# Laser-probing the rotational cooling of molecular ions by electron
collisions

Ábel Kálosi<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, D-69117 Heidelberg, Germany  Columbia Astrophysics
Laboratory, Columbia University, New York, NY 10027, USA  Manfred Grieser
Robert von Hahn  Ulrich Hechtfischer  Now at: ASML Nederland B.V., Veldhoven,
The Netherlands  Claude Krantz  Present address: GSI Helmholtz Centre for
Heavy Ion Research, 64291 Darmstadt  Holger Kreckel  Damian Müll  Daniel Paul
Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg,
Germany  Daniel W. Savin  Columbia Astrophysics Laboratory, Columbia
University, New York, NY 10027, USA  Patrick Wilhelm  Andreas Wolf  Oldřich
Novotný  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, D-69117
Heidelberg, Germany

###### Abstract

We present state-selected measurements of rotational cooling and excitation
rates of CH+ molecular ions by inelastic electron collisions. The experiments
are carried out at the Cryogenic Storage Ring, making use of a monoenergetic
electron beam at matched velocity in combination with state-sensitive laser-
dissociation of the CH+ ions for simultaneous monitoring of the rotational
level populations. Employing storage times of up to 600 s, we create
conditions where electron-induced cooling to the $J=0$ ground state dominates
over radiative relaxation, allowing for the experimental determination of
inelastic electron collision rates to benchmark state-of-the-art theoretical
calculations. On a broader scale, our experiments pave the way to probe
inelastic electron collisions for a variety of molecular ions relevant in
various plasma environments.

Electron collisions with molecular ions are fundamental processes in partially
ionized molecular gas, ranging from astrophysical environments and planetary
atmospheres to low temperature plasma used in advanced technologies [1, 2].
Despite their fundamental relevance, detailed experimental data on the effect
of electron collisions on the internal excitation of the molecular ions are
essentially non-existent. Moreover, to enable computationally tractable
calculations, most state-of-the-art theoretical methods require significant
approximations [3], such as the Coulomb-Born (CB) [4, 5, *chu_erratum_1975, 7,
8] and R-matrix methods [9, 10, 11, 12, 13, 14, 15, 16], which in this context
have remained without experimental verification. For rotationally inelastic
collisions between electrons and molecular ions, in particular, theoretical
calculations have not yet converged [4, 5, *chu_erratum_1975, 7, 17, 8, 9, 18,
10, 11, 19, 12, 13, 14] and we are unaware of any previous state-resolved
experimental cross section data that can guide the theory.

For illustration, we consider the case of molecular ions in cold astrophysical
environments [20, 21]. They are mostly observed [22] by radio-astronomy via
transitions between the lowest rotational states, requiring knowledge of the
inelastic collision processes that contribute to rotational excitation and de-
excitation. At the relevant temperatures of 10–100 K, the dominant partners
for inelastic collisions are hydrogen atoms or molecules, and electrons [19,
14]. Electronic collisions are relevant even at the generally small electron
density in the interstellar medium (typically $\sim$ 10-4 of hydrogen) because
of their higher rate coefficients ($>10^{-6}$ cm3 s-1 at 10–100 K for dipolar
molecular ions). For ions such as HCO+, electron collisions are found to even
dominate the inelastic rates [19].

To represent, in particular, the astrophysical conditions, relevant laboratory
studies should ensure the low-density, binary-collision regime. Hence, very
sensitive detection methods are required. Moreover, the internal molecular
excitation must be probed for only a few populated rotational levels,
corresponding to the cold astrophysical temperatures. We present such a
measurement using methylidyne (CH+) cations circulating in a cryogenic storage
ring. There, the rotational level populations are subject to radiative cooling
in the cryogenic environment and to controlled electron collisions in a
merged, velocity-matched beam of electrons. We probe the evolution of the
level populations as a function of storage time by resonant photodissociation.
At effective temperatures of $\sim$ 20 K for the radiation field and $\sim$ 26
K for the electrons, we reach effective electron densities substantially above
the critical density needed for collisional transition rates to dominate over
the radiative ones. Hence, rotational cooling by inelastic electron collisions
outperforms radiative cooling, leading to $J=0$ as the dominant rotational
level after a much shorter storage time than for radiative cooling alone.
Using laser diagnostics, the electron-induced changes of the rotational
populations are measured and compared to theory. This confirms recent
predictions for the low-temperature inelastic rate coefficients of CH+ and
demonstrates that, in combination with suitable rotational diagnostics,
cryogenic storage rings offer a platform to measure rotationally inelastic
electron-collision rates for a wide range of molecular ions.

Merged-beams studies in room-temperature storage rings [23] demonstrated the
effect of electron collisions on rotational excitation of HD+ by measuring the
fragmentation energies from dissociative recombination (DR) events. However,
individual angular momenta $J$ could not be resolved and the effect of
electron collisions on the rotational populations was far from dominant.
Cryogenic storage rings for molecular ions [24, 25] now enable state-selective
studies for the lowest $J$-levels of elementary diatomic ions. These studies
addressed spontaneous radiative relaxation of rotational excitation in a low-
temperature blackbody field for CH+ [26] and OH- [27, 28,
*schmidt_erratum_2018]. Moreover, the rotational dependence of the low-energy
DR rate was measured in electron collision experiments on HeH+ [30]. The
present work extends these studies by combining laser probing of the
rotational level populations of CH+ ions with the collisional rotational
cooling of these ions by a merged electron beam.

The experiments are performed in the cryogenic storage ring CSR [25] at the
Max Planck Institute for Nuclear Physics in Heidelberg, Germany. A 280-keV
beam of CH+ ions with a broad $J$ distribution is produced in a Penning ion
source, electrostatically accelerated, and injected into the CSR, where the
ions then circulate with a mean storage lifetime of $\sim$ 180 s. For
rotational diagnostics, they are illuminated in a straight section of the ring
with nanosecond laser pulses from an optical parametric oscillator (OPO) with
a tunable wavelength near 300 nm. At certain wavelengths, CH+ ions in specific
low-$J$ levels can be preferentially addressed by resonant photodissociation,
as already demonstrated at CSR [26]. Neutral H atoms from the
photodissociation leave the closed ion orbit at a downstream electrostatic
deflector and are counted on a microchannel plate (MCP) detector [Fig. 1(a)].

On each revolution, the ions interact with the velocity-matched, collinear
electron beam of the CSR electron cooler. The electron cooler delivers a
continuous, nearly monoenergetic electron beam of $\sim$ 20 eV, guided by an
axial magnetic field (10 mT in the interaction region) to yield a 10-mm
diameter, nearly homogeneous current density profile in the collinear overlap
region with the ion beam. The electron density is $7.0(6)\times 10^{5}$ cm-3
(here and below, the number in parentheses is the one-sigma uncertainty of the
final digit) and the relative velocities are mainly transverse to the beam
direction with an effective temperature $T_{\perp}\sim{}$26 K
[$k_{B}T_{\perp}=2.25(25)$ meV with $k_{B}$ being the Boltzmann constant;
further discussed below]. The overlap region is housed in a biased drift tube
where the electrons are decelerated to 11.8 eV, thus reaching the same
velocity as the stored CH+ ions over an effective matched-velocity overlap
region of 0.79(1) m. The ion beam is centered within the electron beam and
reaches an effective full width at half-maximum (FWHM) diameter of $<$ 6 mm
after 10 s of phase-space cooling by the electrons [31]. The DR rate and its
dependence on the electron–ion collision energy are measured using the MCP
detector. The peak in the DR rate at the lowest achievable electron–ion
collision energy serves for matching the average ion and electron beam
velocities.

Figure 1:  (a) Schematic of the storage-ring section for laser probing and
merged-beam electron–ion interaction. (b) Theoretical near-threshold
photodissociation cross sections of CH+ ions [26] in $J\leq 3$, weighted with
a typical $J$ distribution during rotational relaxation in the CSR and
convolved with the laser lineshape (FWHM 6.1 cm-1). Horizontal brackets:
wavenumber ranges between the thresholds for C+(${}^{2}P_{1/2}$), lower, and
C+(${}^{2}P_{3/2}$), upper, for the indicated $J$. Thick vertical lines mark
the wavenumbers $\bar{\nu}_{i}$ ($i=0\ldots 2$) and $\bar{\nu}_{b^{\prime}}$
where laser probing signals were accumulated. The probing wavenumbers are
Doppler shift corrected, as explained in the supplemental material [31].
Individual $J$ contributions (broken curves) add up to the total probed signal
(full curve). The magnified contribution of $J=3$ is shown up to the onset of
its continuum level only.

CH+ [31] has an $X^{1}\Sigma^{+}$ ground state with a rotational constant of
$\sim$ 13.9 cm-1 and a dissociation energy $D_{0}=32\,946.7(11)$ cm-1 [32]. A
metastable $a^{3}\Pi$ state with $\sim$ 1.2 eV excitation energy and measured
lifetime of $\sim$ 7 s [39] is strongly populated in the injected ion beam
[31]. At a storage time $t\gtrsim 20$ s, radiative relaxation has progressed
along the vibrational cascades (all having radiative lifetimes $\tau_{r}<0.1$
s) while the rotational cascade then has essentially depopulated the $J\geq 4$
states [$\tau_{r}<2$ s for $X^{1}\Sigma^{+}(v=0)$]. The population in the
metastable $a^{3}\Pi$ levels was monitored in situ by storage-time-dependent
molecular-fragment imaging making use of the unique signature of DR from these
levels. Their fractional populations were found to be $<0.1$ for $t>25$ s and
$<0.05$ for $t>40$ s [31]. For $t>20$ s, we probe the populations of the
remaining levels, $X^{1}\Sigma^{+}(v=0,J\leq 3)$, via Feshbach resonances in
near-threshold photodissociation [26, 51, 32]. In this process, optical
transitions excite the $A^{1}\Pi$ electronic state which correlates to the
atomic channel H $+$ C+(${}^{2}P_{3/2}$) with a fine-structure excitation of
63.42 cm-1 [52] and supports vibrational energy levels above the H $+$
C+(${}^{2}P_{1/2}$) channel. These levels dissociate by spin–orbit and
rotational coupling while being long-lived enough to cause distinct resonances
[32] in the near-threshold photodissociation spectrum.

Considering the initial CH${}^{+}(X^{1}\Sigma^{+},v=0,J)$ levels, the
photodissociation yield as a function of the photon energy has characteristic
resonances for each $J$ [Fig. 1(b)]. In our laser probing scheme, we use three
photon wavenumbers $\bar{\nu}_{i}$ controlled by the OPO laser tuning, each
situated near a $J$ resonance peak for $J\leq 2$. Gated count rates $R_{i}$
were accumulated in time windows synchronized with the arrival times of H
atoms produced by the $<$ 10-ns laser pulses (repetition rate 20 s-1), where
the arrival times were evaluated as the time difference between the trigger
signal of a laser pulse and the detection time of a corresponding counted
particle. The background count rate $R_{b}$ on the MCP from CH+ collisions
with residual gas molecules was recorded in the breaks between the
synchronized gates. The wavenumbers were cycled and stayed $\sim$ 4 s at each
$\bar{\nu}$. The cycle included a further wavenumber $\bar{\nu}_{b^{\prime}}$
below the $J=3$ threshold to probe a rate $R_{b^{\prime}}$ for all
backgrounds, including any laser-induced one. The resonant laser-induced rates
were normalized to the measured laser pulse energy $\epsilon_{L}$ and to
$R_{b}$ (proportional to the stored ion number), yielding the normalized
signals $S_{i}=(R_{i}-R_{b^{\prime}})/\epsilon_{L}R_{b}$. Based on updated
molecular theory [26] and on a precise study of the resonance structure for
our experimental parameters [31], these normalized signals can be expressed as
$S_{i}=C\sum_{J}p_{J}\sigma_{J}(\bar{\nu}_{i})$ with the known cross sections
$\sigma_{J}(\bar{\nu}_{i})$ and an overall normalization $C$. This relation is
solved for the relative rotational populations $p_{J}$ [31]. With electrons
absent, we also performed finely spaced scans of $\bar{\nu}$ to align the
experimental peaks with the $J$-dependent theoretical spectra. Moreover, the
spectra from laser scans at $t=14$–28 s were fitted with a linear combination
of the calculated $\sigma_{J}(\bar{\nu})$ to derive populations
$\tilde{p}_{J}$ ($J\leq 3$) [31] that we consider as the initial populations
at $t=21$ s for the rotational levels in the $X^{1}\Sigma^{+},v=0$ ground
state.

Figure 2:  Measured and modeled CH+ rotational populations without electron
interaction. Data $p_{J}$ at $t>40$ s (filled symbols and one-sigma
uncertainties) were determined from the signals $S_{J}$ at the $J$-specific
laser-probing wavelengths ($J\leq 2$) averaged over the probing time windows
indicated at the top (horizontal bars). Specialized laser scans (see the text)
yielded populations $\tilde{p}_{J}$ (unfilled symbols) at $t=21$ s (arrow) for
$J\leq 3$. With these as initial conditions, the radiative model was fitted to
the later $p_{0}$ and $p_{2}$ ($p_{1}$ follows as $1-p_{0}-p_{2}$) varying
$T_{\rm low}$ and $\epsilon$. Light colored bands: one-sigma uncertainties of
the fitted model populations (including the $\tilde{p}_{J}$ uncertainties).
Figure 3:  Laser-probed CH+ rotational populations $p_{J}$ (symbols) with
velocity-matched electrons applied in the shaded time windows. Thin full line:
fitted $p_{J}(t)$ for radiative relaxation only from Fig. 2. The other lines
show modeled evolutions starting from the $p_{J}(t=28~{}{\rm s})$ measured
here. Short-dashed: radiation only; dash-dotted: depletion by $J$-dependent DR
also included; long-dashed line: additionally including inelastic collisions
using cross sections $\sigma_{J,J^{\prime}}^{\rm CB}(E)$; thick full line:
instead, using the $R$-matrix results [12] for the inelastic collisions cross
sections. The red line and shaded area show the estimated maximum $J=3$
population. Data points with one-sigma uncertainties.

As shown by the symbols in Fig. 2, relaxation by radiative transitions alone
was probed for ten windows of $t$ up to radiative equilibrium. The deduced
$p_{J}(t)$ were fitted by the corresponding $t$-window averages of $p_{J}$
from rate-equations which included spontaneous radiative decay as well as the
transitions induced by a modeled blackbody radiation field of the CSR. In this
radiative model [31], the spectral intensity is composed of a cryogenic
component at a temperature $T_{\rm low}$ with a $1-\epsilon$ fraction of the
Planck blackbody spectrum plus a small fraction $\epsilon$ of a room-
temperature (300 K) blackbody spectrum from radiation leaks into CSR. With the
fitted parameters $T_{\rm low}=14.6(21)$ K and $\epsilon=1.03(28)\times
10^{-2}$, the radiative model well reproduces the measured $p_{J}(t)$ and is
used to predict the $J$ populations in the stored CH+ ions when collisional
cooling by the electrons is absent. Based on the in situ monitoring, the
remaining small population in the metastable $a^{3}\Pi$ state is neglected.
Assuming a purely thermal radiation field, its effective temperature
$T_{r}^{\rm eff}$ can be derived directly from the asymptote of the population
ratio between the $J=0$ and 1 levels, $(p_{1}/p_{0})_{\rm eq}=0.39(5)$, which
yields $T_{r}^{\rm eff}=19.6(11)$ K.

The effect of low-energy electron collisions was then measured (Fig. 3) by
merging the velocity-matched electron beam with the CH+ ions. Phases with
electrons on and off were alternated, with the electron-off phases serving as
laser-probing time windows. This results in a duty cycle for the inelastic
electron interaction of $\sim$ 50%. The MCP detector was activated for the
laser-probing only and turned off when the electrons were on, as the high DR
event rate would saturate the MCP detector.

Compared to electrons off, the $J=0$ level now becomes populated much faster.
At only 30 s of storage, $p_{0}\approx 0.3$ versus $p_{0}<0.1$ without
electrons. Moreover, the asymptote of the $J=1$ and 0 population ratio,
representing the equilibrium between rotational cooling and excitation, rises
significantly to $(p_{1}/p_{0})_{\rm eq}=0.53(13)$. Both observations reveal a
strong collisional effect and directly show that in the low-temperature regime
the critical electron density for the CH+ $J=0\to 1$ transition is similar to
or lower than our ring-averaged electron density $\bar{n}_{e}=2.06(21)\times
10^{4}$ cm-3. The larger collisional value of $(p_{1}/p_{0})_{\rm eq}$
indicates a somewhat higher electron temperature as compared to $T_{r}^{\rm
eff}$.

With the time dependence of the results in Fig. 3, we quantitatively test the
electron-collision rate coefficients. We analyze the evolution of the $p_{J}$
starting at $t=28$ s, when the $J=3$ relative population has dropped to
$\lesssim 0.05$ and a three-level model ($J\leq 2$) is adequate. Even
disregarding inelastic collisions, already DR as a potentially $J$ dependent
reactive process can influence the $p_{J}$. Therefore, in addition to the
radiative rates, we must account for the variation of the $p_{J}$ due to
differences between the DR rate coefficients for the lowest $J$ levels. These
were obtained in a separate run within the same experiment, measuring the time
dependence of the merged-beams $J$-averaged DR rate coefficient
$\bar{\alpha}_{\rm DR}^{\rm mb}(t)$ in the velocity-matched beams over 100 s
of storage. These measurements required the electron energy to be rapidly
alternated from velocity-matching condition to higher values (0.2–4 eV
collision energy). This led to modified $p_{J}(t)$ values that, during the
times of detuned electron velocity, were again laser-monitored. Correlating
the changes of these $p_{J}(t)$ with those of $\bar{\alpha}_{\rm DR}^{\rm
mb}(t)$, differences between the $J$-specific DR rate coefficients could be
derived [31]. The $p_{J}(t)$ modeled with radiative transitions plus
$J$-selective depletion by DR, but still excluding inelastic collisions, are
separately shown in Fig. 3.

To include the inelastic collisions, cross sections $\sigma_{J,J^{\prime}}(E)$
as functions of the collision energy $E$ are used [31] to determine merged-
beams inelastic rate coefficients $\alpha_{J,J^{\prime}}^{\rm mb}$, thus
adding the related $J\to J^{\prime}$ transitions to the rate equations for
$p_{J}(t)$. We perform this step for $\sigma_{J,J^{\prime}}(E)$ calculated in
the CB approximation [4, 5, *chu_erratum_1975, 8, 31] as well as from the most
recent $R$-matrix calculations [12]. The latter are generally higher than the
CB results by a factor of $\sim$ 1.6. The collision energy distribution used
in the determination of $\alpha_{J,J^{\prime}}^{\rm mb}$ is modeled for a set
of electron temperatures $k_{B}T_{\perp}$ between 1.5 meV and 3.0 meV [31].
The modeled equilibrium population ratio $(p_{1}/p_{0})_{\rm eq}$, largely
governed by $\alpha_{1,0}^{\rm mb}/\alpha_{0,1}^{\rm mb}$, depends
significantly on $k_{B}T_{\perp}$, but only weakly on the theoretical cross-
section size. This size effectively cancels in the ratio $\alpha_{1,0}^{\rm
mb}/\alpha_{0,1}^{\rm mb}$ as the cooling and excitation cross sections are
related through detailed balance. Comparing [31] the model with the measured
$p_{J}$ at $t=142$ s supports the experimental value of
$k_{B}T_{\perp}=2.25(25)$ meV [$T_{\perp}=26(3)$ K].

Figure 4:  (a) Population ratios $p_{1}/p_{0}$ corresponding to Fig. 3
(electrons applied in shaded time windows) with modeled evolutions starting
from the measured populations at $t=28$ s. Short-dashed: radiation only; dash-
dotted: depletion by $J$-dependent DR also included; full line: additionally
including inelastic collisions with the simplified
$\hat{\sigma}_{J,J^{\prime}}(E)$ and fitting for $\hat{\sigma}^{0}_{0,1}$. (b)
Plasma rate coefficients $\alpha_{1,0}(T)$ and $\alpha_{0,1}(T)$ as
constrained by the fitted $\hat{\sigma}^{0}_{0,1}$ (full lines and shaded one-
sigma uncertainty), compared to $R$-matrix theory [12] (coalescing dots) and
to the CB calculation (dashed line).

The data measured at intermediate times for $p_{0}(t)$ and $p_{1}(t)$ can then
be compared to the model calculations employing the experimental $T_{\perp}$.
This probes the absolute values of the inelastic rate coefficients, dominantly
$\alpha_{1,0}^{\rm mb}$ and $\alpha_{0,1}^{\rm mb}$. As seen in Fig. 3 the
data are remarkably well reproduced when using the $R$-matrix cross sections;
slightly worse agreement is found for the CB approximation.

Comparison with theory is also possible considering the thermal rate
coefficients implied by the experiment. For the relevant collision energies,
the $J$-changing cross sections for an energy level pair spaced by $\Delta
E_{J,J^{\prime}}$ are well approximated by a simplified, single-parameter
expression. For excitation ($J^{\prime}>J$), the approximation is
$\hat{\sigma}_{J,J^{\prime}}(E)=\hat{\sigma}^{0}_{J,J^{\prime}}\Delta
E_{J,J^{\prime}}/E$ (where $E>\Delta E_{J,J^{\prime}}$) with the threshold
value $\hat{\sigma}^{0}_{J,J^{\prime}}$ as only parameter. For de-excitation,
detailed balance then yields, at all energies,
$\hat{\sigma}_{J^{\prime},J}(E)=\hat{g}\hat{\sigma}^{0}_{J,J^{\prime}}\Delta
E_{J,J^{\prime}}/E$ with $\hat{g}=(2J+1)/(2J^{\prime}+1)$. An experimental
value for $\hat{\sigma}^{0}_{0,1}$ can be obtained from the time dependence of
the ratio $p_{1}/p_{0}$ starting at $t=28$ s [Fig. 4(a)]. A fit of the
parameter in the simplified model yields $\hat{\sigma}^{0}_{0,1}=1.07(23)_{\rm
fit}(40)_{\rm syst}\times 10^{4}$ Å2, where the first uncertainty stems from
the fitting and the second one from the uncertainties in $T_{\perp}$, the
initial condition at $t=28$ s, and $\bar{n}_{e}$. (Simplified
$\hat{\sigma}_{J,J+1}(E)$ and $\hat{\sigma}_{J+1,J}(E)$ for other level pairs
were included in the model with their $\hat{\sigma}^{0}_{J,J+1}$ matching Ref.
[12], but reducing them to 0 changed the fit result by only $<0.02\times
10^{4}$ Å2.) The simplified cross sections with the experimental
$\hat{\sigma}^{0}_{0,1}$ and the thermal electron energy distribution at a
kinetic temperature $T$ served to obtain the plasma rate coefficients
$\alpha_{1,0}(T)$ and $\alpha_{0,1}(T)$, shown in Fig. 4(b). The corresponding
$R$-matrix results [12] and the CB approximation lie well within the
experimental one-sigma uncertainty range of $\pm$43%.

This demonstrates the use of a merged-beams electron target in a cryogenic
storage ring for quantitative probing of $J$-changing electron collisions
between the lowest rotational levels in small molecular ions. The experimental
method for rotational state probing used here, employing $J$-dependent
resonances in single-step photodissociation, generally cannot be transferred
to other ion species. However, effort is underway [53] to develop
$J$-dependent photodissociation schemes for additional small ions in cryogenic
storage rings. For most systems, such schemes will include intermediate laser
excitation steps similar to the resonant multiphoton dissociation process
earlier realized in ion traps [54]. It will then be possible to extend the
measurements to other systems crucial for spectrocopic probing of the
interstellar medium, such as HCO+. The CSR merged-beams setup offers more
options for controlling and probing the electron collisions, beyond what has
been realized in the present survey. Thus, future experiments may focus on
precisely probing the change of $p_{J}$ in a given electron interaction period
or apply detuned average velocities of the electron and ion beam. This will
allow electron impact excitation and de-excitation cross sections of molecular
cations to be investigated in more detail, over a wider range of electron
collision energies relevant not only for astrophysical environments, and also
for molecular plasma conditions in general.

###### Acknowledgements.

Financial support by the Max Planck Society is acknowledged. 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 C.J. Williams for the NIST close coupling code used in
the resonant photodissociation calculations and A. Faure for supplying the
$R$-matrix cross sections underlying the rate coefficients published in Ref.
[12].

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