# Experimental determination of the dissociative recombination rate
coefficient for rotationally-cold CH+ and its implications for the diffuse
cloud chemistry

Daniel Paul Max-Planck-Institut für Kernphysik, 69117 Heidelberg, Germany
Columbia Astrophysics Laboratory, Columbia University, New York, NY 10027, USA
Manfred Grieser Max-Planck-Institut für Kernphysik, 69117 Heidelberg, Germany
Florian Grussie  Max-Planck-Institut für Kernphysik, 69117 Heidelberg, Germany
Robert von Hahn  Max-Planck-Institut für Kernphysik, 69117 Heidelberg, Germany
Leonard W. Isberner I. Physikalisches Institut, Justus-Liebig-Universität
Gießen, 35392 Gießen, Germany  Max-Planck-Institut für Kernphysik, 69117
Heidelberg, Germany  Ábel Kálosi Max-Planck-Institut für Kernphysik, 69117
Heidelberg, Germany  Columbia Astrophysics Laboratory, Columbia University,
New York, NY 10027, USA  Claude Krantz Max-Planck-Institut für Kernphysik,
69117 Heidelberg, Germany  Holger Kreckel Max-Planck-Institut für Kernphysik,
69117 Heidelberg, Germany  Damian Müll Max-Planck-Institut für Kernphysik,
69117 Heidelberg, Germany  David A. Neufeld Department of Physics & Astronomy,
Johns Hopkins University, Baltimore, MD 21218, USA  Daniel W. Savin Columbia
Astrophysics Laboratory, Columbia University, New York, NY 10027, USA  Stefan
Schippers I. Physikalisches Institut, Justus-Liebig-Universität Gießen, 35392
Gießen, Germany  Patrick Wilhelm Max-Planck-Institut für Kernphysik, 69117
Heidelberg, Germany  Andreas Wolf Max-Planck-Institut für Kernphysik, 69117
Heidelberg, Germany  Mark G. Wolfire Department of Astronomy, University of
Maryland, College Park, MD 20742, USA  Oldřich Novotný Max-Planck-Institut für
Kernphysik, 69117 Heidelberg, Germany

(August 31, 2022)

###### Abstract

Observations of CH+ are used to trace the physical properties of diffuse
clouds, but this requires an accurate understanding of the underlying CH+
chemistry. Until this work, the most uncertain reaction in that chemistry was
dissociative recombination (DR) of CH+. Using an electron-ion merged-beams
experiment at the Cryogenic Storage Ring, we have determined the DR rate
coefficient of the CH+ electronic, vibrational, and rotational ground state
applicable for different diffuse cloud conditions. Our results reduce the
previously unrecognized order-of-magnitude uncertainty in the CH+ DR rate
coefficient to $\sim\pm 20\%$ and are applicable at all temperatures relevant
to diffuse clouds, ranging from quiescent gas to gas locally heated by
processes such as shocks and turbulence. Based on a simple chemical network,
we find that DR can be an important destruction mechanism at temperatures
relevant to quiescent gas. As the temperature increases locally, DR can
continue to be important up to temperatures of $\sim$600\text{\,}\mathrm{K}$$
if there is also a corresponding increase in the electron fraction of the gas.
Our new CH+ DR rate coefficient data will increase the reliability of future
studies of diffuse cloud physical properties via CH+ abundance observations.

Diffuse molecular clouds (381), Laboratory astrophysics (2004), Reaction rates
(2081), Electron recombination reactions (2262)

##  1 Introduction

The methylidine cation $\mathrm{CH}^{+}$ was the first molecular ion detected
in the interstellar medium (Douglas & Herzberg, 1941). In the cold and dilute
environment of diffuse clouds it radiatively relaxes to an internal
temperature of $T_{\mathrm{ex}}\sim$3\text{\,}\mathrm{K}$$ and is
predominantly found in its vibrational $v=0$ and rotational $J=0$ ground state
(Godard & Cernicharo, 2013).

Despite its early detection, the high abundance of $\mathrm{CH}^{+}$ in
diffuse clouds still remains an astrochemical puzzle (Indriolo et al., 2010).
This is because the only known production mechanism via C+,

$\mathrm{C}^{+}+\mathrm{H_{2}}\rightarrow\mathrm{CH}^{+}+\mathrm{H},$ (1)

is endothermic by $\sim$0.4\text{\,}\mathrm{e}\mathrm{V}$$ (Gerlich et al.,
1987) and cannot proceed at quiescent diffuse cloud kinetic temperatures111We
distinguish between the kinetic temperature $T_{\mathrm{k}}$ of the gas in the
cloud, which characterizes the velocity distributions for all particles, and
the excitation temperature $T_{\mathrm{ex}}$, which describes the internal
state-population distribution of the CH+ molecules. The kinetic temperature is
sometimes also called the translational temperature. of
$T_{\mathrm{k}}=$40-130\text{\,}\mathrm{K}$$
($k_{\mathrm{B}}T_{\mathrm{k}}=$3-11\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$,
where $k_{\mathrm{B}}$ is the Boltzmann constant; Shull et al. 2021)222In
principle, CH+ can also form via
$\mathrm{C}^{2+}+\mathrm{H_{2}}\rightarrow\mathrm{CH}^{+}+\mathrm{H}^{+},$ but
this reaction is negligible because of its experimentally-known low rate
coefficient of
$\lesssim$2\text{\times}{10}^{-12}\text{\,}\mathrm{c}\mathrm{m}^{3}\,\mathrm{s}^{-1}$$
and the low abundance of C2+ compared to C+ in diffuse clouds (Plašil et al.
2021 and references therein)..

A number of mechanisms have been proposed to address the CH+ abundance puzzle,
which drive Equation (1) by locally heating the gas or by producing a relative
velocity between ions and neutrals that increase the effective temperature of
the reaction. Such mechanisms include: Alfvén waves (Federman et al., 1996),
turbulent mixing (Lesaffre et al., 2007; Valdivia et al., 2017),
magnetohydrodynamic shocks (Lesaffre et al., 2013), turbulent dissipation in
vortices (Godard et al., 2014), ion-neutral drift (Valdivia et al., 2017;
Moseley et al., 2021), and magnetohydrodynamic turbulence in clouds (Moseley
et al., 2021). These processes are hypothesized to locally heat the gas or
raise the effective temperature to greater than $200\text{\,}\mathrm{K}$
($k_{\mathrm{B}}T_{\mathrm{k}}=$17\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$)
and predict CH+ abundances that can differ from one another by several orders
of magnitude. One aim of our work here is to reduce the uncertainties in the
CH+ chemical kinetics data so that remaining discrepancies between models and
observations can be more reliably used to constrain the astrophysical
properties of the diffuse gas and not be attributed to a lack of understanding
of the underlying chemical kinetics.

Using $\mathrm{CH}^{+}$ observations to constrain which of these mechanisms is
driving the observed abundances, requires an accurate understanding of the
underlying $\mathrm{CH}^{+}$ chemistry. Reaction (1) has been studied both
experimentally (Hierl et al., 1997) and theoretically (Wu et al., 2021). The
dominant destruction reactions are (Myers et al., 2015)

$\displaystyle\mathrm{CH}^{+}+\mathrm{H}\ \ \,$ $\displaystyle\rightarrow$
$\displaystyle\ \mathrm{C}^{+}+\mathrm{H_{2}},$ (2)
$\displaystyle\mathrm{CH}^{+}+\mathrm{H_{2}}\ $ $\displaystyle\rightarrow$
$\displaystyle\ \mathrm{CH_{2}}^{+}+\mathrm{H},$ (3)
$\displaystyle\mathrm{CH}^{+}+\mathrm{e^{-}}\ $ $\displaystyle\rightarrow$
$\displaystyle\ \mathrm{C}+\mathrm{H}.$ (4)

The rate coefficients for hydrogen abstraction via Reaction (2) and
dissociative recombination (DR) in Reaction (4) have been calculated
theoretically (e.g., Faure et al. 2017). We are unaware of any theoretical
calculations for Reaction (3) more sophisticated than the Langevin value (Kim
et al., 1975). Experimentally, Plasil et al. (2011) and Gerlich et al. (2011)
measured the rate coefficients for Reaction (2) at $12-100\text{\,}\mathrm{K}$
and Reaction (3) at $50\text{\,}\mathrm{K}$, respectively, approximating
diffuse cloud environments. However, we are unaware of any cryogenic
measurement for the DR process, Reaction (4).

In diffuse clouds, DR represents the largest uncertainty among all chemical
reactions that form and destroy $\mathrm{CH}^{+}$. Calculations of the DR
cross section are difficult due to the large number of intermediate states
involved in the dissociation dynamics (Chakrabarti et al., 2018; Mezei et al.,
2019). Experimentally, single-pass merged-beams experiments have been
performed by Mitchell & McGowan (1978), where the ions were likely to be
electronically, vibrationally, and rotationally excited. The only DR rate-
coefficient results for electronically and vibrationally relaxed ($v=0$) ions
were obtained at the room-temperature Test Storage Ring (TSR; Amitay et al.
1996). However, the energy resolution in this measurement was
$\sim$20\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$, i.e., much higher than the
collision energies in quiescent gas of a diffuse cloud. Additionally, the ions
occupied a broad range of rotational states with only $\sim 6\%$ in the $J=0$
state. Given that recent experimental work has found that the $J$-specific DR
rate coefficient can vary by over an order of magnitude (Novotný et al.,
2019), the uncertainty in the experimentally derived CH+ rate coefficient from
the above two CH+ works is likely to be a previously unrecognized order of
magnitude. This makes the DR process the most uncertain reaction in the
chemistry forming and destroying $\mathrm{CH}^{+}$ in diffuse clouds.

With the development of the Cryogenic Storage Ring (CSR; von Hahn et al.
2016), it has become possible to perform electron-ion merged-beams collision
studies with internally cold molecular ions. Recent CSR DR experiments with
$\mathrm{HeH}^{+}$ revealed a strong $J$-dependence of the rate coefficient
(Novotný et al., 2019), demonstrating the importance of probing rotationally
cold ions. In this work, we use the CSR to create an electronically and
rovibrationally cold ($v=0$, $J=0-2$) ensemble of $\mathrm{CH}^{+}$ molecules,
measure DR with an order-of-magnitude improved collision energy resolution of
$\sim$2\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$, and extract a rate
coefficient for the rovibrational $\mathrm{CH}^{+}$ ground state appropriate
for diffuse cloud conditions, thus refining the most uncertain reaction in the
$\mathrm{CH}^{+}$ diffuse cloud chemistry.

##  2 Experimental

Figure 1: (a) Schematic of the CSR experimental setup for DR measurements of
CH+ and probing of its internal level populations. See text for details. (b)
Measured relative $J$-level populations in the CSR for storage times of
$60-80\text{\,}\mathrm{s}$ with statistical one-sigma error bars. (c) Relative
$J$-level populations predicted in TSR for a thermal equilibrium at
$T=$300\text{\,}\mathrm{K}$$.

In order to determine the DR rate coefficient of the
$X^{1}\Sigma^{+}(v=0,J=0)$ ground state of $\mathrm{CH}^{+}$, we have used the
experimental setup sketched in Figure 1(a). We stored and phase-space cooled a
$\mathrm{CH}^{+}$ ion beam inside the CSR, prepared internally cold ions,
utilized _in situ_ diagnostic methods to follow time-dependent changes of the
internal level populations, and finally measured DR with well-defined mixtures
of only a few $J$ levels to infer the rate coefficient for the
$\mathrm{CH}^{+}$ ground electronic, vibrational, and rotational state.

Phase-space cooling333Commonly also called electron cooling in literature of
the $\mathrm{CH}^{+}$ ion beam was applied to reduce its diameter and energy
spread (Poth, 1990). The stored ion beam was overlapped with a larger
diameter, nearly mono-energetic electron beam in the CSR electron cooler
section (Figure 1(a)). Phase-space cooling was reached by matching the
electron and ion beam positions and velocities for the first
$21\text{\,}\mathrm{s}$ of storage, before the DR measurements began. Phase-
space cooling was also regularly applied during the collision experiments to
maintain the beneficial ion-beam properties.

Internal cooling of the initial electronically and rovibrationally hot ion
beam, was achieved by radiative decay and by inelastic collisions with the
merged, nearly monoenergetic, velocity-matched electron beam. Radiative
cooling proceeded by interactions with the $\sim$20\text{\,}\mathrm{K}$$ two-
component blackbody radiation field in the CSR (Kálosi et al., 2022). This
dominated the initial internal cooling of highly excited states. After
$21\text{\,}\mathrm{s}$ of storage, most ions were in their ground electronic
state $X^{1}\Sigma^{+}$ with a rovibrationally cold distribution of $v=0$ and
$J=0-2$. Further rotational cooling was driven by inelastic electron
collisions and advanced towards an effective internal ion temperature of
$\sim$26\text{\,}\mathrm{K}$$ (Kálosi et al., 2022).

Figure 2: Merged-beams DR rate coefficient (blue line) for the CH+ ground
state $X^{1}\Sigma^{+}(v=0,J=0)$ with the statistical one-sigma uncertainties
(blue shaded area). The one-sigma uncertainty of the absolute scale is $\pm
13\%$. The room-temperature TSR results of Amitay et al. (1996), which have a
$\pm 50\%$ absolute scaling systematic uncertainty, are shown by the gray
squares. The symbols and lines on top of the figure indicate the full width at
half maximum (FWHM) energy resolution $\Delta E$ of the CSR and TSR
measurements. The divergence of the CSR from the TSR data for
$E_{\mathrm{d}}\gtrsim$10\text{\,}\mathrm{e}\mathrm{V}$$ is due to
contribution from dissociative excitation (see text). The data behind the TSR
and CSR graphs are both available. The latter are part of Figure 5(b).

We monitored the internal cooling _in situ_ by probing $J$-level-resolved
photodissociation of $\mathrm{CH}^{+}$ with an optical parametric oscillator
(OPO) laser at a tunable wavelength near
$\sim$305\text{\,}\mathrm{n}\mathrm{m}$$ and measuring the resulting H atoms
on a neutral-fragment detector (Figure 1(a); Kálosi et al. 2022). A combined
radiative and collisional equilibrium was reached by
$t\sim$70\text{\,}\mathrm{s}$$, with the $J=0$ and $J=1$ levels dominating the
rotational populations (Figure 1(b)). In contrast to the room-temperature TSR
rotational distribution (Figure 1(c)), in the present work over $40\%$ of the
ions populate the $J=0$ level. In addition to the $X^{1}\Sigma^{+}$ ground
state, there is a small fraction of ions ($<13\%$ at $21\text{\,}\mathrm{s}$)
in the metastable $a^{3}\Pi$ state, which decays with a lifetime of
$\sim$10\text{\,}\mathrm{s}$$ and which we monitored by its unique DR
fragmentation pattern. In this way, we obtained full knowledge about the time-
dependent populations of all contributing rotational levels
($p_{J}(t)[J=0-2]$) and the metastable state ($p_{\mathrm{m}}(t)$), see
Appendix A.

DR rate-coefficient measurements on the internally cold $\mathrm{CH}^{+}$ ions
used the electron beam from the electron cooler as a merged-beams collision
target. Neutral C and H atoms resulting from DR in the interaction region of
the electron cooler were collected downstream by our neutral-fragment
detector. From the measured count rate we determined the merged-beams DR rate
coefficient $\alpha^{\mathrm{mb}}$, related to the DR cross section $\sigma$
via

$\alpha^{\mathrm{mb}}=\langle\sigma v_{\mathrm{c}}\rangle.$ (5)

Here, $\sigma(E)$ and the collision velocity $v_{\mathrm{c}}(E)$ are functions
of the collision energy $E$ and $\langle...\rangle$ designates the mean value
over the full distribution of $v_{\mathrm{c}}$ inside the electron-ion overlap
region. This distribution depends on the transverse and longitudinal
temperatures in the rest frame of the electron beam. We achieved
$k_{\mathrm{B}}T_{\perp}=$2.0^{+1.0}_{-0.5}\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$
and
$k_{\mathrm{B}}T_{\parallel}\sim$0.5\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$,
respectively, corresponding to a collision energy resolution of
$\sim$2\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ at matched velocities. Here
and throughout, all uncertainties are quoted at an one-sigma confidence level
with ${}^{+}_{-}$ for asymmetric uncertainties and numbers in parenthesis for
symmetric uncertainties.

In order to study the energy dependence of the DR process, electron energies
were scanned by adjusting the potential of several drift tubes in the merged-
beams interaction section (Figure 1(a)). This yielded
$\alpha^{\mathrm{mb}}(E_{\mathrm{d}})$, where the detuning energy
$E_{\mathrm{d}}$ is defined as the collision energy of ideal, mono-energetic
electron and ion beams, given by the drift-tube potentials. Details on the
electron-ion collision setup and on the absolute scaling of
$\alpha^{\mathrm{mb}}(E_{\mathrm{d}})$ are given in Appendices B and C,
respectively.

The astrophysically relevant $\mathrm{CH}^{+}$ DR rate coefficient was
inferred from our storage-time-dependent measurements of
$\alpha^{\mathrm{mb}}(E_{\mathrm{d}},t)$. For storage times of
$t=$21-100\text{\,}\mathrm{s}$$, we used the experimentally determined
internal level populations to extract the merged-beams rate coefficient
$\alpha^{\mathrm{mb}}_{J=0}(E_{\mathrm{d}})$ for the pure $J=0$ state
(Appendix D). From this, we generated the DR kinetic rate coefficient
$\alpha^{\mathrm{k}}_{J=0}(T_{\mathrm{k}})$ for the $J=0$ state (Appendix E),
needed for diffuse cloud chemistry at a gas kinetic temperature
$T_{\mathrm{k}}$.

##  3 Results

###  3.1 Merged-beams rate coefficient

The rate coefficient $\alpha^{\mathrm{mb}}_{J=0}(E_{\mathrm{d}})$ in Figure 2
decays by two orders of magnitude in the energy range from
$10^{-5}\text{\,}\mathrm{e}\mathrm{V}$ to
$10^{-2}\text{\,}\mathrm{e}\mathrm{V}$ and shows various resonances in the
region of $0.03-2\text{\,}\mathrm{e}\mathrm{V}$. These resonances can be
attributed to capture into neutral Rydberg states of the CH molecule and
subsequent coupling to a dissociative state (Amitay et al. 1996, Mezei et al.
2019). At energies beyond $\sim$3.6\text{\,}\mathrm{e}\mathrm{V}$$, a non-
resonant DR channel opens up and leads to an increased DR rate coefficient
(Amitay et al. 1996; Chakrabarti et al. 2018). In addition, $\mathrm{CH}^{+}$
dissociative excitation (DE) sets in at a threshold of
$\sim$3.6\text{\,}\mathrm{e}\mathrm{V}$$ (Bannister et al. 2003; Chakrabarti
et al. 2017), which we cannot distinguish from DR owing to our experimental
setup.

Comparing our data to the room-temperature TSR data of Amitay et al. (1996),
which had rotational populations close to the simulation shown in Figure 1(c),
we obtain information on the DR rotational level dependence at
$E_{\mathrm{d}}>$0.03\text{\,}\mathrm{e}\mathrm{V}$$ and on measurement-
related effects. At $0.03-2\text{\,}\mathrm{e}\mathrm{V}$, the resonant
structures and absolute values agree well between the two data sets, despite
the differing $J$-level populations. This indicates the absence of major
$J$-dependent resonances at detuning energies above
$0.03\text{\,}\mathrm{e}\mathrm{V}$. This conclusion is corroborated by the
close match of the CSR results for the $J=0$ and $J=1$ merged-beams rate
coefficients for $E_{\mathrm{d}}>$0.03\text{\,}\mathrm{e}\mathrm{V}$$
(Appendix D). At $E_{\mathrm{d}}\gtrsim$3.6\text{\,}\mathrm{e}\mathrm{V}$$,
due to a measurement-related effect, our data are enhanced compared to the TSR
results, where it was possible to discriminate between DR and DE. In addition,
our collision geometry imposes a rate coefficient baseline of
$\sim$6\text{\times}{10}^{-9}\text{\,}\mathrm{c}\mathrm{m}^{3}\,\mathrm{s}^{-1}$$
to the data, which becomes visible at
$E_{\mathrm{d}}>$2\text{\,}\mathrm{e}\mathrm{V}$$ (Appendix C). Both effects
are accounted for in the determination of the $\mathrm{CH}^{+}$ DR kinetic
temperature rate coefficient (Appendix E).

Due to the improved energy resolution of the CSR depicted on top of Figure 2,
we were able to probe the diffuse cloud collision-energy regime
($0.001-0.02\text{\,}\mathrm{e}\mathrm{V}$), where no experimental data have
previously been available. Our results reveal a remarkable increase in the DR
rate coefficient towards lower detuning energies. This feature is considerably
stronger than the $E_{\mathrm{d}}^{-1/2}$ behavior that would result from a
typical $E^{-1}$ cross section dependence of non-resonant DR (Mitchell, 1990).
This can be attributed either to a constructive resonance near
$E_{\mathrm{d}}=$0\text{\,}\mathrm{e}\mathrm{V}$$ or a destructive resonance
near $E_{\mathrm{d}}=$0.01\text{\,}\mathrm{e}\mathrm{V}$$. We also note that
at $E_{\mathrm{d}}=$0.012\text{\,}\mathrm{e}\mathrm{V}$$ the $J=0$ rate
coefficient is lower by a factor of $5.4^{+5.0}_{-2.0}$ compared to $J=1$.
However, we find no significant deviation for
$E_{\mathrm{d}}<$0.008\text{\,}\mathrm{e}\mathrm{V}$$ and
$E_{\mathrm{d}}>$0.02\text{\,}\mathrm{e}\mathrm{V}$$ (Appendix D).

Figure 3: CH+ DR kinetic temperature rate coefficient and its relevance for
the CH+ destruction in diffuse interstellar clouds. (a) The thick solid line
shows the experimentally determined $J=0$ kinetic temperature rate coefficient
of this work. The one-sigma error band in gray represents the quadrature sum
of all systematic uncertainties. The dashed lines indicate rate coefficients
from the UMIST (McElroy et al. 2013, upper line) and KIDA (Wakelam et al.
2012, lower line) astrochemistry databases. The calculated $J=0$ rate
coefficient of Mezei et al. (2019) is drawn as dotted line. (b) Diffuse cloud
parameter ranges where DR is the dominant CH+ destruction process (colored)
for a standard electron fraction
($\chi_{\mathrm{e}}=$1.6\text{\times}{10}^{-4}\text{\,}$$) and for enhanced
values of $\chi_{\mathrm{e}}$ as given. One-sigma error bars for
$f_{\mathrm{H_{2}}}=1$ and $f_{\mathrm{H_{2}}}=0$ are shown at the top and
bottom of the figure, respectively. They were calculated from the combined
systematic uncertainties for the rate coefficients $\alpha_{\mathrm{H}}$
($\sim 50\%$, Plasil et al. (2011)), $\alpha_{\mathrm{H_{2}}}$ ($\sim 20\%$,
Gerlich et al. (2011)), and $\alpha^{\mathrm{k}}_{J=0}$ ($\sim 20\%$, this
work).

###  3.2 Kinetic temperature rate coefficient

To obtain the DR kinetic temperature rate coefficient
$\alpha^{\mathrm{k}}_{J=0}(T_{\mathrm{k}})$ for the pure $J=0$ state, we
converted the $\alpha^{\mathrm{mb}}_{J=0}(E_{\mathrm{d}})$ data (Figure 2) as
described in Appendix E. The result in Figure 3(a) shows that our improved
experimental energy resolution enables us to reliably determine the kinetic
rate coefficient down to $T_{\mathrm{k}}\sim$10\text{\,}\mathrm{K}$$. With the
broad range of collision energies probed, we can provide data up to
$T_{\mathrm{k}}\sim$40\,000\text{\,}\mathrm{K}$$, close to the temperature
equivalent of the dissociation threshold
$D_{\mathrm{0}}/k_{\mathrm{B}}\sim$47\,000\text{\,}\mathrm{K}$$ (Hechtfischer
et al., 2002). The kinetic rate-coefficient shape can be approximated by a
$T_{\mathrm{k}}^{-0.5}$ behavior with two broad peaks at low and high
temperatures of $T_{\mathrm{k}}\sim 10$ and $\sim$8000\text{\,}\mathrm{K}$$,
respectively. These can be attributed to the maxima in the merged-beams rate
coefficient at $E_{\mathrm{d}}\lesssim$0.005\text{\,}\mathrm{e}\mathrm{V}$$
and $E_{\mathrm{d}}=$0.2-2\text{\,}\mathrm{e}\mathrm{V}$$ in Figure 2,
respectively. A discussion about the uncertainties displayed as an error band
is presented in Appendix E.

Figure 3(a) compares our experimental results to values in the UMIST (McElroy
et al., 2013) and KIDA (Wakelam et al., 2012) astrochemistry databases. Here,
the UMIST entry cites a review by Mitchell (1990) reporting the results of a
single-pass merged-beams experiment by Mitchell & McGowan (1978), where
multiple electronic, vibrational, and rotational states were likely to be
involved. Thus, those results are not expected to represent diffuse cloud
conditions. The KIDA rate coefficient is referred to as estimated. Both
database rate coefficients follow a simple behavior near
$T_{\mathrm{k}}^{-0.5}$ and clearly miss the temperature-dependent features
seen in our data. In the temperature regime for quiescent gas in diffuse
clouds, the new CSR results exceed the database values by up to a factor of 6.
Conversely, for gas heated by shocks or turbulence, our data are up to a
factor of $2.5$ lower.

We also provide a comparison to multichannel quantum defect theory
calculations for the CH${}^{+}(J=0)$ DR cross section (Mezei et al., 2019),
which we converted into a kinetic rate coefficient as shown in Appendix E.
Compared to their results, our data are significantly larger at low
temperatures. In particular, in the temperature regime of diffuse clouds
($T_{\mathrm{k}}=$40-130\text{\,}\mathrm{K}$$) theory underestimates the DR
rate coefficient by up to a factor of $7$. For heated gas at
$T_{\mathrm{k}}\sim$1000\text{\,}\mathrm{K}$$, the two results match but begin
to diverge again for $T_{\mathrm{k}}\gtrsim$3000\text{\,}\mathrm{K}$$, where
our data exceed theory by up to a factor of 7.

##  4 Astrochemical implications

Understanding the CH+ abundance in diffuse clouds has been limited, in part,
by uncertainties in the relevant chemistry. Our measurements of the
CH${}^{+}(J=0)$ DR rate coefficient reduce its uncertainty from above an order
of magnitude to $\sim 20\%$ and thus provide a significant improvement in the
reliability of the CH+ chemistry. Remaining discrepancies between observations
and models can now be more solidly used to constrain the physical properties
of these clouds. In Appendix F we provide analytical representations of our
kinetic temperature rate coefficient results for ready incorporation into
astrochemical databases.

Based on our measured DR rate coefficient, we can determine the relative
importance of the CH+ destruction mechanisms given by Reactions (2)-(4). A
temperature-dependent rate coefficient $\alpha_{\mathrm{H}}(T_{\mathrm{k}})$
for Reaction (2) with pure CH${}^{+}(J=0)$ ions was derived from experimental
data by Plasil et al. (2011) (Equation (6a) of their work), which we
extrapolate up to $T_{\mathrm{k}}=$600\text{\,}\mathrm{K}$$. For CH+
destruction by molecular hydrogen (Reaction (3)), we use the rate coefficient
$\alpha_{\mathrm{H_{2}}}=$1.2\text{\times}{10}^{-9}\text{\,}\mathrm{c}\mathrm{m}^{3}\,\mathrm{s}^{-1}$$
from Smith & Adams (1977) and Gerlich et al. (2011), which were obtained for
fully thermal temperatures, i.e., gas kinetic and excitation temperatures, of
$T_{\mathrm{k}}=T_{\mathrm{ex}}=$300\text{\,}$$ and $50\text{\,}\mathrm{K}$,
respectively. Given the match of the rate coefficients at different
$T_{\mathrm{k}}$, we assume that
$\alpha_{\mathrm{H_{2}}}($10\text{\,}\mathrm{K}$\leq
T_{\mathrm{k}}\leq$600\text{\,}\mathrm{K}$)=$1.2\text{\times}{10}^{-9}\text{\,}\mathrm{c}\mathrm{m}^{3}\,\mathrm{s}^{-1}$$.

To compare destruction by atomic and molecular hydrogen collisions to
destruction by DR, we define a critical DR rate coefficient
$\alpha_{\mathrm{cr,DR}}$ at which the CH+ destruction by both processes is
balanced. Diffuse clouds are characterized by their electron densities
$n_{\mathrm{e}}$, their atomic and molecular hydrogen number densities,
$n(\mathrm{H})$ and $n(\mathrm{H_{2}})$, and related quantities: the hydrogen
nucleus number density $n_{\mathrm{H}}=n(\mathrm{H})+2n(\mathrm{H_{2}})$, the
molecular fraction $f_{\mathrm{H_{2}}}=2n(\mathrm{H_{2}})/n_{\mathrm{H}}$, and
the electron fraction $\chi_{\mathrm{e}}=n_{\mathrm{e}}/n_{\mathrm{H}}$. The
critical DR rate coefficient is then given by

$\alpha_{\mathrm{cr,DR}}(T_{\mathrm{k}})=\frac{(1-f_{\mathrm{H_{2}}})\alpha_{\mathrm{H}}(T_{\mathrm{k}})+f_{\mathrm{H_{2}}}\alpha_{\mathrm{H_{2}}}(T_{\mathrm{k}})/2}{\chi_{\mathrm{e}}}.$
(6)

Destruction by DR dominates over that by hydrogen collisions if
$\alpha^{\mathrm{k}}_{J=0}>\alpha_{\mathrm{cr,DR}}$.

Combining Equation (6) with the condition
$\alpha^{\mathrm{k}}_{J=0}>\alpha_{\mathrm{cr,DR}}$, we calculated the diffuse
cloud parameters ($T_{\mathrm{k}}$, $f_{\mathrm{H_{2}}}$), at which CH+
destruction is mainly due to DR. The result is presented in Figure 3(b).
Assuming that $\chi_{\mathrm{e}}=$1.6\text{\times}{10}^{-4}\text{\,}$$ matches
the relative C+ abundance (Sofia et al., 2004), we find that for cold and
mainly atomic gas ($T_{\mathrm{k}}\lesssim$50\text{\,}\mathrm{K}$$,
$f_{\mathrm{H_{2}}}\lesssim 0.05$) DR is the dominant destruction process.
Above this temperature, hydrogen abstraction becomes dominant. In case of
higher molecular fractions ($f_{\mathrm{H_{2}}}\gtrsim 0.1$), we find that for
gas temperatures of $T_{\mathrm{k}}>$40\text{\,}\mathrm{K}$$ destruction by DR
is negligible compared to hydrogen abstraction. Given the assumptions made for
$\alpha_{\mathrm{H}}$ and $\alpha_{\mathrm{H_{2}}}$, the uncertainties
displayed in Figure 3(b) are likely underestimated. For example, no lower
limit for the $J=0$ rate coefficient $\alpha_{\mathrm{H}}$ could be determined
by Plasil et al. (2011), which might lead to a higher relevance of DR in fully
atomic clouds, beyond the results of Figure 3(b).

The results in Figure 3(b) can be put into the context of the CH+ abundance
puzzle in diffuse clouds. In quiescent gas the increased DR kinetic rate
coefficient determined in our work leads to a factor of $\sim 2$ enhancement
in the destruction of CH+ than previously assumed. However, because of the
local heating mechanisms required to produce CH+ and the high reactivity of
the molecules, which prevents them from surviving long enough to flow into
cold regions of the cloud, very little CH+ abundance is expected in cold gas.
Instead, the CH+ chemistry is expected to take place at
$T_{\mathrm{k}}\gtrsim$1000\text{\,}\mathrm{K}$$ (Moseley et al., 2021), where
our data indicate that DR can be neglected compared to destruction by hydrogen
collisions, even when the electron fraction is increased to
$\chi_{\mathrm{e}}=10^{-2}$.

Our measurements also indicate that destruction by DR can be enhanced in
regions with higher cosmic ray ionization rates or lower than average gas
density, where increased electron fractions are found. This is the case in the
Galactic center (Le Petit et al., 2016) or in supernova remnants (Priestley et
al., 2017). Figure 3(b) contains the boundaries for the DR dominated CH+
destruction regime for enhanced electron fractions. We emphasize that these
boundaries depend on the behavior of $\alpha_{\mathrm{H}}$ and
$\alpha_{\mathrm{H_{2}}}$ only for
$T_{\mathrm{k}}\lesssim$600\text{\,}\mathrm{K}$$, but not above. Figure 3(b)
demonstrates that increasing the electron fraction by factors of 6 and 60
compared to quiescent gas in diffuse clouds extends the DR dominated
temperature regime to $T_{\mathrm{k}}\sim 100$ and $500\text{\,}\mathrm{K}$
respectively, even for fully molecular clouds. Thus, in environments with
increased electron fractions and
$T_{\mathrm{k}}\lesssim$500\text{\,}\mathrm{K}$$, the results of this work are
likely to be important for our understanding of the corresponding CH+
chemistry. However, for heated regions in diffuse clouds at
$T_{\mathrm{k}}\gtrsim$1000\text{\,}\mathrm{K}$$, destruction by hydrogen
collisions remains the dominant process, even when considering an electron
fraction of $\chi_{\mathrm{e}}\sim 10^{-2}$.

##  5 Conclusions

With this work we have determined the rate coefficient for the most uncertain
reaction in the CH+ chemistry of diffuse clouds, namely the DR kinetic
temperature rate coefficient. By mimicking diffuse cloud temperatures inside
the CSR and conducting a high-resolution electron-ion merged-beams experiment,
we have obtained the gas kinetic temperature dependence of the DR rate
coefficient for the CH${}^{+}(X^{1}\Sigma^{+},v=0,J=0)$ state, involving all
temperatures from quiescent to locally heated gas. We further characterized
the implications of our experimental results on diffuse cloud chemistry, based
on a simple reaction network that includes the experimental rate coefficients
for the competing CH+ reactions with atomic and molecular hydrogen. By
reducing the uncertainty for the DR rate coefficient from above an order of
magnitude to $\sim\pm 20\%$, our findings will improve the reliability of
future studies for local heating mechanisms in diffuse clouds based on CH+
abundance observations. DR experiments at the CSR with other key molecular
ions are in development and could lead to further improvements in our
understanding of chemical and physical properties of interstellar clouds.

## Acknowledgements

Financial support by the Max Planck Society is acknowledged. D. P., A. K., and
D. W. S. were supported in part by the U.S. National Science Foundation
Division of Astronomical Sciences Astronomy and Astrophysics Grants program
under AST-1907188. We thank Zs. J. Mezei and I. F. Schneider for providing us
their theoretical DR cross sections from Mezei et al. (2019) for comparison of
kinetic temperature rate coefficients in this work.

##  Appendix A Electronic, vibrational, and rotational level population
evolution

In order to extract $J$-specific DR rate coefficients, one needs the storage-
time evolution of all contributing CH+ electronic, vibrational, and rotational
level populations. We have experimentally determined these populations _in
situ_ for the conditions during our DR measurements (Section 2, Appendix B).

We injected into the CSR a CH+ ion beam with a variety of excited states
populated. Most of the electronically excited states are known to radiatively
decay with a lifetime of $\lesssim$10\text{\,}\mu\mathrm{s}$$ (Huber &
Herzberg, 1979). The exception to this is the metastable $a^{3}{\Pi}$ state,
which lacks a spin-allowed transition to the $X^{1}\Sigma^{+}$ ground state
and can only decay via singlet-triplet mixing (Hechtfischer et al., 2007). Due
to different coupling constants for individual $(v,J,e/f)$ states ($e/f$ being
the parity), the $a^{3}{\Pi}$ decay is expected to involve a variety of
different lifetimes (Hechtfischer et al., 2007). Experimentally, an average
lifetime for all contributing states was found to be
$\sim$7\text{\,}\mathrm{s}$$ (Amitay et al., 1996).

Vibrationally excited states of the $X^{1}\Sigma^{+}$ and $a^{3}{\Pi}$ states
radiatively relaxed within the first $3\text{\,}\mathrm{s}$ of storage in the
CSR. For the $X^{1}\Sigma^{+}$ state, the longest living $v=1$ state decays
with a lifetime of $0.7\text{\,}\mathrm{s}$ (Folomeg et al., 1987); while for
the $a^{3}{\Pi}$ state, after $0.5\text{\,}\mathrm{s}$ the $v=1$ population
amounts to less than $1\%$ compared to $v=0$ (Hechtfischer et al., 2007).
Thus, after $t=$3\text{\,}\mathrm{s}$$ we expect only $X^{1}\Sigma^{+}(v=0)$
and $a^{3}\Pi(v=0)$ in the stored ion beam.

We experimentally characterized the radiative decay of the metastable
$a^{3}{\Pi}(v=0)$ state by storage-time-dependent CH+ DR measurements. Our
neutral-fragment detector (Figure 1(a)) is capable of imaging the impact
positions of neutral C and H fragments. We recorded the distances between
coincident C and H fragments from the DR events for a detuning energy of
$E_{\mathrm{d}}=$0\text{\,}\mathrm{e}\mathrm{V}$$ and over storage times of
$3-70\text{\,}\mathrm{s}$. The spatial distribution of these distances is due
to the energy released in the exothermic DR reaction and is used to determine
the kinetic energy of the C and H fragments. This kinetic energy release (KER)
is $>$4\text{\,}\mathrm{e}\mathrm{V}$$ for the $X^{1}\Sigma^{+}(v=0)$ ground
state and $<$1\text{\,}\mathrm{e}\mathrm{V}$$ for the metastable
$a^{3}\Pi(v=0)$ state (Amitay et al. 1996). The very different KER values
enable us to follow the relative DR contributions from the $a^{3}\Pi(v=0)$
state vs. storage time and model the metastable storage-time-dependent
population $p_{\mathrm{m}}(t)$ (see Kálosi et al. 2022 for further details).
We obtained a best match to our data with a two-component decay, involving a
short-lived component (lifetime:
$\tau_{\mathrm{m}}=$10.1(10)\text{\,}\mathrm{s}$$) and a long-lived component
(lifetime $\gg$100\text{\,}\mathrm{s}$$), which we consider to have a constant
population within the storage times probed, using the equation

$p_{\mathrm{m}}(t)=p_{\mathrm{m},t_{0}}e^{-(t-t_{0})/\tau_{\mathrm{m}}}+p_{\mathrm{m},\infty}.$
(A1)

We found $p_{\mathrm{m},t_{0}}=$0.10(3)\text{\,}$$ at
$t_{0}=$21\text{\,}\mathrm{s}$$ and
$p_{\mathrm{m},\infty}=$0.025(7)\text{\,}$$.

Within the $X^{1}\Sigma^{+}(v=0)$ state, rotational cooling of $J\geq 3$
levels was dominated by radiative deexcitation in the
$\sim$20\text{\,}\mathrm{K}$$ two-component blackbody radiation field of the
CSR. Radiative lifetimes of these levels are below $5\text{\,}\mathrm{s}$
(Kálosi et al., 2022). Inelastic collisions between our electron beam and the
CH+ ions, with purely matched velocities
($E_{\mathrm{d}}=$0\text{\,}\mathrm{e}\mathrm{V}$$) for the initial
$21\text{\,}\mathrm{s}$ of beam storage, accelerate this cooling process via
collisional deexcitation. Excitation of levels $J=0-2$ into $J\geq 3$ by
inelastic electron collisions is negligible since it requires collision
energies $>$10\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ (Kálosi et al., 2022).
Such collisions are rare for $E_{\mathrm{d}}=$0\text{\,}\mathrm{e}\mathrm{V}$$
with a collision energy resolution of
$\sim$2\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$. Thus, at the starting time
of our DR measurements ($t=$21\text{\,}\mathrm{s}$$) any contribution from
$J\geq 3$ levels can be neglected.

We monitored the rotational population evolution for
$X^{1}\Sigma^{+}(v=0,J=0-2)$ within our DR measurement interval of
$t=$21-100\text{\,}\mathrm{s}$$ by probing near-threshold photodissociation of
CH+ with our OPO laser (Figure 1(a)). This process has been studied
theoretically and experimentally (Hechtfischer et al. 2002; O’Connor et al.
2016). There is an associated resonant structure for photon wavenumbers of
$\sim$33\,000\text{\,}\mathrm{c}\mathrm{m}^{-1}$$, lying between the
$\mathrm{C}^{+}(^{2}P_{1/2})+\mathrm{H}(^{2}{S})$ and
$\mathrm{C}^{+}(^{2}P_{3/2})+\mathrm{H}(^{2}{S})$ atomic fine structure
levels. These resonances can be assigned to individual initial
CH${}^{+}(X^{1}\Sigma^{+},v,J)$ levels. By probing the photodissociation cross
section at four different photon energies, three of them on top of the
strongest $J=0-2$ peaks and one for background characterization, we determined
the relative rotational populations $\hat{p}_{J}(t)[J=0-2]$ within the six
storage-time windows probed in the DR measurements (Appendix B). This method
for _in situ_ determination of rotational populations was previously applied
at the CSR for characterizing radiative cooling by O’Connor et al. (2016) and
was also utilized for measuring inelastic electron collision cross sections by
Kálosi et al. (2022). Details on the experimental setup and measurement
parameters used in the present work can be found in Kálosi et al. (2022).
Here, we applied a lower electron density and a more complex electron
collision energy scheme, involving not only collisions at
$E_{\mathrm{d}}=$0\text{\,}\mathrm{e}\mathrm{V}$$ but also at higher detuning
energies, which were required to obtain the DR spectrum. Thus, we have
slightly different rotational populations compared to Kálosi et al. (2022).

Figure 4: Relative level-population evolution of CH+ ions in the CSR during DR
measurements. The markers represent laser-probed rotational populations
$\hat{p}_{J}$ with statistical one-sigma uncertainties. The lines indicate
modeled populations $p_{J}$ from experimentally benchmarked radiative
rotational cooling (Kálosi et al., 2022) and using the $J$-dependent inelastic
electron-collision cross sections for rotational (de-)excitation from Hamilton
et al. (2015). Initial populations for the model were fixed to the measured
values at $t=$23\text{\,}\mathrm{s}$$ (leftmost markers). The metastable
$a^{3}\Pi$ population evolution ($p_{\mathrm{m}}$, dash-dotted line) was
modeled with Equation (A1). All $J$ populations were normalized such that
$\sum_{J}\hat{p}_{J}+p_{\mathrm{m}}=1$ and $\sum_{J}p_{J}+p_{\mathrm{m}}=1$.

Figure 4 summarizes the results for the evolution of the metastable
$a^{3}\Pi(v=0)$ and $X^{1}\Sigma^{+}(v=0,J=0-2)$ populations with storage time
in the CSR. In the storage-time interval $t=$21-100\text{\,}\mathrm{s}$$ of
our DR measurements, the inferred population of the $a^{3}\Pi(v=0)$ state is
below $13\%$. The data for the rotational populations of the
$X^{1}\Sigma^{+}(v=0)$ state indicate that the $J=2$ population decays on a
similar time scale as the metastable state but with about a factor of 2 higher
population compared to the metastable state. For $t>$40\text{\,}\mathrm{s}$$
more than $70\%$ of the ions occupy the $J=0$ and $J=1$ levels.

In order to determine the rotational level populations needed to derive the
$J$-specific DR rate coefficients, we used the combined radiative-cooling and
inelastic-collision rotational-cooling models, which yield results in good
agreement with the measured rotational populations, as can be seen in Figure
4. The combined model is described in detail by Kálosi et al. (2022) and uses
the calculated inelastic collision cross sections from Hamilton et al. (2015).
The model was found to be in good agreement with experimental data (Kálosi et
al., 2022). Here, we adjusted the model to the inelastic-collision energy
scheme used in this work. The resulting modeled populations $p_{J}(t)$ are
indicated as lines in Figure 4 and are in agreement with our measured
populations $\hat{p}_{J}(t)$. The model populations are used in the analysis
of the $J$-specific DR rate coefficients (Appendix D). We also used the model
to estimate the contribution of $J\geq 3$ levels for storage times of
$t>$21\text{\,}\mathrm{s}$$, where collisions at
$E_{\mathrm{d}}>$0\text{\,}\mathrm{e}\mathrm{V}$$ were applied with an $\sim
18\%$ duty cycle (see Appendix B). The collisional excitation into $J\geq 3$
resulted in populations $p_{J\geq 3}<2\%$. We thus neglected the $J\geq 3$
levels for further analysis.

We note that in the analysis of the $a^{3}\Pi(v=0)$ population the rotational
levels in this electronic state were not considered individually. For the
determination of level-specific DR rate coefficients (Section 2, Appendix D),
we justify this simplification by the small relative population of
$a^{3}\Pi(v=0)$ ($\lesssim 10\%$), which we found has only a minor effect on
the rate coefficient analysis for the $J=0$ and $J=1$ levels.

##  Appendix B Electron-ion collision experiments

Merged-beams electron-ion collisions in storage rings are frequently used for
obtaining collisional rate coefficients of electronic and vibrational ground
state ions (e.g., Amitay et al. 1996, Tanabe et al. 1998, Geppert et al. 2004,
Novotný et al. 2013). However, these experiments were limited to room-
temperature environments and, thus, included a broad range of rotational
levels of the target molecule. With the commissioning of the CSR, it is now
possible to perform merged-beams DR experiments for rotationally cold ions
(Novotný et al., 2019). In this work we use this setup for measuring the CH+
DR rate coefficient (Figure 1(a)).

The CH+ ion beam was produced in a Penning ion source and accelerated towards
ground potential from an $\sim$280\text{\,}\mathrm{k}\mathrm{V}$$ high voltage
platform (von Hahn et al., 2016). About $10^{5}-10^{6}$ ions with a kinetic
energy of $\sim$280\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$$ were stored
inside the CSR, where they traversed its circumference of
$C_{0}=$35.12(5)\text{\,}\mathrm{m}$$ with a revolution frequency of
$f_{\mathrm{r}}=$57.98(1)\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$. The
storage lifetime of the ions inside the CSR enabled us to perform collision
experiments up to $t=$100\text{\,}\mathrm{s}$$.

We used the CSR electron cooler for both phase-space cooling of the ion beam
and merged-beams DR experiments (Section 2). The electron beam was generated
from a room-temperature photocathode (Orlov et al., 2005) and accelerated to
an energy of $\sim$20\text{\,}\mathrm{e}\mathrm{V}$$ outside of the CSR
(Figure 1(a)). Over the entire electron cooler section, from the cathode to
the collector, the low-energy electron beam was guided by solenoidal magnetic
fields, encountering sequentially field strengths of
$200\text{\,}\mathrm{m}\mathrm{T}$, $\sim 100\text{\,}\mathrm{m}\mathrm{T}$,
$10\text{\,}\mathrm{m}\mathrm{T}$ and $\sim 100\text{\,}\mathrm{m}\mathrm{T}$.
Accordingly, we used a magnetic field gradient with a factor of 20 to
adiabatically expand the electron beam (Novotný et al., 2013), resulting in a
diameter increase to $10.2(5)\text{\,}\mathrm{m}\mathrm{m}$ and a transverse
temperature reduction from
$k_{\mathrm{B}}T_{\perp}=$25\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ (room
temperature) to
$k_{\mathrm{B}}T_{\perp}=$2.0^{+1.0}_{-0.5}\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$,
which matches previous observations (Kálosi et al., 2022). In the merged-beams
interaction region, where the magnetic field strength was
$10\text{\,}\mathrm{m}\mathrm{T}$, we controlled the electron beam laboratory-
frame energy $E_{\mathrm{e}}$ by adjusting the potential of a set of drift
tubes (Figure 1(a)).

Phase-space cooling of the ion beam was established by matching electron and
ion velocities (Poth, 1990). The corresponding electron energy $E_{0}$ is
given by

$E_{0}=\frac{1}{2}m_{\mathrm{e}}f_{\mathrm{r}}^{2}C_{0}^{2}\sim$11.79\text{\,}\mathrm{e}\mathrm{V}$,$
(B1)

where $m_{\mathrm{e}}$ is the electron mass. We applied phase-space cooling to
the stored CH+ ion beam for $21\text{\,}\mathrm{s}$. During this time window,
the Gaussian density profile of the ion beam decreased from an effective FWHM
diameter (Kálosi et al., 2022) of $>$8\text{\,}\mathrm{m}\mathrm{m}$$ to an
equilibrium of $5.2(5)\text{\,}\mathrm{m}\mathrm{m}$. Thus, in our DR
measurements more than $90\%$ of the ions were enclosed within the electron
beam.

For the collision experiments at $t=$21-100\text{\,}\mathrm{s}$$, we detuned
the electron beam laboratory energy $E_{\mathrm{e}}$ (with
$E_{\mathrm{e}}=E_{\mathrm{0}}$ for phase-space cooling) by adjusting the
drift-tube potential. The resulting detuning energy $E_{\mathrm{d}}$ as
plotted in the merged-beams rate coefficient spectra is derived from

$E_{\mathrm{d}}=\left(\sqrt{E_{\mathrm{e}}}-\sqrt{E_{0}}\right)^{2}.$ (B2)

In order not to significantly alter the properties of the phase-space cooled
ion beam, we did not dwell at detuned velocities for more than
$100\text{\,}\mathrm{m}\mathrm{s}$, but instead switched regularly between
matched and detuned velocities. Our standard switching cycle included
$100\text{\,}\mathrm{m}\mathrm{s}$ at
$E_{\mathrm{d}}=$0\text{\,}\mathrm{e}\mathrm{V}$$ (phase-space cooling);
$25\text{\,}\mathrm{m}\mathrm{s}$ at a single measurement energy
$E_{\mathrm{d}}=E_{\mathrm{meas}}$; and $25\text{\,}\mathrm{m}\mathrm{s}$ at a
reference energy
$E_{\mathrm{d}}=E_{\mathrm{ref}}\sim$3.4\text{\,}\mathrm{e}\mathrm{V}$$, where
the electron-induced rate is small compared to background from residual gas.
Between each energy step we implemented an
$\sim$5\text{\,}\mathrm{m}\mathrm{s}$$ waiting time for voltage stabilization.
For each value of $E_{\mathrm{d}}$, we recorded the number of collisions
resulting in neutral particles on our neutral-fragment detector (Figure 1(a)),
which is based on microchannel plates (MCPs). We also recorded the duration
spent at each energy step in order to determine the count rates
$R_{\mathrm{meas}}(E_{\mathrm{d}})$ and $R_{\mathrm{ref}}$ for measurement and
reference energy, respectively. The rate at matched velocities
($R_{\mathrm{cool}}$) was not measured since when probing
$E_{\mathrm{d}}>$0.03\text{\,}\mathrm{e}\mathrm{V}$$ with a typically high
number of ions in CSR, $R_{\mathrm{cool}}$ would saturate our detector, owing
to the high rate coefficient at
$E_{\mathrm{d}}=$0\text{\,}\mathrm{e}\mathrm{V}$$ (Figure 2). We avoided this
saturation by switching off the front MCP voltage in the matched-velocity
phases.

Energy-dependent merged-beams DR rate coefficients are calculated from the
measured count rates. We first subtract the residual gas background, measured
at $E_{\mathrm{ref}}$, to determine the electron-induced signal

$R_{\mathrm{e}}(E_{\mathrm{d}})=R_{\mathrm{meas}}(E_{\mathrm{d}})-R_{\mathrm{ref}}.$
(B3)

The rate coefficient is then given by

$\alpha^{\mathrm{mb}}(E_{\mathrm{d}})=\frac{R_{\mathrm{e}}(E_{\mathrm{d}})}{\eta_{\mathrm{d}}(E_{\mathrm{d}})\zeta
n_{\mathrm{e}}(E_{\mathrm{d}})N_{\mathrm{i}}\hat{l}_{0}/C_{0}},$ (B4)

where $\eta_{\mathrm{d}}(E_{\mathrm{d}})$ is the detection efficiency for DR
events with our neutral-fragment detector, $\zeta$ is the fraction of ions
enclosed by the electron beam ($\zeta=1.000^{+0}_{-0.001}$ for the absolute
scaling measurement in Appendix C), $n_{\mathrm{e}}(E_{\mathrm{d}})$ is the
electron density, $N_{\mathrm{i}}$ is the number of ions in the CSR, and
$\hat{l}_{0}=$0.79(1)\text{\,}\mathrm{m}$$ is the effective electron-ion
overlap length (Kálosi et al., 2022). Here, the electron density is calculated
from the measured electron beam current and radius.

The detection efficiency $\eta_{\mathrm{d}}$ for DR events is determined by
two factors. First, the front MCP plate of our detector has a counting
efficiency for single particles of $p_{\mathrm{d}}=$0.593(15)\text{\,}$$. This
value was obtained in an independent experiment by evaluating the ratio of
single-to-double particle hits resulting from DR reactions of CF+, where
nearly all neutrals arrived at the detector surface in coincident pairs (C and
F). From the value of $p_{\mathrm{d}}$, we calculate an intrinsic DR event
detection efficiency, which is the summed probability to detect either one DR
fragment or both fragments, as

$\hat{\eta}_{\mathrm{d}}=2p_{\mathrm{d}}(1-p_{\mathrm{d}})+p_{\mathrm{d}}^{2}=$0.834(12)\text{\,}$.$
(B5)

Second, taking the geometry into account, the broad spatial distribution of
CH+ DR fragments results in a fraction of H atoms arriving outside the
detector area. This fraction depends on the contributing DR final channels
and, thus, on $E_{\mathrm{d}}$. Using the $p_{\mathrm{d}}$ value from above,
we were able to infer the combined intrinsic and geometric detection
efficiency $\eta_{\mathrm{d}}(E_{\mathrm{d}})$ for DR events from measurements
of the ratio for single-to-double particle hits
$r_{\mathrm{s/d}}(E_{\mathrm{d}})$ via (Paul, 2021)

$\eta_{\mathrm{d}}(E_{\mathrm{d}})=p_{\mathrm{d}}+\frac{p_{\mathrm{d}}-p_{\mathrm{d}}^{2}}{p_{\mathrm{d}}(2+r_{\mathrm{s/d}}(E_{\mathrm{d}}))-1}.$
(B6)

We measured $r_{\mathrm{s/d}}(E_{\mathrm{d,s}})$ at a few specific detuning
energies $E_{\mathrm{d,s}}$ and obtained values of
$\eta_{\mathrm{d}}(E_{\mathrm{d,s}})\in[0.74,\hat{\eta}_{\mathrm{d}}]$ with
statistical uncertainties at all $E_{\mathrm{d,s}}$ below $0.01$. When
evaluating Equation (B4), we interpolated $\eta_{\mathrm{d}}(E_{\mathrm{d}})$
between the measured $E_{\mathrm{d,s}}$. Given the range of possible
$\eta_{\mathrm{d}}$ values, we expect the systematic uncertainty of this
interpolation to be $<5\%$ and, thus, to be negligible compared to our
absolute scaling uncertainty (Appendix C). See Paul (2021) for further details
on the DR event detection efficiency analysis.

The storage-time dependence of the merged-beams rate coefficient
$\alpha^{\mathrm{mb}}(E_{\mathrm{d}},t)$ is found using Equation (B4) for
certain storage-time windows, where we consider the storage-time dependences
of $R_{\mathrm{e}}(t)$ and $N_{\mathrm{i}}(t)$. Since we did not continuously
measure the ion number, we use the detected background rate
$R_{\mathrm{b}}(t)$ from collisions of CH+ with residual-gas particles as a
proxy for the ion number. This rate is inferred from
$R_{\mathrm{b}}(t)=R_{\mathrm{ref}}(t)-R_{\mathrm{dark}}$, where
$R_{\mathrm{dark}}\sim$50\text{\,}\mathrm{H}\mathrm{z}$$ is the experimentally
determined dark-count rate of our detector. We can then calculate a relative
merged-beams rate coefficient

$\alpha^{\mathrm{mb}}_{\mathrm{rel}}(E_{\mathrm{d}},t)=\frac{1}{\eta_{\mathrm{d}}(E_{\mathrm{d}})\zeta
n_{\mathrm{e}}(E_{\mathrm{d}})\hat{l}_{0}/C_{0}}\frac{R_{\mathrm{e}}(E_{\mathrm{d}},t)}{R_{\mathrm{b}}(t)},$
(B7)

for each storage-time window within a given measurement set. We evaluate
Equation (B7) for each measurement set separately, and combine different sets
by scaling the results so that they match at commonly measured ranges for
$E_{\mathrm{d}}$ and $t$. Finally, we determine the dependence
$N_{\mathrm{i}}(R_{\mathrm{b}})$ for a single set to obtain the absolute rate
coefficient in Equation (B4) (see Appendix C).

The energy resolution in our merged-beams rate coefficient spectra, as shown
in Figure 2, is dominated by the velocity spread within the electron beam,
which exceeds that of the phase-space cooled CH+ ion beam. The corresponding
probability distribution $f_{\mathrm{mb}}(E;E_{\mathrm{d}})$ of collision
energies $E$ for a single detuning energy $E_{\mathrm{d}}$ is influenced by
several factors (see Figure 1(a)): the angles between the electron and ion
beams when merging and de-merging the two beams; the electron energy
transition from outside to inside the drift tubes; the transverse temperature
of the electron beam of
$k_{\mathrm{B}}T_{\perp}=$2.0^{+1.0}_{-0.5}\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$;
and the longitudinal temperature of the electron beam
$k_{\mathrm{B}}T_{\parallel}(E_{\mathrm{e}})$, which itself varies with the
laboratory-frame energy of the electron beam ($E_{\mathrm{e}}$), i.e., over
the full overlap section. This variation is due to the kinematic compression
effect when accelerating a particle beam, and due to relaxation effects within
the beam that minimize the potential energy through Coulomb interactions
(Paul, 2021).

To account for all these effects, we have developed an analytic model,
starting from electron emission by the photocathode, and accounting for
subsequent acceleration and deceleration steps, to calculate
$k_{\mathrm{B}}T_{\parallel}(E_{\mathrm{e}})$. Furthermore, we use a Monte-
Carlo simulation in order to model the full collision geometry. An example for
a resulting energy distribution function $f_{\mathrm{mb}}(E;E_{\mathrm{d}})$
is found in Kálosi et al. (2022). In Appendix E we use
$f_{\mathrm{mb}}(E;E_{\mathrm{d}})$ to derive the DR cross section from our
merged-beams rate coefficient. The CSR energy resolutions displayed in Figures
2 and 5 correspond to the FWHM of $f_{\mathrm{mb}}(E;E_{\mathrm{d}})$.

##  Appendix C Absolute scaling of the merged-beams rate coefficient

Absolute scaling of the merged-beams DR rate coefficient requires knowledge of
the number of ions stored inside the CSR. Since we could not measure the ion
number at all storage times, we used the storage-time dependence of the
background rate $R_{\mathrm{b}}(t)$ as a proxy for the ion number evolution
(Appendix B). The proportionality factor

$S_{\mathrm{b}}=R_{\mathrm{b}}(t)/N_{\mathrm{i}}(t)$ (C1)

was evaluated in two separate measurements. First, we calibrated the voltage
induced by the ion beam on a capacitive current pickup in the CSR (PU-C; von
Hahn et al. 2016) to the ion current $I$. Second, we determined the averaged
ion number $\langle N_{\mathrm{i}}\rangle$ from $I$ in a small storage-time
window $t=$21-21.5\text{\,}\mathrm{s}$$, together with the averaged residual
gas background rate $\langle R_{\mathrm{b}}\rangle$.

For the ion current calibration of the PU-C, we compared the induced voltage
to a direct ion current measurement using a Faraday cup, which is located in
front of the CSR. A fast-switched ($<$50\text{\,}\mathrm{n}\mathrm{s}$$)
electrostatic deflector was used to create an $\sim$20\text{\,}\mu\mathrm{s}$$
long square-wave pulse of CH+ ions from the ion source. The square wave was
alternately dumped into the Faraday cup to determine the ion current and
injected into the CSR to detect the corresponding induced voltage on the PU-C
in the first revolution. We verified that the Faraday cup collection
efficiency was essentially unity ($\pm 2\%$). By moving the ion beam across
the Faraday cup aperture, we obtained a flat plateau in the ion current. We
prevented the loss of secondary electrons by applying a negative voltage of
$-250\text{\,}\mathrm{V}$ to an aperture placed directly in front of the
Faraday cup, where further increase of the voltage did not change the measured
ion current. Additionally, we verified that there is an efficient transport
into the CSR by reducing the beam emittance with a set of
$4.5\text{\,}\mathrm{m}\mathrm{m}$ apertures in the injection beamline and
repeating the calibration. The lower emittance beam is expected to be injected
and transported to the pickup similarly or more efficiently than our usual ion
beams. We obtained calibration factors agreeing within $1\%$ for the two
beams. A similar agreement was obtained when reducing the ion current by a
factor of $\sim 10$, which indicates sufficient linearity for the calibration.
We estimate the final uncertainty in the ion current calibration of the PU-C
to be $10\%$, mainly due to the stability of the ion source between square
wave pulses.

With the calibrated PU-C, we were able to determine $S_{\mathrm{b}}$ from
Equation (C1) in a separate measurement. For this, the ion beam was stored
inside the CSR and bunched by applying an AC voltage with a frequency of
$\sim$400\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ with the CSR rf-bunching
system (von Hahn et al., 2016), which is located in the left section of the
CSR in Figure 1(a). This voltage caused the stored ions to split into seven
distinct bunches. When passing the PU-C, each bunch induced a mirror charge.
The corresponding time-dependent voltage was converted into an ion current
$I(t)$. The absolute ion number in the storage-time interval
$t=$21-21.5\text{\,}\mathrm{s}$$ is then given by

$\langle
N_{\mathrm{i}}\rangle=\frac{1}{Zen_{\mathrm{r}}}\int_{$21\text{\,}\mathrm{s}$}^{$21.5\text{\,}\mathrm{s}$}I(t)\mathrm{d}t,$
(C2)

where $Z=1$ is the charge state of CH+, $e$ is the elementary charge, and
$n_{\mathrm{r}}\sim 30000$ is the number of revolutions the ions made in the
CSR during the measurement time. Finally, we calculate the proportionality
factor between ion number and residual-gas-induced detector count rate using
Equation (C1).

The absolute CH+ DR rate coefficient for energies of
$E_{\mathrm{d}}=$0.06-0.29\text{\,}\mathrm{e}\mathrm{V}$$ was determined
simultaneously with the bunched-beam ion-number measurement and extended to a
storage-time interval of $t=$21-100\text{\,}\mathrm{s}$$. After the bunched-
beam ion-number measurement, we removed the AC voltage from the rf system and
obtained a continuous ion beam at $t>$21.5\text{\,}\mathrm{s}$$, similar to
all other DR measurement sets. The rate coefficient
$\alpha^{\mathrm{mb}}(E_{\mathrm{d}}=$0.06-0.29\text{\,}\mathrm{e}\mathrm{V}$,t=$21-100\text{\,}\mathrm{s}$)$
was then calculated from Equation (B4), using
$N_{\mathrm{i}}(t)=R_{\mathrm{b}}(t)/S_{\mathrm{b}}$.

Next, we put the relative rate coefficient for all measurement sets (Appendix
B) onto an absolute scale by matching
$\alpha^{\mathrm{mb}}_{\mathrm{rel}}(E_{\mathrm{d}}=$0.06-0.29\text{\,}\mathrm{e}\mathrm{V}$,t=$21-100\text{\,}\mathrm{s}$)$
to
$\alpha^{\mathrm{mb}}(E_{\mathrm{d}}=$0.06-0.29\text{\,}\mathrm{e}\mathrm{V}$,t=$21-100\text{\,}\mathrm{s}$)$.
The total uncertainty of the absolute scaling we estimate to be $13\%$, where
the main contributions stem from the ion current calibration mentioned above
($10\%$) and the electron density uncertainty ($8\%$).

A small additive correction of
$\delta\alpha^{\mathrm{mb}}=$6.1\text{\times}{10}^{-9}\text{\,}\mathrm{c}\mathrm{m}^{3}\,\mathrm{s}^{-1}$$
is applied to the absolute scaled rate coefficient due to the fact that
$R_{\mathrm{ref}}$ (see Appendix B) was due to both residual-gas-induced
events and a small contribution from DR. We determined
$\delta\alpha^{\mathrm{mb}}$ in a separate experiment by measuring the
absolute rate coefficient at $E_{\mathrm{ref}}$ using

$R_{\mathrm{e}}(E_{\mathrm{ref}})=R_{\mathrm{ref}}-R_{\mathrm{res}}.$ (C3)

instead of Equation (B3). Here, $R_{\mathrm{res}}$ stands for the pure
residual-gas-induced rate, which we determined by switching off the electron
beam in a fast manner, similar to the fast energy adjustment described in
Appendix B. The correction $\delta\alpha^{\mathrm{mb}}$ is due to DR reactions
in the electron-ion beam-overlap section outside of the drift tubes in Figure
1(a). Thus, it occurs at all $E_{\mathrm{d}}$ in the merged-beams rate
coefficient (Figure 2). The collisions outside the drift-tube section take
place at collision energies of $E\sim$1.2\text{\,}\mathrm{e}\mathrm{V}$$,
located on the flank of the DR rate coefficient peak in Figure 2, and are
included in the collision energy distribution
$f_{\mathrm{mb}}(E;E_{\mathrm{d}})$. The offset $\delta\alpha^{\mathrm{mb}}$
is accounted for when calculating the DR cross section in Appendix E.

##  Appendix D Evaluation of level-resolved merged-beams rate coefficients

Figure 5: (a) Merged-beams DR rate coefficient of CH+ for two different ion-
beam storage-time windows in the CSR. Relative rotational populations in the
CSR (see Figure 4) are shown in the inset. The gray areas highlight the
storage times for the measurements shown. (b) Comparison of the $J=0$ and
$J=1$ rate coefficients calculated using a MCMC method (see text). In both
panels, statistical one-sigma uncertainties are indicated by the error bars
and bands, while the absolute scaling uncertainty is $\pm 13\%$. The data
behind panel (b) are available.

Independent control over the internal ion-beam levels and electron collision
energies in the CSR enables us to determine rotational level-resolved DR
merged-beams rate coefficients, as was first demonstrated by Novotný et al.
(2019). Here, we use a similar method to evaluate the CH+ DR merged-beams rate
coefficient of the $X^{1}\Sigma^{+}(v=0,J=0,1)$ levels.

The evaluation of $J$-specific DR rate coefficients relies on storage-time-
dependent DR measurements and knowledge of all internal level populations in
each storage-time window. Here, we have binned the absolute DR rate
coefficient ($\alpha^{\mathrm{mb}}(E_{\mathrm{d}},t)$) into six storage-time
intervals

$\displaystyle t[s]\in\\{$ $\displaystyle 21-25,$ $\displaystyle 25-30,30-40,$
(D1) $\displaystyle 40-60,$ $\displaystyle 60-80,80-100\\},$

which are denoted as $t_{k}$ $(k\in\\{1,...,6\\})$ in the following. Two
examples for the rate coefficients in different time windows are shown in
Figure 5(a). The corresponding rotational ($p_{J}(t_{k})[J=0-2]$) and
metastable ($p_{\mathrm{m}}(t_{k})$) populations within these storage-time
windows are taken from the modeled lines in Figure 4 and are highlighted in
the inset of Figure 5(a). Here, we used the averaged values within each
storage time window $t_{k}$ for further analysis. The relation between level-
specific rate coefficients
$\alpha^{\mathrm{mb}}_{J/\mathrm{m}}(E_{\mathrm{d}})$ and the storage-time-
dependent ones is given by the linear equation system

$\alpha^{\mathrm{mb}}(E_{\mathrm{d}},t_{k})=\sum_{J=0}^{2}\alpha^{\mathrm{mb}}_{J}(E_{\mathrm{d}})p_{J}(t_{k})+\alpha^{\mathrm{mb}}_{\mathrm{m}}(E_{\mathrm{d}})p_{\mathrm{m}}(t_{k}).$
(D2)

Using our six storage-time-dependent results, one obtains an overdetermined
system for evaluating the four level-specific rate coefficients. In principle,
such a system could be solved by least-squares minimization. However, a least-
squares approach is not suited for including the boundary conditions
$\alpha^{\mathrm{mb}}_{J/\mathrm{m}}(E_{\mathrm{d}})\geq 0$, which are
important for the $J=2$ and metastable levels, which both have a small but
non-zero population. As an alternative, we use a Markow Chain Monte Carlo
(MCMC; Hogg & Foreman-Mackey 2018) method for the statistical analysis, which
is based on the Bayesian inference principle and is capable of embedding the
required boundary conditions. The result of the MCMC algorithm is the
posterior probability distribution $p(\alpha^{\mathrm{mb}}_{J/\mathrm{m}}|D)$
for the rate coefficients based on the observed data $D$. The posterior is
calculated from

$p(\alpha^{\mathrm{mb}}_{J/\mathrm{m}}|D)=Ze^{-\chi_{\nu}^{2}/2}p(\alpha^{\mathrm{mb}}_{J/\mathrm{m}}),$
(D3)

where $Z$ is a normalization constant, $e^{-\chi_{\nu}^{2}/2}$ is the log-
likelihood function, and $p(\alpha^{\mathrm{mb}}_{J/\mathrm{m}})$ is the prior
probability distribution for the rate coefficients. The reduced chi-squared
$\chi_{\nu}^{2}$ for $\nu$ degrees of freedom enters the log-likelihood
function and is calculated from the rate coefficient data
$\alpha^{\mathrm{mb}}(t_{k})$ as

$\chi_{\nu}^{2}=\frac{1}{\nu}\sum_{k=1}^{6}\left(\frac{\alpha^{\mathrm{mb}}(t_{k})-\sum_{J/\mathrm{m}}\alpha^{\mathrm{mb}}_{J/\mathrm{m}}p_{J/\mathrm{m}}(t_{k})}{\Delta\alpha^{\mathrm{mb}}(t_{k})}\right)^{2},$
(D4)

where $\Delta\alpha^{\mathrm{mb}}(t_{k})$ are the statistical one-sigma
uncertainties of our measured $\alpha^{\mathrm{mb}}(t_{k})$. We account for
the detuning energy dependence of the rate coefficients
$\alpha^{\mathrm{mb}}(E_{\mathrm{d}},t_{k})$ by running a separate MCMC
analysis for each measured detuning energy. MCMC algorithms are designed for
sampling the log-likelihood function efficiently and are thus well suited for
multidimensional parameter spaces.

For this analysis we used the MCMC python package emcee (Foreman-Mackey et
al., 2013). We applied the above mentioned boundary conditions and set the
initial prior to a uniform probability distribution as

$p(\alpha^{\mathrm{mb}}_{J/\mathrm{m}})=\left\\{\begin{array}[]{ll}Z_{0}&0\leq\alpha^{\mathrm{mb}}_{J/\mathrm{m}}\leq$10^{-5}\text{\,}\mathrm{c}\mathrm{m}^{3}\,\mathrm{s}^{-1}$,\\\
0&\text{otherwise},\\\ \end{array}\right.$ (D5)

where $Z_{0}$ is a normalization constant. Using our measured
$\alpha^{\mathrm{mb}}(E_{\mathrm{d}},t_{k})$ led to a high statistical scatter
of the results. We avoided this by constructing a moving average of our
$\alpha^{\mathrm{mb}}(E_{\mathrm{d}},t_{k})$ and
$\Delta\alpha^{\mathrm{mb}}(E_{\mathrm{d}},t_{k})$ data over the experimental
FWHM energy resolution (see bars on top of Figure 5) and using these average
quantities in Equation (D4). We justify this averaging procedure for the MCMC
input data by the fact that any level-specific features in the data must be
visible over a range of detuning energies larger than our energy resolution.

The MCMC results are the probability distributions
$p(\alpha^{\mathrm{mb}}_{J/\mathrm{m}}|D)$ for the individual rate
coefficients. For $\alpha^{\mathrm{mb}}_{J=0,1}$ we obtain sufficiently narrow
distributions and are able to evaluate their most probable values and one-
sigma uncertainties $\Delta\alpha^{\mathrm{mb}}_{J=0,1}$. These results are
presented in Figure 5(b) as lines and error bands, respectively. We note that
the MCMC analysis was only applied to the rate coefficient data up to
$E_{\mathrm{d}}=$3.6\text{\,}\mathrm{e}\mathrm{V}$$, corresponding to the rate
coefficient minimum before the onset of the direct DR/DE peak in Figure 2. At
higher energies we do not expect any $J$-dependent features due to the non-
resonant character of the DR and DE process (Section 3). Hence, at these
energies, we set
$\alpha^{\mathrm{mb}}_{J=0,1}(E_{\mathrm{d}})=\alpha^{\mathrm{mb}}(E_{\mathrm{d}},$21-100\text{\,}\mathrm{s}$)$
and
$\Delta\alpha^{\mathrm{mb}}_{J=0,1}(E_{\mathrm{d}})=\Delta\alpha^{\mathrm{mb}}(E_{\mathrm{d}},$21-100\text{\,}\mathrm{s}$)$,
leading to the narrowed error band above
$E_{\mathrm{d}}=$3.6\text{\,}\mathrm{e}\mathrm{V}$$. For the $J=2$ and
metastable levels, the MCMC analysis does not yield significant results. Due
to their low population, a broad range of values is accepted for
$\alpha^{\mathrm{mb}}_{J=2}$ and $\alpha^{\mathrm{mb}}_{\mathrm{m}}$.

##  Appendix E Obtaining kinetic temperature rate coefficients from merged-
beams rate coefficient data

The conversion of our CH+ $(J=0,1)$ DR merged-beams rate coefficients
$\alpha^{\mathrm{mb}}_{J}(E_{\mathrm{d}})$ into kinetic temperature rate
coefficients $\alpha^{\mathrm{k}}_{J}(T_{\mathrm{k}})$, as shown in Figure
3(a), is achieved in a two-step process. First, we deconvolve444The terms
convolution and deconvolution in this section are not used in a rigorous
mathematical meaning. They refer to an integral transformation of the cross
section to a rate coefficient (convolution; Equation (5)) and its inverse
(deconvolution). Both transformations are described in detail by Novotný et
al. (2013). the merged-beams rate coefficients into cross sections
$\sigma_{J}(E)$, using our merged-beams collision energy resolution function
$f_{\mathrm{mb}}(E;E_{\mathrm{d}})$ (Appendix B). Second, we convolve the
resulting cross sections with a Maxwell-Boltzmann distribution to obtain the
kinetic rate coefficients.

In the deconvolution process we make use of the relation between cross
sections and merged-beams rate coefficients (Equation (5)) that can be
explicitly expressed as

$\alpha^{\mathrm{mb}}_{J}(E_{\mathrm{d}})=\frac{l_{0}}{\hat{l}_{0}}\int_{0}^{\infty}\sigma_{J}(E)\sqrt{\frac{2E}{m_{\mathrm{e}}}}f_{\mathrm{mb}}(E;E_{\mathrm{d}})\mathrm{d}E,$
(E1)

where $l_{0}=$1.123(14)\text{\,}\mathrm{m}$$ is the total geometrical overlap
length of electrons and ions. Using the deconvolution method of Novotný et al.
(2013), we can generate a cross section histogram $\sigma_{i}$ that is binned
in the energy domain. This method relies on fitting the contents of the
individual cross section bins such that the discrete integral version of
Equation (E1) reproduces the merged-beams rate coefficient data best.

Here, we have improved the procedure of Novotný et al. (2013) in two ways.
First, we use a more detailed collision energy distribution
$f_{\mathrm{mb}}(E;E_{\mathrm{d}})$ that involves not only the geometry of our
drift-tube collision setup (Figure 1(a)) but also the variation of the
electron beam longitudinal temperature
$k_{\mathrm{B}}T_{\parallel}(E_{\mathrm{e}})$ (Appendix B). Second, we model
the energy dependence of the cross section for
$E<$0.5\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$, where our energy resolution
does not allow us to experimentally determine any dependence, as

$\sigma_{0}(E)=\sigma_{0,\mathrm{c}}/E,$ (E2)

instead of assuming a constant value. Here, $\sigma_{0,\mathrm{c}}$ is fitted
in the deconvolution process. Equation (E2) accounts for the possibility of
direct DR at low collision energies and includes the diverging nature of the
DR cross section towards $E=0$ (Mitchell, 1990). We note that assuming instead
a constant cross section for $E<$0.5\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$
does not influence the resulting kinetic rate coefficient significantly.

The kinetic rate coefficients $\alpha^{\mathrm{k}}_{J}(T_{\mathrm{k}})$ are
calculated from a convolution of the DR cross sections with a Maxwell-
Boltzmann distribution of temperature $T_{\mathrm{k}}$ as

$\alpha^{\mathrm{k}}_{J}(T_{\mathrm{k}})=\int_{0}^{\infty}\sigma_{J}(E)\sqrt{\frac{2E}{m_{\mathrm{e}}}}\sqrt{\frac{4E}{\pi(k_{\mathrm{B}}T_{\mathrm{k}})^{3}}}e^{-E/k_{\mathrm{B}}T_{\mathrm{k}}}\mathrm{d}E.$
(E3)

Due to the discrete character of our cross sections, we evaluate Equation (E3)
as a sum instead of an integral to generate the CSR kinetic rate coefficient
curves (e.g., Figure 3(a)).

The kinetic rate coefficients are effected by several uncertainties.
Statistical uncertainties of the merged-beams rate coefficients (see error
band in Figure 2) were propagated through the deconvolution and convolution
procedure into $\alpha^{\mathrm{k}}_{J}(T_{\mathrm{k}})$ and are negligible
compared to the systematic uncertainties. We account for the systematic
uncertainties discussed below and added them in quadrature to generate the
kinetic rate coefficient error bands (see Figure 3(a)). The absolute scaling
uncertainty of $\alpha^{\mathrm{mb}}_{J}$ ($\pm 13\%$; Appendix C) propagates
directly into $\alpha^{\mathrm{k}}_{J}(T_{\mathrm{k}})$ and is independent of
temperature. It dominates the total uncertainty for
$$650\text{\,}\mathrm{K}$<T_{\mathrm{k}}<$20\,000\text{\,}\mathrm{K}$$. We
also ran our deconvolution and convolution procedures for different
longitudinal ($k_{\mathrm{B}}T_{\parallel}$) and transverse temperatures
($k_{\mathrm{B}}T_{\perp}$) of our electron beam. Here, the influence of
$k_{\mathrm{B}}T_{\parallel}$ turns out to be negligible. For
$k_{\mathrm{B}}T_{\perp}$, we vary it between extremes of
$1.5\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ and
$3\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$, where
$k_{\mathrm{B}}T_{\perp}=$1.5\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ results
in a common decrease of all kinetic rate coefficient values and
$k_{\mathrm{B}}T_{\perp}=$3\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ results
in a common increase. The corresponding uncertainties are the main
contribution for $T_{\mathrm{k}}<$650\text{\,}\mathrm{K}$$ and result in a
broadening of the error band towards lower temperatures. Finally, we evaluate
the influence of the merged-beams rate coefficients at high energies
$\alpha^{\mathrm{mb}}_{J}(E_{\mathrm{d}}>$3.6\text{\,}\mathrm{e}\mathrm{V}$)$,
where we cannot distinguish between DR and DE (see Section 3). To account for
any possible DE contribution, we determine the kinetic rate coefficient for
$\alpha^{\mathrm{mb}}_{J}(E_{\mathrm{d}}>$3.6\text{\,}\mathrm{e}\mathrm{V}$)$
scaled to $0$. The corresponding uncertainty becomes dominant for
$T_{\mathrm{k}}>$20\,000\text{\,}\mathrm{K}$$, leading to the broadening of
the lower uncertainty limit with increasing temperature in Figure 3(a).

We also note that for constructing the theoretical kinetic rate coefficient
curve in Figure 3(a) we used the calculated cross section data from Mezei et
al. (2019), which were provided to us by the authors. We convolved their cross
section into a kinetic rate coefficient via Equation (E3), similar to the
treatment of our data. The result is shown as dotted line in Figure 3(a).

##  Appendix F Analytic representations of the $J=0,1$ DR kinetic temperature
rate coefficients

The results for the CH${}^{+}(J=0,1)$ DR kinetic temperature rate coefficients
can be incorporated into astrochemical databases or used as input for chemical
network simulations. For this purpose, we provide two different analytic
representations for the rate coefficient and its one-sigma error bands (Figure
3(a)).

First, we model our data, as well as the upper and lower one-sigma error
boundaries, with the fit function proposed by Novotný et al. (2013),

$\displaystyle\alpha^{\mathrm{k}}_{J}(T_{\mathrm{k}})[\mathrm{cm^{3}\,s^{-1}}]$
$\displaystyle=$ $\displaystyle
A\left(\frac{300}{T_{\mathrm{k}}[\mathrm{K}]}\right)^{n}$ $\displaystyle+$
$\displaystyle
T_{\mathrm{k}}[\mathrm{K}]^{-1.5}\sum_{r=1}^{4}c_{r}\exp\left(-\frac{T_{r}}{T_{\mathrm{k}}[\mathrm{K}]}\right),$

where $A$, $n$, $c_{r}$, and $T_{r}$ are the fit parameters. Their resulting
values are presented in Tables 1 and 2 for the $J=0$ and $J=1$ levels,
respectively. The relative deviations of the listed fit functions from the
rate coefficients and their lower and upper error bounds are below $2.3\%$ at
all temperatures. The representation in Equation (F) explicitly considers
features (i.e., broad peaks or dips resulting from multiple resonances in the
cross section) in the rate coefficient by the summation term, where each $r$
stands for one of the $N_{r}=4$ features. This enables an appropriate modeling
of the data with a low number of fit parameters ($2(N_{r}+1)$). Additionally,
Equation (F) is a continuously differentiable function, which is generally
desired for chemical modeling purposes.

Table 1: Fit parameters for the CH+($J=0$) kinetic temperature rate coefficient and its lower and upper error bound in Figure 3(a), using Equation (F) as the fit function. Parameter | Rate coefficient | Lower error limit | Upper error limit
---|---|---|---
$A$ | $2.43\text{\times}{10}^{-7}\text{\,}$ | $2.07\text{\times}{10}^{-7}\text{\,}$ | $2.94\text{\times}{10}^{-7}\text{\,}$
$n$ | $0.740\text{\,}$ | $0.719\text{\,}$ | $0.761\text{\,}$
$c_{1}$ | $0.642\text{\,}$ | $-0.327\text{\,}$ | $0.660\text{\,}$
$c_{2}$ | $2.36\text{\times}{10}^{-2}\text{\,}$ | $1.73\text{\times}{10}^{-2}\text{\,}$ | $2.85\text{\times}{10}^{-2}\text{\,}$
$c_{3}$ | $-2.58\text{\times}{10}^{-3}\text{\,}$ | $-2.48\text{\times}{10}^{-3}\text{\,}$ | $-3.09\text{\times}{10}^{-3}\text{\,}$
$c_{4}$ | $-1.13\text{\times}{10}^{-3}\text{\,}$ | $-1.10\text{\times}{10}^{-3}\text{\,}$ | $-1.29\text{\times}{10}^{-3}\text{\,}$
$T_{1}$ | $112\,000\text{\,}$ | $151\,000\text{\,}$ | $107\,000\text{\,}$
$T_{2}$ | $12\,000\text{\,}$ | $12\,400\text{\,}$ | $11\,800\text{\,}$
$T_{3}$ | $941\text{\,}$ | $1050\text{\,}$ | $919\text{\,}$
$T_{4}$ | $220\text{\,}$ | $234\text{\,}$ | $215\text{\,}$
Table 2: Same as Table 1 but for CH+($J=1$). Parameter | Rate coefficient | Lower error limit | Upper error limit
---|---|---|---
$A$ | $2.22\text{\times}{10}^{-7}\text{\,}$ | $2.51\text{\times}{10}^{-7}\text{\,}$ | $2.62\text{\times}{10}^{-7}\text{\,}$
$n$ | $0.733\text{\,}$ | $0.613\text{\,}$ | $0.762\text{\,}$
$c_{1}$ | $1.02\text{\,}$ | $-0.183\text{\,}$ | $1.07\text{\,}$
$c_{2}$ | $3.41\text{\times}{10}^{-2}\text{\,}$ | $-7.36\text{\times}{10}^{-3}\text{\,}$ | $4.20\text{\times}{10}^{-2}\text{\,}$
$c_{3}$ | $-2.56\text{\times}{10}^{-3}\text{\,}$ | $-2.36\text{\times}{10}^{-3}\text{\,}$ | $-2.65\text{\times}{10}^{-3}\text{\,}$
$c_{4}$ | $-1.02\text{\times}{10}^{-3}\text{\,}$ | $-6.82\text{\times}{10}^{-4}\text{\,}$ | $-1.10\text{\times}{10}^{-3}\text{\,}$
$T_{1}$ | $108\,000\text{\,}$ | $65\,600\text{\,}$ | $104\,000\text{\,}$
$T_{2}$ | $11\,900\text{\,}$ | $2300\text{\,}$ | $11\,700\text{\,}$
$T_{3}$ | $1260\text{\,}$ | $587\text{\,}$ | $1160\text{\,}$
$T_{4}$ | $264\text{\,}$ | $161\text{\,}$ | $255\text{\,}$
Table 3: Fit parameters for the CH+($J=0$) kinetic temperature rate coefficient and its relative uncertainty in Figure 3(a), using Equations (F2) and (F3) respectively, as fit functions for different temperature ranges. Parameter | Temperature range
---|---
| $10-69.6\text{\,}\mathrm{K}$ | $69.6-469\text{\,}\mathrm{K}$ | $469-3920\text{\,}\mathrm{K}$ | $3920-19900\text{\,}\mathrm{K}$ | $19900-40000\text{\,}\mathrm{K}$
$A$ | $1.77\text{\times}{10}^{-7}\text{\,}$ | $1.16\text{\times}{10}^{-7}\text{\,}$ | $2.03\text{\times}{10}^{-8}\text{\,}$ | $3.08\text{\times}{10}^{-7}\text{\,}$ | $1.70\text{\times}{10}^{-10}\text{\,}$
$\beta$ | $-0.915\text{\,}$ | $-1.13\text{\,}$ | $7.28\text{\times}{10}^{-2}\text{\,}$ | $-0.689\text{\,}$ | $0.756\text{\,}$
$\gamma$ | $2.94\text{\,}$ | $-4.59\text{\,}$ | $-569\text{\,}$ | $2410\text{\,}$ | $-26\,300\text{\,}$
$F_{0}$ | $1.23\text{\,}$ | $1.23\text{\,}$ | $1.28\text{\,}$ | $1.06\text{\,}$ | $6.81\text{\times}{10}^{-10}\text{\,}$
$g$ | $0.657\text{\,}$ | $0.300\text{\,}$ | $31.7\text{\,}$ | $-28.8\text{\,}$ | $-6480\text{\,}$
Table 4: Same as Table 3 but for CH+($J=1$).  Parameter | Temperature range
---|---
| $10-73.3\text{\,}\mathrm{K}$ | $73.3-681\text{\,}\mathrm{K}$ | $681-2680\text{\,}\mathrm{K}$ | $2680-18500\text{\,}\mathrm{K}$ | $18500-40000\text{\,}\mathrm{K}$
$A$ | $1.84\text{\times}{10}^{-7}\text{\,}$ | $1.29\text{\times}{10}^{-7}\text{\,}$ | $2.36\text{\times}{10}^{-8}\text{\,}$ | $2.77\text{\times}{10}^{-7}\text{\,}$ | $4.51\text{\times}{10}^{-11}\text{\,}$
$\beta$ | $-0.846\text{\,}$ | $-0.958\text{\,}$ | $-2.69\text{\times}{10}^{-2}\text{\,}$ | $-0.632\text{\,}$ | $1.07\text{\,}$
$\gamma$ | $2.18\text{\,}$ | $-12.7\text{\,}$ | $-646\text{\,}$ | $2400\text{\,}$ | $-29\,100\text{\,}$
$F_{0}$ | $1.21\text{\,}$ | $1.19\text{\,}$ | $1.22\text{\,}$ | $1.17\text{\,}$ | $1.19\text{\times}{10}^{-11}\text{\,}$
$g$ | $0.866\text{\,}$ | $2.35\text{\,}$ | $15.2\text{\,}$ | $-0.451\text{\,}$ | $-7720\text{\,}$

The second representation uses Arrhenius-Kooij functions, as are typically
used in astrochemistry, combustion chemistry, and other chemical models and
databases. However, with one such function, we cannot model all the
experimentally observed features in the rate coefficient correctly. Instead,
we provide a representation by $m$ piecewise-joined Arrhenius-Kooij equations
($T_{\mathrm{k}}\in[T_{i},T_{i+1}]$, $i\in\\{0,...,m\\}$) with

$\alpha^{\mathrm{k}}_{J}(T_{\mathrm{k}})[$\text{\,}\mathrm{c}\mathrm{m}^{3}\,\mathrm{s}^{-1}$]=A_{i}\left(\frac{T_{\mathrm{k}}[\mathrm{K}]}{300}\right)^{\beta_{i}}e^{-\frac{\gamma_{i}}{T_{\mathrm{k}}[\mathrm{K}]}},$
(F2)

where $A_{i}$, $\beta_{i}$, and $\gamma_{i}$ are the fit parameters. Following
KIDA conventions, the uncertainty is described by a log-normal factor
$F_{J}=\exp(\Delta\alpha^{\mathrm{k}}_{J}/\alpha^{\mathrm{k}}_{J})$ for each
interval with

$F_{J}(T_{\mathrm{k}})=F_{0,i}\exp\left(g_{i}\left(\frac{1}{T_{\mathrm{k}}[\mathrm{K}]}-\frac{1}{300}\right)\right),$
(F3)

where $F_{0,i}$ and $g_{i}$ are the fit parameters. We construct the data for
$F_{J}(T_{\mathrm{k}})$ as average of
$F_{J,\mathrm{high}}(T_{\mathrm{k}})=\exp(\Delta_{\mathrm{high}}\alpha^{\mathrm{k}}_{J}/\alpha^{\mathrm{k}}_{J})$
and
$F_{J,\mathrm{low}}(T_{\mathrm{k}})=\exp(\Delta_{\mathrm{low}}\alpha^{\mathrm{k}}_{J}/\alpha^{\mathrm{k}}_{J})$,
where $\Delta_{\mathrm{high}}\alpha^{\mathrm{k}}_{J}$ and
$\Delta_{\mathrm{low}}\alpha^{\mathrm{k}}_{J}$ correspond to the upper and
lower one-sigma errors, respectively (see Figure 3(a)).

For astrochemical calculations, a continuous behavior of the functions at the
interval borders is typically desired. However, this is not guaranteed in
Equations (F2) and (F3). It can be shown that a continuous behavior of the fit
functions can be achieved by introducing the following dependences of the
$A_{i}$ and $F_{0,i}$ parameters

$\displaystyle A_{i}$ $\displaystyle=$ $\displaystyle
A_{0}\prod_{j=1}^{i}\left(\frac{T_{j}}{300}\right)^{\beta_{j-1}-\beta_{j}}e^{\frac{\gamma_{j}-\gamma_{j-1}}{T_{j}}},$
(F4) $\displaystyle F_{0,i}$ $\displaystyle=$ $\displaystyle
F_{0,0}\exp\left(\frac{g_{i}-g_{0}}{300}\right)\prod_{j=1}^{i}\exp\left(\frac{g_{j-1}-g_{j}}{T_{j}}\right).\qquad$
(F5)

Inserting Equations (F4) and (F5) into Equations (F2) and (F3), we obtain the
continuous fit functions

$\displaystyle\alpha^{\mathrm{k}}_{J}(T_{\mathrm{k}})[$\text{\,}\mathrm{c}\mathrm{m}^{3}\,\mathrm{s}^{-1}$]$
$\displaystyle=$ $\displaystyle
A_{0}\left(\frac{T_{\mathrm{k}}[\mathrm{K}]}{300}\right)^{\beta_{i}}e^{-\frac{\gamma_{i}}{T_{\mathrm{k}}[\mathrm{K}]}}$
$\displaystyle\times$
$\displaystyle\prod_{j=1}^{i}\left(\frac{T_{j}}{300}\right)^{\beta_{j-1}-\beta_{j}}e^{\frac{\gamma_{j}-\gamma_{j-1}}{T_{j}}},$

and

$\displaystyle F_{J}(T_{\mathrm{k}})$ $\displaystyle=$ $\displaystyle
F_{0,0}\exp\left(\frac{g_{i}-g_{0}}{300}\right)\exp\left(g_{i}\left(\frac{1}{T_{\mathrm{k}}[\mathrm{K}]}-\frac{1}{300}\right)\right)$
(F7) $\displaystyle\times$
$\displaystyle\prod_{j=1}^{i}\exp\left(\frac{g_{j-1}-g_{j}}{T_{j}}\right),$

for the rate coefficient and uncertainty fits, respectively. We note that for
the first temperature interval ($i=0$) the product terms evaluate to unity and
that the functions show the same dependence on $T_{\mathrm{k}}$ as given by
Equations (F2) and (F3), respectively.

For the fit with Equation (F), we let the interval borders $T_{i}$ vary as
free parameters and subsequently fixed them for the fit with Equation (F7).
The resulting fit parameters are used to calculate the $A_{i}$ and $F_{0,i}$
parameters via Equations (F4) and (F5). All resulting parameters for Equations
(F2) and (F3), using $m=5$ temperature intervals, are listed in Tables 3 and 4
for the $J=0$ and $J=1$ levels, respectively. The relative deviations of the
fit functions to the rate coefficients and error bounds are below $3.2\%$ at
all temperatures.

The Arrhenius-Kooij representation of our rate coefficient in Equation (F2)
suffers from two disadvantages. First, the piecewise function is not
continuously differentiable. Second, for each feature in the rate coefficient,
one interval has to be added to the function in order to model the data
correctly. This introduces $3(N_{r}+1)$ fit parameters for $N_{r}$ features.
We recommend that chemical modelers use Equation (F) to represent the kinetic
temperature rate coefficient because it is continuously differentiable and
requires fewer fit parameters.

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