Dissociation and radiative stabilization of the indene cation: The nature of the C–H bond and astrochemical implications Open Access

Polycyclic aromatic hydrocarbons (PAHs) have long been thought to be ubiquitous in the interstellar medium (ISM). This is based on the infrared emission bands observed by astronomers at wavelengths coincident with their characteristic vibrational transition energies.¹ However, these bands are common to PAHs as a class and have proven difficult to assign to specific PAH molecules. It was only in the past few years that indene,^2,3 2-cyanoindene,⁴ and two isomers of cyano-naphthalene⁵ were identified in space in the Taurus molecular cloud, TMC-1 by comparing astronomical microwave spectra with known rotational emission lines. Very recently, several larger cyano-functionalized PAHs have been identified by the same methods.^6,7 Interestingly, the observed interstellar abundance of indene (C₉H₈, Fig. 1), the only polycyclic pure hydrocarbon identified to date, was more than four orders of magnitude higher than predicted by astrochemical modeling.⁴ These models neglect radiative cooling and assume that PAHs are rapidly broken down into linear fragments following ionizing collisions with small cations including C⁺ and $H3+$⁠.⁵ Several recent studies have examined the spectroscopy and stability of indene^8,9 and its primary degradation product, indenyl, under astrophysically relevant conditions.^10,11

A previous study¹² on 1-cyanonaphthalene by some of the present authors found that the energized cation is efficiently stabilized by recurrent fluorescence (RF)—the emission of optical photons from thermally excited electronic states.^13,14 This rapid radiative cooling closes some of the destruction pathways included in astrochemical models that underpredicted the abundance of that molecule in TMC-1 by six orders of magnitude.⁵ Numerous laboratory studies have confirmed that RF, rather than infrared photon emission from vibrational relaxation, is the primary radiative stabilization mechanism for PAH cations,^12,15–22 with few exceptions.²³ 

We present a study of the unimolecular dissociation and radiative stabilization of the indene radical cation, $C9H8+$⁠, Ind⁺. The dissociation rate coefficient and activation energy are determined from time-resolved measurements of kinetic energy release distributions of an ensemble of vibrationally excited ions isolated in a cryogenic ion-beam storage ring. RF rate coefficients are calculated based on ab initio molecular dynamics simulations. Master equation modeling including dissociation, RF, and infrared vibrational radiative cooling processes closely reproduces the measured absolute dissociation rate of the stored ion ensemble. We find that Ind⁺, in contrast to fully sp²-hybridized PAH cations, cools mainly by IR photon emission, with at most a small contribution from RF. Nevertheless, we find that Ind⁺ can be expected to be radiatively stabilized following charge-exchange reactions between C⁺ and neutral indene, one of the most important destruction channels for PAHs included in astrochemical models of molecular clouds.

Experiments were conducted using the DESIREE (Double ElectroStatic Ion Ring ExpEriment) cryogenic ion-beam storage ring infrastructure located at Stockholm University^24,25 using procedures similar to earlier experiments on cyano-naphthalene.¹² Indene (Sigma-Aldrich, $>$99%) vapor was introduced into an electron cyclotron resonance (ECR, Panteknik Monogan) ion source. The ECR source produces molecular ions with high internal energies. Positive ions were extracted from the source, accelerated to 86 keV, and those with m/z = 116 were selected with a bending magnet. Ion beams were stored using electrostatic deflectors in the storage ring depicted in Fig. 1. Neutral fragments emitted from vibrationally hot ions in the lower straight section of the ring (see Fig. 1) impinged on a 75 mm diameter Z-stack microchannel plate detector (Photonis) with a phosphor screen anode (phosphor type P24). The position and time of each detected fragment were recorded using a Timepix3 hybrid pixel detector (Amsterdam Scientific) located outside of the DESIREE vacuum chamber. At the end of each storage cycle, the beam was dumped into a Faraday cup coupled to a fast amplifier for current normalization.

To analyze the kinetic energy released during dissociation, the event data from the Timepix were first binned according to time after ionization, with a bin width equal to the revolution period of the ions around the ring. Two dimensional histograms of the detector images for each revolution were centered and azimuthally integrated using the PyAbel package.²⁶ The resulting time-dependent radial intensity distributions were then further averaged over time windows with widths linearly increasing with storage time to improve signal-to-noise at late times where the count rate is low. The radial intensity distribution includes contributions from neutral fragments emitted along the straight section of the storage ring and projected onto the detector plane (see Fig. 1). We fit analytic kinetic energy release (KER) distribution models to this final projected distribution by first applying a matrix transformation from energy to radial distance considering, for each energy bin, the range of radii over which the differential intensity will be spread given the finite length of the straight section. The projection of this spherically symmetric distribution onto the detector plane is achieved with a one-dimensional forward Abel transform. For more information on the analysis, see the  Appendix.

A photo-activation technique was employed to probe the evolution of the internal energy distribution on longer timescales, after the initially hot ions have all either dissociated or been radiatively stabilized. In these experiments, light from a wavelength-tunable optical parametric oscillator laser system was overlapped collinearly with the stored ion beam, as shown in Fig. 1. Following rapid internal conversion and intramolecular vibrational redistribution, photo-activation increases the vibrational energy of the excited ions by a known, fixed amount (the photon energy). Photo-activated ions are thus re-heated to energies above the dissociation threshold, and neutral fragments are again emitted and detected using the same procedures detailed above for the source-heated ions.

Quantitative studies, by the present collaboration as well as others, have generally found that Eq. (3) drastically underestimates RF cooling rates when the parameters E_j and $AjRF$ are estimated using quantum chemical methods such as TD-DFT.^15,19 It was previously reported that this discrepancy could be narrowed by consideration of Herzberg–Teller (HT) vibronic coupling, which greatly increases the transition probability of the lowest-energy optical transition of PAHs.¹² HT simulations of open-shell systems remain a challenge for computational chemistry programs, and recently, a more generally applicable method based on molecular dynamics simulations has been introduced.¹¹ 

Here, we performed temperature-dependent ab initio molecular dynamics (AIMD)³¹ at the ωB97X-D/cc-pVTZ level of theory,^32,33 with canonical sampling through velocity rescaling.³⁴ Trajectories at given initial temperatures, converted to vibrational energies according to the caloric curve based on the state density, were run for 1 ps in 0.5 fs steps, with the vertical transition energies and oscillator strengths for the D_j ← D₀ states computed at each step. The RF rate coefficients for fixed thermal energies E were then calculated and time-averaged for each trajectory.

Dissociation energies for H-loss from the sp³ site were performed using the DSD-PBEP86-D3(BJ)/def2-TZVPP and CCSD(T)/cc-pVTZ theories.

Our calculated RF rate coefficients, based on our AIMD simulations, are presented in Fig. 2. The solid curves are fits of Eq. (3) to the points that are time averaged values from AIMD trajectories with given initial temperatures. The fit parameters E_j and f_j0 are given in Table I, along with experimental and calculated values³⁵ from Chalyavi et al. Prior to fitting, the D_n state energies in the AIMD simulations were scaled by a factor of 1.03 so that the resulting value of E₂ agrees with the 0–0 transition energy Chalyavi et al. measured using He-tagging messenger spectroscopy.³⁵ Chu et al. reported³⁶ a nearly identical energy of this transition at 17 379.3 cm⁻¹ for bare Ind⁺, while Nagy et al. found³⁷ 17 249 cm⁻¹ in a Ne matrix. Significantly, the oscillator strength of the D₁ ← D₀ transition is an order of magnitude greater in our AIMD simulation than in the TD-DFT calculations of Chalyavi et al., leading to a commensurately increased RF rate coefficient.

The velocity distribution of the neutral products, integrated over the first eight revolutions of the stored ions in DESIREE (t = 56–242 μs after ionization), is shown in Fig. 3. The distribution is bimodal with the dominant component with higher mean velocity being due to H-loss from Ind⁺. We attribute the low-velocity component to C₂H₂-loss from ¹³C-indenyl (⁠$13C12C8H7+$⁠), which has the same mass-to-charge ratio as our ion of interest (⁠$12C9H8+$⁠). Owing to its fully sp²-hybridized structure, the indenyl cation [Ind-H]⁺ has a higher dissociation energy for H-loss than Ind⁺ and was found to decay exclusively by C₂H₂-loss in previous experiments at DESIREE.¹¹ Given that the dissociation threshold energy for C₂H₂-loss from Ind⁺ is significantly higher than for H-loss, we do not expect these channels to be competitive on such long timescales.³⁸ We can also rule out sequential fragmentation processes in which stored Ind⁺ first loses an H atom to form [Ind-H]⁺ and then fragments again by C₂H₂-loss. This is because ions with such high internal energies as to undergo multi-fragmentation would not survive to the point of mass selection by the bending magnet.

To handle the [Ind-H]⁺ contamination in our analysis of the KER distributions, a separate measurement was made of all-¹²C [Ind-H]⁺ under the same experimental conditions, giving the shape parameters needed to fit the combined distributions in Fig. 3. As the dissociation of [Ind-H]⁺ is quenched much more rapidly than that of Ind⁺ (see Sec. III D), the parameters for Ind⁺ were obtained by integrating the KER distributions over the 25–50 ms time window, as shown in Fig. 4.

The kinetic energy released in H-loss reactions of hydrocarbons has long been known to be anomalously large,³⁹ and the shape of the KER distribution is notoriously difficult to model.⁴⁰ The reaction generally shows no reverse activation barrier, and indeed, no energy gap in the KER distribution is observed. However, the emission of low-energy (⁠$<$0.05 eV) H atoms is notably suppressed, skewing the distribution beyond what can be readily accommodated by otherwise successful models.³⁹ In their study of the benzene cation, Gridelet et al. argue that the high KER is due to the difference in electronic configurations of the reactant and product states.⁴⁰ The non-adiabatic character of the reaction requires passage through a conical intersection to a dissociative state of the appropriate symmetry, which Gridelet et al. show can qualitatively be understood as tunneling through a modified centrifugal barrier.

Following Gridelet et al., the dissociation pathway of H-loss from Ind⁺ is illustrated in Fig. 5. The ground D₀ state of Ind⁺ crosses a dissociative D* state, which becomes the ground S₀ state of [Ind-H]⁺, at a conical intersection. The D₀ potential energy curve is drawn as a Morse potential with the frequency of the aliphatic CH stretch and the D* as a centrifugal potential for the H⋯[Ind-H]⁺ system. The crossing is treated as a potential energy barrier with the shape of an inverted parabola of height ΔE above the asymptotic energy of the D*/S₀ state. The activation energy E_a is the difference between this asymptotic energy and the minimum of D₀.

From the fit shown in Fig. 8, we find the activation energy E_a = 2.9(1) eV and the pre-exponential factor A = 3.3(5) × 10¹⁴ s⁻¹. It is clear that the simple Arrhenius model is not an ideal description of the measured dissociation rate coefficient. This may be attributed to both experimental challenges, such as the separation of the Ind⁺ and [Ind-H]⁺ contributions to the KER distribution, and to the questionable applicability of classical kinetic theory to complex quantum tunneling processes. However, the resulting parameters are in line with expectations and, as will be shown in Secs. III D–III F, reproduce other experimental observables.

Our activation energy agrees with the 2.6(4) eV found by West et al. for Ind⁺ using a photo-electron photo-ion coincidence technique.³⁸ A weighted average of threshold energies for sp³-hybridized H atom losses from six different PAH cations determined by similar methods give E_a = 2.4(4) eV.^38,46,47 Our own quantum chemical calculations found dissociation energies for H-loss from Ind⁺ of 2.8 eV at both the DSD-PBEP86-D3(BJ)/def2-TZVPP and CCSD(T)/cc-pVTZ levels of theory, in good agreement with our measured value.

The pre-exponential factor is close to the “universal” Gspann value of 1.6 × 10¹⁵ s⁻¹ for the evaporation of atomic and molecular clusters.³⁹ In an RRKM framework, our value of A corresponds to an activation entropy of $ΔS1000 K‡=14(1)$ J K⁻¹ mol⁻¹, consistent with a loose transition state. West et al.³⁸ give $ΔS1000 K‡=−2±38$ J K⁻¹ mol⁻¹ for Ind⁺ and an average value^38,46,47 for sp³-hybridized H-losses of $ΔS1000 K‡=44±20$ J K⁻¹ mol⁻¹.

Our experimentally determined microcanonical dissociation rate coefficient k^diss(E) = k_m(t) is plotted in Fig. 9 as a function of the vibrational energy of the reactant, E = E_tot(T_p) + E_a, along with our calculated radiative cooling rate coefficients k^IR(E) and k^RF(E).

The total neutral product count rate R(t) for Ind⁺ ions stored in DESIREE is shown in Fig. 10 (blue curve). For the first few milliseconds after injection, the rate follows a power-law R(t) ∝ t⁻¹, indicative of a broad internal energy distribution.⁴⁸ After about 10 ms, the rate begins to deviate from the power law trend, as radiative cooling becomes competitive with dissociation. After 100 ms, a constant rate is observed due to dissociation of stored ions induced by collisions with the residual gas in the ring.

The total count rate R(t) is divided into contributions from H-loss from $C9H8+$ (green symbols in Fig. 10) and C₂H₂-loss from the contaminant $C1213C8H7+$ (blue symbols) according to fits of the KER distribution as in Fig. 3. The count rate for a separate measurement of pure ¹²C indenyl (⁠$C9H7+$⁠), scaled to that of the C₂H₂-loss contamination signal, is included for comparison (orange curve). It is clear that H-loss from Ind⁺ is the dominant contribution to R(t), and the C₂H₂-loss channel will therefore be neglected in the remaining analysis.

Results of our master equation simulations compared to the measured dissociation rate Γ(t) are presented in Fig. 11. Simulations were run for a range of temperatures T₀, specifying the initial, Boltzmannian distribution of vibrational energies of the ensemble. The detector efficiency η_det was also taken as a free parameter. The simulation that best reproduced the measured data had T₀ = 1650(40) K and η_det = 0.113(1). This temperature is comparable to that obtained for other PAHs using the same experimental and modeling techniques, and the derived detector efficiency is reasonable for hydrogen atoms with $≈740$ eV kinetic energy in the lab frame as in the present experiment.^19,49

Two alternative simulation results are also presented in Fig. 11. In the first, to test our hypothesis that vibrational motion enhances the RF rate, we turn off RF in the simulation. The best fitting simulation under these conditions [T = 1800(100) K and η_det = 0.101(2)] diverges from the measured rate after a few tens of milliseconds. This shows that RF plays a role in the simulated cooling dynamics despite the RF rate coefficient being lower than for IR cooling in the energy window probed in our experiment. A second scenario leaves RF in place but increases the IR cooling rate by a factor of 1.5. Differences in the IR rates on this scale are sometimes noted between different levels of theory.³⁰ Again, the best fitting simulation [T = 1670(40) K and η_det = 0.108(1)] diverges from the measured data, but in the opposite direction. Thus, an overestimate of the RF rate coefficient could be compensated for by an underestimate of the IR rate coefficient. We conclude that RF makes, at most, a small contribution to the radiative stabilization of indene cations.

Only a fraction of the ions injected into the DESIREE storage ring decay by the dissociation process discussed so far. Most will continue to circulate while cooling by emission of thermal radiation. The evolution of the internal energy distribution of the stored ions g(E, t) can be tracked with laser probing techniques.^19,21,51,52 Figure 13 shows the yield of neutral fragments S as a function of time following laser excitation t_after with hν_laser = 3.75 eV (330 nm) and 3.49 eV (355 nm) photons, for a laser firing time of t = 0.6 s.

The mean energies $Ē$ of the Boltzmann distributions fit to the laser-induced decay curves are plotted in Fig. 14. Good agreement is generally found between these fitted energies and the mean energy of the ensemble in our master equation simulation, which best fits the spontaneous dissociation rate (solid curve in Fig. 14).

The energy loss rate due to radiative cooling is computed as the time derivative of $Ē$⁠, plotted in Fig. 15. Good agreement is found with the calculated total vibrational radiative power $PIR=∑sKsIRhνs$⁠, supporting the accuracy of our calculated rate coefficients, which, as discussed in Sec. III D, imply a minor role for RF in the radiative stabilization of Ind⁺.

Astrochemical models routinely underestimate the observed abundances of small PAHs in TMC-1.^2,5 The main destruction mechanism included in these models is collisions with the cations H⁺, He⁺, C⁺, $H3+$⁠, and HCO⁺, which are assumed to lead inexorably to complete degradation of the PAHs into small linear fragments.⁵ These models may be improved by including more complete product breakdown reactions. In Fig. 16, we give a breakdown diagram for indene following ionizing collisions based on the dissociation and radiative cooling rate coefficients for Ind⁺ (Fig. 9) and [Ind-H]⁺ from Ref. 11. At the low temperatures (∼10 K) prevalent in TMC-1, the atomic cations will undergo electron transfer at large distances with negligible translational-to-vibrational energy transfer,⁵³ leaving the ionized PAH with a maximum internal excitation energy equal to the difference in the ionization potentials (the molecular cations will lead to proton transfer). Note that these maximal energy transfers, given as vertical dashed lines in Fig. 16, are extremely unlikely in distant electron-transfer reactions. With its relatively low ionization potential (11.26 eV), charge transfer forming carbon atoms occurs at shorter distances and will lead to greater heating of the Ind⁺ product. Collisions with C⁺ are thus considered the most important PAH-destruction channels in many astrochemical models.^54,55 However, given the 8.14 eV ionization potential of Ind,³⁸ such collisions yield a maximal energy transfer of 3.12 eV. This is unlikely to induce dissociation of Ind⁺, which is stabilized by radiative cooling up to a critical energy of 3.6 eV, where the dissociation and radiative rate coefficients are equal. Charge exchange reactions with protons could transfer sufficient energy to induce H-loss from Ind⁺, yielding [Ind-H]⁺, although, as stated above, such high high-energy transfers are rather unlikely in distant electron-transfer reactions. Collision with He⁺ could potentially transfer sufficient energy to further break down [Ind-H]⁺. However, the second ionization potential of Ind, at 21.8(1) eV,⁵⁶ is low enough that formation of Ind²⁺ may be competitive.

It seems clear that collisions between PAHs and atomic cations in molecular clouds will primarily produce radiatively stabilized PAH cations. Previous studies have found that these cations do not react with H₂ and react mainly by association with H, N, and O atoms to form larger cationic molecules.^57,58 Destruction of PAH cations in dark clouds is expected mainly in recombination reactions with free electrons⁵⁹ or mutual neutralization with anions including PAH anions.⁶⁰ However, given that neutral PAHs may also be radiatively stabilized by RF,¹¹ it is not obvious that extensive fragmentation should be expected following recombination.

Through analysis of the kinetic energy released upon H atom loss from indene cations, we determine a dissociation rate coefficient consistent with results of an earlier photodissociation study.³⁸ The shape of the KER distribution is consistent with a qualitative model of tunneling through an intersection of potential energy surfaces.^40,41 More experiments on H-loss reactions for other hydrocarbons are needed to assess how general this phenomenon might be.

Using our best estimates of the vibrational and electronic radiative cooling rate coefficients, combined with the dissociation rate coefficient deduced from the present experiments, we find a limited role for recurrent fluorescence in the stabilization of indene cations. This is noteworthy as RF has been found to be the dominant cooling mechanism of nearly all PAH cations investigated experimentally to date.^12,15–22 Still, we conclude that the IR radiative cooling rate is sufficiently high to prevent dissociation following ionizing collisions with carbon cations. This may partly explain the high observed abundance of indene in interstellar clouds,³ demonstrating the importance of radiative stabilization of small PAHs in astrochemical reaction networks.

The supplementary material is available for this article in the form of a spreadsheet listing the computed vibrational frequencies and infrared intensities, the vibrational level density, the caloric curve, the calculated rate coefficients for vibrational and electronic radiative cooling, and the fitted dissociation rate coefficient of Ind⁺.

This work was supported by Swedish Research Council Grant Nos. 2023-03833 (H.C.) and 2020-03437 (H.Z.), Knut and Alice Wallenberg Foundation Grant No. 2018.0028 (H.C. and H.Z.), EPSRC Grant No. EP/W018691 (J.N.B.), Olle Engkvist Foundation Grant No. 200-575 (M.H.S.), and Swedish Foundation for International Collaboration in Research and Higher Education (STINT) Grant No. PT2017-7328 (J.N.B. and M.H.S.). We acknowledge the DESIREE infrastructure for provisioning of facilities and experimental support and thank the operators and technical staff for their invaluable assistance. The DESIREE infrastructure receives funding from the Swedish Research Council under the Grant Nos. 2021-00155 and 2023-00170. This article is based upon work from COST Action CA21126—carbon molecular nanostructures in space (NanoSpace), supported by COST (European Cooperation in Science and Technology).

The authors have no conflicts to disclose.

M. H. Stockett: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). A. Subramani: Formal analysis (equal); Investigation (equal); Writing – review & editing (equal). C. Liu: Formal analysis (equal); Investigation (equal); Writing – review & editing (equal). S. J. P. Marlton: Formal analysis (equal); Investigation (equal); Writing – review & editing (equal). E. K. Ashworth: Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Resources (equal); Writing – review & editing (equal). H. Cederquist: Funding acquisition (equal); Resources (equal); Writing – review & editing (equal). H. Zettergren: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). J. N. Bull: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

A^diss

Pre-exponential factor determined from k_m(t) and T_p [Eq. (9)]

E_a

Activation energy determined from k_m(t) and T_p [Eq. (9)]

E_e

Energy of decaying ions [Eq. (11)]

E_tot(T)

Caloric curve computed from vibrational level density ρ(E)

$Ē$

Mean energy of stored ion ensemble prior to laser activation. Determined from the fit of Eq. (19) to S(t_after)

hν_laser

Laser photon energy

k^diss(E)

Model dissociation rate coefficient [Eq. (7)] using fitted values of A^diss and E_a

k^IR(E)

Vibrational (infrared) emission rate coefficient [Eq. (2)] from SHC model

k_m(t)

Most probable rate coefficient observed at time t [Eq. (10)]

k^RF(E)

Recurrent fluorescence rate coefficient [Eq. (3)] using parameters from AIMD simulations

R(t)

Measured spontaneous dissociation rate

T₀

Canonical temperature characterizing stored ion ensemble. Determined from fitting R(t) with results of master equation simulations

T_e

Microcanonical emission temperature of decaying ions $≈(Tr+Tp)/2$

T_p

Microcanonical temperature of product with energy E − E_a. Determined from fits of Eq. (5) to measured KER distributions

T_r

Microcanonical temperature of reactant with energy E

Γ(t)

Absolute, per particle dissociation rate [Eq. (15)]

ΔE

Barrier height. Determined from fit of Eq. (5) to measured KER distribution at late times

A^diss

Pre-exponential factor determined from k_m(t) and T_p [Eq. (9)]

E_a

Activation energy determined from k_m(t) and T_p [Eq. (9)]

E_e

Energy of decaying ions [Eq. (11)]

E_tot(T)

Caloric curve computed from vibrational level density ρ(E)

$Ē$

Mean energy of stored ion ensemble prior to laser activation. Determined from the fit of Eq. (19) to S(t_after)

hν_laser

Laser photon energy

k^diss(E)

Model dissociation rate coefficient [Eq. (7)] using fitted values of A^diss and E_a

k^IR(E)

Vibrational (infrared) emission rate coefficient [Eq. (2)] from SHC model

k_m(t)

Most probable rate coefficient observed at time t [Eq. (10)]

k^RF(E)

Recurrent fluorescence rate coefficient [Eq. (3)] using parameters from AIMD simulations

R(t)

Measured spontaneous dissociation rate

T₀

Canonical temperature characterizing stored ion ensemble. Determined from fitting R(t) with results of master equation simulations

T_e

Microcanonical emission temperature of decaying ions $≈(Tr+Tp)/2$

T_p

Microcanonical temperature of product with energy E − E_a. Determined from fits of Eq. (5) to measured KER distributions

T_r

Microcanonical temperature of reactant with energy E

Γ(t)

Absolute, per particle dissociation rate [Eq. (15)]

ΔE

Barrier height. Determined from fit of Eq. (5) to measured KER distribution at late times

Here, we describe the implementation of matrix transforms used to compute the following:

1. the total density distribution of the Newton sphere consisting of contributions from neutral particles emitted with a given kinetic energy distribution along a linear segment normal to the image plane, and

2. the forward Abel transform between the Newton sphere density distribution and the radial intensity distribution in the image plane.

These matrix transforms are used to fit model kinetic energy release (KER) distributions P(ɛ) to measured radial intensity distributions P(r) measured with a two-dimensional microchannel plate detector after linear translation of the expanding Newton sphere from the point of reaction to the detector plane. This approach is preferred as the measured P(r) distributions have well-defined uncertainties given by counting statistics, while inverting the transforms inevitably introduces new errors.

We define our energy coordinate vector ɛ_j (j = 0, 1, …, n) to span the range from $εjmin(ρ0)$ to $εjmax(ρn)$⁠. We choose the ɛ_i to increase quadratically, to maintain the scaling of Eq. (A1).

In Fig. 17, we show the results of several transformations of a model KER distribution into integrated Newton sphere density distributions using $T$ matrices with varying lengths of the emitting segment L_SS. The model distribution is of the form of Eq. (5) using the parameters found from the fit to the late-time KER distribution of Ind⁺ (Fig. 4). The actual value of L_SS in DESIREE is 960 mm, leading to a density distribution that is notably distorted relative to that from a point source.

Unfortunately, the $T$ matrix is ill-conditioned and cannot be inverted. In order to plot KER distributions, it is thus necessary to perform a pseudo-calibration by Eq. (A1) with L = L_mid. As shown in Fig. 18, this results in distorted KER distributions for emitting segments of finite length. We reiterate that all numerical results are obtained by fitting of the radial intensity in the detector plane, which, because it only requires the forward transform, is not affected by this distortion.

In Fig. 19, we show a comparison of forward Abel transforms using the complete matrix transform $RA$ and the PyAbel package. Both start from the model distribution P(ρ) from Fig. 17 (for L_SS = 960 mm). For PyAbel, the Hansen–Law algorithm was employed, with other options giving nearly identical results. The differences in the resulting P(r) distributions are smaller than the experimental uncertainties.

With the given choices of δ and $A00$⁠, the transform matrix is readily inverted using the standard Moore–Penrose algorithm. By reduction of dimensionality and cancellation of counting uncertainties by azimuthal integration prior to inversion, the matrix implementation of the reverse Abel transform outperforms the benchmark PyAbel routine. As described in Sec. II A, the uncertainty in the density distribution P(ρ) is found by inverting and averaging 128 P(r) distributions with intensities normally distributed according to the measured distribution and its counting statistics. Applying this same approach using PyAbel, which operates on the full 2D image rather than the radial intensity distribution, takes 60 times longer than the matrix method. As shown in Fig. 20, the PyAbel result is also noisier.