# The fast non-ferric kicker system for the Muon $g-2$ Experiment at Fermilab

A.P. Schreckenberger1<EMAIL_ADDRESS>Department of Physics, Boston
University, Boston, MA 02215, USA  Department of Physics, University of
Illinois, Urbana, IL 61801, USA  D. Allspach  D. Barak  J. Bohn  C. Bradford
Fermi National Accelerator Laboratory, Batavia, IL 60510, USA  D. Cauz  INFN,
Sezione di Trieste e G.C. di Udine, Trieste, Italy  Università di Udine,
Udine, Italy  S. P. Chang  Department of Physics, KAIST, Daejeon 34141,
Republic of Korea  Center for Axion and Precision Physics Research, IBS,
Daejeon 34051, Republic of Korea  A. Chapelain  CLASSE, Cornell University,
Ithaca, NY 14853, USA  S. Chappa  S. Charity  R. Chislett  Department of
Physics and Astronomy, University College London, London WC1E 6BT, UK  J.
Esquivel  C. Ferrari  A. Fioretti  C. Gabbanini  INFN, Sezione di Pisa, Pisa,
Italy  Istituto Nazionale di Ottica, CNR-INO, S.S. A. Gozzini, via Moruzzi 1,
56124 Pisa, Italy  M. D. Galati  L. Gibbons  J. L. Holzbauer  Department of
Physics and Astronomy, University of Mississippi, University, MS 38677, USA
M. Incagli  C. Jensen  J. Kaspar  CENPA, University of Washington, Seattle, WA
98195, USA  D. Kawall  Department of Physics, University of Massachusetts,
Amherst, MA 01003, USA  A. Keshavarzi  Department of Physics and Astronomy,
University of Manchester, Manchester M13 9PL, UK  D. S. Kessler  B. Kiburg  G.
Krafczyk  R. Madrak  A. A. Mikhailichenko2, H. Nguyen  K. Overhage  S. Park
H. Pfeffer  C. C. Polly  M. Popovic  R. Rivera  B. L. Roberts  D. Rubin  Y. K.
Semertzidis  J. Stapleton  C. Stoughton  E. Voirin  D. Wolff

###### Abstract

We describe the installation, commissioning, and characterization of the new
injection kicker system in the Muon $g-2$ Experiment (E989) at Fermilab, which
makes a precision measurement of the muon magnetic anomaly. Three Blumlein
pulsers drive each of the 1.27-m-long non-ferric kicker magnets, which reside
in a storage ring vacuum (SRV) that is subjected to a 1.45 T magnetic field.
The new system has been redesigned relative to Muon $g-2$’s predecessor
experiment, and we present those details in this manuscript.

FERMILAB-PUB-21-069-AD-E

accepted by Nucl. Inst. and Meth. in Phys. Res. Sec. A

11footnotetext: Corresponding author. Email<EMAIL_ADDRESS>Deceased.

##  1 Overview

Based at Fermilab (FNAL), the Muon $g-2$ Experiment (E989) will measure the
muon magnetic anomaly ($a_{\mu}$) to high precision [1]. E989 is the successor
experiment to Brookhaven National Laboratory (BNL) E821, which measured
$a_{\mu}$ to a precision of 540 parts-per-billion (ppb) using the same 1.45 T
storage ring magnet [2]. The BNL experiment is described in Refs. [3, 4, 5]
where an apparent discrepancy between the theoretically calculated and
experimentally measured values of the magnetic anomaly was observed. To
clarify this discrepancy between the Standard Model prediction and the BNL
E821 result, the FNAL experiment will make a more precise measurement of
$a_{\mu}$ using a larger data set and improved systematic treatment. The first
result from the Muon $g-2$ Experiment is presented in detail in Refs. [6, 7,
8, 9]. This paper describes the design, operation, and characterization of the
new non-ferric fast kicker magnets that deflect the incident muon beam onto
storable orbits within the ring.

Figure 1: An idealized sketch of the beam geometry before and after the kicker
pulse. $x_{c}$ is the 77 mm displacement of the inflector aperture relative to
the central radius of the muon storage region, R=7112 mm. $\beta\approx$ 10.8
mrad. Modified from [10].

The Fermilab accelerator complex produces a polarized muon beam for the
experiment’s consumption by colliding protons with a nickel-based target.
Pions are created in these collisions that eventually decay to produce the
$\mu^{+}$ particles that the experiment uses for physics data collection. The
muons enter the storage ring vacuum (SRV) through a superconducting inflector
magnet [11] that is aligned to the tangent of the ring. The inflector’s
interior aperture is displaced 77 mm from the central radius of the storage
region. Consequently, the injected muons cannot be stored without an
appropriate kick.

A series of three 1.27-m-long non-ferric aluminum kicker electromagnets are
placed a quarter of the betatron wavelength from the inflector to reduce the
impact of the ring magnetic field when they are pulsed with $3-4$ kA. The
result of the localized perturbation moves muons onto stable orbits that
facilitate a measurement of $a_{\mu}$. Likewise, the use of non-ferric
materials was explicitly selected to not affect the $a_{\mu}$ measurement,
which is sensitive to magnetic field sources. Beam arrives in a bunch train of
120-ns pulses at an instantaneous rate of 100 Hz (11.4 Hz average), and each
bunch propagates around the ring every 149.2 ns. The kicker current must
support the full duration of each beam pulse and ideally ceases before the
muons make one full revolution of the ring. To address these timing
specifications, Blumlein pulse forming networks (PFNs) were selected to drive
the kicker currents at FNAL. This pulsing system and the kicker plate geometry
provide fundamental differences from the BNL E821 infrastructure.

Figure 1 summarizes the effect of the kicker magnets on the beam geometry.
Muons enter the ring through the inflector at $\theta=0$ on a trajectory that
would not facilitate beam storage. At around $\theta=\pi/2$, muons cross the
central storage radius at an angle $\beta\approx$ 10.8 mrad. With a roughly
1.1 kG-m integrated field perturbation, the kicker magnets compensate for the
$x_{c}$ displacement and beam is stored. The following sections will expand
upon this simple picture by describing the various mechanical and electrical
components of the kicker system. In addition, we describe upgrades where
appropriate to emphasize the differences between the Run-1 (Jan. to Jul. 2018)
and Run-2 (started Mar. 2019) assemblies. The system will also be
characterized, and both the impact on muon storage and the kicker magnetic
field will be quantified. The concluding studies on beam dynamics shown in
Subsection 3.4 stress the importance of the upgrade and characterization
efforts described in this manuscript. We demonstrate the relationship between
kicker system performance, muon storage, and stored-beam quality, all of which
motivated the design choices made by the experiment.

##  2 Kicker Design

Figure 2: Flow diagram of the primary kicker subsystems. From left to right,
the diagram outlines the generation of the kicker pulses from their origins in
the charging supplies to when current traverses the kicker plates. The
relevant subsections for each component are labeled in the curved rectangular
boxes.

In addition to the three Blumlein PFNs, the kicker magnets rely upon a set of
peripheral subsystems to generate the magnetic field perturbation needed to
store muon bunches in the SRV. These subsystems include: HV power supplies,
capacitor banks, transformers, silicon oil supplies, castor oil pumps,
resistive loads, Fluorinert chillers, low-conductivity water (LCW)
connections, SRV feedthroughs, controls, and trigger timing modules. The
following four subsections will mechanically and electrically detail the major
elements of the FNAL kicker. The order of the subsections is arranged to
follow the kicker pulse chronologically from the initial charge until the
current traverses the plates in the SRV. Figure 2 shows a flow diagram of the
main kicker subsystems following that same chronology from left to right, and
the associated subsections detailing each component are labeled.

###  2.1 HV Supplies, Capacitor Banks, and Transformers

A suite of HV supplies, capacitor banks, and three transformers creates the
backbone of the kicker charging circuit. Prior to the start of Run-2, three
Lambda 802L supplies initiated the charging of the kicker system. In the
current configuration, seven Lumina Power Inc. CCHP-6000 power supplies, rated
for 1 kV (6 kW) operations, were installed to increase the peak charging
current over the 802L units. Each of the kicker magnets has an associated
capacitor bank, which receives charge from two Lumina supplies connected in
parallel. The seventh supply serves as a hot spare. Upon receiving the
charging signal from the controls rack, the HV supplies deliver current to the
capacitor banks for a period of approximately 9 ms.

The three capacitor banks consist of five 50 $\mu$F Genteq 97F5211S
capacitors. The elements are connected in parallel to produce 250 $\mu$F of
total capacitance, though the option to expand up to 500 $\mu$F is available.
After the banks are charged, another control signal triggers a silicon
controlled rectifier (SCR) to permit the flow of current from the selected
bank to the allocated single-phase transformer.

Upon design for Run-1, the expected gains of the transformers were 1:85. Each
transformer is submerged in silicon oil and housed in an aluminum vessel. In
each vessel, we mounted a capacitive divider that reads out the voltage from
the secondary winding. It is from these monitors that we obtain the voltages
placed on the PFNs, which are critical measurements for calibrating the
amplitude of the kicker pulses.

Beamline studies conducted during Run-1 repeatedly indicated that the kickers
were operating at a lower voltage than the initial calibration suggested. An
audit of the system revealed two transformer system characteristics that
differed from specifications: 1) the gains of the transformers were not 1:85,
and 2) the capacitive dividers had gains that differed by as much as 43.3$\%$
from the expected ratio of 5000:1.

The final calibrations presented in Subsection 3.1 mitigate the variances due
to divider behavior and cable capacitance by mapping the responses of the
capacitive dividers in the transformer vessels to the response of a calibrated
resistive divider. Additional testing performed just prior to the start of
Run-2 verified the earlier findings. We summarize the results of these
transformer tests in Table 1.

Table 1: Voltage and inductance measurements of the K2 and K3 Blumlein charging transformers. These observations verified the calibration effort that is described in Subsection 3.1 by directly obtaining the gain ratios of the transformers and their capacitive divider monitors. Diagnostic | K2 Test 1 | K2 Test 2 | K3 Test 1
---|---|---|---
Leakage inductance | 183$\mu$H | 180$\mu$H | 230$\mu$H
Primary inductance | 100mH | 100mH | 100mH
Input voltage | 11.3Vpp | 23.4Vpp | 22.2Vpp
Output voltage | 0.99kVpp | 2.12kVpp | 2.08kVpp
Ratio | 1:87.8 | 1:90.6 | 1:93.7
Secondary input voltage | 0.99kVpp | 2.20kVpp | 2.02kVpp
Secondary output voltage | 0.35Vpp | 0.74Vpp | 0.58Vpp
Capacitive divider ratio | 2834.3:1 | 2973.0:1 | 3458.9:1

###  2.2 Blumlein Pulsers

Triaxial Blumlein pulsers were introduced in the Muon $g-2$ Experiment as a
major improvement over the underdamped RLC PFN utilized in BNL E821. Invented
in 1937 by Alan Blumlein, Blumlein pulsers provide several advantages when
compared to a standard transmission line [12, 13]. As demonstrated by the
illustration of a Blumlein pulser in Fig. 3, the switching element in this
type of PFN can be grounded, which simplifies triggering electronics and
offers an additional layer of safety. Another advantage of the Blumlein PFN is
that this system yields a pulse with a peak voltage equal to that of the power
supply output. A standard transmission line would require double the power
supply voltage to produce the same voltage drop across the load. Furthermore,
with a properly matched load impedance, reflections from the unterminated
conductors are exploited in Blumleins to promptly end the electrical pulse. In
BNL E821, the RLC PFN generated a primary pulse that exceeded 450 ns and
engulfed the cyclotron structure of the muon beam [1, 10]. We selected the
Blumlein PFN design specifically to improve the timing characteristics of the
kicker pulse, the duration of which is shown in Eq. 1 (where $L$ is the length
of the Blumlein).

$\tau=2L\frac{\sqrt{\mu_{r}\varepsilon_{r}}}{c}.$ (1)

Figure 3: The simplest circuit diagram for a Blumlein pulser using series
coaxial lines. The load, Z, is attached on the ground conductor at the
midpoint of an unterminated line with length 2D.

The triaxial Blumelin configuration used by the experiment can be represented
through a topological modification of the circuit shown in Fig. 3. The process
is illustrated in Fig. 6, and an expanded detailing of the Blumlein assembly
is provided in Fig. 6. Each 9.35-m-long Blumlein consists of three concentric
cylindrical aluminum conductors (small, medium, large) that are approximately
3 mm thick. The characteristic impedance of the Blumlein ($Z_{0}^{B}$) is set
by the geometry of the conductors, specifically the conductor radii, and the
relative permittivity ($\varepsilon_{r}$) of any dielectric medium used in the
system. The Muon $g-2$ Blumleins are filled with castor oil, which has a
measured relative permittivity of 4.57. The castor oil also serves as a
cooling medium for the PFN switches. The fluid is circulated through the
Blumleins by pumps connected to each pulser with copper piping.

Figure 4: Topological modification of the series coaxial Blumlein model shown
in Fig. 3 to the triaxial configuration used at Fermilab. V represents the
charging voltage of the Blumlein. K is the switching element, and Z is the
load impedance.

Figure 5: Representation of the triaxial Blumlein assembly with the kicker
magnet, M, and resistive load, Z. The load impedance is connected to the small
and large Blumlein conductors through a coaxial cable. The medium Blumlein
conductor is charged to voltage V. The switching element, K, is a thyratron
that facilitates the short between the medium and large conductors.

Figure 6: Top: Picture of all three 9.35-m-long Blumlein PFNs installed in the
Muon $g-2$ experimental hall. Bottom: Illustration of the Blumlein assembly
with all three conductors. The attached CX1725X thyratron is visible, and the
connection of the 5 k$\Omega$ resistor is depicted immediately to the left of
the thyratron. At the left of the diagram, a prototype output cable assembly
is drawn. A modified design with parallel cable sockets, capable of seating up
to four coaxial cables, is currently used.

Equation 2 is the expression for the characteristic impedance of the
Blumleins.

$Z_{0}^{B}=\frac{60\Omega}{\sqrt{\varepsilon_{r}}}\ln{\frac{R}{r}},$ (2)

where $\varepsilon_{r}=4.57$, $R$ is the inner radius of the larger conductor
(either $R=R_{large}^{inner}=71.12mm$ or $R=R_{medium}^{inner}=53.72mm$), and
$r$ is the outer radius of the smaller conductor (either
$r=r_{medium}^{outer}=56.77mm$ or $r=r_{small}^{outer}=43.18mm$). From these
possibilities, Equations 3$-$4 are generated, yielding the following impedance
calculations. In the ideal case, the Blumlein impedance would be equal to half
of the load impedance ($Z/2\approx 6.25\Omega$).

$\displaystyle Z_{0}^{Bsm}$ $\displaystyle=$
$\displaystyle\frac{60\Omega}{\sqrt{4.57}}\ln{\frac{53.72}{43.18}}=6.13\Omega,$
(3) $\displaystyle Z_{0}^{Bml}$ $\displaystyle=$
$\displaystyle\frac{60\Omega}{\sqrt{4.57}}\ln{\frac{71.12}{56.77}}=6.33\Omega.$
(4)

On one end of each Blumlein PFN, a thyratron is seated as the switching
element between the medium and large conductors. The output cable from one
transformer connects on this end of the Blumlein, as seen in Fig. 6. The
ground of the transformer cable attaches directly to the large conductor of
the Blumlein while the high voltage core connects to the medium Blumlein
conductor through a 5 k$\Omega$ resistor.

When a thyratron shorts the conductors of a Blumlein, it initiates the current
pulse that drives the kicker magnet. Given the voltage and impedance
characteristics of the kicker system, a peak current exceeding 4 kA is
achievable. We installed E2V Technologies CX1725X thyratrons on the Blumleins.
These thyratrons hold off peak anode voltages up to 70 kV. They are capable of
conducting currents up to 15 kA, and their maximum allowed repetition rate is
2 kHz, which exceeds the instantaneous delivery rate of the muon beam by a
factor of 20. The CX1725X model uses a tetrode layout, meaning that the
triggering mechanism is controlled via two grids. One grid receives a $10-25$
A current signal to ionize the deuterium gas inside the thyratron. The other
grid is held at a -0.2 kV potential to keep the thyratron in a non-conductive,
inhibited state. When the discharge trigger is received from the timing
system, a $+$2.0 kV pulse is delivered to the same grid as the inhibit. This
moves the thyratron into its conductive state.

With all of these elements described, we can now construct the illustration in
Fig. 6. Here, the diagram depicts the design of all three Blumlein conductors,
the connection of the thyratron, the attachment of the input 5 k$\Omega$
resistor, and a prototype output cable assembly. Also pictured in this figure
is an image of the three Blumlein PFNs mounted on the wall of the experimental
hall.

###  2.3 Resistive Loads and SRV Feedthroughs

The Blumlein PFNs deliver pulses to the kicker magnets through electrically-
conducting, SRV feedthroughs. Outside the vacuum region, resistive loads
electrically connected to these elements facilitate the necessary current to
produce the desired kicker magnetic field.

Two load designs were used in the experiment. The first iteration was used
throughout the commissioning phase and Run-1. A schematic of the resistive
load topology is included in Fig. 8. In this design, four Times Microwave
AA5966 coaxial cables from the Blumlein PFN were independently connected to 50
$\Omega$ resistors housed in an aluminum containment vessel. The resistors
were also attached to the central, high voltage terminal of the vacuum
feedthrough, which mates to the inner kicker plate through an aluminum lead.
The return path follows the outer kicker plate until a lead carries the
current through the load containment vessel back to the braids of the coaxial
cables. The idealized result of this arrangement is the production of a 12.5
$\Omega$ load that is tuned to the impedance of the Blumlein pulser.

We developed the second iteration of the load to improve the strength of the
kicker field for Run-2. Here, the AA5966 central conductors connect to a
common bus that securely holds eight 100 $\Omega$ resistors in parallel with
four RC elements, each consisting of a 50 $\Omega$ resistor in series with a
0.9 nF capacitor. Another bus connects the load to the high voltage input of
the vacuum feedthrough. The Run-2 assemblies also possess capacitive dividers
that serve as another diagnostic tool. A schematic of this load topology is
included in Fig. 8.

We observe both electrical and mechanical improvements in the new load design.
By using RC components, we increase the maximum amplitude of the kicker magnet
current by roughly 10$\%$. Consequently, the kicker system can achieve the
same peak field perturbation at a reduced Blumlein charging voltage. This
effect can be understood through a basic knowledge of capacitor behavior. At
some time $t_{0}$, the capacitors act like closed conductors, which reduces
the impedance to 6.25 $\Omega$. As the capacitors reach $Q_{max}$, the load
resistance asymptotically approaches 12.5 $\Omega$.

Figure 7: Schematic of the resistive load topology used in Run-1. This design
consisted of independent cable-resistor connections to form a 12.5 $\Omega$
element. B represents a cable from the Blumlein PFN. Each R is a 50 $\Omega$
resistor, and F is the high voltage terminal of the vacuum feedthrough.

Figure 8: Schematic of the resistive load topology used in Run-2. Each
Blumlein cable, B, connects to a common bus in this design. P represents eight
100 $\Omega$ resistors connected in parallel. R represents a 50 $\Omega$
resistor, and C is a 0.9 nF capacitor. The P and RC elements are installed in
two concentric rings to minimize inductance. F is the high voltage terminal of
the vacuum feedthrough. Figure 9: SPICE simulation of the kicker magnet
current at approximately 30 kV operation. We show current waveforms for both
of the resistive load topologies.

Using SPICE [14], an open-source circuit simulator, we modeled the kicker
system as lump elements and produced the comparison of the resistive load
topologies displayed in Fig. 9. Here, we simulate that the peak amplitude of
the current in the kicker magnet increases by 11.6$\%$, which agrees with our
initial estimate.

Figure 10: Technical design drawing of the Run-2 load assembly [15]. Element
19 represents the eight 100 $\Omega$ resistors. Elements 20 and 21 comprise
the RC component of the newly developed topology. The connection to the SRV
feedthrough is displayed in the leftmost portion of the image (Elements
10$-$13, 15). The sockets for the input cables from the Blumlein PFN are shown
in the right side of the image (Elements 2, 4$-$6). Figure 11: Technical
design drawing of the SRV HV feedthrough [16]. The HV conductor passes through
a central aperture where it connects to the resistive load on the top side of
the drawing and the SRV lead on the bottom side of the drawing. Left: Exterior
dimensions of the Macor ceramic. Right: Interior dimensions of the Macor
feedthrough. The ceramic is hatched.

A detailed design drawing of the Run-2 resistive load is provided in Fig. 10.
The RC components described in this section are represented by Elements
19$-$21 in the figure. In both load designs, the interiors of the housing
vessels were filled with Fluorinert that served as a cooling fluid. Attached
plumbing facilitated the flow of Fluorinert to chiller pumps located outside
the magnetic storage ring. Considering the shape of the kicker current pulse
and the accelerator duty cycle, the resistive loads will emit approximately
300 W. Each kicker assembly has a dedicated chiller that provides 200 W of
cooling power. The additional heat load is dissipated through radiative blocks
on the aluminum housings that are actively cooled with flowing low-
conductivity water. We record measurements of the Fluorinert temperature in
each of the chiller units, which yield feedback regarding the status of the
loads. At nominal operation, we observe temperatures ranging from 22∘C$-$25∘C
from the readout of these sensors. We installed additional failsafe mechanisms
directly onto each of the load housings. If the temperature exceeds 62.8∘C at
a resistive load, temperature-sensitive switches provide a veto trigger to
prevent further pulsing.

The SRV feedthrough component of the assembly consists of a central HV
conductor shrouded by a Macor ceramic insulator. The vacuum is sealed using
two Viton o-rings seated between the insulator and the vacuum flange. At
various times, we used indium seals in place of one of the o-rings, but the
Viton-only configuration met the mechanical and electrical specifications of
the experiment. We include the technical design drawings of the Macor vacuum
feedthrough in Fig. 11.

###  2.4 Kicker Magnets

Figure 12: Elevation view of the SRV in the kicker region. We superimpose
projections of the E821 kicker plates, the new kicker plates, the trolley
chassis, and the Macor rails used by the trolley. The trolley requires
clearance to pass by the plates to map the magnetic field in the muon storage
region, and the kickers are electrically disabled when the trolley is
deployed.

Figure 13: Elevation view of the SRV in the kicker region at FNAL. The Run-1
ceramic standoffs (tan), aluminum brackets (black), and kicker plates (cyan)
are depicted along with radial and angular dimensions. The trolley rails
(green) and the outline of the trolley shape (pink) are also illustrated.

Figure 14: OPERA simulation of the kicker-induced magnetic field inside a
symmetric (rectangular) vacuum chamber. We display the By component of the
field at y$=$0 in the beam storage region. Figure 15: Isometric diagram of the
SRV feedthrough, high voltage leads, and upstream end of the kicker magnet.
This depiction represents the system prior to the start of Run-1. Aluminum
sleds and Teflon sockets replaced the prototype socket shown. The beam storage
region is displayed at left. We illustrate the kicker plates, current return,
and support structures. The pictured ceramic standoffs are still used.
However, we implemented a fluted design to opportunistically address breakdown
points.

The kicker magnets each consist of a pair of curved aluminum plates that are
1.27 meters in length. Here, we exhibit another design difference between the
E821 and E989 kicker systems. From the profile presented in Fig. 14, we
illustrate the changes to the kicker region of the SRV with the notable
difference coming from the geometry of the Muon $g-2$ Experiment’s kicker
magnets. By utilizing curved plates, the new kicker system is better optimized
for efficiency than the E821 counterpart. The curved profile produces a more
uniform magnetic field perturbation around the center of the beam storage
region. Initial simulations performed for the experiment’s technical design
report suggested that the modified plate geometry would produce a midplane
field of 65 G/kA. This translates to an 86$\%$ improvement in the
field:current efficiency when compared to the E821 plate geometry. Details
regarding the design specifications of the kicker system are described in
Chapter 12 of Ref. [1]. As part of the system characterization effort, a
measurement of the kicker-induced magnetic field was performed, and the
results of that analysis are included in Subsection 3.3.

To support the redesigned profile, the 0.737-mm-thick kicker plates are
suspended from vacuum chamber cages using riveted aluminum brackets and
ceramic standoffs. Additional riveted bridges, located at the ends of the
plates, serve as the electrical connections for every pair. Several challenges
arose during the implementation of the new geometry. The experiment uses a
collection of mobile nuclear magnetic resonance (NMR) probes to map the
magnetic field around the azimuth of the SRV. The probes are contained in a
chassis, and the device is commonly referred to as the trolley. In E821, the
kicker magnet assembly served as the rails for the trolley as it traversed the
kicker region, but at Fermilab, a dedicated rail is required. The radius of
curvature for the plates is 4.445 cm, which we determined by requiring
clearance for the trolley to pass through the kicker region. The diagram in
Fig. 14 depicts only the FNAL plate geometry. In this figure, we illustrate
the support framework of the kicker magnet, relevant cylindrical dimensions,
and the trolley.

Current from the Blumlein PFNs is delivered to the kicker magnets through high
voltage leads that connect to the SRV feedthrough. The isometric view in Fig.
15 illustrates the kicker magnet assembly at the high voltage attachment
point. This diagram shows the conditions prior to the start of Run-1. The SRV
feedthrough is shown on the right side of the figure along with the connection
to the leads. This design had to be modified because we determined that it was
insufficient during system commissioning. Charge would continually accumulate
on the insulators, and thus, the disk became a breakdown point of the system.
In Run-2, we utilized aluminum sleds and smaller Teflon sockets to guide the
electrical contacts of the leads. These devices have smoother surfaces and
greater clearances to suppress the potential for electrical breakdowns.

The beam storage region is visible on the left side of Fig. 15. The upstream
portion of the kicker plates is drawn with the current return, several
aluminum supports, and some ceramic standoffs. During Run-1, we recorded
breakdowns across the ceramic standoffs that held the high voltage (innermost)
plate. To address this issue, we implemented a fluted standoff design that
increases the effective distance between the aluminum bracket and the SRV
walls. Opportunistic installation of the modified standoffs has proven
effective at increasing the kicker system uptime. As with the previous
figures, the trolley rails and chassis outline are also depicted.

We performed a more recent simulation of the kicker-induced magnetic field
using OPERA Version 18R2 [17]. In addition to the dimensions listed in Fig.
14, the model included the value of $2.5\times 10^{7}$ S/m for the
conductivity of aluminum. We present the results of the simulation in Fig. 14,
where the computed midplane magnetic field is 62 G/kA. Additional studies
evaluated the impact of asymmetric (scalloped) vacuum chambers, which led to a
2.4$\%$ increase in the midplane magnetic field (63.4 G/kA), and we also
provide a preliminary measurement for the kicker plate inductance of 0.8
$\mu$H. This value does not include other sources of inductance in the system.
We present a more thorough study of the kicker system impedances in Subsection
3.2.

###  2.5 Controls and Timing

The controls and timing systems facilitate two fundamental interactions with
the kicker. Several instruments establish links between operators, monitoring
hardware, and the kicker charging circuit, whereas other elements parse
information from the injection accelerator to the kicker circuit and data
acquisition electronics.

To control the operating voltage of the kicker circuit, a LabJack T7 unit
produces three reference voltages, $(V^{K1}_{ref},V^{K2}_{ref},V^{K3}_{ref})$,
that set the output amplitudes of the HV supplies described in Subsection 2.1.
A thorough calibration that maps the reference voltages to Blumlein charging
voltages is described in the System Characterization section (Section 3) of
this manuscript.

Triggers for the kicker circuits are introduced via Clock and Commands Center
(CCC) logic. CMS FC7 boards (FPGA-based hardware for generic data acquisition
and control) produce optical signals that are utilized by the kicker control
instruments in the experimental hall.

The first signal initiates the charging of the capacitor banks and establishes
a negative bias on the thyratron grids to keep the switches in their open
states. With NIM logic, the charging signal is split to produce a delayed
signal for the capacitor bank SCRs, which commence the charging of the
Blumlein PFNs.

Additional optical signals provide discharge triggers for the three thyratron
control chassis. Upon receipt, 2 kV potentials are placed on the second
thyratron grids that force the switches into their conductive states. This
action drives current through the magnet load. Adjustment at the nanosecond
level facilitates the precise timing of the kicker pulses relative to the
injected beam. This additional flexibility aided beam storage optimization
efforts. The CCC also triggers both synchronous and asynchronous data
acquisition of the kicker monitoring devices, which are essential for quality
assurance, data analysis, and safety.

Furthermore, both the LabJack and CCC controls are connected to custom FPGA
hardware that registers anomalous electrical waveforms. Information pertaining
to this specific feature of the controls hardware is described in the
following subsection.

###  2.6 Monitors

Table 2: Monitoring systems for the Muon $g-2$ Experiment’s non-ferric kicker. Source and function are also included.  Monitor | Source | Function
---|---|---
PFN charging voltages | Capacitive dividers | Calibration
|  | Timing
Pickup coil voltages | Coaxial loops | Spark detection
|  | Pulse analysis
|  | Timing
Resistive load voltages | Capacitive dividers | Diagnostics
Hall audio | Microphones | Spark detection
|  | (PFN, cables)
Vacuum pressure | SRV gauges | Spark detection
Thyratron currents | HV supply | Operations
Castor oil flow | PFN flow gauges | Operations
Castor oil temp. | PFN temp. gauges | Operations
Fluorinert flow | Chillers | Operations
Fluorinert temp. | Chillers | Operations

Various monitors installed on components of the kicker system collect data
that serve both operational and analytical purposes. In particular, electrical
waveforms pertaining to the Blumlein PFN charging voltages, load voltages, and
lead currents produce information that affects analysis efforts and operation
decisions. Additional readouts are directly fed into the controls rack such
that triggers can be vetoed under unfavorable hardware conditions. Monitors of
fluid levels, flows, and temperatures, for example, act as binary-style
switches that provide input for operations. A comprehensive list of important
system monitors is included in Table 2. For the remainder of this subsection,
we focus on the monitoring of critical electrical waveforms and discuss their
contributions in greater detail.

During production running, we obtained essential measurements by monitoring
voltages from the PFN charging systems and a trio of pickup coils installed
near the SRV leads. Each custom-made, single-loop coil is constructed using
copper coaxial line with a Teflon dielectric, and the areas of the loops are
approximately 3.14 cm2. From the PFN voltage waveforms, we monitored
information regarding the trigger and timing controls, the health of the
Blumlein pulsers, and the kick amplitude. An example from the data is shown in
Fig. 16, where the PFN voltage is measured through a capacitive divider
attached to the secondary winding of the charging transformer. Key information
pertaining to the trigger system can be deduced through this waveform [18].
The SCR mentioned in Subsection 2.1 initiates the rising edge of the pulse,
and the thyratron entering a conductive state generates the sharp voltage drop
at around t = 0.52 ms. Any absence or time deviation in either of those
features indicates a potential issue with the trigger controls. Likewise,
variations from the typical shape displayed in Fig. 16 hint at potential
electrical breakdowns inside the Blumlein PFN.

We also analyzed and calibrated the strength of the kick amplitude by using
the magnitude of the voltage drop in the PFN charging voltage data. We present
a more rigorous description of the calibration analysis in the following
section.

Figure 16: Measurement of the Kicker 1 Blumlein PFN charging voltage taken
from the secondary winding of the K1 transformer. The sharp drop in voltage at
around t = 0.52 ms comes from the thyratron switching element entering the
conductive state. This transition is what generates current in the load.
Kicker 1 was operating at 47.75 kV at the time of observation, which can be
derived from the magnitude of the voltage drop induced by the thyratron
switch.  Figure 17: Readout voltage from a pickup coil located near the leads
of a kicker magnet. The shape of the waveform depends on the current pulsing
through the leads. We use this signal as the primary method to detect sparks,
as anomalous currents significantly change the pulse structure.  Figure 18: A
typical voltage measurement across one of the resistive loads. Data are read
out through a capacitive divider with a ratio of roughly 10000:1. The vertical
axis units are arbitrary as there is an additional scaling factor generated by
the data acquisition hardware.

The readout of the pickup coil offers another important suite of monitoring
information. At a fundamental level, the pickup coil data are related to the
current on the high voltage leads that connect an SRV feedthrough to the
plates of a kicker magnet. Subsequently, the pickup coil facilitates
measurements of the electrical pulses that actually perturbate the magnetic
field in the beam storage region. While we do not use this monitor to
calibrate the magnetic field strength due to miscellaneous noise sources, the
integrated pickup coil signal is related to the magnetic field generated by
the kicker system.

We utilize the pickup coil signals in the optimization of the kicker timing.
From this waveform, see Fig. 17, we determine when each kicker produces its
maximum field. This event corresponds with the first zero crossing in the
pickup coil pulse, which should ideally align with the arrival of the muon
beam. With this information, the trigger controls can be adjusted accordingly.
Fits performed on data collected during kicker timing scans indicate that a
tolerance of $\pm 5$ ns satisfies the experiment’s beam storage requirements.

The primary function of the pickup coils is to serve as spark (electrical
breakdown) monitors. Any spark in the kicker system impacts the current such
that a notable shape change in the waveform occurs. For example, a spark in
the SRV will produce a large distortion on the shape shown in Fig. 17, and
corresponding vacuum behavior is usually recorded. A breakdown in the Blumlein
PFN will not only distort the PFN charging voltage waveform, but it will also
significantly shift the timing of the pulse in the pickup coil data.
Consequently, we feed the outputs from the pickup coils into dedicated boards
in the controls rack. If the boards register an anomalous shape change in the
signal, they automatically veto subsequent charging to prevent damage to the
system.

The shape of the pickup coil waveforms, specifically the reflections in the
tail structure, generated additional analyses. The inductances of the kicker
magnets and HV leads produce the reflections seen in the readouts. This
finding poses a significant challenge for two reasons. First, we opted to use
the Blumlein PFNs given the timing specifications of the experiment, and the
load inductance affects that performance. Second, we required that eddy
currents generated by the kicker system become negligible less than 20 $\mu$s
after muon injection [1]. The presence of reflections in the current waveform
yields the possibility that the criterion is not being met. In the following
section, we describe measurements of the load impedances and magnetic field
perturbations to quantify the system characteristics and potential effects.

For Run-2, we installed a new monitor as part of the upgrade to the resistive
load. Capacitive dividers facilitate direct voltage measurements from the
loads during normal operations. From the waveform in Fig. 18, we can deduce if
there are any bad connections in the line, degraded cables, or malfunctioning
components through variations from the typical shape. Under normal conditions,
we expect to observe a negative voltage pulse that is generated by one of the
Blumlein PFNs.

##  3 System Characterization

During the commissioning phase, several fundamental measurements were made to
characterize the kicker system. These observations can be partitioned into
categories pertaining to calibration, assembly performance, field study, and
beam dynamics.

###  3.1 Calibration

The purpose of the kicker calibration is to relate the references from the
slow controls to the voltages on the Blumleins at the time the thyratrons
conduct ($t_{thyratron}$). The reference is commonly referred to as the set
point voltage ($V_{ref}$) and has a range of $0-10$ volts. As mentioned
previously, the voltage on a Blumlein is measured from a capacitive divider
attached to the secondary winding of the associated transformer. However, it
was determined that the gains and the behaviors of the capacitive dividers
differed from transformer to transformer.

We temporarily installed a calibrated, resistive divider onto the input cable
to the Blumlein PFN. At several values of $V_{ref}$, we recorded the voltage
waveforms at the calibrated divider and the transformer’s capacitive divider.
When the thyratrons conduct, a sharp voltage drop in these waveforms is
observed. Fitting the magnitude of $\Delta V$ at $t_{thyratron}$ as a function
of $V_{ref}$ yielded linear calibration formulas for all three kickers that
are valid between operating voltages of $10-60$ kV. Likewise, we obtained the
calibrations of the capacitive dividers through these measurements.

During Run-1, the electronics and software controls underwent multiple
iterations that altered the $V_{ref}$ scales and behaviors. In addition, we
observed nonlinearity in the kicker response when pulsing at progressively
higher voltages, and we determined that the Lambda HV supplies could not
uniformly charge the PFN when pulsing at the nominal beam repetition rate.
These findings motivated the redesign for Run-2 and generated a different
approach for calibrating the kicker system operating voltage during the first
production run. Through beam dynamics and muon storage studies, we estimate
the total kick strength was between 125 and 137 kV during the Run-1 data sets.

Equations 5$-$7 provide the Blumlein charging voltages (in kV units) for the
respective kicker magnet. These equations apply only to the system
configuration established at the start of Run-2.

$\displaystyle V^{II}_{K1}(V_{ref}^{K1})$ $\displaystyle=$ $\displaystyle
7.26[kV/V]\cdot V_{ref}^{K1}-0.59[kV],\hskip 5.0pt$ (5) $\displaystyle
V^{II}_{K2}(V_{ref}^{K2})$ $\displaystyle=$ $\displaystyle 6.78[kV/V]\cdot
V_{ref}^{K2}-0.38[kV],$ (6) $\displaystyle V^{II}_{K3}(V_{ref}^{K3})$
$\displaystyle=$ $\displaystyle 7.18[kV/V]\cdot V_{ref}^{K3}-0.81[kV].$ (7)

###  3.2 Assembly Performance

In conjunction with the Run-2 upgrades, we made considerable efforts to
quantify the performance of the refurbished and redesigned kicker assemblies.
Prior to the construction and installation of the new resistive loads, we
connected two of the Blumlein PFNs to a test load and recorded the electrical
current responses at several kicker operating voltages.

The test load consisted of four 50 $\Omega$ resistors in parallel and a T&M
Research Products Inc. A-4-0025 current viewing resistor (CVR). To account for
any parasitic capacitance and inductance, a Keysight impedance analyzer was
used to fit the load response to an RLC circuit model. We selected a simple
circuit topology in which the resistor and inductor are connected in series,
and the capacitor is connected in parallel. Consequently, we can represent the
test load in simulation software with a 12.76 $\Omega$ resistor, a 65 nH
inductor, and a $6-25$ pF capacitor. The used CVR model has a 2.5 $m\Omega$
resistance, 8 ns rise time, and 48 MHz bandpass.

We again utilized SPICE to model the Blumlein PFNs and test load as lump
elements. The simulation results are shown in Fig. 19 along with the data
collected when operating Kickers 1 and 2 at approximately 30 kV and 20 kV,
respectively. Space constraints in the experimental hall prohibited measuring
the test load response to Kicker 3.

The data agree reasonably well with the results from simulation. Uncertainties
in the analyzer RLC fit and the absence of lossy elements in the SPICE model
explain the small discrepancies between the data and simulation waveforms.
There is an additional uncertainty in the modeling of the thyratrons, and
their rise times have been tuned in the SPICE simulation to reflect some minor
differences in the thyratron behaviors. Another source of tuning comes from
inductive elements in the thyratron portion of the model circuit. We selected
thyratron inductances (20 nH for Kicker 1 and 15 nH for Kicker 2) by using the
amplitudes of the undershoots in the tails of the data waveforms as
constraints. We also used the results shown in Table 1 to characterize the
transformer attributes in the simulation.

The results presented in Fig. 19 established that the refurbished Blumlein
PFNs behave as expected, as no irregularities in the pulse shape were found.
We built further confidence by considering the pulse durations. The Blumlein
PFNs measure roughly 9.35 meters. Recalling Eq. 1, we compute an expected
pulse duration of 133.3 ns, which is typically ascribed to the full width at
half maximum (FWHM) of the waveform. Taking the $\tau_{FWHM}$ measurements
from data and simulation, we construct Table 3. The idealized, simulated, and
observed values agree to better than 5$\%$. Uncertainties in our measurements
of the relative permittivity of castor oil, and the presence of stray
inductances absent in Eq. 1, explain these differences.

Figure 19: Current through a test load when driven by the Kicker 1 and Kicker
2 Blumlein PFNs. Kicker 1 was operated at approximately 30 kV, and Kicker 2
was operated at approximately 20 kV. SPICE model simulations of the PFN and
test load produced the additional waveforms in the figure.

We also note the shape variation between the waveforms in Figs. 9 and 19. The
broader current pulse observed in Fig. 9 is produced by the larger inductance
of the kicker magnets compared to that of the test load. Reflections observed
in the pickup coil data (see Subsection 2.6) further corroborate this point,
which motivated the following characterization analysis.

Using a Keysight analyzer, we measured the impedances of the kicker load
circuits. These series circuits consist of the resistive loads, SRV
feedthroughs, and kicker magnets. Impedance data were recorded for driving
frequencies ranging from 1 kHz $-$ 5 MHz. We selected this frequency domain
based on the behavior of the Blumlein PFNs and the pickup coil readouts, from
which the rise time of the waveforms provide an upper bound for the relevant
frequencies.

Table 3: Blumlein pulse durations ($\tau$) taken at the FWHM of the current waveforms shown in Fig. 19. Comparison to idealized expectation is also shown. Kicker | $\tau_{ideal}$ | $\tau_{simulation}$ | $\tau_{data}$
---|---|---|---
K1 | 133.3ns | 137.6ns | 137.8ns
K2 | 133.3ns | 136.8ns | 139.0ns

Figure 20: The real impedance component of a series circuit consisting of the
resistive load, SRV feedthrough, and kicker magnet. A Keysight impedance
analyzer drove the circuit over a 1 kHz $-$ 5 MHz range for these
measurements. The response is shown for all three kickers.

Figure 21: The imaginary impedance component of a series circuit consisting of
the resistive load, SRV feedthrough, and kicker magnet. A Keysight impedance
analyzer drove the circuit over a 1 kHz $-$ 5 MHz range for these
measurements. The response is shown for all three kickers.

Figure 21 shows the real component of the impedance as measured in the three
kicker loads. First, we found that the impedance converged to 12.5 $\Omega$
$-$ 13.0 $\Omega$ at low frequencies. This behavior fits expectations when
considering the topology of the system. At low frequencies, the inductive
kicker magnet has a negligible impact on the total impedance, and the
capacitive network inside the resistive load will be fully charged at a time
$t<<T_{analyzer}$, where $T_{analyzer}$ indicates the period of the driver
oscillator. In this case, the parallel network of eight 100 $\Omega$ resistors
provides all of the load impedance, and the circuit resistance should converge
to 12.5 $\Omega$. The measurements in this frequency regime were consistent
with the resistor tolerances.

At higher frequencies in the 1 kHz $-$ 5 MHz domain, we observe a minimum in
the real component of the impedance that demonstrates the measurable effect of
the capacitive network in the Run-2 load topology. The minimum in all of the
measurements shown in Fig. 21 occurs at approximately 3 MHz. We employ
rudimentary circuit knowledge to understand the location of these minima.
Considering only the RC components of the load, the associated timing constant
of 45 ns corresponds to a frequency of 3.5 MHz. Likewise, the resonant
frequency of the LC components in the load is around 2.3 MHz. The actual
circuit consists of non-lump elements and parallel paths that complicate the
picture. However, these two frequencies straddle the minima at similar values
of Re(Z) and can be looked at as conservative bounds on the minima positions.
Furthermore, we observe $\approx$75-ns rise times for the two Blumlein pulsers
in Fig. 19, which corresponds to a similar underlying frequency of 3.$\bar{3}$
MHz. Overall, this specific measurement provides tangible evidence that the
goal of the load topology transition discussed in Subsection 2.3 has been
realized.

Figure 21 shows the imaginary component of the kicker load impedance in the 1
kHZ $-$ 5 MHz frequency domain. We observe notable behavior at higher
frequencies, where the inductance of the kicker magnet yields the dominant
contribution to the total impedance of the load. At 3 MHz, the measured load
inductance is between 1.4 $\mu$H and 1.5 $\mu$H. The characterization
corroborates several findings from simulation and data. The inductance of the
load is responsible for the distortion of the main current pulse when
comparing Figs. 9 and 19. Furthermore, the generated impedance mismatch with
the Blumlein PFN yields reflections that are clearly observed in the pickup
coil waveforms, Fig. 17. The experiment has made considerable efforts to
understand the impact of the reflections on the data, particularly under the
purview of beam simulation.

###  3.3 Field Study

As mentioned in Subsection 2.6, several monitors record electrical waveforms
from which the Muon $g-2$ Experiment can assess the status of the kicker
systems. These devices are crucial for operations, but they do not directly
measure the magnetic field. We constructed a Faraday magnetometer to evaluate
the kicker-induced field amplitude and waveform shape in a similar manner to
the procedure described in Section 4 of [10].

Figure 22: Diagram of the magnetometer apparatus once installed in the kicker
magnet region. From left to right, the principal features of this diagram
include the kicker magnet, the TGG cylindrical housing, the magnetometer arm,
the interface with the SRV, and the optical bench.

The magnetometer exploits the physics of the Faraday Effect and Malus’s Law.
The former of these principles, described by Eq. 8, states that certain
materials can change the polarization angle of linearly polarized light in the
presence of a parallel magnetic field. In addition to the field strength, the
change in polarization angle depends upon the Verdet constant ($v$), which
quantifies the strength of the Faraday Effect in the material, and the
distance light travels through that medium ($d$).

$\Delta\theta=vBd.$ (8)

In this analysis, we used a Terbium gallium garnet (TGG) crystal as the
Faraday rotator, which has a Verdet constant of approximately 126 [rad
$T^{-1}m^{-1}$] for 639 nm light. Laser light was set incident on a linear
polarizer to establish an initial polarization for the beam, which we define
here to be $\theta_{0}=0$. The light then passed into the SRV through the arm
of the magnetometer before a 45∘ mirror directed the laser light down into the
crystal cylinder, which was placed along the midplane of the kicker region. A
second mirror, situated at the bottom of the crystal housing, reflected the
laser light back through the crystal and out the magnetometer arm.

Upon exiting the SRV region, the laser light passed through a second polarizer
with $\theta_{p}=\pi/4$, where $\theta_{p}$ is relative to $\theta_{0}$. This
particular choice for $\theta_{p}$ comes from consideration of Malus’s Law,
Eq. 9.

$I=I_{0}\cos^{2}\theta,$ (9)

where the intensity of light passing through a polarizer is related to the
initial intensity ($I_{0}$) and the angle of polarization ($\theta$). From the
derivative, Eq. 10, we find that angular perturbations around
$\theta_{p}=\pi/4$ yield the largest changes in the measured intensity, and
thus, this configuration optimizes the signal sensitivity. Further adjustments
to $\theta_{p}$ were routinely applied. For example, data taking while the
1.45 T ring field was active necessitated a shift to account for the Faraday
rotation generated by the static field.

$\frac{dI}{d\theta}=-2I_{0}\cos\theta\sin\theta.$ (10)

Table 4: Magnetometer measurements of the midplane magnetic field in comparison to the expectation from simulation. The mean baseline voltage and mean detector voltage response are also included where relevant. Note that subsamples were subject to different gain settings, which explains the change in absolute scale observed between sources. Source |  $\overline{V}_{B=0}$ (mV) |  $\overline{\Delta V}$ (mV) | $|B_{midplane}|(Gauss)$
---|---|---|---
37-Hour | $767.0\pm 43.0$ | $109.3\pm 5.3$ | ${\bf 217.4\pm 16.3}$
Scope | $817.8\pm 81.7$ | $121.0\pm 3.4$ | ${\bf 225.6\pm 23.7}$
Simulation | $-$ | $-$ | ${\bf 217.1\pm 16.6}$

Combining Eqs. 8 and 9, we arrive at the expression shown in Eq. 11, where $l$
is effectively the length of the TGG crystal. The expression for the kicker-
induced magnetic field is derived and shown in Eq. 12, where we make use of
the $B=0$ condition to relate the field strength to the laser intensity
observed when the kicker magnet is off ($I_{B=0}$) and the maximum change in
intensity induced by the magnet pulse ($\Delta I$).

$\displaystyle I$ $\displaystyle=$ $\displaystyle
I_{0}\cos^{2}(\pi/4+\Delta\theta),$ (11) $\displaystyle=$
$\displaystyle\frac{I_{0}}{2}(1-\sin(2\Delta\theta)),$ $\displaystyle=$
$\displaystyle\frac{I_{0}}{2}(1-\sin(2vBd)),$ $\displaystyle=$
$\displaystyle\frac{I_{0}}{2}(1-\sin(4vBl)).$ $\displaystyle B$
$\displaystyle=$ $\displaystyle\frac{1}{4vl}\sin^{-1}(1-2I/I_{0}),$
$\displaystyle I(B=0)=I_{0}/2,$ $\displaystyle\ \therefore B$ $\displaystyle=$
$\displaystyle\frac{1}{4vl}\sin^{-1}(-\Delta I/I_{B=0}).$ (12)

We measure the signal through the use of a Thorlabs PDA10A(-EC) Si-amplified
fixed gain detector, which receives the laser light after it passes through
the second polarizer. The detector produces a voltage waveform that is
directly proportional to the input power, as expressed in Eq. 13.

$V=R(\lambda)G_{t}\frac{R_{load}}{R_{load}+R_{s}}P,$ (13)

where $R(\lambda)$ is the responsivity of the detector at a given laser
wavelength, $G_{t}$ is the transimpedance gain, $R_{load}$ and $R_{s}$
represent resistances added to the output of the detector, and $P$ is input
power [19]. The input power is the only dynamic variable in this equation.
Thus, we arrive at the final form for the expression of the kicker-induced
magnetic field as a function of the detector voltage by exploiting the
relationship in Eq. 14.

$\frac{\Delta V}{V_{B=0}}=\frac{\Delta P}{P_{B=0}}=\frac{\Delta I}{I_{B=0}}.$
(14) $\displaystyle\therefore B$ $\displaystyle=$
$\displaystyle\frac{1}{4vl}\sin^{-1}(-\Delta V/V_{B=0}).$ (15)

A diagram of the physical apparatus is included in Fig. 22. The optical bench
upon which the laser, polarizers, and detector were mounted is illustrated on
the right side of the image. The arm of the magnetometer, made of aluminum and
PEEK (Polyether ether ketone), passes into the SRV through a special flange.
On the left side of the figure, the profile of the kicker magnet is shown
along with the cylindrical TGG housing. We used a 1.3-cm-long crystal with a 5
mm diameter for this measurement.

With the magnetometer installed into the SRV and the TGG crystal inserted
between the kicker plates, we operated Kicker 1 at 47.75 kV. Additional data
were collected during the ramping of the main ring magnetic field, which
confirmed the calibration constant of the TGG crystal shown in Eq. 16.

$vl\approx(1.64\times 10^{-4}\pm 1.36\times 10^{-6})\frac{rad}{G}.$ (16)

Three data sets were collected for the magnetometer studies presented here
that include the crystal calibration, a small sample read out with a Tektronix
oscilloscope, and a 37-hour sample acquired through the primary DAQ.

The detector readout generated the voltage waveform in Fig. 23, which displays
$\Delta V(t)$ over the duration of the kicker pulse. $V_{B=0}$ was obtained by
measuring the DC-coupled voltage produced by the laser prior to kicker magnet
operation. Combining the circuit and field models, we predict that a $217.1\pm
16.6$ G midplane magnetic field will be produced by a Blumlein charging
voltage of 47.75 kV. Table 4 contains a summary of the study, from which we
conclude that the kicker-induced magnetic field produced by a 47.75 kV charge
was $217.4\pm 16.3$ G.

Figure 23: Example voltage waveform produced by the detector of the
magnetometer. We extract the peak values of $\Delta V$ over many of these
traces to acquire the $\overline{\Delta V}$ provided in Table 4.  Figure 24:
Flow diagram of the 2PD magnetometer assembly used to assess magnetic field
transients induced by kicker operation.

Figure 25: Kicker-induced magnetic field in mG (solid black) vs. time as
determined by the 2PD assembly with the static ring field at 0 T (left) and
1.45 T (right). We include the fit result from the fiber magnetometer analysis
(dashed red) in the time domain of the muon precession measurement. The dotted
blue lines correspond to a hypothetical 100-ppb shift on the measured value of
$\omega_{a}$ produced by a transient magnetic field.

Additional magnetometer analyses were conducted to assess the impact of
potential, unwanted kicker-induced magnetic fields at time scales much larger
than the kicker pulse duration. Both induced eddy currents and line
reflections could yield fields capable of perturbing the measurement of
$a_{\mu}$. First, a second photodiode and a polarizing beam-splitter were
added to the configuration described in this section to produce the 2PD
assembly. Through the acquisition of two orthogonally polarized data sets,
subtraction methods increased the sensitivity of the measurement in the time
domain $t>>149$ ns. However, we discovered that ambient sources introduced
mechanical vibrations in the magnetometer arm that hindered efforts to reach
the desired level of precision.

We also deployed a compact magnetometer apparatus that made use of optical
fibers, inline beam-splitters, and two TGG crystals to improve the sensitivity
to small fluctuations in the magnetic field. A full description of this
magnetometer assembly, its calibration and underlying sensitivities (to both
magnetic field sources and backgrounds), and the subsequent analysis are
provided in Section VIII of [8]. That study concluded that:

$\frac{\Delta\omega_{a}}{\omega_{a}}=[-27\pm 37]ppb,$ (17)

where the expression gives the fractional pull on the anomalous precession
frequency ($\omega_{a}$) driven by the non-optimal behavior of the kicker
pulse. Similar to the BNL E821 result in [10], the magnitude of this
systematic fits well within the FNAL experiment’s long-term precision goal of
140 ppb.

A supplementary result to the fiber magnetometer study emerged following
improvements made to the 2PD assembly. Two Thorlabs PD36A2 diodes, with a 15
kV/A transimpedance gain and a 1 MHz bandwidth, were installed onto the
aluminum optical bench and terminated into high-impedance waveform digitizers.
The flow diagram provided in Fig. 24 illustrates the setup of the 2PD
magnetometer assembly. To further assess and verify the impact of mechanical
vibrations on the system, we measured the lateral position and amplitude of
the incident light in a dedicated systematic test using an additional set of
Thorlabs PDP90A2 diodes.

By adjusting the input laser light polarization, half wave plate, and laser
optics, we optimized the 2PD assembly for operation in two static ring field
conditions: 0 T and 1.45 T. The sensitivity of the 2PD assembly is roughly
46.9 mV/mT, which can be compared to the more sensitive 68.3 mV/mT level
achieved by the fiber magnetometer. Figure 25 shows the kicker-induced field
measurements for each of the selected ring conditions. The dotted blue lines
at $\pm$15 mG correspond to a hypothetical $\pm$100-ppb shift on the measured
value of $\omega_{a}$, and the solid black lines show the measurements of the
kicker-induced fields. The primary kicker pulse drives the narrow, negative
spikes at t = 0 ms. The Blumlein charging current produces the positive
$130-160$ mG perturbations prior to t = 0, which have no impact on the
assessment of $a_{\mu}$ as they occur before the muon measurement period. We
observed considerably more activity in the tail structure when the ring field
operated at 1.45 T, which stemmed from Lorentz forces producing mechanical
vibrations in the kicker plates.

Within the 0.03$-$0.70-ms time window critical for the analysis of muon
precession, we measured the average field:

$B_{2PD}=[-5.1\pm 0.3_{stat}\pm 0.5_{cal}\pm 7.8_{vib}]mG,$ (18)

where the additional terms represent uncertainties due to statistics,
calibration, and system vibration. We assigned the vibration term by examining
the RMS of the signal $0.5-1.2$ ms before the kicker pulse. The fit result
from the fiber magnetometer analysis (dashed red lines) is also displayed in
Fig. 25. We found agreement between the two approaches, further indicating
that transient fields induced by the kicker system did not have a detrimental
impact on the Muon $g-2$ measurement.

###  3.4 Beam Dynamics

We conclude the system characterization with a brief study of beam dynamics.
As a critical ring system, the kicker magnets have a significant impact on the
physics data. In particular, there exist relationships between the kicker
performance, muon storage, and beam-related systematics that warrant
discussion.

From a statistics standpoint, we note the correlation between the amplitude of
the field perturbation (the kick strength) and the number of muons that are
successfully stored in the SRV. The significant redesign of the non-ferric
kickers between E821 and E989 was motivated by several factors. In addition to
the timing characteristics of the Blumlein PFNs, muon storage optimization
served as an engineering driver.

Following the expressions in Eq. 19 [1], the integrated field perturbation
($Bdl$) is directly proportional to the desired displacement angle ($\beta$).
Thus, an optimal perturbation exists for the geometry of the experiment’s beam
injection apparatus. As shown in Fig. 1, $\beta\approx 10.8$ mrad for both
E821 and E989. The produced value of $Bdl$ in the SRV is dependent upon the
charging voltage of the PFN, meaning that there exists an ideal operating
voltage about which the muon storage is maximized. Electrical breakdowns in
the E821 circuit prevented the experiment from reaching the storage maximum
during operation [1]. Consequently, the E989 kicker was designed to provide
better storage efficiency.

$\displaystyle\frac{d\vec{p}}{dt}$ $\displaystyle=$ $\displaystyle
q\vec{v}\times\vec{B},$ $\displaystyle d\vec{p}$ $\displaystyle=$
$\displaystyle qvB\hat{r}dt,$ $\displaystyle p\beta\hat{r}$ $\displaystyle=$
$\displaystyle qBdl\hat{r},$ $\displaystyle\therefore Bdl$ $\displaystyle=$
$\displaystyle\frac{p\beta}{q}.$ (19)

We include two strength scans to quantify the muon storage at various kicker
operating voltages. The first data collection occurred at the start of Run-2
once other ring systems were installed and updates to the data acquisition
software were functional. The second collection period occurred during Run-3
after the 4x50 $\Omega$ AA5966 cable configuration was replaced with a 3x32
$\Omega$ Dielectric Sciences 2264 cable arrangement.

Figure 26: Muon storage, presented in arbitrary units, as a function of the
total kick strength in kV. We performed this preliminary scan early in Run-2
as part of the conditioning phase.

Figure 27: Muon storage, in $e^{+}$ candidates/fill units, as a function of
the total kick strength in kV. We performed this scan during Run-3. The blue
points represent data collected over 20-minute-long runs at each voltage
setting. The red fit line and associated band show the muon storage predicted
by this data set. The black circles display the observed muon storage when we
average the storage metric over multiple 30-second-long samples. The
quadrupoles operated at 18.2 kV during this test, and the inflector magnet
current was 2763 A.

Figure 28: The mean equilibrium radii of the stored muon beam as a function of
the total kick strength. The data are represented by the blue points. A fit to
the data (red line) and associated uncertainty band are also included in the
figure. We find that the beam is centered by a total kick amplitude of
approximately 161 kV.

The results of the first study, displayed in Fig. 26, yielded the expected
behavior given the range of the search. The preliminary assessment in the
design report placed the optimal voltage at approximately 165 kV, at which
point we expected to observe a maximum in the muon storage distribution. The
voltage of interest, represented as the total kick voltage, is the sum of the
three individual kicker voltages, and it is proportional to the total induced
magnetic field observed by the muons. Starting at 120 kV, we saw an increase
in the muon storage as the voltage was raised. At around 140 kV, a change in
the rate of rise was noted, suggesting that we were approaching the optimal
set point. The test was stopped at 155 kV due to concerns for equipment
safety. Despite the limitation, this result represented the first evidence
from both E821 and E989 that the muon storage turnover point was being
approached.

Figure 28 displays the results of the Run-3 strength scan. Through Run-2, and
up until this test, the total kick strength was fixed to 142 kV to reduce the
breakdown frequency of the AA5966 cables. Following the installation of the
2264 cables, we were able to improve operating conditions by studying the
newly accessible portions of the parameter space. The muon storage
distributions were analyzed using two methods, both of which are displayed in
the figure. In the first method, we averaged the number of positrons per fill
over multiple 30-second intervals at each kicker set point. This approach
sacrificed overall statistics, but it facilitated on-the-fly data quality
assessments that mitigated the impact of potential beamline variability. The
second technique used Gaussian fits over the full data set to determine the
positrons per fill observed in 20-minute-long runs at each kicker voltage
setting. Both methods yield comparable results and shift the predicted muon
storage maximum from 165 kV to $175-182$ kV. We can explain this shift by
considering the expression in Eq. 19, the actual measurements of the kicker-
induced magnetic field shown in Subsection 3.3, and the corresponding
simulations. $175-182$ kV charging amplitudes will generate angular
displacements of $9.7<\beta<10.4$ mrad. We found these calculations to be
consistent with the 10.8 mrad specification when considering additional
elements that were not included in the design report, such as beam injection
and the resistive load topology.

From a systematics standpoint, kicker performance primarily affects the radial
distribution of the stored muon beam and coherent betatron oscillations (CBO).
Investigation of the radial distribution in Run-1 served as the main evidence
that the kicker system had been operating at a lower voltage than expected.
The finding was the genesis for many of the upgrade efforts described in this
manuscript. The culmination of this work came in Run-3 with the installation
of the 2264 cables, which also facilitated a thorough mapping of the radial
muon distributions and CBO amplitudes at set points ranging from the Run-2
operating point up to the centering position.

The equilibrium radii at various kicker set points were determined using a
Cosine Transform Technique as described in Ref. [20]. These data (blue points)
are shown in Fig. 28 along with a fit and shaded uncertainty band. We find
that the optimal kicker strength for centering the stored muon beam is
approximately 161 kV (when the electrostatic quadrupoles operate at 18.2 kV
and the inflector magnet current is 2763 A). The 161 kV set point became the
primary kicker operating mode as a consequence of this study.

The measurement of $a_{\mu}$ is affected by electric fields in the muon
storage region [9] that modify the observed precession frequency. By centering
the beam, we reduce the impact of this E-field effect by producing a more
ideal muon momentum distribution. Furthermore, the impact of the CBO is
greatly reduced by shifting to 161 kV operations. Over the course of the Run-3
scan, we observed a decrease in the CBO amplitude from around 10 mm to 4 mm.
We expect this shift will improve the quality of the physics data. For Run-1,
the experiment performed rigorous studies to determine the impact of the
kicker system performance on the measurement of the muon magnetic anomaly. In
Subsection 3.3, we described the effect on the magnetic field measurement
detailed in Ref. [8]. Sections II and III of Ref. [9] provide more information
pertaining to the calculation of the systematic uncertainties driven by beam
dynamics considerations.

In summary, we have characterized all elements of the Muon $g-2$ Experiment’s
non-ferric kicker and measured the system impact on several beam dynamics
properties. While we have yet to observe the muon storage maximum with respect
to the kicker operating voltage, we achieved the ideal mean equilibrium radius
during the Run-3 test and confirmed that the upgrade efforts led to improved
conditions for our analyses.

##  4 Conclusions

The fast non-ferric kicker system developed for the Muon $g-2$ Experiment
facilitated the efficient capture of muons for the production of multiple
large data sets. Compared to BNL E821, these new data collectively contain
roughly nine times the number of decay positron events, from which we will
produce the most precise measurement of the muon magnetic anomaly to date.
During these operations, we discovered several limitations in the kicker
system that motivated upgrades between Run-1 and Run-2. System
characterization studies presented in this manuscript demonstrated the
effectiveness of those improvements along with general reliability. We
observed that the non-ferric kickers behaved within expectation, and we have
determined that remaining limitations do not significantly impact the quality
of the physics result.

Any future experiment making use of similar technology should revisit the
nature of load inductance. The characteristics of the magnet load produced an
impedance mismatch in the line that generated unwanted reflections. While this
proved to not be detrimental to the physics goals of the experiment,
significant resources were invested to quantify the effects and develop
potential solution strategies. Invariably, these considerations constructed a
complex parameter space spanning the number of kicker magnets, the inductance
of each load, potential circuit capacitance, the pulse width, necessary
infrastructure, and cost.

## Acknowledgments

We thank our Muon $g-2$ collaborators and Fermi National Accelerator
Laboratory (Fermilab), a U.S. Department of Energy, Office of Science, HEP
User Facility for the resources provided. Fermilab is managed by the Fermi
Research Alliance (FRA), acting under Contract No. DE-AC02-07CH11359. The
authors are supported by the Department of Energy HEP and NP offices (USA),
National Science Foundation (USA), Horizon 2020 research and innovation
programme (EU) under the Marie Skłodowska-Curie grant agreements No. 690835
MUSE and No. 734303 NEWS, Istituto Nazionale di Fisica Nucleare (Italy),
IBS-R017-D1 of the Republic of Korea, UK NSF, and UK Science and Technology
Facilities Council under separate grants.

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