# Studies of Beam Intensity Effects in Fermilab Booster Synchrotron

Jeff Eldred, Valeri Lebedev, Kiyomi Seiya, Vladimir Shiltsev
<EMAIL_ADDRESS>Fermi National Accelerator Laboratory, PO Box 500, MS339,
Batavia, IL 60510,USA

###### Abstract

Detrimental beam dynamics effects limit performance of high intensity rapid
cycling synchrotrons (RCS) such as the 8 GeV proton Fermilab Booster. Here we
report the results of comprehensive experimental studies of various beam
intensity dependent effects in the Booster. In the first part, we report the
dependencies of the Booster beam intensity losses on the total number of
protons per pulse and on key operational parameters such as the machine tunes
and chromaticities. Then we cross-check two methods of the beam emittance
measurements (the multi-wires proportional chambers and the ionization profile
monitors). Finally we used the intensity dependent emittance growth effects to
analyze the ultimate performance of the machine in present configuration, with
the maximum space-charge tuneshift parameter $\Delta Q_{SC}\sim 0.6$, and
after its injection energy is upgraded from 0.4 GeV to 0.8 GeV.

##  I INTRODUCTION

High-energy high-intensity proton beams are widely used for a broad spectrum
of scientific research, including particle colliders, muon spectroscopy,
crystallography of biological molecules, neutrino physics, and neutron
scattering. Over the past four decades, the power of such beams has increased
by about three orders of magnitude with average doubling time of about four
years [1, 2]. Growing demands of the physics research call for more powerful,
more productive and more sophisticated proton accelerators. For example, at
present, the leading accelerators for neutrino research are rapid cycling
synchrotron (RCS) facilities J-PARC in Japan which has reached 515 kW of the
30 GeV proton beam power, and the Fermilab Main Injector delivering up to 766
kW of 120 GeV protons, while the physics needs call for the next generation,
higher-power, megawatt and multi-MW-class superbeams facilities [3].

Further progress in the proton beam power needs improvements in the
accelerator technology [4] and addressing problems related to desired beam
pulse structure, beam losses, and the lifetimes of beam targets [5, 6, 7]. Of
particular challenge are issues associated with the beam dynamics, such as
space-charge effects, instabilities, emittance growth, halo formation and
losses, collimation, accumulation of secondary charges, linear and non-linear
optics optimizations, etc - see comprehensive analysis in Refs.[8, 9, 10, 11,
12]. Advanced beam instrumentation, analytical methods and diagnostics [13,
14, 15, 16], and experimental beam studies are critical for adequate
understanding of the beam dynamics.

Here we present results of the experimental studies of high intensity beam
dynamics in the Fermilab 8 GeV proton Booster RCS, carried out as part of the
Summer 2019 Booster Studies program [17]. In Sec.II we give a brief
description of the accelerator. Next Sec.III is devoted to the beam loss
diagnostics and measurements of the machine transmission efficiency and its
dependence on the total number of protons per pulse, tunes and chromaticities.
The beam emittance diagnostics and measurements, including analysis of the
intensity dependent emittance growth, are presented in Sec.IV. Finally, we
discuss general scaling laws and ultimate performance of the machine now and
after upcoming upgrade in Sec.V.

Figure 1: Schematics of the Fermilab Booster synchrotron. Sectors, each
consisting of four combined function magnets, are numerated 1 to 24. Indicated
are locations of the Ionization Profile Monitors (IPMs), Multi-Wire (MWs) beam
profile monitors in the 8 GeV proton transport line and the 400 MeV $H^{-}$
beam injection line.

##  II FNAL BOOSTER SYNCHROTRON

The complex of Fermilab proton accelerators includes a 750 keV $H^{-}$ RFQ,
400 MeV $H^{-}$ pulsed normal-conducting RF linac, 8 GeV proton Booster
synchrotron, 8 GeV Recycler storage ring that shares tunnel with 120 GeV
proton Main Injector synchrotron, and a 3.1 GeV muon Delivery Ring [18, 19].
About 16 km of beamlines connect the accelerators, bring the beams to fixed
targets and to experiments for high energy particle physics research at the
Intensity Frontier. There are plans to further increase the facility power
from the current world leading level of $\sim$750 kW of average 120 GeV beam
power on the neutrino target to over 1.2 MW at the start of the LBNF/DUNE
experiment [20] in the second half of the 2020’s via replacement of the
existing 400 MeV normal-conducting Linac with a modern CW-capable 800 MeV
superconducting RF linear accelerator (PIP-II, see [21]) and corresponding
beamline for injection into the Booster. There are also several concepts to
further double the beam power to $>$2.4 MW after replacement of the existing 8
GeV Booster synchrotron [22, 23].

Parameter |  | Comments
---|---|---
Circumference, $C$ | 474.20 m |
Inj.energy (kin.), $E_{i}$ | 400 MeV |  $\beta_{i}$=0.701, $\gamma_{i}$=1.426
Extr.energy (kin.), $E_{f}$ | 8 GeV |  $\beta_{f}$=0.994, $\gamma_{f}$=9.526
Cycle time, $T_{0}=1/f_{0}$ | 1/15 s | 20,000 turns
Harmonic number, $h$ | 84 |
RF frequency, $f_{RF}$ | 37.77-52.81 MHz | inj.-extr.
Max. RF voltage, $V_{RF}$ | 1.1 MV |
Trans. energy, $E_{tr}$ | 4.2 GeV |  $\gamma_{tr}$=5.478, at $t$=17ms
No. of cells, magnets | 24, 96 |  $FOFDOOD$, 96o/cell
Total intensity, $N_{p}$ | 4.5$\cdot 10^{12}$ |  $N_{b}$=81 bunches
Rms norm.emitt., $\varepsilon_{x,y}$ | 2.0 $\pi\,\mu$m | 12 $\pi\,\mu$m for 95 %
$\beta$-functions, $\beta_{x,y}$ | 33.7/20.5 m | max.
Dispersion, $D_{x}$ | 3.2 m | max.
Table 1: Main operational parameters of the Fermilab Booster.

The Fermilab Booster [24] is a 474.2 m circumference, alternating-gradient,
rapid-cycling synchrotron containing 96 combined-function magnets – see Fig.1.
Together with capacitor banks, these magnets form a resonant network ($Q=40$)
and get excited with a 15-Hz biased sine wave. Beam acceleration ramp from 0.4
GeV at injection to 8.0 GeV at extraction is 33.3 ms long - half of the magnet
cycle period – and contains about 20 000 turns. Correspondingly, all the
parameters of the machine and beam significantly vary in the cycle – from the
currents in all correctors (trim dipoles, trim quads and skew quads,
sextupoles and octupoles) to RF frequency, voltage and phase (see Fig.2), from
the betatron and synchrotron tunes and chromaticities to proton beam
intensity, positions, sizes, emittances, bunch length and energy spread, etc.
Main parameters of the Booster are given in Table 1. Without going into
details of the Booster high intensity operation and interface with other
machines in the complex which can be found in Refs.[25, 26, 27], here we only
briefly outline main processes which occur at injection, transition crossing
and extraction.

Figure 2: Booster ramp: kinetic energy (black), RF voltage(blue) and frequency
(red).

The Booster receives 400 MeV $H^{-}$ beam of 201 MHz bunches from the Linac
while it is close to minimum of the magnetic field ramp. $H^{-}$ particles are
stripped of two electrons when pass through a thin foil and resulting protons
are accumulated over many turns in the ring (the scheme known as charge
exchange injection [28, 29]). Correspondingly, the total injected and
accelerated beam intensity $N_{p}$ scales with the Linac current, which is
typically $\sim$25 mA, and the total number of injection turns $N_{turns}$,
approximately as $N_{p}=0.34\cdot 10^{12}\cdot N_{turns}$, e.g., about
$4.8\cdot 10^{12}$ for a typical 14-turn injection. The duration of the beam
injection also scales with the number of turns as 2.2$\cdot N_{turns}$
[$\mu$s]. The beam is injected with RF voltage close to zero and then
adiabatically captured over about 300 $\mu$s by the RF system [30, 31]. Right
after injection and in the following several milliseconds, the high intensity
protons beam is subject of the strongest space-charge forces, characterized by
the space-charge parameter $\Delta Q_{SC}\geq 0.5$ (see below). Transverse and
longitudinal ring impedances are large [10, 32, 33] and the Booster operation
requires simultaneous, fast and idiosyncratic adjustment of orbits, optical
functions [34], tunes, chromaticities and many other machine parameters on top
of changing energy, RF voltage and frequency (see Fig.3).

Figure 3: Measured Booster beam characteristics during the acceleration cycle:
(top) the horizontal (blue) and vertical (red) betatron tunes $Q_{x,y}$;
(bottom) (top) the horizontal (blue) and vertical (red) chromaticities
$Q^{\prime}_{x,y}$.

The transition occurs at about 17 ms into the Booster cycle (after injection),
at the beam energy of 4.2 GeV. The Booster is currently operated without a
dedicated $\gamma_{t}$-jump system, though the current of trim quadrupoles, RF
system voltage and RF frequency curves have been tuned to minimize the losses
and control the longitudinal emittance, which somewhat grows from its initial
95% value of about 0.08 eVs [35, 36, 37].

The rapid acceleration in the Booster requires large accelerating voltage. To
inject the Booster beam efficiently into the Recycler for slip-stacking, it is
desirable to rotate the beam in longitudinal phase-space so to reduce the
momentum spread [38]. The Booster bunch rotation is performed via quadrupole
excitation of the synchrotron oscillation as the RF voltage is modulated at
twice the synchrotron frequency and this drives a longitudinal quadrupole
resonance. Once the beam energy is close to the extraction energy we perform
snap bunch rotation, i.e., at about 2 ms before the end of the cycle, the RF
voltage is increased slowly to 650 kV to increase the energy spread of the
bunches and dropped down rapidly to 130 kV. This gives the required small
energy spread for the beam for slip stacking in the downstream accelerators.

Figure 4:  Booster beam losses in the acceleration cycle: (top) the BCHG0
intensity monitor signal (blue, left axis) and the S06 BLM readings (red,
right axis) for nominal operational 14 turns injection intensity; (bottom) the
same for higher intensity 19 turns injection.

The overall average Booster beam loss limit has been administratively set to
$W$=525W, i.e., 35J per cycle with 15 Hz beam cycles. Such level allows us to
maintain all elements in the Booster tunnel without excessive radiation
exposure and corresponds to either 13% beam loss at injection energy or 1.2%
at the transition, or 0.6% at extraction for nominal intensity of about
4.5$\cdot 10^{12}$ protons per pulse. As illustrated in Fig.4, at nominal
intensity, with 14 turns injection, the losses mostly occur at injection, but
the loss at transition becomes dominant at higher intensities. In general, the
beam loss induced radiation is the most important and most challenging factor
limiting the performance of high intensity RCSs [4].

##  III BEAM LOSSES

###  III.1 Beam loss diagnostics

The main diagnostic of the total circulating Booster beam intensity is the
ACNET [39] (Fermilab global accelerator control system) channel B:CHG0, that
uses the signal from the beam current toroid properly mixed and averaged with
the RF waveform signal. The raw B:CHG0 toroid intensity data are quite
reliable and accurate to better than a fraction of one percent at relatively
stable machine and beam parameters, but require correction of the systematic
effects early in the cycle, when the RF frequency, the bunch length and the
bunch structure are quickly changing in time. The required correction was
established using a cross calibration of the intensity loss $\Delta N_{p}$
reported by B:CHG0 and the power loss signal measured by the most appropriate
beam loss monitor BLMS06 - see Fig.4. The BLM at S06 is located between two
Booster collimators, and in regular operation registers the most significant
loss flux over the cycle. The BLM signal represents an integral of a
proportional chamber with decay time of $\sim$200 ms. This signal is reset at
the beginning of each cycle. As one can see BLMS06 peaks for the first time at
10 ms into the cycle with the amplitude proportional to the power loss
integrated well over the initial lossy period of the Booster cycle. The loss
monitor signals at 10 ms are compared to the reported changes in the toroid
signal B:CHG0 and it was found that the latter overestimates $O$(5%) losses at
small intensities by about 2% but that difference disappears (mutual linearity
gets restored) at higher intensities and higher fractional losses - see
details in [40].

Figure 5: Booster RW monitor traces for the bunch beam current profiles right
before (dashed blue) and 40 turns after (solid red) the extraction gap
clearing.

###  III.2 Fractional beam loss

Our studies reveal three kinds of the Booster beam intensity losses : 1) minor
intensity-independent loss at injection $\sim$(1-2)% due to the so-called
”notch clearing”; 2) significant $O(5\%)$ loss shortly after the injection
that can be attributed to space-charge effects as it strongly depends on the
intensity, the working point tunes and chromaticities; and 3) threshold-like
increase of the losses $O(5\%)$ during and after the transition above
$N_{p}=6\cdot 10^{12}$.

The notch clearing. The Booster extraction kicker rise time is about 70 ns
long, and to reduce the losses at extraction one requires an empty gap in the
beam structure. The beam gap is created by removing 3 out of 84 possible
bunches at the lowest feasible beam energies. First, a recently built laser
system [41] is used to create the notch within a Linac beam pulse, immediately
after the RFQ at 750 keV, where activation issues are absent. The beam with
such a gap is injected into the Booster. The laser notch system is not 100%
efficient and, in addition, some particles slip into the gap before the
injection RF capture and as the result there is 400 MeV beam in the gap – as
illustrated by Fig.5. These particles are cleared out at approximately 150
turns after the injection by a kicker pulse that removes 1.4$\pm$0.4% of the
total beam intensity. The particles are directed to a special in-line beam
dump which intercept them with efficiency much better than the efficiency of
the collimation system, so these losses are excluded from the following
analysis of the intensity dependent effects. Ongoing improvements to the laser
notcher power and laser interaction cavity are expected to greatly reduce the
number of unwanted particles in the notch gap.

The largest, and of the most operational concern, intensity dependent losses
which take place over the first 8 ms after the injection are presented in
Fig.6. As one can see, the losses quickly grow with $N_{p}$ – solid line in
Fig.6 is for the fit

$\frac{\Delta N_{p}}{N_{p}}=0.01+0.07\cdot\Big{(}{N_{p}\over{7\cdot
10^{12}}}\Big{)}^{3}\,.$ (1)

The non-zero intercept at small $N_{p}$ might indicate either a different
mechanism of constant losses at low intensity or an insufficient measurement
accuracy.

Figure 6: Intensity-dependent fractional Booster beam intensity loss at
injection vs total number of protons.

The beam intensity losses at the transition crossing were found to be below
$\lesssim 0.5$% up to a threshold at about $N_{p}=6\cdot 10^{12}$ and then
quickly reach 7% at $N_{p}=7\cdot 10^{12}$ [40]. The characteristic dependence
and underlying physics mechanism is quite different from what is observed at
the injection. The intensity dependence at transition is most likely related
to the longitudinal beam loading and the voltage induced by large longitudinal
impedance of the Booster laminated vacuum chamber interior [42, 32, 36]. We
also have to note that for optimal transmission efficiency, the transition
crossing has to be retuned for higher intensities, while during our studies we
operated with the RF and other machine parameters at the transition optimized
for nominal operational intensity of $N_{p}=4.5\cdot 10^{12}$ .

Altogether, in our study the fractional beam intensity losses over the entire
Booster cycle are about $4.2\pm 0.5$% for the nominal intensity while at the
record high injected intensity $N_{p}=7\cdot 10^{12}$ they are about 15%.
Obviously, such losses are not acceptable for routine operation within the
administrative beam power loss limit.

###  III.3 Tune scans

To better understand the nature of the intensity loss phenomena we have
studied the Booster transmission efficiency dependence on the chromaticities
and tunes - the $Q^{\prime}_{x,y}$ and $Q_{x,y}$ scans. Fig.7 shows the
dependence of the losses over the first 1 ms of the Booster cycle ($\sim$450
turns) at the nominal operational tunes $Q_{x,y}$= 6.78/6.88 but at three
different chromaticity settings – the nominal one $Q’_{x,y}$=-4/-16, and then
at $Q’_{x,y}$=-12/-12, and at $Q’_{x,y}$=-20/-20. The fractional losses were
calculated out of the B:CHG0 signal, corrected for the systematic error at
lower intensities and with the extraction gap clearing loss subtracted,
following the method presented in the preceding section.

The results presented in Fig.7 clearly show significant increase of the losses
with the chromaticity. Taking for simplicity the same functional dependence on
intensity as in Eq.(1), the chromatic dependence of the loss data is
consistent with

$\frac{\Delta N_{p}}{N_{p}}=0.013+0.10\cdot\Big{(}{N_{p}\over{7\cdot
10^{12}}}\Big{)}^{3}\Big{(}{Q^{\prime}_{avg}\over 10}\Big{)}^{1.9\pm 0.2},$
(2)

with $Q’_{avg}=(|Q^{\prime}_{y}|+|Q^{\prime}_{y}|)/2$ denoting the average
chromaticity. There is a lower limit on operational chromaticity that depends
on the intensity and is usually associated with the need to maintain the
coherent beam stability. Correspondingly, the low chromaticity operation is
possible only at low intensities. Notably, the strong dependence of the losses
at injection on the chromaticity, presumably due to the space-charge effects,
$\Delta N_{p}\propto Q’^{2}$, is similar to the incoherent beam losses due to
parasitic beam-beam effects observed in the Tevatron collider [43].

Figure 7: Transmission efficiency 1 ms after injection for different
chromaticities $Q^{\prime}_{x,y}$ vs $N_{p}$.

Figure 8: Tune scans of the transmission efficiency over the first millisecond
after injection: (top left) at the $N_{p}=0.95\cdot 10^{12}$ and
$Q^{\prime}_{x,y}$=-6/-6; (top right) $N_{p}=0.95\cdot 10^{12}$ and
$Q^{\prime}_{x,y}$=-20/-20; (bottom left) $N_{p}=4.3\cdot 10^{12}$ and
$Q^{\prime}_{x,y}$=-4/-16; (bottom right) $N_{p}=4.3\cdot 10^{12}$ and
$Q^{\prime}_{x,y}$=-20/-20.

The tune scans were carried out under six different conditions: i) low
intensity and low chromaticity: $N_{p}=0.95\cdot 10^{12}$,
$Q^{\prime}_{x,y}$=-6/-6; ii) low intensity and high chromaticity:
$N_{p}=0.95\cdot 10^{12}$, $Q^{\prime}_{x,y}$=-20/-20; iii) high intensity and
medium chromaticity: $N_{p}=4.3\cdot 10^{12}$, $Q^{\prime}_{x,y}$=-4/-16; iv)
high intensity and high chromaticity: $N_{p}=4.3\cdot 10^{12}$,
$Q^{\prime}_{x,y}$=-20/-20; v) medium intensity and medium chromaticity:
$N_{p}=2.6\cdot 10^{12}$, $Q^{\prime}_{x,y}$=-12/-12; vi) medium intensity and
high chromaticity: $N_{p}=2.6\cdot 10^{12}$, $Q^{\prime}_{x,y}$=-20/-20. Note,
that vertical and horizontal tunes and chromaticities varied only for the time
period of 2 ms after the injection, and for the rest of the Booster cycle,
they stayed as for the routine operational cycles.

The results of the first four are presented in Fig.8. One can see that, in
general, an increase of either the chromaticity or intensity or both leads to
reduction of the available tune space for low loss operation and generally
lower optimal transmission efficiencies. Table 2 summarizes the findings.

$N_{p},10^{12}$ |  $Q^{\prime}_{x,y}$=-6/-6 |  $Q^{\prime}_{x,y}$=-12/-12 |  $Q^{\prime}_{x,y}$=-20/-20
---|---|---|---
0.95 | 1.5%, 0.68/0.84 |  | 1.2%, 0.69/0.88
| 3.5%, 0.82/0.81 |  | 2.6%, 0.77/0.88
4.3 |  | 3.0%, 0.74/0.87 | 7.0%, 0.77/0.88
|  | 7.0%, 0.80/0.90 | 16%, 0.77/0.88
Table 2: Optimal working points for various injected intensities and injection
chromaticities: first line in each box – the minimal intensity loss 1 ms after
injection (raw B:CHG0 data, uncorrected for the B:CHG0 systematic errors and
the notcher gap cleaning) and the optimal horizontal and vertical tunes
$Q_{x}/Q_{y}$; the second line – same for the entire Booster cycle (at
extraction).

The tune scan data reveal stronger sensitivity of the losses to the vertical
tune than to the horizontal one. For example, 14-units increase of the
chromaticity from -6 to -20 at $N_{p}=0.95\cdot 10^{12}$ resulted in the
reduction of the 90% transmission tune area by $dQ_{y}$=0.05 in vertical plane
while $dQ_{x}$=0.02 – see – see Fig.8 a) and b). Similarly, the change of the
chromaticity from -12 to -20 for $N_{p}=4.3\cdot 10^{12}$ led to shrinkage of
the 90% transmission tune area by $dQ_{y}$=0.1 and $dQ_{x}$=0.05, as depicted
in Fig.8 c) and d). That is indicative of a stronger resonance in the vertical
than the horizontal.

##  IV BEAM EMITTANCE EVOLUTION

###  IV.1 Beam emittance diagnostics

In the Booster, there are two types of instruments to measure beam sizes and
therefore, transverse emittances – the multi-wires (MWs) and the ionization
profile monitors (IPMs).

Vertical and horizontal MWs are installed in the extraction beam line and,
therefore, can measure only the emittances of the extracted Booster beam.
There are 48 wires in each instrument, spaced by 1 mm. The focusing optics
function at the MW location are $\beta_{x}$=16.2m, $\beta_{y}$=25.9m and
$D_{x}$=-1.65m. Statistical rms error of the MW emittance measurement is about
0.05 mm mrad.

IPMs operate by collecting ions created after the ionization of residual
vacuum molecules by high energy charged particle beams [14, 16], which are
then guided to a detector by a uniform external electric field $E_{\rm ext}$.
The detector consists of many thin parallel strips, whose individual signals
are registered to make the beam profile signal ready for processing. Two IPMs
– vertical and horizontal, are installed in the Booster at the location with
$\beta_{x}$=6.0m, $\beta_{y}$=20.3m and $D_{x}$=1.8m [44]. The electric field
of about 2.4 kV/cm is formed by application of $V=24$kV extracting voltage
over $D=103$mm gap. The MCP based ion detector employs an array of parallel
thin anode strips spaced 1.5 mm apart. IPMs are very fast and report the
average rms beam sizes (determined by the Gaussian fits of the profiles) on
every Booster turn. Note, that the Booster IPMs do not employ external
magnetic fields to keep the trajectories of the secondaries parallel to the
electric field.

Contrary to the MWs, the IPMs exhibit intrinsic dependence on the proton beam
intensity as the proton space charge fields lead to transverse expansion of
the cloud of ions on its way from its origin (in the proton beam) to the IPM
detector plate - see Fig.9. Comprehensive theory of the IPM operation is
developed in [45]. The rms size of the measured profile in the IPM
$\sigma_{m}$ is related to the original proton beam size $\sigma_{0}$ as:

$\sigma_{m}=\sigma_{0}\cdot h(N_{p},\sigma_{0},D,V,d),$ (3)

where the expansion factor $h$ can be approximated as:

$h\approx
1+F\Big{(}\frac{2\Gamma(1/4)U_{SC}Dd^{1/2}}{3V\sigma_{0}^{3/2}}\Big{)}\big{(}1+t_{b}/\tau_{0}\big{)}.$
(4)

Here $V$ is the IPM extracting voltage (typically, 24 kV in our case) and $D$
is its HV gap (103 mm), $d$ is the distance for ions to travel from the beam
orbit to the IPM collection plate, the space-charge potential for the proton
beam with current $I$ is $U_{SC}$ = 30[V]$I/\beta_{p}$ and gamma function
$\Gamma$(1/4)$\approx$3.625. The numerical factor $F$ is equal to 1 in the
case of unbunched DC proton beam with Gaussian transverse current
distribution, and $F=2\sqrt{2}/\Gamma(1/4)\approx 0.78$ for uniform
distribution with radius $a=2\sigma_{0}$. In the case of Booster, with modest
expansion $h\leq 2$, one can neglect minor corrections due to somewhat unequal
horizontal and vertical beam sizes, but should account the correction factor
$(1+t_{b}/\tau_{0})$ due to the bunch structure of the Booster proton current.
There $t_{b}$ is the bunch spacing (about 19 ns at the end of the Booster
cycle) and characteristic time for an ion to leave the beam
$\tau_{0}=\sqrt{2MD\sigma_{0}/ZeV}$, where $M$ and $eZ$ are the ion’s mass and
charge, that is about 22 ns for typical IPM and beam parameters at the end of
the cycle.

Figure 9: Comparison of the measured rms IPM vertical beam sizes at extraction
for different beam intensities with the rms sizes measured by the MWs and
recalculated to the IPM location.

Also very important are intensity independent effects leading to the IPM
profile smearing such as the initial velocities of the ions, finite separation
between the individual IPM charge collection strips, angular misalignment of
the IPM long and narrow strips with respect to the high energy proton beam
orbit, etc [45]. These effects are monitor-specific, they add in quadrature
and can be determined in comparison of low intensity beam sizes measured by
the IPM and by the MWs $\sigma^{2}_{T}=\sigma^{2}_{m,IPM}-\sigma^{2}_{m,MW}$
at $N_{p}\rightarrow 0$. For the Booster IPMs it was found that such
instrumental smearing is $\sigma^{2}_{T}=2.8\pm 0.1$ mm2 and correspondingly
modified Eq.(3) $\sigma^{2}_{IPM}=\sigma^{2}_{T}+h^{2}\sigma^{2}_{MW}$
describes the Booster IPM data at extraction with some 5% accuracy [46].

The original proton beam size $\sigma_{0}$ can be found from the measured and
correspondingly corrected IPM value of
$\sigma^{*}=\sqrt{\sigma^{2}_{m,IPM}-\sigma^{2}_{T}}$ by reversing the
equation $\sigma^{*}=\sigma_{0}(1+cN_{p}/\sigma_{0}^{3/2})$, where $c$ is
deducted from Eqs.(3,4), if other parameters, such as $\sigma_{T},N_{p},d$ and
the IPM voltage $V$ and gap $D$, are known. A simple practical algorithm gives
better than $\pm$5% approximation over the entire range of the Booster beam
intensities:

$\sigma_{0}\approx\frac{\sigma^{*}}{(1+cN_{p}/{\sigma^{*}}^{3/2})(1+\alpha
c^{2N_{p}^{2}}/{\sigma^{*}}^{2})}\,.$ (5)

For the highest beam intensity in our studies $N_{p}=6\cdot 10^{12}$ the
factor $cN_{p}$=2.53 mm3/2 and the fitting coefficient $\alpha\approx 0.4$
[46]. Eq.(5) can now be used to find out proton beam size over the entire
Booster cycle, i.e., not just for the values measured at extractions - see
Fig.10. There, the black lines for the raw (uncorrected) rms vertical and
horizontal beam sizes $\sigma_{m,IPM}$ as measured by the IPM at each of 20
thousand turns of the Booster acceleration cycle; the pink and green lines
represent the beam sizes $\sigma^{*}$ corrected for the intensity independent
smearing $\sigma_{T}$; and, finally, the true proton rms beam size
$\sigma_{0}$ reconstructed following the above algorithm Eq.(5)are represented
by the red and blue lines. One can see that the overall beam size correction
is about 10-15% early in the Booster acceleration cycle when the rms beam size
is about 5 mm. At the end of the cycle, with proton energy increased from 400
MeV to 8 GeV, the proton beam size is almost a factor of about 1.8 smaller
than it appears in the IPM and accounting for the space-charge expansion
$h(N_{p},D,V,d)$ is the most important. The reconstructed beam sizes
$\sigma_{0}$ at the end of the acceleration cycle match well the extracted
beam size measured by the MWs.

Figure 10: An example of reconstruction of vertical and horizontal rms proton
beam size in the 33 ms (20000 turns) acceleration cycle of the Fermilab 8 GeV
Booster synchrotron with the total beam intensity of $N_{p}=4.6\cdot 10^{12}$:
time dependence of the original IPM data $\sigma^{2}_{m,IPM}$, the data
corrected for smearing effects $\sigma^{*}$ and the same data after additional
correction for the space-charge expansion $\sigma_{0}$ with parameters $D=103$
mm, $d=$52 mm, $V=$24 kV - see text and Eqs.(4, 5).

As can be seen in Fig.10, the horizontal rms beam size exhibits significant
oscillations with twice the synchrotron frequency after the transition. Such
oscillations arise from the mismatch between longitudinal focusing of bunch
fields before and after transition and the effect gets bigger with intensity.
Horizontal IPM is located at small beta-function and high dispersion, so,
compared to the betatron size, the dispersive contribution is large
$D_{x}(\delta p/p)\geq\varepsilon_{x}\beta_{x}/(\beta_{p}\gamma_{p})$ and
variations in the momentum spread $(\delta p/p)$ result in about $\pm 20$%
oscillations in $\sigma_{x}$. Booster is well decoupled and, consequently, the
vertical dispersion is small. Therefore a little perturbation observed at
transition is most probably due to reaction of the the IPM profile expansion
factor $h$ to variation of the bunching factor $B_{f}=\sqrt{2\pi}/\sigma_{s}$
\- the ratio of the peak to average proton beam current - which peaks at the
transition - see Fig.11. The effect is small but becomes more pronounced at
higher intensities (see Refs.[45] below).

Figure 11: Booster bunching factor over the accelerator ramp (inversely
proportional to rms proton bunch length.)

###  IV.2 Beam emittance vs intensity

The injected Linac $H^{-}$ beam comes with the rms normalized transverse
emittances $\varepsilon\sim 1\pm 0.2$ $\pi$ mm mrad [47]. Due to small
injection errors, optics mismatch and multiple scattering in the foil the
initial emittance of the proton beam circulating in the Booster gets to about
$\sim 1.2\pm 0.2$ $\pi$ mm mrad. The latter effect correlates with the total
beam intensity. Indeed, in the process of multi-turn charge exchange
injection, each passage of the stripping foil leads to the emittance growth
of:

$\Delta\varepsilon_{x,y}\approx\beta_{p}\gamma_{p}{\beta_{x,y}\over 2}{l\over
X_{0}}\Big{(}\frac{13.6\,\rm{MeV}}{\beta_{p}pc}\Big{)}^{2}[1+0.0038\ln{(l/X_{0})}]\,.$
(6)

For the Booster carbon foil thickness $l$=380 $\mu$g/cm2, radiation length
$X_{0}$=42.7 g/cm2 and momentum $p$=953 MeV/c that gives 0.0032 $\pi$ mm mrad
per turn in the horizontal plane and 0.011 $\pi$ mm mrad per turn in the
vertical plane. During the injection the beam moves across and off the foil,
so the effective number of turns is about ($N_{turns}$+29)/2 turns [48]. Given
that the total circulating beam intensity $N_{p}$ scales linearly with
$N_{turns}$, the estimated emittance increase at the end of injection grows
with the intensity and for the nominal $N_{turns}$=14 turns injection
$\Delta\varepsilon_{y,foil}\approx 0.24\pi$ mm mrad and
$\Delta\varepsilon_{x,foil}\approx 0.07\pi$ mm mrad.

Figure 12: Booster beam emittance measured by MWs at extraction vs the total
proton intensity.

Measured Booster beam emittances at extraction exhibit strong dependence on
the total proton intensity $N_{p}$ as shown in Fig.12. The MWs data taken at
the nominal operational chromaticities at injection $Q’_{x,y}$=-4/-16 show
strong dependence on the beam intensity and for both planes can be
approximated as:

$\varepsilon_{extr}\mathrm{[\pi\,mm\,mrad]}\approx
1.7+2.1\cdot\Big{(}{N_{p}\over{7\cdot 10^{12}}}\Big{)}^{4\pm 0.3}\,,$ (7)

The emittance growth is strongly dependent on the chromaticity, too, and at
the nominal intensity $N_{p}=4.5\cdot 10^{12}$ the emittance increases from
about 2.1 $\pi$ mm mrad to some 2.4 $\pi$ mm mrad (vertical) and 2.7 $\pi$ mm
mrad (horizontal) if the operational chromaticity at the first ms after the
injection is changed from $Q’_{x,y}$=-4/-16 to -20/-20. The beam emittance
increase with intensity and chromaticity strongly correlates with the
intensity losses - see Eqs.(1, 2) and Figs.6, 7.

The IPM rms beam sizes $\sigma_{0,x,y}(t)$ measured over the Booster
acceleration cycle from injection to extraction and properly corrected
following the analysis of preceding Sec.IV.1 and Eq.(5) can be used for the
emittance calculations
$\varepsilon_{y}=(\beta_{p}\gamma_{p})\sigma^{2}_{0,y}/\beta_{y}$,
$\varepsilon_{x}=(\beta_{p}\gamma_{p})(\sigma^{2}_{0,y}-D^{2}_{x}(\delta
p/p)^{2})/\beta_{x}$, where $\delta p/p$ is the rms energy spread. Of course,
all the factors are now time dependant: the relativistic factors
$\beta_{p}(t)$ and $\gamma_{p}(t)$ are well known - see, e.g. Fig.2; the beta-
functions $\beta_{x,y}$ at the IPM locations vary in the cycle within
$\sim$10%. Calculations of Booster emittance take all these effects into
account, they agree with the MWs data at extraction to within 10% and exhibit
no emittance growth at the lowest intensities - see [46].

Figure 13: Evolution of the IPM vertical emittance in the Booster cycle at
different intensities $N_{p}$ from 0.5$\cdot 10^{12}$ (2 turns injection) to
6.2$\cdot 10^{12}$ (20 turns injection). All the data are smoothed by a 100
turn running window averaging.

The resulting Booster IPM vertical beam emittance evolution over the
acceleration cycle is shown in Fig.13 for a wide range of intensities $N_{p}$
from 0.5$\cdot 10^{12}$ (2 turns injection) to 6.2$\cdot 10^{12}$ (20 turns).
For example, the emittance $\varepsilon_{y}(t)$ at the operational intensity
of $N_{p}=4.3\cdot 10^{12}$ is shown in black line in Fig.13. Shortly after
injection is it about 1.3$\pm$0.1 mm mrad and evolves to 1.9$\pm$0.1 mm mrad
at extraction – in an good agreement with the MW emittance data shown in
Fig.12. In general, one can see that up to about $N_{p}=3.7\cdot 10^{12}$ (12
turns injection, pink line) the emittance is not growing much in the cycle and
is about 1.4-1.6 mm mrad. Above that intensity the emittance evolution
exhibits several features: i) fast growth over the first 2000-3000 turns, ii)
steady growth for the rest of the cycle; iii) spikes at the time of transition
and minor oscillations afterwards; and iv) 5-10% variations at the end of the
cycle. The last two effects are presumably instrumental. Significant variation
of the bunching factor at the transition shown in Fig.11 does affect the IPM
profile expansion $h$ – see Eq.(4). At the end of the acceleration cycle, the
proton beam position in the IPMs varies over the last 2000 turns by as much as
6 mm in the horizontal plane, thus, affecting the IPM profile expansion factor
$h_{y}$ in the vertical plane that scales with the distance from the beam
orbit to the IPM collection plate as $d_{x,y}^{1/2}$ \- see Eq.(4), while
bunch rotation in longitudinal phase-space prior the extraction at the very
last hundreds of turns results in a smaller momentum spread and longer bunches
- again, leading to minor variation of the IPM expansion factor $h$.

The most dominant are the first two effects. Fig.14 shows how they depend on
the beam intensity. The fast rms vertical emittance growth over the first 3000
turns is most probably due to record strong proton space-charge effects (see
below) and scales approximately as:

$\Delta\varepsilon_{y,3000}\mathrm{[\pi\,mm\,mrad]}\approx
0.17+0.61\cdot\Big{(}{N_{p}\over{7\cdot 10^{12}}}\Big{)}^{2}\,.$ (8)

Slow emittance increase is roughly linear in time over the next 16000 turns
and gets as big as 1 mm mrad, or 30% of the emittance, at $N_{p}=6.2\cdot
10^{12}$ and can be approximated as :

$\Delta\varepsilon_{y,3000-19000}\mathrm{[\pi\,mm\,mrad]}\approx
1.85\cdot\Big{(}{N_{p}\over{7\cdot 10^{12}}}\Big{)}^{4}\,.$ (9)

The nature of that effect might be related to the multipacting of electrons in
the beam and/or beam-induced vacuum activity but that still needs to be
confirmed [49].

Figure 14: Vertical rms emittance growth vs Np : (red circles) over the first
3000 turns; (black squares) from 3000 to 19000 turns. The data points are
calculated for the IPM emittance values averaged over five hundred turns
0-500, 3000-3500, 19000-19500. The error bars indicate estimated statistical
uncertainty. Red and black solid line are for the approximations Eq.(8) and
(9), respectively.

###  IV.3 Space-charge tune shift

Space-charge tuneshift parameter $\Delta Q_{SC}$ is a commonly used figure of
merit for beam dynamics. It is equal to [4]:

$\Delta
Q_{SC}=\frac{N_{p}r_{p}B_{f}}{4\pi\varepsilon\beta_{p}\gamma_{p}^{2}}R\,,$
(10)

where $N_{p}$ is the total intensity assuming that the bunches fill all RF
buckets, $r_{p}$ is the classical proton radius, $B_{f}$ is the bunching
factor (the ratio of the peak to average bunch current), $\varepsilon$ is the
normalized rms beam emittance, $\beta_{p}$ and $\gamma_{p}$ are relativistic
Lorentz factors. Factor $R\lesssim 1$ accounts for unequal average beam size
ratio around the ring and, e.g., for the vertical plane it is equal to
$\big{<}2/(1+\sigma_{x}/\sigma_{y})\big{>}$. The tuneshift is negative but we
omit the minus sign for simplicity. In operational circular rapid cycling
accelerators, the space-charge parameter usually does not exceed 0.2-0.4 to
avoid beam losses [50, 4, 51].

Fig.15 shows the vertical SC tuneshift parameter $\Delta Q_{SC}(t)$ calculated
for the Fermilab Booster acceleration cycle on base of measured
$N_{p}(t),\varepsilon_{y}(t),B_{f}(t)$ and known $\beta_{p}(t)$ and
$\gamma_{p}(t)$. One can see that the calculated space-charge parameter
quickly grows after the injection due to fast bunching early in the
acceleration cycle, then falls down due to acceleration and emittance
increase, and exhibits some temporary increase at the transition before ending
at $\sim$0.01 prior to extraction. The maximum Booster tuneshift parameter
peaks at about 1 ms after injection to $\Delta Q_{SC}\simeq 0.65$ and stays
above 0.3 until about 6 ms ($\sim$3000 turns). Naturally, the corresponding
incoherent space-charge tune spread does not easily fit the available tune
space between most dangerous resonances, such as half-integer ones, and that
results in strong resonant excitation of the proton dynamics and eventual
particle losses at the machine aperture. At the highest beam intensity studied
$N_{p}=6.2\cdot 10^{12}$ the maximum space-charge tuneshift parameter $\Delta
Q_{SC}$ peaks at $\simeq 0.75$.

Figure 15: Calculated vertical space-charge tuneshift parameter for the
Booster cycle with $N_{p}=4.6\cdot 10^{12}$. (Shaded area indicates 10%
uncertainty, mostly due to the IPM emittance calculations).

##  V SUMMARY, DISCUSSION

The phenomena of the Booster beam intensity losses, transverse emittance
growth and longitudinal beam excitation at the transition are found to be very
much intensity dependent and strongly interconnected. Our experimental studies
indicate that the losses that occur early in the cycle are most probably due
to the space-charge driven vertical beam size expansion that leads to the flux
of protons on the collimator-limited machine aperture with characteristic
acceptance $A_{y}\simeq 20-25\,\mathrm{[\pi\,mm\,mrad]}$[46]. Both the
fractional beam loss Eqs.(1, 2) and the emittance growth Eq.(8) highly
nonlinearly depend on the total circulating beam intensity $N_{p}$.

Beam losses at and after the transition crossing while small under conditions
of optimal operational tune up the nominal intensity $N_{p}=4.6\cdot 10^{12}$
exhibit a threshold-like behavior above $6\cdot 10^{12}$. IPM measurements
indicate small vertical beam size at the transition while the horizontal size,
dominated by the dispersive contribution $D_{x}(\delta p/p)$, becomes very
large, so it is natural to assume that the proton losses at the transition end
up at the horizontal aperture.

Booster is a lynch pin of the Fermilab proton accelerator complex and its
transmission efficiency has always been subject of continuous monitoring and
several studies for almost four decades - see, e.g. [52, 53, 54]. Naturally,
the Booster performance at present significantly exceeds that of the past -
compare current transmission $\sim$95% at $N_{p}=(4.5-5)\cdot 10^{12}$ with
that of $\sim$(70-75)% at $N_{p}=(3.5-4)\cdot 10^{12}$ in 1980s and 1990s.
What makes this work distinct is that for the first time we have performed a
simultaneous comprehensive analysis of the proton loss dependence on the total
beam intensity, machine tunes and chromaticities, and, also, performed a
cross-calibration of the toroid intensity monitor and the beam loss monitors.

Nonlinear dependence of the extracted Booster beam emittance on $N_{p}$ was
reported in many previous measurements with MultiWire profile monitors - see,
e.g.[52, 55, 56]. The Booster Ionization Profile Monitors operating in the ion
collection mode without external magnetic field are known to be extremely
valuable tools for fast beam size diagnostics of the circulating beam during
acceleration [57, 44, 58]. At the same time, strong space-charge forces of the
high intensity proton beam lead to significant, factor of 2 or more, expansion
of the rms beam size reported by the IPMs compared to the original proton beam
size. To be certain that we properly account that and other, intensity-
independent, effects, we - in addition to following the theoretical recipes of
Ref. [45] as described in Sec.IV.1 \- has performed a systematic calibration
of the IPM with the MultiWires measurements. As the result, we have achieved
$O$(10)% accuracy in the beam emittance reconstruction using the measured IPM
rms beam sizes, experimentally determined intensity-independent instrumental
dispersion $\sigma_{T}$, known beam intensity $N_{p}$ and the IPM parameters
such as extracting electric field $E_{\rm ext}=V_{0}/D$ and the distance $d$
from the beam orbit to the IPM detector.

Our data indicate existence of two phenomena leading to the proton beam
emittance growth: i) the space-charge driven expansion over the first few
thousand turns; and ii) steady emittance increase over the rest of the
acceleration cycle. Both effects grow faster than linear with the proton beam
intensity, resulting in the final (extracted) beam emittance having a
significant component $\propto\,N_{p}^{4}$ \- see Eq.(7), which also dependent
on the machine chromaticity at the injection energy. An attempt of a similar
analysis of the Booster IPM profiles has been undertaken in [15]. It was based
on the phenomenological approximations for the IPM profile space-charge
induced expansion effects developed in [58], and reported space-charge driven
emittance growth early in the cycle was found to be a) significantly, 3-4
times bigger than reported here; and b) scaling approximately linear with
intensity $\propto N_{p}$. These observations also show improbably large
normalized rms emittances of high intensity beams prior to extraction $\sim
10\mathrm{[\pi\,mm\,mrad]}$, thus, directly contradicting our MultiWires
measurements and what we know about the Booster acceptance - that make us to
suspect a systematic error in the IPM data analysis. Notably, the steady
emittance increase during the acceleration cycle was observed, too, though
again, more than twice of we report here in Eq.(9).

Our experimental studies of the Booster losses and emittance evolution augment
similar investigations at other high-intensity proton synchrotrons, see, e.g.,
[59, 60, 61, 62, 63, 64]. They are of great importance to predict the machine
operational conditions in the era of upcoming new 800 MeV injector (PIP-II)
and have to be continued. These studies and foreseen future operation with up
to 50% higher beam intensities would greatly benefit from improved accuracy of
the fast beam intensity diagnostics and of the IPM emittance analysis. The
losses at transition are believed to result from complicated 3D dynamics
including transverse and longitudinal coherent instabilities. Future
investigations of the dependence on the RF voltage, chromaticities, tunes and
other machine parameters will help to better identify the transition loss
origin. ESME [65] and BLonD [66] simulations are also underway to study the
impact of $\gamma_{t}$ jumps on transition in the PIP-II era.

Strong intensity dependence of the space-charge induced losses in the first
few thousands turns after the injection is also of concern. Indeed, let us
consider an RCS accelerator, like the Fermilab Booster, operationally limited
by the uncontrolled radiation level in accelerator enclosures at $W=f_{0}\int
E_{k}dN_{p}$ (here $f_{0}$ is the cycle rate and $E_{k}$ is the kinetic energy
of the lost particle) - typically it is about 1 W per a meter of machine
circumference. Under such limit, the tolerable fractional beam intensity loss
is

${\Delta N_{p}\over N_{p}}=\frac{W}{(1-\eta)N_{p}E_{k}f_{0}}\,,$ (11)

where $\eta$ is the efficiency of the collimation system that directs the
losses into dedicated beam absorbers or dumps. Obviously, the losses should
get smaller with the increase of beam intensity, energy and power. On the
contrary, many beam physics phenomena, such as, e.g., repelling forces of the
proton beam’s own space-charge lead to increase of beam sizes and particle
losses at higher beam intensities. Let us, following Eq.(1), consider the case
when the space-charge induced losses grow with the space-charge tuneshift
$\Delta Q_{SC}$ as:

${\Delta N_{p}\over N_{p}}\propto\alpha\Delta Q_{SC}^{\kappa},$ (12)

where $\alpha$ is a machine dependent constant and the exponent $\kappa$ is
about 3. Then one gets from Eqs.(11, 13) the maximum operational intensity
within the loss power limit $W$ :

$N_{p}^{max}\propto\Big{(}{W\over
1-\eta}\Big{)}^{1\over\kappa+1}\Big{(}{\varepsilon\over
B_{f}}\Big{)}^{\kappa\over\kappa+1}\frac{\gamma_{p}^{{2\kappa-1/2}\over\kappa+1}}{\big{(}\alpha
f_{0}\big{)}^{1\over\kappa+1}}\,\,.$ (13)

From that, one can see that there are several paths to the increase of the
maximum intensity [18] but none leads to direct one-to-one increase of
$N_{p}^{max}$. For the case of the PIP-II Booster and assuming $\kappa=3$,
Eq.(13) anticipates that a 3-fold improvement in collimation system efficiency
(e.g. from $\eta=0.55$ to 0.85) [67] leads to a 31/4 increase in
$N_{p}^{max}$, or 32%. For the two-fold increase in Booster injection energy
$E_{k}$ from 400 MeV to 800 MeV with a new PIP-II linac [21], Eq.(13) predicts
that one should be able to safely increase $N_{p}^{max}$ by 41% from the
current operational value, except the increase in cycle rate $f_{0}$ from 15
Hz to 20 Hz under the same plan will cut the expected benefit to just 31%.
Between the two improvements (neglecting any beneficial effects of PIP-II
injection painting), Eq.(13) at $\kappa=3$ favors the 44% increase in
intensity from 4.5$\cdot 10^{12}$ to 6.5$\cdot 10^{12}$ with the proposed
improvements for the PIP-II Booster.

More broadly, Eq.(13) evaluates strategies for improving intensity in other
high-intensity rings. Flattening the longitudinal bunch current profile, e.g.
by using additional 2nd or 3rd harmonics RF systems, leads to reduction of the
bunching factor and the factor of two smaller $B_{f}$ could lead to 1.68 times
higher maximum intensity. Acceleration of twice larger emittance beams would
give about the same effect, but it usually not possible within available
machine aperture of existing machines. At ultimate intensities, significant
promise in loss reduction lies in improved beam dynamics that would make
$\alpha$ and $\kappa$ smaller, for example by injection “painting” to make the
space-charge forces more uniform, by compensation of the most detrimental
resonant driving terms (including enforcement perfect periodicity in machine
focusing optics), via the space-charge compensation using electron lenses
[68], or by implementation of the non-linear integrable optics [69]. For the
latter two topics in particular, an R&D program is underway at the Fermilab
IOTA facility [70].

## Acknowledgements

We would like to thank C.Y. Tan, C. Bhat, Yu. Alexahin, A. Burov, S.
Nagaitsev, W. Pellico and R. Thurman-Keup for numerous discussions on the
topics of this study and S. Chaurize, V. Kapin and K. Triplett for their
invaluable help with experimental Booster beam studies. In addition, the
Summer 2019 Booster beam study campaign involved N. Eddy, C. Jensen, J.
Larson, and H. Pfeffer of Fermilab, H. Bartosik, N. Biancacci, M. Carla, A.
Saa Hernandez, A. Huschauer, F. Schmidt of CERN, D. Bruhwiler, J. Edelen of
the Radiasoft SBIR company and V. Kornilov of GSI. We greatly appreciate their
fruitful cooperation and the spirit of international beam physics
collaboration.

Fermilab is supported by U.S. Department of Energy, Office of Science, Office
of High Energy Physics, under Contract No. DE-AC02-07CH11359.

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