target studies is a uniform distribution on
the circlular endface, but this is hard to achieve in practice with a proton extraction line.
The parabolic distribution, however, is the projection into the plane of a 4D ‘waterbag’
phase-space distribution, which is often used as a model of proton bunches.
The incoming beam was assumed to be parallel in the simulation, as typical proton
drivers have beams with too small an angular divergence to greatly affect the resultsof a hadron shower calculation in a block of material. For instance, the ISS baseline
proton driver [24] has a beam with a 100% un-normalised emittance of 175 mm·mrad
just after injection at a kinetic energy of 200 MeV, which becomes 10.3 mm·mrad at the
10 GeV extraction energy, scaling with 1/(βγ). At a beam focus or ‘betatron waist’, thisemittance is the product of the beam radius and maximum particle angle (x0 ). Setting
the radius to the 1 cm of the reference rod gives a beam divergence of 1.03 mrad. Evenat the lowest considered energy of 2.2 GeV this is only 3.77 mrad, which would produce a0.75 mm transverse offset in a 20 cm rod and little difference in interacting path length.
MARS can add a magnetic field as would be present in the rod in this situation. However, this may change in value if the magnetic design of the front end is to be optimised,so one field would not be correct for all simulations. The field strength also has little
effect on the pion production, since over the width scale of the rod it produces only aslight curvature of the pion trajectories of interest. For instance, a typical pion spirallingon an R = 5 cm radius helix produced at the centre of an r = 1 cm radius rod will haveits velocity vector rotated about the rod axis by θ ' r/R by the time it exits the rod.
The displacement at the rod’s surface will be to leading order 21 θr but the difference in59

path length ∆l, important in particle production is given by ∆l/r = 61 θ2 plus higher order1
so the differences in interacting path length due to the magnetic
terms. Here, 16 θ2 ' 150
field actually fall below the statistical errors in the Monte Carlo simulations.
The above simplifications mean that these pion production results are applicable quite
generically for pion production from protons hitting a cylindrical target and the optima
and trends discovered may be useful outside the context of the neutrino factory.

4.1.1

Pion Capture Probability Grid

To evaluate the number of ‘useful’ pions produced by the target, the pions leaving its
surface have their longitudinal and transverse momenta (pL , pT ) recorded and plots suchas figure 4.2 show this momentum distribution and the later behaviour of the pions in themuon front end. The MARS code was not available to run on the distributed optimisation network, so it was not practical to run MARS in conjunction with every front end
investigated. Instead, a pion distribution is simulated through two representative muon
front ends, each producing a probability function for pion survival as a function of theirinitial (pL , pT ) coordinates.
This makes optimisation of the target much quicker since a simple postprocessing step
can now turn a target pion distribution into a figure for useful pions, rather than requiring
tracking to be performed each time. The effect is similar to using momentum ‘cuts’ but
is based more closely on the actual behaviour of the particles in the representative muon
front ends.
Figures 4.2 and 4.3 display the fate of pions as a function of their initial (pL , pT )coordinates in two front ends chosen for their contrasting design principles. Figure 4.2
uses RF phase rotation to a low energy spread from section 8.3, whereas figure 4.3 uses
a magnetic chicane buncher and linac from section 8.2. Since decay to a muon occurs
in the front end, the end result is not deterministic: the range of possible decay muonsoriginating from a single pion causes the blurring of the coloured regions in the figures.
An additional small amount of blurring will come from any correlations of the behaviour
with the pion’s starting position on the target surface because reducing the initial fivephase-space variables (assuming axial symmetry) into two (pL , pT ) idealises the rod as apoint source. However, the plots can still be used to produce a method for counting usefulpions if they are treated as showing survival probability as a function of initial momentum.
To obtain probability figures, the plots were binned into 30 × 30 MeV/c squares andthe ratio of survived muons (white points on the plots) to the total number of initialpions was calculated in each one. In fact, since MARS generates particles with statisticalweights, the ratios were calculated between sums of weights for the appropriate pions:
X
wi
Pab =

Transmitted muons i:(pL ,pT )i ∈Square(a,b)

X

=
wi

Tab
,
Wab

Original pions i:(pL ,pT )i ∈Square(a,b)

where the numerator and denominator of this expression define Tab and Wab .
60

Figure 4.2: Scatter plot of the initial transverse and longitudinal momenta of pions at thetarget, coloured by their fate in a particular phase rotation front end. If the pion decays,
the fate of its daughter muon determines the colour.

Figure 4.3: Using a front end with a chicane and pre-acceleration linac in figure 4.2 insteadof a phase rotator gives different loss mechanisms but the white area of pion momenta
that decay into transmitted muons is similar, with a slight difference in central energy.
61

Figure 4.4: Probability grids derived from the particle acceptance data in figures 4.2 and4.3. Lighter blocks represent higher probabilities of pions in that block decaying intotransmitted muons; black is zero probability, white would be 100%.
The resulting matrices of survival probability [Pab ] are shown in graphical form infigure 4.4. Both lattices had a few squares with this probability exceeding 90%, so there
is a subpopulation of pions that will nearly always produce successful muons regardlessof random decay effects or emission position on the rod.
Unfortunately the low statistics in some squares, particularly near the pT = 0 axis
where the true phase space volume represented by the square is smaller, generate unwanted noise and fluctuations in the probability map. Smoothing the values between
adjacent squares appears to be one way of mitigating this. As it happens, often a squarewith bad statistics that is by chance nearly empty (Wab is small) has neighbours withcomparatively good statistics, which should be given more weight when calculating an
average. Empirically, the following scheme produced a substantial improvement in theprobability maps:
P̃ab =

Tab + 0.1(Ta−1,b + Ta+1,b + Ta,b−1 + Ta,b+1 ).
Wab + 0.1(Wa−1,b + Wa+1,b + Wa,b−1 + Wa,b+1 )The resulting smoothed probability matrices [P̃ab ] are plotted in figure 4.5.
Table 4.2 gives the numerical values of the probability grid squares for the phaserotation lattice. This grid is used to define useful pions in the remainder of this chapter;that is, if pion i is in square (ai , bi ), thenX phaserot
Useful pion weight =P̃ai biwi .
i

The range of pions transmitting muons with more than 50% probability corresponds
roughly to the region with total momentum 250 ≤ |p| ≤ 450 MeV/c and transverse mophaserot
mentum pT ≤ 200 MeV/c. The effect of using the chicane/linac grid instead of [P̃ab]is demonstrated in section 4.1.2

62

Figure 4.5: The tracking-derived grids in figure 4.4 smoothed to remove some aberrant
values originating from statistical noise.

Table 4.2: Probability of successfully transmitting a muon through the phase rotation
front end, as a function of bins in the (pL , pT ) space of the original pion emitted from thetarget surface. The bins are 30 MeV/c in size and the lower end values are given in the
top table row for pL and the left-hand column for pT , with zero transmission outside theregion shown. Probability values are in percent.
MeV/c

0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480

0 - - - 30 - - - 60 - - - 90 - - - 120 - - - 150 - - - 180 - 0.1 - 0.4
210 0.1 1.4 0.3 1.7
240 - 0.2 - 0.2
270 - - - -

0.4
2.9
2.6
1.6
0.1

- 6.0 58.8 54.2 91.9 92.6 92.3 92.3 76.8 15.6
- 1.1 14.2 45.2 47.6 64.6 79.9 90.5 86.3 65.7 13.0 0.8- 0.8 12.2 49.3 78.7 85.3 77.5 88.5 62.7 51.4 9.2 0.5
0.2 6.9 28.6 49.4 74.4 74.5 77.4 80.6 39.7 17.0 1.8
1.5 14.4 19.4 42.6 64.8 70.0 66.3 67.0 41.9 20.5 1.5
2.1 5.8 27.5 47.0 53.5 65.3 59.4 53.8 26.5 5.0 0.2
5.6 12.0 24.2 41.5 45.3 49.7 49.1 38.2 17.2 1.8
6.6 15.0 18.5 20.9 26.3 41.4 34.9 18.6 6.7 0.5
4.5 2.6 3.5 9.7 13.9 20.1 20.7 4.0 0.6
0.5 0.1 0.1 1.1 3.4 1.6 1.5 1.5 0.2-

63In a constant B field and the absence of any other forces, the path of a charged particle
will be a helix with a certain (constant) radius that depends on the momentum of theparticle transverse to the magnetic field direction and the strength of that field. This is
known as the Larmor radius and can be calculated directly from the equations of motion
given in section 2.1 to be
pT
rLarmor =,|q||Bz |where q is the charge of the particle. Using MeV/c and units of the elementary charge
gives a handier version of this formula for this sort of calculation:rLarmor [cm] ='pT [MeV/c] (108 /c)|q [e]| |Bz [T]|pT [MeV/c]
3|Bz [T]|for a singly-charged particle.

The solenoid containing the target has |Bz | = 20 T and q = ±e for a pion, so the helix
radius that the pT = 200 MeV/c cut-off point corresponds to is rLarmor = 3.34 cm, using theexact version of the formula above. This is the largest helix radius at which more than half
the pions are transmitted through the first solenoid. The centre of the helix will not be thesolenoid axis: instead, the location the pion was emitted from the target surface will be at
one side of the helix and the furthest excursion will be 2rLarmor = 6.67 cm from this point.
This is approaching, though not equal to, the 10 cm radius of the solenoid bore but theparticles are now sufficiently far off-axis to risk encountering the non-linear fringe field of
the solenoid on the way out, in which case the helix follows the diverging field lines to larger
radii rather than the solenoid axis, leading to loss. The largest possible transmittable helix
would have a diameter of 11 cm for the standard 1 cm radius target (that is, from the far
side of the target to the inner bore), so we would expect a complete removal of particles
with rLarmor ≥ 5.5 cm. This corresponds to pT ≥ 330 MeV/c and the transmitted particles
disappear completely just below this value, with 300 ≤ pT < 330 MeV/c being the first
empty row of bins, just below those shown in table 4.2.

4.1.2

Difference in Yield Between Probability Grids

In the last section, two different probability grids were generated and it was stated thatthe phase rotation grid would be used for all subsequent yield evaluation rather than the
one derived from the chicane/linac design. As evidence for this choice, some of the yieldgraphs presented in section 4.2 are evaluated here using both grids but with all otherattributes kept the same; in fact they come from the same MARS particle output file.
Specifically, the standard target with parameters in table 4.1 was used and the proton
beam energy varied from 2.2 GeV to 120 GeV using the primary values in table 3.2.
Figure 4.6 shows the results for just the positive pions. The chicane/linac design has
an overall lower efficiency than the phase rotator, for reasons discussed in the tracking
results of chapter 8, which is one reason for it to be excluded. Another question would
be whether the two different designs show different qualitative behaviour: as the purposeof the target optimisation is to find the location of the yield peak in such parameters as
64

Figure 4.6: Comparison of useful π + yields calculated with two different probability grids
over a range of energies. 1σ statistical error bars are shown.

Figure 4.7: Comparison of useful π − yields calculated with two different probability grids.
1σ statistical error bars are shown.

65

proton energy, rod length and so on, the absolute yield matters less than the position (andshape) of the optimum. To compare the shapes without the yield difference, the dottedline on the graph is a scaled version of the chicane/linac results. The scaling factor chosen
was the average ratio between the two original yield series.
Figure 4.7 shows the same three lines for the negative pions, which have a slightly
differently-shaped graph to the positives. In both cases, the scaled chicane/linac line stays
very close to the phase rotation line, with a maximum difference of 10% for low protonenergy (and low-yielding) simulations producing negative pions. These differences are
only 2–3σ in terms of the statistical error on each data point but are probably systematic
because they have the same sign in groups of nearby energies. However, the overall shapesof the graphs and locations of the optimum ranges are still similar, so the phase rotation
grid is used as a representative figure of merit considering it both represents a higher-yield
front end and evaluates design choices similarly to the chicane/linac grid.

4.2

Proton Energy and Rod Material

This section contains what are probably the most important findings about pion yield
gained from the MARS15 studies. Section 3.1 described how the proton energy is a key
parameter that has a large effect on the design of the proton driver, so the optimal energy
must be determined before design work can continue. The proton energy is varied in thesimulations here and the results plotted to show the location and shape of this optimum.
Although high-Z (Z = atomic number) targets from period 6 of the periodic table have
conventionally been the choice for the neutrino factory, this study also tried alternative
materials at intermediate and low-Z. Specifically, at high Z, the tantalum of the standard
UK solid target configuration at the time (table 4.1) and the liquid mercury used in US
Study II [25] were both run. For an intermediate-Z target, copper was chosen, morebecause it was the densest element in period 4 of the periodic table than for its suitability
as a real target: it melts at perhaps too low a temperature to be considered for a solid
target with radiative cooling. The objective was to change Z without changing the otherparameters affecting the hadronic showering behaviour (primarily density) by too much.
Carbon in the form of graphite is an establised low-Z target material that is finding
applications in neutrino superbeam experiments [19]. Boron is a few percent denser (2.34
vs. 2.26 g/cm3 ) but is not so commonly used, thus carbon is the element chosen fromperiod 2.
There is still a significant difference in density between these elements so a legitimate
comparison must give them each a different length of target so the protons will have thesame probability of interacting. The relevant parameter that determines how frequently
protons undergo nuclear interactions with path length in a material is its hadronic interaction length. This is used in table 4.3 to find a reference target length for each material
that is used in the simulations of varying proton energy that follow. The original 20 cm
tantalum rod corresponds to 1.79 hadronic interaction lengths.
The table also includes molybdenum (a candidate element from period 5 that was not
simulated) and tungsten, which later became the preferred solid target due to superior
66

Table 4.3: Hadronic interaction length data from the Matprop tool at SLAC [58] allows
reference lengths for other materials to be scaled from the 20 cm tantalum target.
Element
Ta
Hg
C
Cu
W
Mo

Z Interaction length (cm) Equivalent to 20 cm Ta (cm) Rounded
73
11.18
20
20 cm
80
13.95
24.96
25 cm
6
36.92
66.05
66 cm
29
15.16
27.12
27 cm
74
9.54
17.07
17 cm
42
14.96
26.76
27 cm

mechanical strength at high temperatures. Its pion production behaviour will not be very
different to the other period 6 elements plotted, which follow each other closely.
Figure 4.8 plots the useful pion yields (as defined in section 4.1.1) for all four materials
over the primary range of energies given in table 3.2, showing each sign of pion separately.
The units are chosen to be proportional to a constant beam power across all energies. The
measure of “useful pions per incoming proton” may seem a sensible choice, but a 100 GeV
proton carries much more kinetic energy and thus can create more hadronic particles thana 3 GeV proton. This is fixed by using the unit of “useful pions per proton·GeV”, where
the fast proton in the previous example would count for 100 p·GeV and the slow one for 3
p·GeV. As the denominator is now just a measure of beam kinetic energy, the numericalyield rate of useful pions can be calculated from the proton beam power P :Useful pion rate [s−1 ] =

P [W]
Y,
109 e

where Y is yield in these units.
There is an overall decreasing trend on all materials and signs above 10 GeV, with moremarked differences in the low-energy behaviour. High-Z targets also fall off at low energy,
exhibiting an optimum, whereas the carbon target continues to rise almost uninterrupted.
This was naturally of interest, so later plots include an extension of the carbon simulation
range to proton energies as low as 800 MeV. Unfortunately, as discussed in section 3.1,
compressing the proton bunch to the order of nanoseconds becomes infeasibly difficultfor energies below about 3.5 GeV, so this carbon peak must remain of theoretical interest
unless breakthroughs are made in proton bunch compression schemes.
To clarify the optimisation, the pion yields of different signs may be summed to givethe total yield as shown in figure 4.9 for each material. An optimal plateau in yield exists
for the high-Z elements in the range 5–15 GeV. The ‘interesting’ increase of carbon yieldat low energy finally reaches a maximum at 1 GeV before falling off. The situation foreach sign individually, as examined in the NuFact conference paper [59] on these results,
gives an optimal range (for high-Z targets) of 5–30 GeV for π + and 5–10 GeV for π − . The
lower end of the plateau at 5 GeV may not be as sharp in reality since MARS changes
between different physics models over the range 3–5 GeV (see figure 5.3 and chapter 5 ingeneral for comparisons between codes and physics models).
67

Figure 4.8: Variation of single-sign useful pion yields with incoming proton energy for
four different materials. Note the log scale for proton energy. Error bars will be omitted
from yield graphs from now on for clarity, but errors will be similar to those in figure 4.6.

Figure 4.9: Total useful yield of pions with any sign as a function of energy, for four
different target materials.
68Figure 4.10: Effective useful yields of pions for CP violation calculations, which arecalculated from results of both neutrinos and antineutrinos.
When the target produces different numbers of positive and negative useful pions andthus ultimately different numbers of neutrinos and antineutrinos of each flavour, it issensible to ask what the impact of this lopsided mix of particles will be on the physics
results. Evaluating the mass differences ∆m2ij relies on measuring the oscillation length
accurately and as particles and antiparticles have the same mass, the statistics from bothneutrinos and antineutrinos undergoing the same oscillation channel may be added. So
for this application the summed figure of merit in figure 4.9 is appropriate. However, theCP-violating angle δ is measured via the relation [60]s12 c12 s23 c23 s13 c213 sin δ ∝ P (ν̄α → ν̄β ) − P (να → νβ ),where sij = sin θij , cij = cos θij and α and β are some nonequal neutrino flavours. Thus
the parameter sin δ is calculated from the difference of two oscillation probabilities, oneof which is measured using the neutrinos and the other with the antineutrinos.
√ The
individual
statistical errors in these probabilities would be proportional to 1/ Y+ and√
1/ Y− , which add in quadrature to make the statistical error in sin δ proportional top
1/Y+ + 1/Y− . This is the same scaling behaviour as if the measurement had comefrom ‘simple’ statistics with an effective yield Yeff = (1/Y+ + 1/Y− )−1 , half the harmonicmean of the two signed yields and less than either of them individually. Yeff is plotted infigure 4.10, where although most of the graphs are a similar shape to the yield sum, thelow-energy carbon peak is diminished because it was made up of a very uneven balanceof positive and negative pions. Positive pions dominate in the low-energy regime, in partdue to the charge excess coming from the incident protons when production of π + π − andsimilar charged pairs is not completely dominant.

69

Figure 4.11: Variation of heating in the 20 cm-long tantalum target with incoming proton
beam energy, which exhibits a minimum around 8 GeV. The kinetic energy of protons
going to the forward beam dump and other selected secondaries are also plotted.

4.2.1

Variation of Heat Deposition with Energy

While figure 3.3 showed the division of incoming proton beam power for the standard
target in table 4.1, this apportionment will change if the incoming protons have a different
kinetic energy. MARS outputs energy deposition data by default in the file MARS.OUT andthis subsection uses that file to examine the results from the energy scan of the tantalum
target in the previous section.
Figure 4.11 plots the results of integrating the energy deposition within the target
to get the total heat deposition at each proton energy. It also shows the variation of
the proton power emitted in the forward cone of 5 degrees from the central axis, which
was the other part of the energy flux that was separated out in figure 3.3’s inner pie
chart. Table 4.4 has the same results in numerical form. As section 3.3 described, theamount of heating in the target is a determining factor in the design, so minimising theheat deposited is an important goal when optimising the proton energy. Fortunately,comparing figure 4.11 with figure 4.9 reveals that the optima for particle production andlow heating lie at almost the same energy. The range for which the summed useful pionyield for tantalum is within 10% of the optimum value is 5–15 GeV, whereas the range forwhich the rod heating is no more than 10% above the minimal value is 6–15 GeV.
Figure 4.12 shows which areas of the rod are heated the most by the incoming proton
beam at various energies. The heating is not uniform, in fact uniform heating would bea fairly bad model of the situation so should only be used as an upper bound on theperformance, such as in section 3.3. The heating density ϑ is plotted in units of kg−1
70Figure 4.12: MARS15 simulations of heat deposition distributions for the standard tantalum rod with four different incoming proton beam energies. The grid cells are 1 mm
squares and the proton beam enters from the left. Heating is measured in Watts per
kilogram (of target material) per Watt of incoming beam.

71

Table 4.4: Table of heat depositions in the rod as a function of proton beam energy. The
figures in kilowatts are scaled for a 4 MW proton source. The rightmost columns givethe power going into the forward-directed protons of concern for the forward beam dump
discussed in section 3.5. Compare with table 4.5.
E [GeV]
2.2
3
4
5
6
8
10
15
20
30
40
50
75
100
120

Heat deposited
24.7%
22.8%
18.8%
14.6%
13.1%
12.8%
12.9%
13.6%
14.7%
16.6%
17.3%
17.8%
18.2%
18.5%
18.6%

kW Protons within 5◦
988.7
14.4%
912.4
15.5%
750.1
16.2%
582.9
18.1%
525.1
21.8%
512.0
23.2%
514.4
24.3%
543.7
27.1%
588.3
29.0%
665.2
29.7%
690.5
29.0%
711.7
28.3%
726.3
26.5%
739.7
26.4%
742.7
26.1%

kW
577.0
619.2
649.8
724.5
872.3
927.0
974.0
1085.91160.41187.91160.21133.11059.01054.11043.4

because the plots are universal for heating that scales with incoming proton power orenergy. Thus the unit may be thought of as either (J/kg)/Jpulse or (W/kg)/Wbeam . The
real heating power H in a volume V may be recovered using the formulaH = Pbeam V ρTa ϑ.
The peak of the heat deposition distribution moves further down the rod as proton energy
increases. At high energies this is because a 120 GeV proton beam of a fixed power
consists of fewer protons each carrying over 100 times its own rest mass in kinetic energy.
This energy is gradually converted into the mass of more hadronic particles in successiveinteractions forming a shower through the rod, in which the kinetic energy of daughter
particles successively decreases as more hadrons are dislodged from nuclei. This meansthat the bulk of the energy deposition is delayed in the case of higher-energy protons
until a large number of slow (<GeV) hadronic secondaries are passing through the rod
material. This happens further down the rod as more generations of the shower are
required to break down the energy in this way. At low energies, the incoming protons are
already of an energy where they produce slow secondaries, a significant number of which
may even stop in the target, so the 2.2 GeV heating distribution is intense and has a peakalmost at the front end of the rod. The overall intensity of heating in the rod is lowest
for intermediate proton energies, reflecting the minimum in figure 4.11 as expected.
The distribution shapes are not so radically different from each other to change theprevious conclusion (derived only from the total heating) about which proton energies are
preferred from the point of view of providing a favourable thermal environment for thetarget, however.
72

4.3

Rod Length

The length of the rod can be varied to values other than the standard lengths in table 4.3.
The optimisation consideration is to increase the rod length so that enough of the protons
interact but not so much that the generated pions will spiral in the magnetic field andbe reabsorbed by the rod. This is, however, dependent on the magnetic field value andrequires evaluation of the pion production at least at the endplane of the target, which
is not what the surface production simulations here with no field give. Instead, they willshow only the production element of the situation, without the reabsorption.
Work on the reabsorption scenario has been done by Harold Kirk [61] and John Back[62], giving similar optimal lengths to those assumed in the reference scenario for materials
in the previous section, which is in turn similar to the ‘conventional wisdom’ of using 2
interaction lengths for such target. Varying the length produces a broad optimum withlittle variation in yield around this point, this is in fact why the author assumed somescaled lengths instead of optimising them first. Thus the simulations in this section justprovide some information about the pion source characteristics along the length of therod and a ‘sanity check’ on the MARS results compared to theory.

Figure 4.13: Useful pion production from tantalum rods of various lengths, excluding
reabsorption, together with inverted exponential fits using the 11.18 cm hadronic interaction length of tantalum as the decay constant. All other simulation parameters are as intable 4.1.
73

Figure 4.13 shows the simulated production of pions weighted by the usefulness criterion developed in section 4.1.1. Without reabsorption, these should fit an exponential
law that for a rod of length l gives
Production = A(1 − e−l/B ).
This model is derived from assuming the incoming protons have an equal probabilityof interacting in each length element of material and when they do interact, the same
number of pions are produced at exactly that point along the rod. The number of protons
remaining thus decays, as does the pion production at any position going along the rod,so the integrated production becomes an inverted exponential. A suitable choice for B
would be the hadronic interaction length given in table 4.3 and this is used for the fit lines
shown on the graph, though B can also be made a free parameter.
With B fixed at 11.18 cm, the best fit values of A for the two signs are A+ = 0.00883
and A− = 0.00948 useful pions per proton·GeV. The RMS deviations of data values fromthe fit are 3.87% for positives and 2.49% for negatives, expressed as percentages of therelevant A value, which sets the scale of the function. As the protons are at 10 GeV, there
are roughly 0.076 useful pions per proton (for l = 20 cm) and thus the effective statistics
are about 7600 pions from the original
105 protons. Therefore the statistical fit error√
would be expected to have size 1/ 7600 = 1.15%, which indicates this data has a small
systematic deviation from the fit. This can be seen by carefully examining graph 4.13,
which shows the fit lines overestimating production early on but underestimating it later.
Allowing the value of B to also become a parameter in the least squares fit improves thesituation somewhat. The two signs now have parameters A+ = 0.00938 π/(p · GeV), B+ =13.83 cm and A− = 0.00980, B− = 12.70 with RMS errors of 1.70% and 1.26% respectively.
This is about as close a fit as the statistics will allow. It should be noted with cautionthat this does not mean the nuclear interaction length is ‘wrong’, just that there are othercomplexities in the pion-generation process taking place in a rod of finite thickness thatare not modelled exactly by the simple exponential fit. For instance there is likely to bea forward emission angle so pions would be offset from their point of production whencounted at the rod surface. For interpolation purposes, the fits with modified B are still
useful.

4.3.1

Variation of Heat Deposition with Length

The simulations of different length targets in the previous section also generated output
files of energy deposition and outgoing particle distributions. This means that the sectorsof the inner pie chart in figure 3.3 can be reconstructed for each length and the dependency
on target length examined. Again, these simulations do not include contributions fromre-entrant pions that have spiralled back into the target due to the magnetic field, and ina similar way the criterion of ‘5 degrees angle of emission’ is measured for protons at therod surface assuming they continue in straight lines.
Figure 4.14 shows the variation of two energy measures with rod length. The amountof heat deposited in the target increases with length but levels off, which can be explained
if only a few particles penetrate to large distances to deposit heat, with the majority of74

Figure 4.14: Variation of some of the heat loads shown in figure 3.3 with length of thetantalum rod. The 20 cm point corresponds to the reference simulation, with the points
on either side retaining the baseline parameters from table 4.1 except for rod length.
energy having left the rod in the form of secondaries (corresponding to the inner blue
segment in figure 3.3). It also looks like it follows an inverted exponential curve as the
production figures did in the last section. Using the 11.18 cm nuclear interaction length
to set the exponential scale again, the best fit is achieved for
H = 0.1575(1 − e−l/11.18 ),where l is the target length in centimetres and H is the heating in the target as a proportionof incident beam power. The RMS error in this fit is 7.28% so it is not particularly good.
Allowing the length scale to vary gives H = 0.1787(1 − e−l/16.75 ) with an error of 2.52%,
which is better but the increase is still not fitted by a simple decaying exponential. The
reason for this can be found in figure 4.12: if one of the central plots (there is no plotcorresponding exactly to the 10 GeV used here) is truncated at a certain length, theheating in a rod of that shorter length will be the integral of the heat deposition over that
smaller volume. The heating density takes a few centimetres to reach its peak, unlike an
inverted exponential that has the highest rate of change at the beginning. So there may
be no simple model for the heat load as a function of length here.
The other line in figure 4.14 counts heat from protons at small angles that have notundergone kinematically severe interactions so emerge either from the end or glancingly
out of the side of the rod having only been slightly scattered. For short rods there
will be a large contribution from protons that have gone from end to end and these arethe ‘uninteracted’ protons used in the exponential model of yield in the last section. The
decrease of this heat measure looks superficially like exponential decay but does not reduce
fast enough at large rod lengths because of an accumulated population of protons that75

Table 4.5: Heat deposition in the target and power in the forward proton beam dump forvarious lengths of tantalum rod. Kilowatt figures are scaled for a 4 MW proton source.
Target length (cm) Heat deposited
5
3.7%
10
7.8%
15
10.8%
20
12.9%
30
15.3%
40
16.2%
50
16.6%

kW Protons within 5◦
147.9
69.8%
311.4
48.8%
431.6
33.9%
514.4
24.3%
611.1
13.9%
648.1
9.7%
662.8
8.1%

kW
2792.91950.11357.3
974.0
556.0
386.6
325.3

have scattered out of the sides of the rod and thus are no longer subject to progressive
absorption by the rod material. A fit will not be attempted for this reason, but the valuesof this and the other heat measure are tabulated in table 4.5.
Both of these heat factors are of concern to the target station design: the heating
for the integrity of the rod itself and the forward protons for their beam dump (seesection 3.3 for more detailed discussion). The amount of heat in the forward protons
varies quite steeply with increased rod length, meaning that if the forward beam dump
becomes difficult, increasing the target length could reduce its heat load by a factor of
2–3×. This would not impact pion yield as steeply, though care must be taken withreabsorption effects not included here.

4.4

Rod Radius

The radius of the cylindrical target could be other than the 1 cm used so far. The advantage of using a wider target is that if the incoming proton beam size expands proportionately, the energy deposition density decreases by the square of the radius, potentiallymaking thermal shock issues much more manageable. The disadvantage is that the thickened target can absorb more of the pions generated in its interior before they exit. To
take advantage of the former point, the simulations in this section have kept the proton
beam radius equal to rod radius as it varies, which also avoids excessive yield loss fromabsorption because protons will be producing pions all the way to the surface.
Output particle sets were obtained from MARS for both low (2.2 GeV) and high
(30 GeV) proton beam energies, for both pion signs, with all other parameters as in table
4.1. The useful pion yield was calculated by the formula used throughout this chapter
and the results plotted in figure 4.15. The different energies have different overall levels ofyield, with a charge asymmetry in favour of positive pions for the 2.2 GeV case since at low
energies the charge excess from positive protons is still visible above the pion charge-pair
production. For radii of 1 cm and above, the graphs show a steady reduction as the radiusincreases (barring some statistical noise), which can be explained by absorption of pions
by tantalum nuclei in the target. A more surprising feature is that the yield also starts
76

Figure 4.15: Effect of varying the radius of the cylindrical tantalum target on useful pionyield. Graphs are plotted for two different incoming proton energies and both pion signs.

Figure 4.16: Useful pion yield variation with radius as a proportion of the r = 1 cm case.
to fall off for very thin rods. This is because tantalum and similar elements are high-Ztargets and charged particles passing through a medium experience multiple Coulomb
scattering with cross section proportional to Z 2 for the material. This cross section is also
multiplied by the material density when calculating the number of interactions, so sincetantalum is both high-Z (Z = 73) and dense (ρ = 16.7 g/cm3 ), it scatters the incoming
protons quite strongly. For thin targets, this scattering is enough to knock a significant
number of protons out of the target material by the time they reach the end 20 cm away,
which results in a decrease in pion production.
Figure 4.16 isolates the radius dependence in the four scenarios by taking the ratiowith the 1 cm situation in each case. It is difficult to optimise further than the 1 cm
radius case; although all points between 0.5 and 1 cm radius have comparable yields, heat
deposition density scaling as the inverse square of radius would be up to 4× as large.

77

Figure 4.17: Useful pion yield as an increasingly-large hole is bored along the axis of thetarget, simulated with a uniform density proton beam (protons going through the hole
are lost).

4.4.1

Rod with a Hole

During the investigation of the target yields, it was suggested that boring a hole along
the axis of the rod might reduce on average the path length in material for emitted pions
and thus decrease reaborption. This would waste protons that go through the hole, but
it was suggested that a hollow phase space distribution could be created for the protons
that minimised the number in this area. Therefore some MARS simulations were run fortargets with various sized holes.
The standard target simulation (table 4.1) was modified in one important way aside
from drilling a hole: the other simulations so far have used a parabolic proton distribution
with a varying density, higher at the centre and decreasing to zero at the edges, used torepresent a standard proton beam from a synchrotron. However, the beam for hollow
targets will be optimised for the purpose, so this is no longer relevant. The simplest
way to investigate any improvement from the hole without having to worry about proton
accelerator issues is to use a uniform distribution, so the proton flux is the same everywhereon the front face of the target, including the hole where a known proportion (rhole /r)2 arewasted and miss the target completely. Thus in figure 4.17 the results for 0 ≤ rhole ≤0.8 cm are plotted for both signs and the low and high proton energies used in the lastsection. This figure on its own is not particularly useful as the curves look similar toscaled graphs of the non-wasted proton beam proportion 1 − (rhole /r)2 and the questionis whether the yield per impacting proton has been improved.

78

Figure 4.18: Useful pions per proton hitting the target rather than the hole, displayed asa ratios to the yield efficiency of the whole-target case.
Figure 4.18 shows the same pion yields divided by the proportion of protons hitting thetarget for each hole radius and normalised to the case with rhole = 0 cm (a whole target).
It shows that the efficiency of the target is, if anything, made worse by the incorporationof a hole, so this possibility for the target design may be excluded. The worsening of theefficiency for the targets with the largest holes could be explained by the out-scatteringof protons from the remaining target, the same effect that made targets with very small
radii inefficient.

4.5

Proton Beam Distribution

The previous subsection (4.4.1) alluded to the fact that the transverse distribution of
the incoming proton beam was not in fact fixed but might be tailored to the target, tothe extent possible, if this would increase the useful pion yield. This beam manipulation
would probably be done with optical elements in the transfer line taking the extracted
beam from the final proton ring to the target. Quadrupole magnets control the beamfocussing in linear terms and can change the proton spot size in both axes (investigated in
section 4.5.1) and higher order magnets such as octupoles can apply nonlinear distortions
to the beam distribution, creating for example a beam with more particles towards theedges than in the circular parabolic case. A uniform beam is one such example and iscompared in section 4.5.2.

4.5.1

Proton Beam RadiusFigure 4.19 shows the variation in useful pion yield when the target radius is kept at
1 cm but the proton beam radius is scaled to 3 mm or 1 mm as well as the full-size case.
There is very little effect on either pion sign, so statistical error bars of one sigma fromthe simulation were plotted. The most significant difference is between the 3 mm and1 cm beam radii for the negative pions and here the difference is 2.9σ in favour of using
a full-sized beam. This would be consistent with the theory that pions coming from thesmaller beam would always have to travel through at least 7 mm of material (6% of a
hadronic interaction length for tantalum) before leaving the rod, so may be reabsorbed
more, but the difference is too small to make a conclusion.
79

Figure 4.19: Effect (or rather, non-effect) on yield of having the incoming proton beamsmaller than the 1 cm radius of a tungsten target cylinder. With 1σ statistical error bars.

Figure 4.20: Comparison of pion yields generated by uniform versus parabolic transverse
proton beam distributions hitting the tantalum target, at two energies. 1σ statisticalerror bars are shown.
80

4.5.2

Circular Uniform Beam vs. Circular Parabolic

Changing the proton beam distribution from the circular parabolic with density ρ(r) ∝1 − (r/R)2 to a uniform density (ρ(r) ∝ 1) beam will mean more particles near theedge of the rod, though the difference will not be as dramatic as between the 3 mm
and 1 cm radii parabolic beams in the last section. Figure 4.20 compares the yields forthe two distributions, for both signs of pion and two proton energies, with one sigma
statistical error bars shown on each value. The difference between the bars normallyfalls within the error bars, the most deviant case being the 30 GeV positive pions where
the circular parabolic distribution yield is larger than the uniform by 3.4σ. As in theprevious subsection, the conclusion is that the difference in yield from changing the proton
distribution is too small to be significant in these simulations and is certainly less than
10%. Quite possibly this is too small to be significant at all in the design.

4.6

Downstream Effects

All the other sections in this chapter consider the optimisation of the target decoupledfrom the rest of the system, based on local variables such as the momenta of the pionsat the rod surface. This section looks at two effects where there is nontrivial coupling
between the proton beam energy on the target and the downstream front-end. Section
4.6.1 examines the contribution of the decay chain of kaons, only produced for higher
proton energies and mostly decaying outside the target. Section 4.6.2 examines how thelocation of beam losses changes in the front end as the proton energy is varied.

4.6.1

Pion Production from KaonsThe charged kaons K + and K − can be produced in a target bombarded by a high-energy
proton beam and have a mean lifetime of 12.4 ns (compared to 26.0 ns for the chargedpion). Since they are produced at relativistic speeds, they will travel several metres
before decaying and so will not decay within the MARS target simulation.

Table 4.6: Principal decay channels of the kaon [27], together with the number of pionand muon products and thus the expected number of eventual muons from each channel.
K + decay toProbability
#µ #π
+
µ νµ
63.39±0.18%
100 +π e νe4.93±0.07%00π 0 µ + νµ3.30±0.06%
10π+π021.03±0.13%
0
1
+ 0 0
π π π
1.757±0.023% 01π+π+π−
5.59±0.05%
0
3
Total
(100%)

81

Mean eventual muons
0.6339±0.0018
00.0330±0.0006
0.2103±0.0013
0.0176±0.0002
0.1677±0.0015
1.0624±0.0028

Figure 4.21: Comparison of the number of pions and kaons produced by the tantalum
rod target at various proton beam energies.
The major decay channels of K + and their respective probabilities are shown in table
4.6. Many of the decay channels produce charged pions or muons, which will contribute to
the neutrino factory yield. The charged pion itself has a 99.98770±0.00004% probabilityof decaying into a muon [27], so effectively the numbers of the two are added together,minus a tiny factor. This arithmetic reveals that on average, a charged kaon will ultimately
produce about 1.06 muons and is therefore roughly equivalent to having a pion in the same
location.
Figure 4.21 shows the total number of charged kaons as well as pions produced by
the target. Below 4 GeV, no kaons are produced at all, whereas at the highest energies
the positive kaons are adding another 6% to the total particle count and the more rarely
produced negative kaons contribute 3%. There is a slight mixing between the signs introduced by the K + → π + π + π − decay but this will only account for 5.6% of the decaysof the few percent of kaons, so is negligible. Thus kaons produce a small but systematicincrease in yields for higher proton beam energies, particularly for positive particles.

4.6.2

Redistribution of Losses

While previous sections dealt with beam transmission, the pions and muons that are lost
will produce radioactive hot spots in the muon front end due to beam loss. The dumpingof protons was discussed in section 3.5 and there will be a large level of general activation
around the immediate target area from neutrons and other particles, but losses frompions and muons present a different problem because they are the intended particle forthe focussing channel and can thus propagate a long way downstream before being lost.
82

Figure 4.22: Losses of pions and muons redistribute themselves through the decay channel
as the proton energy on the target is varied. Values are plotted as a fraction of the initial
number of pions from the target; the total is slightly more than 100% due to decays suchas K + → π + π + π − .
Therefore the Muon1 code was run on a particular pion decay channel and phaserotation system, specifically the optimal lattice from the PhaseRotC optimisation discussed
in section 8.3, with a logging mechanism to count the weight of lost particles along theaxis of the channel. Based on their location, the losses were put into categories andplotted as proportions of initial pions in figure 4.22, which also includes small strips at
the bottom to represent on- and off-energy particles reaching the end of the beamline(the off-energy particles will be lost even further downstream). ‘Backwards’ losses arethose from pions travelling in the wrong z direction, so they hit the beam plug before
the S1 20 T solenoid (D1 refers to the first drift space before S2, the lower field solenoid
that follows). ‘Reimpacted’ losses are those that are reabsorbed in the tantalum cylinder
target according to a simple absorption-length model, though [62] has a full study of this.
This exercise was repeated for all the proton energies for which MARS output particle
sets were available (see [63] for the complete report). Kinematically it might be expectedthat the pion angular distribution will change as the forward proton momentum increases
and this is consistent with the trends appearing in the graph. At low proton energies more
(up to 61%) of the pions are lost within the first solenoid (S1) and target area, whereas
at high energies this decreases to 28% but the losses in the decay channel excluding S1
have more than doubled from 23% to 55%. So the increased boost from the proton
is producing pions at more forward angles that are lost further down the decay channel.
Beyond the decay channel, the amount of remaining beam at each stage exhibits a humped
shape with an optimum in the mid-energies and this coupling to proton energy no longer
changes, indicating that the beam has been cut down to an accepted region where only83the magnitude of yield within the region is important and not its precise distribution.
This suggests the beamline beyond the end of the decay channel is decoupled fromthe choice of proton beam energy other than by the scaling in pion yield, which means
optimisation of those later sections can be done independently of the proton driver choice.

4.7

Summary of Optimal Target Parameters

The values in table 4.7 summarise many considerations about the performance of thetarget. The proton energy, target radius and lack of a hole are chosen to optimise pionyield, with the important proton energy checked again with a different code in the next
chapter. The proton beam radius and rod angle have a weak effect on yield (plots for rod
angle are shown on the final page of [62]), so the beam radius is chosen to reduce thermal
shock and the rod angle to mitigate the proton hot spot on the beam dump. The proton
distribution is likely fixed at roughly circular parabolic by the proton accelerator itself
but this was found not to be a disadvantage for pion yield.
There remain many feasible choices for the target material, as the lines in figure 4.9
achieve comparable peak performances for achievable proton energies (this excludes the
1 GeV carbon peak). This report assumes a high-Z material similar to tantalum or tungsten, while [61] examines the optimisation in the carbon target case and figures for carbon
are also given. Section 4.3 shows the exponential levelling off of yield with increasing number of hadronic interaction lengths in the target but does not account for reabsorption ofpions hitting the target for a second time, since this requires tracking of pions outside thetarget including a magnetic field. As the magnetic field will be varied in a later optimisation using Muon1, it is unknown at the point of generating the initial pion distribution.
However, other studies including [61, 62] find that the peak yield is for practical purposesvery close to the ∼1.8 interaction lengths given in table 4.3.
Table 4.7: Recommended neutrino factory target parameters, assuming a solid target.
Component

Parameter
Energy (Ek )Proton beam Radius
Distribution
Material
LengthTargetRadius
Shape
Angle

Recommended value
5–15 GeV
Equal to target radius
Circular parabolic
Tantalum used here (others possible)
20 cm for Ta, ∼60 cm for carbon
1 cm for high-Z, 1.25 cm for carbon
Solid cylinder with no hole
14 mrad to accelerator axis

84

Source§4.2, §5.1, [59]§4.5.1§4.5.2§4.2
Table 4.3, [61]§4.4, [61]§4.4.1§3.5, [62]Chapter 5
Benchmarking Pion Production
When predicting pion production figures in the regime of the neutrino factory, it is important to realise that before the HARP experiment [4], there was no systematic attempt tomeasure pion yields in this energy domain. Thus the MARS15 cross section models had
to be fitted to what little data there was available, with some interpolation based on how
proton cross sections behave (see the MARS15 manual [64] section on hadron production).
GEANT4 was in a similar situation. Section 5.1 sees how close these different models areto each other and 5.2 attempts a comparison of MARS15 with recent HARP data.

5.1

Comparison with GEANT4

Without a code available that incorporates the HARP data, comparing with the resultsof another code will roughly gauge the uncertainties present in the current models. The
GEANT4 code [3] is also commonly used by particle physicists to model the interactionof particles with matter, so it was used as a comparison point for MARS.
Results in this section are from a collaboration with K. Walaron who wrote a morecomprehensive report about benchmarking GEANT4 with MARS15 on this problem [65].
The findings were also presented in a paper [59] and the author’s talk at the NuFact’05
conference.
The standard target problem specified in table 4.1 was reposed as a GEANT4 input
data set and run for the same range of energies as the MARS15 simulations in section
4.2. The generated pions were weighted by the process defined in section 4.1.1 and theTable 5.1: The proton energy ranges over which pion yield is near-optimal, as predicted
by MARS15 and GEANT4 for each sign of pion.
Code
Optimal π +MARS15 5–30 GeVGEANT4 4–10 GeV

85

Optimal π −5–10 GeV
8–10 GeVFigure 5.1: The results from two different Monte Carlo codes in predicting the yield ofuseful pions from the target as a function of incoming proton energy.
resulting figure of ‘useful pions’ plotted on the same graph in figure 5.1. Although theform of the graph with an optimum at intermediate proton energies is similar between
the two codes, there is a factor of 1.35–3.05 (average 1.85) difference in absolute yield
between the two, meaning that the predictive accuracy of these codes is still at a fairly
rudimentary level in this situation. Both codes were run using default hadronic model
settings, the effect of changing these in GEANT4 is explored in the next section.
Fortunately, the conclusions relevant to the neutrino factory about the optimal energy
for the proton driver are not contradictory between the two codes and are summarised intable 5.1. The energy range of 8–10 GeV is safely within the optimal range for both signs,
according to both of the codes used here. This is as good a recommendation as possibleusing tools at this stage of development and was the one given to the International Scoping
Study [24] in 2006 when defining their baseline parameter set for the neutrino factory.

5.1.1

Energy Cutoffs in Hadronic Models

GEANT4 provides several modes called ‘use cases’ for selecting the models used at different energies for hadronic particle production models. Multiple models may be necessary
because cross sections involving protons cannot be calculated from first principles andmust use either a simplified physical model or a fit from measured data, both of which
may only be applicable in a certain energy range.
86Figure 5.2: Comparison of all the available hadronic modes of GEANT4 together withMARS15 results for the total pion yield from the standard tantalum target.

87Figure 5.3: Diagram of energy cutoffs between hadronic models used in different GEANT4‘use cases’. The MARS15 model is linearly mixed from the cascade-exciton model(CEM2003) at 3 GeV to the MARS inclusive hadron production model at 5 GeV.
To see the effect of use cases on the results, all the possible choices were run for a range
of energies and the total (uncut) pion yield plotted in each case, resulting in figure 5.2.
It shows that the binary cascade (BIC) models diverge from the rest of the predictions,
including MARS, as the proton energy decreases. The Bertini cascade (BERT) models
also have a different shaped energy dependence but it is not as severe. These results
are provided for information only here, as a detailed dissection of the GEANT4 code’s
hadronic sector is beyond the scope of this thesis.
Figure 5.3 shows how MARS15 and the various GEANT4 modes use different models at
different energies; the 25 GeV cutoff in GEANT4 may not be as sharp as depicted becausesome difference at 10 GeV was found when changing the high-energy component in figure5.2. The ramp between 3 and 5 GeV in MARS is also noteworthy when interpreting theyield graphs earlier in this chapter, as ‘corners’ in the plots at those values may be an
artefact of the random mixing which the code makes between the two models, which uses
proportions of each model that vary linearly in energy from zero to 100% over that energy
range. Extrapolating the pure model lines from ≤ 3 GeV and ≥ 5 GeV in an example
such as figure 4.7 would probably form two lines that do not meet, showing again thatthese models are not much better than a factor of 2 accurate in prediction of absolutepion yields.

88

5.2

Comparison with Early HARP Data

The HARP experiment [4] was designed to measure the production cross sections of π +and π − from high-energy protons on various solid targets of different atomic number (Z).
The applications for this data include studies of superbeams, neutrino factories and muon
colliders, all of which generate a tertiary beam (of muons or neutrinos) from the pionsthat are produced in large numbers (compared to other possible parent particles) whenhadrons hit a solid target. Importantly, the experiment was also designed to measured2 σ
the double-differential cross section dpdθ, which may be integrated across an area in (p, θ)
space to retrieve a partial cross section σ for the range of interest. This level of detail
enables a full comparison with the output of Monte Carlo code such as MARS15.
Table 5.2: The four proton momenta used in the HARP experiment, converted to kinetic
energies in GeV used for protons throughout the rest of this chapter.
p [GeV/c] Ek [GeV]
3
2.205
5
4.149
8
7.117
12
11.098HARP used protons of four different energies extracted from the CERN PS (Proton
Synchrotron). The energies used are actually specified exactly in terms of the proton
momentum in GeV/c but the GeV kinetic energy equivalents are also listed in table 5.2.
This range is a subset at the low end of the energy range studied in the rest of thischapter. HARP had two detectors: a forward spectrometer that detects pions of low
angle and relatively high energies and a Large Angle Spectrometer (LAS) that detects
generally slower pions at higher angles. The locations of the detectors’ apertures in(pL , pT ) momentum space is plotted in figure 5.4, which also highlights the region of usefulpion yield for the neutrino factory front end. Results from the forward spectrometer onan aluminium target are reported in [66] but this momentum range is of more interestfor superbeams and misses the neutrino factory zone completely. The LAS overlaps with
about half of the useful yield region, missing the areas below its lower θ limit of 0.35
radians. HARP LAS results have been reported for a tantalum target in [67], which isrelevant for the solid neutrino factory target (and of a similar Z to a mercury jet) andforms the basis of the comparison with MARS15 in this section.
The HARP tantalum target is a cylinder and was replicated in a modified MARS
simulation using the parameters in table 5.3. The short length of 5.6 mm is to produce
approximately 5% of a hadronic interaction length in tantalum so that it behaves approximately as a ‘thin target’ (or foil) allowing extraction of the cross section without
too much interference from cumulative absorption, energy loss or scattering of protons
or pions. The dimensions were derived from measurements of the target recorded in thedata sheet at http://www.hep.shef.ac.uk/research/harp/ta5.html.

89Figure 5.4: The pion momentum space plot of figure 4.2 with the acceptance rangesof the two HARP detectors superimposed. The Large Angle Spectrometer (LAS) is thecentral area and the forward spectrometer extends beyond the right edge of the plot
to |p| = 6500 MeV/c. The density of successful pions (white) leaving the channel ishighlighted as grey squares, note this is different from the probability of transmission
plotted in figure 4.5.

Table 5.3: Parameters of the MARS15 simulation of the (small) tantalum cylinder usedin the HARP 5% interaction length study.
Code Used
Target Material
Target Shape
Target Dimensions
Target Mass
Protons Per Monte Carlo Run
Proton Beam Profile

MARS15
Tantalum
Cylindrical
5.6 mm length × 15.1 mm radius
3
66.195 g (ρharp
Ta = 16.59 g/cm derived)
106
Parallel pencil beam at centre of target endface, 1% RMS Gaussian momentum spread.

90

The full range of results across all pion momentum and angular bins, for both pion signs
and all four incoming proton energies are plotted in figures 5.5 and 5.6. Converting the
MARS output, which is in terms of weighted numbers of pions in various bins for a known
number of protons, into a cross section requires knowledge of the number density ρN oftantalum nuclei in the simulated target. The density of tantalum is ρmars= 16.68 g/cm3
Ta
and its atomic mass is A = 180.9, soρmarsρN = NA Ta = 5.551 × 1022 cm−3 ,Awhere NA is Avogadro’s number. Next, given the path length l in centimetres and theobservation that a total weight w of pions were produced from N incoming protons in thesimulation, the cross section may be estimated asw
σ [barns] =.
N lρN 10−24
This formula is also correct if w and σ are only counted for pions within a particular zoneof momentum space. If the zone is a small region such as one of the HARP momentum
bins, the double differential cross section in that bin can be estimated by dividing σ by
the area of that bin in (p, θ) space.
Some more consolidated (and easier to interpret!) graphs are presented in figure 5.7,
which combines the forward and backward parts of the HARP detector, and in figure
5.8, which looks purely at the ratio between the signs in only the forward hemisphere.
Figure 5.7 shows that MARS15 is underestimating the production of pions in backward
directions by a factor of 2 or more, whereas the forward direction has a rather better fit,
though MARS produces a peculiar kink around p = 200 MeV/c in the 3 GeV/c protoncase. Figure 5.8 shows that MARS is producing proportionately more negative pions thanthe HARP data suggests is the case. The HARP data shows an excess of positive pions,increasing at low energies, while the MARS15 data is more equal but also tends to include
more positives as the proton energy decreases.
Finally, the overall pion cross section may be plotted as a function of proton kinetic
energy, to provide a link with the results of section 4.2. This is not quite a ‘pion yield’
because the target here is much shorter than any neutrino factory would use, but the crosssection in the relevant momentum zone may be compared across proton energies as a guide.
The HARP report [67] also performed this analysis and chose the zone with θ < 0.95 radand 250 ≤ |p| < 500 MeV/c to represent the neutrino factory, which encompasses theregion on figure 5.4 where the acceptance overlaps with the detector. This is not totallycorrect because the yield from the most forward pions is missing, but being aware of thiscaveat, the comparison graph may still be plotted, giving figure 5.9 where MARS15 has
used the same cut as HARP. The lines at the top of the graph are total yields in theforward hemisphere of LAS (excluding the p ≥ 700 MeV/c bin as in the HARP report).
The author’s interpretation of figure 5.9 is as follows. In total, MARS produces too
many π − (by up to 24%) and too few π + (by up to 16%) at all energies, but the agreement
at the ∼20% level is encouraging given the large discrepancy with GEANT4 discussed
in section 5.1. When the cross sections are cut nearer to the domain of interest, thepositive pion value never deviates by more than 13% between MARS and measurements,
whereas the situation for π − becomes worse, with MARS always overestimating the yield,
sometimes by as much as 44% in the case of negative pions at 4 GeV.
91

Figure 5.5: π + cross sections in all bins of the HARP Large Angle Spectrometer (black)
together with MARS15 simulated values (red). Compare with figure 23 of the HARP
report [67]. Numbers in the top-right of graphs are the angular bins in milliradians.
MARS error bars are 1σ statistical; HARP error bars are 1σ but also include systematics.

92

Figure 5.6: π − cross sections in all bins of the HARP Large Angle Spectrometer (black)
together with MARS15 simulated values (red). Compare with figure 24 of the HARP
report [67]. MARS error bars are 1σ statistical; HARP error bars are 1σ but also include
systematics.

93

Figure 5.7: HARP measurements of forward (left) and backward (right) pion production
cross sections for π + (top) and π − (bottom), comparable with figure 26 in [67]. MARS15
simulation predictions for the same quantities are in red, with 1σ statistical error bars.
HARP 1σ error bars also include systematics.

94

Figure 5.8: Ratio of negative over positive pions in the forward region of the HARP Large
Angle Spectrometer. HARP measurements are in black and MARS15 simulation results
in red. This is comparable with figure 27 of [67]. MARS error bars are 1σ statistical;
HARP error bars are 1σ but also include systematics.

95

Figure 5.9: Pion cross section per GeV of proton energy as a function of the proton
beam energy. Results are shown both for the entire forward LAS acceptance with |p| <700 MeV/c and for a cut with θ < 0.95 rad and 250 ≤ |p| < 500 MeV/c in the area of
the neutrino factory acceptance. This plot is comparable to figure 30 of [67] without theθ-only cut. MARS error bars are 1σ statistical; HARP error bars are 1σ but also include
systematics.
The energy dependence characteristics for both MARS and HARP data suggest thatthe optimal proton energy is around 5 GeV, lower than previously concluded, with aslightly faster drop off as the energy increases. Note however that the exclusion of forward
pions with θ < 0.35 rad from the calculation may systematically bias towards lower protonenergies where the lack of forward momentum allows more pions to escape with larger
angles. The fact that even the MARS results now favour this lower energy suggests thischange may be due to the difference in cuts rather than a difference between the codeand reality. Additionally, the slope between 5 GeV yields and those at 10 GeV was notvery steep in previous MARS results, so it could be reversed by quite a small effect. Oneenergy dependence feature that is definitely different between code and reality is how the
MARS results fall much more rapidly than HARP when going from 5 to 3 GeV/c proton
momentum. This is due to the interpolation between hadron models that kick in for
MARS at about this energy, as illustrated in figure 5.3.
Overall, the agreement between MARS15 and HARP data in the pion momentum andproton energy ranges of interest, with the correct target material, is very encouraging
for positive pions (deviations of less than 13% in these figures of merit), whereas MARS
overestimates negative pion production here by about 30%, which should be taken into
account in future. Later releases of MARS and GEANT will hopefully integrate moresophisticated fits from the HARP data and allow more accurate evaluation, perhaps at
the 10% level or better, of the neutrino factory pion yield.
96Chapter 6
Muon Energy Loss and Scattering
Algorithm
The beam in the neutrino factory front end is larger and more divergent than those foundin conventional accelerators. The emittance of a beam (defined in section C.4) is a measureof the beam size and divergence that is invariant under optical elements such as magnets,
so can only be reduced via other means. D. Neuffer in 1983 therefore proposed muon
ionisation cooling [68, 69], originally intended for muon colliders, that uses the energy
loss of muons in a low-Z material to reduce the emittance. However, the material also
causes scattering of the muons, which increases the emittance. After presenting analytic
approximations of the resultant cooling rate along a section of beamline, the main resultof this chapter is an accurate computer model of scattering using cross-sections of themuon–LH2 interaction that can be integrated into tracking codes.
Although this cooling model has been integrated into Muon1, it was found that there
exist promising front ends without cooling (see chapter 8), provided a muon linac is used
for further acceleration. Thus the main results of this thesis do not include muon coolingbut performance of the cooling algorithm on a 30 cm LH2 block is presented in this chapter.

6.1

Principle of Muon Cooling

The muons from the decay channel have transverse momenta of up to 200 MeV/c, whereas
beams in conventional accelerators have particles with much lower pT , which can behandled with weaker magnets. For example the ISIS [50] proton accelerator covers theenergy range 70–800 MeV (of the same order as the muons) but the pT of the protons
does not exceed 30 MeV/c. Optically, pT can be decreased by defocussing the beam to alarger spatial size but the muon beam is already very large (∼30 cm width) in the phaserotation channel, so that method of decreasing pT is not an option.
Liouville’s Theorem prohibits a system of pure electric and magnetic fields from reducing the amount of position-momentum phase space occupied by a beam with a verylarge number of particles. Without going into detail, this is because these systems can bemodelled as a Hamiltonian system and Liouville’s Theorem is a mathematical consequence97

of Hamilton’s equations [70].
For muon cooling and other applications where this phase space volume must be
reduced, exceptions to the conditions of this theorem must be found. One way is to noticea beam is not a continuum of infinitely-many particles and statistical effects may be used.
A famous example of this is stochastic cooling of protons [71], which relies on the small
deviation of the centroid of the beam due to statistical fluctuations to create an opposing
electric field, cancelling a small amount of the beam spatial spread and over very many
revolutions reducing the beam size in a ring significantly. This method becomes slower
in proportion to the square root of the longitudinal density of particles, typically taking
seconds for proton beams, so would be unsuitable for the neutrino factory’s high-intensity
muon beam given the muon’s short lifetime. †Another class of exceptions to Liouville’s Theorem is when the dynamics are not fully
represented by a Hamiltonian system. Introducing an additional process, very often aninteraction with matter, can give systems of this sort. One example is ‘charge-stripping
injection’ in which an ion such as H− encounters a thin foil and has its electrons stripped
off, giving a proton. This is an irreversible process as protons reencountering the foil
are just scattered slightly. The effect is that many copies of an original H− beam may
be superimposed on the same phase space for protons because they come from outside
the protons’ Hamiltonian system using the foil; superimposition of the two different-sign
beams is easily done using a dipole magnet because they bend in opposite directions.
Scattering is also a non-Hamiltonian process as the scattering kick given to each particle
is random and not a function of its canonical coordinates: this increases the phase spacevolume and results in beam ‘heating’.
The primary mechanism used in ionisation cooling is the energy loss of particles as theycollide with those of matter, causing the matter to heat up slightly. This is a member ofthe class of non-Hamiltonian exceptions to Liouville’s Theorem. It may not seem obviousthat deceleration in matter is a non-Hamiltonian system, as it can be described by smoothdifferential equations under some circumstances. Assuming the material is homogenousand − ∂H
, must not depend on x or t.
and stationary, the formulae for ẋ and ṗ, that is ∂H
∂p
∂x
∂g
∂H
If ∂p is only a function of p then H = f (p) + g(x, t). Now ṗ = − ∂H
= − ∂x(x, t), but this∂xmust also depend only on p, thus is a constant, which is inconsistent with decelerationof particles in all directions because ṗ must be negative for positive p and vice versa. So
energy loss in materials is a possible candidate for non-Liouvillian reduction of the phasespace volume (‘cooling’) of the beam.
The reduction of momentum regardless of its direction is the important feature ofenergy loss in the context of a cooling system. Removing energy from a nearly-parallel
beam is equivalent to putting it through a decelerating electric field, which is known to
be Liouvillian, so the important effects can only appear when the beam is put throughthe energy absorber with a range of angles. These are shown as effects on the particle’s
momentum vector in figure 6.1: the particle loses energy in its direction of travel, experi†

The statistical exception to Liouville’s Theorem can also allow emittance growth: repeated nonlinear
forces (or dynamical distortions) acting on the beam can eventually cause the continuous volume in phasespace containing the beam to become so twisted and intermixed with a larger external volume that it
is no longer well-represented by the limited number of particles it contains. At this point, the beam isconsidered to have grown in phase space volume and fills a larger region with a lower density.

98Figure 6.1: Muon momentum is reduced in magnitude (step 1) along the direction of travelby some material before being replenished by forward momentum by an RF cavity (step 3).
This would reduce the pT of all muons if it were not for scattering in the material (step 2)that counteracts the reduction when pT is small enough. Diagram adapted from [74].
ences some scattering (an inevitable material effect that heats the beam) and then can bereaccelerated in a phase-space-conserving way by the electric field of an RF cavity. Noticethat for sufficiently large incident particle angles (relative to the RF axis), or sufficiently
small amounts of scattering, this results in a beam in which all particles have momentum
vectors more parallel to the axis than before. Because the non-Liouvillian absorber ispresent, this no longer comes at the cost of an enlarged beam in positional coordinates,
though it does come at the cost of magnets to provide large angles in the absorber.
The US Feasibility Study II [25] performed a detailed investigation of a channel ofabsorbers, magnets and RF cavities to achieve muon ionisation cooling, resulting in theirdesign of a cooling unit cell shown in figure 6.2. As the Z of the absorber material
increases, scattering increases faster than the energy loss, meaning that the optimal material to use is liquid hydrogen (gaseous hydrogen not being dense enough to have an effectwithin a practical length). This choice implies a complex cryogenic system and containment for the hydrogen, which must also be contained with low-Z (but not liquid) windows,
beryllium being chosen there. To avoid a net deceleration of the beam, the amount of
RF must be balanced to that of absorber and the channel must also be relatively short to
reduce muon decays. Finally, the focussing magnets must be added, solenoids being usedhere for their axial symmetry and short focal length for relatively low-energy beams.
The LH2 design is the one being tested in the MICE experiment [72]. However, themore recent ISS [24] and IDS [73] designs have used lithium hydride (LiH) absorbers,
trading off some cooling performance in the higher Z material in exchange for a simpler
cooling cell with less cryogenics. It is also possible to reclaim some of the lost performanceby having thinner beryllium windows on the RF cavities because the windows are no longer
structural: they are supported by the LiH absorber material.
99

Figure 6.2: Cross section of the cooling lattice cell from US Study II [25], containing
four-cell RF cavities with liquid hydrogen absorbers between. The surrounding solenoid
coils can be seen as rectangular blocks.

6.2

Cooling Rates from Linearised Optics

The effect of the beam passing through matter may be approximated while staying within
the linear optics model described in appendix C. A single transverse plane (x, x0 ) will beconsidered and ‘emittance’ in this section will usually be the normalised RMS emittanceof that plane. As explained in section 6.1, a cooling channel normally consists of reacceleration after the energy-absorbing material: thus the choice of normalised emittance, whichis constant under acceleration. If the un-normalised emittance had been chosen instead,
it would appear to decrease in the accelerating cavities but increase in the material fromscattering, though the net effect would be the same as the material-only effect on thenormalised value, provided the beam ends the channel with the same energy it started
with.
The useful part of material interaction with regard to cooling is the energy loss per< 0, which can be derived from the Bethe–Bloch formula [75].
unit length of material dE
dz
With pure energy loss, x0 stays the same while x continues to evolve as in a drift space,so the only effect on εnrms is via the normalising factor βγ.
dεnrmsdzd
un d(βγ)[βγεunrms ] = εrmsdz
dzd(βγ) dE
1 d(βγ) dE
= εun
= εun
rms
rms
dE dz
mc2 dγ dz
unnεrms dEεdE== 2 rms 22
βmc dzβ γmc dz
=

loss

100

=

εnrms
dE
2
2
β (mc + Ek ) dz

Here, Ek is the kinetic energy of the particle and m is its rest mass (dE is identical for
kinetic or total energy). The formula shows that the negative dEof the material causes adz
reduction in emittance, which cannot go below 0 due to the presence of εnrms on the righthand side. If the particle were kept at a constant Ek (and β) by compensating acceleration,
the predicted cooling without scattering in a uniform material can be evaluated by defining
the cooling rate k > 0:1 dεnrms
dE/dz
d ln εnrms
= n
= 2
dz
εrms dz
β (mc2 + Ek )⇒ εnrms (z) = εnrms (0)e−kz .

−k =The Bethe–Bloch curves give a point of minimum energy loss at approximately βγ = 4.
This is of interest mainly because the energy loss does not change much with particle
energy around this point, so beams with energy spread do not undergo nonuniform deceleration, to first order. For muons this corresponds to Ek = 330 MeV and β = 0.970, a little
higher than the energy in the muon front end. In liquid hydrogen at this point, the energy= −28.2 MeV/m given its density
loss is 3.97 MeV per gram/cm2 of material [27], or dEdz
of only 0.071 g/cm3 . For graphite, the energy loss is 1.74 MeV per g/cm2 [76] and the den= −383 MeV/m.
sity varies depending on grade but is approximately 2.2 g/cm3 , giving dE
dz−1
The cooling rates are therefore kLH2 = 0.0687 m and kgraphite = 0.935 m−1 . At βγ = 2.5
in liquid hydrogen the energy loss per unit length is a few percent higher but kLH2 increases to 0.120 m−1 , mainly through the reduction of Ek to 179 MeV, so the energy lostis a greater proportion of the muon’s kinetic energy. 179 MeV is very near the front end
energy. Continuing to decrease Ek would theoretically increase the cooling rate arbitrarily, but the benefits are lost from the nonuniform deceleration experienced far from theBethe–Bloch minimum and scattering in the material, discussed next.
Collisions with the electrons and nuclei in a material cause frequent random changes
to the incoming particle’s momentum. In the case of muons, these collisions are almost
entirely electromagnetic in nature (as opposed to pions that can also undergo hadronic
processes). For the purposes of the linear model, these will be approximated by a continuous random walk in which hx02 i increases linearly with distance. This generates anincrease in beam emittance. By the Central Limit Theorem, the process will produce a
Gaussian distribution if continued indefinitely from any starting distribution and a Gaussian distribution is the only one that will emerge unchanged apart from an increase invariance. It is also a small-angle approximation that neglects large-angle scatters that
occur in rare head-on collisions with nuclei and will usually lead to the muon being lost.
The starting point is a formula [77] for the RMS angular deflection (projected into
one transverse plane) of a particle emerging from a thickness z of material with radiation
length X0 :


r
z
z
13.6 MeV/c(q/e)
1 + 0.038 ln,
θrms =βp
X0
X0
where q/e is the charge of the particle in electron units and p is the forward momentum.
This comes from a fit to the more comprehensive Molière theory of scattering; the logarithmic term in brackets to the right is not compatible with the simple random walk model101

but fortunately has values within a few per cent of unity for thicknesses of similar order to
X0 . For liquid hydrogen, X0 = 8.90 m and for graphite, 0.193 m, which are similar to thecharacteristic cooling lengths 1/k defined previously, so this approximation is likely to begood for cooling systems. Neglecting this term and using the small-angle approximation
θ ' x0 , the outgoing mean square muon angle isx02(13.6 MeV/c)22= θrms =(q/e)2
2 2
β p

z
,
X0

which increases linearly with z by
d hx02 i(13.6 MeV/c)2 (q/e)2=.
dz
β 2 p2 X0
Recalling the definition of 2D RMS emittance from section C.5, the normalised emittanceunder consideration can be written in terms of beam momentsqunnεrms = βγεrms = βγ hx2 i hx02 i − hxx0 i2 .
The scattering process considered separately to energy loss does not affect the relativistic
factors β and γ; nor does it immediately (without drift) affect the beam size hx2 i or thecorrelation hxx0 i because the scatters are uncorrelated with x position. Only hx02 i remains
as a relevant variable and the effect of scattering on emittance may be evaluated directly
using the chain rule:
 n 
dεrms
dεnrms d hx02 i=
dz scatterd hx02 i dz
q
dd hx02 i2
2 i hx02 i − hxx0 ihx= βγ
d hx02 idz
02
2d hx i
hx2 i d hx02 i
hx i= βγ un.
= βγ qdz
2εrms dz
2
2
02
0
2 hx i hx i − hxx i
Recalling the definition of the Twiss function = hx2 i /εun
rms and using the earlier formula
02
for d hx i /dz,
 n βγ (13.6 MeV/c)2 (q/e)2
dεrms(13.6 MeV/c)2 (q/e)2 γ==
dz scatter2
β 2 p 2 X0
2βp2 X0(13.6 MeV/c)2 (q/e)2 γ(13.6 MeV/c)2 (q/e)2==.
2β 3 γ 2 m2 c2 X0
2β 3 (mc2 + Ek )mX0
Combining these results gives the overall change in emittance in a uniform absorbing
material with both scattering and energy loss [78]:
 n 
 n 
dεnrms
dεrms
dεrms
εnrms
dE (13.6 MeV/c)2 (q/e)2
=
+
= 2
+
.
dz
dz loss
dz scatter β (mc2 + Ek ) dz
2β 3 (mc2 + Ek )mX0
This consists of a negative part that is proportional to εnrms and a positive part that isconstant. Ignoring kinematic factors, the negative (cooling) term is maximised by choosing
a very negative dEand the positive (heating) term is minimised by choosing a large X0
dz102and small in the cooling medium. Note that the is actually a function of z dependingon the optics of the beam in the absorber, but with many absorbers it can be replaced byan averaged constant value.
The cooling equation can be solved to give an offset exponential:⇒dεnrms= −kεnrms + hdz
εnrms (z) = (εnrms (0) − h/k) e−kz + h/k= (εnrms (0) − εeq ) e−kz + εeq ,where the cooling rate k is defined as before, h is the heating term and the equilibrium
emittance(13.6 MeV/c)2 (q/e)2
εeq = h/k =2βmX0 (−dE/dz)
is the value that εnrms converges to as z → ∞. If εnrms starts below εeq then the scattering
dominates the cooling from energy loss and the beam emittance grows. Note that (q/e)2 =1 for muons.
For solids and liquids, it is the reaccelerating RF systems and not the energy loss inin a channel can be. It is also
the material that determine how large the average dE
dzpossible to cool using a large amount of gas but this time higher pressures (∼100 atm)
are needed to prevent the slow energy loss from making the cooling channel impracticallylong. When the beam emittance is much larger than εeq , the average energy loss rate is
the dominating factor in how rapidly the cooling progresses, so the channel is optimised
for the limited apertures of the RF cavities to allow as much reacceleration as possible.
Liquid hydrogen is often cited as a good cooling material [72] because it has the lowest
X0 of any solid or liquid, the low scattering allowing beams to have relatively low x0 in
the absorbers. To optimise the performance of cooling in the final, low-emittance stages,
εeq can be reduced by choosing materials with the highest possible product X0 (−dE/dz),
which favours slightly higher-Z materials such as graphite. At this point the emittanceis assumed to be small enough that optics with high x0 (low ) in the absorbers can beproduced without the beam later becoming too large for the RF apertures.

6.2.1

ExampleThe best yielding front end from chapter 8 produces a beam with RMS normalised emittance εin = 14812 mm·mrad when projected into either transverse phase space plane(table 8.3). Suppose one wishes to add cooling to this front end until the beam canbe transmitted by an FFAG-based accelerator that has a normalised acceptance of only
Aout = 15000 mm·mrad in each plane, instead of a muon linac. Transverse focussing optics
suitable for cooling channels are described in detail in [79]; here the absorber and beta
function from the start of the Study II [25] cooling channel are used.
The acceptance places a hard cut on particles outside the ellipse of π ptimes that area,
whereas the ellipse corresponding to the RMS emittance contains only 1− 1/e = 39.35%
of a beam with a Gaussian distribution. After more than one cooling length in a coolingchannel, the beam will become very close to a Gaussian, so that distribution is assumed
103

here. If 95% of the beam is required to be within the 2D ellipses of both planes, it issufficient to ensure that 97.5% is within each one. The amount of a 2D Gaussian outside n2sigma is e−n /2 and setting this lost amount to 2.5% gives n = 2.716. Since Aout = n2 εout ,the required normalised RMS output emittance εout = 2033 mm·mrad is derived.
The central beam momentum in the channel is 200 MeV/c, corresponding to a kinetic
energy Ek = 120.5 MeV. Each 2.75 m cell contains a 35 cm thick liquid hydrogen absorber,plus enough RF cavities to restore the lost beam energy. The beta function minimum
where the absorber is located is = 47 cm. Using the data for LH2 given previously
in this section and X0,LH2 = 8.81 m, calculated from Tsai’s formula [80], the equilibrium
RMS emittance is
εeq =(13.6 MeV/c)2 0.47 m= 0.001873 m = 1873 mm · mrad,2(0.884)(105.7 MeV/c2 )(8.81 m)(28.2 MeV/m)which is definitely small enough for this example. The length of cooling material required
can be found by setting εnrms (0) = εin and εnrms (z) = εout :1εin − εeqnn−kz
εrms (z) = (εrms (0) − εeq ) e+ εeq ⇒ z = ln.
kεout − εeq
For the muon energy considered, k = 0.159 m−1 and the log term is ln(80.7) = 4.39,
giving a total material length of zLH2 = 27.53 m required for cooling. To get this thickness
requires 79 of the 35 cm Study II absorber units and therefore 79 cooling lattice cells of
2.75 m length each, giving a total cooling channel length of 217.25 m.
Finally, note that since βγ = 1.89, the input beam’s 99% normalised emittance of
97455 mm·mrad corresponds to an un-normalised emittance εun99 = 51485 mm·mrad at thestart of the cooling channel. p
The radius of the beam at the absorber may be calculated
εunusing the beta function: r =99 = 15.6 cm. The absorbers specified in Study II havea window diameter of 360 mm (r = 18 cm), which is large enough for the beam from thefront end.

6.3

Using the ELMS Scattering Model for Liquid Hydrogen

A model of the interaction of muons with matter is required in order to simulate the coolingchannel in Muon1. These interactions consist of different types of collisions by the muons
with atomic electrons and nuclei, each of which transfers a certain amount of transverse
and a certain amount of (backward) longitudinal momentum to the muon. These also
depend on the incoming muon momentum,
so abstractly a material M can be described by
RR
a function FM (∆pL , ∆pT ; p) where
FM (∆pL , ∆pT ; p)d∆pL d∆pT gives the frequency ofcollisions (in inverse metres) along the path of a muon with momentum p. Integrating overa specific range of (∆pL , ∆pT ) momentum kicks gives the partial frequency of collisions
in that range. This function is related simply to the double differential cross-section by
the molecular number density ρM of the material:
FM = ρM

d2 σM.
d∆pL d∆pT

104

(a) Magnitudes of the delta function elements shown as colours.

(b) Location of the soft/hard collision cut.

Figure 6.3: Elements of the ELMS [81] muon–LH2 cross-section for a 200 MeV/c muonon a log-log plot. The diagonal lines with a sharp downward bend at the end (‘nuclear’and ‘electron’) would be half-ellipses on a linear graph.

105

The ELMS software [82] was developed at Oxford for the purpose of tracking muons ina realistic manner through a block of LH2 material. Previous treatments condensed many
physical processes into typically just two: longitudinal momentum loss of the muon as itis retarded by the material and multiple scattering, often modelled in the assumption ofa large number of soft collisions, producing a Gaussian random walk in transverse momentum. ELMS uses photoabsorption cross-sections in conjunction with atomic physics
to determine the full form of FM (reported in [81]), in which scattering and energy loss
are no longer independent processes but instead each collision will cause a little of each.
A log-log plot of the function FLH2 (for liquid hydrogen) with p = 200 MeV/c is shownin figure 6.3. The dotting in some regions is because FM is a probability distribution andis approximated in ELMS by a sum of delta functions: colours in the plot represent themagnitude of each of these and are also on a log scale.
Given this input, tracking a muon through the material is just a matter of sampling
the probability distribution and applying the momentum kick at each collision. Theseoccur as a Poisson process along the path, with rate equal to the integral of FM for thecurrent muon momentum. Unfortunately, this rate is 9.7×105 m−1 for a 200 MeV/c muon,
which leads to slow tracking since this is typically 2500 collisions per Muon1 timestep.

6.4

Accelerated Scattering Tracking Algorithm

To increase the speed of the scattering model, Muon1 separates the relatively rare ‘hard’collisions, in which the muon momentum and direction can change dramatically, from thefrequently occurring softer collisions. For soft collisions, both the paraxial approximation(small angles) and the Central Limit Theorem (frequent collisions with no long tails in
distribution) apply, meaning that the evolution of the muon momentum in the intervals
between successive hard collisions can be treated as a Gaussian continuous random walk.
This is exactly what simpler models such as the one used in section 6.2 do everywhere
but here it has higher accuracy because the statistically ill-conditioned events have been
removed. The expected momentum loss, transverse scattering variance and ‘straggling’variance (statistical variation of dE/dz), which determine the Gaussian ∆p distribution,
can be reconstructed from 1st and 2nd moments of FM (∆pL , ∆pT ; p), where the integrals
are performed over the soft region below the cut:
ZZ
d hpL i
1 dE==
∆pL FM d∆pL d∆pT(momentum/energy loss);
vz dz
dz
ZZ
dVar(pT )=(∆pT )2 FM d∆pL d∆pT(multiple scattering);
dz

2
ZZ
dVar(pL )d hpL i2=(∆pL ) FM d∆pL d∆pT −(straggling).
dz
dzHere, Var(x) is the variance operator defined as h(x − hxi)2 i = hx2 i − hxi2 , a quantitythat adds linearly when independent random variables such as the ∆p from individual
scatters are added to each other. Note that d hpT i /dz = 0 by symmetry.
A suitable cut between soft and hard collisions (shown in figure 6.3(b)) was chosen106Figure 6.4: A pencil beam of muons is simulated entering an LH2 absorber. (top)Transverse view of 100 simulated muon paths, looking into the absorber. (bottom) The
same paths seen from the side, on a much larger scale (figures in metres).

Figure 6.5: Distributions of energy loss and exit angle for 200 MeV/c muons simulated
through 30 cm of LH2 . The green plot is 104 muons run with individual ELMS collisions,
red points are 106 muons using the faster method. 1σ statistical error bars are shown.
107

Figure 6.6: Distributions of energy loss and exit angle for one million 200 MeV/c muons
simulated through 30 cm of LH2 . The red plot uses the faster ELMS-based method developed in this chapter; the blue plot is ICOOL [30] on default settings. 1σ statistical error
bars are shown.
for the integrals. The tracking algorithm models the occurrence of hard collisions as a
Poisson process in path length, calculating those collisions exactly and using the Gaussian
momentum changes from soft collisions on the segments between. The new algorithm
performed 1200 times faster than modelling every collision individually. Muon tracks
such as those shown in figure 6.4 are produced, with distributions coming out of theabsorber that are so far indistinguishable from complete tracking, as shown in figure 6.5.

6.4.1

Comparison with ICOOL

The same problem with 30 cm of liquid hydrogen was run using the code ICOOL [30],which also includes muon scattering and energy loss models. Figure 6.6 shows the result, with ICOOL agreeing with the ELMS-based method for scattering but having a
differently-shaped tail in the energy loss distribution. It should be noted that ICOOL
includes several options for these physics models but the defaults were selected here. The
input file for this ICOOL run is listed in appendix D.

6.5

Integration with Muon1

The issue of combining sporadic events like collisions with the otherwise smooth Runge–
Kutta tracking of a particle curving along its trajectory has been touched on briefly insection 2.5.3, which gives a procedure for finding the collision point with an isosurface.
The exact intersection point is found by a root-finding algorithm in t but the position of
the particle at each of the various t values tested by this algorithm has to be calculated
by invoking the Runge–Kutta algorithm for a partial timestep.
Particle decays are another type of sporadic event and the time distribution of these108

was described in section 2.4, without implementation details. Here Muon1 takes the easy
option: since the timescale of pion decay is much larger than the 0.01 ns tracking timestep,rounding the decay time to the next timestep is not considered a bad approximation anddecays are performed at the end of timesteps for any particles whose internal proper timeclock exceeds their next scheduled decay time. Quantitatively, the error in this method is
to increase the average pion lifetime by less than 1 part in 5000.
The reason for this recap about sporadic events is that the scattering tracking algorithm described in previous sections introduces quite frequent ones, which the Muon1
tracking logic must handle properly. In figure 6.4 the hard scatters are just visible asblack dots marked along the purple trajectories and they occur with a similar frequency
to the ∼3 mm Muon1 timesteps. The soft scatters also need to be added as a Gaussian
perturbation to every substep inside a material.
To describe explicitly the hierarchy of steps and substeps required to take accountof all four effects on the particle’s motion (the three above plus bending from magnetic
fields included in the Runge–Kutta integration), the actual stack of functions called by
Muon1 is presented herein. The stack roughly reflects the size scales at which different
phenomena take place, mnemonically: f ields ≺ scattering ≺ geometry ≺ decays.
• Simulation_advance is the top-level function called by the main loop of Muon1
repeatedly during a simulation. It advances the whole beam by one timestep, including all physics effects. In fact, because Muon1 detects the number of CPUspresent on a computer and splits the beam into an equal number of particles for
each, this function is mostly thread-synchronisation and wrapper code, spawning
copies of Simulation_advance_t in parallel.
• Simulation_advance_t is a single-threaded function that advances a range of particles by one timestep including all physics effects. For each particle, it calls collideparticle repeatedly if necessary, advancing the particle through one contiguous
geometry region at a time until its timestep is over. At this point, decayparticle
is called on it.
– decayparticle checks if a particle is due to decay by its internal proper timeclock and if so, uses the algorithm described in section 2.4.3 to generate its
decay products. It calls no further functions.
• collideparticle tracks a particle to the end of its current geometry region, including magnetic fields and scattering but not decays. The main purpose is to determine
the first surface the particle hits, which marks where it leaves the region. The function loops through all components potentially in the vicinity of the particle (as per
section 2.5.1) and calls the collision function for each, which varies from component
to component; this in turn usually calls scatterparticle_isocd a number of times
to see if the particle’s path hits any of the isosurfaces described in section 2.5.3.
• scatterparticle_isocd determines where a particle hits an arbitrary isosurface,supplied as a function pointer argument. It uses a hybrid algorithm of Newton–
Raphson root finding, root bracketing and bisection (in awkward cases) to find thevalue of t when the particle is on the surface. It is important to note that every109position of the particle used by this algorithm is calculated by a call to scatterparticle over a substep, so it takes full account of magnetic fields and scattering.
• scatterparticle advances a particle by a given δt through a known material. If
the material is vacuum, this passes control directly to advanceparticle, otherwise
it uses the algorithm described in section 6.4 to calculate a sequence of steps between
successive ‘hard’ collisions. The intervening steps are each a call to advanceparticle with the ‘soft’ Gaussian scattering contribution added on to the momentum
afterwards.
• advanceparticle is the core Runge–Kutta routine within Muon1 and tracks a particle through a magnetic field over a specified timestep using the algorithm in section
2.2. It does not take account of material scattering, geometry or particle decays.
The required B field values at various points are obtained by summing the contributions from components suspected to be near the particle, each calculated by thatcomponent’s magnetic field function (if any).

110Chapter 7
Design and Optimisation System
The accelerator design must be specified to the Muon1 code so that the components
and magnetic fields can be correctly placed in the simulation space and the algorithms
described in chapter 2 applied. Section 7.1 describes the general format of Muon1 lattice
files, which can be used to specify either a single accelerator or an optimisation range ofdesigns. After simulation, designs are scored with the figures of merit in section 7.2 andlogged in result files described in section 7.3, where they can be used as input for the
genetic algorithm (section 7.4) that selects new designs to simulate. As this process of
repeated simulation takes a large amount of computing time, it has been deployed as a
distributed computing project (section 7.5) where the spare CPU time of volunteers fromthe Internet is used.

7.1

Muon1 Lattice Files

Accelerators are specified to Muon1 via ‘lattice files’, which are text files of a special form
placed in the lattices directory, located in the same folder as the program muon1.exe.
Lattices are simulated with a command such as muon1 -c LatticeName, which will
run lattices\LatticeName.txt on the command-line. A full list of the command-line
switches available is displayed by invoking the muon1 -? help switch.
Instead of providing an exhaustive reference to the lattice file format, which may evolve
over time, this section will give a series of examples that show the general capabilities ofthe system.

7.1.1

A Simple Beamline of Components

In its basic form, a lattice file defines a set of components of different types having different
physical attributes such as length, magnetic field, etc. These components are then placed
successively along a theoretical reference axis for the accelerator built out of a sequenceof straights and circular arcs. After the components have been defined, the order of thecomponents along the beamline is given, as shown in the lattice file below.
111

{
A Simple Beamline of Components
Curly bracket sections may also be used as comments - in this casebe careful to respecify the component line format before use.
}
{Drift Length}
D1 0.25
D2 0.5
RD1 0.32{Solenoid Field Length Radius}
S0 20 0.45 0.1
S1 -3.97 0.4 0.15
{RF Frequency/MHz Radius Voltage/MV Phase}
BRF1 31.4285 0.15 2.25 -90◦
BRF2 201.14 0.325 0 -90◦
RRFA 3.1428 0.2 0.8562 -180◦
RRFB 201.14 0.2 0.5 -67.633◦{Dipole Length Angle Field HalfGap EdgeIn EdgeOut HalfWidth}
IB 1 -8◦ -0.123608 0.325 4◦ 0◦ 0.5{RodSource Timespread Axial File X Y Z RodAngle}
TantalumRod 3 3 pi_plus_2G2eV.csv 0 0 288e-3 0
TantalumRod,S0,D1,S1,D2,S1,D1,BRF1,D1,S1,D1,BRF2,D2,IB,
RD1,RRFA,RD1,RRFB;

Here, blocks of component definitions are introduced by a format line of the form
{Type Attrib1 Attrib2...}, after which lines of the form Name Value1 Value2... will
create a component of type ‘Type’ called ‘Name’ with attribute Attrib1 equal to Value1
and so on. Importantly, components are not inserted into the beamline when they aredefined and the order of definition does not matter. The beamline order is defined by thecomma separated list of component names at the end of the file, which is terminated bya semicolon. Components may be included multiple times, once, or not at all.
The attributes Length and Angle are special in that, regardless of the type of component they are an attribute of, they determine how the reference axis of the machine isbuilt up. A length without an angle just extends the axis by that distance each time thecomponent is included. A length L and angle θ will include a circular arc of curvatureradius L/θ for that length, with negative angles producing reverse bends.
112

All values are in SI units by default, but Muon1 recognises values with units written
on the end (such as nanoseconds or degrees in this file) as well as scientific ‘e’ notation(288e-3). The units may also be included in the format definitions to save writing themfor each component, as MHz is here.

7.1.2

Cell Algebra

For machines with repeated beamline units, writing a straight list of all the components
could be unnecessarily long or just undescriptive. To eliminate this redundancy, larger
cells may be defined as blocks of the existing components with lines such as the ones
shown below.
DFB1:QDT1,CLM,QFT1,DM3,BEND,DM3;
RCS2:8*(2*DFB1,3*DFB2,DFB0,DFB1,DF0,2*DF1);
A1:2*(ACHR,-ACHR);
FlipA1:~A1;
Each line starts with the new cell name, followed by a colon and the semicolonterminated definition of that cell. This has the same format as the unprefixed list ofmachine components at the end of the file. In both cases, the operators of multiplication(*), negation (-) and mirror reflection (~) are available and can operate to any level withbracketing on the cells already defined. Multiplication repeats an identical sequence ofcells the specified number of times; negation inserts the cells in reverse order and mirror reflection inserts the cells in the correct order but with the Angle attribute of eachcomponent changed in sign.

7.1.3

Optimisable Variables

The examples so far have specified a single accelerator. To optimise, the program requiresa parameterised range of accelerators. Optimisation variables may be added by changing
the definition of a component in the manner below.
{Solenoid Length Radius Field}
S1 [0.2,0.45] 0.1 [0,20]This defines a solenoid S1 whose length may be varied from 0.2 to 0.45 metres with afield from 0 to 20 Tesla. Its radius, however, is fixed at 0.1 metres since this attribute is
specified as a single value rather than an interval range in square brackets. The replacement of a single value by a [min,max] bracket is all that is required to turn a point
design into an optimisation ready for running.
The variables are named automatically and these names will reappear in the result
files discussed in section 7.3. The variable name is the component name followed by the113first letter of the attribute name, all in lowercase, which would be s1l and s1r here. Moreletters from the attribute name are added as necessary if ambiguities arise.
It should be stressed that if multiple copies of S1 are included in the beamline, they
will all be identical. To have another solenoid with the same ranges for optimisation
but potentially different values, which would generate additional optimisation variables,
it would have to be defined separately with a different name like S2.

7.1.4

Example with Scripting Language

In more complex optimisations, the attributes of some components may have to be varied
in a correlated way (for example to exclude impractical combinations), or the sequence ofcomponents in the beamline may change according to an optimisable parameter. Muon1
includes a C-like scripting language that can be used within the lattice files, with occurrences of the # symbol switching in and out of the script language on any given line. To
show the various uses for this, the full lattice file for the PhaseRotB optimisation (a decay
channel plus phase rotation system) is explained here.
{
PhaseRotB/~270
SolenoidsOnly (i.e. decay channel 30m long) plus 31.4MHz phase rotation
Solenoids now allowed up to 90cm long
Bugfix of PhaseRotA
May 21, 2004}
!MINVER 4.41
{Parameter Value}
t_max 500e-9
n_decays 3
The initial comment block has the explanation and date in it. !MINVER is an optional
control command that states the minimum version of Muon1 needed to run this lattice,important for the distributed project where users might not upgrade their software quickly
enough. If running in distributed computing mode, the program searches for other compatible lattices to run instead. The last two lines are ‘parameters’, which are defined in
the same way as components and determine general features of the simulation. Here theystate that the simulation will terminate after 500 ns and each unstable particle (pions
here) will undergo three multiple decays as described in section 2.4.1. The termination
time is needed because some slow muons may be created in magnetic bottles near thesolenoids and will orbit forever unless the program is stopped.
{RodSource Timespread Axial File X Y Z RodAngle}
TantalumRod 3 1 ’pi_plus_2G2eV.csv’ 0 0 [0,0.45] [0,0.5]114

{Synchronous-particle t_start Energy Z}
SyncMuon #TantalumRod.Timespread?+5:0#ns 180MeV #TantalumRod.Z#
Two components are defined: a RodSource of pions, with optimisable angle and Z
position, and a Synchronous-particle that is tracked to provide reference crossing times
for the timing of the RF components. Note how the File attribute of the TantalumRod isa string, telling Muon1 to load the particles from the file datafiles\pi_plus_2G2eV.csv.
The attributes of SyncMuon contain the first example of the scripting language in use.
Its start time is set to be +5 ns if the rod particles have been loaded with a timespreadin emission and zero otherwise. Its Z position is set equal to the Z offset of the rod.
#z=0; ok=1;#for (n=1;ok;n++)#{{Solenoid Length Radius Field}# if (n==1) #S1 [0.2,0.45] 0.1 [0,20];#;# else {S[n]# [0.2,0.9] [0.1,0.4] #if (n<=4) #[-5,5];#; else #[-4,4];#;}# z+=S[n].Length;# if (z>=30) ok=0;# if (ok)# {{Drift Length Bore}#D[n]# [0.5,1] 0.5#
z+=D[n].Length;# }#}#MaxSol=n-1;
DecayChannel: #for (n=1;n<=MaxSol;n++) {S[n] if (n<MaxSol) {#,#D[n]#,#}}#;
The construction of the decay channel is a more complex example, with most lines
being scripted. There is a for loop and a z counter that is increased as solenoid and drift
components are defined until it reaches 30 m in length. Each of the solenoids has optimisable length, field and (with the exception of S1) radius and the drifts have optimisable
length too. The notation D[n] is a crude array implementation and would be equivalent
to a variable called D3 if n = 3, for example. The script variable MaxSol records thenumber of the last solenoid and then the list of components in the DecayChannel cell isprinted by another for loop, putting drifts after all but the last solenoid.
{Parameter Value}
PhaseRotCells [20,40]; #// 30 in original#PhaseRotCells=int(PhaseRotCells);#for (n=1;n<=PhaseRotCells;n++)115

#{{Parameter Value}
PD#n# [0.5,1]{Drift Length Bore}
PD#n#a #n==1?PD[n]*2/3-0.25:PD[n]*2/3# 0.15
PD#n#b #PD[n]/3# 0.15
{Solenoid Length Radius Field}#PS[n]# [0.2,0.9] 0.15 [-4,4]{RF Frequency/MHz Radius Voltage/MV Phase}#PRF[n]# 31.4285 0.15 [0,#(PD[n]+PS[n].Length)*2.25/1.4#] [-180◦ ,+180◦ ]#}PhaseRot: #for (n=1;n<=PhaseRotCells;n++) {PD#n#a,PRF#n#,PD#n#b,PS#n if (n<PhaseRotCells) #,#;#}#;The definition of the phase rotation channel starts with another use of ‘parameters’ inlattice files, that is to create an optimisation variable that is not tied to a specific attributeof a component but becomes a variable in the script that influences other things. Here,
PhaseRotCells determines the number of RF-solenoid periods in the phase rotator and ofcourse has to be rounded to an integer before use. The ‘for’ loop that follows is similar inprinciple to the decay channel one, with some additional algebra to position the RF kickcomponent two thirds of the way through the drift, which becomes split into two drifts
named like PD3a, PD3b. Another notable use of scripting is to impose an upper limit on
the RF peak voltage of 2.25 MV per 1.4 m of drift length available to install the 31.4 MHz
resonator into. This being based on a CERN design [83] where the large resonator also
enclosed the solenoid, the solenoid’s length is added to the drift available for this purpose.
{Finish-aperture radius MinEnergy MaxEnergy}
END 0.5 #180-23#MeV #180+23#MeV
TantalumRod,SyncMuon,DecayChannel,PhaseRot,END;
The final component to be defined is a Finish-aperture to collect the successful
muons that have finished the channel in the kinetic energy range 180 ± 23 MeV. The fileends with the definition of the entire accelerator in terms of the cells defined already.

7.2

Figures of Merit

The transmission of muons, as a percentage of the particle weight of the original pions,is used to score an accelerator design once the simulation has finished. A simulation will
finish either when there are no more active particles remaining, or the t_max time limit
has been reached. In the first of these cases, all the particles will then be either marked
as ‘lost’ or ‘finished’, whereas in the second some will still be ‘active’. From the point of116

view of the simulation, ‘lost’ and ‘finished’ particles are treated exactly the same except
with a different label: they both have stopped moving.
So how does a particle acquire the special ‘finished’ status? Currently this is doneusing the Finish-aperture component as at the end of the example in the previous
section. It produces a disc of the given radius, perpendicular to the beamline, that acts
as a blockage. If energy bounds are not specified, then all particles hitting the disc are
stopped and marked ‘finished’. If, as in the example, there is an energy range, thenparticles outside that range are also stopped, but marked as losses.
P At the beginning of the simulation, Muon1 normalises all the particle weights so thati wi = 1. Thus after finishing, the percentage transmission score is simply
X
T = 100
wi .
finished muons i

However, for more difficult optimisations, a random element of the design space may be sopoor that almost all such designs will give a zero score. It is useful to be able to distinguishbetween such designs to see which are nearer to being successful, so that the optimiser
may climb out of the area of poor designs on its own without having to be seeded withan already good design. Thus two additional scores are defined to deal with the cases ofbeam not being transmitted through the whole design and beam being transmitted but
mostly of the wrong energy.
The figure for a design that transmits less than a proportion  = 0.1% of the beam
is defined by finding the distance along the beamline where at least  weight of muons
survived. That is the largest distance s, measured along the beamline axis, for which theweight of stopped muons with si ≥ s is at least . Call this distance s and the total
length of the beamline l. Then the distance score is defined as
D = s /l.
It ranges from near zero where almost all the particles stop in the early accelerator, tonearly one when the fraction of surviving muons is only reduced to below  right near theend.
The figure for designs that transmit less than  of the particles in the right energy range [Emin , Emax ] is determined from the energies Ei of the muons that hit theFinish-aperture. The ratio by which the energies are wrong, as a weighted average, isgiven byXX
wi (1 − Ei /Emin ) +wi (1 − Emax /Ei )W =Ei <EminEi >EmaxX.

wimuons on apertureThis can range from values near zero when most of the muons have almost the correctenergy, to values of nearly one when the majority of the energies are very far outside thespecified range.
The above figures T , D and W are blended together with linear ramps to make asmooth overall score S that increases with the performance of the design. If T /100 ≥ ,117the transmission score is assumed to have enough statistics to be used on its own, whereas
in the region /2 < T /100 <  it is linearly blended into the ‘fallback’ score S1 , as follows:
when T /100 ≥  T
T /100−/2
−T /100T + /2 S1
/2 < T /100 < 
S=
/2

S1
T /100 ≤ /2
If more than  weight of muons survive the channel, regardless of their energy, thefallback score rates the design between 0 and −1 depending on how wrong their energies
are. If fewer than /2 get to the end, the design is scored as D − 2, which ranges between−2 and −1 depending on how far the muons survived down the channel. In between, linear
interpolation is again used. Let f be the raw transmission fraction of muons, disregarding
energies (so f ≥ T /100). Then the fallback score is defined as:when f ≥ 
 −W
f −/2−f(−W ) + /2 (D − 2)
/2 < f < 
S1 =
 /2
D−2
f ≤ /2
The combined score S as defined above is the one used in the Muon1 program when it
outputs results. Note that for any value above 0.1 it is just equal to the muon transmissionin percent. Lower yielding designs are given scores in a range down to −2.

7.3

Muon1 Result (and Queue) Files

In order for the optimiser to learn from previously simulated designs, the variables andscore from each simulation are logged to one or more result files. Generally these will
be results.txt and results.dat in the same directory as the Muon1 executable. All
result files are sequences of records, one added for each simulation, which have the sametwo-line format as the example below.
#gen=8;d10l=075;d11l=237;d12l=013;d13l=000;d14l=011;d15l=000;d16l=024;...
s8l=999;s8r=999;s9f=978;s9l=999;s9r=999;tantalumrodr=000;tantalumrodz=660;
0.365180 (368.8 Mpts) [v4.41f] {FD19E68427264FC7F20D451C} <PhaseRotB>
The first line can be very long so has been split for clarity. It contains a series ofvariable=value; pairs with the variable names coming from the lattice file as describedin section 7.1.3. The value has been (linearly) normalised to three digits with 000 corresponding to the lowest and 999 the highest allowable value in the range. Accuracies
of more than three significant figures were considered to be difficult to engineer, so by
restricting the accuracy this representation prevents some degree of over-optimisation.
Variables whose name begins with a # character are not from the lattice, they have been118

added for other logging purposes, for instance #gen is automatically added to record which
genetic algorithm mode produced the design (see section 7.4 for details). Hashed variables
are standard integers that are not limited to three digits.
The second line always starts with the score that the design specified by the first line
achieved. It must also contain the name of the lattice in angle brackets, <PhaseRotB>
here, which specifies that the variables on the first line should be used with the lattice
file lattices\PhaseRotB.txt. Other data is also usually supplied: in round brackets theamount of calculation required for the simulation in mega-particle-timesteps (Mpts) andthe version of Muon1 in square brackets. This is useful for logging the overall amountof CPU activity on a distributed network of computers as well as benchmarking, whilethe version code can be used to filter out erroneous results in the case of a buggy Muon1
release. The hex code in curly brackets is a checksum produced by the program to makeit more difficult to flood the distributed project with ‘fake’ results. The user may also
add their own comments at the end of the second line, which are ignored by the program.
Normally on completion of a simulation, the two-line result is appended to both thefiles results.txt and results.dat, with the files being created if they do not yet exist.
The reason for this duplication is that when running as part of the distributed computing
network, results.txt contains only the buffer of results that have not yet been sent andis deleted after a successful upload. results.dat contains a complete history of results
produced and may even be augmented with extra results generated on other machines, toserve as a knowledge base for the optimiser to generate new designs from.
A third file queue.txt may also contain results in a similar format and this has two
purposes. Firstly, it acts as the method for a user to input specific designs (from a lattice’s
optimisation range) to test. These will specify the first line with the normalised values
for the variables and the lattice tag on the second line, together with the string TEST inplace of the score, since it has not been calculated yet. This form is shown in the example
below, although the first line will often be much longer in reality.
s8l=916;s8r=896;s9f=895;s9l=909;s9r=976;tantalumrodr=000;tantalumrodz=446;
TEST <ChicaneLinacB>
If a lattice has no optimisable variables, it is not worth preparing a queue.txt file in thisway before running Muon1 because specifying the lattice on the command-line will provide
sufficient information† . The second purpose for the queue file is to recheck high-scoring
results for both statistical accuracy and reliability against computer errors. Specifically, if
a result has the highest score achieved so far for its lattice, Muon1 will run it an additional
four times, storing the intermediate results in the queue.txt file and then finally add aresult with an averaged score (excluding the highest and lowest) to results.txt andresults.dat. In these cases, the intermediate result may have a comma-separated listof scores in place of the single score normally found at the start of the second line.
Whatever the usage mode, the top result in the queue file is processed first and removed
when completed, with the rest of the file being moved up.
†

The reader may wonder what happens if an optimisable lattice is run from the command-line without
a queue file. The answer is that Muon1 is always in continuous optimisation mode by default, so will
choose either a random design (if results.dat does not exist) or continue optimising the designs basedon the work accumulated in results.dat so far.

119

7.4

Genetic/Evolutionary Algorithms

When given an optimisable lattice, Muon1 has the task of choosing values for a potentiallylarge number (often hundreds) of variables. Genetic and evolutionary algorithms are
suitable for problems of large dimensionality where the optimum can be found without
necessarily being given a nearby starting point. This is because they effectively perform asearch on the design space that starts randomly and becomes focussed in higher-yielding
areas as accumulated results reveal where these areas are. ‘Brute force’ searches where agrid in the search space is fully explored are not practical for high-dimensional problems
because even if each variable is only sampled at 2 values, there are 2D combinations fora D-dimensional grid, which exceeds 1030 with D = 100, for example. This means that
‘intelligent’ forms of brute force searching that subsequently use finer grids on areas withgood results also cannot be used because even the initial coarse grid is too time-consuming
to search. Another common optimisation method is to start from an initial design andevaluate the sensitivity of all parameters to small variations, then adjust these in the
directions that increase the score. Unfortunately when the figure of merit is particle
yield from a tracking simulation, the score has statistical noise that overwhelms the small
differences in score produced by the small variations used to gauge the sensitivity.
The genetic algorithm terminology originated in early computer simulations of biological evolution [84]. These first examples of genetic algorithms followed a very close
analogy with DNA replication, generating a ‘population’ of initially random binary strings(genomes), then performing artificial selection (as opposed to natural selection) to eliminate a large proportion of them before the remaining good strings were allowed to ‘mate’in a process that mixes blocks of the parent binary strings together (genetic crossover)
and occasionally inverts some of their bits (mutation). These offspring then form the population for the next generation and the process repeated many times, with the removalof the least-good designs biasing the distribution of strings towards areas in which thehigher scoring designs are to be found.
In Muon1, the ‘genome’ is just the first line of the result as described in section
7.3, containing the normalised values for all the variables that specify the particular
accelerator design within a lattice’s optimisation range. ‘Artificial selection’ is performedon the grounds of higher scores (calculated as in section 7.2) being better and ‘mating’ by
a variety of methods including analogues of the original mutation and genetic crossover.
However, the original formulation of the genetic algorithm is often inefficient in realengineering problems and requires some adaptation as reviewed in [85]. For Muon1, thepresence of discrete generations means the designs and results from each generation are
destroyed apart from their influence on the gene pool of the next generation. When eachresult is scored on the basis of a simulation that can take an hour or more to complete,
this is considered wasteful. Instead, all the results are logged permanently but operations
that generate new designs, such as mutation or crossover, are based on a subselection
highly biased towards the higher scoring designs. This means low-scoring designs remainin the result log but hardly ever procreate.
The biased selection algorithm in Muon1 works as follows. A real number α ∈ [0, 1.25]
is chosen uniformly at random and is used to define the subsample size s = bN α c where120

N is the total number of results in the results.dat history file. Random results are
examined from this file s times over and the one having the highest score is selected foruse. Methods such as crossover require two input genomes so will call this algorithm twice,or more if the second genome comes back the same as the first, until two differing ones
are selected. The advantage of allowing values α > 1 and hence s ≥ N is that nearly allof the results will be examined, so there is a finite probability of picking the very highest
result in the file, which does not decrease appreciably with N . Allowing values of α near 0means it is still possible for merely ‘average’ results to get picked, although with a reduced
probability.
A second major modification to the binary genetic algorithm is needed because it has
no concept of continuously-varying parameters, instead it would treat their floating point
representation as just another binary string and produce ‘corrupt’ values that may be
nowhere near either of the two parent values after a bit-wise crossover. Although this
could be seen as introducing a little more mutation into the system and therefore harmless, Muon1 takes the approach of using continuous parameters and analogous operations
to crossover (where values themselves remain intact, like individual bits) or mutation(where increments are chosen from a continuous distribution of values). This continuous
parameter system also naturally allows other ways of generating new genomes, such as
taking a weighted average of two sets of variables, an operation that has no analogue using
binary strings.
With these modifications, Muon1’s algorithm can more accurately be described as partof the broader class of ‘evolutionary algorithms’, which use the concept of learning froma pool of past results but do not impose a discrete genome analogous to DNA base pairs.
The full implementation contains several ways of generating new results from those
selected from the results.dat file. As mentioned in section 7.3, each result is labelled
via the #gen field to denote how it was created. These different methods (known as ‘trial
types’) and their #gen values are described in the list below.
• Random (#gen=0) does not use any previous results and sets all variables to uniform
random values. This is called when no results are in the results.dat history file,or with a decreasing probability N −0.4 when it contains N results.
• Mutate (#gen=1) and Crossover (#gen=2) function roughly as described in theoriginal genetic algorithm.
• Interpolate (#gen=3) takes two results and produces a weighted average of theirparameters. A random number on the [0,1] interval determines how much weight isgiven to the first genome with the balance made up by the second.
• Extrapolate (#gen=4) is the same but uses numbers outside the [0,1] range toextend beyond one of the results’ locations in design space.
• MuSpherical (#gen=5) produces a mutation according to a spherically symmetric
distribution of offsets, if the variables are considered the axes of the design spacewith the Euclidean metric.
• MuOne (#gen=6) is like Mutate but only changes the value of a single variable.
121• Extreme (#gen=7) is designed for the common situations when the optimal valuesof many parameters are at the extremal end of the allowed range (e.g. highest
magnetic or accelerating field). It moves some of the variables towards the end of
the range.
• LocalGrad (#gen=8) estimates the local gradient of the score function from a clusterof the nearest results to a selected one and produces a suggested improved design
by moving ‘uphill’ a certain distance from there.
• TopoSmooth (#gen=9) uses the numbering that often appears in variable names toinfer which apply to adjacent components (numbers differing by one) and smooths
out each variable’s value by averaging it with that of its neighbours.
• CellDouble (#gen=10), CellDelete (#gen=11) and CellCommute (#gen=12) also
use the numbering of the variable names to perform the functions of duplicatinga beamline cell, deleting a cell and swapping (commuting) two adjacent cells witheach other.
The effect of adding these extra trial types to the algorithm was investigated with twobenchmark problems.

7.4.1

Sine Wave Interpolation Test

The first benchmark problem was designed to have a figure of merit that depends on alarge number of variables in a smooth but correlated way. A suitable problem is the interpolation of a sine wave with a linear spline L(x) using the variables (x1 , y1 ) . . . (x10 , y10 )with xn ∈ [0, 2π] and yn ∈ [−1, 1] as the positions of the interpolating nodes. The score
would then beZ
2π(sin(x) − L(x))2 dx,

10 −0

so that better spline approximations increase towards 10. Since the performance of the
optimiser in a single run would depend on the random values chosen during various operations, a program was written to run the optimiser many times (with different random
seeds) on the same problem and accumulate statistics on its average behaviour. Figure
7.1 shows this during operation for the sine interpolation problem.
After running the optimisation test problem 100 times, each of which was terminatedeither when the score reached 9.99 or when 10000 results had been generated, some averagevalues (including error bars) were derived that show the true performance of the optimiser
regardless of the random numbers used. The two most important statistics collected were
the percentage of times the optimisation reached the target score before the 10000 result
cutoff and the average number of iterations taken to get there in the cases where it did
succeed. These are plotted in figure 7.2 for all 26 = 64 combinations of the trial types
with #gen=1 through 6. When multiple trial types were used, they were selected withequal probability for every non-random result. In the case where no trial types were used,
only random results were generated, explaining the zero percent success rate in that case122Figure 7.1: Genetic algorithm test run of the sine wave interpolation benchmark problem.
The green graphic at the top shows successive best linear spline approximations to thesine wave with high scoring lines (≥ 9.98) in yellow and the final result (first above 9.99)
in black. Below is a graph of the generally increasing scores of the results over time.
(the optimisation problem was not so easy that a million random guesses would produce
a success!)
The success rate and average convergence time data can be amalgamated into a single
figure of merit to represent the number of optimiser convergences expected in a million results (106 ∗ (success probability)/(average time)) and this is what is shown in figure 7.3. The general trend is that optimiser performance is patchy and often failing toconverge when only a few trial types are combined and more robust when there are many.
Crossover and interpolation get nowhere without adding other types because they only
produce results ‘between’ those few random ones already produced. Adding a mutate typeto anything guarantees at least some level of convergence, though spherical mutation is
generally poor and single-parameter mutation is so good it almost beats all other combinations on its own (though with a big error bar). However, most mutation combinations
are moderately enhanced by adding crossover or extrapolation.
123

Figure 7.2: Success rate and average convergence time for the optimiser on the sine wave
interpolation test problem with all possible combinations of the first six trial types being
switched on and off. Error bars are one standard deviation.

Figure 7.3: Figure of merit for optimiser performance running on the sine interpolation
test problem, for each possible subset of trial types used (1σ statistical errors shown).

124

Figure 7.4: Success rate and average convergence time for the optimiser on the sequential
parameters test problem, for each possible subset of trial types used (1σ statistical errors
shown).

7.4.2

Sequential Parameters Test

A second benchmark problem was devised to test the optimiser in rather less smoothsituations, including discontinuities and situations where some parameters become ‘slack’unless activated by the proper choice of others. The latter situation of sequential parameters may occur when good choices in the early part of a beamline are required for theparticles to propagate far enough to be affected by the variables controlling components
further on. To represent this, some ‘goal’ values g1 . . . g10 are chosen randomly by the computer and kept secret from the optimiser, which will make guesses as vectors (x1 , . . . x10 )with xn , gn ∈ [0, 1]. If |xn − gn | ≤ 0.05 for all n, the score is 10 and the successful optimisation is terminated. Otherwise let xk be the first value such that |xk − gk | > 0.05
and put the score equal to k + 1 − |xk − gk |. This increases from slightly more than zerofor guesses where x1 is totally wrong, increasing fractionally as the guess becomes moreaccurate and jumping above each integer n once xn becomes close enough to the goal.
The success rates and average optimisation iterations for this second problem are plottedin figure 7.4 and show a significantly different pattern to that of sine wave test problem,
though sharing the same feature that combining more trial types is generally better.
A single figure of merit can be produced for the sequential parameters test problem
in the same way as for the sine interpolation test. To show the situations in which the
optimiser performs well whichever sort of problem it is given, the geometric mean of
the figures of merit for both problems is plotted in figure 7.5, the geometric mean being
chosen so that a complete fail on either problem will ensure a zero score overall. The highperformance of the MuOne trial type alone has become less significant than it was in thesine wave test, with the best overall performance coming from the combination of it withcrossover and extrapolation. However, the mix of all six types still performs three-quarters
125Figure 7.5: Combined figure of merit for optimiser performance across two different test
problems, with 1σ error bars, as a function of the set of trial types used.
as well as anything else and comes with the insurance that if another, totally different,
sort of problem was attempted that responds well to, for example, MuSpherical, thenthat algorithm would still be available. Muon1 by default runs with all these trial types
active, together with LocalGrad and some of the ‘topological’ trial types but with a lower
probability. The weighting of each type can also be customised in the config.txt file by
the user.
An illustration of the distribution of results generated by having all these different
methods of evolution active at the same time is figure 7.6. This is projection of one
user’s results.dat file onto the two principal axes of variation in the design space, whenrunning the SolenoidsTo15cm lattice that optimises of the pion decay channel. Some
diverging evolutionary pathways are clearly visible by virtue of the fact the user had
several computers working independently for a period. The progression of another such
evolution will be examined more closely in the next subsection; the resultant accelerator
designs for all optimisations will be described in chapter 8.

126

Figure 7.6: Projection of a user’s result evolution through the design space. Principal
component analysis (PCA) was performed to determine the two axes of largest variance inthe results in the high (∼100) dimensional design space, which were used to plot each result
as a point in two dimensions. Black/purple indicates the highest scoring results, throughto blue for the lowest. Notable features include: clustering around the highest scores;
bridging between two particular high scoring points in search of an optimum between;a diffuse blue ball of randomly-generated results at the left; and six separate pathways
optimising out from the blue ball, caused by the program being run from scratch on six
separate computers for a period, each one going its separate way.

127

Figure 7.7: Progression of successive best scores attained by the PhaseRotEb1 lattice.

7.4.3

Example Lattice Optimisation

The evolution of all the parameters in the PhaseRotEb1 lattice is plotted in this subsection.
How all these lattices differ from each other will be discussed in chapter 8 but for now it
suffices to say that PhaseRotEb1 optimises the decay channel and phase rotation sectionsof the muon front end, aiming for muons transmitted into an energy window of 180 ±23 MeV. This optimisation is particularly easy to display because the later sections of the
decay channel and the whole phase rotation are made from identical cells, which cuts down
the number of variables significantly. Figure 7.7 plots the progress of the optimisation over
the two month period it was running on the distributed network. Only the successive bestscores received by the central server are plotted, so the very earliest results with scores
between −2 and −1 are missed out, as by the time the participating computers uploaded
their first batch of ∼50 results they had already achieved scores above −1.
The rate of increase in the muon yield slows down over time because the higher a scorealready is, the more difficult it is to generate a result that exceeds it. Despite saving on
independent variables in later parts of the channel, PhaseRotEb1 is still an optimisationof 48 variables. This means any sort of exhaustive search or guarantee that the eventual
design is the best one in the whole design space is out of the question, since the design
space consists of 100048 = 10144 points. The optimisation only sampled 5.89×105 of these,
while finding a design much better than the same number of random (or systematicallyspaced) samples would, though it is very much unknown how many points exceed theyield found.
128

Figure 7.8: Evolution of variables concerning the target rod and its enclosing solenoid (S1)in the PhaseRotEb1 lattice. The 0–999 range represents 0–20 T for the S1 field, 20–45 cm
for the S1 length, 0–0.5 radians for the rod rotation angle and 0–45 cm Z displacementsof the rod centre into the solenoid (measured from the front coil face).

To see how the variables had to be tuned to achieve increases in score, they are allplotted in groups on the seven graphs that follow (figures 7.8–7.14). Score is a more significant variable to plot against than time as it does not depend on the speed of computer
calculation in any way.
Different types of variables are optimised while the score is in different ranges. To
achieve a score above zero, at least 0.1% of the beam must finish within the correctenergy range, so the first parameter to become close to its final optimised value is the RFphase used in the phase rotation on figure 7.10. For the transmission to reach 1%, thefields of the solenoids in the late decay channel and phase rotation (purple and black on
figure 7.11) were given the same sign and the configuration of the target rod was changed
so that the solenoid had a high field and length with the rod being more parallel to its
axis. The discontinuity in many of the parameters at once around a score of 0.9 could be
the optimiser jumping out of its current local minimum to another, as the evolution of
results in another minimum overtook the first. Evolutionary algorithms are often put inthe category of ‘global optimisers’ for this sort of reason, though as pointed out before, atruly global optimiser for completely general score functions does not exist.
Between 1 and 2% yield the main adjustments were to the voltage in the RF cavities
and the length of the phase rotation section (figure 7.9), which were both increased. At
a score of about 2.2 on figure 7.13 another jump occurred in which all the drifts in the129

Figure 7.9: Evolution of the number of cells in the decay channel and phase rotation
sections of the PhaseRotEb1 beamline.

Figure 7.10: Evolution of the RF parameters in the phase rotation section of thePhaseRotEb1 beamline.
130Figure 7.11: Evolution of solenoid fields in the PhaseRotEb1 lattice.

Figure 7.12: Evolution of solenoid lengths in the PhaseRotEb1 lattice.
131

Figure 7.13: Evolution of drift lengths in the PhaseRotEb1 lattice.

Figure 7.14: Evolution of solenoid radii in the PhaseRotEb1 lattice.
132

decay channel were shortened to the minimum allowed length of 0.5 m. From here up to
the final high of 3.355% most of the optimisation is concerned with gradually adjusting
various parameters, in ways that appear to become quite correlated above 3%; one clear
example is the anticorrelation of the rod Z offset and rotation in figure 7.8. The only
further ‘global-type’ jumps are in solenoid number 4’s length at 2.9% (figure 7.12) andfield at 3.2% (figure 7.11), changes that presumably improve the optics accepting thebeam from the target. These could likely be the result of a MuOne trial type in which only
one parameter is changed, in this case to a value with higher yields.

7.5

Distributed Computing Project

To obtain the computing power required to run a 3D Monte Carlo simulation for eachscore evaluation within an optimisation, Muon1 can be run on many computers at once ina configuration known as ‘distributed computing’ that does not require particularly fast
network connections between them. In fact, participating in such projects has become apopular pastime for many members of the public, as evidenced by the number listed on
hub sites such as [86]. The additional features required to turn a scientific distributed
program into one usable by the public are:
• A project website where the program can be downloaded (Muon1’s is at [5]).
• Availability for a popular operating system, Windows in Muon1’s case.
• Ease of installation: here, Muon1 simply requires extracting from its archive andonce run will perform any additional setup by itself, downloading data from theInternet as appropriate.
• Smooth multi-tasking with other applications. Most such programs are designed torun at a very low task priority level, so only use ‘spare’ CPU cycles available when
the user is not making demands of their computer. Muon1 also can work invisiblyin the background.
• Some kind of feedback on the user’s contribution or progress in the calculation.
This can take the form of an overall scoring system measured in calculation units
appropriate to the project and/or a graphical interface while a single calculation isrunning. Muon1 supports both of these and the calculation units used are megaparticle-timesteps, or Mpts, measured as 106 Runge-Kutta steps.
Once downloaded from [5] and installed on a computer, the behaviour of Muon1 (in its
default distributed computing mode) is summarised by figure 7.15. This process combines
all the elements described in previous sections of this chapter. The ‘results storage’ isthe results.dat file and the ‘results buffer’ is results.txt. The reason Muon1 does
not require computers to be connected by fast network connections is that individual
simulations are run entirely on a single computer and will be so repeatedly until a large
enough batch of results has been collected in the results buffer to be sent. This system also
copes with intermittent connectivity by continuing to generate and store results suggested
133

Figure 7.15: Cycle of operation of the Muon1 client program on a user’s machine.

by the optimiser until a connection is available. Computers periodically check back withthe central website to update the list of lattice files they should be working on.
As described so far, this system would be one of separate optimisations happening
independently on many computers. However, another thing that Muon1 tries to download from the website is a ‘sample file’ for each optimisation, which contains a selection(typically 100) of the best results generated throughout the whole network. This provides
a narrow channel of communication between the different installations, but it is effective:
new high-scoring results and their variants are propagated across the network within hours
because higher scoring results are far more likely to be selected out of results.dat forvariation using the genetic algorithm.
Although most of Muon1’s calculation is done with the program running in invisiblemode (it is taking 95% of the CPU while I am writing this paragraph), a graphical modeis also provided which gives the fullscreen display shown in figure 7.16. This was originally
written to help with the debugging and monitor the progress of the tracking code but isalso now used by many project users as a screensaver.
The project as a whole also needs a ‘back end’ to receive and store results fromthe many users (typically more than 100 will be running the program at once). Thisadditional infrastructure is represented in figure 7.17. Users have two methods to upload
results: earlier versions of Muon1 used an FTP script but since some networks block
FTP, an upload via HTTP (the protocol for viewing normal web pages) was added later.
These go to an intermediate stage of a few FTP servers (also operated by volunteer
users) that are provided as redundancy so the Muon1 clients always have somewhere to
upload to. Every hour, the database computer at RAL retrieves all new results from theseintermediates as part of a script that collects and counts the results submitted by each134

Figure 7.16: Muon1’s 3D display of particles and components while running in graphicalmode. This view shows the rod within the first solenoid and the second solenoid (shownas grey wireframe cylinders). Particles are mostly pions at this point and are coloured
according to their energy, red being low and blue high. The brown annuluses on theleft-hand side are additional geometry added to stop backwards pions from flying off into
empty space (which wastes simulation time).

135

Figure 7.17: Flow of data through the Muon1 distributed project network.

user. This script also verifies the checksum of each result and removes any duplicateduploads. Lastly, the script publishes a table of the total number of results and Mpts
that each user has computed on each optimisation, together with their highest result
score on each optimisation. Users are identified by the filename given to their results
during upload, which is taken from the user.txt they create in the Muon1 directory (ifuploads are switched on and no user.txt file is found, there is a GUI prompt to enter theusername). A second script on the database computer periodically counts through all the
results for each active optimisation and generates a ‘sample file’ (as mentioned earlier)that is uploaded to the website.
With hundreds of active users, a considerable number of simulation results can begenerated over time. Log files enabled the cumulative totals over most of the project
lifetime to be plotted in figure 7.18. Observe that the number of results does not increaseas smoothly as the amount of calculation. This is because simulations from different
lattices can vary in length depending on how much beamline they simulate. Even resultsfrom the same lattice can vary significantly early on, as a low-scoring result in which mostof the beam is lost will require less calculation than one where much of the beam travels
further. A dip in the number of results is visible due to repeat removal and maintenance:
the whole database currently stands at 183 GB and went through several software revisions
to cope with the increasing size.
The particle-timesteps axis has much larger figures on it, by a factor of 108 –109 . Thiscan be understood since each simulation has ∼104 active particles undergoing 104 –105
timesteps of 0.01 ns each, the total 100–1000 ns time corresponding to beamline lengthsof the order of 100 m.
136

Figure 7.18: Total simulation results received and calculation performed across all optimisations run on the distributed computing network.
Users’ scores and the progress of current optimisations are shown on the Muon1 website, a screenshot of part of this is shown in figure 7.19. The left-hand scatter plot shows
result score against the number Mpts used to produce it, with a strong diagonal featurefrom the correlation of calculation time with result score mentioned earlier. This plot
was originally produced as a curiosity but became part of the site when it was foundthat plotting Mpts as a second variable to score often separates different ‘evolutionary
tracks’ so it becomes clear when a new design branch has been taken. It also shows up
any erroneous results as these will very often have the wrong amount of calculation for agiven score.
The distributed optimisation project allowed many different lattice files corresponding to ranges of possible muon front end designs to be uploaded to the project website
and subsequently optimised. When the best muon yield achieved for a particular lattice
stopped increasing for a long period of time (over a week), the lattice was retired from thewebsite and the highest-scoring result extracted. Chapter 8 analyses the highest-scoring
results from eight lattices covering a variety of front end configurations.

137

Figure 7.19: Summary graphs for each optimisation and users’ best results are updated
hourly on the Muon1 website [5]. From left to right the graphs are: a scatter plot (calculation time against score) for all results of this lattice; history of successive best scores, total
results and calculation; and a zoom into the last week of the middle plot. The scatter plot
can be clicked for a more detailed version and the table below shows the current ten users
with the highest submitted scores, along with their amounts of calculation and numberof results submitted from this lattice.

138

Chapter 8
Results from Optimisation
This chapter describes the designs resulting from optimisation on the distributed computing network using the Muon1 code. Several different optimisation ranges of accelerators(‘lattices’) were submitted to the network, starting with just the decay channel and later
including phase rotation and linac sections. These optimisation ranges are specified intables and the best designs are shown with plots of beam behaviour and transmission
figures. Full parameters for these designs are listed in appendix E.

Figure 8.1: Lattices described in sections of this chapter, showing output energy criteria.

139

Table 8.1: Optimisation ranges and best muon yield for the SolenoidsOnly lattice.
Decay channel (to 30 m):
TargetAngle
Z offset
S1
LengthRadius
Field
Solenoid
LengthRadiusField (S2–S4)
Field (S5–)
DriftLengthProton beam energy:
2.2 GeVBest yield: µ+ /π µ+ /p·GeV
16.676%0.012652
22.873% including pions

8.1

0–0.5 radians
0–0.45 m
0.2–0.45 m
0.1 m (fixed)
0–20 T0.2–0.6 m
0.1–0.4 m−5–5 T−4–4 T
0.5–1 mMuon energy criterion:Anyµ+ /s (4 MW proton beam)
3.16×1014

Decay Channel

The first piece of the muon front end to be optimised consisted of the target and decay
channel, with the target enclosed in the first (stronger) solenoid S1 and the remaining
solenoids centred on the same axis, in the direction the proton beam (and the majority ofthe pions) will be travelling. The lattice file for this was named SolenoidsOnly because
it contains just the decay channel and no RF.
Table 8.1 defines the optimisation ranges of all the components in this lattice. The
earlier optimisations, such as this one, used a MARS15 data set of pions generated froma 2.2 GeV proton beam because this was the energy of the CERN SPL design [40] at thetime. Later on this switched to a 10 GeV proton beam, primarily because of the findingsof the International Scoping Study in 2006 [24].
The target rod may be rotated by up to half a radian (probably an excessive angle
but a suitable upper bound) and moved in Z so that its centre could be anywhere fromthe entrance to the exit of S1. Most optimisations chose a near-zero angle for the rod
and placed it near the centre of S1, which was given the maximal allowed length of 45 cm.
SolenoidsOnly is unusual in that it is the only optimisation examined that assigned a
relatively large angle, 95.6 milliradians, to the rod. The figure of merit for this lattice
is total weight of muons leaving the channel, with no energy cuts at all. Primarily it isconcerned with the transverse focussing and magnetic capture from the target. However,since the decay channel is only 30 m long, a significant number (∼27%) of undecayedpions still remain in the beam, so the decay-channel-only lattices in this section have thetransmission including the pions also printed for reference.
Ranges for the variables concerned with solenoids have been chosen to avoid impracti140

Figure 8.2: Longitudinal phase space evolution (top) and beam envelope plot (bottom)
for the optimised design from the SolenoidsOnly lattice.

cally large field volumes. Only S1 may have a field up to 20 T and the subsequent solenoids
are limited to 4 or 5 T. Because of its high field a much tighter bore is anticipated forS1 (10 cm radius), which cannot be increased. The remaining solenoids with only one
quarter the field strength can have radii up to four times as big, which keeps the stored
energy the same. Drifts of at least 50 cm are imposed between solenoids for independentcryostats, diagnostics and maintenance access, this is unlike Study II [25] and others inwhich the area near the target is a monolithic module. The solenoid-drift pattern in thechannel repeats until a distance of 30 m is reached, where the channel is terminated andthe muons within the final solenoid bore counted.
Figure 8.2 shows the behaviour of the beam from the target in this optimised decay
channel. The beam maintains a constant size through most of the channel, coming from a
regular focussing structure. All but two of the solenoids have been assigned the maximum
possible radius, field and length and all drifts have been given the shortest allowed length.
This means the technical constraints on the design are the primary limitation and suggests
the yield would increase if solenoids were allowed to be larger, longer, higher in field or
closer together. The exceptions are the third solenoid, which is only 48.3 cm long (60 cmmax) and the fourth solenoid, which has a field of 4.24 T (5 T max) and a length of
55.3 cm. The effects of these non-regularities are visible in the beam envelope followingthat point. It is likely these are genuine optima related to the matching from the initial
pion distribution because of these solenoids’ proximity to the target and the fact that all
the other solenoids have reliably been set at very close to maximal parameters.
Figures 8.4 and 8.5 show the RMS optical functions and emittances calculated fromthe beam in this design. The values are obtained by calculating moments of the particle
distribution as it passes through planes of constant z spaced along the accelerator. Since
only a finite number of particles are tracked, there is some statistical error visible for
instance between the x and y planes for the and functions: these should be identicalfor a cylindrically symmetrical machine. Figure 8.4 also shows very low at the location
141

Figure 8.3: Pions (green) decaying to muons (purple) in the optimised design from the
SolenoidsOnly lattice.

Figure 8.4: Optical functions in the optimised SolenoidsOnly design.

142

Figure 8.5: Emittance evolution in the optimised SolenoidsOnly design.

of the target, followed by negative indicating a diverging beam, as would be expected.
The functions then settle into a periodic cell, with an upward drift of that may be
explained because is not a ‘normalised’ quantity and therefore varies with overall beamenergy. Here, as the slightly higher-energy pions decay into muons, the average energy
gradually reduces and unnormalised quantities therefore increase.
Figure 8.5 shows an artifact of the logging scheme whereby particles caught in magnetic
bottles in the gaps between the solenoids are counted multiple times, making the remaining
particle weight (and 4D emittance) appear to increase in these regions. Ignoring this, thenormalised 4D emittance increases slightly in the middle section of the channel from pion
decays, before becoming invariant as would be expected. The 2D emittances projected
into the x and y planes oscillate above the 4D values because coupling between the planes,
which is induced by solenoids, increases the projected emittances (intuitively: the beam
appears bigger because information about its correlations has been lost).
The longitudinal phase space plots in figure 8.2 show the drift from the initial near
instantaneous (1 ns RMS) spread of pion energies, with lower energy particles lagging
behind more in time. This lag is nonlinear owing to the large energy spread involved.
Figure 8.3 shows the same thing only with the beam coloured by composition, with thetwo particle species given different colours, instead of total intensity. The overall decay ofpions to muons is visible in the transverse plot while the longitudinal plot shows that thepions tend to have slightly higher kinetic energies than their decay products at a given
position in the beam. It also illustrates the time dilation phenomenon with higher-energy
pions taking longer to decay than the slow ones near the bottom of the plot.
143

Figure 8.6: Pions in the initial (pL , pT ) momentum space coloured green if they aretransmitted through the decay channel and red otherwise.

The decay channel on its own does not attempt to select particles by energy. Figure 8.6
shows the momentum distribution of pions from the target, coloured according to whether
they survive the decay channel. The solenoid fields and bores dictate an upper limit on
pT and there is no upper limit on pL as a particle of high enough speed will always survive
through a linear channel if it has a line of sight to the exit. While the reason for losses withpL < 0 (backwards pions) is obvious, the poor survival for pL less than about 200 MeV/cis because low forward momentum allows the pion to be reabsorbed in the target rod
after spiralling in the 20 T field. Particles with very low pL are also more easily diverted
by the radial component of solenoid fringe fields, which can exert a force in the negative
Z direction when the particle has a suitable transverse velocity. The force will reduce
the pL of the particle further, increasing the angle between its direction of motion andthe accelerator axis; in extreme cases, particles can be reflected backwards (the magnetic
bottle effect); losses become more probable in either situation.
Because the 40 cm radius of the decay channel is too large for most acceleration systems, an optimisation named SolenoidsTo15cm was started that forced the final solenoid(and aperture) to be 15 cm in radius. Table 8.2 shows the resulting yields, which on firstsight appear encouraging given the cross-sectional area of the accelerator is reduced by
86% but the yield only by 33%. However, this lattice is an example of an optimiser finding
an inadequacy in the figure of merit. The final solenoid’s length has been reduced to theminimum possible here, which means its short length can accommodate a betatron focusof the beam. That is, although the beam is small in the final solenoid where the figureof merit is counted, the smallness has come at the expense of increased x0 values and the144Table 8.2: Optimisation ranges and best muon yield for the SolenoidsTo15cm lattice.
Decay channel (to 30 m) as for SolenoidsOnly (table 8.1) except:
Solenoid
Length
0.2–0.6 m
Radius
0.1–0.4 m
Radius (final solenoid) 0.15 m (fixed)
Field (S2–S4)−5–5 T
Field (S5–)−4–4 T
Proton beam energy:
Muon energy criterion:
2.2 GeV
Any++Best yield: µ /π µ /p·GeVµ+ /s (4 MW proton beam)
11.185%0.008486
2.12×1014
15.756% including pions

Figure 8.7: Longitudinal phase space evolution (top) and beam envelope plot (bottom)
for the optimised design from the SolenoidsTo15cm lattice.

145

Figure 8.8: Optical functions in the optimised SolenoidsTo15cm design.

Figure 8.9: Emittance evolution in the optimised SolenoidsTo15cm design.
146

Table 8.3: Normalised emittances of output beams from all optimised lattices in thischapter when projected into the x–x0 or y–y 0 plane (choice of plane does not matter as
the beam is cylindrically symmetric).
(mm·mrad)SolenoidsOnly
SolenoidsTo15cm
ChicaneLinacB
PhaseRotB
PhaseRotC
PhaseRotEb6a
PhaseRotLinac1
Linac900Removable6c2

RMS
34076
18417
8216
9972
10208
11592
13108
14812

75%
90%
95%
99%
100%
93666 177399 239913 343130 496991
52940 84057 106017 156766 315720
24031 37154 45431 59919 91079
29677 42770 51120 64029 84877
30366 44289 52548 65661 98812
34266 50288 60103 79165 114378
38696 58298 69791 88635 134122
44028 66225 78091 97455 140681

beam would blow up again in the next cell. The formation of the focus can be seen in thetransverse plot on figure 8.7, towards the end of the channel.
Figures 8.8 and 8.9 also show the focus preceeded by a region of converging beamwith positive and the effect of beam scraping, which decreases the 4D emittance as thehighest amplitude particles are removed. However, the 4D emittance also exhibits growth
between the final two solenoids because the abrupt change in optics means the beam ismismatched there.
Although of limited use otherwise, this optimised design does confirm the nontriviality of the non-regular matching section from the target observed in SolenoidsOnly.
SolenoidsTo15cm has found a different optimum with all of S2–S5 deviating from maximal parameters in some way and S4 having a flipped field. The channel then continues
with the same regular structure as in SolenoidsOnly, with maximal solenoid fields, untila few cells before the final focus (with solenoid S26 being given a flipped field and several
non-minimal drift lengths). It is encouraging that the optimiser has produced ‘matching
sections’ in a similar way to a human designer, even though every parameter was separately variable so it was equally possible to put such a section in the middle of the channelaway from the boundary conditions at the ends.
In front ends with no cooling, the beam emittance will generally increase with the yield,
since there is no way to pack more particles into a smaller emittance. The values for all
the designs examined in this chapter are given in table 8.3 with various amounts of thebeam contained in the ellipse. Since the next component after the front end will be a linac
using quadrupoles, projected 2D emittances are given here instead of the 4D amplitudeused for solenoid channels, as the dynamics of the two planes will become uncoupled
after this point. As energy increases, the un-normalised emittance corresponding to thesevalues will fall, permitting more conventional structures further down the muon linac. For
example, above 8.7 GeV a beam of normalised emittance 80000 mm·mrad may fit throughan ILC 1.3 GHz module of bore radius 35 mm and length 1256 mm. Lower frequencies
with larger bores and shorter modules can fill the gap from 0.9–8.7 GeV following on fromsomething like the Linac900Removable6c2 option.

147

Table 8.4: Optimisation ranges and best muon yield for the ChicaneLinacB lattice.
Decay channel (to 30 m) as for SolenoidsOnly (table 8.1) except:
Solenoid
Length
0.2–0.9 m
Radius
0.1–0.4 m
Field (S2–S4)−5–5 T
Field (S5–)−4–4 T
Chicane: as specified in [37] (fixed)
Linac:
Structure
Number of cells
60–120
Solenoid
Length
0.2–0.9 m
Radius0.15 m (fixed)
Field−4–4 T
Drift
Length
0.5–1 mRF
Frequency
88 MHz (fixed)
Voltage0–(Ldrif t + Lsol )×4 MV/m
Phase
−180–180◦
Proton beam energy: Muon energy criterion:
2.2 GeV
400±100 MeV++Best yield: µ /π µ /p·GeVµ+ /s (4 MW proton beam)
1.400%
0.001062
2.65×1013

8.2

Chicane and 400 MeV Muon Linac

This section examines an attempt to undo the distance–energy correlation that appears in
the decay channel using a bending magnet chicane as shown in figure 8.10 and describedin [37]. This idea from Grahame Rees was implemented in Muon1 with field maps andput on the distributed optimiser as the lattice ChicaneLinacB. Once the distance–energy
correlation is undone, the rebunched beam is captured in an 88 MHz RF solenoidal linac to400 MeV using cells similar to those in [83]. This frequency is favoured because experimental work at CERN (also described in the reference) has verified operation of such cavities
up to the 4 MV/m gradient used in this optimisation. It is different to the 201 MHz usedin many other studies [25, 24, 73] because it attempts to capture the muons in a single
bucket rather than splitting into a train of bunches spaced at a higher RF frequency.
Table 8.4 shows which parameters were optimised in the decay channel and linac eitherside of the chicane. The maximum length allowed for solenoids in the decay channel has
been increased from 0.6 to 0.9 m because these solenoids were almost always given the
maximal length in the decay channel optimisations. The field and radius were also maximal but these would present much higher cost and engineering implications to increase
than the length. The result is that ChicaneLinacB has a nontrivial optimum for solenoid
length, about 83 cm, that is no longer at the end of the range, with the field converging
to 3.25 T as well. Periodic solenoid channels exhibit resonances that cause particles to belost around certain energies determined by the amount of focussing per cell. To increasetransmission, it is likely that the amount of focussing (solenoid field and length) has been148

Figure 8.10: The structure of the magnetic bending chicane lattice (top) and muons ofenergy 120–270 MeV tracked through its field maps in Muon1 (bottom), adapted from[37]. Placement relative to the end of the decay channel is also shown.
149

Figure 8.11: Longitudinal phase space evolution (top) and beam envelope plot (bottom)
for the optimised design from the ChicaneLinacB lattice.

Figure 8.12: Pions (green) decaying to muons (purple) in the optimised design from the
ChicaneLinacB lattice.
optimised to place these resonance energies far from the bulk of the particle distribution.
The linac section has a similar structure to the decay channel only now the solenoidradii are fixed at 15 cm to not go outside the RF bore. The drifts of ≥50 cm now become
crucial for inserting the RF gaps, with the lattice assuming the RF is capable of 4 MV/maverage gradient along the entire channel. For the figure of merit, muons are now only
counted within a band of 400±100 MeV kinetic energy, a range that is chosen to containthe RF bucket of the muon accelerator [37].
Figure 8.11 shows how the RF bucket forms and moves up to within the target energy
bands. It also shows how the optics of the magnetic chicane put a severe cut on the
momentum of particles transmitted: compare the drifted (2nd ) longitudinal distribution to
the much smaller (3rd ) one after the chicane, which is unwrapped in this plot and appears
as a section of large transverse amplitudes. This leads to the chicane/linac design having
150

Figure 8.13: Particle transmission as a function of distance for all eight designs discussed
in this chapter.
a low transmission of 2.65×1013 muons per second, approximately one quarter of the 1014
required for the neutrino factory to make 1021 neutrinos of each type in a 107 second
operating year. The large loss is caused by the limited acceptance of the chicane, partly
longitudinal but more importantly in handling particles with large transverse amplitudes.
Examining the behaviour of undecayed pions in this system (figure 8.12) shows thatthey are mostly lost in the chicane because they have the wrong momentum. A greener tailof pions can be seen exiting the bottom of the transverse plot immediately after the chicane
begins, though a few with similar energies to the muons survive through it. Almost no
pions (one macro-particle in this simulation) remain at the 400 MeV energy. This plot also
highlights a feature of the decay channel that the losses between the solenoids are almost
entirely muons because these have the unpredictable directions resulting from pion decay.
The original pion beam gets scraped into shape within the first ten solenoids meaningalmost no pion losses after that.
Work moved on from the chicane idea after this optimisation because of the large
losses. Figure 8.13 shows how the eight optimised designs discussed in this chapter retain
their beam. Large losses are always present at the end of the wider, 30 m-long decay
channel but the chicane takes the transmission three or four times lower than comparable
points in other designs. Dog legs on the end of these graphs show the final reduction inyield from wrong-energy muons being rejected.

151

Table 8.5: Optimisation ranges and best muon yield for the PhaseRotB lattice.
Decay channel (to 30 m): as for ChicaneLinacB (table 8.4)
Phase rotator:
Structure
Number of cells
20–40
Solenoid
Length
0.2–0.9 m
Radius0.15 m (fixed)
Field−4–4 T
Drift
Length
0.5–1 mRF
Frequency
31.4285 MHz (fixed)
Voltage0–(Ldrif t + Lsol ) × 45
MV/m
28
Phase
−180–180◦
Proton beam energy: Muon energy criterion:
2.2 GeV180±23 MeV++Best yield: µ /π µ /p·GeVµ+ /s (4 MW proton beam)
3.332%
0.002528
6.31×1013

8.3

Phase Rotation

The large energy spread of the beam in the decay channel causes problems for a lot of
conventional accelerator designs, as focussing varies greatly between particles. To address
this problem, a phase rotation section uses RF cavities to rotate the slanted longitudinal
distribution (seen in figure 8.2) so that more of the particles finish in a smaller energy
band, usable by systems downstream. It can achieve this mainly because after the drift
in the decay channel, there is a correlation between arrival time and velocity, so if the
RF voltage is made decelerating for the head of the bunch containing high energies andaccelerating for the tail, this will cancel much of the variation. This is only effective overa limited range because of the sinusoidal nature of the RF voltage variation, thus ideallya phase rotation section is optimised to put this range in the densest region of muons.
It can also be said that phase rotation is doing the opposite of the bending chicane insection 8.2, which tried to reduce the variation in time of arrival, while the energy spread
was kept necessarily the same as a consequence of using a purely magnetic system. Flighttimes were varied as a function of energy in the chicane, whereas RF systems can varyenergies as a function of time of arrival.
The PhaseRotB optimisation specified in table 8.5 has the same decay channel as
ChicaneLinacB and a phase rotation section similar to the linac from that lattice except
using a lower frequency. The aim is not to produce an accelerating RF bucket but tohave the RF period comparable to the spread in muon time of arrival from the decay
channel. A frequency of 31.4285 MHz was chosen for compatibility reasons with the Rees
cooling ring [36] but is a good choice irrespective of the downstream system (the frequency
was not optimised because that requires parameterising the RF gradient limit, which is
2.25 MV per 1.4 m of beamline here). The goal kinetic energy spread of 180±23 MeV
was also inherited from the ring’s acceptance but the important point is that it is a lot
152

Figure 8.14: Longitudinal phase space evolution (top) and beam envelope plot (bottom)
for the optimised design from the PhaseRotB lattice.
Table 8.6: Optimisation ranges and best muon yield for the PhaseRotC lattice.
Decay channel (to 30 m): as for ChicaneLinacB (table 8.4)
Phase rotator: as for PhaseRotB (table 8.5)
Proton beam energy: Muon energy criterion:
10 GeV180±23 MeV++Best yield: µ /π µ /p·GeVµ+ /s (4 MW proton beam)
3.042%
0.003435
8.57×1013

narrower than that for the 400 MeV linac in ChicaneLinacB and has a central energy near
the mean of the particles in the decay channel. It provides a figure of merit that rewards
narrowness in energy spread without requiring net acceleration or deceleration.
With the bending chicane removed, the muon transmission increased from 2.65 to
6.31×1013 muons per second into a proportionately narrower energy range. Figure 8.14
shows the resulting design and how the RF section flattens the energy distribution to get
more muons into the goal range. In fact, even more can finish in this range if there issome slight over-flattening, resulting in the S-shaped energy distribution in the rightmost
plot. The transverse plot shows that beam sizes in the RF section are similar to thosein the ChicaneLinacB linac but many more particles have been retained, the regions ofincreased density represented by purple and red colours.
It was at this point that new results regarding the optimal proton beam energy on thetarget (see section 4.2) became available. As well as being integrated into the Scoping
Study [24] recommendations, the new preferred value of 10 GeV was used in the Muon1
optimisations by replacing the old pion data file by a MARS calculation at that energy
instead of 2.2 GeV. The data files were both generated with the same version MARS15.
The PhaseRotC lattice changes the proton beam energy to 10 GeV but leaves everything else from the PhaseRotB optimisation the same, as shown in table 8.6. It therefore
153

Figure 8.15: Longitudinal phase space evolution (top) and beam envelope plot (bottom)
for the optimised design from the PhaseRotC lattice.
provides a direct comparison of the effect of changing the proton beam energy inclusiveof the front end to the end of phase rotation being optimised. Perhaps surprisingly, theefficiency in terms of muons per original pion has decreased to only 0.913 times of thatwith the 2.2 GeV beam but the real yield in µ+ /p·GeV has increased by a factor of 1.359.
This difference comes from the fact that more pions were produced per Watt of proton
beam power in the target at 10 GeV, while the decrease in relative efficiency comes fromproportionately more pions being produced in areas of the momentum phase space where
they are not captured. The high energy tail is a likely candidate for the location of thiswastage, since more kinetic energy is available from the 10 GeV protons in early collisions to produce high energy (multi-GeV) pions, which will not be captured by the phaserotation system.
The resulting yield of 8.57×1013 muons per second from PhaseRotC is becoming close
to the 1014 neutrino factory requirement, though the later bunching and acceleration
systems are at this point unspecified. The optimised design (listed in appendix E) has
only subtle differences to that for PhaseRotB, leaving most of the yield improvement
accounted for by the proton beam energy change.
Figure 8.15 shows that the transverse profile of the beam is very similar to thePhaseRotB channel in figure 8.14. The only visible difference being that PhaseRotC has
a greater density of particles present in the first ∼5 solenoids of the decay channel near
the target, as would be expected if the target produced more pions that were later lost.
The pion/muon breakdown is shown in figure 8.16, with large pion losses visible between
solenoids S2 and S3; the continuation of pion to muon decay in the phase rotation sectionis also apparent.
Both of the PhaseRotB and PhaseRotC lattices had almost every attribute of eachcomponent in the beamline variable in the optimisation. This led to a large number ofvariables, 279 in the case of PhaseRotB and also a slow pace of optimisation: these lattices
were still increasing in score very gradually after as much as 6 months running on the
distributed network while individual components were tuned to gain minor increases in154

Figure 8.16: Pions (green) decaying to muons (purple) in the optimised design from thePhaseRotC lattice.

Table 8.7: Optimisation ranges and best muon yield for the PhaseRotEb6a lattice.
Decay channel as for ChicaneLinacB (table 8.4) except:
Structure
Number of cells
15–50
Blocks
6 after S9Phase rotator as for PhaseRotB (table 8.5) except:
Structure
Number of cells
6–60
Blocks6
Proton beam energy: Muon energy criterion:
10 GeV180±23 MeV++Best yield: µ /π µ /p·GeVµ+ /s (4 MW proton beam)
3.480%
0.003930
9.81×1013

155

Figure 8.17: Longitudinal phase space evolution (top) and beam envelope plot (bottom)
for the optimised design from the PhaseRotEb6a lattice.
yield. It was expected, then, that a reduction in the number of variables could quicken thepace of convergence. This was attempted in the lattice PhaseRotEb6a, presented in table
8.7, using the technique of ‘blocking’ some of the many cells in the late decay channel andphase rotation section together.
The blocking works by observing that long homogenous sections of beamline are usually
optimised to regularly repeating cells with almost the same parameters as each other eventhough they are controlled be independent variables in the optimisation. Sections with RF
may understandably incorporate some variation through the channel as the longitudinal
phase space shape of the beam changes, however on local scales (i.e. without significant
drift) the effects of RF are nearly additive in voltage, so a slow variation in cell parameters
is expected to suffice. Only special sections such as matching near the target require
further cell-to-cell adaptation. So with a blocked lattice, the majority of the beamline
will be specified as blocks of identical cells that share the optimisation variables for only
one cell. The PhaseRotEb6a lattice (other lattice variations were tried but it worked thebest) splits the late decay channel starting from solenoid S10 and the whole of the phaserotation section each into 6 blocks of equal length, the overall length of these sections
being controlled by the decaycells and phaserotcells variables. Allowing the decay
channel to vary in length is new but neither section can shorten to the point where eachblock consists of less than one cell. The number of optimisation variables was reduced
threefold to 93.
The yield from this lattice was a further improvement to 9.81×1013 muons per second,
essentially meeting the required rate. Figure 8.17 shows how the optimiser has chosen adecay channel of just under 30 m long without being forced to this length as in previous
lattices. The only constraint was that the number of solenoids in the decay channel had
to be between 15 and 50. This design also tapers the radii of the solenoids earlier on,
something that may be favourable from a cost perspective. The overall technique of thephase rotation remains similar but with a slightly longer time interval of particles ending
up in the goal energy range, leading to the increased yield.

156

8.3.1

Alternative Design for Two Signs

As shown in figure 3.3 and explained throughout chapter 4, both positive and negative
pions are produced in the target in similar numbers. These will decay to both signs ofmuon in the decay channel, the motion of the negative particles being the mirror imageof the positive ones provided the field remains axially symmetric. This can be proved byinspecting the cross product in the Lorentz magnetic force law F = q(v × B) under achange of sign in vx , Bx and q and confirming the result is a change in the sign of Fx only.
Thus the decay channel will work similarly on both signs of particle.

Figure 8.18: Effect of the phase rotator optimised for only one sign of muons (red) on
the negative sign (blue).
The same is not true of the RF phase rotation sections because these will decelerate
negative particles in the time periods when positive particles are accelerated. The resultof a single-bunch phase rotation section (from lattice PhaseRotC) acting on both signs ofmuon is shown in figure 8.18. The positive muons have been rotated to a narrow energy
range but the variation in energy of the negative muons has been doubled because the RF
rotation has been acting in the wrong sense. This results in the very poor longitudinaltransmission of negative muons (9.4%) into the goal energy band, as shown at the topof the figure. The results in this section were calculated with a fast 1D-only code soefficiencies do not include transverse losses.
157

Figure 8.19: Proposed system for rotating both signs of muons with the same RF, eventhough they do not start with a 180◦ phase advance between them.

If the negative muons could somehow be delayed by 180◦ of the RF phase, then the
E fields experienced by them would be the exact opposite of those for the positive muons
and identical longitudinal dynamics would result, provided only one RF frequency wasused in the channel. Time of flight differences could be produced by a dipole sending
one sign of muons onto longer path-length trajectories but this is technically difficult
with such a high-emittance beam. Another option that does not change the transverse
dynamics, using purely RF, is illustrated in figure 8.19. This technique notices that a timeof flight difference between the signs can be obtained after a drift section by accelerating
one sign and decelerating the other, which is exactly what an on-peak RF cavity would
do to the initial mixed sign beam. Once the two signs are suitably far apart the RF will
no longer produce the opposite effect on each and there exists a phase that will reverse
the acceleration and deceleration of the two signs while still rotating each bunch in thecorrect direction. This corresponds to a phase difference not equal to 180◦ in which thebunches are on the same sign of the RF voltage waveform but experience opposite slopes(so the effect will be opposite net accelerations with the same rotation direction).
One-dimensional simulations of such a scheme were attempted with the same input
distribution of muons from the end of the decay channel and gave results such as shownin figure 8.20. In this scheme the longitudinal transmission of the two signs is more equaland the average transmission of both is higher. The disadvantage is that more complex
phase space manipulations require a much longer RF channel, 183 m instead of the 42 m
for the single-sign design, which would result in increased cost.
158

Figure 8.20: Improved phase rotation forming bunches of both signs of muons.

159

8.4

Phase Rotation and 400 MeV Linac

Using RF phase rotation instead of a chicane and changing the proton beam energy to
10 GeV improved greatly on the ChicaneLinacB design, but so far the goal energy has
been 180±23 MeV, which does not include any overall acceleration of the beam. A more
direct comparison could be made if a linac were added and the goal energy range changed
back to the 400±100 MeV used in the chicane and linac design.
Table 8.8: Optimisation ranges and best muon yield for the PhaseRotLinac1 lattice.
Decay channel as for ChicaneLinacB (table 8.4) except:
Structure
Number of cells
10–50
Blocks1 after S9Phase rotator as for PhaseRotB (table 8.5) except:
Structure
Number of cells
1–60
Blocks1
Linac as for ChicaneLinacB (table 8.4) except:
Structure
Number of cells
60–120Blocks
1
Proton beam energy: Muon energy criterion:
10 GeV400±100 MeV++Best yield: µ /π µ /p·GeVµ+ /s (4 MW proton beam)
6.944%
0.007840
1.96×1014

The PhaseRotLinac1 lattice, summarised in table 8.8, simply adds the 88 MHz linac
from ChicaneLinacB onto the end of the existing phase rotation design. The linac now
can inherit the blocking technique from the phase rotation (in PhaseRotEb6a) to reduce
the number of variables. Setting the number of blocks to just one per section proved tobe the most successful of various combinations tried, so the late decay channel, phaserotation and linac now each have a single cell that repeats throughout the structure.
The optimum (printed in appendix E) has an intermediate length decay channel with
39 solenoids covering a distance of just over 50 m, a maximal length 60 cell phase rotator
but a minimal length 60 cell linac. The 88 MHz linac has become minimal in one additional
way: all the RF voltages have been set to zero! This means that the resulting design,
shown in figure 8.21, performs all its longitudinal manipulation using the 31.4 MHz cavities
intended for phase rotation. This is possible because no constraint was ever imposedthat theses cavities should be used at bunching phase rather than for net acceleration or
deceleration. The 88 MHz linac section could therefore be removed to reduce the length ofthe design shown to under 140 m and give a marginal increase in yield by avoiding somemuon decay. This rejection of the higher RF frequency is likely because once the hybrid
phase rotation/acceleration is able to move the beam into the target energy range on its
own, imposing 88 MHz RF just produces corrugations that overall move beam outside therange. For a longer linac, having fully formed bunches is a necessity and this happens in
the 900 MeV linac optimisation of section 8.5.
160

Figure 8.21: Longitudinal phase space evolution (top) and beam envelope plot (bottom)
for the optimised design from the PhaseRotLinac1 lattice.
The yield of 1.96×1014 muons per second is nearly twice the requirement for the neutrino factory. This has come about mainly by widening the energy range to be similar to
ChicaneLinacB, which was intended to hold tall 88 MHz RF bucket(s) that are narrowin time. By contrast, this optimum has filled the band with a wide distribution of times
using the lower frequency; any linac built beyond this design would have to use a similarly low frequency to catch this distribution in a single bunch, the alternative being a
rebunching section at a higher frequency (like 201 MHz as used in [25]), which would add
costs in terms of muon decays.

8.5

Direct 900 MeV Linacs

The goal energy range of 400±100 MeV, while useful for a comparison with ChicaneLinacB,
is large compared to the energy ranges accepted by the downstream accelerators. Basedon the ISS report [24] section 5.2, the first recirculating linac is assumed to accept an input
kinetic energy of 0.9 GeV with a momentum spread ∆p/p of 7%. This translates into akinetic energy range of 830.41–969.64 MeV, roughly 140 MeV wide instead of 200 MeV.
A longer linac with this goal energy is specified in optimisation Linac900Removable6c2,
detailed in table 8.9. The phase rotation at 31.4 MHz and linac at 88 MHz are still present
but the linac can be made much longer. All sections of the beamline can now be removed
completely by the optimiser (apart from the first solenoid!) in the pursuit of higher yields.
The result this time is that the decay channel has been reduced to just four solenoids
and the phase rotation segment has been removed entirely. Figure 8.22 shows that thisdesign does all the work using the 88 MHz linac structure, which has been given the
maximum allowed number of cells. Unlike PhaseRotLinac1 in the previous section, thislinac forms something much more resembling a conventional RF bucket in the longitudinal phase space. The yield of 1.07×1014 muons per second also meets the neutrino
factory requirement, although the linacs beyond this design must have a wider bore to161

Table 8.9: Optimisation ranges and best muon yield for the Linac900Removable6c2 lattice.
Decay channel as for ChicaneLinacB (table 8.4) except:
Structure
Number of cells
1–50
BlocksUp to 6 after S9Phase rotator as for PhaseRotB (table 8.5) except:
Structure
Number of cells
0–60
BlocksUp to 6Linac as for ChicaneLinacB (table 8.4) except:
Structure
Number of cells
0–240
BlocksUp to 6RF
Phase
−360–360◦
Proton beam energy: Muon energy criterion:
10 GeV
900MeV (k.e.) ±7% ∆p/p++
Best yield: µ /π µ /p·GeVµ+ /s (4 MW proton beam)
3.786%
0.004274
1.07×1014

Figure 8.22: Longitudinal phase space evolution (top) and beam envelope plot (bottom)
for the optimised design from the Linac900Removable6c2 lattice.

162

Figure 8.23: Pions (green) decaying to muons (purple) in the optimised design from the
Linac900Removable6c2 lattice.
accommodate an uncooled beam with normalised emittance as given in table 8.3.
This yield is lower than PhaseRotLinac1 both because of the stricter final energy criterion and because the muons have to survive a nontrivial amount of acceleration. However,the 31.4 MHz combined phase rotation and acceleration approach used in PhaseRotLinac1
could likely not be extended to higher energies because a stable RF bucket was neverformed. This is why the entire linac was optimised again with the new energy goal andnot extended from the previous system.
The lack of decay channel means that this accelerator copes with multiple species(pions and muons) at the same time. Figure 8.23 colours the beam by composition andgives an illustration of how this works in practice, with the pions occupying a higher
kinetic energy level bucket than the muons. This is because the charged pions have a rest
mass of 139.57 MeV/c2 compared to the muons’ 105.66 MeV/c2 , which means to have thesame velocity (equivalently β or γ) they must be given 32% more kinetic energy. The
velocity is what determines the RF phases when the particles arrive at the cavities so
scaling up the kinetic energies in this fashion ensures correct phasing for both species,
ignoring transverse path length effects and the fact the RF voltage will be less effective
compared to the energy the pions already have in rotating the contents of the bucket.
Fortunately the bucket only undergoes a fraction of a rotation before most of the pions
have decayed so the effect of this discrepancy in bucket synchrotron frequency is small.
Removing the decay channel may not be practical to the extent the optimiser has
done, since the area in the first few solenoids near the target has particularly high losses
and radiation and this may not be conducive to stable operation of the RF cavities.
It is noticeable in the transverse plot of figure 8.22 that the beam does not undergo any
adiabatic shrinkage in x as it is accelerated. Given the relation between un-normalised andnormalised emittance, this may seem counterintuitive, but the shrinkage has happened
elsewhere: figure 8.24 shows the distribution of x0 in the beam along the length of the linac.
Reduction of x0rms is visible from the 10 m point until the end of the linac, with the change
in distribution shape in the 4th and 6th sixths of the linac caused by flipped-sign solenoids
163

Figure 8.24: Plot of the x0 distribution of the beam against z for the optimised designfrom the Linac900Removable6c2 lattice.
being used there. Reduction before about 10 m is not considered genuinely adiabatic
because it is mostly beam losses in the decay channel, where there is no acceleration.
This behaviour comes about because the magnetic field is constant throughout the
linac and geometrically the RF acceleration will reduce x0 by increasing pz but leave x
and px alone. In synchrotrons the beam shrinks in both x and x0 because the focussing
structure and Twiss parameters are kept constant by raising magnetic fields in proportion
to the beam momentum. With linacs there is more flexibility and the emittance may be
traded off between xrms and x0rms by varying the focussing field. Higher focussing, best
attained with quadrupoles for higher energies, favours smaller xrms .
Figure 8.25 visualises this same effect as an increase in along the channel. Notice thereduction of un-normalised 4D emittance in figure 8.26 as the beam is accelerated, while
normalised emittance stays constant once all the pions have decayed. The 2D emittancesin this plot oscillate above the 4D values because of x–y correlation caused by the periodic
solenoid fields. The sharp decrease of emittance at the beginning is due to losses as thebeam enters the narrower linac section very soon after the target.
The component listing in appendix E shows that the first four of the six linac blocks
have roughly the same RF parameters but the fifth turns off the RF voltage almost
entirely. Then the final block switches it back on but at a completely different phase. The
effect of this is the ‘lopsided’ rotation of the particles in the bucket shown in figure 8.22,
with the different RF phase in the final sixth of the linac rotating particles distributed
along a diagonal in phase space into the goal band. This nonconstant phase could allow
a longer bunch of particles to finish in the goal band than would normally be expected,by shearing the distribution to reduce momentum spread and increase time spread.
Overall then, the optimised design from Linac900Removable6c2 is a muon front end(without cooling) and initial acceleration system to 900 MeV, producing over 1014 muons
per second of a single sign. Optimisation of a two-sign system similar to that in section 8.3.1 is possible but not yet attempted with the full Muon1 code, so it is not known
how many opposite sign muons can be accelerated. For comparison, the ISS baseline [24]includes a cooling channel so ends with an RMS emittance of 7400 mm·mrad, which isabout half of the 14812 mm·mrad produced here (see table 8.3 for more data). The
ISS baseline transmits 0.0083 µ+ /p·GeV compared to 0.0043 in this design, though thisvalue varies strongly with the linac energy range: for example PhaseRotLinac1 produces
0.0078 µ+ /p·GeV in a range based on an 88 MHz linac rather than the ISS RLAs.

164

Figure 8.25: Optical functions in the optimised Linac900Removable6c2 design.

Figure 8.26: Emittance evolution in the optimised Linac900Removable6c2 design.

165

Chapter 9
Conclusion
This thesis has described the design of a pion production target and muon front end forthe neutrino factory, which meets the 1014 muons per second criterion derived from 1021
muons per operational year [1]. In the process, the effect of target design variations on
useful pion yield has been characterised and a design system using a distributed genetic
algorithm has been created to produce entire accelerator designs automatically. Therefore
the work can be interpreted from the viewpoint of the three disciplines below.
Accelerator design. The baseline neutrino factory design [24, 25] has been wellstudied but includes components such as the muon cooling channel and a 6 m long magnet
surrounding the target (figure 3.7) that may prove challenging to build. The designs inthis thesis have no muon cooling and enforce 50 cm gaps between all solenoids, which
sacrifices some muon yield, in exchange for a target area that can be taken apart andfewer beam windows to contain cooling material. The solid target option has also beenused instead of the liquid mercury jet. The designs presented have taken an acceptable
number of muons to 900 MeV, so it is suggested that the muon linac continues all the way
to 25 GeV, providing a much simpler design than the baseline (figure 1.3) at a length andcost that is not necessarily much greater, as explained at the end of section 1.4.
Tracking of this design has been done with the author’s Muon1 code that is fully 3D,
includes particle decays and uses a particle set generated by MARS15 [2] from the target
rather than an idealised beam distribution. Section 8.3.1 presents some promising work
using a 1D code that shows adaptations of this design can accelerate both signs of muon.
Target design. The recommended design parameters for the pion production target
are given in section 4.7. However, many of these ‘optimal’ values are surrounded byquite broad ranges in which the pion yield is almost as high. Thus, readers interested indesigning a target station may find the graphs in sections 4.2–4.5 more useful since they
allow other engineering constraints (of which there will be many) to be traded off against
performance. The considerations that can be identified at the current level of design detailare discussed in chapter 3, including how the proton accelerator interfaces via the target
with the rest of the neutrino factory. However, the chapter can only be an introduction
to the problem of target design, which is a field of study in its own right [87].
The target designs were compared using calculations from the material interaction

166code MARS15. However, hadronic interactions are not yet a ‘solved problem’ in particle
physics and large discrepancies can remain between the empirical models available. Hencechapter 5 attempts to quantify the degree of uncertainty by comparing the MARS15 resultswith GEANT4 [3] and data available from the HARP experiment [4]. The codes agree
quite well on the optimal proton energy for the neutrino factory, though the absolute pionyields from GEANT4 are too high by a factor of two. MARS15 is shown to have better(25% level) agreement with the HARP data in the region available for comparison.
Scientific computing. The search for high-yielding muon front end designs wasmade as automatic as possible by the use of computer optimisation. The user necessarily
has to specify the figure of merit and boundary conditions such as the particle source,
practical range of sizes for components and general structure of the beamline, with therest of the problem being recast as maximising a function. The maximisation is done bya genetic algorithm, which requires many simulations, so the code was made to run ona distributed computing network to perform the large amount of work in parallel. The
general description of a range of accelerators can be written in the Muon1 lattice fileformat and submitted to this network and optimisation will commence. The optimiser
has been relatively successful on problems with hundreds of variables and the designs
presented in chapter 8 are all examples of this.
Notably, the distributed network consisted mostly of volunteers who downloaded theclient program from the project’s website [5], with a few of the author’s computers also
running. The project has more than 100 users active at any one time, with thousands
having run the program at some point and over 60 million muon front end designs havebeen simulated. This demonstrates the enthusiasm of the public for interacting with
scientific projects, even complex ones like the neutrino factory.

Summary and Implications
The design of the front end in section 8.5 together with a muon linac to 25 GeV provides
a useful complement to the current main thread of neutrino factory design work being
pursued in the International Design Study (IDS) [73]. Neutrino factory design work
globally has progressed through several stages, initially concentrating on increasing themuon yield as much as possible, leading to US Feasibility Study II [25], with a relatively
aggressive design for a cooling channel using liquid hydrogen. The next stage focussed
more on maintaining an acceptable muon yield while reducing the estimated hardware
cost, thus came the lithium hydride cooling channel of the International Scoping Study(ISS) [24] and the replacement of the high-energy end of the recirculating linac (RLA) byan FFAG, which provided more turns through the same RF cavities and therefore reduced
the number of cavities required to achieve a set level of acceleration.
Unfortunately, yield vs. cost was not the only trade-off being made during these design
revisions. The rearrangement of the acceleration system to allow more passes per RF
cavity has come at the cost of a much more complex machine. The current IDS baseline
contains four muon accelerators of three different types (linac, RLA1, RLA2, FFAG),
compared to the two types (linac, RLA) found in Study II. The RLAs require magnet
switchyards at each end, all of which are different from each other, and the FFAG performs
167

an unconventional form of RF acceleration that relies on small time-of-flight differencesin the orbit during acceleration.
It is possible that all of the components in the current IDS baseline design will workwithout major faults. However, the fact that complexity increases risk should not beignored. That is partly the motivation of the proof-of-principle experiments MICE [72],
EMMA [88], FETS [89] and MERIT [39], which test various parts or analogous models
for the muon cooling channel, FFAG, proton driver linac and target, respectively. At
the present point in time, the risks of the neutrino factory design cannot be addressed
properly without admitting the possibility that one or more of those experiments may
produce a negative result (or identify problems in scaling up to a full-size or full-powersubsystem).
The design in this thesis, while not the best in muon yield or possibly cost, relies
on fewer unconventional technologies than the current baseline and is simpler. No muon
cooling or FFAG is involved at any point. Large, low-frequency RF cavities similar to thosein cooling channels are used, but there is no scattering material or cryogenics involved,
and the cavities are placed between solenoids rather than at the peak field, which has
been shown to reduce cavity performance [90]. Linacs are proven technology and will notdiffer greatly for muons instead of protons, the only requirement being a larger bore at
the low energy end, which other neutrino factory designs also have. The pion production
target is unavoidably difficult because of the high powers involved but this thesis suggests
how it could be reduced to the problem of a conventional solid target, provided the solid
targets can be moved for each pulse.
In conclusion, achieving the 1014 muons per second goal with the simplified designpresented here provides an important piece of risk management for the neutrino factory
project. If the MICE or EMMA experiments point to problems with their novel technologies, fall-back designs that work without the corresponding subsystems are feasible.
The solid target option provides an alternative to MERIT’s liquid target (although it has
its own distinct set of challenges) and the range of acceptable proton driver energies in
section 4.2 provide additional flexibility to evaluate and make use of high-power proton
facilities that may be built in the future.
Future WorkFor the neutrino factory to be realised, a great deal (hundreds of person-years) of detailed
engineering will have to be done. There are also some important conceptual questions
that must be resolved first: what sort of target (solid or liquid) is desired? What are theimplications for cost of a simplified design such as the one in this thesis compared to thebaseline, which is shorter but more complex?
In terms of the Muon1 code and its associated optimisation work, it would be interesting to put the cooling components into the optimiser, or allow the solenoids near thetarget to overlap, to see if this produces a muon yield as high as the current IDS baseline.

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174

Appendix A
MARS.INP Template File
The token $energy in the file below is replaced by a number in GeV for each energy
required. MARS uses kinetic energy and requires that this ends in a decimal point if it
is a whole number (for instance 6. for 6 GeV but 2.2 for 2.2 GeV). The token $hole isalways replaced by 0. in simulations where the rod is solid.
Tantalum Rod Pion Production [TEMPLATE]/home/csf/brooks/mars15/dat
C GUI switch
CTRL 0C Generate 100000 events
NEVT 100000
C 0.01mm boundary precision, 2cm initial trial step
SMIN 0.001 2.
C Beam is protons(1), parallel gaussian(2) [now modified in BEG1 subroutine]
IPIB 1 2
C 0.5cm RMS transversely (so rod is 2sigma), 1ns RMS longitudinally
BEAM 0.5 0.5 5=29.9792458
C Energy in GeVENRG $energy
C Two materials used: tantalum and vacuum
NMAT 2MATR ’TA’ ’VAC’
C Use ’R-sandwich’ type geometry (default)
INDX 2=F
C Two R-sections: vacuum(#2) to $hole cm radius, tantalum(#1) to 1cm radius,
C ending at z=20cm
NLTR 2RSEC $hole 1. 51=10 10 101=2 1
175

ZSEC 20. 1251=200
C We record particles at three surfaces, 1cm out, z=0-20cm;
C and r=0-1cm, z=0cm and z=20cm to fort.83 (81..90)
NSUR 3
RZTS 1. 1. 0. 20. 83.
0. 1. 0. 0. 83.
0. 1. 20. 20. 83.
STOP

176

Appendix B
Modification to User Subroutine
BEG1 in m1504.f
This is the section of the long Fortran file ‘m1504.f’ that has been modified to generate
the correct input proton distribution.
*

PARAMETER (CLIGHT=29979245800.D0)
PARAMETER (PI=3.141592653589793238D+00)
C- - - - - - - - - - - - - - - - - - - - - - - - - - - C+++ INSERT YOUR SOURCE TERM HERE +++C
READ()...,W1,...
C
W=W*W1
R=SQRT(1.D0-SQRT(RAND())) ! SJB: 1cm radius circular parabolic beam
TH=RAND()*PI*2.D0
X=R*COS(TH)Y=R*SIN(TH)C++++++++++++++++++++++++++++++++++++
RETURN
END

177

Appendix C
The Linearised Beam Optics Model
To understand and quantify the performance of cooling channels, it is informative to
make the small-angle, or paraxial approximation to the transport of the muon beam, inwhich the optical elements have linear (and thus easy to analyse) effects. This approachalso reveals the Twiss parameters † [91, 92] widely used in accelerator lattice design.

C.1

Equations of Motion in z

Section 2.2 shows how the transport of particles in the most general case can be describedin terms of an augmented phase space vector s(t) = (x, y, z, γvx , γvy , γvz , t) for eachparticle and how it changes over time as given by a differential equation of the formṡ = f (s). If the particles are assumed to be travelling forwards so that z increases
monotonically with t, it is possible to restate the problem with z as the independent
variable instead of t, using a different set of phase-space variables u(z) including thetransverse positions and velocities, forward velocity (or energy) and the time of arrivalt as a new dynamical variable that depends on z. For a non-straight accelerator, thistreatment uses curvilinear coordinates such that x = y = 0 represents a specified reference
path through the machine, on which z is equal to the path length and the three axes remain
orthogonal. Non-zero x and y are measured on planes of constant z perpendicular to thereference path. These conventions mean that the mapping u(z0 ) → u(z1 ) fully describes
the behaviour of the physical piece of accelerator for which z0 ≤ z ≤ z1 (rather than the
behaviour through a specific time interval). This mapping can be obtained from solving
the new equation of motion in the form u0 = g(u), prime denoting differentiation withrespect to z, noting that cases where ż ≤ 0 are outside the scope of this method.
Just as in section 2.1, there is some choice of how to parameterise the phase space,a sensible option being u = (x, y, t, x0 , y 0 , Ek , z) where Ek is the kinetic energy of thedtparticle, in place of the peculiar t0 = dz
= v1z coordinate that would otherwise havebeen used. Here, the form of g will be derived in cartesian coordinates only, because thetransformation to the curvilinear coordinates of choice is standard mathematics and would†This name is historically inaccurate and they should be called Courant–Snyder parameters. However,Twiss is the more common usage and avoids confusion with the ‘Courant–Snyder invariant’.

178yield very long equations in the general case. This requires expressing the z-derivativeof each component of u in terms of known constants and other components of u. The
constants in this case are the speed of light c and the mass m and charge q of the particle.
dx
= x0
dz
dy
= y0
dz
1
dt
=
dz
vz
p
1 + x02 + y 02=v
p(mc2 + Ek ) 1 + x02 + y 02p=
c 2Ek mc2 + Ek2
This final expression will be written as v1z as it appears frequently in the other terms. The
derivatives of the primed variables need some manipulation to relate them to the time
derivatives of γv given in section 2.1.
1 dx0
dx0
=
dz
vz dt


1 d γvx=
vz dt γvz
1d(γvx )
0 d(γvz )
=
−x
γvz2
dt
dt
d
(γvi ) = mq (Ei + (v × B)i ).
It is now possible to substitute the Lorentz force law dt

dx0q=(Ex + vy Bz − vz By − x0 (Ez + vx By − vy Bx ))
dz
mγvz2
q
q
0
=
(x0 y 0 Bx − (1 + x02 )By + y 0 Bz )
(E
−
x
E
)
+
x
z
2
mγvz
mγvz


2
qc
1
1 0 0
0
02
0
=
(Ex − x Ez ) + (x y Bx − (1 + x )By + y Bz )mc2 + Ek vz2vz
The derivative of y 0 can be calculated in a similar way, provided care is taken with theindices.


dy 0
qc2
1
1
0
02
0 0
0
=
(Ey − y Ez ) + ((1 + y )Bx − x y By − x Bz )dz
mc2 + Ek vz2
vz
Finally, the rate of change of energy comes only from the electric field.
dEk
1 dEk
=
dz
vz dt
1
=
q(v · E)vz
= q(x0 Ex + y 0 Ey + Ez )Taken together with dz/dz = 1, these expressions define u0 = g(u). The E and B fields
also depend on z, which is why the independent variable z was appended to the definitionof the vector u(z) so as to close the system, analogously to the 7-element s(t).
179

C.2

Transfer Matrices

Given an initial condition u(z0 ) and a final position in the beamline z1 ≥ z0 , the firstorder system of differential equations u0 = g(u) derived in the last section may be solved
to give u(z1 ) = m(u(z0 )) for some new function m. This sort of function is known asa mapping function because it can be applied to the initial phase space location to map
directly from one point in the accelerator to another. As t was included in the definitionof the phase space vector u, this even works for time-varying electromagnetic fields since
E and B are still uniquely specified by the phase space point. However, mappings may
generally not be used when there are interactions of the particles with each other, suchas ‘space charge’ when the Coulomb interaction from other particles in the beam enters
the definition of E to a significant extent. This is because u no longer contains all the
relevant information about the state of the system.
Differential equations defined by a nonlinear function such as g will usually produce anonlinear mapping function m that is even more complicated. A key technique for dealing
with this complexity is to linearly approximate m by considering small perturbations froma ‘central’ point u0 at the beginning of the accelerator section: u(z0 ) = u0 + δu0 , whereterms in (δu0 )2 and above may be neglected. The result of the mapping is now
u(z1 ) = m(u(z0 )) = m(u0 + δu0 )= m(u0 ) + (δu0 · ∇)m(u0 ) + O((δu0 )2 )' u1 + δu1 ,
u1 = m(u0 )δu1 = (δu0 · ∇)m(u0 )assuming m is smooth enough to be Taylor expanded and defining the output central
point and its perturbation term as shown. The components of this output perturbation
are given by(δu1 )i =X=

X(δu0 )j ∂j mi (u0 )jAij (δu0 )j ,Aij = ∂j mi (u0 )jwhere the matrix of constants A is defined so that in matrix product notation,
δu1 = Aδu0 .
This matrix A is formally known as the Jacobian of the function m at the point u0 .
Since perturbations in the input or output z are zero by the definition of the problem, the
7th row and column can be removed and the remaining 6 × 6 matrix is called a (linear)transfer matrix in accelerator terminology [93, 94]. It completely codifies the first order
perturbative behaviour of particles near the trajectory that starts at u0 and ends at u1 .
For beams, the central point is usually chosen to be the barycentre of the beam distribution, for example u0 = hu(z0 )i, which ensures that the δu0 values are ‘small’ forparticles near the centre of the beam, or for the whole beam if it and the nonlinear terms
in m are small enough.
180

C.3

Beam Shape and Covariance Matrices

Now that the actions of accelerator sections on particles close to the central point havebeen represented to first order as matrices, it makes sense to find a compatible model ofthe (sufficiently small) beam itself that these can act on. Consider first a vector of randomvariables ζ such thatRζ ∼ ζ
ζi2 = 1for all 1 ≤ i ≤ 6, where angle brackets take the expected value of a random variable,∼ means ‘identical in distribution’ and R is any 6 × 6 rotation matrix. These formulae
imply that the ζi all have unit variance and the distribution of ζ is invariant underrotations of the 6-dimensional space, thus the probability density function has the formf (|ζ|). By symmetry arguments, these also imply thathζi = 0hζi ζj i = 0

for all i 6= j, with the first identity coming from considering R to be various 180◦ rotations
that change the sign of some ζi and the second from considering a 90◦ rotation in the i–jplane, which changes ζi ζj to −ζj ζi . This means that the ζi are not linearly correlated.
Note that there are many possible choices for the distribution ζ that satisfy the conditions:
one is a unit Gaussian distribution, in which case all the ζi are independent and identically
distributed; another is a uniform distribution on the interior of a 6-dimensional sphere, inwhich case they are still identically distributed but no longer independent.
The model of the beam shall be thatδu ∼ Sζfor some 6 × 6 matrix S that defines the beam shape, where δu is still the phase space
offset from the central point (its 7th entry is set to zero and removed) but interpreted now
as a random variable statistically representing all particles in the beam. This distribution
transforms through the section of the accelerator with transfer matrix A follows:δu0 ∼ S0 ζ⇒

δu1 ∼ AS0 ζ,

so S1 = AS0 is the transformation rule for shape matrices S.
Although shape matrices are a natural way of defining beam shape (and useful for
generating specific distributions computationally starting from a normalised generator forζ), they are not uniquely determined by the beam itself sinceRζ ∼ ζ

⇒SRζ ∼ Sζ,

meaning that the shape matrices SR and S generate the same beam for any rotation
matrix R. This situation appears elsewhere in physics where there is a gauge group,for example the electromagnetic vector potential Aµ can be chosen in multiple ways todescribe the same physical situation. It would be useful at this point to find the ‘beam
equivalent’ of the physical field F µν that is uniquely determined by measurable parameters.
The measurable parameters in this case are expectations of functions of the beam hf (δu)i.
181

If the central point is the beam barycentre then hδui = hSζi = S hζi = 0, so the next
simplest measurables are the quadratic functions, or covariances, equal to
Cij = hδui δuj i .
These may be aggregated into a matrix and as the expectation operator treats matrix elements separately, the right-hand side can be expressed as a vector outer product, allowing
some useful manipulation:
C =
=

δuδuT
Sζ(Sζ)T = Sζζ T ST

= S ζζ T ST .
The elements of ζζ T are just ζi ζj , so by the unit variance and zero linear correlation
properties of ζ, ζζ T = I, the 6 × 6 identity matrix. This gives the primary relation
between C and S:
C = SST .
That C is gauge-invariant can be checked by replacing S by SR and noting that RRT = I
for rotation matrices. Using the transformation law S1 = AS0 for shape matrices, theaction of a section of accelerator on the covariance matrix can be derived.
C1 = S1 ST1 = AS0 ST0 AT = AC0 AT

C.4

Un-normalised and Normalised Emittances

Section 6.1 discussed Liouville’s theorem: the fact that Hamiltonian systems conserve
phase space volume. It will be useful to find a measure of phase space volume for theδu ∼ Sζ model of the (sufficiently small) beam developed so far. The distribution may
not have a sharp edge in phase space because the random vector ζ could be somethingsuch as a Gaussian that never exactly goes to zero. However, the variances hζi2 i are defined
to be unity, so in this case it is possible to take the distribution of ζ as the definition of‘unit phase space volume’. Thus the distribution of particles δu will have volume det S inthese units, as the determinant of a matrix is the factor by which it scales volumes. This
concept of phase space volume is known as un-normalised RMS emittanceεunrms = det S;
it can also be defined relative to the measurable covariance matrix using properties of thedeterminant:√2
undet C.
det C = det SST = det S det ST = (det S)2 = εun
⇒
ε
=
rms
rms
Note that the number of dimensions of δu in the phase space considered is often
reduced to two or four, rather than the full six, in order to see the emittance effects ina subset of the variables of the system. This works best when coupling between pairs ofphase space variables is weak: the longitudinal pair (t, Ek ) are often only weakly coupled182

to the transverse pairs (x, x0 ) and (y, y 0 ), so there exist varieties of (2D) longitudinalemittance. In a channel consisting only of quadrupole and dipole magnets, the transverse
pairs are also decoupled from each other and 2D transverse emittances can be used foreach pair of variables (or phase space plane). Solenoids, however, mix the x and y planes,
so 4D transverse emittance better represents the beam.
In cases where a beam without holes has a defined edge in phase space, that is theprobability density function f (ζ) = 0 for |ζ| > R but f (ζ) > 0 for |ζ| < R, the actual
volume of phase space occupied by the beam (however diffuse it may be in some regions)can be calculated. In a D-dimensional phase space this is given by
V = SD RD det S,
where SD is the volume of a D-dimensional unit sphere, useful values being S2 = π,S4 = π 2 /2 and S6 = π 3 /6. By convention, the factor of SD is usually dropped, leading to
the definition of un-normalised 100% emittance (also called geometric emittance)√
DD
det C.
εun
=
R
det
S
=
R
100
For a known model distribution such as a uniform or parabolic beam, the value of R may
be calculated from the requirement that the ζi have unit variance; note that in these casesthe ζi are not independent, so a D-dimensional integral is required. Beams with holes(f (ζ) = 0 for some |ζ| < R) are also possible but there is no convention on whether theholes are counted in the total emittance.
Liouville’s theorem states that phase space volume is conserved provided the system
is described by a Hamiltonian and the correct measure of phase space volume is used.
The emittances discussed so far use the variables x0 and y 0 , which only conserve phasespace volume if the beam is not accelerated or decelerated. To make the theorem valid in
general, x0 and y 0 must be replaced by the ‘canonical conjugate momenta’ in the x and y
directions. The end result of this calculation (described in [95]) is a just rescaling of the
un-normalised emittances by the relativistic factor βγ for each transverse plane:
εn2D = βγ εun
2D

εn4D = (βγ)2 εun
4D ,giving a quantity known as the normalised emittance. This scaling applies to the RMS,
100% and other percentile variants of the emittance similarly. Note that if Liouville’s
theorem is obeyed, the incoming and outgoing emittances of a section of accelerator are
the same, so det S1 = det S0 , which places a restriction on the transfer matrix that
det A = 1.
The units of emittance depend on the number of phase space dimensions considered.
For a single plane such as (x, x0 ) (2D emittance), x can be measured in millimetres andx0 in milliradians, so mm·mrad is a commonly used unit. The basic SI unit would be
metre-radians, or even just metres since radians are dimensionless, but this is a verylarge unit; however, mm·mrad is sometimes written as µm. For the two transverse planes
taken together (4D emittance) the unit becomes (mm·mrad)2 . In the longitudinal plane(t, Ek ), the two variables are already conjugates in the Hamiltonian sense, so longitudinalemittance is always normalised and is commonly stated in units of eV·s (electronvoltseconds). This makes the full unit for 6D emittance (mm·mrad)2 ·eV·s. Sometimes whenparticles are tracked in the time domain, 6D emittances in (mm·mrad)3 appear because
now the z direction is treated exactly like the other two.
183

C.5

Twiss Parameters

The covariance matrix C consists of measurable quantities that determine the shapematrix of the beam up to internal rotations S → SR. It is symmetric because Cij =Cji = hδui δuj i, so in the case of 2D phase space for a single axis (δu = (x, x0 )), there
are three, not four, measurables: hx2 i, hxx0 i and hx02 i. The overall size of the beam has
already been quantified by RMS emittance satisfying
22εun
rms = det C = x2x02 − hxx0 i ,so it is possible to produce a scale-invariant matrix of the remaining measurable quantities
that only relate to the beam shape:det(C/εunrms ) = 1 
hx2 i /εun
hxx0 i /εun
un
rms
rms
=
C/εrms =02
unhxx0 i /εun−
rms hx i /εrms−
,and . Note that thesewhere the last line defines the beam Twiss parameters ,definitions mean , ≥ 0. Because C only had three independent variables and now one
has been removed, the Twiss parameters only have two degrees of freedom between them,
the unit determinant condition producing the constraint− 2 = 1.
An upright beam is defined as being one in which = 0, or equivalently hxx0 i = 0
and is informative for understanding what the Twiss parameters mean (see figure C.1). A
beam will pass through this configuration, in which there is (instantaneously) no overall
expansion or contraction because the outermost particles have x0 = 0, multiple times ina focussing system. p
Upright beams typically occur at all the maxima and minima of
the beam size xrms = hx2 i within the focussing channel. In the upright case the Twiss
parameters and emittance are
εun
rms

= 0,
p
=
hx2 i hx02 i = xrms x0rms ,
s
hx2 i
hx2 i
xrms
= un =
= 0 ,
02
εrms
hx i
xrms

(definition of upright beam)= 1/ .

(by

−

2

= 1)

The RMS emittance emerges as the product of the RMS semi-axes of the phase space
ellipse, which is consistent with being proportional to area. The function, perhaps themost commonly used Twiss parameter, is the ratio by which the beam RMS size exceeds
the RMS angle of a particle x0rms . It has dimensions of length and is equal to the distancethe beam would have to drift for the RMS change in particle x during the drift√ to beequal to the original xrms , after which an elliptical phase space beam would be 2 times
the original total size. This is the characteristic length of a focus from optics.
p
The definition of = hx2 i /εun
εun
rms (regardless of ) can be rearranged as xrms =rms ,unmeaning that for a constant εrms ,determines spatial beam size. Thus if a beam isrequired to be small in a certain region, the only pure optical technique available is to
reduce , as the emittance can only be changed by non-Liouvillian processes.
184

Figure C.1: RMS ellipse shapes in phase space, showing Twiss parameters.
When the beam is not upright, 6= 0, with negative implying an expanding beamand positive a contracting one. If a linear magnetic field is applied such that it reduces
x0 for positive x, this leads to an increase in and the beam is focussed in that plane;a magnetic gradient of the opposite polarity would reduce and defocus the beam. The
Twiss parameters can also be used in multidimensional systems, typically by projecting
the phase space into a single 2D plane and examining the RMS ellipse shape, though if
there is correlation of the distribution between the different axes this may make the beam
appear larger than it is.
Note that similar constructions apply to the 100% sizes of beams that are ellipticalin phase space, with the Twiss parameters being the same but xmax , x0max being somemultiple of the RMS values. Twiss parameters can also be found as a function of length
in a periodic section of accelerator, from the condition that the input and output beam
ellipse shapes in phase space must match. These are known as matched Twiss parametersand have the property that a beam with those values of , and should be matched(finish with the same distribution it started with) in that accelerator structure provided
its emittance is small enough that the linear paraxial model holds.

185

Appendix D
ICOOL Input File
Below is the for001.dat input file for simulating a pencil beam of 200 MeV/c muons
through 30 cm of LH2 . It uses the default ICOOL scattering model settings and wasadapted from the drif.f01 example included in the package. Particle output is written
to for009.dat. The npart parameter may be reduced from 1000000 for fewer particles.
Drift space example&cont npart=1000000 nprnt=3 prlevel=1 /&bmt /
1 2 1. 1
0.
0.0
0.
0.
0.
0.200
0.
0.
0.
0.
0.
0.000
0
&ints /
&nhs /&nsc nscat =1 /-0.10 5e-3 40 1 2-0.10 10e-3 20 2 2&nzh /&nrh /&nem /&ncv /SECTIONOUTPUTSREGION0.30
1 0.003
1 0. 0.10
NONE
0. 0. 0. 0. 0.
LH
CBLOCK0. 0. 0. 0. 0.
ENDSECTION

0. 0. 0. 0. 0.

!mean: x y z px py pz!sigs

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

186Appendix E
Listings of the Optimised Designs

Table 9.1:
Muon1
SolenoidsOnly.
RodSource
TantalumRod
Solenoid
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
S16
S17
S18
S19
S20
S21
S22
S23
S24
S25
S26

File
‘pi plus 2G2eV’
Length
0.45
0.5996
0.48308
0.55275
0.6
0.59920.59960.6
0.6
0.59920.5996
0.59880.59920.6
0.59880.6
0.59920.59960.6
0.6
0.59960.6
0.6
0.6
0.59880.6

component

RodAngle
0.0955956
Radius0.1
0.39970.39940.4
0.4
0.4
0.39940.39970.4
0.39820.4
0.39940.4
0.39970.4
0.4
0.39970.4
0.4
0.39970.39970.3985
0.4
0.4
0.397
0.4
187

listingZ
0.24865Field
20
54.95996
4.23924
3.99199
3.96797
3.97598
4
4
3.97598
3.97598
3.991993.991994
4
4
3.991994
4
4
3.991994
3.975983.98398
3.96797
4

of

cell

continued from previous page
S27
0.59880.39973.99199
S28
0.59720.4
3.95996
S29
0.52873
0.39853.98398
Drift
Length
Bore
D1
0.5
0.5
D2
0.5
0.5
D3
0.5
0.5
D4
0.5
0.5
D5
0.5045
0.5
D6
0.5
0.5
D7
0.5
0.5
D8
0.5
0.5
D9
0.5
0.5
D10
0.5
0.5
D11
0.5
0.5
D12
0.5
0.5
D13
0.5
0.5
D14
0.5
0.5
D15
0.5
0.5
D16
0.5
0.5
D17
0.5
0.5
D18
0.5
0.5
D19
0.5
0.5
D20
0.5
0.5
D21
0.51101
0.5
D22
0.5
0.5
D23
0.5
0.5
D24
0.5
0.5
D25
0.5005
0.5
D26
0.5
0.5
D27
0.5
0.5
D28
0.504
0.5
Beamline order: TantalumRod, S1, D1, S2, D2, S3, D3, S4, D4, S5, D5, S6, D6, S7, D7,
S8, D8, S9, D9, S10, D10, S11, D11, S12, D12, S13, D13, S14, D14, S15, D15, S16, D16,
S17, D17, S18, D18, S19, D19, S20, D20, S21, D21, S22, D22, S23, D23, S24, D24, S25,
D25, S26, D26, S27, D27, S28, D28, S29

188

Table 9.2:
Muon1
SolenoidsTo15cm.
RodSource
TantalumRod
Solenoid
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
S16
S17
S18
S19
S20
S21
S22
S23
S24
S25
S26
S27
S28
Sfinal
Drift
D1
D2
D3
D4
D5
D6
D7
D8
D9
D10
D11

File
‘pi plus 2G2eV’
Length
0.448750.6
0.421020.33894
0.52673
0.59920.598
0.59399
0.59880.59880.59520.6
0.6
0.6
0.589990.6
0.59640.6
0.59640.59520.59399
0.59720.5964
0.596
0.596
0.6
0.6
0.6
0.2
Length0.5
0.5
0.5
0.65415
0.66667
0.573070.5
0.5
0.5
0.50801
0.5

component

RodAngle
0.0145145
Radius0.1
0.325230.4
0.3976
0.39910.39910.39640.39790.39790.39580.4
0.4
0.39970.387090.4
0.3976
0.39970.4
0.39880.39670.4
0.39850.39550.39910.39459
0.39730.39820.31952
0.15
Bore
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
189

listingZ
0.22252
Field
20
5
5
-5
3.94394
3.79983.943944
3.96797
3.86386
3.97598
3.95195
43.88789
3.887893.89593.935943.91191
3.90393.89593.9359443.90393.97598
3.79179-3.61562
3.45546
4
4

of

cell

continued from previous page
D12
0.5
0.5
D13
0.5
0.5
D14
0.53153
0.5
D15
0.5
0.5
D16
0.5
0.5
D17
0.5
0.5
D18
0.5
0.5
D19
0.51001
0.5
D20
0.51602
0.5
D21
0.5
0.5
D22
0.5
0.5
D23
0.5
0.5
D24
0.56356
0.5
D25
0.506510.5
D26
0.776280.5
D270.65766
0.5
D28
0.5
0.5
Beamline order: TantalumRod, S1, D1, S2, D2, S3, D3, S4, D4, S5, D5, S6, D6, S7, D7,
S8, D8, S9, D9, S10, D10, S11, D11, S12, D12, S13, D13, S14, D14, S15, D15, S16, D16,
S17, D17, S18, D18, S19, D19, S20, D20, S21, D21, S22, D22, S23, D23, S24, D24, S25,
D25, S26, D26, S27, D27, S28, D28, Sfinal

190

Table 9.3:
Muon1
ChicaneLinacB.
RodSource
TantalumRod
Solenoid
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
S16
S17
S18
S19
S20
S21
S22
S23
S24
LS1
LS2
LS3
LS4
LS5
LS6
LS7
LS8
LS9
LS10
LS11
LS12
LS13
LS14
LS15
LS16
LS17

RodAngle
0
Length
0.39494
0.846750.710110.794190.87477
0.83904
0.83273
0.841840.83694
0.829930.83904
0.832030.820120.820120.835540.83904
0.819420.829930.830630.76056
0.826430.827830.835540.82292
0.82152
0.842540.86426
0.88318
0.82853
0.833430.860760.890190.83203
0.822220.826430.822220.819420.83904
0.841840.842540.86076

component

Z
0.2009Radius0.1
0.36847
0.375080.37988
0.36877
0.369070.378080.369070.39309
0.37658
0.39069
0.37147
0.36366
0.38018
0.39610.37327
0.37538
0.376280.369070.372070.36246
0.37658
0.37147
0.26306
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
191

listing

of

cell

File
‘pi plus 2G2eV.csv’Field
19.9199
4.81982
4.09914.07908
3.29533.27928
3.1992
3.111113.167173.17518
3.17518
3.15115
3.15115
3.15916
3.18318
3.25526
3.23924
3.391393.43143
3.33534
3.39943.439442.67868
2.18218
2.74274
3.143143.23123
3.27928
3.38338
3.27127
3.2953
3.24725
3.28729
3.391393.36737
3.35936
3.27127
3.135143.17518
3.27127
3.22322

continued from previous page
LS18
0.83273LS190.8019LS200.80681LS210.81662LS220.83764LS230.85025LS240.84324LS250.83904LS26
0.82853LS27
0.83343LS28
0.83343LS290.82503LS30
0.83133LS310.83834LS32
0.83834LS330.83203LS34
0.84955LS35
0.83694LS36
0.83343LS370.83343LS380.83554LS39
0.84184LS400.82923LS410.82853LS420.84114LS430.86917LS440.85866LS450.89089LS460.84464LS47
0.81592LS480.84254LS490.85095LS50
0.82993LS510.84464LS520.82993LS530.82082LS540.83273LS55
0.84535LS560.85235LS570.85936LS58
0.86637LS59
0.87127LS600.87057
Drift
Length
D1
0.5
D2
0.5

0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
Bore
0.5
0.5
192

3.207213.29533.31932
3.311313.215223.19923.35135
3.36737
3.367373.30333.28729
3.19119
3.28729
3.311313.45546
3.527533.30333.17518
3.19119
3.19119
3.191193.21522
3.24725
3.479483.27127
3.27127
3.24725
3.1992
3.223223.27928
3.19119
3.15916
3.06306
3.17518
3.21522
3.23924
3.1992
3.15115
3.32733
3.223223.135143.16717
3.24725

continued from previous page
D3
0.5
D4
0.5
D5
0.50501
D6
0.5
D7
0.5
D8
0.5
D9
0.5
D10
0.5
D11
0.5
D12
0.5
D13
0.5
D14
0.5
D15
0.503
D16
0.5025
D17
0.5
D18
0.5
D19
0.5
D20
0.5
D21
0.5
D22
0.5
D23
0.5
O2
0.2144O1
0.4713
EDGE
0
HBF
0.176
HBD
0.176
HalfHBD
0.088
∼HBF
0.176
∼HBD
0.176
∼HalfHBD
0.088
LD1a
0.0833333
LD1b
0.16667LD2a0.33333LD2b0.16667LD3a0.33333LD3b0.16667LD4a0.33333
LD4b
0.16667LD5a0.33333
LD5b
0.16667
LD6a0.33333
LD6b
0.16667LD7a0.33333
LD7b
0.16667LD8a0.33333
LD8b
0.16667

0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
1
1
1
1
1
1
1
1
1
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
193

continued from previous pageLD9a
0.33333
LD9b
0.16667
LD10a
0.33333
LD10b
0.16667
LD11a
0.33333
LD11b
0.16667
LD12a
0.33333
LD12b
0.16667
LD13a
0.33333
LD13b
0.16667
LD14a
0.33333
LD14b
0.16667
LD15a
0.33333
LD15b
0.16667
LD16a
0.33333
LD16b
0.16667
LD17a
0.33333
LD17b
0.16667
LD18a
0.33333
LD18b
0.16667
LD19a
0.33333
LD19b
0.16667
LD20a
0.33333
LD20b
0.16667
LD21a
0.33333
LD21b
0.16667
LD22a
0.33333
LD22b
0.16667
LD23a
0.33333
LD23b
0.16667
LD24a
0.33333
LD24b
0.16667
LD25a
0.33333
LD25b
0.16667
LD26a
0.33333
LD26b
0.16667
LD27a
0.33333
LD27b
0.16667
LD28a
0.33333
LD28b
0.16667
LD29a
0.33333
LD29b
0.16667LD30a
0.33333LD30b
0.16667
LD31a
0.33333
LD31b
0.16667

0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
194

continued from previous pageLD32a
0.33333LD32b
0.16667LD33a
0.33333LD33b
0.16667
LD34a
0.33333
LD34b
0.16667
LD35a
0.33333
LD35b
0.16667
LD36a
0.33333
LD36b
0.16667LD37a
0.33333
LD37b
0.16667
LD38a
0.33333
LD38b
0.16667
LD39a
0.33333
LD39b
0.16667
LD40a
0.33333
LD40b
0.16667
LD41a
0.33333
LD41b
0.16667
LD42a
0.33333
LD42b
0.16667
LD43a
0.33333
LD43b
0.16667
LD44a
0.33333
LD44b
0.16667
LD45a
0.33333
LD45b
0.16667
LD46a
0.33333
LD46b
0.16667LD47a
0.33333
LD47b
0.16667
LD48a
0.33333
LD48b
0.16667
LD49a
0.33333
LD49b
0.16667
LD50a
0.33333
LD50b
0.16667
LD51a
0.33333
LD51b
0.16667
LD52a
0.33333
LD52b
0.16667
LD53a
0.33333
LD53b
0.16667LD54a
0.33333
LD54b
0.16667

0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
195

continued from previous page
LD55a
0.33333
LD55b
0.16667LD56a
0.33333
LD56b
0.16667LD57a
0.33333
LD57b
0.16667LD58a
0.33333
LD58b
0.16667LD59a
0.33333LD59b
0.16667LD60a
0.33333
LD60b
0.16667
Rectangular-aperture x1
B1A
-0.6
B2A
-0.6
∼B1A
-0.6
∼B2A
-0.6
BMagnetFile1
L2_FIELD‘l2y0.txt’∼L2_FIELD‘l2y0.txt’
RFFrequency
RF1
88 MHzRF2
88 MHzRF3
88 MHzRF4
88 MHzRF5
88 MHzRF6
88 MHzRF7
88 MHzRF8
88 MHzRF9
88 MHzRF10
88 MHzRF11
88 MHzRF12
88 MHzRF13
88 MHzRF14
88 MHzRF15
88 MHzRF16
88 MHzRF17
88 MHzRF18
88 MHzRF19
88 MHzRF20
88 MHzRF21
88 MHzRF22
88 MHzRF23
88 MHzRF24
88 MHzRF25
88 MHz

0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
x2
0.6
0.4
0.6
0.4
File2
‘l2y5.txt’‘l2y5.txt’Radius
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
196

y1
-0.15
-0.15
-0.15
-0.15
File3
‘l2y10.txt’‘l2y10.txt’
Voltage
5.28079 MV
5.31104 MV
5.45706 MV
4.30324 MV
3.88851 MV
5.00805 MV
4.6857 MV
5.44943 MV
5.20546 MV
4.10299 MV
3.84518 MV
4.06064 MV
5.03466 MV
3.93535 MV
4.15313 MV
3.94565 MV
3.66139 MV
3.59131 MV
3.70632 MV
3.78307 MV
3.86418 MV
4.02229 MV
3.86558 MV
4.30806 MV
3.75306 MV

y2
0.15
0.15
0.15
0.15
File4
‘l2y13.txt’‘l2y13.txt’
Phase-64.6847◦-63.964◦-49.9099◦-42.7027◦-51.7117◦-44.1441◦-56.3964◦-69.7297◦-76.2162◦-70.4505◦-53.8739◦-54.2342◦-44.8649◦-39.0991◦-36.2162◦-33.3333◦-28.2883◦-22.5225◦-30.0901◦-35.8559◦-43.7838◦-52.7928◦-45.9459◦-50.6306◦-47.7477◦

continued from previous pageRF26
88 MHzRF27
88 MHzRF28
88 MHzRF29
88 MHzRF30
88 MHzRF31
88 MHzRF32
88 MHzRF33
88 MHzRF34
88 MHzRF35
88 MHzRF36
88 MHzRF37
88 MHzRF38
88 MHzRF39
88 MHzRF40
88 MHzRF41
88 MHzRF42
88 MHzRF43
88 MHzRF44
88 MHzRF45
88 MHzRF46
88 MHzRF47
88 MHzRF48
88 MHzRF49
88 MHzRF50
88 MHzRF51
88 MHzRF52
88 MHzRF53
88 MHzRF54
88 MHzRF55
88 MHzRF56
88 MHzRF57
88 MHzRF58
88 MHzRF59
88 MHzRF60
88 MHz

0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15

197

4.05873 MV
3.96693 MV
4.39406 MV
4.18596 MV
3.90204 MV
4.21195 MV
3.97616 MV
3.52008 MV
4.85243 MV
4.20218 MV
3.67328 MV
4.09507 MV
4.786 MV
1.81061 MV
0.65464 MV
3.67041 MV
4.30132 MV
4.72014 MV
5.32583 MV
5.5023 MV
5.29782 MV
4.56816 MV
4.35957 MV
3.84054 MV
3.78611 MV
3.89799 MV
3.82338 MV
3.66498 MV
4.22632 MV
4.09933 MV
4.25064 MV
4.10937 MV
4.16338 MV
3.94223 MV
3.79754 MV-45.5856◦-45.5856◦-44.8649◦-40.9009◦-43.7838◦-45.2252◦-45.5856◦-50.991◦-62.5225◦-59.6396◦-61.4414◦-54.955◦-50.2703◦-40.5405◦-39.0991◦-38.3784◦-36.9369◦-36.2162◦-32.973◦-37.2973◦-38.018◦-38.018◦-42.7027◦-26.8468◦-34.4144◦-31.5315◦-32.6126◦-29.009◦-23.2432◦-19.6396◦-37.2973◦-47.3874◦-71.8919◦-152.973◦-70.0901◦

continued from previous page
Beamline order: TantalumRod, S1, D1, S2, D2, S3, D3, S4, D4, S5, D5, S6, D6, S7, D7,
S8, D8, S9, D9, S10, D10, S11, D11, S12, D12, S13, D13, S14, D14, S15, D15, S16, D16,
S17, D17, S18, D18, S19, D19, S20, D20, S21, D21, S22, D22, S23, D23, S24, 2*(O2,O1, EDGE, B1A, HBF, HBF, HBF, HBF, B1A, EDGE, O1, B2A, HBD, HalfHBD, L2_FIELD,
HalfHBD, HBD, B2A, O1, EDGE, B1A, HBF, HBF, HBF, HBF, B1A, EDGE, O1, O2), 2*(O2, O1,
EDGE, ∼B1A, ∼HBF, ∼HBF, ∼HBF, ∼HBF, ∼B1A, EDGE, O1, ∼B2A, ∼HBD, ∼HalfHBD,∼L2_FIELD, ∼HalfHBD, ∼HBD, ∼B2A, O1, EDGE, ∼B1A, ∼HBF, ∼HBF, ∼HBF, ∼HBF,∼B1A, EDGE, O1, O2), LD1a, RF1, LD1b, LS1, LD2a, RF2, LD2b, LS2, LD3a, RF3, LD3b,
LS3, LD4a, RF4, LD4b, LS4, LD5a, RF5, LD5b, LS5, LD6a, RF6, LD6b, LS6, LD7a, RF7,
LD7b, LS7, LD8a, RF8, LD8b, LS8, LD9a, RF9, LD9b, LS9, LD10a, RF10, LD10b, LS10,
LD11a, RF11, LD11b, LS11, LD12a, RF12, LD12b, LS12, LD13a, RF13, LD13b, LS13,
LD14a, RF14, LD14b, LS14, LD15a, RF15, LD15b, LS15, LD16a, RF16, LD16b, LS16,
LD17a, RF17, LD17b, LS17, LD18a, RF18, LD18b, LS18, LD19a, RF19, LD19b, LS19,
LD20a, RF20, LD20b, LS20, LD21a, RF21, LD21b, LS21, LD22a, RF22, LD22b, LS22,
LD23a, RF23, LD23b, LS23, LD24a, RF24, LD24b, LS24, LD25a, RF25, LD25b, LS25,
LD26a, RF26, LD26b, LS26, LD27a, RF27, LD27b, LS27, LD28a, RF28, LD28b, LS28,
LD29a, RF29, LD29b, LS29, LD30a, RF30, LD30b, LS30, LD31a, RF31, LD31b, LS31,
LD32a, RF32, LD32b, LS32, LD33a, RF33, LD33b, LS33, LD34a, RF34, LD34b, LS34,
LD35a, RF35, LD35b, LS35, LD36a, RF36, LD36b, LS36, LD37a, RF37, LD37b, LS37,
LD38a, RF38, LD38b, LS38, LD39a, RF39, LD39b, LS39, LD40a, RF40, LD40b, LS40,
LD41a, RF41, LD41b, LS41, LD42a, RF42, LD42b, LS42, LD43a, RF43, LD43b, LS43,
LD44a, RF44, LD44b, LS44, LD45a, RF45, LD45b, LS45, LD46a, RF46, LD46b, LS46,
LD47a, RF47, LD47b, LS47, LD48a, RF48, LD48b, LS48, LD49a, RF49, LD49b, LS49,
LD50a, RF50, LD50b, LS50, LD51a, RF51, LD51b, LS51, LD52a, RF52, LD52b, LS52,
LD53a, RF53, LD53b, LS53, LD54a, RF54, LD54b, LS54, LD55a, RF55, LD55b, LS55,
LD56a, RF56, LD56b, LS56, LD57a, RF57, LD57b, LS57, LD58a, RF58, LD58b, LS58,
LD59a, RF59, LD59b, LS59, LD60a, RF60, LD60b, LS60

198

Table 9.4: Muon1 component listing of cell PhaseRotB.
RodSource
TantalumRod
Solenoid
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
S16
S17
S18
S19
S20
S21
S22
S23
PS1
PS2
PS3
PS4
PS5
PS6
PS7
PS8
PS9
PS10
PS11
PS12
PS13
PS14
PS15
PS16
PS17
PS18

RodAngle
0
Length
0.435240.81592
0.741640.720620.844640.85375
0.87898
0.88669
0.89299
0.89299
0.89229
0.89229
0.89299
0.89159
0.88949
0.89790.885990.9
0.88458
0.894390.891590.890890.891590.890190.890890.890890.89369
0.89229
0.892290.89510.89510.894390.894390.89650.894390.89580.8986
0.89369
0.894390.89299
0.8965Z
0.26802
Radius0.1
0.37688
0.38288
0.39550.39459
0.39520.397
0.39670.39520.39550.3952
0.394290.39429
0.39399
0.39520.39520.3952
0.394290.39489
0.39520.39580.39730.39219
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15

File
‘pi plus 2G2eV.csv’Field
19.8599
4.88989
4.75976
4.90991
3.93594
3.90393.83984
3.807813.91191
3.93594
3.91992
3.91191
3.90393.91992
3.91992
3.94394
3.91992
3.91191
3.93594
3.82382
3.75175
3.57558
3.61562-2.96697-3.02302-3.04705-3.07107-3.07107-3.07908-3.06306-3.08709-3.07107-3.03904-3.04705-3.07107-3.07107-3.05506-3.07908-3.0951-3.07107-3.06306

199

continued from previous page
PS19
0.893690.15
PS20
0.891590.15
PS21
0.892290.15
PS220.89510.15
PS23
0.893690.15
PS24
0.892990.15
PS25
0.892290.15
PS26
0.892290.15
PS270.89580.15
PS280.894390.15
PS290.889490.15
PS30
0.89229
0.15
Drift
LengthBore
D1
0.50050.5
D2
0.812810.5
D3
0.513010.5
D4
0.50801
0.5
D5
0.5
0.5
D6
0.5
0.5
D7
0.5
0.5
D8
0.501
0.5
D9
0.504
0.5
D10
0.5005
0.5
D11
0.5
0.5
D12
0.5
0.5
D13
0.5
0.5
D14
0.5
0.5
D15
0.5
0.5
D16
0.5
0.5
D17
0.5
0.5
D18
0.5
0.5
D19
0.5
0.5
D20
0.5
0.5
D21
0.5
0.5
D22
0.5
0.5
PD1a
0.0833333
0.15
PD1b
0.166670.15
PD2a0.333330.15
PD2b0.166670.15
PD3a0.333330.15
PD3b0.166670.15
PD4a0.333330.15
PD4b0.166670.15
PD5a0.333330.15
PD5b
0.166670.15
PD6a0.33333
0.15-3.0951-3.08709-3.08709-3.07908-3.08709-3.11111-3.13514-3.15916-3.16717-3.11111-2.94294-3.12713

200

continued from previous pagePD6b0.166670.15
PD7a0.333330.15
PD7b0.166670.15
PD8a0.333330.15
PD8b
0.166670.15
PD9a0.333330.15
PD9b
0.166670.15
PD10a0.333330.15
PD10b
0.166670.15
PD11a
0.333330.15
PD11b
0.166670.15
PD12a0.333330.15
PD12b
0.166670.15
PD13a
0.333670.15
PD13b
0.166830.15
PD14a0.333330.15
PD14b
0.166670.15
PD15a
0.333330.15
PD15b
0.166670.15
PD16a
0.333330.15
PD16b
0.166670.15
PD17a
0.333330.15
PD17b
0.166670.15
PD18a
0.333330.15
PD18b
0.166670.15
PD19a
0.333330.15
PD19b
0.166670.15
PD20a
0.333330.15
PD20b
0.166670.15
PD21a
0.333330.15
PD21b
0.166670.15
PD22a
0.333330.15
PD22b
0.166670.15
PD23a
0.333330.15
PD23b
0.166670.15
PD24a
0.333330.15
PD24b
0.166670.15
PD25a
0.333330.15
PD25b
0.166670.15
PD26a
0.333330.15
PD26b
0.166670.15
PD27a
0.333330.15
PD27b
0.166670.15
PD28a
0.333330.15
PD28b
0.166670.15
PD29a
0.333330.15
201continued from previous pagePD29b
0.166670.15
PD30a
0.333330.15
PD30b
0.16667
0.15
RFFrequency
RadiusPRF1
31.4285 MHz 0.15
PRF2
31.4285 MHz 0.15
PRF3
31.4285 MHz 0.15
PRF4
31.4285 MHz 0.15
PRF5
31.4285 MHz 0.15
PRF6
31.4285 MHz 0.15
PRF7
31.4285 MHz 0.15
PRF8
31.4285 MHz 0.15
PRF9
31.4285 MHz 0.15
PRF10
31.4285 MHz 0.15
PRF11
31.4285 MHz 0.15
PRF12
31.4285 MHz 0.15
PRF13
31.4285 MHz 0.15
PRF14
31.4285 MHz 0.15
PRF15
31.4285 MHz 0.15
PRF16
31.4285 MHz 0.15
PRF17
31.4285 MHz 0.15
PRF18
31.4285 MHz 0.15
PRF19
31.4285 MHz 0.15
PRF20
31.4285 MHz 0.15
PRF21
31.4285 MHz 0.15
PRF22
31.4285 MHz 0.15
PRF23
31.4285 MHz 0.15
PRF24
31.4285 MHz 0.15
PRF25
31.4285 MHz 0.15
PRF26
31.4285 MHz 0.15
PRF27
31.4285 MHz 0.15
PRF28
31.4285 MHz 0.15
PRF29
31.4285 MHz 0.15
PRF30
31.4285 MHz 0.15

Voltage
2.2074 MV
2.2197 MV
2.2197 MV
2.21744 MV
2.21297 MV
2.21297 MV
2.2309 MV
2.22865 MV
2.21183 MV
2.21407 MV
2.21966 MV
2.21407 MV
2.22832 MV
2.2275 MV
2.21296 MV
2.21407 MV
2.21409 MV
2.2219 MV
2.21969 MV
2.21186 MV
2.21969 MV
2.21743 MV
2.22193 MV
2.23874 MV
2.21521 MV
2.21745 MV
2.22528 MV
2.2208 MV
2.20852 MV
2.22417 MV

202

Phase
-114.414◦-113.333◦-107.928◦-113.694◦-112.973◦-112.252◦-113.694◦-114.414◦-114.054◦-113.694◦-113.333◦-113.694◦-112.973◦-113.694◦-113.694◦-112.252◦-114.054◦-111.892◦-111.892◦-112.973◦-114.414◦-113.333◦-83.0631◦-104.324◦-107.568◦-112.973◦-114.054◦-112.252◦-112.973◦-112.613◦

continued from previous page
Beamline order: TantalumRod, S1, D1, S2, D2, S3, D3, S4, D4, S5, D5, S6, D6, S7, D7,
S8, D8, S9, D9, S10, D10, S11, D11, S12, D12, S13, D13, S14, D14, S15, D15, S16, D16,
S17, D17, S18, D18, S19, D19, S20, D20, S21, D21, S22, D22, S23, PD1a, PRF1, PD1b,
PS1, PD2a, PRF2, PD2b, PS2, PD3a, PRF3, PD3b, PS3, PD4a, PRF4, PD4b, PS4, PD5a,
PRF5, PD5b, PS5, PD6a, PRF6, PD6b, PS6, PD7a, PRF7, PD7b, PS7, PD8a, PRF8, PD8b,
PS8, PD9a, PRF9, PD9b, PS9, PD10a, PRF10, PD10b, PS10, PD11a, PRF11, PD11b, PS11,
PD12a, PRF12, PD12b, PS12, PD13a, PRF13, PD13b, PS13, PD14a, PRF14, PD14b, PS14,
PD15a, PRF15, PD15b, PS15, PD16a, PRF16, PD16b, PS16, PD17a, PRF17, PD17b, PS17,
PD18a, PRF18, PD18b, PS18, PD19a, PRF19, PD19b, PS19, PD20a, PRF20, PD20b, PS20,
PD21a, PRF21, PD21b, PS21, PD22a, PRF22, PD22b, PS22, PD23a, PRF23, PD23b, PS23,
PD24a, PRF24, PD24b, PS24, PD25a, PRF25, PD25b, PS25, PD26a, PRF26, PD26b, PS26,
PD27a, PRF27, PD27b, PS27, PD28a, PRF28, PD28b, PS28, PD29a, PRF29, PD29b, PS29,
PD30a, PRF30, PD30b, PS30

203

Table 9.5: Muon1 component listing of cell PhaseRotC.
RodSource
TantalumRod
Solenoid
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
S16
S17
S18
S19
S20
S21
S22
S23
PS1
PS2
PS3
PS4
PS5
PS6
PS7
PS8
PS9
PS10
PS11
PS12
PS13
PS14
PS15
PS16
PS17
PS18

File
‘pi plus 10GeV.csv’
Length
0.44274
0.81592
0.753550.72062
0.808210.85025
0.876180.887390.88669
0.890190.88949
0.88949
0.890190.890190.883180.87968
0.88108
0.882480.9
0.88949
0.88669
0.85866
0.887390.87758
0.878280.878280.87828
0.89299
0.88178
0.883180.882480.882480.88038
0.894390.89510.89299
0.887390.885290.887390.88669
0.88529

RodAngle
0
Radius0.1
0.37898
0.4
0.389190.39099
0.39580.394590.39489
0.39309
0.3952
0.39279
0.39249
0.39189
0.39129
0.39520.39520.39159
0.39159
0.39159
0.39219
0.3964
0.39670.38859
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15

204

Z
0.26396
Field
19.9199
4.839844.86987
4.88989
3.87988-3.81582
3.85586
3.847853.91191
3.88789
3.879883.87988
3.871873.87988
3.855863.86386
3.84785
3.89593.81582
3.77578
3.66366
3.591593.5996-3.05506-3.04705-3.06306-3.1031-3.08709-3.1031-3.0951-3.06306-3.07107-3.08709-3.07908-3.08709-3.06306-3.07107-3.0951-3.11111-3.08709-3.08709

continued from previous pagePS19
0.88318PS20
0.88038
PS21
0.88108PS22
0.88178
PS23
0.88388PS24
0.88178
PS25
0.87968
PS26
0.87968
PS27
0.89299
PS28
0.89229PS29
0.87828
PS30
0.87828
Drift
LengthD10.5
D20.80881D3
0.51602D4
0.50601
D5
0.503
D6
0.5
D7
0.501
D8
0.5
D9
0.5025
D10
0.5
D11
0.5
D12
0.5
D13
0.5
D14
0.5
D15
0.5
D16
0.5
D17
0.5
D18
0.501
D19
0.5
D20
0.504
D21
0.50651
D22
0.50501PD1a0.0833333
PD1b
0.16667PD2a0.33333PD2b0.16667
PD3a0.33333PD3b0.16667
PD4a0.33333
PD4b
0.16667
PD5a0.33333
PD5b
0.16667
PD6a0.33333

0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
Bore
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
205

-3.07908-3.08709-3.0951-3.07107-3.11111-3.15115-3.16717-3.21522-3.20721-3.14314-3.04705-2.96697

continued from previous pagePD6b0.16667PD7a0.33333
PD7b0.16667PD8a0.33333
PD8b
0.16667PD9a0.33333
PD9b
0.16667
PD10a0.33333
PD10b
0.16667
PD11a
0.33333
PD11b
0.16667
PD12a0.33333
PD12b
0.16667
PD13a0.33333
PD13b
0.16667
PD14a0.33333
PD14b
0.16667
PD15a0.33333
PD15b
0.16667
PD16a
0.33333
PD16b
0.16667
PD17a
0.33333
PD17b
0.16667
PD18a
0.33333
PD18b
0.16667
PD19a
0.33333
PD19b
0.16667
PD20a
0.33333
PD20b
0.16667
PD21a
0.33333
PD21b
0.16667
PD22a
0.33333
PD22b
0.16667
PD23a
0.33333
PD23b
0.16667
PD24a
0.33333
PD24b
0.16667
PD25a
0.33333
PD25b
0.16667
PD26a
0.33333
PD26b
0.16667
PD27a
0.33333
PD27b
0.16667
PD28a
0.33333PD28b
0.16667
PD29a
0.33333

0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
206

continued from previous pagePD29b
0.16667PD30a
0.33333PD30b
0.16667
RF
Frequency
PRF1
31.4285 MHz
PRF2
31.4285 MHz
PRF331.4285 MHz
PRF431.4285 MHz
PRF531.4285 MHz
PRF631.4285 MHz
PRF731.4285 MHz
PRF831.4285 MHz
PRF931.4285 MHz
PRF10
31.4285 MHz
PRF11
31.4285 MHz
PRF1231.4285 MHz
PRF13
31.4285 MHz
PRF14
31.4285 MHz
PRF15
31.4285 MHz
PRF16
31.4285 MHz
PRF17
31.4285 MHz
PRF18
31.4285 MHz
PRF19
31.4285 MHz
PRF20
31.4285 MHz
PRF21
31.4285 MHz
PRF22
31.4285 MHz
PRF23
31.4285 MHz
PRF24
31.4285 MHz
PRF25
31.4285 MHz
PRF26
31.4285 MHz
PRF27
31.4285 MHz
PRF28
31.4285 MHz
PRF29
31.4285 MHz
PRF30
31.4285 MHz

0.15
0.15
0.15
Radius
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15

207

VoltagePhase
2.16521 MV -115.495◦
2.16631 MV -116.216◦
2.17518 MV -114.414◦
2.19513 MV -115.495◦
2.17151 MV -115.495◦
2.19627 MV -154.775◦
2.16957 MV -115.495◦
2.17069 MV -115.856◦
2.19738 MV -115.495◦
2.15629 MV -116.577◦
2.1894 MV -115.135◦
2.22192 MV -112.973◦
2.18271 MV -114.054◦
2.17393 MV -113.694◦
2.16172 MV -113.694◦
2.15831 MV -113.694◦
2.1996 MV -113.333◦
2.20184 MV -113.333◦
2.16067 MV -111.171◦
2.15629 MV -113.333◦
2.20182 MV -81.2613◦
2.15848 MV -113.333◦
2.1907 MV -113.694◦
2.16737 MV -112.613◦
2.15298 MV -113.333◦
2.16407 MV -113.333◦
2.21857 MV -111.892◦
2.21521 MV -112.252◦
2.16409 MV -111.532◦
2.21509 MV -111.892◦

continued from previous page
Beamline order: TantalumRod, S1, D1, S2, D2, S3, D3, S4, D4, S5, D5, S6, D6, S7, D7,
S8, D8, S9, D9, S10, D10, S11, D11, S12, D12, S13, D13, S14, D14, S15, D15, S16, D16,
S17, D17, S18, D18, S19, D19, S20, D20, S21, D21, S22, D22, S23, PD1a, PRF1, PD1b,
PS1, PD2a, PRF2, PD2b, PS2, PD3a, PRF3, PD3b, PS3, PD4a, PRF4, PD4b, PS4, PD5a,
PRF5, PD5b, PS5, PD6a, PRF6, PD6b, PS6, PD7a, PRF7, PD7b, PS7, PD8a, PRF8, PD8b,
PS8, PD9a, PRF9, PD9b, PS9, PD10a, PRF10, PD10b, PS10, PD11a, PRF11, PD11b, PS11,
PD12a, PRF12, PD12b, PS12, PD13a, PRF13, PD13b, PS13, PD14a, PRF14, PD14b, PS14,
PD15a, PRF15, PD15b, PS15, PD16a, PRF16, PD16b, PS16, PD17a, PRF17, PD17b, PS17,
PD18a, PRF18, PD18b, PS18, PD19a, PRF19, PD19b, PS19, PD20a, PRF20, PD20b, PS20,
PD21a, PRF21, PD21b, PS21, PD22a, PRF22, PD22b, PS22, PD23a, PRF23, PD23b, PS23,
PD24a, PRF24, PD24b, PS24, PD25a, PRF25, PD25b, PS25, PD26a, PRF26, PD26b, PS26,
PD27a, PRF27, PD27b, PS27, PD28a, PRF28, PD28b, PS28, PD29a, PRF29, PD29b, PS29,
PD30a, PRF30, PD30b, PS30

208

Table 9.6:
Muon1
PhaseRotEb6a.
RodSource
TantalumRod
Solenoid
S1
S2
S3
S4
S5
S6
S7
S8
S9
SB1
SB2
SB3
SB4
SB5
SB6
PS1
PS2
PS3
PS4
PS5
PS6
Drift
D1
D2
D3
D4
D5
D6
D7
D8
D9
DB1
DB2
DB3
DB4
DB5
DB6
PD1a
PD1b
PD2a
PD2b

File
‘pi plus 10GeV.csv’
Length
0.42698
0.674370.31211
0.46206
0.2007
0.66386
0.464160.75846
0.52372
0.78298
0.78999
0.787890.77598
0.778080.794190.62883
0.629530.64074
0.63584
0.63724
0.63724
Length0.5
0.670170.50.50050.522520.5
0.5
0.52653
0.593090.5
0.5
0.50501
0.501
0.5
0.5
0.33333
0.166670.33367
0.16683

component
RodAngle
0.023023
Radius0.1
0.29940.332130.282880.276880.317420.26787
0.261560.25586
0.208110.207510.207210.205410.211410.2045
0.15
0.15
0.15
0.15
0.15
0.15
Bore
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.15
0.15
0.15
0.15
209

listingZ
0.22162Field
20
4.57958
3.37838
3.46847
1.26126-2.55856-2.7027
2.582584-3.44745-3.5035-3.52753-3.55155-3.55956-3.52753
3.89593.98398
3.959964
43.99199ofcellcontinued from previous pagePD3a0.333330.15
PD3b0.166670.15
PD4a0.335670.15
PD4b0.167830.15
PD5a0.333330.15
PD5b
0.166670.15
PD6a0.33333
0.15
PD6b
0.16667
0.15
RFFrequency
RadiusVoltagePhase
PRF1
31.4285 MHz
0.15
1.68525 MV -143.604◦
PRF2
31.4285 MHz
0.15
1.79067 MV -141.802◦
PRF3
31.4285 MHz
0.15
1.72139 MV -130.631◦
PRF4
31.4285 MHz
0.15
1.72294 MV -111.171◦
PRF5
31.4285 MHz
0.15
1.75452 MV -91.7117◦
PRF6
31.4285 MHz
0.15
1.73074 MV -81.6216◦
Beamline order: TantalumRod, S1, D1, S2, D2, S3, D3, S4, D4, S5, D5, S6, D6, S7,
D7, S8, D8, S9, D9, 3*(SB1, DB1), 3*(SB2, DB2), 2*(SB3, DB3), 3*(SB4, DB4), 3*(SB5,
DB5), SB6, DB6, SB6, 7*(PD1a, PRF1, PD1b, PS1), 7*(PD2a, PRF2, PD2b, PS2), 7*(PD3a,
PRF3, PD3b, PS3), 7*(PD4a, PRF4, PD4b, PS4), 7*(PD5a, PRF5, PD5b, PS5), 6*(PD6a,
PRF6, PD6b, PS6)

210

Table 9.7:
Muon1
PhaseRotLinac1.

component

listingofcell

RodSource
File
RodAngle Z
TantalumRod ‘pi plus 10GeV.csv’ 0.02002
0.12928
Solenoid
LengthRadiusField
S1
0.437490.1
20
S2
0.729030.23904
4.92993S30.39059
0.27297-5
S4
0.7956
0.34414-1.48649
S50.72482
0.39973.55155
S6
0.85375
0.37357
1.66967
S7
0.482380.26426-3.65566
S8
0.88108
0.3997-2.75876S90.70170.25255-4SB10.864960.1979-4
PS1
0.8993
0.15
4
LS1
0.9
0.15
-4
Drift
Length
Bore
D1
0.5
0.5
D2
0.55455
0.5
D3
0.7973
0.5
D4
0.5
0.5
D5
0.891390.5
D6
0.9985
0.5
D7
0.5
0.5
D8
0.910410.5
D90.83884
0.5
DB1
0.5
0.5
PD1a
0.333330.15
PD1b0.166670.15
LD1a0.333330.15
LD1b0.16667
0.15
RFFrequency
RadiusVoltagePhase
PRF1
31.4285 MHz
0.15
2.24887 MV 41.6216◦
RF1
88 MHz0.15
0 MV
156.577◦
Beamline order: TantalumRod, S1, D1, S2, D2, S3, D3, S4, D4, S5, D5, S6, D6, S7,
D7, S8, D8, S9, D9, 29*(SB1, DB1), SB1, 60*(PD1a, PRF1, PD1b, PS1), 60*(LD1a, RF1,
LD1b, LS1)

211

Table 9.8:
Muon1 component
Linac900Removable6c2.

listingofcell

RodSource
File
RodAngle Z
TantalumRod ‘pi plus 10GeV.csv’ 0.00500501 0.14144
Solenoid
LengthRadius
Field
S1
0.45
0.1
20
S2
0.8986
0.192195S30.30440.38979
4.83984
S4
0.82643
0.39885LS10.8993
0.15
-4
LS2
0.9
0.15
-4
LS3
0.8993
0.15
-4
LS4
0.9
0.15
3.93594
LS5
0.9
0.15
-3.95195
LS6
0.89089
0.15
3.91992
Drift
Length
Bore
D1
0.5
0.5
D2
0.5
0.5
D3
0.5
0.5
D4
0.50150.5
LD1a0.333330.15
LD1b0.166670.15
LD2a0.355690.15
LD2b0.177840.15
LD3a0.333330.15
LD3b0.166670.15
LD4a0.333330.15
LD4b
0.166670.15
LD5a
0.335670.15
LD5b0.167830.15
LD6a0.333330.15
LD6b
0.16667
0.15
RFFrequency
RadiusVoltagePhase
RF1
88 MHz0.15
5.59159 MV-11.5315◦
RF2
88 MHz0.15
5.72839 MV
0◦RF3
88 MHz0.15
5.59159 MV-5.76577◦
RF488 MHz0.15
5.57197 MV
10.0901◦RF5
88 MHz0.15
0.0337178 MV 0◦
RF6
88 MHz0.15
5.25726 MV-139.099◦
Beamline order: TantalumRod, S1, D1, S2, D2, S3, D3, S4, D4, 40*(LD1a, RF1, LD1b,
LS1), 40*(LD2a, RF2, LD2b, LS2), 40*(LD3a, RF3, LD3b, LS3), 40*(LD4a, RF4, LD4b,LS4), 40*(LD5a, RF5, LD5b, LS5), 40*(LD6a, RF6, LD6b, LS6)212