Francesca Gulminelli

Anthea F Fantina

The equation of state of neutron stars and the role of nuclear experiments

Introduction

Neutron stars (NSs) are unique laboratories to probe matter in extreme conditions, not accessible in terrestrial laboratories. Indeed, with a mass of about 1-2 solar masses and a radius of about 10 km, NSs have densities spanning several orders of magnitude, exceeding in their core the density found inside atomic nuclei. While the properties of the outer layers consisting of a (solid) crust of nuclei embedded in an electron gas, together with a neutron gas in the denser region (possibly on top of a nuclear "pasta" mantle), and those of the outer core made of homogeneous asymmetric nuclear matter, can, to some extent, be constrained by nuclear physics experiments and theory, the nature of the NS inner core formed of strongly interacting matter at very high density, still remains largely unknown [START_REF] Haensel | Neutron Star I: Equation of State and Structure[END_REF].

The remarkable advances of multi-messenger astronomy on different dense-matter astrophysical sources has very recently led to the first quantitative measurements of various properties of NSs such as the mass-radius relation [START_REF] Riley | [END_REF][3][4] and the tidal polarizability [5]. These new observations, as well as the plethora of upcoming results from the LIGO/Virgo collaboration and those expected from the next generation of detectors, raise hopes in the next future to answer fundamental open questions such as the structure and degrees of freedom of baryonic matter in extreme density conditions, the occurrence of phase transitions, and the presence of deconfined matter in the core of NSs. In turn, these new powerful probes of ultra-dense matter have boosted the experimental and theoretical research on nuclear matter, and specifically the study of the nuclear-matter equation of state (EoS), that is, the functional relation P(rB) between pressure and mass-energy density of baryonic matter.

This direct connection between astrophysical measurements and the microphysics of dense matter is due to the fact that, under the unique hypothesis of validity of general relativity, there is a one-to-one correspondence between any static observable of mature and slowly rotating NSs and the underlying EoS. In particular, the Tolman-Oppenheimer-Volkoff (TOV) equations of hydrostatic equilibrium allow to associate the EoS to a unique relation between the mass M and the radius R of the NS [START_REF] Haensel | Neutron Star I: Equation of State and Structure[END_REF]. This means that simultaneous measurements of mass and radius of different NSs would give a direct measure of the pressure of baryonic matter at given densities (more massive stars being associated to higher densities), thus a measurement of the EoS.

Unfortunately, the most precise measurements of NS masses concern binary systems, while radii have been recently inferred from the pulse profile modelling of X-ray data from (isolated) millisecond pulsars, making the observational mass-radius correlation relatively loose.

An especially interesting complementary observable is given by the tidal polarizability L, which measures the deformability of a compact object under the gravitational influence of a second body. The bijective correspondence between the EoS and the variation of the tidal polarizability with the star mass is known since the sixties, and a first measure of this parameter was provided by the LIGO/Virgo collaboration in the celebrated gravitational-wave event GW170817 [5].

Modelling the neutron-star equation of state

The extraction of the dense-matter composition and degrees of freedom is complicated by the fact that there is no ab-initio calculation of dense matter in the nucleonic nor in the hadronic or partonic sectors, and effective models must be used instead (see [6] for a review). In order to extract the nuclear EoS from the gravitational waveforms, agnostic inference is therefore used such as to explore all the possible shapes of the P(rB) functional [7]. The standard method consists in parametrizing the EoS with piecewise polytropes, subject to general physics constraints such as causality, the respect of the unitary limit, and the asymptotic properties of lattice QCD. Many alternative powerful techniques have also been proposed, such as spectral function expansions, parametrized sound speed models, as well as non-parametric inference based on gaussian processes ( [7] and references therein). The domain is evolving fast, and this collective effort is leading to a model-independent prediction of the different NS properties, based on constraints given by the various astrophysical observations. However, this considerable progress is not sufficient to pin down the microscopic structure and properties of the ultra-dense baryonic matter composing the NS. Indeed, whereas a given macroscopic M(R) or L(M) curve is univocally associated to a microscopic P(rB) functional, identical EoSs can be obtained under different hypotheses on the underlying microphysics. Indeed, whatever the effective model employed, the energy density of baryonic matter depends on the different conserved charges of the strong interactions, namely the baryonic, charge, and strange density. On the other hand, the gravitational pressure only imposes the total energy density. The relative proportion of the different densities is determined by weak interactions, which are in equilibrium inside the NS, but the condition of chemical equilibrium explicitly depends on the (model dependent) energy functional.

Therefore, a way to extract information on the structure and properties of dense matter is to use EoS parametrizations that cover the parameter space of effective nuclear models, under the different hypotheses on the effective degrees of freedom at high density. Presently, the tighter constraints are obtained in the case of purely nucleonic models, that are routinely optimised and calibrated over a plethora of experimental nuclear data. Parametrizing the EoS with a functional that can explore the complete parameter space of nucleonic EoSs is known as nucleonic meta-modelling [8]. It consists in introducing a flexible energy functional able to reproduce existing effective nucleonic models as well as interpolate between them. The parameter space being regulated by our present theoretical and experimental knowledge of nuclear physics, this technique allows to predict the astrophysical observables with controlled uncertainties. Moreover, it can be used as a null hypothesis to infer from the astrophysical data the presence of exotic non-nucleonic degrees of freedom. Nucleonic meta-modelling consists in a density or Fermi momentum expansion of the energy per particle of infinite nuclear matter composed of neutrons and protons with respective densities nn and np. In particular, a Taylor expansion around the saturation density n0 allows the separation of the low-order derivatives, which are better determined by nuclear theory and experiments, from the high-order ones, which contain the highest uncertainties [9]. Schematically, we can write, at the order N:

𝑒 ! "𝑛 " , 𝑛 # % = 𝑡 $% + ∑ & '! ! ')* "𝑣 ' +, + 𝑣 ' +-𝛿 . % , "/" ! 0" ! - ' , (1)
where we have singled out the dominant degenerate Fermi gas contribution 𝑡 $% "𝑛 " , 𝑛 # %, and defined the total baryonic density 𝑛 = 𝑛 " + 𝑛 # and the neutron-proton asymmetry 𝛿 = " " /" # " . The isoscalar (isovector) coefficients 𝑣 ' +, (𝑣 ' +-) can be one-to-one connected to the so-called empirical parameters [6,9], given by the successive derivatives of the energy functional at saturation:

𝑋 ' +,(+-) = 3 $ 4 !(&'() * 3" $ / ")" ! ,6)* , (2)
where 𝑒 * ! is the energy of symmetric 𝑛 " = 𝑛 # matter, and the density-dependent symmetry energy is defined as 𝑒 ,78 ! = 𝜕 . 𝑒 ! /𝜕𝛿 ' 3 6)* . For realistic applications, the expansion Eq. ( 1) is slightly complexified, introducing the extra density dependence induced by the effective mass, a low-density correction enforcing the zero-density limit [9], and a specific treatment of the NS crust [10]. This ensures a unified EoS treatment, meaning that the same nuclear model is applied in the different regions of the NS, which is crucial to avoid ad-hoc EoS matching that may introduce unphysical effects in the NS modelling [11].

The low-order empirical parameters (up to k=2) are relatively well constrained by ab-initio nuclear theory and low-energy nuclear experiments. On the theory side, an impressive progress has been made in the chiral perturbation theory of nuclear interactions, with a diagrammatic expansion which allows a reliable estimation of the uncertainties due to the missing terms [12]. On the experimental side, the low-order empirical parameters have been extracted by a systematic comparison of density functional approaches with a large set of high-precision nuclear observables such as masses, skins, electric dipole polarizability, isobaric analog states systematics, flows in heavy-ion collisions, collective modes (see [6,9,13] for a review, and references therein). A (non-exhaustive) compilation of these different constraints is reported in Fig. 1, in what concerns the symmetry energy at saturation 𝑋 * +-= 𝐽 and its slope 𝑋 & +-= 𝐿. The tighter constraints are provided by nuclear theory, but it is interesting to observe that nuclear phenomenology gives independent uncertainty intervals for the empirical parameters which agree with the ab-initio modelling. Among the different parameter evaluations reported in Fig. 1, the only one which is in principle fully model-independent is the one based on the parityviolation electron scattering measurement of the 208 Pb skin by the PREX experiment [14]. This measurement seems to suggest values of J and L considerably higher than the other estimations. However, the systematic errors are so large that no clear conclusion can be drawn at this stage.

Figure 1. Slope of the symmetry energy L versus the symmetry energy coefficient J, as extracted from different experimental and microscopic ab-initio theoretical constraints (see [6] for original references of the constraints).

If chiral effective field theory is today the most powerful constraint on the neutron-matter EoS, the role is inverted concerning the empirical parameters 𝑋 ' +, characterizing symmetric matter. For those parameters, the precision of EoS measurement from the combined analysis of different nuclear experiments is remarkable. This is due to the fact that only nuclei between driplines can be probed in terrestrial experiments, and the properties of extremely neutronrich nuclei are not yet fully explored. The lowest-order parameters, namely the saturation energy 𝑋 * +, ≡ 𝐸 * and the associated saturation density n0, are very precisely measured at the 2% level by nuclear mass and radii measurements, and the compressibility 𝑋 . +, ≡ 𝐾 * is presently extracted at the 10 % level from refined analyses of giant monopole resonance data. All these different constraints can be incorporated in a nucleonic meta-modelling representation of the nuclear EoS such as the one of Eq. ( 1) with the standard tools of Bayesian inference [9,10]. To ensure that models are physically viable up to the extreme densities in the core of NSs, additional requirements are considered, namely the causality of the EoS and its capability of sustaining massive NSs (above 1.97 Msun) like the precisely measured pulsar J0348+0432. The resulting most general EoS compatible with these constraints is represented in Fig. 2. This estimation agrees with the constraint extracted from an agnostic analysis of the gravitational-wave event GW170817 [5], also reported in Fig. 2. This shows a nice convergence between our micro-and macro-physics understanding of compact objects. We can also see that the knowledge on nuclear matter that can be extracted from laboratory experiments and ab-initio nuclear theory is very compelling, and it could be worthwhile to include it in future astrophysical data analyses to improve their predictive power. Some first studies, where the nuclear physics constraints are directly included in the data analyses, start to be proposed [15]. It is also interesting to observe that several popular models currently used to interpret nuclear data fall outside the range of the posterior estimation of the pressure in Fig. 2. This is because most nuclear models have been optimised to reproduce specific properties of nuclear data, notably in nuclear structure, where a very limited range of densities and isospin asymmetries are explored. Therefore, their extrapolation to beta-equilibrated very neutron-rich matter in a large density domain might not be compatible with the combined set of constraints.

Are non-nucleonic degrees of freedom excluded?

The unique hypothesis of the nucleonic meta-modelling is that the relevant degrees of freedom inside the core of a NS are given by protons and neutrons (supplemented by electrons and muons). In this respect, the EoS estimation of Fig. 2 can be considered reliable, since it is reasonable to suppose that baryonic matter should be purely nucleonic up to around twice the saturation density. Despite considerable uncertainty exists about the possible presence of hyperons in the core of NSs, the high mass of the lightest lambda hyperon imposes that the threshold density for its possible appearance be above about 2n0 [16]. Concerning the possible presence of deconfined matter, agnostic modelling of a first-order phase transition including constraints from LIGO/Virgo and NICER data seems to exclude the possibility of a transition below ~2𝑛 * [17]. If we believe our nuclear physics constraints, the only possible scenario allowing a violation of the EoS in Fig. 1 would be the absolute stability of quark matter, which, if true, would lead to the existence of pure quark stars [START_REF] Haensel | Neutron Star I: Equation of State and Structure[END_REF]18]. This shows the huge challenge connected to an increased precision of the measurement of the NS properties at moderate densities from future astrophysical observations.

The situation is different when we analyse observables that imply an integration over the whole density profile of the NS. For the mass-radius relation (Fig. 3, upper panel), the parameter space compatible with nucleonic degrees of freedom is large, because densities as high as ~6 -8 𝑛 * can be reached in the core of the most massive NSs, and the associated EoS is dominated by the high-order empirical parameters 𝑋 ' +,(+-) (with k>2). Those parameters are virtually unconstrained by nuclear physics experiments, which probe densities close to saturation where their influence is too small and low-order parameters dominate. Concerning ab-initio nuclear theory, the power counting associated to the diagrammatic expansion breaks down when the short-distance structure cannot be neglected, thus those calculations, though extremely powerful at low density, cannot be safely extrapolated beyond ~1.5 𝑛 * [12]. These limitations of both nuclear theory and experiments lead to an increased uncertainty in the EoS at high density. Despite these large uncertainties, the nuclear physics prediction of the M(R) relation is tighter than the present astrophysical measurements (see Fig. 3). Combining the whole set of experimental and theoretical filters is definitely important for a reliable prediction of astrophysical observables.

In the case of the M(R) relation, the hypothesis of nucleonic degrees of freedom implicit in the meta-modelling is certainly restrictive. Indeed, even if the exact location of the deconfinement transition towards quark matter is essentially unknown, if strange quark matter is not absolutely stable such a transition is expected at high density [18]. In particular, if this transition is strongly first order, the parameter space of the EoS increases, leading to a much larger spread of the possible M(R) relation ( [19] and references therein). Because of the phenomenological nature of the effective models used for the quark phase, settling the parameter space associated to a phase transition to quark matter is a very hard task. However, the parameter space associated to the absence of such a transition is much more limited (see Fig. 3). This means that the nucleonic meta-modelling can be used as a null hypothesis to set the existence of deconfined matter in the core of NSs, provided that the observations are sufficiently precise to challenge the nucleonic predictions.

The first steps in this direction come from the recent measurement of the tidal polarizability parameter L in compact-binary mergers [5]. The compatibility of the posterior distribution of the L parameter shown in Fig. 3 (lower left panel) with the LIGO/Virgo gravitational-wave measurement at the 90% level indicates that there is no compelling evidence that exotic matter is present in the NS core. However, the shape of the probability distribution predicted by lowenergy nuclear physics constraints is very different from the one inferred from the GW analysis. New astronomical observations will hopefully produce more stringent constraints in the next future. For example, if a NS as massive as 2.5 Msun were measured, this would lead to a clear incompatibility between the L measurement and the purely nucleonic hypothesis, as shown in Fig. 3 (lower right panel), thus providing a more convincing evidence of the presence of deconfined matter in compact stars.

Future constraints from nuclear-physics experiments

In the absence of such astrophysical evidence, an alternative path to challenge the nucleonic hypothesis would consist in putting more stringent constraints on the nucleonic metamodelling prediction from improved laboratory measurements. A reduced error bar in the neutron skin measurement such as that performed by the PREX collaboration [14] would greatly constrain the low-order empirical parameters in the isospin sector, thus allowing better extrapolations to higher density, and reduced uncertainties in the astrophysical predictions. The same is true for measurements from relativistic heavy-ion collisions, which are a unique experimental probe of densities of the order of 2n0. Only symmetric matter is probed in such experiments, but even a loose constraint on the higher-order parameter 𝑋 0 +, ≡ 𝑄 * would greatly reduce the present uncertainty in the nucleonic extrapolation of the high-density EoS. Encouraging results in this sense have been recently presented by the HADES collaboration [20].

To conclude, even if the composition, properties and degrees of freedom of ultra-dense matter such as can be found in the core of NSs is still largely unknown, we are presently facing an extremely exciting and rapidly evolving scientific epoch. For the first time, terrestrial nuclear experiments can be connected to astronomical observations, and it makes no doubt that, together with the amazing recent progress of ab-initio theoretical modelling, they will further contribute in a next future to unveil the remaining mysteries in the properties of matter under extreme conditions.

Figure 2 .

 2 Figure 2. Marginalized posterior for the pressure P versus baryon mass-energy density rB. The blue dark (light) shaded region shows the 1s (2s) confidence intervals. Some popular nucleonic models are represented as lines. The vertical black bar corresponds to the constraint inferred from GW170817 in [5]. Figure adapted from [10].

Figure 3 .

 3 Figure 3. Upper part: marginalized posterior for the NS mass-radius relation. The blue shaded regions show the 68%, 95% and 99% confidence intervals. Lower part: Marginalized posterior for the tidal polarizability parameter of the two components of GW170817. Dark (light) shaded regions represent the 50% (90%) confidence interval with the Bayesian estimation; black dashed lines show the confidence regions as obtained in [5]. The left and right panels correspond to different constraints on the maximal NS mass. The results for the same popular EoS models as in Fig.2 are also represented. Figures adapted from [10].


Sam Bailey

Tzany Kokalova

Martin Freer

Carl Wheldon

Robin Smith

Joseph Walshe

Neven Soić

Lovro Prepolec

Vedrana Tokić

Francisco Miguel Marqués

Lynda Achouri

Franck Delaunay

Marian Parlog

Quentin Deshayes

Beatriz Fernández-Dominguez

Bertrand Jacquot

David Blaschke

The identification of α-clustered doorway states in 44,48,52 Ti using machine learning

A novel experimental analysis method has been developed, making use of the continuous wavelet transform and machine learning to rapidly identify α-clustering in nuclei in regions of high nuclear state density. This technique was applied to resonant scattering measurements of the 4 He( 40,44,48 Ca,α) resonant reactions, allowing the αcluster structure of 44,48,52 Ti to be investigated. Fragmented α-clustering was identified in 44 Ti and 52 Ti, while the results for 48 Ti were less conclusive, but suggest no such clustering.

Introduction

Experimental studies of physical systems are often concerned with answering simple questions: Does the Higgs boson exist? Can we observe gravitational waves? Ideal experiments are designed whereby the results depend on the answer to these questions, and so by making such measurements these answers can be inferred. It is, however, often also the case that these fundamental properties are just one of many complex and independent parameters that affect the experimental data. The other parameters could be anything from other fundamental physical constants, which are perhaps unknown or known to poor precision, to experimental effects such as the detector resolution and efficiency. Therefore, in order to answer the 'interesting' questions, one must first answer many 'uninteresting' questions about the meaa e-mail: samcbailey90@gmail.com b e-mail: t.kokalova@bham.ac.uk (corresponding author) surements, and in fact often it is these uninteresting questions which dominate the efforts of researchers in their fields.

In this article we present a novel technique which uses machine learning [START_REF] Duda | Pattern Classification[END_REF] to bypass the difficult and uninteresting parts of the analysis, and address the fundamental questions directly. Machine learning refers to a set of numerical algorithms which allow computers to learn patterns and make predictions without encoding those patterns explicitly. These techniques have exceptional analytical potential, and have been used to great effect in a plethora of fields, for example to perform image analysis and facial recognition [START_REF] Jia | 29th Chinese Control And Decision Conference (CCDC)[END_REF], to understand the sentiment of a paragraph of text [START_REF] Tang | [END_REF], to automatically identify interesting events in high energy physics experiments, such as the LHC [4], and to automatically distinguish between true gravitational wave signatures and those produced by non-astrophysical noise in LIGO data [5].

Here the fundamental question we wish to address is: given an experimental energy spectrum produced by the resonant scattering of a nucleus with 4 He, is α-clustering observed in the structure of the compound nucleus formed in this reaction? Alpha-clustering is the phenomenon whereby protons and neutrons form sub-structures within the nucleus, and it can usually be ascribed to specific nuclear energy levels, known as α-clustered states. This has been shown to play a pivotal role in dictating the properties and interactions of light nuclei [6,7], yet it has not been observed to the same extent in heavy nuclei. It is tempting, therefore, to suggest that systems which contain few nucleons are more likely to form cluster structures than those composed of many nucleons, and efforts to understand this trend 123 have led to considerable experimental and theoretical work investigating α-clustering in medium mass nuclei, some of which is detailed in Ref. [START_REF] Ohkubo | Alpha-clustering and molecular structure of mediumweight and heavy nuclei[END_REF]. It is unclear, however, whether the reduction of experimentally observed α-cluster structures in heavy nuclei truly reflects a shift in structural preference away from α-clustering, or whether experimental difficulties which arise with increasing nuclear mass have concealed the cluster structures in this region.

One experimental difficulty which is unique to heavier systems is the increasing nuclear level density. This leads to more complex experimental spectra, and also means that α-clustered states often serve as doorway states [9] in the α decay-channel, and as such, rather than searching for a single α-clustered state, one must instead search for groups of fragmented states all sharing the strength of the original clustered state. Usually the analysis of these experimental spectra requires the extraction of the properties of all of the energy levels which are populated in the reaction, and then the energy levels are compared with a theoretical nuclear model in order to ascertain whether or not they exhibit signs of α-clustering. However, the significant increase in the complexity of the spectra means that unambiguously extracting all of the energy levels is a very challenging prospect, and is often the primary obstacle when analysing experimental data in this mass region.

In this scenario, the uninteresting properties are the energy levels, which are difficult to extract and the majority of which will correspond to non-clustered structures. So rather than attempting to extract the states, in this article a technique is developed which simulates many spectra, each time assuming a unique and random combination of nuclear states, but in each case controlling for whether or not an α-clustered structure is present. Machine learning is then employed to learn the differences between spectra which do or do not contain α-clustered states, independent of the properties of the other states in the spectrum. This algorithm can then be applied to the measured data to ascertain the existence of α-clustering.

In this article this technique is employed to examine the evolution of α-clustering in titanium isotopes. Previous work on 44 Ti has identified a range of α-clustered states, many of which have been shown to be fragmented [10][11][12]. These observations are in agreement with predictions made by αcluster model calculations [8, ch. 2] and a deformed basis Antisymmetrised Molecular Dynamics calculation [9], indicating a good understanding of the underlying α-cluster structure. There has, however, been comparatively little work done to investigate similar structures in neutron rich titanium isotopes. Analyses of α-transfer reactions have indicated that the degree of α-clustering in titanium isotopes decreases with increasing nuclear mass, both in the ground state [13,14] and in excited states [11], however, a measurement of 48 Ca(α,α) elastic scattering shows significant resonant structure [15]. This may be indicative of α-clustered states in 52 Ti above the α-decay threshold, however no formal analysis was performed on this measurement.

The present work investigates 44,48,52 Ti by measuring the resonant scattering reactions, 4 He( 40,44,48 Ca,α). This allows the degree of α-clustering above the α-decay threshold to be compared consistently between the three isotopes, and is an ideal testing ground for a novel machine learning technique as 44 Ti can be used to test the reliability of the procedure, as it is already well understood, before the technique is applied to the neutron rich isotopes.

Experimental measurements and results

The 4 He( 40,44,48 Ca,α) measurements were made using the Thick Target Inverse Kinematics (TTIK) technique [16]. The reaction chamber was filled with 4 He gas, which acted firstly as a medium to smoothly decrease the energy of the calcium ions as they travel through the chamber via electronic interactions, and secondly as the target for the desired nuclear reactions. This allows a measurement to be made of the entire excitation spectrum without changing the beam energy. The scattered α-particles were measured using two 1mm thick Double-sided Silicon Strip Detectors (DSSDs), placed at the opposite end of the reaction chamber to the beam entrance in the E-E configuration. This ensured that the measurements consisted purely of α-particles and allowed the measurements to be made at a scattering angle of 180 • in the centre-of-mass frame. The measured spectra are shown in Fig. 1, and more details on the experimental work can be found in Ref. [17,[START_REF] Bailey | New Analytical Techniques for the Investigation of Alpha Clustering in Nuclei[END_REF].

A crucial aspect of the TTIK technique is that the measured spectra are in fact a convolution of the true excitation function with the experimental resolution. This serves to reduce the height of any resonances which are much narrower than the experimental resolution. This behaviour can severely hinder the analysis of TTIK spectra if the experimental resolution is poor, however, if it is small enough such that it only impacts states which are too narrow to be considered α-cluster candidates, and it does not cause neighbouring states to become indistinguishable, then it can be considered a useful property as its only effect will be to remove nonclustered states from the spectra. In the present work, REX [START_REF] Curtis | [END_REF], a Monte-Carlo simulation of thick target resonant scattering experiments, was used to calculate the experimental resolution as 45 keV at Full Width Half Maximum.

Alpha clustered doorway states model

The cross-section, dσ/d , of the resonant reactions measured in this work can be calculated directly from the energy levels in the compound nucleus using R-matrix theory [20].

Fig. 1 Measurements of the differential cross-section, made at a centre-of-mass scattering angle of 180 • , of the 4 He( 40 Ca,α) (top), 4 He( 44 Ca,α) (middle) and 4 He( 48 Ca,α) (bottom) reactions. The resonances in these measurements are relevant to the structure of 44 Ti, 48 Ti and 52 Ti respectively It is, therefore, possible to simulate dσ/d by first generating a set of 'non-clustered' energy levels, and then optionally coupling these levels to an α-clustered doorway state. The simulated spectra are generated from the energy levels using the Simplified R-Matrix [21], and classified as either non-clustered (no α-clustered doorway states), or clustered (one α-clustered doorway state). Many clustered and nonclustered spectra were generated, each time with a unique and random set of energy levels.

The Simplified R-Matrix calculates dσ/d for reactions where all initial and final state nuclei are spin-0. The cross-section is calculated as a function of excitation energy, E x , and centre-of-mass scattering angle, θ , from the excitation energies, E λ , orbital angular momenta, L λ , partial decay widths, λμ , and total decay widths, λ , of the energy levels, where the energy levels are indexed by λ and the decay channels are indexed by μ. This is written explicitly as

dσ d = δ μμ f b (E x , θ) - i 2k μ λ (2L λ + 1) × λμ λ (e 2iβ λ (E x ) -1)e 2iφ L λ P L λ (cos θ) 2 , ( 1
)
where

β λ (E x ) = arctan λ /2 E λ -E x (2)
and f b is the background amplitude,

k μ = 2m μ E c.m. /h, E c.m.
is the centre-of-mass energy of the system, m μ is the reduced mass, P L λ is a Legendre polynomial of order L λ and φ L λ is the partial wave phase shift. The partial wave phase shifts exist only in the simplified version of the R-matrix to account for the behaviour of the interference between the resonances and the background amplitude. In this work they were randomised between 0 and π to account for all possible types of interference.

In practice the cross-section is not measured as a continuous quantity, and instead is measured in a finite number of excitation energy bins. In order to ensure that the simulations match the experimental data the cross section was calculated discretely for each experimental bin, dσ/d n , where E x n and θ n are the excitation energy and scattering angle of the bin respectively. Additionally, the background amplitude was defined by fitting a smoothing spline to the experimental spectra which approximated the background, and sampling this at E x n . Finally the simulated cross-section was convoluted with the experimental resolution, and noise was added based on the experimental signal to noise ratio, in order to make the simulations as directly comparable to the measured spectra as possible.

The non-clustered energy levels were simulated by generating a set of shell-model like energy levels, known as class-I energy levels and indexed by λ I , characterised by ensuring that the levels adhere to the appropriate statistical distributions (described below) indicative of the shell model.

The partial widths, λ I μ , for each decay channel μ were constructed to follow Porter-Thomas statistics [22] by Gaussianly distributing the reduced widths, γ λ I μ , with a mean of 0 and variance given by γ 2 μ . The partial widths are calculated from the reduced widths using where P μL λI is the penetrability through the combined Coulomb and centrifugal barrier, and L λ I is the orbital angular momentum in channel μ. The penetrability was calculated from the regular and irregular Coulomb wavefunctions [START_REF] Thompson | Nuclear Reactions for Astrophysics[END_REF].

The values of γ 2 μ dictate the average strength of each decay channel. In these simulations they were chosen by defining the mean square ratio to the Wigner limit for single particle decays, θ 2 sp , and the ratio to the single particle strength for each decay channel, R μ/sp . The Wigner limit, γ 2 μw , is a theoretical upper bound on the reduced width. Written formally, this gives

γ 2 μ = R μ/sp θ 2 μ γ 2 μw . ( 4
)
For all of the spectra in this work the only open decay channels are the proton, neutron and α channels. Since the proton and neutron decays are both decays to single particles, R p/sp , R n/sp ∼ 1, however, one would expect average αdecay strength to be weaker than the proton and neutron strengths for purely shell-model type states as the α-particle is a more complex particle, and so R α/sp < 1.

The excitation energies, E λ I , and spins and parities, J π , were generated such that the nearest neighbour state spacings of states with the same J π followed the Wigner distribution [START_REF] Wigner | [END_REF], defined as

P w (D J π ) = π D J π 2 D J π 2 exp - π D 2 J π 4 D J π 2 , ( 5
)
where D J π is the mean nearest neighbour state spacing for states with the same J π , and is calculated from the overall mean state spacing, D , using the Gaussian cutoff factor from the Fermi-gas model [START_REF] Thompson | Nuclear Reactions for Astrophysics[END_REF],

D J π = D 2J + 1 2σ 2 spc 1 2πσ 2 spc exp - J (J + 1) 2σ 2 spc , ( 6
)
where the spin cutoff factor σ spc is defined by assuming that the nucleus is a rigid rotating sphere. The clustered spectra were generated by coupling an αclustered doorway state, known as a class-II state, to the set of class-I states, to produce a set of compound states, indexed by λ. The class-II state was assumed to exist in a highly deformed secondary minimum in the deformation potential energy surface, and was characterised as being α-clustered by a large ratio to the Wigner limit in the α-channel, θ 2 II,α , and zero decay widths in all other channels. Its spin and parity, J π II , were randomised, and its excitation energy, E II , was randomised uniformly within the measured energy range.

The coupling between the class-I and class-II states was based on the work by Bjørnholm and Lynn [25] for the treatment of fission isomers. The compound states were generated by solving the eigenvalue equation

E I H c H T c E II ⎡ ⎣ C (I) λ C (II) λ ⎤ ⎦ = E λ ⎡ ⎣ C (I) λ C (II) λ ⎤ ⎦ , ( 7
)
where E I is a diagonal matrix containing E λ I , E λ is the excitation energy of the compound state and C (I) λ and C

(II) λ are the coefficients which produce the compound state from the class-I and class-II states. The matrix H c is a 1 × N I matrix, where N I is the total number of class-I states. The elements of H c are 0 for class-I states which have a different J π to the class-II state, and otherwise are taken from a normal distribution, centred on 0 with a variance given by H 2 c . This ensures that the class-II state only couples to class-I states of the same J π , and the use of a normal distribution is justified in Ref. [25] to account for the random behaviour of the overlap between the class-I and class-II state wavefunctions. The value of H 2 c defines the strength of the coupling, and, therefore, the number of class-I states which will couple significantly to the doorway state, known as the fragmented states. However, the number of fragmented states depends also on the state spacing of the class-I states. Therefore, N c is defined for each clustered spectrum, which is directly proportional to the expected number of fragmented states, and from this H 2 c is defined as

H 2 c = N c D J π II 2 π . ( 8
)
The reduced width amplitudes of the compound states are calculated from C (I) λ and C

(II) λ as

γ λμ = λ I C (I) λ λ I γ λ I μ + C (II) λ γ II ,μ . ( 9
)
An ensemble of spectra, containing an equal number of clustered and non-clustered spectra, were generated using this model. The input parameters, D , θ 2 sp , R α/sp for both types of spectra and additionally θ 2 II,α , N c , E II , J π II for the clustered spectra, were randomised within sensible ranges to ensure that all reasonable scenarios were accounted for. Choosing the ranges for each of these parameters is akin to choosing a prior distribution in Bayesian statistics. The ranges used and their justifications are given in Table 1, and an example of the clustered and non-clustered spectra produced are shown in Fig. 2. This spectrum ensemble was used as 'training data' to train a Random Forest Classifier (RFC) to classify spectra as either clustered or not clustered, where each spectrum is characterised by a set of 'features' calculated from dσ/d n . More details on the RFC are given in Sect. 4.

The features used were calculated from dσ/d n using a combination of the Continuous Wavelet Transform (CWT) [START_REF] Van Den | Wavelets in Physics[END_REF] and a Principle Component Analysis (PCA) [START_REF] Abdi | [END_REF]. It 123

R α/sp <20%
Chosen to ensure that the α-channel was significantly reduced compared with the proton and neutron channels for the shell model type states E II -Chosen to be within the experimentally measured energy range for each measurement J π II ≤ 7 - All higher spins have a negligible contribution to the measured spectra due to the large centrifugal barrier. Furthermore only natural parity states were allowed since the entrance channels for all of the measurements were composed of spin-0 nuclei

N c 2
Chosen empirically to ensure that the doorway state couples to more than one class-I state, but remains suitably localised-not coupling to all states in the spectrum. This value was not randomised since the randomisation of the state spacings and coupling matrix elements was already sufficient to produce a variety of state fragmentations

θ 2 II,α 0.2-1
Chosen to ensure a large alpha component for the class-II state, indicative of an α-clustered structure Fig. 2 An example of a simulated clustered spectrum (top) and nonclustered spectrum (bottom). The position of the doorway state is indicated by the dotted line. These spectra were calculated for 44 Ti, using the parameters:

D = 0.05 MeV, θ 2 sp = 0.035, R α/sp = 5%, N c = 2, E II = 12 MeV, J π II = 3 -and θ 2 II,α = 0.2.
Clustering likelihood: 4% vs 89% was shown in Refs. [17,[START_REF] Bailey | New Analytical Techniques for the Investigation of Alpha Clustering in Nuclei[END_REF] that the CWT is an effective tool for the identification of α-clustered doorway states from TTIK measurements. The CWT calculates wavelet coefficients, W ,nm , from dσ/d n by folding it with an appropriately chosen wavelet, (E). The wavelet is scaled by δE m , known as the scale parameter, which allows features in the spectrum to be expanded as a function of scale. The wavelet coefficients are calculated as

W ,nm = 1 √ δE m ∞ -∞ dσ d ( ) -E x n δE m d , ( 10
)
where is a dummy variable used to facilitate the integration, and in practice the integral was calculated numerically using the trapezoidal rule. In this work the complex Morelet wavelet [START_REF] Van Den | Wavelets in Physics[END_REF], which can be thought of as a windowed Fourier transform, was used. This is defined formally as

(E) = d √ π 1/2 exp (-i2π E) exp - E 2d 2 , ( 11
)
where d defines the size of the window, and in this work d = 0.8 MeV. In this case δE m is the equivalent of the period in a typical Fourier transform, and W ,nm is similar to a Fourier transform coefficient, but localised at E x n . In this work 70 values of δE m were used, uniformly spaced between 0 and 1 MeV. The CWTs of the 4 He( 40,44,48 Ca,α) spectra are shown in Fig. 3. In this work the magnitude of the wavelet coefficients, W ,nm , are used and the phases are discarded, as it was observed that the phases contained little useful information regarding the α-clustered nature of the spectrum. It would, however, be inefficient to use W ,nm directly in the RFC as they are not orthogonal, with large correlations between neighbouring values of W ,nm , and a large number of coefficients are required to adequately characterise a spectrum, which leads to an unnecessarily computationally intensive analysis. Instead a PCA is performed on W ,nm as a form of dimensionality reduction. This constructs a new set of orthogonal features from W ,nm , chosen to ensure that the largest fraction of the variance in the original feature set is retained in the fewest possible features. In this case 300 PCA features were used, which accounted for 99.3% of the variance in the W ,nm feature set. More details on the PCA algorithm can be found in Ref. [START_REF] Abdi | [END_REF]. The PCA algorithm is very sensitive to the initial distributions of the features, and 123 Fig. 3 The CWTs of the measured spectra, 4 He( 40 Ca,α) (top), 4 He( 44 Ca,α) (middle) and 4 He( 48 Ca,α) (bottom). In each case the heatmap shows the magnitude of W ,nm as a function of δ E m and E x n works optimally when these are approximately normally distributed and normalised to a mean of 0 and a variance of 1. In order to accomplish this, the logarithm was taken of W ,nm , and the logged values were independently normalised to have a mean of zero and unit variance across the training data. The PCA was then performed on these normalised log wavelet coefficients.

The result of this process is a set of PCA features, PCA k , which each correspond to a certain W ,nm distribution. Some examples of these distributions are shown in Fig. 4 for k = 0, 1, 2, 20, and an example of the stages of producing the PCA variables from a raw spectrum are shown in Fig. 5. The ensemble of PCA k for all of the simulated spec- The consequence of using PCA features as opposed to directly using the W ,nm features is that they much more naturally describe the overall properties of the spectrum than they do the properties of individual resonances within the spectrum. For example it is evident from Fig. 4 that PCA 0 represents the average amplitude of the resonances throughout the spectrum, relative to the amplitude of the noise in the spectrum, and PCA 1 represents whether or not the average resonant amplitude increases or decreases throughout the spectrum. The higher order PCA variables then begin to account for the shapes of the resonances, the spacings between the resonances and the widths of the resonances, however, these properties are all merged by the PCA algorithm, obscuring the properties of individual resonances. While this may lead to a reduction in the sensitivity of this algorithm to the more subtle effects of α-clustering on the spectra, the dominant effects ought to still be captured by the PCA features.

Machine learning

A RFC [START_REF] Boulesteix | Wiley Interdiscip[END_REF] is an ensemble machine learning method, which combines many randomised decision trees to produce a more robust and sophisticated classification than is possible using a single decision tree. Each tree is randomised by training it on a random subset of the training data, and at each node in Fig. 5 The 4 stages of feature engineering used to prepare the spectra for the RFC. Left to right: the raw spectrum, the CWT of the spectrum, the normalised logarithm of the CWT, the PCA of the normalised CWT.

For each stage of processing the top plot shows an example spectrum, and the bottom plot shows the distribution of some randomly chosen features across the entire training data set the tree the optimal splitting criterion is chosen from a subset of the available features.

The RFC classifies a spectrum by allowing the individual decision trees to perform the classification independently, and then averaging the results. This method produces a pseudo-likelihood that the spectrum is clustered, L c , which is calculated as the fraction of the decision trees which predict that the spectrum is clustered. It is possible to calibrate the pseudo-likelihood to give the true likelihood that the spectrum is clustered, L c . This calibration was performed by calculating L c for every spectrum in the training data via fivefold cross-validation [31], which splits the training data into 5 segments and then trains the RFC on 4 of those, before using it to calculate L c for the spectra in the 5th segment. This process is repeated, leaving out each of the segments one at a time, until L c has been calculated for every spectrum in the training data. All of the clustered and non-clustered spectra were then binned separately as a function of L c , producing two histograms, N c n and N nc n respectively, with bin centroids at L c,n . The true clustering likelihood was then calculated from these histograms as the fraction of the spectra in each bin that are clustered, given formally as

L c,n = N c n N c n + N nc n . ( 12
)
Finally a logistic function was fit to L c,n as a function of L c,n , producing the continuous function L c (L c ), under the constraints that L c (0) = 0 and L c (1) = 1. This function was then used to convert between L c and L c , an example of which is shown in Fig. 6. Five-fold cross-validation was also used to tune the RFC hyper-parameters by calculating the percentage of the cross-validated classifications which were correct, known as the classification accuracy. The hyper-parameters that were tuned were the total number of decision trees which compose the RFC, and the minimum number of events which may be contained within a single node of a decision tree. The optimal values chosen were 1000 decision trees and 75 events respectively. While traditional RFCs use fully grown Fig. 7 The sensitivity of the RFC as a function of the number of class-II states in the spectrum decision trees, rather than limiting them by defining a minimum number of events per node, it was found in this work that fully grown trees sometimes overfit to the training data, producing unreliable results.

In addition to the classification accuracy, two other quantities were used to assess the quality of the RFC, the fraction of the clustered spectra which were classified correctly (sensitivity) and the fraction of the non-clustered spectra which were classified correctly (specificity). These are often also referred to as the True Positive Rate (TPR) and True Negative Rate (TNR) respectively. These are used, in addition to the accuracy, to probe the behaviour of the RFC in the following section.

Results

Three RFCs were produced, one each for 44 Ti, 48 Ti and 52 Ti, with cross-validated classification accuracies of 76%, 77% and 79% respectively, sensitivities of 74%, 78% and 81% respectively and specificities of 77%, 76% and 79% respectively. It is interesting to observe the dependence of the sensitivity of the RFCs on some key simulation parameters, as the sensitivity can be treated as a measure of how easy it is to observe an α-clustered doorway state. The three RFCs all behaved similarly, so one can assume that the conclusions drawn here are applicable to all three measurements, and only the results for 44 Ti are presented.

Firstly it was important to ascertain that the RFCs were capable of identifying α-clustered states in spectra containing more than one, despite being trained only on spectra with a single α-clustered state. The sensitivity of the RFC was plotted as a function of the number of class-II states in the spectra in Fig. 7, demonstrating that the sensitivity increases with the number of class-II states in the spectrum. This is to be expected for a sensible RFC since if there are many class-II states present it becomes less likely that the RFC will miss all of them. Fig. 8 The sensitivity of the RFC as a function of θ 2 II,α , fit with a Gaussian process using a Matern kernel (line). The shaded region indicates a 1σ confidence interval

The sensitivity was calculated as a function of θ 2 II,α by binning the training data uniformly into 40 θ 2 II,α bins and calculating the sensitivity independently for each bin. These values were then smoothly interpolated using a Gaussian process with a Matern kernel, which assumes that the data points ought to be correlated highly with those close in θ 2 II,α , and uses the magnitude of the errors on the data points to infer the smoothness of the interpolation and the size of the confidence interval. The data and the Gaussian process fit are shown in Fig. 8. Below θ 2 II,α ∼ 0.25 the sensitivity decreases, while above it plateaus. This indicates that if the α-clustered doorway state one is attempting to observe has a large ratio to the Wigner limit in the α-channel, above 0.25, it is much easier to observe than if one attempts to observe a similar state with a smaller θ 2 II,α . This is a sensible result, as states with small α-widths will look similar to class-I states, and, therefore, be more difficult to identify.

Finally the sensitivity was calculated for each J π II , as a function of E II . This is plotted in Fig. 9, and shows that at high energies, low-spin doorway states are difficult to observe, and conversely at low energies high-spin doorway states are difficult to observe. This is because the resonant amplitude is proportional to (2J + 1) 2 , which amplifies high-spin states, however, the increased centrifugal barrier for high spin states dramatically decreases their penetrability factor and, therefore, their decay widths. Therefore, at low energies, where the barrier penetrability is especially dominant, the high spin states are difficult to populate, whereas at high energies they are populated and their increased amplitude dominates the spectrum, obscuring the low-spin resonances.

Upon their application to the experimentally measured data, the RFCs predicted clustering likelihoods of 92%, 41% and 83% respectively, indicating that it is very likely that 44 Ti and 52 Ti contain at least one α-clustered doorway state and unlikely that 48 Ti does. This is consistent with previous observations of α-clustered doorway states in 44 Ti [10][11][12], as well as with a previous analysis of these data, which iden-123 Fig. 9 The sensitivity of the RFC as a function of the excitation energy of the class-II state, for each J π of the class-II state (data points with error bars). The values are fit smoothly using a Gaussian process with a Matern kernel (solid line), and a 1σ confidence interval is shown (shaded region). The Gaussian process fits are compared in the bottom-right plot tified doorway states in 44 Ti and 52 Ti but not in 48 Ti by examining the characteristic CWT scales of these measurements [17,[START_REF] Bailey | New Analytical Techniques for the Investigation of Alpha Clustering in Nuclei[END_REF].

Next, the sensitivity of these results to the ranges used to produce the ensemble of training spectra was investigated. The upper and lower limits of D and θ 2 sp , the lower limit of θ 2 II,α , the upper limit of R α/sp and the value of N c were all varied, and new training ensembles were generated, to which new RFCs were fit and clustering likelihoods were recalculated for each isotope. The clustering likelihoods are plotted as a function of the parameter limits in Fig. 10. Firstly, while L c is almost completely insensitive to the choice of limits on D , it does exhibit a dependence on the other parameter limits, to varying degrees of severity. The clustering likelihood decreases slightly for all isotopes as both θ 2 sp limits increase. This is because as these limits increase, the average widths of the non-clustered resonances increases, reducing the difference between clustered and nonclustered spectra. The clustering likelihood also decreases for all isotopes as the lower limit on θ 2 II,α increases. Increasing this limit effectively increases the threshold at which a state is considered α-clustered, and consequently the clustering likelihood ought to naturally decrease as this increases and the criteria for α-clustering gets harsher. It is also the case that the clustering likelihoods increase for low values of R α/sp . This is because the value of R α/sp dictates the average size of the non-clustered resonances. If the simulated resonances in the non-clustered spectra are all very small, then any resonances in the measured spectra will produce a large clustering likelihood. Finally, it can be seen from the clustering likelihoods as a function of N c that while fragmented states are observed in 44 Ti and 52 Ti, if one looks for non-fragmented α-clustered states instead (i.e. small values of N c ), then the clustering likelihood falls below 0.5 for all three isotopes, indicating none are observed. This is consistent with the expectation that if α-clustered states exist in this mass region, they ought to behave as doorway states. Overall however, while there are some small variations in L c for extreme values of the parameters, the fundamental results that 44 Ti and 52 Ti contain α-clustered doorway states, while 48 Ti does not, are preserved, indicating a robust analysis.

It is possible to calculate the relative importances of each PCA parameter, which indicates which parameter has the most influence over the resulting classification. This is calculated by evaluating the average 'height' of each parameter in the decision trees, and assuming that the most important parameters are those that are used earlier (or higher). These importances are plotted in the lower panel in Fig. 11. It is clear that the importance is highest for the lowest order PCA variables, suggesting that it's the overall group properties which contribute most significantly to the classification, for example the average resonance amplitudes, and the higher order terms are not as important. This demonstrates that the RFC is predicting the existence of α-clustered doorway states by examining the average resonant amplitude observed in the spectra, and how the resonant amplitude varies as a function of excitation energy.

It is also possible to calculate the contribution each PCA feature makes to L c , L c,k , such that L c = 0.5+ k L c,k . For example, a negative contribution for a given parameter means 123 Fig. 10 The clustering likelihood for 44 Ti (red), 48 Ti (green) and 52 Ti (blue), as a function of the limits used for the training data. In each case one limit is varied, and the others are held constant at their default values given in Table 1. In each plot the horizontal black dashed line indicates L c = 0.5, and the vertical black dotted line indicates the default parameter value that parameter represents a swing towards not clustered, and a positive contribution represents a swing towards clustered. These clustering likelihood contributions are plotted for each nucleus in the bottom panel of Fig. 11. These values can be used to assess exactly how the RFCs made the classification decisions for 44,48,52 Ti. In all three cases PCA 0 contributes negatively, indicating that alone the average amplitude of the resonances is not large enough to demonstrate the existence of an α-clustered doorway state. However, in the cases of 44 Ti and 52 Ti PCA 1 makes a very large positive contribution to L c . It is clear from looking at the spectra that both of these nuclei have large resonances at low excitation energies, and so it seems reasonable to conclude that the existence of large resonances at low excitation energies is indicative of αclustered doorway states in 44 Ti and 52 Ti. Note this work has used a binomial classification system, where the result must be one of two results (clustered or not) which could introduce a systematic bias. In future work it could be generalised to a multinomial classification problem, where predictions are attempted if the data are (A) shell model, (B) alpha clustered, (C) alpha clustered and coupled to shell model, (D) …etc., with a different class for each nuclear structure or model to

Discussion

To summarise, by training an RFC to evaluate the differences between spectra simulated either with or without αclustered states, α-clustering has been identified in 44 Ti and 52 Ti. The results for 48 Ti are less conclusive, but tentatively suggest that α-clustering is not present in this energy region. If one searches for a single α-clustered state in the spectra, rather than sets of fragmented α-clustered states indicative of a doorway state, then none of the measurements return a positive result, indicating that the α-clustered structures observed in 44 Ti and 52 Ti act as doorway states. This suggests that the doubly-magic nature of 40 Ca and 48 Ca is particularly important for the existence of α-clustered states.

The use of machine learning here has allowed these conclusions to be drawn without requiring the extraction of the individual spins, parities, energies and widths of the nuclear energy levels. This is very powerful, as it is likely that those parameters could not be robustly extracted from the current measurements alone, yet using this technique it was still possible to quantitatively answer the crucial, fundamental questions of α-clustering in this mass region.

It is important to note that the combination of the PCA and the RFC here constituted quite a 'blunt' machine learning algorithm, since it effectively focused only on the average resonant amplitude of the measurements and ignored the more subtle features such as the state spacing and the resonance shapes. It may be possible to improve upon the results shown here by employing a more sophisticated machine learning technique, such as convolutional neural networks, which have been used with great success for image analysis in other fields [32].

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Fig. 4

 4 Fig. 4 Examples of some PCA variables for 44 Ti. A heatmap colour scale is used to highlight the shape of the PCA components. See text for details

Fig. 6

 6 Fig. 6 An example of the likelihood calibration. The logistic function fit is shown in red, L c,n are shown in blue. The dashed black line indicates L c = L c

Fig. 11

 11 Fig. 11 Top panel: The relative importance of the PCA features for the classification of α-clustered spectra, calculated based on the average height at which each feature is used in the decision trees. Bottom panel: The contribution each feature makes to the overall clustering likelihood

Table 1

 1 The parameter ranges used to produce the ensemble of spectra Chosen empirically based on the measured spectra, and is consistent with the state spacings measured in TTIK measurements of α-scattering from other medium mass nuclei[START_REF] Norrby | Alpha-cluster structures in medium light nuclei[END_REF]

	Parameter	Range	Justification
	D	40-60 keV
	θ 2 sp	0.02-0.05	Chosen to generously encompass the value extracted from 44 Ca( p, p) measurements, θ 2 p = 0.034
			[27]

λ I μ = 2P μL λI γ 2 λ I μ , (3)

Acknowledgements

The authors would like to thank the STFC (Grant No. ST/J000140/1) and gratefully acknowledge that the research leading to these results has received funding from the European Union's Seventh Framework Programme under Grant Agreement No. 262010 (ENSAR). The GANIL facility is thanked for the delivery of the beams and the beamline preparation. ECT* Trento has supported this work and this infrastructure is part of a project that has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 824093.

Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors' comment: The spectra here are simulated. The related data are addressed by publication [17], so the present data availability statement is valid.]