Ionic photofragmentation cross sections of the HS⁺, H₂S⁺ and HCl⁺ molecular ions near the 2p threshold

Jean-Paul Mosnier*^a, Eugene T. Kennedy^a, Denis Cubaynes^bc, Jean-Marc Bizau^bc, Ségolène Guilbaud^c and Stéphane Carniato^d
aSchool of Physical Sciences, Dublin City University, Dublin 9, Ireland. E-mail: jean-paul.mosnier@dcu.ie
^bSynchrotron SOLEIL, L'Orme des Merisiers, Saint-Aubin, BP 48, F-91192 Gif-sur-Yvette, Cedex, France
^cInstitut des Sciences Moléculaires d'Orsay, UMR 8214, Université Paris-Saclay, Rue André Rivière, Bâtiment 520, F-91405 Orsay Cedex, France
^dLaboratoire de Chimie Physique-Matière et Rayonnement (LCPMR), UMR 7614, Sorbonne Université, Campus Pierre et Marie Curie, 4 place Jussieu, F-75005 Paris, France

First published on 13th August 2025

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1 Introduction

The Auger decay of free atomic and molecular species upon the absorption of short wavelength (ionising) photons is a fundamental process in nature with relevance and applications in many different fields such as plasma physics,¹ astrophysics,^2,3 medical physics⁴ and the analytical, surface and material sciences, e.g. ref. 5. Due to the strengthening of the Coulomb potential, the systematic study of the photoionisation behaviour of ionised atomic and molecular species^6,7 is of equal fundamental interest. In the case of molecular ions, the non-spherical nature of the molecular field together with the coupling of the nuclear degrees of freedom will add complexity compared to the atomic case as seen in unique spectral features such as the molecular shape resonance.⁸

While for atomic ions there exists a large corpus of experimental and theoretical photoionisation data, this is not the case for molecular ions. This is largely due to the significant challenges posed to both the experimental techniques and the theoretical models, namely (1) the production of sufficiently sensitive molecular ion beam or trap photoionisation apparatus and, (2) the accurate description of highly excited molecular states, respectively. In spite of these difficulties, the field of soft X-ray absorption in molecular ions has witnessed marked progress over the past ten years with combined experimental and theoretical modelling studies for, e.g., N₂⁺,⁹ NO⁺,¹⁰ H₃O⁺,¹¹ C₃H₃⁺¹² or O₂⁺.¹³ This is in parallel with a body of work on the structure and dynamics of molecular hydride ions: HC⁺, HO⁺, and HSi⁺,¹⁴ HI⁺,¹⁵ HSi⁺, H₂Si⁺, H₃Si⁺,¹⁶ H_yN⁺ (y = 0 to 3),¹⁷ HBr⁺,¹⁸ HN⁺¹⁹ and HF⁺.²⁰ Further, the Auger decay rates and fragmentation dynamics of H₂C⁺ and H₃C⁺ were investigated theoretically in ref. 21.

It is well known that diatomic molecular hydrides and their ions play key roles in the evolution of the interstellar medium and are, thus, invaluable probes (tracers) of many chemical and radiative processes in molecular clouds, e.g. ref. 22–27. In this work, we present for the first time experimental and theoretical absorption cross sections for the sulfanylium HS⁺ and chloroniumyl HCl⁺ diatomic molecular hydrides. These were discovered in the interstellar gas in 2011²⁸ and 2012,²⁹ respectively. We also report experimental cross sections for the sulfaniumyl H₂S⁺ molecular ion which, to our knowledge, remains hitherto unidentified in the interstellar gas.

Another point of interest concerning the inner shell photoionisation of molecular hydrides is that the initial core hole and its subsequent decay are necessarily localised on the heavy element. If the non-radiative decay leads to dissociation of the original parent molecular ion, the fragments are imparted additional kinetic energy, i.e. the phenomenon of kinetic energy release or KER. As the KER mirrors the Auger decay,³⁰ related experimental and theoretical studies are key to the understanding of the interplay between electronic and nuclear decay processes in molecular photoionisation. This has been an extensively researched field for the past nearly sixty years and is still actively explored, see e.g. ref. 20 and 30–38 and all the references therein.

In this work, we report absolute photofragmentation cross sections for the HS⁺, H₂S⁺ and HCl⁺ molecular ions in the photon energy regions of the 2p threshold. The specific experimental aspects of our photoion yield measurements are detailed in the experimental section. This is supplemented with a discussion (in Appendix), based on an analytical approach, of some physical aspects of KER that are pertinent to our cross section measurements carried out using a fast ion beam. In Section 4 Analyses and discussions, the experimental results are analysed with the help of ab initio density functional theory (DFT) and post Hartree–Fock configuration interaction (CI) theoretical calculations. Comparisons of the results with other relevant isoelectronic atomic and molecular species are also presented in this section.

2 Experimental details and presentation of results

2.1 Experimental procedures and details

A detailed description of the merged-beam MAIA (Multi-Analysis Ion Apparatus) apparatus as well as the general experimental procedures used in the present work are given in ref. 39 and our more recent works, e.g. ref. 16 and 40. Some specific details pertaining to the present work are given below and in Table 1. The HS⁺ or HCl⁺ sample (parent) molecular ions were produced from H₂S or HCl gas, respectively, leaked into a permanent magnet electron cyclotron resonance ion source (ECRIS). The latter was excited by a 12.36 GHz microwave power supply and operated at the powers indicated in Table 1. For each of the photoionisation experiments, the individual beam of the HS⁺ or HCl⁺ parent ion was extracted at −4 kV producing a velocity v_4kV m s⁻¹. A magnetic filter and electrostatic deflector, tuned to v_4kV and a given nuclear mass, then selected and guided the sample ion beam (¹H³²₁₆S⁺ or ¹H³⁵₁₇Cl⁺, respectively HS⁺ or HCl⁺ in the rest of the paper) to the interaction region. The latter is a 0.57 m in length and 0.06 m in diameter cylinder subtending a solid angle of 8.7 × 10⁻³ sr and an exit angle of 6.0° at the (ion) entrance apex. The background pressure was ≃2 × 10⁻⁹ mbar for all the experiments. In the interaction region the parent ion beam was merged with the counter-propagating, left-handed circularly polarised, synchrotron radiation (SR) beam produced by an undulator and monochromatised by a 600 l mm⁻¹ grating. The effects of the superposition of the short wavelength synchrotron radiation reflected in the second and third diffraction orders of the grating were evaluated and found to have negligible contributions on the final cross section results. In order to capture the greatest possible number of tagged ions (see below), an electrostatic (Einzel) lens of matched aperture follows the polarised cylinder.

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After the interaction region, the S²⁺ or S³⁺, and Cl²⁺ or Cl³⁺, photo-ion signals were counted with a microchannel plate following mass/charge analysis by a combination of a bending electro-magnet and an electrostatic deflector. A high-voltage (tag) bias of −1 kV was applied to the interaction region in order to distinguish the photo-ions produced within the interaction region from those produced outside the region. In consequence, the singly charged parent positive ion beam saw its kinetic energy (KE) of 4 keV increased to 5 keV as it penetrated the interaction volume. The value of the speed of the HS⁺ and HCl⁺ parent ions in the interaction region is thus set to and respectively.† Calibration of the apparatus has provided the following relationships between the electro-magnet current I_A, the resulting magnetic field strength B_G and the kinetic energy E_K of the ion (mass m) under analysis after the interaction region:

	(1)

where q′ is the final charge state of the fragment. B_G, I_A, m and E_K are in units of Gauss, amp, amu and eV, respectively.

By measuring the photon and ion beam parameters, their overlap volumes (the form factor in Table 1), using calibrated photon and ion detectors and subtracting noise, it was possible to obtain the measured cross sections on an absolute basis.³⁹ Finally, photon energy scans of the photo-ions count rates were carried out to map out the energy dependence of the photoabsorption cross sections. The photon energies were calibrated using a gas cell and known argon reference lines⁴¹ and could be determined to within an accuracy of 40 meV. The relative uncertainty in the measured absolute cross sections is generally within 15%.³⁹

2.2 Principle of the measurements and results

Upon absorption of a photon creating an inner-shell vacancy, the parent molecular ion will emit one (or several) Auger electron(s) and, simultaneously or sequentially, dissociate into atomic fragments. The KE of the fragments will necessarily include a contribution from the kinetic energy release (KER) inherent in the series of relaxation processes leading to nuclear dissociation of the parent molecular ion, see e.g. ref. 33. E_K in eqn (1) comprises the KER value for the decay/dissociation process at play. This KE change is also accompanied by a change in the direction of travel of the fragments, away from the initial direction of the parent ion beam. These processes are key to our measurements because they underpin our analyses of the ionic fragment yield data and subsequent estimation of photo-absorption cross sections. In the Appendix Section at the end of the paper, we provide background details on the dynamics of formation of the dissociation fragments using the basic equations that express conservation of momentum and energy. We, thus, calculate, as a function of KER, both the expected KE changes and dissociation angles of the fragments with respect to the corresponding original values of the parent molecular ion beam. Notably, we show there that these additional,‡ purely molecular, effects do not affect our X²⁺ and X³⁺ fragments yield measurements in the specified experimental conditions, with the exception of the unlikely case of very large KER values of many tens of eV's.

More specifically in the present case, following the absorption of a photon of energy in the region of the X 2p threshold, the HX⁺* parent molecular ion may either (1) directly dissociate and the resulting X⁺* ion single- or double-Auger decay to X²⁺ or X³⁺, or (2) resonant/Auger decay with subsequent multiple dissociation routes of the HX⁺*/HX²⁺2p⁻¹ mono/dication leading to X²⁺ or X³⁺ fragments§ or (3) decay radiatively (with possible subsequent unimolecular dissociation, e.g. HX⁺* → S + H⁺ + hν′). Therefore, an appropriate summation of the measured cross sections for the total yield of the X⁺, X²⁺ and X³⁺ ions will provide a fair approximation of the photo-absorption cross section of the HX⁺ molecular ion if pure radiative processes are ignored. We present these results for the X²⁺ ions, X³⁺ daughter ions¶ of the HS⁺ and HCl⁺ parent ions in Fig. 1 and 2, respectively. These data represent the total contributions from the multiple core excitation/ionisation/dissociation routes (those leading to H⁻ are not considered) as follows, where ē is a free electron:

	(2)

Similar results are presented in Fig. 3 for the triatomic H₂S⁺ molecular ion. The X⁺ photo-ion yield cannot be obtained in the present work as the 1 amu mass difference between the X⁺ fragment and HX⁺ parent ions is too small to be safely distinguished by our demerger/analysing apparatus. Therefore, the fragmentation routes ending on stable S⁺ or Cl⁺ atomic ions (italicised top line of eqn (2)) are not amenable to experimental measurement. The total approximate photo-absorption cross sections, i.e. the sum of the single and double ionisation channels, are shown in Fig. 8(b), (c) and 11(b) for HS⁺, H₂S⁺ and HCl⁺, respectively.

In other experiments, carried out independently, we have measured the photoionisation cross sections of the S⁺⁴⁰ and Cl⁺⁴² atomic ions. If it is assumed that the 2p photoionisation cross section for the HS⁺ and HCl⁺ molecular hydrides has a strong atomic character at photon energies markedly above threshold, then we may fruitfully compare the two sets of atomic and molecular data. At 190 eV photon energy, the S⁺ continuum cross section is 3.4 Mb⁴⁰ which is 18% larger than the value obtained from Fig. 8(b). In the case of the chlorine species, at the photon energy of 260 eV,⁴² this value is 15%. As these differences are well within the combined error bars of the atomic and molecular measurements, they cannot be uniquely attributed to the contributions to the total molecular cross section of other (than X²⁺ and X³⁺) ionisation/fragmentation channels. For the triatomic H₂S⁺ species, using the same normalisation as for HS⁺, the S²⁺ and S³⁺ cross sections are only 23% and 1% of the atomic S⁺ values, leaving a potential contribution of 76% for the other ionisation/fragmentation channels. This seems compatible with the new physical conditions appearing in the case of the photoionisation of a triatomic molecule, viz. the number of possible fragmentation/ionisation routes following electronic relaxation is significantly increased leading to a larger selection of charge/mass ratios for the fragments.

In Sections 3 and 4, the experimental data for the HS⁺ and HCl⁺ diatomics are analysed in the light of ab initio density functional theory (DFT) and post Hartree–Fock multiconfiguration self-consistent field theoretical calculations of the absorption oscillator strengths in the region of the 2p threshold. As similar calculations are not available for the H₂S⁺ triatomic molecule, the experimental data are discussed qualitatively without the support of theoretical simulations.

3 Theoretical and computational aspects

3.1 Calculation of electronic energies

The open-shell triplet HS⁺ and doublet HCl⁺ molecules belong to the C_∞v point group. Their valence-shell configurations are 3sσ²3pσ²3pπ² and 3sσ²3pσ²3pπ³, respectively. All the calculations in this work were performed with the GAMESS(US) computer package.⁴³ Initial geometries were optimised using density functional theory (DFT) at the B3LYP level, which combines the Becke three-parameter hybrid exchange functional⁴⁴ with the Lee–Yang–Parr gradient-corrected correlation functional⁴⁵ including third-order Douglas–Kroll⁴⁶ scalar relativistic corrections. The augmented correlation-consistent polarised core–valence quintuple-zeta (aug-cc-pCV5Z) basis set^47,48 was employed, with additional diffuse (4s, 4p, 4d) functions added for sulfur and chlorine atoms. The corresponding shapes and energy ordering of the topmost occupied and lowest unoccupied valence molecular (MO) orbitals are shown in Fig. 4 and 5, for HS⁺ and HCl⁺, respectively.

The structural and spectroscopic properties of HS⁺ and HCl⁺ differ significantly from those of their HS and HCl neutral parents. Upon ionisation, both species exhibit slightly elongated bond lengths: 1.3694 Å for HS⁺ and 1.3233 Å for HCl⁺, compared to 1.34–1.36 Å for HS and 1.2746 Å for HCl. This elongation is indicative of a reduced electron density in the bonding region, consistent with the removal of an electron from a bonding or weakly antibonding orbital. Dipole moments increase upon ionisation, from 0.974 D (HS) to 1.2531 D (HS⁺), and from 1.08 D (HCl) to 1.2479 D (HCl⁺), reflecting the enhanced polarisation and charge separation in both the cationic states. The fundamental vibrational frequencies are red-shifted relative to the neutral species: from approximately 2600 cm⁻¹ (HS) to 2517.66 cm⁻¹ (HS⁺), and from 2940 cm⁻¹ (HCl) to 2614 cm⁻¹ (HCl⁺). This decrease corroborates the weakening of the bond strength upon removal of one valence electron. When comparing the two cations, HCl⁺ exhibits a shorter bond length and a larger vibrational frequency than HS⁺. This is consistent with the stronger H–Cl⁺ bond and is likely driven by the higher electronegativity of chlorine relative to sulfur.

3.2 Simulation of absorption cross section spectra

The final-state energies and absorption intensities in the regions of the sulphur and chlorine (2p) L-edge were calculated in the equilibrium geometry of their respective ground state. The intensities (oscillator strengths) calculations were performed taking into account spin–orbit coupling (SOC) effects at the Breit–Pauli level of theory and using a post-Hartree–Fock configuration interaction approach restricted to single-excitations (CI-S), as implemented in the GAMESS code. These effects are intrinsic to the sulfur and chlorine 2p core levels, where spin–orbit splitting leads to energetically distinct 2p_3/2 and 2p_1/2 components, with differences for sulphur and chlorine of ≃1.2 and 1.6 eV, respectively. The inclusion of such coupling is essential for accurately modelling the fine structure observed in core-level spectra. The spin–orbit coupling calculations were performed on variational CI wavefunctions, using optimized Hartree–Fock [HS²⁺]/[HCl²⁺]2s⁻¹ molecular orbitals (MOs) as initial guess orbitals for the SOC-CI calculations. The core-excited reference S/Cl 2s⁻¹ configurations were chosen to avoid the artificial preferential orientation effects that can arise in spin–orbit computations for S(2p)/Cl(2p)_x,y,z⁻¹. Given that the S(2s)/S(2p) and Cl(2s)/Cl(2p) orbitals are each (independently) nearly close in energy, equivalent relaxation effects are expected.

To account for the nuclear motion caused by core-level excitation of the low-lying excited states of interest, numerical gradients were calculated for the core-excited states in the ground state geometries. The intensities of the vibrational progressions were calculated within the Franck–Condon approximation.

The theoretical treatment of H₂S⁺ and more generally that of X₂A molecules, particularly in the context of core-excited or ionized states, is highly nontrivial. The main reasons for this are as follows. First, the molecule exhibits near-degenerate bound and dissociative electronic states upon both valence and core excitations, especially for S 1s⁻¹ and S 2p⁻¹ states. This leads to strong non-adiabatic couplings and conical intersections along the S–H stretching coordinates which complicate the construction of reliable potential energy surfaces and require multi-configurational approaches to capture the interplay between the electronic and nuclear dynamics, see ref. 49 for H₂S and ref. 50 for H₂O. Secondly, experiments have shown that core excitation into the lowest unoccupied orbital induces dissociation faster than Auger decay, a hallmark of ultrafast dynamics. The detailed vibrational structure in the resonant Auger spectra of the S*H fragment has been resolved,⁵¹ indicating the involvement of multiple vibrationally excited levels during dissociation. To accurately reproduce such features requires a precise modelling of the dissociation pathway, including geometry-dependent vibrational state calculations and timing derived from core-hole lifetimes which are essentially used as internal stopclocks.⁵¹ Given these complexities, i.e. non-adiabatic couplings, spin–orbit interactions in the S 2p manifold, ultrafast dissociation and dense vibronic structures, we believe that the extension of our current model to H₂S⁺ largely exceeds the methodological scope of the present work. It is thus deferred to future work as part of a dedicated, molecule-specific, study.

4 Analyses and discussions

4.1 HS⁺ photo-absorption cross sections

4.2 HCl⁺ photo-absorption cross sections

In the following, we shall adopt a pattern similar to Section 4.1 for the order of the discussion of the main features of the HCl⁺ data. Some of the details will therefore not be repeated again in this section and the reader will easily make reference to Section 4.1 for cross-comparisons.

5 Conclusions

Monochromatised SOLEIL synchrotron radiation (hν) was used in a photon-ion merged beam apparatus to measure the absolute cross sections of the HS⁺ + hν → S²⁺, HS⁺ + hν → S³⁺, HCl⁺ + hν → Cl²⁺, HCl⁺ + hν → Cl³⁺, H₂S⁺ + hν → S²⁺ and H₂S⁺ + hν → S³⁺ ionic photofragmentation reactions over the photon energy regions spanning the energies of the 2p ionisation thresholds of sulphur (∼180 eV) and chlorine (∼220 eV). The absorption oscillator strengths (f-values) of the HS⁺ and HCl⁺ 2p core excitations to valence and Rydberg states were computed using extensive ab initio density functional theory (DFT) and post-Hartree–Fock configuration interaction calculations including spin–orbit coupling. Very good agreements were obtained between the HS⁺ and HCl⁺ synthetic photoabsorption cross section spectra generated from the f-values and the summed (S²⁺ + S³⁺) and (Cl²⁺ + Cl³⁺) photofragmentation cross section spectra, respectively. Detailed comparison between the theoretical and experimental results enabled the electronic configuration assignment of some of the core-hole states, notably in the 2p⁻¹3d_σ spectral regions that contain several sharp resonances. Comparison of photoionisation data along the isoelectronic series starting at the united atom, i.e. the Ar⁺, HCl⁺, H₂S⁺ and Cl⁺, HS⁺ series, as well the effects of protonation along the S⁺, HS⁺, H₂S⁺ and Cl⁺, HCl⁺ series, were considered. Common features in all the molecular spectra along these series included (1) an intense and broad 2p → σ* resonance, indicative of strongly dissociating upper states and (2) a marked atomic character for the 2p → 3d_σ resonances.

Conflicts of interest

There are no conflicts to declare.

Data availability

The primary data that support all the findings of this study are available in the published article. The manuscript is deposited in DORAS, the DCU Research Repository, at the following https://doras.dcu.ie/31270. Additional data are available from the corresponding author upon any reasonable request.

Appendix

The absorption of a single photon of energy E in an inner electron subshell will excite the parent molecule (HX⁺ in this work) to an autoionising resonance level. The subsequent decay of the latter by electron emission (Auger process) will usually leave the doubly-ionised molecule in a dissociative final state with excess internal energy ε_int above the asymptotic final potential energy of the fragments ε^H_int and ε^X_int (HX → H + X fragmentation is assumed). This excess energy is retrieved in the form of kinetic energy imparted to the dissociation fragments. The sum of the KE for all the fragments constitutes the kinetic energy release (KER), i.e. ε_int − (ε^H_int + ε^X_int) = E_KER. If the fragments are all electrically charged, the final step of this process – the release of the fragments into the continuum of kinetic energies – has historically been interpreted as a coulombic explosion.³¹ See eqn (2) for the ionisation channels where Coulomb explosion is possible in the present case. The analyses presented below are valid for neutral or charged fragments.

The kinematics of the dissociation fragments is determined solely by the laws of conservation of momentum and energy and similar to the spontaneous disintegration of a heavy particle into fragments of smaller masses.^20,61,62 In the centre of mass (CM) frame of reference,|| the fragments move away from each other with equal and opposite momenta p₀ such that E_KER = p₀²/2μ where μ is the reduced mass of the two fragments 1/μ = 1/m_H + 1/m_X and p₀ = m_Hv^H₀ = m_Xv^X₀ with v^H₀ and v^X₀ the speed of the H and X fragments in the CM system, respectively. If the velocity of the parent molecular ion, i.e. the sample beam, in the laboratory (lab) frame is ^HX_C then the following vector equations hold (see Fig. 14 of ref. 61). The dissociation geometry in the laboratory frame of reference is shown in Fig. 12 together with some of the key symbols used in the analytical treatment of the dissociation.

H − HXC = H0 and X − HXC = X0,	(3)

where ^H and ^X are the velocities of the X and H fragments in the lab frame, respectively. Solving these equations for v^H or v^X ≡ v yields,

	(4)

where V_C ≡ V^HX_C and θ ≡ θ^H or θ^X is the angle at which the X or H dissociation fragment moves away from the direction _C of the parent ion beam.** If V_C > v₀ (in the present case V_C ≫ v₀), the fragment can only move forward at any angle θ such that 0 ≤ θ ≤ θ_max (see Fig. 14 of ref. 61) with sinθ_max = v₀/V_C. The behaviour of θ_max for the fragments (daughters) produced following dissociation of the 1HS⁺ and HCl⁺ parent ions is presented using solid lines in Fig. 13 as a function of E_KER. The range of E_KER values, from a fraction of an eV to several ten's of eV's, is typical for the inner-shell photoionisation processes in hydrides we are considering in the present work, e.g. ref. 20 and 21. Both daughters cannot simultaneously dissociate at their respective θ_max value. If, for argument's sake, the X fragment dissociates at θ^X_max then the ¹H daughter will dissociate at an angle . The value of is then obtained from considerations of the geometry of the dissociation process in the CM frame⁶¹ as follows. The deviation angle θ^X₀ of the heavy fragment in the CM frame corresponding to its maximum deviation angle in the lab frame θ^X_max is such that

cosθX0 = −sinθXmax with θH0 + θX0 = π	(5)

The H deviation angles in the lab frame can now be obtained from

	(6)

We have calculated these angles and their values are represented as broken lines in Fig. 13. For a given pair of daughters at a given value of E_KER, the sum (solid line + broken line) of the angular values gives the maximum (recoil) angle between the directions of motion of the two fragments as they move away from one another. The speed of the H atom in the lab frame can now be obtained by substituting the value of θ^H in eqn (4). We see from Fig. 13 that, as expected from the significant mass difference between the fragments, θ^X_max is only a fraction of a degree while it is several ten's of degrees for the H fragments. From the value of the construction parameters of our apparatus (see Section 2), it is evident that all the heavy fragments will be transmitted through to the mass analysis/detection stage, while this will not be the case for most of the hydrogen fragments.

The dissociation process necessarily includes a change of the kinetic energies of the fragments, away from the initial KE value of the parent molecular ion, that depends on the E_KER value. It is thus a purely molecular effect and in addition to the KE changes that will experience the photoionised ion when travelling from the high-voltage biased interaction region to the field-free region preceding the entry into the magnetic analyser region. We discuss below these changes in the context of our experimental conditions. As we have just shown that the fragments can dissociate at any angle θ such that 0 ≤ θ ≤ θ_max with sinθ_max = v₀/V_C, we calculate the possible values of the fragments KE for the two possible extreme values of θ = 0 and θ = θ_max. This is done readily by substituting these values in eqn (4) and converting to KE values. At θ = 0°, we can have

	(7a)

or

	(7b)

From Fig. 14 and 15, we see that the KE of one of the fragments either increases or decreases when that of the other fragment conversely decreases or increases, as a function of increasing KER. Similar behaviour is observed when the fragments dissociate at their maximum angle θ_max (see solid lines in Fig. 13). In which case we have

	(8a)

or

	(8b)

The positive root in eqn (4) has been kept to obtain eqn (8a) and (8b) to ensure adequate conservation of the total energy, i.e. KE_HX + E_KER = KE_X + KE_H. The values of θ^H and θ^X to use in eqn (8a) and (8b) are obtained from the ¹H and X broken lines on Fig. 13, respectively.

Focusing now on the heavy fragments in Fig. 14 and 15, we see that the effect of KER is to distribute homogeneously (quadratically from eqn (4)) their kinetic energies between the minimum and maximum possible values of and respectively, about the KE_X value of . We take ³²S for a E_KER value of 10 eV†† as an example. Using eqn (7) and (8) and the data from Fig. 13 gives KE_max and KE_min values of (4848 + 75) = 4925 eV and (4848 − 75) = 4773 eV‡‡ in the interaction zone, respectively. Using eqn (1) and assuming single ionisation, KE values of 2925 eV and 2773 eV, after the tagging/interaction zone, would correspond to a magnetic field difference on the exit electro-magnet analyser of ∼23 G or, equivalently, a current difference of ∼0.7 A. We emphasise that this is an ideal, calculated value approximating the overall broadening effect of the KER on the ions final KE value prior to entering the analyser/demerger magnet.

From Fig. 16, we see that the value of 0.7 A is noticeably smaller than the measured resolution parameter of ∼1.5 A ≡ 40 G ≡ 260 eV FWHM of the analysing electro-magnet set for detection of the ³²S²⁺ ions. The measured widths of the curves of Fig. 16 comprise all the -always present- instrumental effects such the fragments refocalisation using ion optics, the fragments emittance at the interaction region exit and the aberrations of the ion optics, for the most part. These independent effects add in quadrature to the KER broadening contribution to produce the final width. Fig. 16 shows this data for the four ionic fragmentation routes, i.e. four different demerger magnet settings, that were used in this work: HS⁺ → S²⁺ and HCl⁺ → Cl²⁺ - the single ionisation/fragmentation channels shown in the top half of Fig. 16- and HS⁺ → S³⁺ and HCl⁺ → Cl³⁺-the double ionisation/fragmentation channels shown in the bottom half of Fig. 16. Thus, we conclude that the bulk of the ion kinetic energies comprised between KE_max and KE_min is in all probability transmitted through to the detection stage for E_KER values of up to a few ten's of eV. For larger E_KER values, this experimental situation would likely become less well verified, resulting in some truncation of the spectrum of transmitted KE_X values. From Fig. 15 and 16, similar conclusions hold for ³⁵Cl.

Acknowledgements

The authors thank the SOLEIL PLEIADES beamline staff John Bozek, Aleksandar Milosavljevic, Christophe Nicolas and Emmanuel Robert for their help and interesting discussions during the experiments.

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Footnotes

† The susbscript C indicates that this is the centre of mass velocity of the molecule prior to dissociation (see Appendix section).

‡ This is meant in reference to similar measurements performed on atomic ion beams.

§ Although higher-order ionisation processes are possible, we limit our discussion to single and double ionisation.

¶ We have found no experimental evidence for the presence of HX2+ or HX3+ at the detection site.

|| The corresponding quantities are indexed with a 0 susbscript.

** More precisely, θ is half the aperture cone angle of the spherical cone of radius v0 centred on the direction C of the parent ion beam.

†† It is not necessary to consider the width of the KER distribution30 in this discussion.

‡‡ Further analytical developments would show that the probability of dissociation with very low recoil angles is small and therefore the range of energies considered in the present example is certainly larger than the actual one. This further supports our conclusions.

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