# Full Scale Proton Beam Impact Testing of new CERN Collimators and Validation
of a Numerical Approach for Future Operation

M. Bergeret,  F-X. Nuiry  M. Calviani  M.A. Fraser  M. Butcher  L-M. Grec  L.
Gentini  A. Lechner  M. Frankl  V. Rizzoglio  S. Burger  A. Cherif

###### Abstract

New collimators are being produced at CERN in the framework of a large
particle accelerator upgrade project to protect beam lines against stray
particles. Their movable jaws hold low density absorbers with tight geometric
requirements, while being able to withstand direct proton beam impacts. Such
events induce considerable thermo-mechanical loads, leading to complex
structural responses, which make the numerical analysis challenging. Hence, an
experiment has been developed to validate the jaw design under representative
conditions and to acquire online results to enhance the numerical models. Two
jaws have been impacted by high-intensity proton beams in a dedicated facility
at CERN and have recreated the worst possible scenario in future operation.
The analysis of online results coupled to post-irradiation examinations have
demonstrated that the jaw response remains in the elastic domain. However,
they have also highlighted how sensitive the jaw geometry is to its mounting
support inside the collimator. Proton beam impacts, as well as handling
activities, may alter the jaw flatness tolerance value by $\pm$ 70 µm, whereas
the flatness tolerance requirement is 200 µm. In spite of having validated the
jaw design for this application, the study points out numerical limitations
caused by the difficulties in describing complex geometries and boundary
conditions with such unprecedented requirements.

##  1 Introduction

CERN is the European Organization for Nuclear Research located in Geneva on
the border between France and Switzerland. Many activities at CERN currently
involve upgrading the world’s largest particle accelerator complex to increase
the luminosity of the Large Hadron Collider (hereinafter “LHC”). In that
framework, a large improvement of the LHC injector chain is being conducted
under the LHC Injector Upgrade Project (hereinafter “LIU Project”) [1]. The
proton beams for the LHC are fast-extracted from the last injector, the Super
Proton Synchrotron (hereinafter “SPS”), at 450 GeV/c through two transfer
lines: TI 2 and TI 8. A machine protection system had been developed at the
time of the LHC’s construction to protect the beam line equipment against
possible proton beam steering errors. By means of collimators, it was intended
to attenuate the damage potential of the proton beams, which is related to the
intensity and emittance, below a safe level corresponding to the damage limit
of accelerator components. The LIU requirements have involved the increase of
the attenuation capacity of the so-called TCDI collimators (“Target Collimator
Dump Injection”) by a factor of 3.5, and has consequently required the
construction of a new hardware: the TCDIL collimators (“Target Collimator Dump
Injection Long”).

###  1.1 TCDIL collimators

The TCDI collimators employed isostatic graphite absorber blocks (density =
1.8 g.cm-3) over a length of 1.2 m to intercept off-trajectory proton beams.
The TCDIL collimators require the installation of low density absorber blocks
(density > 1.7 g.cm-3) over a length of 2.1 m, for which material selection
has been driven by a beam impact testing experiment [2].

As shown in Figure 1, each collimator contains a pair of sub-mechanical
assemblies (hereinafter “jaws”), which hold the 2.1 m-long series of absorbing
blocks. A flatness tolerance of 200 µm is requested over the common inner
absorber plane for precise positioning around the proton beams [3]. To do so,
the jaws employ two patterns: A 2.1 m-long grade 316L stainless steel
stiffener (hereinafter “backstiffener”) that governs the overall jaw geometry,
and a grade 304L stainless steel clamping system with compression springs,
clamps and compression plates, that holds the absorbing blocks on the
backstiffener. The jaws are then installed in an ultra-high vacuum tank on two
cylindrical shafts going through a set of fixed and oblong GLIDCOP® AL-15
guiding plates, which are linked to an actuation system with four independent
translational degrees-of-freedom. The jaws can therefore translate in the
transverse direction with adjustment of their angular orientation with respect
to the beam trajectory. A static vacuum pressure of $5.0\times 10^{-9}$ mbar
is required with a maximum total ougassing flux of $2.0\times 10^{-7}$
mbar.L.s-1. Six collimators will be installed at specific positions in each of
the two transfer lines TI 2 and TI 8.

Figure 1:  Exploded view of a horizontal TCDIL collimator showing the jaws
linked to the actuation system by means of cylindrical shafts. Detailed view
of the jaw by outlining the assembly parts and the 200 µm flatness requirement
over the common active absorber surfaces.

###  1.2 Jaw design requirements

The jaw design is challenging as it must withstand direct impacts of LIU
proton beams without jeopardizing the 200 µm flatness tolerance requirement.
Table 1 states the characteristics of the standard LIU proton beam. An
extensive analysis [3] studied the different accidental scenarios causing beam
mis-steering, and has concluded that the jaw absorbers would intercept proton
beams between 0 mm and 6.9 mm from the inner absorber plane. This distance is
named the impact parameter, as seen in Figure 1.

In case of a direct proton beam impact, the particles interact with the
absorber nuclei to locally generate extreme heat. In addition, the primary
nuclear interactions spread a shower of secondary particles that deposits
additional energy in the different stainless steel parts of the assembly. Such
a fleeting event, in the order of microseconds, induces a thermal burst that
turns into a complex structural response with the excitation of many dynamic
modes. Numerical models have been defined in ANSYS® to study the jaw
structural response, especially in the worst case scenario, and its design
studied accordingly. However, despite the increasing accuracy of numerical
approaches, uncertainties related to the model assumptions remained. An
experimental validation was therefore required.

Table 1:  Proton beam requirements for standard LIU and maximum proton beam capabilities at the HiRadMat facility. The intensity of a standard LIU beam is 76 % higher. Beam Parameters | Standard LIU | HiRadMat
---|---|---
Proton energy [GeV] | 450 | 440
Number of bunch | 288 | 288
Bunch intensity | $2.3\times 10^{11}$ | $1.3\times 10^{11}$
Pulse length [µs] | 7.8 | 7.2
Horizontal beam size at 1$\sigma$ [mm] | 0.405 | 0.313
Vertical beam size at 1$\sigma$ [mm] | 0.647 | 0.313

###  1.3 Proton beam impact testing

CERN’s HiRadMat facility [4] uses a dedicated experimental area, which can be
irradiated by SPS proton beams at a momentum of of 440 GeV/c. The facility is
especially convenient thanks to a wide flexibility of beam intensities and
beam optic parameters. It is commonly used for research and development of
beam intercepting materials and validation of equipment [2, 5, 6, 7]. Table 1
shows the maximum HiRadMat irradiation capabilities.

A HiRadMat beam impact testing experiment was hence developed to assess the
jaw design subjected to proton beam impacts as required by the LIU project.
The so-called HiRadMat 44 experiment (hereinafter “HRMT 44”), had to both (i)
confirm that the jaw geometry is not severely affected in the worst case
scenario, and (ii) provide experimental online results to enhance and validate
the numerical models. The experiment is of considerable importance to offer
reliable collimator equipment and guide future maintenance strategies.
Finally, HRMT 44 was conceived also as an opportunity to (iii) validate the
integrity of a new absorbing material as performed in [2]. The Novoltex®
Sepcarb® 054-62 is a 3D carbon-carbon composite manufactured by ArianeGroup
and has thermo-mechanical properties similar to the already validated Naxeco®
Sepcarb® 358-02 [8].

This study will present the numerical approach that has been used to analyze
the structural response of impacted jaws. The model assumptions will
especially be scrutinised. Then, the experimental set-up will be developed
with a focus on the instrumentation and its role. Afterwards, the experimental
procedure and the online results will be shown for comparison with the
numerical results. Finally, the post-irradiation examinations will assess the
jaw integrity after being subjected to the proton beam impacts.

##  2 Numerical analysis of proton beam impacts in the collimator jaw

The thermo-mechanical description of the interaction between a proton beam and
a collimator jaw relies on several physical problems and engineering
methodologies. For that purpose, finite element methods, and related software
platforms such as ANSYS® Workbench, offer an increasing opportunity to define
and solve 3D models with complex geometries, loads and responses. This section
examines the jaw structural responses, especially in the worst case scenario,
after presenting the required simulation workflow.

###  2.1 Simulation workflow

As shown in Figure 2, the simulation workflow that leads to the structural
problem integrates two successive and uncoupled steps. The physical problem,
or simulating the nuclear interactions and transports of particles in media,
is a prerequisite solved by the FLUKA [9] Monte Carlo code. It provides the
heat density generated in the collimator jaw according to pre-defined beam
characteristics. The thermal problem can be solved thereafter by an ANSYS®
transient thermal model, which calculates the temperature distribution at the
end of the pulse and its evolution related to conduction, convection and
radiation processes. The spatial and time dependent temperature distributions
can finally be imported in the ANSYS® transient structural model.

The severity of the beam impacts depends on the beam characteristics
summarized in Figure 2. The intensity, energy and pulse length are defined by
the LIU beam requirements. It has been demonstrated that the jaw structural
response is not sensitive (within a factor of two) to the beam spot sizes,
defined as the standard deviations ($\sigma$) of the beam protons. Smaller
beam sizes only induce more focused loads in the absorbers due to the primary
particle interactions. Hence, the principal variable input of the simulation
workflow is the beam impact parameter. “Deep impacts” lead to higher total
energy deposition in the stainless steel backstiffener, and consequently, the
deepest beam impact parameter (6.9 mm, see section 1.2) embodies the highest
risk of affecting the jaw integrity.

Figure 2:  Workflow diagram of the numerical models defined in FLUKA and
ANSYS® for studying the jaw thermo-mechanical response in case of a proton
beam impact. The principal variable input is the beam impact parameters (in
green). The ANSYS® Transient Structural model is based on two main assumptions
(in purple) that must be confirmed by the experimental results (in red).

###  2.2 Structural response

The duration of a beam induced heat load is of the order of a few microseconds
as it corresponds to the pulse length, as seen in Table 1. As shown in Figure
3, the backstiffener’s peak temperature shortly ramps in 7.8 µs to a quasi-
adiabatic thermal state, just like the absorbers and the other stainless steel
parts. Such a fleeting event partly prevents thermal expansion because of an
inertia effect, and gives rise to a dynamic response that can be thought of as
various mode shapes of the jaw being excited with different characteristic
times. The model is consequently set up across three phases, delimited by
dotted lines in Figure 3.

Figure 3:  Full scale 3D geometry of the jaw defined in the ANSYS® transient
structural model. Two frictional contacts model the connection between the jaw
guiding plates and the cylindrical shafts with a friction coefficient of µ =
0.1. Temperature dependent linear elastic material properties are implemented.
Finally, the simulation set up can describe three phases with distinct
structural modes, delimited by dotted lines in a characteristic peak
temperature evolution in the backstiffener after the “deepest standard LIU
proton beam impact”. Figure 4:  Preliminary 2D-plane-strain model in ANSYS®
transient structural of a jaw cross section with the higher backstiffener
temperature profile. In case of the deepest standard LIU proton beam impact,
the peak temperature in the backstiffener is of 108 °C. The right plot shows
how the short duration thermal load creates larges compressive-to-tensile
x-normal stresses.

In the first phase, the purely thermal stresses, or quasi-static stresses
created by a non-uniform temperature field, are superposed with high-frequency
dynamic modes (with characteristic times of a few microseconds) with elastic
and potentially elastic-plastic total stresses in the backstiffener. A
preliminary 2D-plane-strain analysis, presented in Figure 4 in the case of the
deepest standard LIU beam impact, had been defined at a characteristic jaw
cross-section with the higher temperature profile in the backstiffener. The
plane strain assumption was hitherto justified by the jaw longitudinal
dimension and by the duration of the heat load that tends to prevent
longitudinal expansion [10]. The plot in Figure 4 highlights how dynamic the
structural response is by comparing the static and transient x-normal stresses
at two points. Strong inertia effects are observed with large compressive-to-
tensile dynamic stresses, reaching -236 MPa at the end of the beam pulse. From
that analysis, the simulation set up uses short time steps able to estimate
the high-frequency dynamic modes up to 100 µs. After that time, the dynamic
stresses are largely damped and no longer show a risk of resulting in plastic
strains.

At a longer time scale, the response is governed by lower frequency modes
(with characteristic times of a few milliseconds) with mode shapes that impel
large bending oscillations. The structural model has to estimate the maximum
deformation amplitudes and ensure that the risk of contact with the ultra-high
vacuum tank is limited. An ANSYS® modal analysis, presented in Figure 5, had
evaluated the fundamental at a frequency of 54 Hz to set up accordingly the
simulation time step in the second phase, which lasts for 1 s. The bending
oscillations are then largely attenuated as the friction at the contact
between the guiding plates and the cylindrical shafts dissipates energy.

Figure 5:  Preliminary ANSYS® modal analysis of the jaw in the thermal state
reached at the end of the proton beam pulse. The fundamental has a frequency
of 54 Hz and governs the response in the second phase.

Afterwards, the jaw structural response reaches a quasi-static state deformed
by thermal strains. As most of the beam energy deposition occurs in the
absorber blocks and in the lower part of the backstiffener, the jaw acquires a
convex geometry with the downstream and upstream sides extending away from the
beam trajectory, as shown in Figure 6. Thermal conduction and energy
dissipation processes finally govern the last phase until the system has
cooled to the initial temperature condition.

Figure 6:  ANSYS® static structural analysis of the jaw response reached after
at the beginning of the third phase (1 s) after the deepest standard LIU
proton beam impact. A deformed geometry results from the thermal strains with
the jaw downstream and upstream sides extending away from the beam trajectory.

###  2.3 Structural model assumptions

A full scale 3D jaw geometry has been defined in the ANSYS® transient
structural model with a symmetry about the y-z plane. The model is based on
two strong assumptions, which are highlighted in purple in Figures 2 and 3 .
First, the implemented material properties are temperature dependent linear
elastic. That is questionable given the large dynamic stresses discussed in
section 2.2. Then, two frictional contacts model the connection between the
jaw guiding plates and the cylindrical shafts with a friction coefficient of µ
= 0.1. The analysis of the HRMT 44 online results has converged to this value,
see section 5.2, although it had been previously estimated based on empirical
studies. A frictional mount brings on the risk of residual deformations after
cooling.

The challenging jaw design requirements, see section 1.2, cannot be guaranteed
with certainty from a numerical approach, especially considering the jaw
dimensions and the design complexity. A HiRadMat beam impact testing
experiment was the opportunity to test a jaw under thermo-mechanical loads
representative of LIU accidental scenarios, and ideally, provide an
experimental validation. On the other hand, experimental online results could
be compared to the numerical results to enhance the accuracy of the model.

##  3 HiRadMat 44 set-up

###  3.1 Test bench

The HRMT 44 test bench is based on a prototype TCDIL collimator. As shown in
Figure 7, two jaw designs are linked to a standard collimator actuation system
with a precision and repeatability of $\pm$ 20 µm [11] controlled by four
stepper motors and four Linear Variable Differential Transformer (hereinafter
“LVDT”) position sensors. The jaws are installed in a primary vacuum tank. The
cover is closed with fast-clamps to speed up the dismounting operations in
view of the post-irradiation examination, hence limiting the personal exposure
to ionizing radiation resulting from proton beam impacts. Two 85 mm-diameter
proton beam windows made of 0.254 mm-thick PF-60 beryllium foil [12] ensure
the vacuum tightness while inducing a minimum interaction with the beam.
Flanges integrate several electrical feedthroughs and a radiation resistant
glass window offers a visual port to the jaws. The vacuum vessel is mounted on
a finely 3D-adjustable support with a fifth translational degree of freedom
controlled by one stepper motor and one LVDT to increase the stroke range of
the jaws, mentioned in section 3.2. The equipment is finally installed on a
standard HiRadMat table, which provides an electrical plug-in interface for a
quick installation in the dedicated experimental area.

Figure 7:  Two jaw designs are mounted inside the HRMT 44 test bench, whose
design is largely based on the TCDIL collimator. On the left with respect to
the beam direction, a TCDIL jaw with an absorber thickness of 37 mm. On the
right, an upgraded TCDIL jaw with a total absorber thickness of 47 mm.

###  3.2 Collimator jaws

The first jaw design, mounted on the left-hand side (L) in the direction of
the beam, is a standard TCDIL jaw. Four 37 mm-thick Sigrafine® R7550 graphite
blocks formed the 2.1 m absorber length. The block lengths are either 510 mm
or 540 mm depending on their position along the jaw. The second jaw design,
mounted on the right-hand side (R), is a slightly modified TCDIL jaw with a
total absorber layer of 47 mm instead of 37 mm as shown in Figure 7. The
additional layer is composed of two 1050 mm-long Sigrafine® R7550 graphite
plates. It increases the relative distance between the proton beam and the
stainless steel backstiffener, and consequently decreases the beam-induced
energy deposition in the backstiffener. The upstream graphite block was
replaced by three 170 mm-long Novoltex® Sepcarb® 054-62 blocks to study
objective (iii), see section 1.3. In comparison with the TCDIL-collimator, the
strokes of the jaws were increased to reach maximum beam impact parameters of
20.5 mm and 24.9 mm in the TCDIL-jaw and the modified version respectively.
The reasons will be detailed in section 4.

###  3.3 Experiment Instrumentation

The experiment counted on extensive instrumentation for real-time data
acquisition and monitoring of several physical values as set out in Table 2.
First, it aimed at accumulating experimental input data of the beam
characteristics, and especially the beam impact parameters, see section 2.1.
To that end, the HRM Beam TV (hereinafter “HRM BTV”) [13, 14] employs a
silicon carbide (SiC) screen that is placed in the beam trajectory roughly 1m
before the experiment. When the beam passes through the screen, the radiator
material emits light, a camera records its intensity and software interpolates
the horizontal and vertical distributions with Gaussian functions. It then
gives online measurements of the beam position, as well as information about
the beam spot sizes with a precision of 100 µm. The Beam Pick-Up Button
Position was installed to measure the relative beam bunch positions, as
presented in [2], which is related to the objective (iii) of the experiment,
see section 1.3.

Table 2:  Instrumentation overview and description of the recorded physical effects. Jaw instrumentation | Quantity | Physical Effect
---|---|---
Interferometer (Optic fiber based) | 3 | Bending oscillations
Linear Variable Differential Transformer (LVDT) | 1 per jaw | Quasi-static cooling
Platinum Resistance Thermometer (PT100) | 3 per jaw | Temperature monitoring
Strain gauges | 14 per jaw | Permanent strains
RadHard camera | 1 | Online visual observation
Beam instrumentation |  |
HiRadMat Beam Television (HRM BTV) | 1 | Beam position and sigma
Beam Pick-Up Button Position (BPKG) | 1 | Beam bunch position

In addition, the instrumentation aimed at accumulating experimental data to
understand the jaw structural behaviour during beam impacting events and to
assess online the uncertainties raised in section 2.3. Figure 8 presents the
jaw’s instrumentation by outlining its positions and giving a concise idea of
the mounting. The interferometers are positioned along the two jaws in three
locations, 170 mm (Identifier: “HSSL1”), 1050 mm (Identifier: “HSSL2”), and
1930 mm (Identifier: “HSSL3”) from the jaw front faces, with the
retroreflector mirrors mounted on the right-hand side jaw (hereinafter “jaw
(R)”) and the optical head on the left-hand side jaw (hereinafter “jaw (L)”).
HSSL stands for High Speed Serial Link, which is the communication protocol
used by the interferometers. They were intended to record y-displacements, see
Figure 7 to offer an image of the second response phase governed by large jaw
bending oscillations. A Linear Variable Differential Transformer (hereinafter
“LVDT”) (Identifier: “LVDT_R” on jaw(R) and “LVDT_L” on jaw (L)) is probing
each jaw at 1975 mm from the front jaw face. Again, they were intended to
record y-displacements, however in this case, to study the third response
phase and eventually detect permanent deformation after cooling. The LVDT coil
assemblies were mounted inside a vacuum flange fixed on the vacuum tank, while
the LVDT core assemblies translated by maintaining contact with the
backstiffener thanks to a compression spring. Seven strain gauge patches were
installed per jaw along the backstiffener face in contact with the absorbers.
The patches are composed of two strain gauges measuring in the transverse and
longitudinal directions, respectively in the x and z directions, to detect
possible irreversible strains after cooling, which would certainly have been
caused by the characteristic high compressive-to-tensile stresses of the first
response phase. Finally, three Platinum Resistance Thermometer (hereinafter
“PT100”) per jaw were monitoring the jaw temperature, without being designed
to measure the backstiffener’s peak temperature.

Figure 8:  Detailed diagram of the jaw instrumentation used in HRMT 44.
Pictures show each instrument design and the mounting position are pointed out
by colored shapes.

The Data Acquisition System (hereinafter “DAQ”) was configured to be able to
appreciate the beam-induced jaw response by the identification of bending
oscillations (with a characteristic time of a few milliseconds) and quasi-
static states (with a characteristic time of a few seconds) during a total
recording time of 600 s. Afterwards, an online supervisory program allowed
acquisitions spread over several days to detect possible permanent
deformations after cooling.

##  4 Experimental procedure

HRMT 44 was focused on recreating the thermo-mechanical load of the worst case
standard LIU beam accidental impact scenario by using proton beam
characteristics set out in Table 1. The limited HiRadMat maximum intensity of
$1.3\times 10^{11}$ protons has been compensated by increasing the beam impact
parameters in order to best fit the backstiffener energy depositions. Indeed,
the test bench allows to reach higher impact parameters in both jaws as
explained in section 3.2. As described in section 2.1, the beam impact
parameter is a sensitive beam characteristic requiring a specific experimental
procedure for precise online measurements. Before the high-intensity (288
bunches) beam impacts, the jaws have been aligned to the real beam trajectory
thanks to low intensity pulses (1 bunch). The procedure, so-called Beam Based
Alignment [15], suppresses mechanical errors between the HRMT BTV and the jaw
active planes. In this way, the measurement error is determined by the
precision of the instrument (100 µm, see section 3.3), the actuation system
(20 µm, see section 3.1), and the jaw flatness (62 µm for jaw (L), 103 µm for
jaw (R)).

###  4.1 High-intensity proton beam impacts

In total, four high-intensity proton beams successfully impacted the jaws with
the conditions set out in Table 3. Deep (R) and deep (L) beams have been
delivered after two “grazed” shots related to objective (iii). As requested,
HRMT 44 obtained large beam impact parameters, 18.60 mm and 16.50 mm
respectively, to compensate the lower maximum HiRadMat intensity in comparison
to the standard LIU. The next section will evaluate the thermo-mechanical
loads.

Table 3:  High-intensity (288 bunches) beams impacted in the HRMT-44 jaws on July, 18 2018. Beam Parameters | Grazed 1 | Grazed 2 | Deep (R) | Deep (L)
---|---|---|---|---
Target | jaw (R) | jaw (R) | jaw (R) | jaw (L)
Impact time | 21:03:24 | 21:39:51 | 22:44:24 | 23:38:13
Intensity | $3.56\times 10^{13}$ | $3.57\times 10^{13}$ | $3.56\times 10^{13}$ | $3.53\times 10^{13}$
Impact parameter [mm] | 0.530 | 0.530 | 18.60 | 16.50
Horizontal Spot Size111Corresponding author. [mm] | 0.450 | 0.420 | 0.420 | 0.430
Vertical Spot Size111Spot sizes measured by the HRM BTV. [mm] | 0.380 | 0.350 | 0.350 | 0.390

###  4.2 Related energy deposition

FLUKA Monte Carlo simulations [9] have been employed in order to infer the
energy density depositions induced by the beam impacts in the HRMT 44 jaws for
the different experimental beam conditions. Figure 9 shows a comparison of the
energy density reached in the backstiffener along the beam orthogonal
projection, and as an evidence, the worst standard LIU proton beam impacts
have been successfully recreated in both jaws. The peak energy depositions
would have reached $3.20\times 10^{8}$ J.m-3 and $1.94\times 10^{8}$ J.m-3
during deep (R) and deep (L) respectively, which approximately corresponds to
maximum increases of temperature of 86 °C and 52 °C in 7.2 µs.

Figure 9:  Energy deposition reached in the jaw backstiffener along the beam
orthogonal projection in the different cases. The proton beams impacted during
HRMT 44 have successfully recreated the worst LIU thermo-mechanical loads in
both jaws.

Nevertheless, a narrower heat “footprint” would have been induced in the HRMT
44 jaws as shown in Figure 10, together with lower total energy deposition.
The coupled effect of lower beam intensities and larger impact parameters has
indeed focused the thermo-mechanical loads. For example, the total energy
deposition induced in the backstiffener and in the absorbers by deep (L) are
141 kJ and 301 kJ, respectively, whereas the worst standard LIU proton beam
would depose 183 kJ and 489 kJ. The relative differences of 23 % and 39 % has
limited the HRMT 44 jaw structural responses, especially by lowering the
bending oscillations and quasi-static deformations. The impacted beams were
however the best possible options, taking into account the current HiRadMat
capabilities.

Figure 10:  Energy deposition reached in the jaw backstiffener planes in
contact with the absorbers. The proton beams impacted during HRMT 44 have
induced more focused thermo-mechanical loads and deposited less total energy
than the LIU worst case scenario. Energy deposition in Jaw (L): (A) HRMT 44
(B) LIU; in jaw (R): (C) HRMT 44 (D) LIU.

##  5 Online results analysis and comparison with the numerical results

Analyzing experimental online results of the jaw structural responses is an
opportunity to validate or enhance the numerical models. As described in
section 2.2, the preliminary analysis had highlighted two main possibilities
for severely affecting the jaw flatness requirement. The following sections
will present the acquired data and draw preliminary affirmations about the
models’ validity. It will focus on the jaw with the initial design, jaw (L),
as the structural behaviors are identical to a certain scale factor related to
the lower energy deposition.

###  5.1 Non-induced plastic strain

Fifteen strain gauges failed during the experiment, most likely because of
intense vibrations and/or rapid heating that may have broken the soldered
electrical connections. The others remain below 200 µmm/mm with amplitudes
that cannot be correlated to beam induced irreversible strains, and
consequently seem to justify the elastic material assumption of the jaw
structural model. This affirmation will be further investigated in the post-
irradiation examinations.

###  5.2 Jaw bending oscillations

Figure 11 presents a time-frequency analysis of the interferometer
displacement signals acquired during the high-intensity beam impact deep (L).
Signal losses occurred in the very first instants up to 40 ms, certainly due
to some internal limitations of the instruments, and have disrupted the
absolute displacement data. Thus, some post-processing has been performed by
cleaning the signal losses and offsetting the mean displacement to zero in
order to clear the incorrect constant term.

The damping properties of the jaw structural response can be appreciated, and
have been used to update the first estimate of the friction coefficient at the
interfaces between the guiding plates and the cylindrical shafts. The
iterative approach has converged to µ = 0.1 to finally give good agreement
between the experimental and numerical results. It also confirms that the jaw
bending oscillations are completely damped 1 s after the proton beam impact.

On the other hand, the Fast Fourier Transform offers a mathematical tool to
identify the dominant vibration modes in the frequency domain. The
experimental response is dominated by two frequencies at 36 Hz and 56 Hz,
observable through the three installed interferometers. The 56 Hz-frequency
mode is correctly predicted by the numerical model, and as discussed in
section 2.2, corresponds to the jaw fundamental. However, the 36 Hz-frequency
mode remains a mystery at this time. It cannot be accounted by a jaw vibration
mode, so the study may further investigate solid modes related to the jaw
support and the actuation system.

Figure 11:  Time-frequency analysis of the interferometer displacement signals
acquired during the high-intensity beam impact in the jaw, deep (L), and
compared to the jaw structural model results. The time displacement plots, on
the left hand side, show similar damping properties. The Fast Fourier
Transforms give a frequency domain representation of the signals, and
therefore highlight the jaw vibration modes.

###  5.3 Residual deformations after cooling

Figure 12 shows the signal recorded by LVDT_L during the high-intensity beam
impact deep (L) during 600 s, which corresponds to the DAQ recording time.
Some post-processing has been required to clean the signal before 1 s as large
disruptions with non-physical amplitudes occurred. Because of the jaw
vibration, the LDVT core was certainly interfering with the LVDT coil
assembly. The plot has been completed by two spread acquisition points at
40,000 s and 53,000 s. In addition, the y-displacement calculated by the
numerical model at the probing spot is plotted in red. The characteristic jaw
bending oscillations of the second response phase are clearly visible, as
explained in section 5.2.

After that phase, as discussed in section 2.2, the jaw acquires a convex
quasi-static geometry caused by thermal strains concentrated in the absorbers
and in the lower volumes of the backstiffener. One second after the proton
beam impact, the signals are comparable in terms of amplitude, 2.515 mm and
2.341 mm for the experimental and numerical results respectively. Afterwards,
the curves diverge as the model predicts a displacement increase until 30 s
and then a continuous decrease, whereas a continuous displacement decrease has
been observed during the experiment. Various phenomena could explain such an
inconsistency, especially the wrong prediction of the sticking-slipping states
at the boundary interfaces between the guiding plates and the cylindrical
shafts, but it requires further investigation. Approximately 4 days were
necessary for the jaw to cool down at the initial temperature conditions. At
the end, permanent displacements have been detected in both signals. The
amplitudes are comparable, 58 µm and 118 µm according to the experimental and
numerical results respectively.

Figure 12:  The LVDT_L signal acquired during the high-intensity beam impact,
deep (L), is plotted in solid blue, whereas the spread acquisitions during
cooling are marked by a blue cross. The y-displacement calculated by the
numerical model at the probing spot is plotted in solid red. Some
inconsistencies are visible in the first instant of the third response phase.

##  6 Post-irradiation examinations

After the experiment was completed, the HRMT 44 test bench was stored for two
months in a dedicated area to reach acceptable residual radioactive dose
rates. The following post-irradiation examinations first aimed at comparing
the jaws’ geometry before and after the high-intensity proton beam impacts by
means of metrology measurements. It aspires to consolidate the online results
analysis in section 5, more specifically to confirm both the absence of
plastic strains in the backstiffener and the slight permanent jaw deformations
after cooling. On the other hand, a visual observation was required to confirm
that no damage had been induced in the absorber blocks in Novoltex® Sepcarb®
054-62.

###  6.1 Metrology measurements

As shown in figure 13, metrology measurements have been performed in different
configurations, starting from the experimental assembly with the jaws still
mounted inside the HRMT 44 test bench to a freely supported jaws, and finally
to a freely supported bare backstiffener. A Zeiss Prismo Ultra MMT machine
with a range of 2400 mm $\times$ 1200 mm $\times$ 1000 mm and a precision of
1.2 + L/500 µm was used for the measurements at an ambient temperature of 20
$\pm$ 1 °C. The analysis also compares the two jaw designs to examine the
benefits brought by the upgraded version.

Figure 13:  The different configurations used for metrology measurements. (A):
Jaws in the HRMT 44 test bench. (B): Jaws hold up on two supports similar to
free mounts. (C): Bare backstiffener hold up on two supports similar to free
mounts.

####  6.1.1 Jaws in the HRMT 44 test bench

Figure 14 presents the metrology measurements carried out while the jaws were
still installed in the HRMT 44 test bench and gives a visual representation of
the jaw forms thanks to flatness geometric characteristics. After the high-
intensity proton beam impacts, two measurement runs were performed within a
period of 1 month, while a CERN internal transport occured in-between. Both
the TCDIL jaw, jaw (L) on the left-hand side, and the upgraded version, jaw
(R) on the right-hand side, had acquired a more convex geometry, which
corresponds to the LVDT online results. The flatness values were respectively
of 62 µm and 103 µm before closing the vacuum vessel, and had reached 124 µm
and 168 µm just after opening it in October 2018. Surprisingly, noticeable
changes had been detected in November, as jaw (L) acquired an even more convex
geometry with a flatness value of 184 µm, and jaw (R) straightened its form up
by reaching a flatness value of 97 µm (very similar to the value measured
before the high-intensity proton beam impacts).

These measurements have highlighted how the jaws can be affected by shocks and
vibrations generated by transportation as well. It seems that the geometry is
sensitive to the connection between the cylindrical shafts and the jaw guiding
plates, which by a stick-slip phenomenon can lead to permanent deformations.

Figure 14:  Visual representation of the jaw flatness characteristics before
and after the high-intensity proton beams, while the jaws were installed in
the HRMT 44 test bench. Two post-measurements have been performed within a
period of 1 month, and show how the jaw geometry is sensitive to proton beam
impacts, as well as transportation.

####  6.1.2 Freely supported jaws

As a second step, the jaws were removed from the HRMT 44 test bench, and
measured with “sliding” contact planes on the guiding plates. The jaw forms
are presented in Figure 15. Both jaws became straighter as the flatness values
go from 134 µm and 133 µm, respectively for jaw (L) and jaw (R) before the
high-intensity proton beam impacts, to 62 µm and 106 µm afterwards.

Figure 15:  Visual representation of the jaw flatness characteristics before
and after the high-intensity proton beams, while the jaws were freely
supported. The proton beam impacts have straightened the jaw forms up.

####  6.1.3 Freely supported backstiffener

As a final step, both jaws were disassembled and metrology measurements have
been performed on the backstiffeners, again with a free mount. The flatness
visual representations in Figure 16 clearly confirm that the backstiffeners
were not affected by the proton beam impacts as the flatness values are very
stable: from 79 µm and 66 µm for backstiffener (L) and backstiffener (R)
before, to 60 µm and 56 µm after, respectively. These measurements seem to
validate the model’s elastic assumption.

Figure 16:  Visual representation of the backstiffener flatness
characteristics before and after the high-intensity proton beams. The proton
beam impacts have not altered the forms, which seems to validate the absence
of irreversible strains and therefore, the elastic material assumption of the
numerical model.

###  6.2 Absorber visual observations

A visual observation of the three Novoltex® Sepcarb® 054-62 absorbers was
performed before and after the HRMT 44 high-intensity proton beam impacts.
Figure 17 shows pictures of the upstream absorber, in which the most critical
thermo-mechanical loads occurred. The comparison has not brought any defect,
cracks or visible deformations to light, and seems to confirm the material
integrity after the beam Grazed 1 and Grazed 2 impacts. The study is detailed
in [8], in which the induced loads are evaluated through a numerical model
using the experimental beam data, especially the beam spot size and the beam
impact parameter, and in which the results are compared to future possible LIU
scenarios.

Figure 17:  Picture of the upstream Novoltex® Sepcarb® 054-62 absorber before
(May 2018) and after (January 2019) the HRMT 44 high-intensity proton beam
impacts. No defect is visible.

##  7 Conclusion

###  7.1 Findings

The HRMT 44 test bench has successfully been impacted by high-intensity proton
beams at CERN’s HiRadMat facility to recreate in two jaws the thermo-
mechanical loads of the worst accidental case scenario considered by the LIU
project. As introduced, the experiment’s objectives were to validate the TCDIL
collimator jaw design, challenging in view of the 0.200 mm flatness tolerance
requirement and to enhance the jaw numerical model.

Firstly, the absence of relevant plastic strains, and consequently the model’s
elastic assumption, has been confirmed by both the experimental online results
and the post-irradiation examinations. They have proved that the jaw core
part, the backstiffener, is not altered by the worst proton beam impacts, and
this in spite of the large dynamic stresses induced by the thermal burst.

Then, the study has highlighted how the jaw geometry, while installed in the
test bench and by extension, in a TCDIL collimator, is sensitive to the
connection between the cylindrical shafts and the jaw guiding plates,
regardless of the jaw design. At that time, frictional contacts model the
connection with a friction coefficient based on the analysis of the jaw
response damping properties. Numerical predictions are however extremely
challenging as they rely on a precise description of stick-slip phenomena,
which actually depends on many external parameters. Experimental estimations
to further validate this hypothesis are consequently suggested.

On the one hand, it seems that even less critical beam impacting events would
induce convex permanent deformations, with an alteration of the flatness value
up to 70 µm. On the other hand, handling activities for transportation,
installation or maintenance, have the capacity, as well, to alter randomly the
jaw characteristics. Depending of the shock and/or vibration direction, the
flatness can either be improved or degraded to an extent of $\pm$ 70 µm.

The manufacturing of the TCDIL collimators will be soon completed, and
installation in the SPS-to-LHC transfer lines is foreseen for 2020.

###  7.2 Further studies

Some open points have nevertheless been highlighted during this study. For
example, an unexpected vibration mode at a frequency of 36 Hz has been
detected through the interferometer online results, whereas the jaw
fundamental frequency is 56 Hz. The quasi-static jaw response during cooling
seems, in addition, to require a deeper understanding of the stick-slip
phenomena at the connection.

Finally, a complex phenomenon has been highlighted by the metrology
measurements of the jaw when removed from its supports. The geometries were
indeed straightened, even if the backstiffener integrity were not affected.

## Acknowledgments

The authors want to acknowledge all the personnel from the different CERN
services involved in the HRMT 44 experiment for the support in the execution
of design (EN-MME and EN-STI), manufacturing (EN-MME and EN-STI), assembly
(EN-STI), installation (EN-HE, EN-EA and EN-STI), execution (BE-OP) and post-
irradiation examination activities (HSE-RP-AS and EN-MME-MM). Their
collaborative efforts have made the experiment a success. In addition, the
authors would like to particularly thanks F. Harden (EN-EA) for her essential
support in the coordination of the HiRadMat facility and C. Bracco (TE-ABT)
for her effort carrying out the Beam Based Alignment of the jaws. Finally, the
authors thank warmly Louisa Catherall for the English proofreading of this
paper.

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Trees

The possible periods of parallel chip-firing games on directed trees were
identified by Prisner prisner. We begin by citing his result.

[[prisner, Corollary 2.3]] $T=1$ for all parallel chip-firing games on
directed acyclic graphs. For completeness, we state the following direct
corollaries of acyclic. $T=1$ for all parallel chip-firing games on directed
paths. $T=1$ for all parallel chip-firing games on directed non-path trees.

##  Cycles

Consider the simple cycle $C_{n}$. We label the vertices of the cycle
$v_{0},v_{1},v_{2},\dots,v_{n-1}$ such that there exists an edge directed from
$v_{i+1}$ to $v_{i}$ for all $0\leq i\leq n-2$ and an edge directed from
$v_{0}$ to $v_{n-1}$. The indices are taken modulo $n.$ We first show that if
all vertices start with at most 1 chip, every vertex always has at most 1
chip. If $c_{0}(v)\leq 1$ for all $v$, then $c_{t}(v)\leq 1$ for all $t$. We
proceed by induction. Clearly, the statement is true for $t=0$. We take as our
inductive hypothesis that $c_{t}(v)\leq 1$ for all $v$. Suppose some vertex
$v$ satisfies $c_{t+1}(v)\geq 2$. Since $\deg^{-}v=1$, we must have
$c_{t}(v)\geq 1$. Then $c_{t}(v)=F_{t}(v)=1$. By the definition of the
parallel chip-firing game, $c_{t+1}(v)\leq
c_{t}(v)-F_{t}(v)\deg^{+}v+\deg^{-}v\leq 1$, which is a contradiction. Thus,
$c_{t+1}(v)\leq 1$ for all $v$. By induction, $c_{t}(v)\leq 1$ for all $v$ and
all $t$. cequalsf allows us to claim that $c_{t}(v)=F_{t}(v)$, since $v$ fires
on round $t$ if and only if $c_{t}(v)=1$. We are now ready to prove the main
theorem about directed cycles. The possible periods $T$ of a parallel chip-
firing game on a directed cycle are $T=i$ for all $i$ such that $i\;|\;n.$ If
$F_{t}(v_{i})=1,$ then $v_{i-1}$ has at least one chip after round $t,$ so
$F_{t+1}(v_{i-1})=1.$ Thus, if $F_{t}(v)=1,$ then $F_{t+n}(v)=1.$ Assume for
the sake of contradiction that there exists some vertex $u$ and some time
$t\geq 0$ such that $F_{t}(u)\neq F_{t+n}(u).$ Since $F_{t}(v)=1$ implies
$F_{t+n}(v)=1$, we must have $F_{t}(u)=0$ and $F_{t+n}(u)=1.$ Let the game
have period $T$. If $F_{t+n}(u)=1,$ then $F_{t+nT}(u)=1$. However, if
$F_{t+nT}(u)\neq F_{t}(u)$, then $T$ cannot be the period. By contradiction,
we have that $F_{t}(v)=F_{t+n}(v)$ for all $v$ and $t$. By cequalsf,
$c_{t}(v)=c_{t+n}(v)$ for all $v$ and $t$. Thus, the period $T$ of a parallel
chip-firing game on a directed cycle must satisfy $T\;|\;n.$ To construct a
game with period $T=i$ for $i\;|\;n,$ place a single chip on each vertex
$v_{j}$ that satisfies $j\equiv 0\pmod{i}$. On turn $t,$ there will be a chip
on each vertex $v_{j}$ that satisfies $j\equiv t\pmod{i}.$ The smallest $T$
for which $c_{v}(t)=c_{v}(t+T)$ is therefore $T=i$.

##  Complete Graphs

We first note as a surprising aside that while there exists a parallel chip-
firing game on any undirected graph with $T=2$, there can be no orientations
of a graph that allow for a parallel chip-firing game with $T=2$. We begin by
citing a lemma from Jiang [jiang]. [[jiang, Proposition 2.5]] For any parallel
chip-firing game on an undirected graph, $c_{T}(v)=c_{0}(v)$ for all $v\in V$
if and only if $\sum_{t=0}^{T}F_{t}(v)$ is constant for all $v\in V$. In
talways2, we show that there exists a parallel chip-firing game on any
undirected graph $G$ such that every vertex fires exactly once in the first 2
rounds. However, parallel chip-firing games on directed graphs behave
differently, as demonstrated by the following lemma. No parallel chip-firing
games on any orientation of $K_{n}$ with $n\leq 4$ have $T=2$. Note that
$K_{2}$ is a path, and $K_{3}$ a cycle. For these cases, see path_par and
cycles, respectively. Then suppose that there exists a game on $G=(V,\,E)$,
some orientation of $K_{4}$, with period 2. Recall that because $G$ must be
strongly connected, each vertex $v$ in $G$ must satisfy $\deg^{+}v>0$ and
$\deg^{-}v>0$. Also, by Lemma LABEL:lost, each vertex must fire at least once
in each period. Since $\deg v=3$, $v$ must be one of two types: $\deg^{-}v=1$
and $\deg^{+}v=2$, or $\deg^{-}v=2$ and $\deg^{+}v=1$. Additionally, since
$\sum_{v\in V}\deg^{-}v=\sum_{v\in V}\deg^{+}v$, there must be two vertices of
each type. Let the two vertices with one out-edge be $v_{1}$ and $v_{2}$, and
let the other two vertices be $u_{1}$ and $u_{2}$. Note that there must exist
an edge between $v_{1}$ and $v_{2}$, and that regardless of its direction, one
of the two vertices must view it as its single out-edge. Thus, there is only
one edge directed from $v_{1}$ or $v_{2}$ to $u_{1}$ or $u_{2}$. Without loss
of generality, let it be directed from $v_{1}$ to $u_{1}$. On the other hand,
there are three edges directed from $u_{1}$ or $u_{2}$ to $v_{1}$ or $v_{2}$.
Since $T=2$, we must have $f_{v_{1}}\leq 2$. However, $f_{u_{1}}\geq 1$ and
$f_{u_{2}}\geq 1$. Then there can be at most two chips that travel from a
vertex in $V=\\{v_{1},\,v_{2}\\}$ to one in $U=\\{u_{1},\,u_{2}\\}$, but at
least three chips that travel from a vertex in $U$ to one in $V$. Thus, it is
impossible for every vertex to have the same number of chips after 2 rounds,
so there are no games on $G$ with $T=2$. Next, let $\tau(n)$ be the maximum
possible period of a parallel chip-firing game on an orientation of $K_{n}$.
We show that $\tau(n)$ is at least asymptotic to $n!$. We conjecture that
$\tau(n)$ is indeed asymptotic to $n!$ It holds that $\tau(n)\sim n!$. We
define the following orientation of $K_{n}$ with vertices
$v_{1},\,v_{2},\,\ldots,\,v_{n}$ to be a _useful_ orientation.

  * •

There exist edges directed from $v_{i}$ to $v_{i+1}$ for each $1\leq i<n$, and
there exists an edge directed from $v_{n}$ to $v_{1}$.

  * •

There exist edges directed from $v_{1}$ to every other vertex except $v_{n}$.

  * •

There exist edges directed from $v_{j}$ to all vertices $v_{k}$ with $1<k\leq
j-2$.

See Figure  for a diagram of a useful orientation of $K_{5}$.

[every node/.style=circle, draw, minimum size=6mm, inner sep=1mm, font=, -¿,
line cap=round, line join=round, thick] (1) at (90:2) 1; (5) at (162:2) 5; (4)
at (234:2) 4; (3) at (306:2) 3; (2) at (18:2) 2; [-¿] (1) – (2); [¡-] (5) –
(4); [¡-] (4) – (3); [¡-] (3) – (2); [-¿] (1) – (4); [-¿] (1) – (3); [¡-] (3)
– (5); [¡-] (2) – (5); [¡-] (2) – (4); [-¿] (5) – (1);

Figure : A useful orientation of $K_{5}$.

The following lemma shows that vertex 3 fires the most in all parallel chip-
firing games on $K_{n}$ with $n\geq 4$. For any $n\geq 4,$ all parallel chip-
firing games on $K_{n}$ satisfy $f_{3}\geq f_{i}$ for all $1\leq i\leq n$.
Since $\deg^{+}v_{3}=1$, it must fire once in each period for every chip
received in that period. Note that $v_{3}$ receives chips from all other
vertices except $v_{4}$. Thus, $f_{3}\geq f_{i}$ for all $i\neq 4$. Also note
that $\deg^{+}v_{2}=1$, and that $v_{2}$ receives chips from $v_{4}$. Then
$f_{2}\geq f_{4}$. Since $f_{3}\geq f_{2}$, we have $f_{3}\geq f_{4}$. Let the
_convergent period_ $T_{c}(G)$ of a graph $G$ be $\min(T)$ over all parallel
chip-firing games on $G$ with at least $c$ chips. Note that $T_{c}(G)=1$ for
any undirected graph $G$ [yunseo]. We will show that the same does not hold
for directed graphs. For a useful orientation of $K_{n}$, the expression
$\lim_{c\rightarrow\infty}T_{c}(K_{n})$ increases at least factorially with
$n$. Since the number of chips on each vertex does not change after a full
period, the following equations must hold.

$\aligned&f_{n}=(n-2)f_{1}\\\ f_{1}+f_{4}+\ldots+f_{n}=f_{2}\\\
f_{1}+f_{j-1}+f_{j+2}+f_{j+3}+\ldots+f_{n}=(j-2)f_{j}\quad\forall j:\;3\leq
j\leq n-2\\\ f_{1}+f_{n-2}=(n-3)f_{n-1}\\\ f_{n-1}=(n-2)f_{n}$ (1)

Since the total number of chips in the game is fixed, System 1 cannot be
independent. We arbitrarily drop the first equation, leaving a system of $n-1$
independent equations in $n$ variables. Let $T_{n}$ denote the smallest
positive integer value of $f_{3}$ which satisfies the above system for a
particular value of $n$. We claim that, for $n\geq 4$,

$T_{n}=T_{n-2}+(n-1)T_{n-1}.$

By definition, each of $f_{1},f_{2},\ldots,\,f_{n}\geq 0.$ By lost, if any
$f_{i}=0$, then we have $T=1$ and that all vertices are forever passive.
However, if $c$ is very large, some vertex must fire on every round. Thus, we
further constrain $f_{i}>0$ for all $i$. Note that for any solution
$(f_{1},f_{2},\ldots,\,f_{n})$, it must be true that $(\lambda f_{1},\lambda
f_{2},\ldots,\,\lambda f_{n})$ is also a solution. Thus, without loss of
generality, we set $f_{1}=1.$ System 1 has now been reduced to

$\aligned&f_{2}-f_{4}-\ldots-f_{n}=1\\\
-f_{j-1}+(j-2)f_{j}-f_{j+2}-f_{j+3}-\ldots-f_{n}=1\qquad\forall j:\;3\leq
j\leq n-2\\\ -f_{n-2}+(n-3)f_{n-1}=1\\\ -f_{n-1}+(n-2)f_{n}=0.$ (2)

System 2 can be written as the following augmented matrix $M_{n}=(A|B)$, where
$A$ is an $(n-1)\times(n-1)$ matrix and $B$ is an $(n-1)$-dimensional column
vector. We denote the element in the $i$th row and $j$th column as
$M_{n}^{i,\,j}$, where $1\leq i\leq n-1$ and $1\leq j\leq n$.

$M_{n}=\amatrix{9}1&0-1-1-1\ldots-1-1-11\\\ -110-1-1\ldots-1-1-11\\\
0-120-1\ldots-1-1-11\\\ \vdots\vdots\\\ 00000\ldots-1n-301\\\ 00000\ldots
0-1n-20$ (3)

We now apply Gauss-Jordan Elimination to transform $M_{n}$ into its reduced
row echelon form. In particular, we seek the value of $T_{n}=f_{3}$, which
will be the largest of the $f_{i}$ by vertex3. We denote the submatrix formed
by omitting the last row and column of $M_{n}$ as $M_{n}[-1]$. Note that
$M_{n-1}$ is very similar to $M_{n}[-1]$. In particular, $M_{n}[-1]$ differs
from $M_{n-1}$ only in the negation of the $(n-1)$th column and the value of
$M_{n}[-1]^{(n-2),\,(n-2)}=0$, whereas we have $M_{n-1}^{(n-2),\,(n-2)}=1$. We
proceed with the following algorithm. We first add each row to the one beneath
it, in order. Thus, we cancel the leading $-1$ terms in each row, putting
$M_{n}$ into row echelon form.

$M_{n}=\amatrix{9}1&0-1-1-1\ldots-1-1-11\\\ 01-1-2-2\ldots-2-2-22\\\
0010-3\ldots-3-3-33\\\ \vdots\vdots\\\ 00000\ldots 01-(n-3)n-2\\\ 00000\ldots
001n-2$ (4)

We next apply the rest of the Gauss-Jordan Elimination algorithm for $M_{n-2}$
on $M_{n}$ to create a new matrix $M_{n}^{\prime}$. The final three columns of
$M_{n}^{\prime}$ will have arbitrary constants, but the rest of
$M_{n}^{\prime}$ will be in reduced row echelon form. Recall that
$M_{n}^{\prime 2,\,n-2}=-M_{n-2}^{2,\,n-2}=-T_{n-2}$. To negate $M_{n}^{\prime
2,\,n-2}$, we note that $M_{n}^{\prime n-2,\,n-2}=1$. Thus, we add the
$(n-2)$th row to the second row $T_{n-2}$ times, putting the $(n-1)$th column
into reduced row echelon form. If it was true that
$M_{n}^{n-2,\,n}=M_{n-1}^{n-2}{n-1}$, we would now have that $M_{n}^{\prime
2,\,n}=T_{n-1}$. However, $M_{n}^{n-2,\,n}$ is instead $M_{n-1}^{n-2}{n-1}+1$.
Therefore, adding the $(n-2)$th row to the 2nd row $T_{n-2}$ times adds an
additional quantity of $T_{n-2}$ to $M_{n}^{\prime 2,\,n}$, so we now have
that $M_{n}^{\prime 2,\,n}=T_{n-2}+T_{n-1}$. We now proceed with the rest of
the Gauss-Jordan Elimination algorithm for $M_{n-1}$ on $M_{n}^{\prime}$ to
create a new matrix $M_{n}^{\prime\prime}$. Now only the $(n-1)$th column will
not be in reduced row echelon form. Recall that $M_{n}^{\prime\prime
2,\,n-1}=-T_{n-1}$. To negate $M_{n}^{\prime\prime 2,\,n-1}$, we note that
$M_{n}^{\prime\prime n-1,\,n-1}=1$. Thus, we add the $(n-1)$th row to the
second row $T_{n-1}$ times, putting the second row completely into reduced row
echelon form. Note that $M_{n}^{\prime\prime
n-1,\,n}=\sum_{i=1}^{n-1}M_{n}^{i,\,n}=\deg^{+}v_{1}=n-2$. Thus,

$M_{n}^{\prime\prime 2,\,n}=M_{n}^{\prime
2,\,n}+T_{n-1}(n-2)=T_{n-2}+T_{n-1}+T_{n-1}(n-2)=T_{n-2}+(n-1)T_{n-1}.$

Since the second row is now in reduced row echelon form, we have

$f_{3}=T_{n}=T_{n-2}+(n-1)T_{n-1}.$

Note that $T_{n}$ is the smallest possible positive value of $f_{3}$, since
$\gcd(f_{1},\,f_{3})=\gcd(1,\,f_{3})=1$. For $n\leq 3$, vertex3 does not hold.
However, we set $T_{1}=T_{2}=T_{3}=1$, which reflects the lowest possible
periods of parallel chip-firing games on $K_{1}$ through $K_{3}$. For $n\geq
4$, we use the above recursive formula. By the OEIS Sequence A058279 [oeis],
$T_{n}\sim O(n!)$. Recall that $f_{3}$ is the number of times that $v_{3}$
fires in the first $T$ rounds. Thus, $T\geq f_{3}$. Since $f_{3}\geq T_{n}$,
we have that $T\geq T_{n}$ for all parallel chip-firing games with enough
chips. Thus,

$\lim_{c\to\infty}T_{c}(K_{n})\geq T_{n}\sim n!.$ $None$

Figure  shows an example of a parallel chip-firing game on $K_{4}$ with
$T=T_{4}=4$. The vector $(f_{1},\,f_{2},\,f_{3},\,f_{4})=(1,\,3,\,4,\,2)$ is
indeed a solution to System 1 for $K_{4}$.

1.5 [every node/.style=circle, draw, minimum size=6mm, inner sep=1mm, font=,
-¿, line cap=round, line join=round, thick] (1) at (0, 1.5) 1; (2) at (1.5,
1.5) 2; (3) at (1.5, 0) 3; (4) at (0, 0) 4; [draw=none] at (0, 2) 1;
[draw=none] at (1.5, 2) 0; [draw=none] at (1.5, -0.5) 2; [draw=none] at (0,
-0.5) 2; [-¿] (1) – (2); [-¿] (2) – (3); [-¿] (3) – (4); [-¿] (4) – (1); [-¿]
(1) – (3); [-¿] (4) – (2); 1.5 [every node/.style=circle, draw, minimum
size=6mm, inner sep=1mm, font=, -¿, line cap=round, line join=round, thick]
(1) at (0, 1.5) 1; (2) at (1.5, 1.5) 2; (3) at (1.5, 0) 3; (4) at (0, 0) 4;
[draw=none] at (0, 2) 2; [draw=none] at (1.5, 2) 1; [draw=none] at (1.5, -0.5)
1; [draw=none] at (0, -0.5) 1; [-¿] (1) – (2); [-¿] (2) – (3); [-¿] (3) – (4);
[-¿] (4) – (1); [-¿] (1) – (3); [-¿] (4) – (2);
1.5 [every node/.style=circle, draw, minimum size=6mm, inner sep=1mm, font=,
-¿, line cap=round, line join=round, thick] (1) at (0, 1.5) 1; (2) at (1.5,
1.5) 2; (3) at (1.5, 0) 3; (4) at (0, 0) 4; [draw=none] at (0, 2) 0;
[draw=none] at (1.5, 2) 1; [draw=none] at (1.5, -0.5) 2; [draw=none] at (0,
-0.5) 2; [-¿] (1) – (2); [-¿] (2) – (3); [-¿] (3) – (4); [-¿] (4) – (1); [-¿]
(1) – (3); [-¿] (4) – (2); 1.5 [every node/.style=circle, draw, minimum
size=6mm, inner sep=1mm, font=, -¿, line cap=round, line join=round, thick]
(1) at (0, 1.5) 1; (2) at (1.5, 1.5) 2; (3) at (1.5, 0) 3; (4) at (0, 0) 4;
[draw=none] at (0, 2) 1; [draw=none] at (1.5, 2) 1; [draw=none] at (1.5, -0.5)
2; [draw=none] at (0, -0.5) 1; [-¿] (1) – (2); [-¿] (2) – (3); [-¿] (3) – (4);
[-¿] (4) – (1); [-¿] (1) – (3); [-¿] (4) – (2);

Figure : An example of a parallel chip-firing game on $K_{4}$ with
$T=T_{4}=4$.

The matrix $M_{n}$ is extremely similar to the Laplacian matrix of $K_{n}$. In
fact, the matrix representation of System 1 is exactly the Laplacian.

##  Complete Bipartite Graphs

Let $\tau(a,\,b)$ be the maximum possible period of a parallel chip-firing
game on an orientation of $K_{a,\,b}$, with $a\leq b$. We show that
$\tau(a,\,b)$ is at least asymptotic to $\min(a,\,b)!$. We conjecture that
$\tau(a,\,b)$ is indeed asymptotic to $\min(a,\,b)!$. It holds that
$\tau(a,\,b)=O(a!)$. Let the partitions of the complete bipartite graph
$K_{a,\,a}$ be $L$ and $R$, where $|L|=|R|=a$. Let $v_{i}\in L$ if $i\equiv
1\pmod{2}$, and let $v_{i}\in R$ otherwise. We define the following
orientation of $K_{a,a}$ to be a useful orientation.

  * •

There exist edges directed from $v_{i}$ to $v_{i+1}$ for each $1\leq i<n$, and
there exists an edge directed from $v_{2a}$ to $v_{1}$.

  * •

There exist edges directed from $v_{1}$ to $v_{2i}$ for all $1\leq i<a$.

  * •

There exist edges directed from $v_{2j+1}$ to all vertices $v_{2k}$ with
$1\leq k<j$.

  * •

There exist edges directed from $v_{2j}$ to all vertices $v_{2k-1}$ with
$1<k<j$.

See Figure  for a diagram of a useful orientation of $K_{4,\,4}$.

1.5 [every node/.style=circle, draw, minimum size=6mm, inner sep=1mm, font=,
-¿, line cap=round, line join=round, thick] (1) at (0, 3) 1; (3) at (0, 2) 3;
(5) at (0, 1) 5; (7) at (0, 0) 7; (2) at (2, 3) 2; (4) at (2, 2) 4; (6) at (2,
1) 6; (8) at (2, 0) 8; [-¿] (1) – (2); [-¿] (2) – (3); [-¿] (3) – (4); [-¿]
(4) – (5); [-¿] (5) – (6); [-¿] (6) – (7); [-¿] (7) – (8); [-¿] (8) – (1);
[-¿] (1) – (4); [-¿] (1) – (6); [-¿] (6) – (3); [-¿] (8) – (3); [-¿] (8) –
(5); [-¿] (5) – (2); [-¿] (7) – (2); [-¿] (7) – (4);

Figure : A useful orientation of $K_{4,\,4}$.

The following lemma shows that vertex 4 fires the most in all parallel chip-
firing games on a useful orientation of $K_{a,\,a}$ with $a\geq 3$. For any
$a\geq 3$, all parallel chip-firing games on a useful orientation of
$K_{a,\,a}$ satisfy $f_{4}\geq f_{i}$ for all $1\leq i\leq 2a$. Since
$\deg^{+}v_{4}=1$, it must fire once in each period for every chip it receives
in that period. Note that $v_{4}$ receives chips from all vertices $v_{2i-1}$
with $1\leq i\leq a$ except $v_{5}$, so $f_{4}\geq f_{2i-1}$ for all $1\leq
i\leq a$. Also note that $\deg^{+}v_{3}=1$ and that $v_{3}$ receives chips
from all vertices $v_{2i}$ with $1\leq i\leq a$ and $i\neq 2$. Thus,
$f_{4}\geq f_{3}\geq f_{2i}$ for all $1\leq i\leq a$ and $i\neq 2$. Finally,
$\deg^{+}v_{2}=1$ and $v_{2}$ receives chips from $v_{5}$, so $f_{4}\geq
f_{3}\geq f_{2}\geq f_{5}$. Using vertex4, we provide a factorial lower bound
for the convergent period of a useful $K_{a,\,a}$ orientation. For a useful
orientation of $K_{a,\,a}$, the expression
$\lim_{c\rightarrow\infty}T_{c}(K_{a,\,a})$ increases at least factorially
with $a$. Since the number of chips on each vertex does not change after a
full period, the following equations must hold.

$\aligned&f_{2a}=(a-1)f_{1}\\\ f_{1}+f_{5}+f_{7}+\ldots+f_{2a-1}=f_{2}\\\
f_{1}+f_{2j-1}+f_{2j+3}+\ldots+f_{2a-1}=(j-1)f_{2j}\quad\forall j:\;2\leq
j\leq a-2\\\
f_{2j-2}+f_{2j+2}+f_{2j+4}+\ldots+f_{2a}=(j-1)f_{2j-1}\quad\forall j:\;2\leq
j\leq a-1\\\ f_{1}+f_{2a-3}=(a-2)f_{2a-2}\\\ f_{2a-2}=(a-1)f_{2a-1}\\\
f_{2a-1}=(a-1)f_{2a}$ (5)

Since the number of chips in the whole game is fixed, System 5 cannot be
independent. We arbitrarily drop the first equation, leaving a system of
$2a-1$ independent equations in $2a$ variables. Let $T_{a}$ denote the
smallest positive integer value of $f_{4}$ which satisfies System 5 for a
particular value of $a$. We claim that, for $a\geq 3$,

$T_{a}>T_{a-1}(a-1).$

By definition, each of $f_{1},f_{2},\ldots,\,f_{2a}\geq 0.$ By lost, if any
$f_{i}=0$, we have that $T=1$ and that all vertices are forever passive.
However, if $c$ is very large, some vertex must fire on every round. Thus, we
further constrain $f_{i}>0$ for all $i$. Note that for any solution
$(f_{1},f_{2},\ldots,\,f_{2a})$, it must be true that $(\lambda f_{1},\lambda
f_{2},\ldots,\,\lambda f_{2a})$ is also a solution. Thus, without loss of
generality, we set $f_{1}=1.$ System 5 has now been reduced to

$\aligned&f_{2}-f_{5}-f_{7}-\ldots-f_{2a-1}=1\\\
-f_{2j-1}+(j-1)f_{2j}-f_{2j+3}-\ldots-f_{2a-1}=1\quad\forall j:\;2\leq j\leq
a-2\\\ -f_{2j-2}+(j-1)f_{2j-1}-f_{2j+2}-f_{2j+4}-\ldots-f_{2a}=0\quad\forall
j:\;2\leq j\leq a-1\\\ -f_{2a-3}+(a-2)f_{2a-2}=1\\\
-f_{2a-2}+(a-1)f_{2a-1}=0\\\ -f_{2a-1}+(a-1)f_{2a}=0.$ (6)

System 6 can be written as the following augmented matrix $M_{a}=(A|B)$, where
$A$ is an $(2a-1)\times(2a-1)$ matrix and $B$ is a $(2a-1)$-dimensional column
vector. We denote the element in the $i$th row and $j$th column as
$M_{a}^{i,\,j}$, where $1\leq i\leq 2a-1$ and $1\leq j\leq 2a$.

$M_{a}=\amatrix{10}1&00-10-1\ldots 0-101\\\ -1100-10\ldots-10-10\\\
0-1100-1\ldots 0-101\\\ 00-1200\ldots-10-10\\\ 000-120\ldots 0-101\\\
0000-13\ldots-10-10\\\ \vdots\vdots\\\ 000000\ldots a-2001\\\
000000\ldots-1a-100\\\ 000000\ldots 0-1a-10$ (7)

We now apply Gauss-Jordan Elimination to transform $M_{a}$ into reduced row
echelon form. In particular, we seek the value of $T_{a}=f_{4}$, which will be
the largest of the $f_{i}$ by vertex4. We denote the submatrix formed by
omitting the last row and column of $M_{a}$ as $M_{a}[-2]$. Note that
$M_{a-1}$ is extremely similar to $M_{a}[-2]$. In particular, $M_{a}[-2]$
differs from $M_{a-1}$ only in the negation of the $(2a-2)$th column. We
proceed with the following algorithm. We first add each row to the one beneath
it, in order. Thus, we cancel the leading $-1$ terms in each row, putting
$M_{a}$ into row echelon form.

$M_{a}=\amatrix{10}1&00-10-1\ldots 0-101\\\ 010-1-1-1\ldots-1-1-11\\\
001-1-1-2\ldots-1-2-12\\\ 0001-1-2\ldots-2-2-22\\\ 00001-2\ldots-2-3-23\\\
000001\ldots-3-3-33\\\ \vdots\vdots\\\ 000000\ldots 1-(a-2)-(a-2)a-1\\\
000000\ldots 01-(a-2)a-1\\\ 000000\ldots 001a-1$ (8)

We next apply the rest of the Gauss-Jordan Elimination algorithm for $M_{a-1}$
on $M_{a}$ to create a new matrix $M_{a}^{\prime}$. The final three columns of
$M_{a}^{\prime}$ will have arbitrary constants, but the rest of
$M_{a}^{\prime}$ will be in reduced row echelon form. Recall that
$M_{a}^{\prime 3,\,2a-2}=-M_{a-1}^{3,\,2a-a}=-T_{a-1}$. To negate
$M_{a}^{\prime 3,\,2a-2}$, we note that $M_{a}^{\prime 2a-2,\,2a-2}=1$. Thus,
we add the $(2a-2)$th row to the second row $T_{a-1}$ times, putting the
$(2a-2)$th column into reduced row echelon form. Note that $M_{a}^{\prime
2a-2,\,2a}=\sum_{i=1}^{2a-2}M_{a}^{i,\,2a}=\deg^{+}v_{1}=a-1$. Therefore, we
add $(a-1)T_{a-1}$ to $M_{a}^{\prime 3,\,2a}$. Note that the only remaining
element of the third row of $M_{a}^{\prime}$ that is not yet in reduced row
echelon form is $M_{a}^{\prime 3,\,2a-1}$. We have $M_{a}^{\prime
3,\,2a-1}=-x$ for some $x$, which we will negate by adding the $(2a-1)$th row
to the third row $x$ times. Importantly, the value of $M_{a}^{\prime 3,\,2a}$
is not reduced. In other words, if $M_{a}^{\prime\prime}$ is the resultant
matrix after the completed Gauss-Jordan Elimination
algorithm,$M_{a}^{\prime\prime 3,\,2a}>M_{a}^{\prime 3,\,2a}>(a-1)T_{a-1}.$
Since $f_{3}=T_{a}=M_{a}^{\prime\prime 3,\,2a}$, we have

$T_{a}>(a-1)T_{a-1}.$

Recall that $f_{4}$ is the number of times that $v_{4}$ fires in the first $T$
rounds. Thus, $T\geq f_{4}$. Since $f_{4}\geq T_{a}$, we have that $T\geq
T_{a}$ for all parallel chip-firing games with enough chips. Thus,

$\lim_{c\to\infty}T_{c}(K_{a,\,a})\geq T_{a}\geq\Omega(a!).$

Matrix 7 is again extremely similar to the Laplacian matrix of $K_{a,\,a}$. As
in the case of the complete graph, the matrix representation of 5 is exactly
the Laplacian. As $c$ goes to infinity, $T_{c}(K_{a,\,b})$ increases at least
factorially with $\min(a,\,b)$. Let $a\leq b$. We consider an orientation of
$K_{a,\,b}$ with $K_{a,\,a}$ as the only sink component. By nonsink, no
vertices outside $K_{a,\,a}$ fire in a parallel chip-firing game on
$K_{a,\,b}$. Thus, any $T$ that can be the period of a parallel chip-firing
game on $K_{a,\,a}$ can be the period of a parallel chip-firing game on
$K_{a,\,b}$.