DISSOCIATIVE RECOMBINATION MEASUREMENTS OF HCl⁺ USING AN ION STORAGE RING

Measurements were carried out at the TSR heavy-ion storage ring of the Max Planck Institute for Nuclear Physics in Heidelberg, Germany (Habs et al. 1989). With a base pressure of ∼10⁻¹¹ mbar, TSR is ideally suited to simulate the two-body collision regime important for interstellar gas-phase chemistry. Details on various aspects of the merged beams technique as used at TSR and the recent developments of the photocathode electron beam and the fragment imaging technique have been described by Amitay et al. (1996), Krantz et al. (2009), and Novotný et al. (2010). Here we discuss only those aspects specific to this study.

HCl⁺ ions were produced in a cold-cathode Penning ion source from a mixture of H₂ and HCl parent gases. Ions were extracted from the source and brought to a kinetic energy of 2.4 MeV using a Pelletron-type accelerator. The magnetically mass-filtered beam of ¹H³⁵Cl⁺ was then selected for and injected into TSR where it was stored for ∼33 s.

The stored ion beam was merged for ∼1.5 m with a nearly monoenergetic electron beam. This beam, which we refer to as the Target, is generated by a photocathode operated at temperatures of around 100 K (Orlov et al. 2004, 2005, 2009), thereby enabling us to perform electron–ion collision studies with high-energy resolution (Sprenger et al. 2004). After ion injection at time t = 0 and continuing until t = 16 s, the electron beam was velocity matched to the ions. During this time elastic collisions of the ions with the low energy spread (i.e., cold) electron beam transferred energy from the circulating ions to the single-pass electrons. This mechanism of electron cooling of the ions, also known as phase-space cooling (Poth 1990), results in reducing the ion beam velocity spread and a reduced ion beam diameter.

The ion beam energy of 2.4 MeV was limited by the highest available magnetic field strength available for deflecting the stored ions along the orbit within TSR. Velocity matching the electrons with the 36 amu HCl⁺ ions resulted in an electron beam energy as low as 36.9 eV. Under these conditions, obtaining a phase-space cooled ion beam requires an electron beam of both high density and low temperature. Such a beam was achieved by utilizing the cryogenically cooled photocathode electron source in the Target which provided an electron density of n_e ≈ 2 × 10⁶ cm⁻³ and a low energy spread as we discuss below. The cooling capability of the Target for high-mass ions was demonstrated previously for a beam of 31 amu CF⁺ ions (Krantz et al. 2009). In the present measurement, in spite of the low electron energy and high ion mass, an ion beam diameter as low as 0.5 mm was obtained. The second electron beam device of the TSR, referred to as the Cooler (Steck et al. 1990), proved to have insufficient cooling force to cool such heavy, slow molecular beams within the available storage time. Therefore the Cooler was not used for the results presented here.

After cooling, the Target electron beam was used as the interaction medium to measure HCl⁺ DR for a range of electron–ion center-of-mass collision energies by varying the laboratory energy of the electron beam. Neutral DR products generated in the Target were not deflected by the first dipole magnet downstream of its position in TSR and continued ballistically until they hit a detector. Data were collected with this detector from t = 16 s to t = 33 s as described in Section 3.3.

The energy spread of the photocathode-generated electron beam is parameterized by effective temperatures T_⊥ and T_||, perpendicular and parallel to the bulk electron beam velocity, respectively. The magnetic field guiding the electrons was higher at the photocathode, compared to the interaction zone, by a factor of ξ = 20 for most of the measurements. This expansion of the magnetic field leads to a lowering of the perpendicular electron temperature (Danared 1993; Orlov et al. 2005). By fitting dielectronic recombination resonances (Lestinsky et al. 2008) and from fragment imaging spectra obtained at similar values of ξ (Stützel 2011), we deduce effective temperatures of $k_{\rm B}T_{{\perp }}=1.65\pm 0.35$ meV and $k_{\rm B}T_{\parallel }= 25^{+45}_{-5}\, \mu$eV for the present measurement, where k_B is the Boltzmann constant. Here and throughout the rest of this paper, all uncertainties are quoted at an estimated 1σ statistical confidence level. Some of the data were also acquired with an expansion factor of ξ = 40, for which a reduced transverse electron temperature of $k_{\rm B}T_{{\perp }}\approx 0.83$ meV is estimated, while k_B T_|| is expected to remain unchanged (e.g., Danared 1993). The relevant temperatures $k_{\rm B}T_{{\perp }}$ and k_B T_∥ enter the data analysis as described in Section 3.4.

The interaction geometry is determined by the shapes of the overlapping electron and ion beams. The electron beam geometry is determined by the magnetic fields in the Target (Sprenger 2003). The electron beam can be treated as a cylindrical body, bent at the ends, with a diameter d_e given by the product of the cathode size and the square root of the expansion factor ξ. This yields d_e = 12.6 mm and 17.8 mm for ξ = 20 and 40, respectively. The cooled ion beam diameter d_i is typically less than 0.5 mm. The electron and ion beam geometries are shown in Figure 1 and are discussed in more detail in the Appendix.

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The fact that the electron beam is significantly wider than the ion beam enabled the complete spatial overlap of the ion beam by the electrons in the interaction region. We optimized the alignment of the beams by minimizing the time needed for phase-space cooling of the beam (Hochadel et al. 1994). Better alignment gives a better overlap, thereby reducing the required cooling time. Observationally, we monitored the necessary cooling time and beam size either using a beam profile monitor (Hochadel et al. 1994) or by examining the center of mass for DR events as projected on the multichannel plate (MCP) imaging detector (Amitay et al. 1996; Krantz et al. 2009).

The bulk electron beam energy in the laboratory frame, E_meas, is needed to calculate the center-of-mass electron–ion collision energy during the measurement. We determine E_meas from the measured cathode voltage by correcting it for the space charge of the electron beam (Kilgus et al. 1992). This correction requires knowing the electron density profile, which we measured with the method described in Sprenger et al. (2004). The electron density is expected to be homogeneous along the beam axis.

The HCl⁺ ions produced in the discharge are expected to initially possess electronic, vibrational, rotational, and fine-structure excitation. We estimate that much of this internal excitation radiatively relaxes during the 16 s of electron cooling. Electronically, the only known HCl⁺ metastable state below the dissociation limit is the A ²Σ⁺ which lies ∼3.6 eV above the X ²Π ground state. Measured and calculated radiative lifetimes for A ²Σ⁺ → X ²Π transitions are shorter than ∼3.4 μs (Pradhan et al. 1991 and references therein). The known radiative lifetimes for vibrational relaxation of the X ²Π electronic ground state span from ∼4.9 ms for v = 1 → 0 to ∼1.3 ms for v = 9 → 8 (Pradhan et al. 1991). Higher vibrational levels are expected to decay even faster. Thus we expect the stored ions to cascade quickly to their ground electronic and vibrational levels.

We predict that the rotational lifetimes are sufficiently short so that during the initial phase most of the rotations J will have radiatively relaxed and come into equilibrium with the ∼300 K black body radiation of the vacuum chamber. We have calculated the rotational radiative lifetimes of X ²Π_3/2(v = 0) for levels ranging from J = 3/2 to J = 51/2. In actuality, we could truncate the calculations at J = 20/2, as the higher levels decay so rapidly that they do not affect the precision of the model. Our approach uses a method similar to that of Amitay et al. (1994). The static dipole moment for HCl⁺ was taken from Cheng et al. (2007). The dominant emission lines are expected to be J → J − 1. Restricting ourselves to these transitions, we have built a relaxation model using the spontaneous radiative decay lifetimes along with stimulated emission and absorption by the 300 K black body radiation. For the initial rotational excitation we have taken a Boltzmann distribution at a temperature of 8000 K. This is approximately the excitation temperature derived for CF⁺ produced in the same ion source (Novotný et al. 2009). After the initial 16 s of ion storage, the predicted average excitation energy exceeds the 300 K equilibrium by only 10%. The excitation energy averaged over the ion population during the t = 16–33 s measurement window exceeds room temperature excitation by only 3%. This predicted level of excitation might be slightly overestimated due to the omission of spin-orbit coupling which may provide extra decay pathways. We have also not accounted for the possible additional acceleration of the rotational cooling from super-elastic ion collisions with electrons (e.g., Shafir et al. 2009).

The one type of excitation that is unlikely to relax during ion storage is the fine-structure splitting of the X ²Π_{3/2 − 1/2}, which amounts to ∼80 meV (Sheasley & Mathews 1973). We are unaware of any published lifetime estimates for this transition. However, for a rough estimate we can use the radiative lifetime of the fine-structure excited J = 1/2 level in the isoelectronic atomic system Ar⁺ with an energy of ∼165 meV. That lifetime is calculated to be 23.7 s (Fischer & Tachiev 2013). Since the radiative decay rate of this magnetic dipole transition scales as the third power of the transition frequency, then leaving aside the details of the transition matrix element, we expect that the lifetime of the corresponding level in the HCl⁺ molecular state will be longer by almost an order of magnitude. Hence, the X ²Π_1/2 radiative lifetime is expected to significantly exceed the ion storage time here.

Data acquisition began after the 16 s period of injection and cooling was completed. During data acquisition the Target beam energy was stepped repeatedly between cooling, measurement, reference, and off (all defined below). The time durations were 35, 25, 25, and 25 ms, respectively. An additional 5 ms delay before each step was added to allow the power supplies to reach the desired voltage. This cycle was repeated 130 times for a total of ∼17 s.

For the first step, cooling, the electrons were velocity matched to the ions at an electron energy of E_cool = 36.91 eV in the laboratory frame. These interleaved cooling steps ensured a constant phase-space spread of the ions during data acquisition.

In the second step, measurement, the electron beam energy was detuned, giving a mean energy of E_meas in the laboratory frame. The nominal center-of-mass collision energy can then readily be calculated from the mean electron and ion velocities in the laboratory frame. This yields what we call the detuning energy

$$\begin{equation} E_{\rm d} = \left(\sqrt{E_{\rm meas}} - \sqrt{E_{\rm cool}}\right)^2. \end{equation} \tag{ 5 }$$

The detector count rate in this step was used to determine the merged beams rate coefficient versus E_d. The detuning energy was changed for each new ion injection.

The third step, reference, was included as a cross check and for normalization. The energy in this step was set to a constant value of E_d = 0.019 eV. The resulting signal was used to monitor the ion beam intensity.

In the last step, off, the electron beam was not admitted into the interaction region. The resulting background signal was due solely to ion interactions with the residual gas inside TSR.

DR events were measured using a 10 × 10 cm² Si surface-barrier detector located ∼12 m downstream of the Target. Fragments from a given dissociation event arrived at the detector with only a few nanoseconds difference in flight times. This is shorter than the detector time resolution and so only one pulse was counted for each dissociation event, independent of the number of products reaching the detector. The high exothermicity of HCl⁺ DR results in relative fragment kinetic energies of up to KER ≈ 8.3 eV. This can cause a large displacement between the fragments when reaching the detector. The resulting positions of some of the H fragments exceeded the size of the detector. The heavier Cl fragments, however, were confined to a narrow cone and 100% of the DR-generated Cl struck the detector. We verified this using an MCP-imaging technique (e.g., Amitay et al. 1996). Thus, to achieve essentially 100% DR event counting efficiency, we derive our signal from the count of detector events independent of the number of fragments detected in each of them.

The detector count rate consisted of both DR and background events. The dominant source of this background was collisions of ions with residual gas. We have corrected for these events by taking the count rate at measurement and subtracting the count rate acquired with the electron beam off. Our approach does not account for potential background due to non-DR, but this is expected to be negligible at the energies studied (Krauss & Julienne 1973; Dalgarno & Black 1976; Larsson & Orel 2008). Additional background due to electron-driven DE forming H⁺ + Cl was not an issue as the measured collision energies were all significantly below the DE threshold of 5.3 eV.

The measured relative merged beams DR rate coefficient is determined by normalizing the recorded DR signal by the electron density and ion number (e.g., Amitay et al. 1996). The electron beam density is well determined by the measured beam current, energy, and geometry (Sprenger et al. 2004). However, the maximum stored HCl⁺ ion current of I_i < 1 μA was too low to be determined directly using a DC current transformer. Instead we used a measurement of the relative ion beam intensity by taking the detector signals from the reference and off steps as proxies for the relative ion beam intensity. This allows us to normalize our measured data to obtain a relative merged beams DR rate coefficient versus E_d. The whole curve was then scaled using an absolute measurement of the merged beams DR rate coefficient at matched electron and ion velocities (i.e., E_d = 0 eV). Here we used an independent method based on precise ion beam storage lifetime measurements with and without the electron beam present in the interaction zone (Novotný et al. 2012). This method requires using the length of the beam interaction region, which was determined from the known beam geometries to be L = 1.570 m.

In the past, researchers have corrected the merged beams DR rate coefficient for the effects from the merging and demerging of the electron beam with the stored ions (Lampert et al. 1996). Below we introduce a new data analysis method which avoids the necessity of this correction.

Using storage ring results, one can generate a DR rate coefficient suitable for a plasma with a Maxwell–Boltzmann collision energy distribution. This involves deconvolving the measured DR data to remove the effects of the experimental electron velocity spread and the beam overlap geometry. Mowat et al. (1995) and Lampert et al. (1996) have presented approximate methods for addressing these two issues. The resulting cross section can then be convolved with a Maxwell–Boltzmann distribution to generate a plasma rate coefficient suitable for astrochemical modeling. Here we have developed a more precise method for deriving the cross section which avoids making the approximations used by Mowat et al. and Lampert et al.

3.4.1. Measured Merged Beams DR Rate Coefficient

Experimentally we have measured a merged beams DR rate coefficient α_mb which is given by

$$\begin{equation} \alpha _{\rm mb}(E_{\rm d}) = \int \sigma (E)\,v\,f_{\rm mb}(E, E_{\rm d}, T_{||}, T_\perp, \boldsymbol{X})\,{dE}. \end{equation} \tag{ 6 }$$

Here σ is the DR cross section, E and v are the center-of-mass electron–ion energy and the relative velocity, respectively, and f_mb is the center-of-mass energy spread taking into account both the electron beam energy spread and the experimental geometry. The relationship between the collision energy and velocity is given by $E=\frac{1}{2}\,\mu \,v^2$, where μ is the reduced mass of the collision system. The large mass difference between the electron and ion allows us to set μ = m_e . The term f_mb is a function of the detuning energy; the electron energy spread, which is given by the effective temperatures T_⊥ and $T_\Vert$; and the overlap geometry between the beams, symbolically represented as X .

For parallel ion and electron beams, the velocity spread can be described by a "flattened" Maxwellian distribution in the center-of-mass frame. This velocity distribution is discussed in more detail by Andersen et al. (1989) and Poth (1990), while the corresponding energy distribution function is described by Schippers et al. (2004). Note that a pre-factor of $\frac{1}{2}$ is missing from Equation (1) given in Schippers et al.

In the merging and de-merging regions of the Target, the energy distribution is additionally distorted due to the beams no longer running co-linearly. Lampert et al. (1996) discuss the corresponding increase of collision energies, assuming that T_⊥ = T_|| = 0. We are not aware, however, of an analytical representation of f_mb which accounts for the effects of both the velocity spread and the full overlap geometry. To address this issue, we have developed a numerical method for describing such an energy distribution which we discuss below.

3.4.2. Limitations of Previous Methods for Extracting a DR Cross Section

The traditional methods for deconvolving the cross section from the merged beams rate coefficient first correct the measured data for the overlap geometry effects using, for example, the method of Lampert et al. (1996). The corrected data are then treated as if the beams are parallel with an energy distribution $f_{\rm mb}^*$, giving

$$\begin{equation} \alpha _{\rm mb}^*(E_{\rm d}) = \int \sigma (E)\,v\,f_{\rm mb}^*(E, E_{\rm d}, T_{||}, T_\perp)\,{dE}. \end{equation} \tag{ 7 }$$

The analytic form of $f^*_{\rm mb}$ is given by Schippers et al. (2004) and the characteristic experimental energy spread in $f_{\rm mb}^*$ corresponds to (Müller 1999)

$$\begin{equation} \Delta E \approx \sqrt{(k_{\rm B}T_{{\perp }}\,\ln {2})^2+16 \ln {2}\;k_{\rm B}T_{\parallel }\,E_{\rm d}}. \end{equation} \tag{ 8 }$$

For high energies, where $E_{\rm d}\gg k_{\rm B}T_{{\perp }}$, the relative width ΔE/E_d decreases to a small value. In that case, $f_{\rm mb}^*$ can often be approximated by a delta function δ(E − E_d). Equation (7) then collapses to $\alpha _{\rm mb}^*(E_{\rm d}) =\sigma (E_{\rm d})\,v$, and the cross section simplifies to

$$\begin{equation} \sigma (E_{\rm d}) = \alpha _{\rm mb}^*(E_{\rm d}) \sqrt{\frac{m_{e}}{2E_{\rm d}}}. \end{equation} \tag{ 9 }$$

The situation is not so simple for $E_{\rm d}\lesssim k_{\rm B}T_{{\perp }}$. This regime is particularly critical for obtaining an accurate plasma rate coefficient at the low temperatures relevant to the cold ISM. At these low detuning energies, the cross section can be approximately derived using a method introduced by Mowat et al. (1995). Their technique makes use of the fact that typically $k_{\rm B}T_{{\perp }}\gtrsim 50\,k_{\rm B}T_{\parallel }$ in ion storage ring experiments. They go on to neglect the longitudinal energy spread by setting k_B T_∥ = 0. This makes ΔE and the shape of $f_{\rm mb}^*$ independent of E_d and one can then deconvolve $\alpha _{\rm mb}^*$ using Fourier analysis.

However, the approach of Mowat et al. raises a number of issues. Their method ignores the fraction of electron–ion collision energies E lying below E_d. This can be seen in Figure 2 which shows the energy distribution function $f_{\rm mb}^*$ using two sets of Target electron temperatures. As a representative detuning energy we have chosen E_d = 1.3 meV, which corresponds to a typical particle energy at molecular cloud temperatures of 10 K. In Figure 2, we also plot $f_{\rm mb}^*$ for the two Target values of $k_{\rm B}T_{{\perp }}$ but with k_B T_∥ = 0. These energy distributions clearly differ from those using a realistic k_B T_∥ = 25 μeV. In $f_{\rm mb}^*(k_{\rm B}T_{\parallel }= 0)$ there is no collision energy population below E_d. For $f_{\rm mb}^*(k_{\rm B}T_{\parallel }= 25\, \mu$eV) and an assumed value of $k_{\rm B}T_{{\perp }}=1.65$ meV (0.825 meV), about 5% (10%) of the distribution is shifted to below E_d. This fraction increases for decreasing $k_{\rm B}T_{{\perp }}/k_{\rm B}T_{\parallel }$ or with increasing E_d, further reducing the validity of setting k_B T_∥ = 0 in order to extract cross sections. Additionally, the potential errors in this method may increase if the cross section at low energies is highly structured or increases rapidly with decreasing energy. By using an energy distribution function which does not accurately represent the experimental conditions, the deconvolutions can accordingly under- or over-estimate the cross section at a given value of E_d.

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3.4.3. New Method for Extracting a DR Cross Section

3.4.4. Derived Plasma DR Rate Coefficient

Using the extracted σ'(E) as a representation of σ(E), we can generate a rate coefficient for a plasma with a thermal Maxwell–Boltzmann distribution f_pl. The resulting rate coefficient as a function of the plasma temperature T is given by

$$\begin{equation} \alpha _{\rm pl}(T) = \int \sigma (E)\,v\,f_{\rm pl}(E, T)\, {dE}. \end{equation} \tag{ 10 }$$

The analytical form of f_pl(E, T) is well known and numerical integration of Equation (10) is straightforward.

The dominant uncertainty in the measured merged beams DR rate coefficient is derived from the ion beam storage lifetime measurements used to put the experimental results on an absolute scale. This error amounts to 11%. Additional minor sources of error arise from the electron density evaluation (∼4%) and the uncertainty on the electron beam geometry within the Target (≲ 2%). The total systematic uncertainty for the absolute scaling is then 12%.

The statistical uncertainty of each data point is given by the counting statistics for the number of signal and background counts. At collision energies E_d < 10 meV, this error amounts to ∼5%. At higher energies, this error increases as the signal and background become comparable. The statistical error is ∼45% at E_d ≈ 0.1 eV and grows up to ∼200% at E_d ≈ 1 eV.

Figure 3 presents our merged beams rate coefficient for DR of HCl⁺. These data are also given in tabular form in Table 1. The error bars show the 1σ counting statistics. The detuning energy ranges from E_d = 12 μeV to 4.5 eV. The data were acquired with ξ = 20 and cover the storage times of 16–33 s.

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Although our model for the rotational relaxation of HCl⁺ discussed in Section 3.1 indicates that most of the ions are in thermal equilibrium with the black body radiation of the TSR chamber, a small amount of rotational relaxation may still occur during the measurement period. To test for any effect of this cooling, we split the acquired data into two sets covering the storage times of 16–24 s and 24–33 s and analyzed these data sets independently. DR rate coefficients typically depend on the rotational excitation of the parent ions (Larsson & Orel 2008). Hence any significant change in rotational population should be observable in the DR rate coefficient spectra. However, to within their statistical uncertainties, the resulting two data sets for α_mb were equal at all values of E_d. This strongly suggests that the rotational excitation of the ions reached equilibrium with the black body radiation of the vacuum chamber during the initial electron cooling phase.

A possible source of concern in storage ring merged beams experiments is that the low energy α_mb can potentially be distorted if the ion beam energy is dragged by the detuned electron beam. The derivation of E_d through Equation (5) rests on the assumption of a constant mean ion beam velocity which is matched to the mean electron beam velocity at cooling. This assumption breaks down if, during the measurement step, the ion beam is dragged toward the detuned electron beam velocity by dynamical friction forces of the same type as those causing electron cooling (Poth 1990). The actual collision energy is then smaller than that deduced from Equation (5). Since low-energy DR for HCl⁺ increases with decreasing energy, this will enhance the measured signal at higher energies over what one would expect if there were no dragging. This will appear in the data as a broadening of α_mb close to 0 eV. In order to exclude such an experimental artifact, we have performed a series of tests described below. Based on these studies, we believe the ion beam dragging is not an issue in our experiment and that the low-energy shape in α_mb is due solely to DR.

The stability of the ion beam energy was tested by varying the duration of the various steps in the data cycle. The standard data cycle was chosen to optimize the duty factor and cooling as well as reduce dragging effects. All of the tests in this paragraph further improved cooling and reduced any potential dragging effects, but at the expense of significantly decreasing the measurement duty factor. The cooling step was lengthened by a factor of ∼3 to provide additional cooling and improve the ion beam stability. The measurement step was shortened from 25 ms to 10 ms to reduce any dragging effects. Additionally, we varied the energy difference between the reference step and the cooling step. If cooling were insufficient this would help to further minimize the dragging of the ion beam energy when at reference. To within the statistical accuracy of the measurement, all of these tests produced an α_mb that agreed with the data obtained under standard measurement conditions. Thus dragging of the ion beam energy does not appear to contribute to the observed low-energy rate coefficient.

To further test the DR origin of the low-energy data, we have measured the behavior of α_mb(E_d) for two different energy spreads. As one reduces the width of the collisional energy distribution, this results in a higher amplitude for α_mb at E_d ∼ 0 eV after averaging over the electron energy distribution. Dragging, on the other hand, modifies only the width of the low-energy DR spectrum, not the peak amplitude. Here we narrowed the experimental energy spread by increasing the adiabatic expansion factor of the magnetic guiding field in the Target electron gun from ξ = 20 to ξ = 40. The transverse temperature is expected to scale as ξ⁻¹. Figure 4 presents both the ξ = 20 and ξ = 40 results. The higher expansion data do indeed show an additionally increased amplitude of the rate coefficient at E_d ≲ 1 meV. The half-width at half-maximum of the data below ∼1 meV also narrowed by ∼17%. This strongly suggests that the observed low-energy rate coefficient is caused by the DR process and not by beam dragging artifacts.

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Still, the higher amplitudes in the low-energy α_mb(ξ = 40) data do not fully exclude a combination of dragging effects mixed with DR. Thus, as an additional quantitative test, we have converted the measured α_mb(ξ = 40) to a model of the rate coefficient expected at the lower expansion factor $\alpha ^{\prime \prime }_{\rm mb}(\xi = 20)$. A real DR resonance, as opposed to a dragging effect, should appear similar in both the converted $\alpha ^{\prime \prime }_{\rm mb}(\xi = 20)$ and the measured α_mb(ξ = 20). To generate $\alpha ^{\prime \prime }_{\rm mb}(\xi = 20)$ we deconvolved the α_mb(ξ = 40) data using the corresponding energy distribution f_mb(ξ = 40) and the fitting procedure described in Section 3.4.3. We then reconvolved the extracted cross section with f_mb(ξ = 20) to obtain the model of the experimental rate coefficient $\alpha _{\rm mb}^{\prime \prime }(\xi = 20)$.

Figure 4 compares the measured rate coefficient α_mb(ξ = 20) and the converted $\alpha _{\rm mb}^{\prime \prime }(\xi = 20)$. The measured α_mb(ξ = 40) and the corresponding model $\alpha ^{\prime }_{\rm mb}(\xi = 40)$ are also displayed. We find satisfying agreement between α_mb(ξ = 20) and $\alpha ^{\prime \prime }_{\rm mb}(\xi = 20)$. The small differences remaining are attributed, in part, to uncertainties in the scaling used to derive $k_{\rm B}T_{{\perp }}$ for ξ = 40 from that at ξ = 20. To first order we expect $k_{\rm B}T_{{\perp }}$ to scale as ξ⁻¹, but there may be small nonlinear terms that we have ignored. These terms may introduce uncertainties in the f_mb(ξ = 40) used which could carry over into the extracted cross section and finally into $\alpha ^{\prime \prime }_{\rm mb}(\xi = 20)$. Additionally, we did not measure α_mb(ξ = 40) at energies of E_d > 20 meV, so above this point we have used the α_mb(ξ = 20) data which may produce small discrepancies at energies E_d ≳ 10 meV.

We have converted the experimental DR rate coefficient α_mb(ξ = 20) to a cross section using a procedure introduced in Section 3.4.3 and described in detail in the Appendix. The resulting σ'(E) is plotted in Figure 5 for the energy range E = 0–4.5 eV. These data are also given in tabular form in Table 2. The lower edge of the first energy bin is set to E = 0 and is not displayed on the logarithmic energy scale of the plot.

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Table 2. Cross Section σ for DR of HCl⁺

Ecenter	Ewidth	σ	$U_{\rm stat}^{\rm lo}$	$U_{\rm stat}^{\rm hi}$	$U_{\rm syst}^{\rm lo}$	$U_{\rm syst}^{\rm hi}$
(eV)	(eV)	(cm2)	(cm2)	(cm2)	(cm2)	(cm2)
4.11(−5)	8.22(−5)	1.06(−11)	−4.01(−12)	+4.72(−12)	−1.57(−12)	+2.44(−12)
1.68(−4)	1.72(−4)	3.93(−12)	−1.37(−12)	+1.30(−12)	−7.11(−13)	+6.36(−13)
4.38(−4)	3.67(−4)	2.58(−12)	−2.93(−13)	+2.83(−13)	−4.25(−13)	+5.74(−13)
9.95(−4)	7.47(−4)	7.96(−13)	−5.71(−14)	+5.37(−14)	−1.30(−13)	+1.41(−13)
2.00(−3)	1.27(−3)	2.09(−13)	−1.58(−14)	+1.73(−14)	−2.93(−14)	+3.12(−14)
3.34(−3)	1.40(−3)	9.76(−14)	−1.05(−14)	+9.87(−15)	−1.34(−14)	+1.24(−14)
4.80(−3)	1.53(−3)	7.26(−14)	−7.22(−15)	+6.96(−15)	−9.92(−15)	+1.24(−14)
6.40(−3)	1.67(−3)	3.12(−14)	−2.79(−15)	+2.90(−15)	−6.08(−15)	+4.16(−15)
8.14(−3)	1.80(−3)	2.11(−14)	−1.76(−15)	+1.77(−15)	−2.65(−15)	+3.13(−15)
1.00(−2)	1.93(−3)	1.38(−14)	−1.38(−15)	+1.26(−15)	−2.15(−15)	+1.76(−15)
1.20(−2)	2.07(−3)	9.14(−15)	−1.18(−15)	+1.25(−15)	−1.16(−15)	+1.60(−15)
1.41(−2)	2.20(−3)	5.46(−15)	−1.09(−15)	+1.03(−15)	−1.37(−15)	+6.81(−16)
1.64(−2)	2.34(−3)	4.50(−15)	−1.08(−15)	+1.06(−15)	−5.63(−16)	+9.45(−16)
1.88(−2)	2.47(−3)	2.52(−15)	−9.71(−16)	+1.01(−15)	−6.19(−16)	+3.16(−16)
2.13(−2)	2.61(−3)	2.74(−15)	−8.99(−16)	+8.60(−16)	−3.39(−16)	+4.66(−16)
2.40(−2)	2.76(−3)	1.17(−15)	−6.68(−16)	+7.78(−16)	−2.48(−16)	+1.56(−16)
2.70(−2)	3.17(−3)	2.00(−15)	−7.12(−16)	+6.15(−16)	−2.48(−16)	+2.42(−16)
3.04(−2)	3.61(−3)	1.56(−15)	−5.30(−16)	+5.20(−16)	−2.06(−16)	+3.16(−16)
3.42(−2)	4.11(−3)	4.87(−16)	−3.25(−16)	+4.94(−16)	−2.24(−16)	+1.23(−16)
3.86(−2)	4.68(−3)	1.43(−15)	−4.14(−16)	+4.04(−16)	−1.79(−16)	+2.47(−16)
4.37(−2)	5.34(−3)	8.28(−16)	−3.17(−16)	+3.16(−16)	−1.53(−16)	+1.01(−16)
4.94(−2)	6.10(−3)	1.04(−15)	−2.66(−16)	+2.90(−16)	−1.25(−16)	+1.65(−16)
5.59(−2)	6.99(−3)	5.03(−16)	−1.68(−16)	+1.64(−16)	−6.54(−17)	+6.06(−17)
6.34(−2)	8.04(−3)	3.85(−16)	−1.36(−16)	+1.34(−16)	−5.67(−17)	+4.66(−17)
7.21(−2)	9.27(−3)	5.77(−16)	−1.29(−16)	+1.33(−16)	−6.93(−17)	+6.94(−17)
8.21(−2)	1.07(−2)	5.59(−16)	−1.13(−16)	+1.13(−16)	−6.80(−17)	+7.65(−17)
9.37(−2)	1.25(−2)	1.69(−16)	−1.05(−16)	+1.38(−16)	−2.03(−17)	+2.09(−17)
1.08(−1)	1.56(−2)	1.61(−16)	−8.57(−17)	+8.94(−17)	−1.94(−17)	+1.95(−17)
1.25(−1)	1.79(−2)	2.19(−16)	−1.01(−16)	+9.69(−17)	−2.74(−17)	+2.68(−17)
1.44(−1)	2.14(−2)	2.72(−16)	−7.82(−17)	+7.19(−17)	−3.28(−17)	+3.27(−17)
1.68(−1)	2.58(−2)	2.71(−16)	−7.67(−17)	+7.47(−17)	−3.25(−17)	+3.25(−17)
1.96(−1)	3.14(−2)	2.27(−16)	−7.02(−17)	+7.46(−17)	−2.73(−17)	+2.73(−17)
2.31(−1)	3.81(−2)	1.53(−16)	−6.10(−17)	+5.90(−17)	−1.84(−17)	+1.83(−17)
2.73(−1)	4.58(−2)	3.33(−17)	−2.49(−17)	+6.86(−17)	−1.16(−17)	+1.18(−17)
3.23(−1)	5.49(−2)	1.16(−16)	−5.75(−17)	+6.35(−17)	−1.41(−17)	+1.41(−17)
3.84(−1)	6.61(−2)	5.35(−17)	−3.70(−17)	+6.46(−17)	−8.89(−18)	+9.05(−18)
4.57(−1)	8.01(−2)	1.25(−16)	−5.76(−17)	+5.73(−17)	−1.53(−17)	+1.53(−17)
5.46(−1)	9.76(−2)	2.37(−17)	−1.80(−17)	+4.98(−17)	−8.58(−18)	+8.71(−18)
6.55(−1)	1.20(−1)	1.20(−17)	−8.68(−18)	+5.59(−17)	−1.38(−17)	+1.39(−17)
7.89(−1)	1.49(−1)	6.36(−17)	−3.76(−17)	+4.13(−17)	−7.74(−18)	+7.71(−18)
9.57(−1)	1.87(−1)	6.11(−17)	−3.22(−17)	+3.29(−17)	−7.35(−18)	+7.35(−18)
1.17(+0)	2.38(−1)	4.60(−17)	−2.84(−17)	+3.22(−17)	−5.55(−18)	+5.57(−18)
1.44(+0)	3.07(−1)	3.98(−17)	−2.58(−17)	+3.45(−17)	−5.12(−18)	+5.12(−18)
1.79(+0)	3.90(−1)	8.88(−17)	−3.60(−17)	+3.95(−17)	−1.07(−17)	+1.07(−17)
2.22(+0)	4.77(−1)	3.21(−17)	−2.12(−17)	+3.86(−17)	−6.75(−18)	+6.76(−18)
2.75(+0)	5.70(−1)	3.82(−17)	−2.48(−17)	+3.00(−17)	−4.78(−18)	+4.79(−18)
3.37(+0)	6.70(−1)	8.93(−17)	−2.76(−17)	+2.97(−17)	−1.08(−17)	+1.08(−17)
4.32(+0)	1.24(+0)	1.02(−16)	−2.64(−17)	+2.49(−17)	−1.23(−17)	+1.23(−17)

Notes. The binned cross section σ is given as a function of bin energy center E_center. Also listed is the energy width E_width of each bin. The asymmetric statistical and systematic uncertainties are given by U_stat and U_syst, respectively. The superscripts "lo" and "hi" give the lower and upper limits for the corresponding uncertainties. The total systematic error in the table includes the 12% scaling error and the uncertainties from $k_{\rm B}T_{{\perp }}$ and k_B T_∥ (see text). The format x(y) signifies x × 10^y .

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Deriving the cross section involves fitting α_mb with a sum of model rate coefficients, each convolved using a single cross-section bin. These sub-functions, $\sigma ^{\prime }_i\,\Psi _i(E_{\rm d})$, are plotted in Figure 3. We plot the corresponding collision energy distribution function f_mb for several values of E_d in Figure 6. To ensure physically meaningful results, we limited the fitting parameters $\sigma ^{\prime }_i$ to positive values only. The fitting yielded a minimum χ²/N_DF = 1.02, where N_DF = 30 is the number of degrees of freedom in the fit.

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To explore the numerical stability of the fitting, we have derived the cross section from a set of 1000 simulated merged beams spectra $\alpha _{\rm mb}^{\rm sim}$. Each data point in a given simulated spectrum was obtained from the sum of the experimental value α_mb(E_d) and an additive shift which was randomly chosen from a Gaussian distribution centered around zero and with a width equal to the 1σ statistical error in α_mb(E_d). Each $\alpha _{\rm mb}^{\rm sim}$ was then fitted to derive a cross section. The cross-section values in each energy bin nearly follow a normal distribution. In some bins, however, up to 8% of the fitted σ_i values converged to zero. We attribute this behavior to the numerical instability for some of the fits, possibly due to converging to a local χ² minimum or because of strong correlations with cross-section values in neighboring energy bins. The mean cross-section values for each bin are displayed in Figure 5. The standard deviations above and below the mean are displayed by asymmetric vertical error bars and reflect the statistical errors propagated from α_mb. The mean cross-section values differ from those obtained directly from the experimental data by only a small fraction of the standard deviations. To test the accuracy of the mean cross section, we have convolved it using Equation (6) to generate a merged beam rate coefficient and compared the result to $\alpha ^{\prime }_{\rm mb}$, which was derived by fitting the experimental data. The difference between the two is less than a fifth of the statistical uncertainties in our experimental data.

Additionally, we have investigated the sensitivity of the cross section to the uncertainties in $k_{\rm B}T_{{\perp }}$ and k_B T_∥. We have used their extreme values $k_{\rm B}T_{{\perp }}= 1.3-2.0$ meV and k_B T_∥ = 20–70 μeV to generate f_mb and derived the cross section from the 1000 model rate coefficients, as discussed above. The mean cross sections were then compared to the one obtained with the most probable electron beam temperatures $k_{\rm B}T_{{\perp }}= 1.65$ meV and k_B T_∥ = 25 μeV. The differences originating from $k_{\rm B}T_{{\perp }}$ and k_B T_∥ were added in quadrature and are displayed by gray bars in Figure 5. The sensitivity of σ'(E) to the electron beam parameters generally decreases with increasing energy. This is due to the decreasing relative energy spread ΔE/E with increasing energy. There is, though, an enhancement of the sensitivity seen in a few bins at E ∼ 0.03 eV. We attribute this to narrow structures in the cross section with spacing comparable to the energy resolution at these energies (of the order of 2.5 meV). The effect of the electron beam temperature parameters is significantly lower than the statistical errors at all energies. Last, the 12% uncertainty on the absolute scaling of α_mb directly propagates to σ'. However, this error is too small to be seen in Figure 5.

We have used the model cross section σ' as a representation of the DR cross section σ and converted it to a plasma rate coefficient using the procedures described in Section 3.4.4. The resulting α_pl(T) is plotted in Figure 7 for a plasma temperature range T = 10–5000 K.

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To propagate the statistical uncertainty from the measured α_mb to the derived α_pl, we follow the technique used for the cross section in Section 4.2. First we created a set of 1000 simulated merged beams spectra $\alpha _{\rm mb}^{\rm sim}$ based on the measured α_mb and the corresponding statistical uncertainty. Each $\alpha _{\rm mb}^{\rm sim}$ was used to derive a cross section which was subsequently convolved with vf_pl to generate a plasma rate coefficient $\alpha _{\rm pl}^{\rm sim}$. The spread in the values of $\alpha _{\rm pl}^{\rm sim}$ at each T closely followed a normal distribution. The Gaussian width of this spread at each T was taken as the 1σ statistical accuracy for α_pl. For T below 400 K these errors are less than 1%. They increase at higher temperatures and reach 8% at 5000 K.

The uncertainties in $k_{\rm B}T_{{\perp }}$ and k_B T_∥ introduce an error in the derived α_pl. The plasma rate coefficient displayed in Figure 7 was derived for $k_{\rm B}T_{{\perp }}=1.65$ meV. We have also evaluated α_pl taking into account the ±0.35 meV uncertainty in $k_{\rm B}T_{{\perp }}$ and repeated the analysis using the limiting values of $k_{\rm B}T_{{\perp }}$. The corresponding respective change of α_pl is ±12% at T = 10 K, decreases to ±8% at 100 K, and is less than ±5% above 1000 K. Varying k_B T_∥ within the estimated range of values has a much smaller effect. At T = 10 K the change in α_pl is $^{+3.5}_{-0.5}$% and drops to less than $^{+0.5}_{-0.1}$% at T ⩾ 50K. In this paragraph, positive (negative) changes in α_pl correspond to increased (decreased) $k_{\rm B}T_\Vert$ and k_B T_∥.

The only other important systematic uncertainty in α_pl is the 12% absolute scaling error propagated from α_mb. We treat this scaling error and those from $k_{\rm B}T_{{\perp }}$ and k_B T_∥ as independent and add them in quadrature. The resulting total systematic error amounts to 17% at 10 K, 14% at 100 K, and less than 13% above 1000 K.

We have fitted our results for ease of use in astrochemical models. We found that neither the two-parameter function commonly used to describe DR plasma rate coefficients (e.g., Florescu-Mitchell & Mitchell 2006) nor the more general three-parameter extension used in astrochemical databases (e.g., Woodall et al. 2007; Wakelam et al. 2012) are able to fit our measured plasma rate coefficient accurately over the entire temperature range from 10 to 5000 K. Such fits do not reproduce our results to within better than 40%. Therefore we propose a modified form of the two-parameter function, namely

$$\begin{equation} \alpha _{\rm pl}^{\rm fit}(T) = A\,\left(\frac{300}{T}\right)^n + B, \end{equation} \tag{ 11 }$$

where

$$\begin{equation} B = T^{-3/2}\sum _{i=1}^4 c_i \exp (-T_i/T). \end{equation} \tag{ 12 }$$

This new function allows for more accurate fits than the previously used fitting functions. The results of fitting Equation (11) to our data over the full temperature range are given in Table 3. The deviations of $\alpha _{\rm pl}^{\rm fit}$ from the data are less than 1% over the full temperature range. It should be emphasized that the internal temperature of the HCl⁺ ions in the experiment yielding these result is expected to lie near 300 K.

The principal difference between our new method for deriving the cross section and the plasma rate coefficient compared to the methods used in previous merged beams DR measurements lies in the conversion of the merged beams rate coefficient to a cross section. Our new method is essentially a one-step process. The traditional method requires two steps: first, it corrects the merged beams rate coefficient for the increased electron–ion collision energy in the merging and de-merging regions while ignoring the electron velocity spread (Lampert et al. 1996) then the corrected rate coefficient is deconvolved to yield a cross section, assuming parallel beams in the interaction zone and zero electron energy spread along the beam axis, i.e., k_B T_∥ = 0 (Mowat et al. 1995). Our approach allows us to estimate the errors arising from the assumptions for each of these two steps.

To test the validity of the approach from Lampert et al. (1996), we have extracted the cross section from α_mb using our new approach and converted it to $\alpha ^*_{\rm mb}$ using Equation (7). The resulting $\alpha ^*_{\rm mb}$ then represents the rate coefficient as it would be measured in a merged beams configuration with parallel beams only and no merging or de-merging sections. Next we have reanalyzed our measured α_mb data using the method of Lampert et al. (1996) to generate $\alpha ^{* \rm L}_{\rm mb}$. Comparing the two data sets, $\alpha _{\rm mb}^*$ and $\alpha _{\rm mb}^{* \rm L}$, at high energies, where $E_{\rm d}\gtrsim k_{\rm B}T_{{\perp }}$, our correction procedure and the one proposed by Lampert et al. (1996) are equal to within their statistical accuracies. However, at lower energies, the traditional correction results in a merged beams rate coefficient that is up to ∼5% higher. Both our toroid correction method and that of Lampert et al. increase the merged beams rate coefficient at these low energies. Therefore, the difference can be interpreted as an over-correction by the older approach. We attribute this difference to the neglected electron velocity spread in the Lampert et al. method.

Next we tested the combined effects of the Lampert et al. (1996) and the Mowat et al. (1995) steps. To do this, we employed our new method for the cross-section derivation with two modifications. First, we used $\alpha ^{* \rm L}_{\rm mb}$ as the input rate coefficient, and second, we generated the collision energy distribution for straight beams $f_{\rm mb}^*$ while setting k_B T_∥ = 0. This latter step simulates the Mowat et al. approximation. We then converted the extracted cross section to the plasma rate coefficient $\alpha ^{\rm M}_{\rm pl}$ using Equation (10). The resulting plasma rate coefficient is larger than α_pl by 2.3% at T = 10 K, 4.2% at T = 100 K, and to 1.7% at T = 1000 K. The spread in these values due to the uncertainties in k_B T_∥ and $k_{\rm B}T_{{\perp }}$ is less than ±0.9%.

The errors in α_pl for HCl⁺ DR which are introduced by the Lampert et al. (1996) and Mowat et al. (1995) approximations are only a few percent, much smaller than the other systematic errors in the measurement. However, larger differences could potentially arise if the measured merged beams rate coefficient were highly structured or decreased more rapidly with increasing energy. Additionally, it is not trivial to implement a fast Fourier transform (FFT) method, such as recommended by Mowat et al. (1995), using DR data on a non-uniform energy grid. Furthermore, the FFT method does not weigh the input data points by their statistical importance. All of these issues are readily accounted for by our new approach. The advantages of our new method are likely to be critical for reliably analyzing the expected highly structured DR data from upcoming Cryogenic Storage Ring (CSR) experiments on rotationally cold molecular ions (von Hahn et al. 2011; Krantz et al. 2011).

Up to E_d ≈ 0.5 eV, the HCl⁺ DR rate coefficient α_mb decreases rapidly with increasing energy. This behavior is typical for low-energy DR spectra (Florescu-Mitchell & Mitchell 2006; Larsson & Orel 2008) yet, as shown in Figure 3, the slope between E_d = 4 and 30 meV is much steeper than the rate coefficient derived from the σ∝E⁻¹ expected for the direct DR process (Bates 1950; Larsson & Orel 2008). As described in Section 2, direct electron capture to a repulsive neutral potential surface is unlikely at these low collision energies. Thus, at these energies, DR is probably dominated by the indirect process and the enhanced rate coefficient most likely results from numerous DR resonances. These are unresolved due to both the energy resolution of the experiment and the natural widths of the resonances.

Based on the collision energies, these resonances probably originate from neutral Rydberg states converging to a rotationally excited HCl⁺ X ²Π core. The exact positions for the resonances cannot be easily determined because of the large number of combinations between the initial ∼300 K distribution of HCl⁺ excited states and the many energetically accessible ro-vibrationally excited levels in the neutral Rydberg system. Still, the steep decrease of α_mb between E_d = 4–30 meV may be attributed to the rotational structure of HCl⁺. As discussed in Section 2, each rotational level of the ion is expected to be the end point for a Rydberg DR resonance series. In Figure 3 we plot the range of such end point energies allowing for ΔJ = 1, 2, and 3 changes of the angular momentum with respect to the initial ionic state. The range of the end points (indicated by the gray bars) is due to the initial rotational excitation of the HCl⁺ ions (using rotational parameters from Sheasley & Mathews 1973; the levels spacings are nearly the same for ions in the X ²Π_3/2 or X ²Π_1/2 state, respectively.) The energy range of the fastest decrease in α_mb matches best the lowest possible change of the angular momentum, i.e., ΔJ = 1. Thus, it appears likely that indirect processes involving rotational excitation of the ∼300 K HCl⁺ ions play an important role for the DR rate at the lowest energies. Correspondingly, rotational excitation rates by low-energy electron collisions of HCl⁺ may be large.

Somewhat more resolved structures appear between 30 and 300 meV. There is the suggestion of a neutral Rydberg series limit at ∼80 meV corresponding to the 3/2 → 1/2 HCl⁺ fine-structure transition (Sheasley & Mathews 1973) and at ∼0.3 eV corresponding to the v = 0 → 1 excitation (Linstrom & Mallard 2013). The structures are blurred partly by the energy resolution of the experiment and partly by the initial rotational distribution of the stored HCl⁺ ions. Further interpretation cannot be given without more detailed calculations.

The increase in α_mb at E_d ≳ 0.6 eV can be attributed to opening of additional dissociation pathways forming either excited Cl (above 0.6 eV) or excited H (above 1.9 eV) or both. Moreover, Rydberg resonances involving the A²Σ⁺ electronically excited HCl⁺ core may also be present. Ion pair production, which can reduce the DR signal as discussed in Section 2, will set in above 1.7 eV. At these energies, the statistical uncertainties in our data prevent us from being able to discern the impact on the DR signal due to the opening up of these channels. DE channels are not accessible in the investigated collision energy range.

The derivation of the DR cross section from the merged beams rate coefficient makes the DR data independent of the experimental configuration. On the other hand, the experimental energy spread together with the statistical quality of the rate coefficient data limits the resolving power of our method for deriving the cross section. We have adjusted the cross-section binning such that it provides good energy resolution while keeping the numerical instabilities small. The interpretation of the derived cross section is therefore somewhat limited and one needs to proceed cautiously in interpreting any structures which have an energy width comparable to the energy binning. Given that caveat, much of the structure discussed above in α_mb can also be seen in the cross section.

Our experimentally derived DR plasma rate coefficient for HCl⁺ displays an unusually steep slope. Existing astrochemical models usually approximate DR of HCl⁺ by (Neufeld & Wolfire 2009)

$$\begin{equation*} \alpha _{\rm pl}^{\rm di}\approx 2.0\times 10^{-7}\times (300/T)^{0.5}\, {\rm cm}^3\,{\rm s}^{-1}. \end{equation*}$$

This is believed to correspond to DR of typical diatomic molecular ions (Florescu-Mitchell & Mitchell 2006). However, some astrochemical databases use a value 1.5 times higher for HCl⁺ (e.g., Woodall et al. 2007), though there is no obvious reason for this. Taking the ratio of $\alpha _{\rm pl}^{\rm }/\alpha _{\rm pl}^{\rm di}$ yields 1.5, 1.1, 0.64, 0.33, and 0.16 at T = 10, 30, 100, 300, and 1000 K, respectively. Such differences are larger than our experimental error bars. Thus we find that the "typical" diatomic DR rate coefficient is incorrect in both magnitude and temperature dependence for HCl⁺. Note, however, that our derived rate coefficient is for an internal temperature of ∼300 K of the HCl⁺ ions.

Our new data suggest that HCl⁺ depletion by DR in the cold ISM (∼10–50 K) is faster than or similar to that currently assumed in existing astrochemical models. However, at higher temperatures the new data display a slower HCl⁺ destruction by DR. As HCl⁺ leads to the formation of H₂Cl⁺, the H₂Cl⁺ abundance will also decrease or increase when using our new data, depending on the temperature. Unfortunately, the kinetic temperatures of the observed environments were not directly derived in the works of Lis et al. (2010), De Luca et al. (2012), and Neufeld et al. (2012). Model calculations (Neufeld & Wolfire 2009) predict that there is an abundance maximum of HCl⁺ in the outer parts of dark interstellar clouds which, at an opacity of A_v ∼ 1, are less dense than the core and with a temperature of ∼500 K are also hotter. In these regions, DR is predicted to clearly dominate the destruction of HCl⁺. Thus, our measured DR rate coefficient (which is about a factor of four lower than the ones used in current astrochemical models) will augment the predicted HCl⁺ abundances in that region, leading to an improved agreement with the observed HCl⁺ densities. It will also affect the H₂Cl⁺ abundance in the outer parts of the clouds (which also predicted an abundance maximum at roughly the same cloud density as the one for HCl⁺ is found), since a lesser efficient competition by the DR of HCl⁺ will allow more H₂Cl⁺ to be formed by reaction of HCl⁺ with H₂.

Neufeld et al. (2012) have explored the effect of lowering the HCl⁺ DR rate coefficient on the discrepancy between the predicted and observed HCl⁺ and H₂Cl⁺ abundances. In their astrochemical model they decreased α_pl of HCl⁺ to $\alpha _{\rm pl}^{\rm di}/10$. The resulting abundances of HCl⁺ and H₂Cl⁺ increased by factors of 2.7 and 1.3, respectively. In both cases the calculated abundances are still far too low to match the greater than 10 times larger observed abundances. The result of their tests suggests that our new HCl⁺ DR data, which are about four times smaller than $\alpha _{\rm pl}^{\rm di}$ at the relevant temperature, cannot fully explain the discrepancy between current astrochemical models and the observations. Still, our new data differ from $\alpha _{\rm pl}^{\rm di}$ not only in magnitude but also in temperature dependence. Therefore the test of Neufeld et al. (2012) does not directly describe the effect of our new DR data and their calculations should be repeated in order to fully evaluate the remaining discrepancy.

We also note that our experimental data are for HCl⁺ rotationally excited to T_rot ∼ 300 K. In the cold ISM, however, the internal excitation is expected to be much lower. In the very low density environment in interstellar clouds, the collision rate is much lower than the typical radiative decay time (Spitzer 1978). Thus T_rot is generally even lower than the kinetic temperature T and most of the molecules are expected to be in their rotational ground state. The response of the DR rate coefficients to T_rot has not yet been studied systematically. The few existing studies on light ions find that the measured merged beams rate coefficient exhibits an increasingly rich resonant structures with decreasing T_rot (e.g., Amitay et al. 1996, Zhaunerchyk et al. 2007, Petrignani et al. 2011, Schwalm et al. 2011). This can be explained by the smaller number of rotational states that contribute with decreasing T_rot. The individual resonant structures thereby become resolved as they no longer overlap with resonances of other, energetically higher states. Low values of T_rot can result in either a lower or higher plasma rate coefficient, depending on the particular shape of the cross section. We expect that future DR measurements at the CSR facility will be able to address this issue by investigating molecular ions rotationally colder than what can be achieved with TSR.