# Stability of ionisation-injection-based laser-plasma accelerators

Simon Bohlen<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  Universität Hamburg, Luruper Chaussee
149, 22761 Hamburg, Germany  Jonathan C. Wood<EMAIL_ADDRESS>Deutsches
Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  Theresa
Brümmer  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg,
Germany  Florian Grüner  Universität Hamburg, Luruper Chaussee 149, 22761
Hamburg, Germany  Carl. A. Lindstrøm  Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  Martin Meisel  Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany  Theresa Staufer  Universität
Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany  Richard D’Arcy
Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany
Kristjan Põder  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607
Hamburg, Germany  Jens Osterhoff  Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany

###### Abstract

Laser-plasma acceleration (LPA) is a compact technique to accelerate electron
bunches to highly relativistic energies, making it a promising candidate to
power radiation sources for industrial or medical applications. We report on
the generation of electron beams from an $80\text{\,}\mathrm{MeV}$-level LPA
setup based on ionisation injection (II) over a duration of 8 hours at a
repetition rate of 2.5 Hz, resulting in 72000 consecutive shots with charge
injection and acceleration. Over the full operation time the moving averages
of the total beam charge of $14.5\text{\,}\mathrm{pC}$ and the charge between
70–$80\text{\,}\mathrm{MeV}$ did not drift on a detectable level. The largest
source of shot-to-shot jitter was in the beam charge (26% standard deviation),
which was most strongly correlated with fluctuations in the plasma density
(3.6% standard deviation). Particle-in-cell simulations demonstrate that this
was chiefly caused by stronger laser self-focusing in higher density plasmas,
which significantly increased the ionised charge along with the emittance of
the beam. The nonlinearity of this process imposes tight constraints on the
reproducibility of the laser-plasma conditions required for a low jitter II-
LPA output if self-focusing plays a role in the laser evolution.

##  I Introduction

Laser-plasma acceleration (LPA) enables the acceleration of electron bunches
with gradients often exceeding
$100\text{\,}\mathrm{GV}\text{\,}{\mathrm{m}}^{-1}$ by exploiting the large
electric fields of a plasma wave Tajima and Dawson (1979). As these fields are
around three orders of magnitude stronger than those provided by conventional
radio-frequency technology, LPAs have long been proposed as future compact
electron accelerators. So far, experiments of applications of these sources
have concentrated on secondary radiation such as the production of undulator
radiation Fuchs  _et al._ (2009), hard x-rays from Thomson or Compton
scattering Compton (1923); Catravas, Esarey, and Leemans (2001); Schwoerer
_et al._ (2006); Ta Phuoc  _et al._ (2012) or bremsstrahlung Bethe and Maximon
(1954); Edwards  _et al._ (2002); Glinec  _et al._ (2005); Döpp  _et al._
(2016), and high resolution tomographic imaging Cole  _et al._ (2015); Wenz
_et al._ (2015); Cole  _et al._ (2018a) using betatron radiation produced by
oscillations of the electron beam within the LPA cavity Rousse  _et al._
(2004); Kneip  _et al._ (2010). Recently, free electron lasing of LPA produced
electron beams was demonstrated for the first time Wang  _et al._ (2021).
The development of these applications has gone hand-in-hand with progress in
realising high quality electron beams via a range of techniques. It has been
shown that electron beams with few per-cent energy spread or less can be
produced Mangles  _et al._ (2004); Geddes  _et al._ (2004); Faure  _et al._
(2004); Wang  _et al._ (2016). Maximum reported energy gains have continued to
increase to as high as $8\text{\,}\mathrm{GeV}$ Wang  _et al._ (2013); Kim
_et al._ (2013); Gonsalves  _et al._ (2019), while nanocoulomb level beams
with $\sim$10\text{\,}\mathrm{pC}\text{\,}{\mathrm{MeV}}^{-1}$$ at
$0.25\text{\,}\mathrm{GeV}$ have been demonstrated Couperus  _et al._ (2017).
Even microcoulomb class beams have been derived from highly energetic laser
drivers Shaw  _et al._ (2021). Sub 1 mm-mrad normalised emittance beams have
been measured Weingartner  _et al._ (2012), while the naturally few tens of
micrometers size of the plasma wakefield, in typical conditions for most
experiments, ensures an electron beam duration of a few tens of femtoseconds
Mangles  _et al._ (2006) to a few femtoseconds Lundh  _et al._ (2011).
Combining the above features leads to the generation of beams with an
estimated 6D brightness close to those produced at modern large-scale linacs
Wang  _et al._ (2016). To date, however, there has been relatively little work
on high repetition rate operation over long timescales or on reproducibility,
which are equally important factors when considering a ‘workhorse’ accelerator
to drive secondary sources, as highlighted in the recently published roadmap
for the field Albert  _et al._ (2021).
Almost all studies of LPAs have focused on results ranging from single shots
to of order 100 shots with only a handful of exceptions: a 100,000 shot run at
$1\text{\,}\mathrm{Hz}$ with a stabilised laser path Maier  _et al._ (2020),
electron beam optimisation using active feedback at $5\text{\,}\mathrm{Hz}$
Dann  _et al._ (2019), and experiments with milli-Joule, kHz class lasers
producing moderately relativistic electron beams He  _et al._ (2015); Rovige
_et al._ (2020); Gustas  _et al._ (2018).

Figure 1: Schematic of the experimental setup. Inset top left: measured plasma
electron density (black), its rms deviation (grey shaded area) and the density
profile used in the simulations (red line).

In this work we report on the generation and acceleration of highly
relativistic electron bunches peaking at an average energy of
$78\text{\,}\mathrm{MeV}$ from an LPA using ionisation injection (II) running
at $2.5\text{\,}\mathrm{Hz}$ for $8\text{\,}\mathrm{h}$. Charge was injected
on every one of 72,000 consecutive shots and no decrease in beam charge was
detected over the run, while the spectral peak dropped gradually by
$1.3\text{\,}\mathrm{MeV}$ per 10,000 shots. This was achieved with no
stabilisation or active feedback, and using a simple gas jet target, in the
setup described in section II. The properties of the electron beams are
presented and discussed with reference to particle-in-cell simulations in
sections III and IV respectively. The charge per shot was the parameter with
the largest variance and was strongly correlated with plasma density. The
trend was reproduced in PIC simulations which showed that density changes
alter the self-focusing of the laser pulse Sprangle, Tang, and Esarey (1987)
and thus the volume from which tightly bound electrons are ionised and
injected in to the LPA. In this way small plasma density fluctuations can
significantly alter the charge and phase space of the injected beam. While
ionisation injection reliably produces electron beams, we show that the plasma
density and laser energy should be very tightly controlled for it to be
considered to be a low jitter injection platform.
An immediate consequence of high repetition rate, reproducible beams would be
an improvement in the average dose rate of relativistic electrons or energetic
photons from secondary sources by multiple orders of magnitude. For the
electron energies reported here, this system could (relative to other all-
optical set-ups) serve as the foundation for a high flux source of few keV
x-rays from betatron radiation Rousse  _et al._ (2004), multi-MeV gamma rays
from bremsstrahlung or sub-100 keV x-rays from Thomson scattering. An
additional application for such beams could be very high energy electrons for
the treatment of deep-seated tumours Kokurewicz  _et al._ (2019); Palma  _et
al._ (2015); Papiez, DesRosiers, and Moskvin (2002); DesRosiers  _et al._
(2008); Subiel  _et al._ (2014), where several nanocoulombs of charge should
be delivered in a per-cent level energy bandwidth in an acceptably short time
for the patient. The naturally ultra-high peak dose rate from LPAs may prove
to be beneficial for treatment via reduced damage to healthy tissue, as shown
in FLASH radiography studies Favaudon  _et al._ (2014); Durante, Bräuer-
Krisch, and Hill (2018). In this case, we will use these electron beams in an
all-optical Thomson scattering setupStaufer  _et al._ (2019) to produce hard
x-rays for use in x-ray fluorescence imaging Cheong  _et al._ (2010); Manohar
_et al._ (2016); Grüner  _et al._ (2018); Schmutzler  _et al._ (2021);
Sanchez-Cano  _et al._ (2021); Kahl  _et al._ (2021), an application in which
the imaging time is shortened proportionally with repetition rate.
Implementations of such high-repetition rate LPA sources at GeV levels would,
for example, lead to a reduction of betatron x-ray tomography scan times from
a day to minutes, with photon energies of multiple tens of keV. Another
important benefit would be a large increase in the signal to noise ratio of
data from electron-photon scattering experiments in the nonlinear regime of
QED, where more precise statistical studies are required to better
discriminate between the available models Cole  _et al._ (2018b); Poder  _et
al._ (2018).

##  II Experimental setup

The experimental setup is depicted in Fig. 1. The electrons were accelerated
by focusing a high-intensity laser into a pulsed, supersonic jet of gas. The
gas jet was opened for $7\text{\,}\mathrm{m}\mathrm{s}$ and the laser was
fired $5\text{\,}\mathrm{m}\mathrm{s}$ after the opening to ensure a
continuous flow regime. The gas was evacuated by a differentially pumped cone
with an opening of $8\text{\,}\mathrm{mm}$ placed $4\text{\,}\mathrm{mm}$
above the nozzle. This enabled the ambient pressure in the interaction chamber
to remain at a constant level of
$2.0\text{\times}{10}^{-4}\text{\,}\mathrm{mbar}$ and the vacuum compressor at
$2.1\text{\times}{10}^{-5}\text{\,}\mathrm{mbar}$ when pulsing the gas jet at
a repetition rate of $2.5\text{\,}\mathrm{Hz}$.
To diagnose the electron beam charge a cavity charge monitor (CCM) was used. A
CCM is a cavity that non-invasively measures the charge of the electron bunch
passing through it from the amplitude of the induced TM01 electromagnetic mode
Lipka  _et al._ (2011); Lipka, Lund-Nielsen, and Seebach (2013). It was placed
$1.1\text{\,}\mathrm{m}$ behind the gas jet. At the end of the beamline the
electron bunch was deflected by a dipole magnet with a length of
$0.5\text{\,}\mathrm{m}$ and a field of $0.14\text{\,}\mathrm{T}$. The
dispersed electrons were detected using a DRZ-type phosphor screen Kurz  _et
al._ (2018); Schwinkendorf  _et al._ (2019) which was imaged with a camera to
measure the energy spectrum of the beam. For further characterisation a DRZ
profile screen could be driven into the electron beam
$1.25\text{\,}\mathrm{m}$ from the gas jet to measure the transverse profile
and pointing of the electron bunch. A vacuum beam pipe between the LPA chamber
and the CCM stopped electrons travelling at angles
$|\theta|>$13.3\text{\,}\mathrm{mrad}$$ from propagating downstream to any of
the diagnostics.
The LPA was driven by a high power titanium sapphire laser system delivering
$170\text{\,}\mathrm{mJ}$ pulses centered on $805\text{\,}\mathrm{nm}$ at
$2.5\text{\,}\mathrm{Hz}$, with an rms energy variation of 1.3%. To maintain
the laser energy at a constant level over the full $8\text{\,}\mathrm{h}$ run
only a single manual increase in the pump laser flashlamp voltage was required
around shot number 10,000. This was the only adjustment made during the run.
The full width at half maximum (FWHM) pulse duration was $26.9\pm
0.2\text{\,}\mathrm{fs}$. This and all errors here represent the standard
deviation unless stated otherwise. The laser was focused with an $f/12$ off-
axis parabola resulting in a 1/$e^{2}$ of intensity radius of $8.0\pm
0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$. After accounting for the energy
not contained in the central focal spot (35.3% of the energy was contained
within the FWHM) the peak normalised vector potential of the pulse was
$a_{0}=$1.41\pm 0.03\text{\,}$$. The laser was not operated at its maximum
repetition rate of $10\text{\,}\mathrm{Hz}$ to keep the energy deposition on
the compressor gratings and beamline optics at a moderate level, as excessive
grating heating has been shown to degrade the spatio-temporal properties of
the compressed pulse Fourmaux  _et al._ (2009); Leroux  _et al._ (2018);
Dresselhaus (2020).
The gas was helium with a 0.5% nitrogen dopant by weight (equivalent to
approx. 7 N2 molecules per 10,000 He atoms) and the backing pressure was
$3.9\pm 0.1\text{\,}\mathrm{bar}$. A dedicated transverse probe beam was used
to produce interferograms of the interaction. After converting the phase shift
from the interferograms to a line integrated plasma density, the 3D plasma
density was found assuming cylindrical symmetry via the Abel inversion. The
peak density at the interaction height of $1.2\text{\,}\mathrm{mm}$ above the
nozzle was $(1.47\pm 0.01)\text{\times}{10}^{19}\text{\,}{\mathrm{cm}}^{-3}$,
where the error is the standard error, and its longitudinal profile is shown
in the inset of Fig. 1. The shot-to-shot variation of the plasma density was
estimated from the standard deviation of the peak phase shift of the probe
laser beam. This is less susceptible to noise which can be amplified during
the Abel inversion, especially close to the laser axis. The resulting rms
uncertainty of 3.6% was close to that of the backing pressure at 2.6%.
The gas species was chosen so that electrons would be injected into the
wakefield via ionisation injection Chen  _et al._ (2006); Oz  _et al._ (2007);
Pak  _et al._ (2010); McGuffey  _et al._ (2010); Clayton  _et al._ (2010). In
this scheme the helium electrons and the first five nitrogen electrons are
ionised by the front of the laser pulse and form the plasma wakefield. The
core electrons of nitrogen are only ionised close to the peak of the laser
pulse and can be trapped inside the wakefield potential. In this experiment
self-focusing of the laser pulse was required to trigger ionisation and
injection.

##  III Stable long-term electron acceleration

Before the main run the LPA was optimised for high charge and reliable
injection by varying the gas jet backing pressure, the valve opening time, the
laser energy and the laser focal position relative to the gas jet. After the
optimisation, the LPA was run at a repetition rate of
$2.5\text{\,}\mathrm{Hz}$ for $8\text{\,}\mathrm{h}$, totalling 72,000 shots.
Electron beams were successfully injected and accelerated on every shot. The
average charge of the electron beams measured by the CCM was constant over the
entire dataset as shown in Fig. 2.

Figure 2: Measured charge as a function of shot number and run time. The black
line shows the 100 shot moving average of the charge with its standard
deviation as grey band. A linear fit of all charge measurements is shown in
red. Inset: a histogram of the injected charge.

The mean charge was $14.5\pm 3.8\text{\,}\mathrm{pC}$, corresponding to a flux
of approx. $2.2\text{\,}\mathrm{nC}$ per minute, adding up to more than
$1000\text{\,}\mathrm{nC}$ for the 8 hour run. A linear fit of the charge as a
function of shot number shows a negligible increase of the mean charge by
$0.05\text{\,}\mathrm{pC}$ after 72,000 shots. The minimum measured charge of
$1.5\text{\,}\mathrm{pC}$ is well above the noise level of
$0.5\text{\,}\mathrm{pC}$ of the CCM, demonstrating injection on every shot.
Note that the CCM measures the charge of all of the electrons passing through
it, while the electron spectrometer only detects electrons with a minimum
energy of $37\text{\,}\mathrm{MeV}$. The number of counts registered on the
electron spectrometer screen increased by 2.1% over the whole run.

Figure 3: Pointing and divergence of the electron beams. (a) Average
2D-divergence of 200 shots on the profile screen and the corresponding
pointing of the single shots (red dots). (b) and (c) 100 shot moving average
of the pointing and divergence (black lines) in the non-dispersive axis, with
their 100 shot standard deviations shown as grey bands. Figure 4: Electron
energy spectra $(\mathrm{d}N/\mathrm{d}E/E)$. (a) As function of shot number
and time (waterfall). (b) Average spectrum with its standard deviation (grey
area) and the standard deviation of normalised spectra indicating fluctuations
independent of charge fluctuations (dotted line).

The 2D electron beam pointing and FWHM divergence were measured using the
profile screen in an initial 200 shot run, shown in Fig. 3a. The pointing and
divergence were monitored in the horizontal axis via the electron spectrometer
throughout the main run, and they are plotted in Fig. 3b and c. The FWHM
divergence of the beams was $6.4\pm
0.4\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$ in the horizontal axis
and $12.3\pm 0.5\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$ in the
vertical axis, which was also the dispersion axis of the electron
spectrometer. The divergence was larger in the polarisation axis of the laser,
typical of ionisation injection experiments McGuffey  _et al._ (2010). The
pointing stability was found to be almost symmetric with a standard deviation
of $1.7\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$ and
$1.3\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$ in horizontal and
vertical axis respectively. The standard deviation of the horizontal pointing
over the entire run was $1.9\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$
with a mean divergence in the same axis of $5.5\pm
0.9\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$, within the error of the
initial measurement.
The measured electron energy spectra ($\mathrm{d}N/\mathrm{d}E/E$, where $N$
is the number of electrons and $E$ their energy) are plotted as a waterfall in
Fig. 4 (a). The average spectrum is shown in Fig. 4 (b) along with its
standard deviation and the standard deviation of the normalised spectra, which
is independent of the charge fluctuation. The figure shows that the spectra
were broadband ranging from the low energy cut-off of the spectrometer up to
approximately $130\text{\,}\mathrm{MeV}$. The mean peak energy, defined as the
energy at which $\mathrm{d}N/\mathrm{d}E/E$ is maximised, was $78\pm
7\text{\,}\mathrm{MeV}$. A slow drift of the peak electron energy over time
towards lower values can be observed in Fig. 4. A linear fit to this shows
that the peak energy decreases by $1.3\text{\,}\mathrm{MeV}$ per 10,000 shots,
falling from $82.3\text{\,}\mathrm{MeV}$ to $72.8\text{\,}\mathrm{MeV}$ over
72,000 shots, while the jitter remained approximately constant at 8–9% over
the run.
While the peak energy, as defined, allows for easy comparison with other works
or simulations, it is not necessarily the most important figure of merit for
some applications. For most quasi-monochromatic secondary sources or radio-
biological studies one may employ an energy selective beam transport system to
take the electron beam from the accelerator to the insertion device or target.
Thus, one wishes to produce a constant charge within some energy bandwidth,
integrated over a number of shots. In this experiment $\mathcal{F}=15.1\pm
1.4$ % of the total charge per shot lay between 70 and
$80\text{\,}\mathrm{MeV}$, denoted $q$. This was constant over the full run
and no resolvable evolution was observed over 72,000 shots. Therefore, the
deviation in $\sum_{N}q_{N}=\mathcal{F}Q$ would be dominated by the
fluctuation of the total charge $Q$, which decreases as $1/\sqrt{N}$ with the
number of integrated shots $N$. For a single shot $q=$2.2\pm
0.6\text{\,}\mathrm{pC}$$. Increasing to a larger number of shots with a
practical example, this system would deliver $1\text{\,}\mathrm{nC}$ in the
range 70-$80\text{\,}\mathrm{MeV}$ in 3 minutes with a dose error of 1.2%. Due
to the long term stability of $Q$ and $\mathcal{F}$, this would not change
perceptibly over $8\text{\,}\mathrm{h}$.

##  IV Analysis of the interaction

From a large dataset it is possible to create meaningful correlations of
outputs against measured inputs. These can be used to identify which parts of
a system should be stabilised to improve its precision Maier  _et al._ (2020).
In Fig. 5 the measured charge and peak energies are correlated with the plasma
density and laser energy. Also shown is the variation in the peak electron
energy with charge. To negate the effects of long term drifts these plots are
of a subset of the first 3600 consecutive shots (5% of the total). They show
that the injected charge was particularly sensitive to the plasma density and
to a lesser extent the laser energy. An increase of electron density from 1.4
to $1.5\text{\times}{10}^{19}\text{\,}{\mathrm{cm}}^{-3}$ doubled the measured
charge.

Figure 5: Correlations of the data using a subset of 3600 shots (5% of total
shots). The sub-plot titles show the correlation coefficients (Pearson’s $r$).
Red markers represent PIC simulation results.

The peak electron energy, however, was almost completely unaffected by changes
in the plasma density or the laser energy within these ranges. Finally, there
is a positive correlation between the peak electron energy and the injected
charge. This is the opposite of what one would naively expect from beam
loading. Note that this correlation is quite strongly influenced by a small
number of exceptionally low charge and peak energy shots. If we exclude all
shots more than $2\sigma$ below the mean in both charge and peak energy the
correlation coefficient reduces from 0.36 to 0.24. Overall it is clear that
the main limiting factor to the stability of the system was the plasma density
fluctuation impacting the amount of accelerated charge per shot.
To further understand these correlations and the physics behind the behaviour
of the system a number of particle-in-cell (PIC) simulations were performed
using the cylindrical quasi-3D code FBPIC Lehe  _et al._ (2016). The window
size was $$40\text{\,}\mathrm{\SIUnitSymbolMicro
m}$\times$30\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ with 40 and 10
cells/$\mathrm{\SIUnitSymbolMicro m}$ in $z$ and $r$ respectively, and moved
in $z$ at the linear group velocity of the laser pulse. Three azimuthal modes
were used. The plasma consisted primarily of a pre-ionised species with a
longitudinal profile shown in the inset of Fig. 1 and a peak density of
$1.47\text{\times}{10}^{19}\text{\,}{\mathrm{cm}}^{-3}$, modelled with 108
particles per cell. An additional species modelled the nitrogen component of
the plasma, pre-ionised to a charge state of 5+, with 48 particles per cell
and the same relative density as in the experiment. The laser was modelled as
an $800\text{\,}\mathrm{nm}$ gaussian pulse in space and time, with otherwise
the same properties as were used in the experiment: $a_{0}=1.41$, a $1/e^{2}$
intensity radius of $8.0\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and a FWHM
duration of $26.9\text{\,}\mathrm{fs}$. Its vacuum focal position was at the
top of the plasma density upramp.
The electron spectrum from the simulation is shown in Fig. 6a, where it is
compared to the experimentally measured spectrum.

Figure 6: PIC simulation results. (a) Normalised experimental electron
spectrum (black) with its standard deviation (grey), the raw simulated
spectrum (blue dashes), and the simulated spectrum propagated to the
spectrometer screen (solid blue line). (b) Simulated mean energy (solid blue)
and rms energy spread (orange) vs. $z$. (c) Simulated charge from II vs. $z$
at $n_{e}\pm(1,2)\sigma_{n_{e}}$ (bounded by the blue and pale blue shaded
areas). The behaviour at $n_{e}$ is shown in black, as it is in all following
plots. The peak area of the laser pulse envelope with sufficient intensity to
produce N6+ is shown for the lowest (dashed line) and highest (solid line)
density simulations in grey, in the region where charge is trapped. (d)
Simulated $a_{0}$ evolution at densities $n_{e}\pm(1,2)\sigma_{n_{e}}$.
Normalised $n_{e}(z)$ is shown in orange. The grey dashed lines are the
$a_{0}$ thresholds required to ionise the core electrons of nitrogen according
to the ADK model Pak  _et al._ (2010). (e) Laser waist $w_{0}$ (blue) and
length $c\tau$ (orange) vs. $z$ at $n_{e}\pm(1,2)\sigma_{n_{e}}$. (f)
Normalised slice emittance in $x$ (blue) and $y$ (orange) of II electrons at
$n_{e}\pm(1,2)\sigma_{n_{e}}$ at $z=1$ mm. In dashed, solid and dotted green
are the corresponding current profiles at $n_{e}-2\sigma_{n_{e}}$, $n_{e}$ and
$n_{e}+2\sigma_{n_{e}}$. In higher $n_{e}$ simulations the $w_{0}$ and $c\tau$
are smaller, while $a_{0}$, $Q$ and $\epsilon_{n}$ are larger.

Also shown is the simulated spectrum that has been tracked to the electron
spectrometer screen. The entrance aperture of the CCM had a radius of
$13.3\text{\,}\mathrm{mrad}$, so injected electrons in the simulation
travelling at angles larger than this were eliminated. To produce a spectrum a
random sample of 4 million unique macroparticles that remained from the
injected beam were propagated individually in 3D to a plane representing the
electron spectrometer screen via the measured magnetic field of the dipole.
Since an imaging spectrometer was not used this allows for a better comparison
between the simulation and experimental results, i.e. it includes the effect
of the divergence on the measured spectrum. An example of this can be seen in
Fig. 6a where the spiky ‘raw’ spectrum, caused by periodic ionisation in the
oscillating laser field, is washed out via the propagation leaving a smooth
spectrum similar to the experiment. The propagation algorithm was applied to
all further spectral analysis.
To calculate the charge from the simulation, electrons with
$\gamma\beta_{z}<11\sim\gamma_{ph}$, the Lorentz factor associated with the
wakefield’s phase velocity, were removed to eliminate the electrons that make
up the plasma wave, as were electrons with a divergence larger than the CCM
aperture. A $15.0\text{\,}\mathrm{pC}$ bunch remained from the simulation,
comprised entirely of II electrons, which is in excellent agreement with the
experimental charge measurement. The peak energy of the beam was
$82.9\text{\,}\mathrm{MeV}$, close to the mean experimental value of
$78\text{\,}\mathrm{MeV}$. Fig. 6a shows that the simulation overestimates the
length of the high energy tail and underestimates the amount of charge at low
energies. Note that the spectra in Fig. 6a are normalised to their maximum
values, principally because we do not have a dedicated measurement of the
charge between 0 and $37\text{\,}\mathrm{MeV}$ from the experiment.
In parts b, d and e of Fig. 6 the dynamics of the interaction are examined to
explain the measured beam properties. Figs. 6d and e show that the laser
quickly self-focuses in the density upramp to a peak $a_{0}$ of approximately
3 at a short distance into the flat top density region. At this point charge
is injected via ionisation injection, which is plotted in Fig. 6c (note that
the charge in this figure is the total II charge trapped in the wakefield, not
propagated to the screen). The electron beam mean energy and rms energy spread
are plotted in Fig. 6b. The rapid initial increase in both energy and energy
spread are a result of continuous injection without significant beam-loading.
After an initial ‘over-focusing’ $a_{0}$ stabilises at 2.7 due to a
combination of slight defocusing and self-compression. This drop in $a_{0}$
causes the rate of charge injection to decrease. The electrons that were
injected early continue to accelerate while the reduced rate of trapping slows
down the increase in the energy spread, beginning the process of reducing the
relative energy spread. Once the laser reaches the density downramp it rapidly
defocuses and no further trapping occurs. Simultaneously the bunch reaches
dephasing, which further reduces the energy spread via phase space rotation.
At the end of the simulation there is a broadband electron bunch which is
nevertheless peaked, with a relative energy spread of approx. 30% compared to
its initial 100% spread. The lack of correlation of peak energy with density
and laser energy in Fig. 5 is attributed to operating close to dephasing,
where the beam is located around the zero-crossing of the wakefield at the end
of the plasma, such that slight under-acceleration in lower densities and
slight over-acceleration at higher densities, relative to the dephasing
length, have a minimising effect.
The physics behind the jitter in the output beam parameters will now be
elucidated with a range of PIC simulations. The laser energy $E_{las}$ and the
plasma density $n_{e}$ were varied independently of each other by 1 and 2
experimental standard deviations ($\sigma_{E}=1.3$% and
$\sigma_{n_{e}}=3.6$%). The resulting charges and peak energies are overlaid
on the experimental correlation data in Fig. 5a–d and are broadly in good
agreement, especially in the case of charge as a function of $n_{e}$. At the
highest density there is a rapid increase in charge in the simulation because
5.5 pC of self-injected (SI) charge passed through the virtual CCM aperture.
In general there is a strong scaling of II charge ($Q_{I}$) with $n_{e}$,
increasing by 1.4 pC (9%) per $\sigma_{ne}$, 3 times more rapidly than
$n_{e}$. The naive expectation might have been a 1:1 correspondence, assuming
there was the same volume of laser ionisation but a higher density of nitrogen
atoms.
To explain this observation, the main source of jitter in this experiment, the
evolution of $Q_{I}$ and the laser $a_{0}$, $w_{0}$ and duration $c\tau$ are
plotted in Fig. 6 c-e. Black lines are from simulations at density $n_{e}$,
and the results from simulations at $n_{e}\pm\sigma_{n_{e}}$ and $n_{e}\pm
2\sigma_{n_{e}}$ are bounded by the dark and pale shaded areas respectively.
In all simulations the initially mismatched laser pulse rapidly self-focuses
to a similar $w_{0}$ and $a_{0}$, which is above the ionisation threshold for
both core electrons of nitrogen. Consequently, charge is injected rapidly and
at a similar rate for all densities. The critical part of the simulation is
between $z=0.6\ \textrm{and}\ 0.9$ mm where the $n_{e}$ fluctuations provide
an order 10% correction to the matched spot size, the spot size that drives a
stable blown out wake Lu  _et al._ (2007), and thus a significant increase in
$a_{0}$ in the high $n_{e}$ simulations. The effect of a higher density, a
smaller matched $w_{0}$ and a higher $a_{0}$ is to _increase_ the area of the
laser pulse with $a_{0}$ above the ionisation thresholds, a proxy for the
amount of nitrogen being ionised. This is demonstrated in Fig. 6c, where from
$z=0.6$ mm the area rapidly diverges between the lowest and highest density
simulations. In all simulations, a higher density resulted in a larger area of
the laser pulse with an above-threshold $a_{0}$. This is why the rate of
charge injection diverges after $z=0.6$ mm in Fig. 6c, with high $n_{e}$
simulations injecting charge at a significantly higher rate than those at low
$n_{e}$. Up until $z=1$ mm there is no large difference in pulse length, and
after this point trapping of II charge is terminated as the laser defocuses in
the downramp. Changes in the matched spot size are the principal cause of the
observed charge jitter. The argument that the charge varies according to the
volume of the focused pulse above ionisation threshold would also easily
explain the correlation between charge and laser energy in Fig. 5b, although
we do not examine this in depth here.
As the volume over which the laser can ionise varies with $n_{e}$, so do the
current profiles and transverse phase spaces of the injected beams. Fig. 6f
plots the normalised slice emittances $\epsilon_{n}$ in $x$ (the laser
polarisation plane in the simulation) and $y$ as well as the current profile
of the beam at $z=1$ mm, the point at which injection is terminated in all
simulations. The laser evolution is clearly imprinted on to the bunch. The
first electrons to be injected are expected to be at the front (positive
$(z-ct)$) and the last at the rear due to the phase advance of electrons in
the wake over time. The longitudinal phase space of the beam is approximately
linear and positively chirped in each case, supporting this lemma. The front
of the beam has the highest current since it was injected first during the
over-focusing of the laser, and it is trailed by the electrons injected during
the matched spot phase where there is a large variation in current with
density. While the expected behaviour of particularly $\epsilon_{n,x}$ is
harder to predict since it is a function of the ionising volume and the field
strength where each electron is ionised, it is clear that substantial
differences exist in the slice emittance for relatively small density changes.
This could, for example, lead to large spot size variations if the electron
beam was focused downstream of the plasma.
While the simulations presented here are specific to our experimental set up,
the general behaviour is likely to extend to any II based LPA reliant on self-
focusing, which are common in the literature. With II it is often the case
that the laser intensity in the plasma is close to the ionisation threshold of
the dopant species, as this is required to ionise in a small volume around the
centre of the wakefield so that a good quality beam can be trapped and
accelerated. Our results suggest that to produce a system based on II with
per-cent level shot-to-shot charge fluctuations, the plasma density and laser
energy must be controlled at the level of a fraction of a per-cent. Otherwise,
the nonlinear process of self-focusing and the threshold process of ionisation
combine to enhance any fluctuations in the inputs. It is likely that this
problem can be mitigated by using a guiding channel based on hollow capillary
waveguides Hansson  _et al._ (2014), or dynamically formed plasma waveguides
based on laser Geddes  _et al._ (2004) or discharge ionisation and heating
Karsch  _et al._ (2007). Each of these comes at the expense of a more
complicated setup and/ or increased risk of damage from the LPA driver, and
will contain other sources of jitter which should be evaluated in an LPA
context.

##  V Conclusions

This work has demonstrated that even with simple targetry and a modest laser
energy, that can be produced with a table-top laser system, an ionisation
injection based LPA producing $\sim 80$ MeV electrons can be run at repetition
rates of multiple Hz and remain stable over 8 hours. This is a significant
step towards the use of LPAs as workhorse machines requiring minimal user
intervention over full day timescales and we envisage wider adoption of these
machines outside the traditional national laboratory or university laser lab
environments.
72,000 consecutive shots all demonstrated charge injection and acceleration,
and the mean charge was robust against drifts (within the measurement error)
over the full run. The peak energy dropped by $1.3\text{\,}\mathrm{MeV}$ per
10,000 shots, from an initial $82.3\text{\,}\mathrm{MeV}$. However, the
average charge per shot in selected energy bands, important for future
applications, was extremely stable. For example we found that the system
accelerated an average of $2.2\text{\,}\mathrm{pC}$ to energies between 70 and
$80\text{\,}\mathrm{MeV}$ with no drift over the full 8 hours. The largest
shot-to-shot fluctuation was in the total beam charge. Correlations of the
measured beam properties revealed that the 3.6% fluctuations in the plasma
density were the principal driver of the shot-to-shot charge jitter, however
they had no discernible effect on the peak electron energy. Particle-in-cell
simulations, in agreement with the data, showed that the relative charge
fluctuation was $\sim 3$ times the density fluctuation, and was mostly a
result of a smaller matched spot size of the laser in the plasma increasing
the volume of the (same energy) laser pulse that was able to ionise the core
nitrogen electrons. This work suggests that to achieve low jitter systems
based on ionisation injection very fine control over the intensity of the
laser pulse in the plasma is required, which may be most readily achieved by
sub-per-cent level reproducibility of the plasma density and laser energy.

###### Acknowledgements.

The authors would like to thank M. Dinter, S. Karstensen, S. Kottler, K.
Ludwig, F. Marutzky, A. Rahali, V. Rybnikov, and A. Schleiermacher for their
engineering and technical support. The authors acknowledge funding from
Helmholtz ARD, Helmholtz ATHENA, the DESY Strategy Fund, and the BMBF
InnovationPool through project PLASMED X. This work was supported by the
Maxwell computational resources at DESY.

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##  Appendix A VLA H i and SEST SID CO ($3-2$) position-velocity maps

Position-Velocity (P-V) maps of our VLA H i and SEST SID CO ($3-2$)
observations, oriented parallel to the major axis of NGC 1365, are presented
in Fig. A.1.

Figure 11: [Top]: VLA H i P-V maps oriented parallel to the major axis of NGC
1365 for different minor axis offsets
($y=-5\arcsec,-2\aas@@fstack{\prime\prime}5,0\arcsec,+2\aas@@fstack{\prime\prime}5$).
The major axis offsets should be subtracted by 15″ to obtain the $x$-offsets
from the optical nucleus, with positive values in the NE direction. [Bottom]:
The SEST SID CO($3-2$) P-V maps oriented parallel to the major axis of NGC
1365 for the same minor axis offsets as in the H i maps, with offsets ($x,y$)
being from the optical nucleus. The velocity offsets are with respect to 1613
km s-1.

##  Appendix B On interstellar water chemistry

###  B.1 Water abundances in NGC 1365 vs Galactic prototype regions

In Table B.1 we compare our current NGC 1365 H2O abundance estimates with
recent Odin results for the Galactic Center Sgr A region (Karlsson et al.
kar13 (2013)), and with Odin and Herschel HIFI results for the local Orion
molecular cloud (to demonstrate observationally determined H2O abundances
caused by the PDR, warm cloud core, Hot Core, bipolar outflow and shock
chemistries which are likely to be present also in the central regions of NGC
1365). For comparison we also list Herschel and Odin/SWAS results for the cold
dense cores of the Galactic molecular clouds DR 21 and W 3 and for the S 140
PDR interface region.

Table 6: Comparison between observed H2O abundances in NGC 1365, our Galactic
Center region (Sgr A) and some nearer Galactic sources.

Source | Region | Abundance vs H2 [$\times 10^{-8}$] |
---|---|---|---
(distance) | (kinetic temperature) |  Odin **footnotemark: $*$ | Herschel
NGC 1365 aafootnotemark: $a$ | absorption ($\approx$20 K) | $5-8$ |
(18.6 Mpc) | warm emission ($40-60$ K) |  $6-600$ssfootnotemark: $s$ |
| shocks (350 K) |  |  $200-10,000$uufootnotemark: $u$ttfootnotemark: $t$
Sgr A ddfootnotemark: $d$ | +50 km s-1 Cloud (60 K) | 5 |
(8.7 kpc) | Red wing of +50 km s-1 Cloud eefootnotemark: $e$ | 130 |
| CND (150 K) fffootnotemark: $f$ | 9 |
| Red wing of CND eefootnotemark: $e$ | 760 |  $100-1300$bbfootnotemark: $b$
| +20 km s-1 Cloud (60 K) | 3 | 4ccfootnotemark: $c$
| Red wing of +20 km s-1 Cloud eefootnotemark: $e$ | 110 |
|  $-$30 km s-1 arm ggfootnotemark: $g$ | 4 |
Orion KL mmfootnotemark: $m$ | PDR interface region (70 K) | 9 h,ih,ifootnotemark: $h,i$ |
(450 pc) | 2′ south core (75 K) | 11 h,ih,ifootnotemark: $h,i$ |
| Compact Ridge (115 K) | 280 jjfootnotemark: $j$ | 260 kkfootnotemark: $k$
| Hot Core (200 K) | 1200 jjfootnotemark: $j$ | 1000 l,kl,kfootnotemark: $l,k$
| Low velocity outflow | 290 jjfootnotemark: $j$ | 480 kkfootnotemark: $k$
| High velocity outflow (vs all H2) | 2200 jjfootnotemark: $j$ |
|  |  | 7000 llfootnotemark: $l$
| High velocity outflow (vs shocked H2] | 10000 hhfootnotemark: $h$ |
|  | 28000 jjfootnotemark: $j$ |
DR 21 | Cool dense core (23 K) |  | 0.1 n,on,ofootnotemark: $n,o$
(1.7 kpc) | Foreground cloud (10 K) |  | 2 n,on,ofootnotemark: $n,o$
| Low velocity outflow |  | 280 n,on,ofootnotemark: $n,o$
W3 IRS 5 | Cool dense core (40 K) | 0.2 ppfootnotemark: $p$ |
(2.3 kpc) |  |  |
S 140 | Clumpy PDR interface with dark cloud (55 K) | 5/0.1 q,rq,rfootnotemark: $q,r$ |
| Low velocity outflow | 40 qqfootnotemark: $q$ |

(∗) based upon ortho-H2O, H${}_{2}^{18}$O and H${}_{2}^{17}$O
($1_{10}-1_{01}$) observations; (a) this paper; (b) from Herschel mapping of
the CND/Sgr A* by Armijos-Abendano et al. (arm19 (2019)); (c) from Herschel
mapping of the +20 km s-1 Cloud by Armijos-Abendano et al. (arm19 (2019)); (d)
from Karlsson et al. (kar13 (2013)), assuming an ortho-to-para ratio of 3; (e)
outflow/shock; (f) Circumnuclear Disk (torus) surrounding the Sgr A* black
hole; (g) foreground spiral arm; (h) Olofsson et al. (olo03 (2003)); (i)
Wirström et al. (wir06 (2006)); (j) Odin spectral scan, Persson et al. (per07
(2007)); (k) Herschel/HIFI spectral scan, Neill et al. (nei13 (2013)); (l)
Melnick et al. (mel10 (2010)); (m) mapped in detail by Odin, Hjalmarson et al.
(hja05 (2005)); (n) based upon para-H2O ($1_{11}-0_{00})$ observations, van
der Tak et al. (tak10 (2010)); (o) assuming an ortho-to-para ratio of 3; (p)
from Odin mapping by Wilson et al. (wil03 (2003)) in close agreement with SWAS
results by Snell et al. (sne00 (2000)); (q) clump/interclump values; (r) from
Odin mapping by Persson et al. (per09 (2009)). (s) the lower abundance
estimates appear at higher densities and can be accommodated by PDR models,
while the high abundances (at a density of $10^{4}$ cm-3) would indicate that
the existing shocks also contribute to the observed ground state H2O emission.
(t) shock model abundance (Flower & Pineau des Forêts flo10 (2010)) in an
effective (accumulated) 10″ size area of the circumnuclear torus (see Sect.
4.4). (u) caused by slow velocity shock chemistry in the case of (here
unconfirmed) dominant FIR excitation (Gonzalez-Alfonso et al. gon10 (2010),
gon12 (2012), gon14 (2014)).

###  B.2 Water chemistry in cold and warm regions

To make our discussion of the interstellar H2O chemistry going on in the
central regions of NGC 1365 more to the point, we here provide a very
simplified summary of the main formation and destruction routes, for cases
where the theoretical models are well established (for details we refer to the
extensive review paper by van Dishoeck et al. dis13 (2013)) and where a sample
of relevant observed H2O abundances have been listed in Table B.1.

####  B.2.1 Low temperature gas-phase water chemistry

The oxygen chemistry is initiated by cosmic-ray (CR) ionization of H2 which
rapidly leads to the formation of the trihydrogen ion (H${}_{3}^{+}$). Next
step is

H${}_{3}^{+}$ \+ O =¿ OH+ \+ H2 ; H2O+ \+ H (1)

followed by

OH+ \+ H2 = H2O+ \+ H (2)

and, subsequently,

H2O+ \+ H2 = H3O+ \+ H (3)

followed by dissociative recombination,

H3O+ \+ e- =¿ H2O + H ($\approx$17%), (4)

but also ($\approx$83%) OH + H2 ; OH + 2H ; O + H2 \+ H.

At low temperatures the main loss of gaseous H2O is via accretion (adsorption,
“sticking”) onto cold dust grain surfaces (where water ice has been observed
to be a main constituent). Only at grain temperatures of 100 K, or higher, the
H2O molecules are efficiently desorbed (evaporated) from the icy grain
surfaces. While the just (crudely) described ion-molecule reaction scheme, for
a standard Galactic CR flux, is capable of producing a gas-phase H2O abundance
$X$(H2O) $\approx 10^{-6}-10^{-7}$ versus H2, the “freeze out” on cold grain
surfaces may reduce this abundance by several orders of magnitude, as is
illustrated in Table B.1 by the observed values of $(1-2)\times 10^{-9}$ in
the DR 21 and W 3 molecular cloud cores.

The expected gas-phase H2O abundance is closely related to the H${}_{3}^{+}$
production. An increase of the CR flux leads to a faster chemistry, and also
results in higher H${}_{3}^{+}$ and H2O abundances (cf. Farquhar et al. far94
(1994); Nilsson et al. nil00 (2000), on-line Figs. 16 and 21). In NGC 1365 we
have already envisioned a probable CR focusing by the observed magnetic field
aligned along the bar. Alternatively, X-ray ionization of H2, rapidly leading
to enhanced H${}_{3}^{+}$ abundance, also would do the same job. X-ray
emission has indeed been observed in NGC 1365, especially in the nuclear
region (from the central AGN, black hole region; see Lindblad lin99 (1999)).
Theory for XDR (X-ray Dominated Region) physics, chemistry has been developed
by Meijerink & Spaans (mei05 (2005)) and has been applied to galaxy nuclei by
Meijerink et al. (mei07 (2007)). Also an increased UV flux from one or more
newly formed massive stars, influencing the surface layers of their mother
molecular clouds and a developing H ii region also is bound to produce an
enhanced H2O abundance (theory by e.g, Hollenbach et al. hol09 (2009), hol12
(2012), who also study the influences of increased CR ionization). In Table
B.1 we have listed observed H2O abundances of $(5-10)\times 10^{-8}$ in the
Orion PDR (Photon Dominated Region) interface layer between the molecular
cloud and the M 42 H ii region, and in the S 140 PDR. The water abundances of
$(2-3)\times 10^{-8}$, observationally estimated for the DR 21 foreground
cloud and the Sgr A $-$30 km s-1 arm, both seen in absorption, also are likely
results of PDR chemistry, but now in lower density regions of low visual
extinction where the general Galactic background UV radiation can penetrate.

####  B.2.2 Ice chemistry

In parallel with the low temperature gas-phase chemistry, there are also
ongoing grain surface reactions. In addition to the formation of H2, which
requires a third body for its efficient formation, there are also a number of
reactions (collisions) between atoms and molecules sticking on the grain
surfaces, and jumping between the potential wells on the surface, hence
forming ices containing H2O, CH3OH, NH3, CO, CO2, and CH4. To contribute to
the gas-phase chemistry, these surface-sticking species need to be desorbed
(evaporated, i.e., converted from solid state to gas-phase constituents). This
desorption will require a grain temperature increase, which may be caused by
absorption of the strong UV light from nearby newly formed stars, which is the
case in PDR chemistry (Hollenbach et al. hol09 (2009)) as well as in Hot Core
chemistry (Rodgers & Charnley rod01 (2001)). According to Table B.1 the
observed H2O abundances are as high as $10^{-5}$ and $3\times 10^{-6}$ in the
high density Orion Hot Core (where $T_{\rm kin}\approx 200$ K) and Compact
Ridge (where $T_{\rm kin}\approx 115$ K) regions, respectively.

####  B.2.3 High temperature water chemistry

Detachment of ice layers of grain-sticking molecules may also be caused by
shocks, as may be the case in low velocity bi-polar outflows from young stars,
supernova shocks and Galactic density wave shocks (cf. Flower & Pineau des
Forêts flo03 (2003); Melnick et al. mel08 (2008)). In Table B.1 we list
observed gas-phase H2O abundances of a few $\times 10^{-6}$ in the Orion and
DR 21 low velocity outflows, and similar values for the red wing emissions
from the Sgr A molecular cloud cores.

In addition to the evaporation of ices we also have to consider some key high
temperature gas phase reactions, that is,

O + H2 = OH + H (5)

and subsequently

OH + H2 = H2O + H. (6)

Shock heating and compression in this way may result in very high post shock
gas-phase H2O abundances (cf. e.g., Bergin et al. ber98 (1998); Flower et al.
flo003 (2003); Kristensen et al. kri07 (2007); Flower & Pineau de Forêts flo10
(2010)), as illustrated in Table B.1 by an observationally determined $X$(H2O)
versus shocked H2 approaching $3\times 10^{-4}$ in the Orion high velocity
outflow, where (almost) all elemental oxygen has been “locked-up” in gas-phase
water. Here we should again remember that outflow driven shock excitation of
H2 has been observed by Galliano et al. (gal12 (2012)) in some of the
circumnuclear torus hot spots of NGC 1365 (i.e., massive stellar clusters with
their associated massive star forming gas clouds).

##  Appendix C Summary of the molecular physics and the interstellar chemistry
of CH and CH+

An energy level diagram for CH is provided by Rangwala et al. (ran14 (2014))
and more detailed ones may be found in Rydbeck et al. (ryd76 (1976)) and
Turner (tur88 (1988)). The rotational levels of the light weight CH molecule
appear in the submm/FIR range and show a doublet spectral line pattern caused
by the splitting of the individual rotational state energies into
$\Lambda$-doublets, resulting from the relative orientations of the orbital
momentum axis of the unpaired $\pi$ electron and the molecular rotation axis
(being parallel or orthogonal, denoted as + or – parity). Magnetic hyperfine
interaction further splits the $\Lambda$-doublet state energies, which was
crucial for the identification of the interstellar radio transitions of CH,
but causes overlapping lines and even difficult blends of emission and
absorption at submm and FIR wavelengths. The observed doublet pattern of the
rotational lines is the result of the quantum mechanical selection rule that a
state parity change is required for an electric dipole transition to take
place.

CH, CH+, and CN were the first interstellar molecules to be identified (in
1937-1941), via their absorption lines of the visual light from background
stars (see the review by Gerin et al. ger16 (2016)). The interstellar
detection of the radio signals at $\approx 3.3$ GHz from the three hyperfine
transitions within the $\Lambda$-doubled ${}^{2}\Pi(N=1,J=1/2)$ CH ground
state was a matter of searches across a wide frequency range, since the
frequencies were only very crudely known from molecular quantum mechanics
applied to optical spectroscopy (Rydbeck et al. ryd73 (1973); Turner &
Zuckerman tur74 (1974)). Subsequently the three CH hyperfine transitions were
extensively observed and analyzed, and it was shown that these lines were weak
masers (amplifying any background emission) in molecular cloud cores, cold
dark clouds as well as in lower density spiral arm clouds (Rydbeck et al.
ryd76 (1976); Hjalmarson et al. hja77 (1977)). The CH abundance versus H2 was
observed to decrease (from $\approx 3\times 10^{-8}$ to $\approx 6\times
10^{-10}$) with increasing cloud density, which may well be expected in
regions where this reactive radical is consumed in the formation of more
complex molecules, and also is consistent with more recent interstellar
chemistry modeling (e.g., Herbst & Leung 1986 a,b). The lower satellite line
of CH at 3264 MHz displayed an interesting anomaly in that its intensity
rapidly increased with increasing FIR continuum emission, which was explained
as “FIR pumping” and resulted from the parity structure of the
$\Lambda$-doubled rotational states (Rydbeck et al. ryd76 (1976)).
Subsequently CH detections in external galaxies could also be done, using the
very sensitive maser receiver from Onsala Space Observatory installed on the
Australian Parkes 64-m radio telescope (Whiteoak et al. whi80 (1980)).

The first astronomical observation of CH in the far-infrared was performed by
Stacey et al. (sta87 (1987)). Using NASA’s Kuiper Airborne Observatory they
were able to detect the $N=1,J=1/2$ ground state $\Lambda$-doublet to the
$N=2,J=3/2$ rotationally excited $\Lambda$-doublet state transitions of CH as
an absorption line doublet at $\approx 149$ $\mu$m (2006.8/2010.8 GHz) against
the FIR thermal dust emission of Sgr B2. These transitions are “always”
expected to be observed in absorption since their critical densities for
collisional excitation are ¿ 1010 cm-3. Stacey et al. estimate the CH
abundance versus H2 to be $\approx 10^{-9}$ for the Sgr B2 (denser) molecular
cloud and $\approx 10^{-7}$ for the (lower density) absorbing clouds in the 3
kpc arm and the expanding molecular ring. They also conclude that the enhanced
3264 MHz CH emission from the Sgr B2 molecular cloud must be primarily a FIR
excitation effect. Recently much more detailed, higher spectral and spatial
resolution, 2 THz CH absorption doublet studies toward a number of Galactic
molecular clouds have been performed from SOFIA (the Stratospheric Observatory
for Infrared Astronomy; Wiesemeyer et al. wie18 (2018)).

The lowest rotationally excited CH ($N=1;J=3/2$) state, only 25.7 K above the
ground state, was detected by means of the very large 1000 foot (300 m)
Arecibo and Green Bank 300 foot (91 m) telescopes in a number of molecular
cloud cores, in terms of four hyperfine transitions within the
$\Lambda$-doublet at $\approx 700$ MHz, all seen in absorption (Ziurys &
Turner ziu85 (1985); Turner tur88 (1988)). An interesting conclusion was that
the CH abundance versus H2 was as high as $2\times 10^{-8}$ in regions of
density $2\times 10^{4}$ cm-3 – an abundance which was too high to be
consistent with chemical models for dense quiescent clouds, but instead
pointed at the action of shock chemistry. In fact, molecular outflows (causing
shocks) have been observed in the regions considered. The results from the
(early) modeling of low velocity shocks ($5-20$ km s-1) in dense (104 cm-3)
gas clouds by Mitchell (mit84 (1984)) is a useful guideline here – and may be
so also in our case of NGC 1365.

The detection of the CH 533/537 GHz transition doublet in the directions of
several massive star-formation regions relied on Herschel HIFI observations,
and revealed intriguing blends of absorptions from lower density (diffuse)
spiral arm foreground clouds and emission from the molecular cloud cores
(Gerin et al. ger10 (2010)). Thanks to the hyperfine splitting, nowadays known
from laboratory spectroscopy, paired with molecular quantum mechanics the
emission/absorption line pattern could be deconvolved. The CH abundance vs H2
in the foreground clouds was determined to be $\approx 3.5\times 10^{-8}$ in
clouds having densities in the range $100-1000$ cm-3 – an abundance consistent
with the results from decades of optical CH absorption line observations, and
precisely modeled by PDR chemistry (originating from Black & Dalgarno bla73
(1973); see Gerin et al. ger10 (2010) and Gerin et al. ger16 (2016) for a
discussion and references). From their SOFIA observations and a careful
analysis of the CH absorption line doublet at $\approx 149$ $\mu$m (from the
ground state, just as the 533/537 GHz transitions) Wiesemeyer et al. (wie18
(2018)) confirm this conclusion, that is, that CH may be used as a reliable
tracer of H2 columns and not only in lower density (diffuse) clouds but also
in denser high mass star formation regions. This is contrary to the previously
mentioned expectations from chemical models for dense quiescent clouds (e.g.,
Herbst & Leung 1986 a,b) and suggests influences of shock chemistry (caused by
bipolar outflows) in the denser star forming clouds (as was concluded already
by Turner tur88 (1988), based upon his detailed analysis of the $\approx 700$
MHz CH absorption line data).

The Herschel HIFI detections by Falgarone et al. (2010a ) of the CH+($1-$0),
and 13CH+($1-0$), absorption lines at 835 and 830 GHz in the directions of the
same star-formation regions also showing CH absorptions, confirmed the long
standing problem that the observed CH+ abundance versus H2 $\approx 5\times
10^{-8}$, i.e. similar to that of CH and several orders of magnitude higher
than the predictions by UV-driven steady state PDR chemistry models. So-called
turbulent dissipation models, where the activation energy ($\approx 4600$ K),
needed for the endothermic reaction C+ \+ H2 = CH+ \+ H, is drawn from
dissipation of turbulence, appear to provide the solution of this dilemma
(TDR; Falgarone et al. 2010a ; see Gerin et al. ger16 (2016) for a discussion
and further references). In their Herschel HIFI observations of CH+ toward the
massive star-formation region DR 21 molecular cloud ridge – one of the most
powerful molecular outflows seen in vibrationally excited H2 – another CH+
surprise appeared, in terms of a strong broad CH+($1-0$) emission line from
the cloud core, together with the expected deep absorption features from the
DR 21 molecular ridge and foreground gas (Falgarone et al. 2010b ). The broad
emission was satisfactorily interpreted by means of a state-of-the-art C-shock
model in dense, UV-illuminated gas (Falgarone et al. 2010b ), while the
absorptions were consistent with TDR modeling results (Falgarone et al. 2010a
).

The importance of intense UV-illumination for the CH+ formation in dense
molecular cloud PDR surfaces became very clear from the Herschel observations
of CH+ in the Orion Bar (Nagy et al. nag13 (2013), nag17 (2017)) and the
mapping of CH+ and CH in an extended region including Orion KL as well as the
Orion Bar (Morris et al. mor16 (2016)). In the Orion Bar the CH+ ($J=1-0$ and
$2-1$) emission lines turned out to be markedly broader than other molecular
lines ($\Delta v\approx 5$ km s-1 vs $\approx 3$ km s-1, as observed by HIFI),
and the strong higher energy lines observed by PACS (up to $J=6-5$ at an upper
state energy of 838 K) and indicate a kinetic temperature of $500-1000$ K.
This very high rotational excitation of CH+ together with the line broadening
are clear signs of “formation pumping”, where the activation energy of 4600 K,
required for the endothermic reaction C+ \+ H2 = CH+ \+ H to take place, is
supplied by vibrationally (and rotationally) excited H2, caused by the intense
UV-illumination in the observed regions. The H2 excited state excess energy
here will cause the high rotational excitation of CH+ as well as line
broadening by additional translational CH+ motion (as discussed in detail by
Nagy et al. nag13 (2013) and theoretically modeled by Godard and Cernicharo
god13 (2013); see also Gerin et al. ger16 (2016) for discussions and
references). In the Orion Bar the observed CH+ population distribution leads
to a column density 40 times larger than the “minimum” column density
calculated from the CH+($J=1-0$) emission line, assuming a “reasonable”
excitation temperature in the range $20-150$ K, and hence an estimated CH+
abundance larger than that of CH (Table 2 of Nagy et al. nag17 (2017)). This
CH+ formation scenario is favored also for the extended Orion molecular cloud
surface where (some) CH also may result from dielectronic recombination of CH+
(Morris et al. mor16 (2016)).


# Can a pure neutron state of matter exist?

Furong Xu  State Key Laboratory of Nuclear Physics and Technology, School of
Physics, Peking University, Beijing 100871, China

††journal: Science Bulletin

Atomic nucleus is a self-organized system composed of protons and neutrons
through the short-range attractive nuclear force. For protons there is a long-
range Coulomb repulsion, while no Coulomb interaction exists between neutrons.
Then, a naive question arises: can a pure neutron system exist? A pure neutron
system seems to be more bound because there is no Coulomb repulsion between
neutrons. However, no bound pure neutron system has been found except in
neutron stars where neutrons are squeezed together by the huge gravitational
force.

From a simple phenomenological macroscopic point of view, an atomic nucleus
can be considered as a droplet with volume, surface and Coulomb energies.
Indeed, the Coulomb repulsion reduces the binding energy of the nucleus, which
means that the Coulomb energy is not conducive to the binding of the nucleus.
However, there exists a quantum term, called the symmetry energy proposed by
Wigner, which arises from the Pauli exclusion of identical particles. In the
symmetry energy term, the asymmetry in the number of neutrons and protons
results in a decrease in the binding energy of the nucleus, which means that
the symmetry energy prefers similar numbers of neutrons and protons to very
different numbers. Therefore, according to the phenomenological liquid drop
model, the Coulomb energy and symmetry energy compete to balance the numbers
of neutrons and protons to make the nucleus lower in energy.

In a microscopic framework of the shell model, neutrons and protons occupy
their respective single-particle levels, and prefer their Fermi surfaces close
to each other making the A-body nucleon system lower in energy. A nucleus can
balance its neutron and proton numbers through the $\beta$ transition in which
a neutron (proton) is converted into a proton (neutron) by the emission of an
electron (positron) accompanied by an antineutrino (neutrino). The neutron and
proton can be approximately treated as the same type of Fermions by
introducing an isospin quantum number, e.g., defining the isospin projection
$t_{z}=1/2$ for neutron and $-1/2$ for proton. Then, the nucleus appears with
an isospin quantum number as well, similarly to the spin number of its angular
momentum, with an isospin projection $T_{z}=\frac{1}{2}(N-Z)$ measuring the
difference in the number of neutrons and protons.

Figure 1 plots the ground-state energies of $A=4$ nuclear systems. We see that
the $A=4$ energy is minimized at $T_{z}=0$, i.e., at the 4He nucleus in which
the neutron number is equal to the proton number. While the $T_{z}=0$ 4He is
well bound, the $T_{z}=-1$ 4Li and $T_{z}=+1$ 4H are unbound resonances which
are above the 3He+p and 3H+n breakup thresholds, respectively. Therefore, a
pure neutron system is not favored, though the Coulomb interaction between
protons is repulsive. The many-body correlations of a nuclear system favor a
certain balance in the number of neutrons and protons.

Nevertheless, it is still possible that pure neutron systems might exist in
the form of resonances. The first pure neutron system that comes to mind may
be the dineutron with two neutrons (2n), which is unbound by only about 100
keV. However, the two neutrons stay in the $l=0$ s-wave with zero centrifugal
barrier, and therefore no 2n resonance is possible. However, a pure four-
neutron (4n) system, called the tetraneutron, was speculated about 60 years
ago, and has been sought ever since. Unfortunately, no firm evidence for a
bound 4n state has been obtained. Whether can the 4n system exist in the form
of resonance? Indeed, an experiment performed in 2016 at RIKEN, using the
double-charge-exchange (DCX) reaction 4He(8He, 8Be) [1], indicates the
possibility of the existence of the tetraneutron resonance, though the
conclusion cannot be definitely made due to the large experimental
uncertainty. Many theoretical attempts have also been made to investigate
possible multineutron systems. Most theoretical investigations do not support
bound multineutrons [2, 3, 4], instead suggesting multineutron resonances [5,
6, 7, 8, 9]. Among the theories, first-principles calculations based on
realistic nuclear forces are fascinating. However, one also needs to well
treat the neutron correlations including the coupling to the continuum [9].

Figure 1:  The ground-state energies of $A=4$ nuclear systems as a function of
the isospin projection $T_{z}$. The calculated energies and resonance widths
from [10] are adopted because available experimental data for 4Li and 4H can
be significantly different from one experiment to another (see Ref. [10]). The
blue dashed line indicates the breakup threshold.

After the DCX reaction 4He(8He, 8Be) experiment done in 2016 [1], an
international collaboration team performed another fascinating experiment
using the knockout reaction 8He(p, p4He), published in Nature recently [11].
In the experiment, the high-energy 8He secondary beam was projected onto a
target of protons. The 8He nucleus has a structure of four valence neutrons
outside the $\alpha$ core (i.e., 4He). In the quasi-elastic knockout reaction,
the $\alpha$ core was removed from 8He by a target proton, leaving a 4n system
moving on. The experiment [11] did not directly measure the 4n system, but
detected the knocked-out $\alpha$-particle and the scattered proton. By the
missing mass reconstruction, the energy spectrum of the 4n system can be
obtained, which gives a well pronounced peak at an energy of
2.37$\pm$0.38(statistical error)$\pm$0.44(systematic error) MeV near the 4n
threshold, suggesting a resonance-like structure of the 4n system [11]. This
suggestion of a 4n resonance is in agreement with the DCX reaction 4He(8He,
8Be) experiment [1], but with much higher experimental precisions. A resonance
width was extracted, $\Gamma$=$1.75\pm 0.22$(stat.)$\pm$0.30(sys.) MeV, which
corresponds to a lifetime of only $(3.8\pm 0.8)\times 10^{-22}$ seconds [11].

Certainly, more experimental and theoretical studies are necessary to further
understand the nature of the observed peak in the missing mass spectrum [11]
and also to exclude other possibilities, e.g., the peak manifesting itself as
a possible coupling among the 4n and other ingredients surrounding the
reaction channel, as commented in Refs. [12, 13]. A direct detection of the 4n
system should be of most value. The measurement of the momentum correlation of
the four neutrons can give more insights into the structure of the 4n system.
Choosing different reaction mechanisms would provide additional information
about the properties of the 4n system. Nevertheless, combined with the recent
state-of-the-art first-principles calculations [6-10], a tetraneutron
resonance is highly likely. Figure 2 displays the ab initio predictions based
on no-core shell model (NCSM) [5, 6] and no-core Gamow shell model (NCGSM)
[9], along with the energies and widths of the 4n peaks obtained in the
experiments [1, 11]. We see that the newest experimental result published in
Nature [11] is in excellent agreement with the theoretical prediction made by
the elaborated ab initio NCGSM [9]. Starting from a fundamental chiral nuclear
force, the NCGSM [9] treated well the coupling to the continuum by using the
complex-momentum Berggren ensemble. The continuum effect is important in the
calculation for a resonant state. The use of natural orbitals includes more
correlations [9].

Figure 2:  Experimental [1, 11] and predicted [5, 6, 9] resonance energy and
width of the 4n system, with the horizontal axis indicating the year when the
experimental or calculated result was published. In the year-2022 experimental
data [11] the narrower green shadow indicates the experimental error in the
width [11], while the year-2016 experiment [1] only estimated an upper limit
on the width, indicated by blue arrows. JISP16, DaeJ16 and N3LO indicate the
realistic nuclear forces used in NCSM [5, 6] and NCGSM calculations [9].

The ab initio calculations [7, 9] have even showed that a three-neutron (3n)
resonance is possible with a lower energy and narrower width than the 4n
resonance. Though it has been commented that the 3n resonance may be less
likely to exist due to the odd number of nucleons and thus weaker binding
[11], one probably should not consider it with the pairing mechanism of even-
number nucleon systems. One might treat it as a whole with full three-body
correlations and three-nucleon force. For the 4n system, a realistic four-
nucleon force may also need to be considered.

Multineutron resonances provide a unique playground for the studies of
neutron-neutron interaction and many-body correlations of pure neutron
systems. One can study neutron-proton and proton-proton interactions by
neutron-proton and proton-proton scatterings, respectively, but neutron-
neutron scattering is not an idea to study the neutron-neutron interaction
because no pure neutron target is available. Further studies of the pure
neutron states of matter are necessary to answer the fundamental questions of
nuclear physics and the impacts, e.g., on the nuclear processes of nuclear
astrophysics.

Acknowledgements

Dr. Jianguo Li at Institute of Modern Physics, Chinese Academy of Sciences, is
acknowledged for preparing the figures and useful discussions. The author
thanks the support from the National Natural Science Foundation of China
(11835001, 11921006, and 12035001).

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