Sohail Shah

Elisabeth Thronsen

Frédéric De Geuser

Constantinos Hatzoglou

Calin D Marioara

Randi Holmestad

Bjørn Holmedal

Frederic De Geuser

On the use of a cluster identification method and a statistical approach for analyzing atom probe tomography data

Keywords: Aluminum alloys, Atom probe tomography (APT), Transmission electron microscopy (TEM), clusters, Pair correlation function (PCF), Natural ageing (NA)

Introduction

Atom probe tomography (APT) has a unique combination of three dimensional (3D) spatial resolution down to the nanometer scale along with high chemical sensitivity [1,[START_REF] Kelly | Atom probe tomography[END_REF]. This enables the study of small precipitates or clusters in metals. Quantification of cluster size and composition at different stages of precipitation is essential and remains at the forefront of research in understanding and designing the next generation of alloys. Several approaches for cluster identification have been developed to define and characterize clustering and precipitation over the years [1,[START_REF] Vaumousse | A procedure for quantification of precipitate microstructures from three-dimensional atom probe data[END_REF][START_REF] Stephenson | New techniques for the analysis of fine-scaled clustering phenomena within atom probe tomography (APT) data[END_REF][START_REF] Hyde | Analysis of radiation damage in light water reactors: comparison of cluster analysis methods for the analysis of atom probe data[END_REF][START_REF] Lefebvre | 3DAP measurements of Al content in different types of precipitates in aluminium alloys[END_REF][START_REF] Felfer | Detecting and extracting clusters in atom probe data: A simple, automated method using Voronoi cells[END_REF][START_REF] Johnson | The atom probe and Markov chain statistics of clustering[END_REF][START_REF] Lefebvre | Application of Delaunay tessellation for the characterization of solute-rich clusters in atom probe tomography[END_REF][START_REF] Samudrala | A graph-theoretic approach for characterization of precipitates from atom probe tomography data[END_REF][START_REF] Zelenty | Detecting clusters in atom probe data with Gaussian mixture models[END_REF][START_REF] Lefebvre | Atom probe tomography: put theory into practice[END_REF][START_REF] Miller | Atom-probe tomography: the local electrode atom probe[END_REF]. In most cases, they consist of three data mining steps; (i) the clustering algorithm identifies solute clusters in the data by considering solute segregation [START_REF] Lervik | Atomic structure of solute clusters in Al-Zn-Mg alloys[END_REF], (ii) an enveloping algorithm searches for atoms of other elements that are within each solute cluster and (iii) an eroding algorithm erodes the matrix-particle interface that may have formed as a result of the enveloping step.

These methodologies generally demand the user to manually determine multiple parameters, leading to inconsistent results and challenges in reproducing them. Also, to correctly identify certain clustering phenomena, a visual inspection of the results is often used to assess the suitability of the method. This too is strongly user dependent and can give inconsistent results. Usually, a well-defined parameter estimation procedure is used as seen for the case of the maximum separation method (MSM) [START_REF] Williams | Defining clusters in APT reconstructions of ODS steels[END_REF] to improve reproducibility and consistency in the results. Studying clustering and precipitation through atom probe is rendered more complicated due to the local magnification effects due to differences in evaporation fields between different phases (matrix and particles) [START_REF] Vurpillot | Modeling atom probe tomography: A review[END_REF]. This effect is most severe for high evaporation field particles [START_REF] Hyde | Analysis of radiation damage in light water reactors: comparison of cluster analysis methods for the analysis of atom probe data[END_REF]. The advantage of cluster identification is that it enables the extraction of local information by providing information on each identified cluster (composition, size).

Statistical approaches such as radial distribution functions [START_REF] De Geuser | 3D atom probe study of solute atoms clustering during natural ageing and pre-ageing of an Al-Mg-Si alloy[END_REF][START_REF] Sudbrack | Direct observations of nucleation in a nondilute multicomponent alloy[END_REF], on the other hand, aim at describing the distribution of solute atoms as a whole and do not consider each cluster separately. This gives element specific averaged information on the cluster state of the matrix, both in terms of amplitude and correlation length. This enables a direct comparison of the degree of clustering between different datasets without the need of user-defined parameter.

Recently, Zhao et al. [START_REF] Zhao | Parameter free quantitative analysis of atom probe data by correlation functions: Application to the precipitation in Al-Zn-Mg-Cu[END_REF] coupled this parameter free statistical approach to an interpretation model enabling a fit of the RDFs in order to extract the composition of second phase particles and estimate volume fraction, number density and particle size. The methodology has been previously applied to Al-Cu-(Li-Mg) [START_REF] Ivanov | A combined characterization of clusters in naturally aged Al-Cu-(Li, Mg) alloys using small-angle neutron and X-ray scattering and atom probe tomography[END_REF] alloys, Al-Mg alloy system [START_REF] Medrano | Cluster hardening in Al-3Mg triggered by small Cu additions[END_REF] and the Al-Zn-Mg [START_REF] Zhao | Parameter free quantitative analysis of atom probe data by correlation functions: Application to the precipitation in Al-Zn-Mg-Cu[END_REF] system. This enables a statistical significance of the results and reproducibility, while giving the average values from the dataset analyzed.

Here, we combine two different approaches: a local cluster identification method, i.e. the isoposition method (IPM) and a statistical methodology based on RDF or rather on pair correlation functions (PCF) [START_REF] Lefebvre | Atom probe tomography: put theory into practice[END_REF][START_REF] Zhao | Parameter free quantitative analysis of atom probe data by correlation functions: Application to the precipitation in Al-Zn-Mg-Cu[END_REF]. On the basis of these two methodologies applied to atom probe datasets of two Al-Zn-Mg-(Cu) alloys, as well as with the help of simulated datasets, we aim at better assessing the robustness of the experimental parameters obtained by APT on very small clusters.

Experimental Section

6 mm thick plates from two commercial alloys, received from Benteler automotives are used in this study. The major difference between the two alloys is that one contains copper and is slightly richer in solute content, while the other is a copper-free alloy. The accurate composition measurements of the two alloys are given in Table 4. Table 1. Alloy composition of the investigated alloys [START_REF] Shah | Effect of cyclic ageing on the early-stage clustering in Al-Zn-Mg (-Cu) alloys[END_REF]. Small blocks of samples sized 3 cm × 3 cm were cut and solution heat treated (SHT) at 480ºC for half an hour in a salt bath followed by a quick water quench. The samples were grounded and polished for micro-hardness measurements using a 1000 gf Vickers indenter. The hardness values reported are an average of 5 individual measurements for each condition. The alloys were allowed to naturally age (NA) for 3 months at room temperature, after which they were used for APT and TEM characterizations.

All samples for APT were prepared by following a standard two-step electropolishing process [START_REF] Lefebvre | Atom probe tomography: put theory into practice[END_REF]. A minor contamination of copper (Cu) was seen as a capping layer at the tip of the APT needle similar as reported by Famelton et al. [START_REF] Famelton | Xenon plasma focussed ion beam preparation of an Al-6XXX alloy sample for atom probe tomography including analysis of an α-Al (Fe, Mn) Si dispersoid[END_REF]. This could be due to the Cu present within the sample itself or the Cu tube holding the APT needle which contaminates the electrolytic solution. The reconstructed volume containing this Cu enriched region was removed from the subsequent data analysis.

APT experiments were carried out on the Local Electrode Atom Probe (LEAP) 5000XS by Cameca instruments. It is a straight flight path instrument with a detection efficiency of 80%. For all analysis, the base temperature was set to 30K and the laser energy is adjusted to obtain an equivalent pulse fraction of 20% of the DC voltage to avoid preferential evaporation (Ref) (i.e. around 100-130 pJ). A detection rate of 0.5% and a pulse frequency of 250 kHz were used for all samples analyzed. Datasets containing 25-35 million ions were collected for each alloy. Reconstructions were made using the structural information according to Gault et al. [START_REF] Gault | Advances in the calibration of atom probe tomographic reconstruction[END_REF] by using the Integrated Visualization and Analysis Software (IVAS) by Cameca Instruments Inc. The Norwegian Atom Probe App software [25] developed by C. Hatzoglou has been additionally used for post processing of the experimental as well as for the simulated data.

The cluster identification method used, i.e., the IPM method, has been developed at the Groupe de Physique des Materiaux (University of Rouen, France) and is based on the chemical concentration (Mg + Zn) and atomic distance criteria. A detailed explanation of the methodology and its criteria estimation can be found elsewhere [START_REF] Hyde | Analysis of radiation damage in light water reactors: comparison of cluster analysis methods for the analysis of atom probe data[END_REF][START_REF] Lefebvre | Atom probe tomography: put theory into practice[END_REF].

In the 7xxx Al alloy system, the clusters exhibit higher atomic density as compared to that of the matrix, and with morphological distortion. This is mainly due to the local magnification effects typically observed due to the difference in evaporation fields between cluster/matrix, previously reported [START_REF] Hatzoglou | A chemical composition correction model for nanoclusters observed by APT-Application to ODS steel nanoparticles[END_REF][START_REF] Lawitzki | Compensating local magnifications in atom probe tomography for accurate analysis of nano-sized precipitates[END_REF] and also seen in the same alloys in a previous study [START_REF] Shah | Effect of cyclic ageing on the early-stage clustering in Al-Zn-Mg (-Cu) alloys[END_REF][START_REF] Thronsen | The evolution of precipitates in an Al-Zn-Mg alloy[END_REF]. However, the cluster dimension along the evaporation direction (Z direction) is less biased by local magnification effects [START_REF] Vurpillot | Modeling atom probe tomography: A review[END_REF][START_REF] Vurpillot | Trajectory overlaps and local magnification in threedimensional atom probe[END_REF], since it is the least biased direction in the APT. A more precise estimate of the size of clusters, as identified by iso-position method, can be based on a directional estimate in the Z-direction, the so-called Extentz parameter [START_REF] Miller | Atom-probe tomography: the local electrode atom probe[END_REF], which for one cluster is defined as

𝐸 𝑧 = |𝑍 𝑚𝑎𝑥 -𝑍 𝑚𝑖𝑛 | (1)
Here 𝑍 𝑚𝑎𝑥 and 𝑍 𝑚𝑖𝑛 are the outermost atoms in the z direction of the identified cluster. Hence, 𝐸 𝑧 is the maximum distance between outermost cluster atoms in this direction. The 𝐸 𝑧 is very similar to the way size measurements from TEM results often are obtained.

Another standard way of estimating size for spherical particles in APT is using the radius of gyration (or diameter), as obtained from a cluster identification methodology. The radius of gyration, is the root mean square of the distance between the center of mass and the coordinates of the constituent atoms classified as a cluster [START_REF] Lefebvre | Atom probe tomography: put theory into practice[END_REF][START_REF] Miller | Atom probe tomography: A technique for nanoscale characterization[END_REF]. Other methodologies like the best fit ellipsoid [START_REF] Karnesky | Best-Fit Ellipsoids of Atom-Probe Tomographic Data to Study Coalescence of Gamma Prime (L1_2) Precipitates in Ni-Al-Cr[END_REF] can also be used to quantify size measurements for non-spherical particles, but are not required in this study. Sizes reported in this study are all based on diameters extracted from the Guinier radius and from 𝐸 𝑧 . The Guinier diameter is simply twice the Guinier radius calculated from the cluster identification. The size and compositional errors are calculated based on the standard deviation. The error in the number density is calculated by dividing the number density by the square root of the total number of clusters.

The PCF method is based on the RDF calculated for each solute (Zn, Mg and Cu) as the central atom. The RDF na-b(r) is defined as the number of atoms of species 'b' at a distance 'r' of an atom 'a'. The RDF enables the computation of the average composition Ca-b (r) of element 'b' at a distance of 'r' of the element given as:

𝐶 𝑎-𝑏 (𝑟) = 𝑛 𝑎-𝑏 (𝑟) ∑ 𝑛 𝑎-𝑏 (𝑟) 𝑏 (2)
If 𝐶 𝑎 ̅̅̅ is the bulk composition of element 'a', then in an isotropic system, 𝐶 𝑎 ̅̅̅ . 𝐶 𝑎-𝑏 (𝑟) is the correlation of the spatial composition of element 'a' and element 'b' given as:

𝐶 𝑎 ̅̅̅ ⋅ 𝐶 𝑎-𝑏 (𝑟) = < 𝐶 𝑎 (𝑟 1 ⃗⃗⃗ ) ⋅ 𝐶 𝑏 (𝑟 2 ⃗⃗⃗ ) > (3)
Where

The 𝛾 𝑎-𝑏 (𝑟) = 0 occurs when there is no correlation, and this is true for large 'r'. The PCF can be rewritten as

𝛾 𝑎-𝑏 (𝑟) = < ∆𝐶 𝑎 (𝑟 1 ⃗⃗⃗ ) ⋅ ∆𝐶 𝑏 (𝑟 2 ⃗⃗⃗ ) > (5)
This highlights the fluctuation in composition (∆𝐶 𝑎 = 𝐶 𝑎 (𝑟 1 ⃗⃗⃗ ) -𝐶 𝑎 ̅̅̅ ) which is the difference from the average, is relevant to the PCF. A normalized correlation function can further be introduced:

𝛾 𝑎-𝑏 (𝑟) = ∆𝐶 𝑎 ⋅ ∆𝐶 𝑏 ⋅ 𝛾 𝑎-𝑏 0 (𝑟) (6)
Where 𝛾 𝑎-𝑏 0 (0) = 1 and 𝛾 𝑎-𝑏 0 = 0 for large 'r'. If we consider only a single type of ion, we obtain the autocorrelation:

𝛾 𝑎-𝑎 (𝑟) = ∆𝐶 𝑎 2 ⋅ 𝛾 𝑎-𝑎 0 (𝑟) (7)
As mentioned by Zhao et al. [START_REF] Zhao | Parameter free quantitative analysis of atom probe data by correlation functions: Application to the precipitation in Al-Zn-Mg-Cu[END_REF], the mean squared composition fluctuation is immune to the local magnification effects due to difference in the electric field required to induce the evaporation phenomenon of the different phases (but not to spatial resolution effects).

If we further assume a two-phase system with a homogenous composition in element 'a', with precipitate composition 𝐶 𝑝 𝑎 in a matrix of composition 𝐶 𝑚 𝑎 and that both share the same atomic volume, the volume fraction of the precipitates can be written as:

𝑓 𝑣 = 𝐶 𝑎 ̅̅̅ -𝐶 𝑚 𝑎 𝐶 𝑝 𝑎 -𝐶 𝑚 𝑎 (8)
The average of the product of compositional fluctuations can be then calculated as:

∆𝐶 𝑎 ⋅ ∆𝐶 𝑏 = 𝑓 𝑣 (1 -𝑓 𝑣 )(𝐶 𝑝 𝑎 -𝐶 𝑚 𝑎 )(𝐶 𝑝 𝑏 -𝐶 𝑚 𝑏 ) (9)
For a single type of ions, this can be simplified as:

∆𝐶 𝑎 2 = 𝑓 𝑣 (1 -𝑓 𝑣 )(𝐶 𝑝 𝑎 -𝐶 𝑚 𝑎 ) 2 = (𝐶 𝑝 𝑎 -𝐶 𝑎 )(𝐶 𝑎 -𝐶 𝑚 𝑎 ) (10)
The value of the PCF at origin ('r'=0) in combination with the matrix composition [START_REF] Geuser | Determination of matrix composition based on solute-solute nearest-neighbor distances in atom probe tomography[END_REF] help extract the precipitate composition and volume fraction without being affected by local magnification effects. The normalized correlation function for a sphere of radius 'R' can also be calculated assuming a homogenous density with a sharp interface with the matrix as [START_REF] Guinier | Small-angle scattering of X-rays[END_REF][START_REF] Philippe | Clustering and nearest neighbour distances in atom-probe tomography[END_REF]:

𝛾 0 𝑠𝑝ℎ𝑒𝑟𝑒 (𝑟, 𝑅) = { 1 - 3𝑟 4𝑅 + 𝑟 3 16𝑅 3 (𝑟 ≤ 2𝑅) 0 (𝑟 > 2𝑅) (11)
Assuming a log-normal size distribution of spheres and integrating Equation Erreur ! Source du renvoi introuvable., the experimental data can be fit to extract the size of the clusters. A 20% dispersity in the log-normal distribution is assumed and a contribution of an excluded volume is also added.

The PCF is plotted as 𝛾(𝑟) ⋅ 𝑟 allowing for an easier comparison of the differences in size between different conditions, as the position of the maximum of 𝛾(𝑟) ⋅ 𝑟 is related to the length of correlation (i.e., to the size of the clusters).

The TEM specimens were prepared by grinding bulk specimens to ~ 100 μm thickness before they were punched out to 3 mm discs and subsequently electropolished using an electrolyte mixture of 1/3 HNO3 and 2/3 C3CH3OH. The temperature was kept at 25±5°C at an applied voltage of 20 V. High angle annular dark-field scanning TEM (STEM) images were collected using a double corrected JEOL ARM200CF operated at 200 kV with convergence semi-angle and inner collector angle 27 and 48 mrad, respectively.

Results

Hardness

The hardness of the supersaturated solid solution (SSSS) for the 7003 alloy starts at around 46 HV while it is higher for the 7046 alloy, close to 61 HV. The hardness progressively increases for both alloys during NA, as shown in Figure 5. The blue circles indicate the condition which was chosen for APT analysis.

The hardness measurements taken after 3 months of NA, revealed a considerable hardness increase to around 115 HV for the 7003 alloy and 135 HV for the 7046 alloy. Further hardness measurements revealed that the hardness for both alloys later saturated at around 125 HV and 155 HV for the 7003 and 7046 alloy, respectively, after one year NA. Nevertheless, 3 months of NA was chosen as the conditions to characterize the microstructure of these alloys.

APT results processed by the IPM methodology

The APT datasets were used to extract smaller region of interest (ROI) that were used for visualizing the clustering, as shown through Figure 2 (a andb) for 7003 alloy and Figure 2(d ande) for 7046 alloy. The IPM was used to extract size distributions, as seen in Figure 2 (c andf) for 7003 and 7046 alloys, respectively. Size measurements in this study refer to the diameter of the precipitates measured using Equation (1) similar to our previous studies [START_REF] Shah | Effect of cyclic ageing on the early-stage clustering in Al-Zn-Mg (-Cu) alloys[END_REF][START_REF] Shah | Precipitation, mechanical properties and early slant ductile fracture in cyclic and naturally aged Al-Zn-Mg alloys[END_REF]. Additionally, the radius of gyration has been used to elucidate the mean diameter. The radius of gyration is the radius of a body having its mass concentrated at a single distance from its center of mass [1,[START_REF] Kelly | Atom probe tomography[END_REF]. The size distributions shown is an average of the size distributions obtained for the two datasets for each alloy.

For both alloys, the size distribution fits well with a log-normal distribution. Table 4 shows the quantitative measurements extracted from the IPM. It is quite evident that the two datasets for each alloy show similar size and composition of the clusters. The average size of the clusters for the 7003 alloy is around 1.8 nm, while it is slightly lower, around 1.5 nm, for the 7046 alloy, as measured by the 𝐸 𝑧 . The average Zn/Mg ratio for the 7003 alloy is around 2 while it is around 1.5 for the 7046 alloy, as indicated in Table 4. The measured ExtentX (Ex) is consistently larger than 𝐸 𝑧 , as seen in Table 2 for all four datasets. This is mainly due to the spatial resolution aspect of APT dominating the local magnification effects and will be discussed in detail in later sections. Since Extent in x and y are very similar, only Ex is reported in Table 2 and Erreur ! Source du renvoi introuvable.. The measured 𝐸 𝑧 is in close agreement with the size measurements by TEM (shown in the later section) and hence is regarded as more relevant in terms of describing the size using the IPM in this case.

APT results processed by the PCF methodology

The calculations of zinc (Zn-Zn), magnesium (Mg-Mg), copper (Cu-Cu) correlations along with the cross correlations for the three elements are shown in Figure 5. The first maximum of the curves, indicative of the size of the objects, is similar for all the correlations. This indicates a single distribution of clusters containing both Zn and Mg (and Cu for 7046). This size is consistently smaller for 7046 than for 7003, in agreement with the IPM results. The amplitude of the Zn-Zn PCF is larger for 7003 than for 7046, while the Mg-Mg PCF amplitude is smaller for 7003 than for 7046. These observations confirm a higher Zn/Mg ratio for the 7003 than for the 7046. The Cu correlations in Figure 3 containing are shown only for the 7046 alloy, since 7003 is a Cu-free alloy. Figure 3(f) shows higher scatter for the Cu-Cu plot due to lower signal from the Cu atoms.

More quantitative results can be obtained from the PCF by fitting the data to a 2-phase modelled dataset (matrix+clusters) containing spherical particles of homogeneous composition and with sharp interfaces (we shall further see that while this might be a reasonable assumption for larger precipitates, it is a coarse approximation for small clusters). The spherical shape is a good approximation being confirmed by the TEM analyses presented in the next section. We further assume a log-normal distribution of the diameter. This model implies that all correlations should be proportional, i.e., they should all have the same shape 𝛾 0 (𝑟), but with a different amplitude Δ𝐶 2 . For more efficient fitting, we thus fitted the shape to the Zn-Zn correlation and assumed it was fixed for the other correlations, with only the amplitude changing. Despite the limitations of the interpretation model (spheres, homogeneous composition, sharp interface), the quality of the fit is excellent, enabling the parameters of the model (size, compositions) to be extracted. The parameters extracted from the fit are shown in Table 3. These parameters show similar trends than the IPM data, in that the 7003 sizes are consistently larger than the 7046, the cluster compositions are in reasonable agreement and the Zn/Mg ratio is close to 2 for 7003 and 1.5 for 7046. There are also systematic discrepancies between the IPM and PCF results, which can be explained by the way these parameters are computed and will be discussed further in the paper.

Cluster observed by HAADF-STEM

Atomically resolved HAADF-STEM was employed to investigate the clusters in the alloys. b). This is in accordance with our previous results on the same alloy [START_REF] Shah | Effect of cyclic ageing on the early-stage clustering in Al-Zn-Mg (-Cu) alloys[END_REF]. To investigate this in a quantitative matter, the diameters of at least 300 GPI zones were measured per condition.

The GPI zones in 7003 had an average diameter of (1.87±0.17) nm, while in 7046 the average diameter was (1.67±0.11) nm.

Size distributions for the 7003 and 7046 alloys are shown in Figure 4(c andd). Here, only the structured GPI zones that could clearly be identified were used to estimate the distribution (around 50-60 GP zones). The unstructured clusters, usually the clusters smaller than 1 nm in diameter, were challenging to include in the statistics due to the uncertainty of overlapping with lower-lying clusters. While this should not impact the trend indicating slightly larger clusters in the 7003 alloy than in the 7046 alloy, it should be kept in mind that it means that the mean diameters estimated from HAADF-STEM are likely overestimated.

The IPM and the PCF methodology applied on simulated datasets

Experimental artefacts associated with an APT dataset include:

(i) A reduced detection efficiency of 80% (similar to the LEAP 5000XS used to analyze the experimental datasets in Figure 2). (ii)

Positioning uncertainty of atoms with different resolutions.

The reduced detection efficiency of 80% was accounted for in the simulated datasets. APT volumes were simulated with box dimensions 20 × 20 × 100 nm 3 . A dataset with a mean particle diameter of 1.8 nm was synthesized, picked up from the learnings of the 7003 alloy datasets. This is labeled as V1.8. A standard deviation of around 20% of the mean diameter and a log-normal distribution were applied. The particles were Zn and Mg enriched but without Cu, like the experimental datasets of the 7003 alloy. Only Al, Zn and Mg elements were used to generate the volumes. The particles introduced into the volumes had a sharp interface. Once all particles have been introduced, a random spatial distortion of each atom position was made to obtain a more realistic dataset as compared to an experimental measurement. Hence, the initially abrupt particle interface was blurred.

First, an 'ideal resolution' of 0.2 nm in depth and 0.5 nm in lateral resolution was applied to the entire dataset. This implies that the position of atoms was randomized by 0.2 and 0.5 nm in the evaporation and lateral directions respectively. The concentration of solute in the particles is set higher than what is extracted from the experimental data. Next, the resolution is degraded until a match in the PCFs is observed when compared with the experimental PCFs. We call this dataset the 'degraded resolution' and labels it V1.8DR.

The number densities of both datasets were set to 1E 25 m -3 . The size distribution for V1.8 is shown in Figure 5(b) while the corresponding PCFs are seen through Figure 5(c-e). The amplitude of the PCF of V1.8 is much greater than other datasets (experimental and simulation) while the amplitude of PCF of V1.8DR is very similar to that of the experimental datasets (7003 D1 and D2). The quantitative data extracted by the IPM for V1.8 and V1.8DR are shown in Table 4. Both these volumes have a higher concentration of solute input in the particles; 20 at.% Zn and 10 at.% Mg. Since both the ideal resolution and the degraded resolution volumes resulted in concentrations much lower than the input concentrations, it is clear from the IPM results that the apparent composition of the clusters is greatly affected by the resolution. Number densities of detected clusters are also highly affected, for both volumes, as seen in Table 4.

When the resolution is degraded, the size measurements for V1.8DR by the IPM are underestimated, as seen in Table 4, using the Ez parameter, being smaller than expected. This is a non-trivial effect, which indicates that a substantial proportion of the interfacial cluster atoms has been attributed to the matrix during IPM. Since the applied spatial resolution is better in Z than in XY, the Ex values are consistently larger than Ez.

Because of the broader extension of the clusters due to the spatial resolution, the PCF methodology computes a larger average size than prescribed by input the dataset. The degraded resolution also strongly affects both the amount of detected solutes in the clusters and their number densities c.f. Erreur ! Source du renvoi introuvable. and Table 4.

The PCF methodology confirms the effect of the degraded resolution with similar results as compared to IPM for V1.8, as seen in Table 4 and Table 5. Most importantly, it should be noticed that V1.8DR seems to mimic the behavior of the 7003: it gives similar results on the detected clusters with both IPM and PCF, the overall shape and amplitude of the PCF curves are similar, and the volume was obtained with initial cluster sizes compatible with HAADF-STEM.

Discussion

The apparent discrepancies in the clusters parameters as seen by the methodologies

We have looked at the clustering state in the 7003 and 7046 alloys by APT (IPM and PCF) and by HAADF-STEM. We found similar trends among the methods: the clusters seem consistently smaller in the 7046 than in the 7003. The Zn/Mg ratio was found to be about 2 for 7003 and 1.5 for 7046 with both APT methods. There are however systematic discrepancies which can be at least partially explained both by the difference in approach between IPM and PCF based methods, and by the nature of APT data. These discrepancies are:

i) The IPM method finds consistently higher solute content in the clusters than the PCF methodology. ii)

The IPM method finds consistently lower number density of clusters than the PCF methodology. iii)

Both APT methods find larger clusters than HAADF-STEM.

These discrepancies should be discussed in the context of very small clusters which do not present a sharp interface with the matrix, partly or entirely because of a finite spatial resolution limiting the positioning accuracy.

It should be kept in mind that the IPM methodology and the PCF methodology are two very different approaches. The IPM is a cluster identification method. Its objective is to label each atom in the dataset as belonging either to the clusters or to the matrix. The result depends on the choice of parameters, which are chosen through an optimization process aiming at minimizing the amount of "false positives" whereby atoms from the matrix are wrongly labeled as clusters. It requires as input an estimate of a minimal size for the clusters. An erosion step guarantees that the clusters do not include a shell of matrix. This ensures that compositions computed on the detected clusters are measured in a condition, neither including too small clusters nor interfacial area of the clusters that includes matrix atoms from a diffuse interface. This is in contrast with the PCF method, which considers the dataset as a whole and assumes that the clusters have a homogeneous composition. Hence, when the cluster/matrix interface is actually diffuse and the cluster size is small, the average composition measured by the PCF fitting method is likely to be smaller, explaining discrepancy i).

Discrepancy ii) is indirectly related to i) and can also be explained by the selective approach (IPM) vs global approach (PCF): the smaller clusters from the dataset may be part of the cluster distribution fitted by the PCF approach, while being labeled as too small to be a cluster by IPM, thus lowering the detected number density. Whether this is a good or a bad thing depends on the situation. The IPM parameters could be set to detect more clusters, but very likely at the cost of an apparent lower cluster composition.

Discrepancy iii) is more informative on the metrology of such small clusters by APT than on the methods themselves. The size histograms obtained by HAADF-STEM are likely to be very good estimates of the size of the clusters. If anything, they might be slight overestimations since the smaller diffuse clusters were left out of the calculation. This means that the clusters apparent size in APT is too large. Since the datasets reconstructions were properly calibrated and since the local magnification effect (higher density in these objects) should have a reverse effect, this can be only attributed to a spatial resolution blurring of the interfaces [START_REF] De Geuser | Metrology of small particles and solute clusters by atom probe tomography[END_REF].

The application of the two methodologies to simulated datasets enabled to better estimate the instrumental effect on the cluster parameters by attempting to create realistic simulated datasets that are compatible with our experimental results.

Studies of Simulated Volumes

Number density measurements

From the ideal resolution dataset 64% of the particles are detected with the IPM, while when applied to the degraded resolution dataset (V1.8DR) only 39% of particles are detected. Hyde et al. [START_REF] Hyde | Analysis of radiation damage in light water reactors: comparison of cluster analysis methods for the analysis of atom probe data[END_REF] found it challenging to detect clusters smaller than 1 nm in diameter using the same methodology, while the detection of clusters was even worse using the MSM, despite having a higher solute concentration contrast between the matrix and the clusters. This is in line with the current observations as demonstrated in Figure 5(b) with the size distribution showing no clusters detected below 1 nm in diameter.

The results for the 'degraded resolution' suggest that the number densities measured are even lower when a lower resolution is accounted for. The spatial distortions in lateral and depth resolution make it further challenging to detect the clusters smaller or close to 1 nm in diameter. To distinguish the smallest particles (with diameter smaller than about 1 nm) from random solute fluctuations becomes problematic, and thus a lot of fine particles are not detected by the IPM. With the degraded resolution, the smaller clusters are simply not there anymore due to the blurring caused by the spatial resolution effects.

The statistical method using the PCF on the other hand, estimates the number density from the volume fraction of particles based on the average volume of particles as 𝑓 𝑣 /⟨𝑉⟩. There is only a slightly higher estimated volume fraction for the cases with degraded resolution. However, the average particle diameter estimated for V1.8DR is higher than for V1.8. Hence, similar number densities are predicted by the PCF methodology for the ideal resolution and the degraded spatial resolution. As already discussed, this difference between IPM and PCF is due to different approaches (selective for IPM and global for PCF).

Size measurements

When considering the ideal resolution dataset (V1.8), we see that both methods (IPM and PCF) overestimate the size of particles by around 28% c.f. Table 4 and Table 5. However, when considering the degraded resolution dataset (V1.8DR), a smaller average cluster diameter is estimated by the IPM (~1.9 nm c.f. Table 4), while the PCF methodology overestimates the size by almost 60% (c.f. Table 5) The IPM mitigates the effect of spatial resolution and blurring of atoms by utilizing the erosion step which removes the interface atoms from the clusters. This in turn reduces the size of clusters and can explain the trend of lower size measurements for V1.8DR. Note that this dataset is very similar to the experimental one in terms of PCF amplitudes and concentration estimates.

The PCF methodology, on the other hand, slightly overestimates the cluster diameter with degraded spatial resolution. This can be explained by the scatter to the atomic positions, which for the PCF methodology leads to an increase in the net volume within each cluster. The volume fraction measured by the PCF methodology also increases for V1.8DR as compared to their ideal resolution counterparts, which explains the increase in the mean diameter estimated by the PCF methodology.

Composition estimates

The simulated volumes V1.8 and V1.8DR had a higher input concentration of solutes in the particles (Zn = 20 at. % and Mg = 10 at. %) than the measured ones. This was mainly done achieve a similar amplitude of the PCFs after spatial degradation as compared to the experimental data. The amplitudes of V1.8DR and of the experimental dataset (7003 D1 and D2) are very similar, as seen in Figure 6.

For both the IPM and the PCF methodology, the Zn and Mg content in the clusters are close to the input composition of particles for V1.8. However, when the spatial resolution is degraded, the Mg and Zn contents in the clusters become underestimated by both methodologies for V1.8DR, as seen in Table 4 and Table 5.

Let us first consider the PCF approach. The amplitude of PCFs for the degraded datasets is smaller than its ideal resolution counterpart, as seen in Figure 6. Note that Equation (10) for the particle compositions involves the bulk composition as well as the matrix composition. 𝐶 𝑝 𝑎 is very sensitive, because of the factor of 1/(𝐶 ̅ -𝐶 𝑚 ) in Equation [START_REF] Samudrala | A graph-theoretic approach for characterization of precipitates from atom probe tomography data[END_REF]. This increases the uncertainty and makes the composition of solute in particles challenging to accurately measure by this method. The computation of the particle composition thus strongly depends on the matrix and bulk compositions used. The values of ∆𝐶 𝑀𝑔 2 , ∆𝐶 𝑍𝑛 2 and ∆𝐶 𝑍𝑛-𝑀𝑔 2 at r=0 can be used in Equation [START_REF] Samudrala | A graph-theoretic approach for characterization of precipitates from atom probe tomography data[END_REF] to extract these as the two unknowns (𝐶 𝑝 𝑀𝑔 𝑎𝑛𝑑 𝐶 𝑝 𝑍𝑛 ), although matrix compositions can be influenced strongly by the lateral scatter.

Although IPM estimated the composition of particles in V1.8 close to the input concentrations, care must be taken, since the particles smaller than about 1 nm in diameter, remain undetected due to the spatial degradation and possibly the particles no longer existing (become a part of the random fluctuation). These small particles then become poorly represented in terms of the statistics. Hence, the composition of these particles has a high uncertainty, and the estimated composition is dominated by the larger particles, which have a defined interface and composition. However, in the synthesized samples considered here, all particles had the same stoichiometry, hence this issue did not affect the estimates in our cases.

The IPM underestimates the composition for the degraded resolution volumes. This is expected, since the lateral scatter can result in the artificial introduction of a significant level of matrix (Al) into the defined particle. A similar trend was seen by Hyde et al. [START_REF] Hyde | A sensitivity analysis of the maximum separation method for the characterisation of solute clusters[END_REF] when using MSM to determine the composition for simulation of trajectory aberrations. This results in a small solute cluster being significantly diluted due to the scatter caused by the degraded resolution parameters. Interestingly, the concentration of solute in particles measured by both IPM and PCF for V1.8DR is very similar to the experimental datasets, as seen in Table 3. Although the solute concentration is underestimated when compared to the input values in the dataset, we see that the degraded resolution (V1.8DR) is indeed a good representation of the actual experimental dataset. This in turn indicates the underestimation of solute concentration in the experimental datasets. It is well known that the local magnification effects cause trajectory aberrations, ultimately leading to the underestimation of solute inside clusters [START_REF] Hatzoglou | Experimental artefacts occurring during atom probe tomography analysis of oxide nanoparticles in metallic matrix: Quantification and correction[END_REF]. We see a similar trend here confirmed by V1.8DR, wherein a higher concentration of solute in particles is heavily underestimated, causing erroneous composition measurements. The Zn/Mg ratio estimated for both simulated volumes is close to two, as seen from Table 4 and Table 5, indicating that the quantification of the amount of Zn and Mg inside particles is affected proportionally and hence, the true ratio is expected to be preserved.

Experimental data interpretation

The analysis conditions were optimized for the experimental data to avoid preferential evaporation. A high pulse fraction and low temperature were used to analyze all datasets [START_REF] Saxey | Correlated ion analysis and the interpretation of atom probe mass spectra[END_REF].

No sign of molecular dissociation [START_REF] Jin | Effects of atom probe analysis parameters on composition measurement of precipitates in an Al-Mg-Si-Cu alloy[END_REF] was seen. Based on the understanding gained from the simulated volumes, the average compositions of GP zones for 7003 and 7046 alloys estimated by both PCF methodology and IPM are underestimated. The apparent particle composition was lower than the input value (20 at. % Zn and 12 at. % Mg) for the simulated volume, along with a reduced measured number density, mainly due to particles below 1 nm not being detected.

In principle, the shapes of the PCF's for Zn, Mg and Zn-Mg should all be the same, since they origin from the same particles (spheres). This is very well reproduced in the simulated volume, as seen in Figure 5(b andc). Both simulated volumes (V1.8and V1.8DR) have the same shape as seen from Figure 5(b andc). They overlap each other and follow the 'pseudobinary' assumption.

The experimental dataset, on the other hand, shows slightly different shapes of the PCF's.

Since we have only a single population of clusters, (GP zones predominantly) we should ideally expect all PCF's to be of the same shape. In particular, the mean size of the GP zones would be slightly smaller if it was based on the fit from Mg rather than that of Zn. Several works on Al-Zn-Mg alloys [START_REF] Engdahl | Investigation of fine scale precipitates in Al-Zn-Mg alloys after various heat treatments[END_REF][START_REF] Stiller | Investigation of precipitation in an Al-Zn-Mg alloy after two-step ageing treatment at 100 and 150[END_REF] have reported this effect, where the extension of the Zn enrichment in precipitates is higher than that of Mg. This can be interpreted as an APT artefact, wherein subtle changes in the evaporation field of the matrix atoms within particles changes the electric field distribution and hence causes these chromatic aberrations [START_REF] Marquis | Chromatic aberrations in the field evaporation behavior of small precipitates[END_REF].

However, the use of the Zn PCF in the calculations cannot solely explain the overestimation of the mean diameter by the PCF methodology. A decrease of the estimated diameter of around 7% is seen when using the PCF of Mg instead of Zn. The spatial resolution is element specific, as we see in the case of PCF's plotted for different elements for the same dataset in Figure 7. This is more of an instrument effect that is not seen in the simulated volumes.

Additionally, this will also affect the results obtained by the IPM. The spatial resolution plays an important role in determining the size, as also pointed out by other authors [START_REF] De Geuser | Metrology of small particles and solute clusters by atom probe tomography[END_REF][START_REF] Marquis | Chromatic aberrations in the field evaporation behavior of small precipitates[END_REF].

It can be seen by comparing Figure 5(a) and Figure 5(b andc) that there is a part of the PCF which is negative. This "dip" is more prominent in the PCF curves from the measured experimental datasets than in the simulated volumes. The interparticle interaction qualitatively gives this effect and the dip can be well reproduced by accounting for an 'excluded volume' [START_REF] Kruglov | Correlation function of the excluded volume[END_REF] in the PCF methodology. This is because two particles cannot overlap. Hence the position of each particle is not independent. However, the dip can also be due to depletion in solute around the particles as part of the solute diffusion, which is also not accounted for. Anyhow, the correction for the dip decreases the estimate of the mean diameter only slightly, by a few percent. Hence, it will neither affect much the estimates of particle compositions. The IPM gives representative size measurements with Guinier diameter, Ex as well as Ez for the simulated volumes as seen in Erreur ! Source du renvoi introuvable. and Table 4. This is not the case for experimental datasets though. TEM measurements agree with the IPM size measurements (Ez) for the experimental datasets. The Ex is overestimated for all four datasets and on average is around 1.7 times the Ez. The considered clusters are low field precipitates giving rise to higher atomic density, hence the opposite was expected, i.e., a contraction in the XY plane and potentially to appear smaller. However, this is not the case. This opposite trend is attributed to the local magnification effects and trajectory overlaps due to the difference in evaporation field between the matrix and particles which are seen in the experimental datasets [START_REF] Vurpillot | Trajectory overlaps and local magnification in threedimensional atom probe[END_REF]. In other words, the spatial resolution is locally worse.

Number density measurements, as highlighted in the above section, is heavily underdetermined by IPM for V1.8. From the TEM results, many disordered clusters smaller than 1 nm are seen, which are not taken into account while estimating the size distribution as seen in Figure 4(c and d) due to the challenge of de-convoluting the overlapping clusters.

Both methodologies have shown reasonable agreements on size and composition measurements. The important understanding from the simulated volumes in an attempt to mimic the experimental datasets showed that both methodologies underestimate the concentration of solute within particles. Features less than 1 nm usually go undetected, and consequently the number densities are underestimated. Care must be taken, since number densities measured by IPM are sensitive to the size of the particles, while the estimate by the PCF is based on the average volume of the particles.

Conclusions

By using a combination of APT data interpretation methodologies, supported by simulated datasets and HAADF-STEM experiments, we have shown that reliable APT measurements of the size and compositions from small clusters issued from the early stages of decomposition in Al-Zn-Mg alloy are challenging. Because of their small size, the particles/clusters are subject to significant distortions due to trajectory aberrations, leading to an increase of their apparent size as well as a decrease of their apparent solute content. While these effects are likely to be system dependent, we have estimated in our case that the clusters solute content dropped 40%. The effect on the apparent size is more complex to evaluate because it is strongly anisotropic, due to the anisotropy of the local magnification effect. It is clear, however, that these effects are most severe for the smaller clusters, so that particular care should be taken when interpreting apparent size and compositions of clusters from APT measurements of naturally age hardened or under aged conditions of AlZnMg aluminum alloys.

Figure 1 .

 1 Figure 1. Evolution of hardness as a function of ageing time for both 7003 and 7046 alloys.The blue circles indicate the condition which was chosen for APT analysis.

Figure 2 .

 2 Figure 2. APT sub-volumes of dimensions 25 × 25 × 70 nm 3 extracted from the APT datasets showing clustering in the 7003 alloy for (a) dataset 1 and (b) dataset 2. Similar cluster volumes for the 7046 alloy are shown for (d) dataset 1 and (e) dataset 2. The average size distribution based on 𝐸 𝑧 of the two datasets of the two alloys is shown in (c) and (f).

Figure 3 .

 3 Figure 3. Pair correlation functions for (a) Zn-Zn, (b) Zn-Mg, (c) Zn-Cu, (d) Mg-Mg, (e) Mg-Cu and (f) Cu-Cu pairs from the APT datasets shown in Figure 2.

Figure 4 .

 4 HAADF-STEM images from (a) 7003 and (b) 7046 showing a dense population of GPI zones. Corresponding size distribution estimation has been done for (c) 7003 and (b) 7046 based on 50-60 clusters. The finer clusters without structure were not included in the statistics due to uncertainty of overlap. a and b show that both 7003 and 7046 exhibit a dense population of GPI zones. Visually, the GPI zones have a larger diameter in the 7003 alloy (Figure 4. HAADF-STEM images from (a) 7003 and (b) 7046 showing a dense population of GPI zones. Corresponding size distribution estimation has been done for (c) 7003 and (b) 7046 based on 50-60 clusters. The finer clusters without structure were not included in the statistics due to uncertainty of overlap. a) than the ones in the 7046 alloy (Figure 4. HAADF-STEM images from (a) 7003 and (b) 7046 showing a dense population of GPI zones. Corresponding size distribution estimation has been done for (c) 7003 and (b) 7046 based on 50-60 clusters. The finer clusters without structure were not included in the statistics due to uncertainty of overlap.

Figure 4 .

 4 Figure 4. HAADF-STEM images from (a) 7003 and (b) 7046 showing a dense population of GPI zones. Corresponding size distribution estimation has been done for (c) 7003 and (b) 7046 based on 50-60 clusters. The finer clusters without structure were not included in the statistics due to uncertainty of overlap.

Figure 5 .

 5 Figure 5. Simulated volume (a) V1.8 showing iso-concentration surface of 7 at.% Zn+Mg to visualize the particles inserted. Size distribution obtained from IPM for (b) V1.8. PCF for (c) Zn-Zn, (f) Zn-Mg and (g) Mg-Mg for both volumes.

Figure 6 .

 6 Figure 6. PCF plots of (a) Zn-Zn, (b) Zn-Mg and (c) Mg-Mg for the simulated and experimental (7003) datasets. The degraded resolution dataset of the simulated volume is a close match in terms of PCF's compared to the 7003 datasets.

Figure 7 .

 7 Figure 7. Normalized PCF for Zn-Zn, Zn-Mg and Mg-Mg obtained from (a) 7003 experimental dataset and simulated volumes (b) V2DR and (c) V3DR.

Table 2

 2

	𝐶(1-𝐶) 𝑁	, where, C is the

. Average (mean) results of cluster analysis by IPM for APT datasets of 7003 and 7046 alloys. Compositions and Zn/Mg ratios are calculated by class size of 0.2 nm the average is computed. The uncertainties are calculated through 2√

Table 3 .

 3

		Mean	Cluster Composition (at.%)		Number
		diameter (nm)	Zn	Mg	Cu	Density (10 24 /m 3 )	Zn/Mg
	7003 D1	2.98	9.76	5.03	0.14	5.9	1.97
	7003 D2	3.15	8.69	4.52	0.11	6.3	1.95
	7046 D1	2.61	7.76	5.34	0.3	13.2	1.47
	7046 D2	3.1	8.02	5.32	0.36	7.7	1.53

Mean cluster size (diameter), composition, number density and Zn/Mg ratio obtained by fitting the PCF methodology data in Figure

5

assuming a log normal distribution of spherical clusters.

Table 4 .

 4 Average (mean) results of particle analysis done by the IPM on simulated volumes.

	Conditi on	Guinier Diameter (nm)	Ex (nm)	Ez (nm)	Cluster Composition (at.%) Zn Mg	Number Density (10 24 /m 3 )	Zn/Mg
	V1.8	2.32 ± 0.24	2.36 ± 1.11	2.24 ± 1.01	16.56 ± 0.52	8.17 ± 0.32	5.65 ± 0.02 2.03 ± 0.08
	V1.8D R	1.93 ± 0.20	2.40 ± 1.012	1.89 ± 0.86	12.92 ± 0.37	6.37 ± 0.21	3.50 ± 0.01 2.03 ± 0.07

Table 5 .

 5 Mean particle diameter, composition, number density and Zn/Mg ratio obtained by the PCF methodology by fitting the data in Figure5, assuming a log normal distribution of spherical particles.

	Condition	Diameter (nm)	Particle Composition (at.%) Zn Mg	Number Density (10 24 /m 3 )	Zn/Mg
	V1.8	2.24	18.76	9.43	6.82	1.99
	V1.8DR	2.90	10.50	5.12	5.14	2.05


I Zanon

E Clément

A Goasduff

J Menéndez

T Miyagi

M Assié

M Ciema

F Flavigny

A Lemasson

A Matta

D Ramos

M Rejmund

L Achouri

D Ackermann

D Barrientos

D Beaumel

G Benzoni

A J Boston

H C Boston

S Bottoni

A Bracco

D Brugnara

G De France

N De Sereville

F Delaunay

P Desesquelles

F Didierjean

C Domingo-Prato

J Dudouet

J Eberth

D Fernández

C Fougères

A Gadea

F Galtarossa

V Girard-Alcindor

V Gonzales

A Gottardo

F Hammache

L J Harkness-Brennan

H Hess

D S Judson

A Jungclaus

A Kaşkaş

Y H Kim

A Kuşoglu

M Labiche

S Leblond

C Lenain

S M Lenzi

S Leoni

H Li

J Ljungvall

J Lois-Fuentes

A Lopez-Martens

A Maj

R Menegazzo

D Mengoni

C Michelagnoli

B Million

D R Napoli

J Nyberg

G Pasqualato

Zs Podolyak

A Pullia

B Quintana

F Recchia

D Regueira-Castro

P Reiter

K Rezynkina

J S Rojo

M D Salsac

E Sanchis

M Siciliano

D Sohler

O Stezowski

Ch Theisen

A Utepov

J J Valiente-Dobón

D Verney

M Zielinska

High-precision spectroscopy of 20 O benchmarking ab-initio calculations in light nuclei

similarity renormalization group ab-initio calculations was performed for the first time in a nucleus far from stability. It was shown that the ab-initio approaches using chiral EFT forces are challenged by detailed high-precision spectroscopic properties of nuclei. The reduced transition probabilities were found to be a very constraining test of the performance of the ab-initio models.

The excited states of unstable 20 O were investigated via γ-ray spectroscopy following the 19 O(d, p) 20 O reaction at 8 AMeV. By exploiting the Doppler Shift Attenuation Method, the lifetime of the 2 + 2 and 3 + 1 states were firmly established. From the γ-ray branching and E2/M1 mixing ratios for transitions deexciting the 2 + 2 and 3 + 1 states, the B(E2) and B(M1) were determined. Various chiral effective field theory Hamiltonians, describing the nuclear properties beyond ground states, along with a standard USDB interaction, were compared with the experimentally obtained data. Such a comparison for a large set of γ-ray transition probabilities with the valence space in medium Introduction.-Nuclear structure studies aim at understanding the properties of atomic nuclei based on nucleons interacting in the nuclear medium by combined strong, electromagnetic and weak interactions. Chiral effective field theory (EFT) provides a framework for nuclear forces based on quantum chromodynamics which, together with ab-initio many-body approaches, allows one to perform first-principle nuclear structure calculations including two-and three-nucleon forces in various regions of the Segré chart [1][2][3][4]. Previous studies of neutron-rich isotopes have proven to be especially suitable to establish advanced theoretical calculations based on chiral EFT forces. In particular, the neutron drip line for oxygen presents a strong anomaly with 24 O being the last bound isotope, whereas theoretical predictions positioned the drip line at doubly-magic 28 O [5][6][7][8][9]. This puzzle was solved by the introduction of chiral EFT three-body forces [10]. These contributions have been studied extensively in subsequent works, especially in comparison with mass [11][12][13][14][15][16], charge radius [17][18][19][20][21][22][23], and electromagnetic moment [24][25][26] measurements of neutron-rich systems. The present challenge is to obtain unambiguous experimental measurement to compare to different ab-initio calculations to improve their accuracy and predictive power. Electromagnetic transition probabilities play a major role in testing the quality of the chiral EFT interaction with ab-initio approaches [4,27], since they are connected to the nuclear wave functions. The comparison between high precision measurements in excited states and state-ofthe-art ab-initio calculations provides a sensitive probe of the nuclear structure details comparable to nuclear masses or charge radii. The isotopic chain of oxygen was identified as an ideal laboratory to benchmark state-of-the-art ab-initio theory [1]. In neutron-rich oxygen, the introduction of three-body forces induces a repulsion between the neutron 1s 1/2 and 0d 3/2 orbitals defining the drip-line [10]. Detailed spectroscopy of these orbitals at the drip-line remains an experimental challenge but relevant information can be obtained in slightly less exotic nuclei, such as 20 O [28]. In this Letter, we present the spectroscopic study of non-yrast states in 20 O using state-of-the-art instrumentation.

Experimental details.-The 20 O nucleus was populated in the 19 O(d, p) 20 O* direct reaction in inverse kinematics, using a pure radioactive beam post-accelerated to 8 AMeV, with an average intensity of 4 × 10 5 pps, delivered by the SPIRAL1 accelerator complex in GANIL and impinged on a deuterated polyethylene target (CD 2 ). Two types of targets were employed in the experiment: a 0.3 mg/cm 2 -thick self-supporting CD 2 target for spectroscopy measurements and a 0.3 mg/cm 2 -thick target deposited on a 24.4 mg/cm 2 -thick Au backing (hereinafter mentioned as CD 2 and CD 2 +Au, respectively). Using the CD 2 target, detailed spectroscopy was performed and the CD 2 +Au was used for measuring the lifetime of the populated excited states using the Doppler Shift Attenuation Method (DSAM) [29]. The measurements were performed in triple coincidence: the beam-like recoils were detected in the VAMOS++ magnetic spectrometer [30] to reject the background coming from fusion-evaporation and fusion-fission events, protons were measured at backward angles by the MUGAST array [31] and, at backward angles, the AGATA array [32] was employed for the detection of the γ rays emitted by the excited nucleus.

The coupling of these three instruments provides a large solid angle for detection of recoiling nuclei, a high precision kinematic reconstruction and a unique sensitivity for γ rays emitted in flight thanks to γ-ray Tracking Algorithms [33], resulting in unprecedented Doppler correction capabilities. This unique combination of direct reaction and state-of-the-art spectrometers allows one to perform a combined charged particle and γ-ray spectroscopy, along with the measurement of sub-picosecond lifetimes. In particular, the MUGAST array allows for the selection on an event-by-event basis of the excited states directly populated in the final 20 O * nucleus and measure its velocity at the reaction moment for each of the populated states to perform a feeding-free, fully controlled and high accuracy lifetime measurement. More details on the experimental apparatus and data analysis procedure can be found in Ref. [31].

Spectroscopic study.-The energy and angle of the protons detected in MUGAST allowed for the reconstruction of the 20 O excitation energy spectrum using NPTool [31,34] (see also the Supplemental Material [35]). The ground state, the 2 + 1 at 1.67 MeV, the 4 + 1 at 3.55 MeV, the 2 + 2 at 4.07 MeV and 3 + 1 states at 5.23 MeV were observed. Moreover, two additional states, already identified in [36] and both tentatively assigned as J π = (2 + 3,4 ), were observed. Excited states above the neutron separation threshold in 20 O were populated [START_REF] Zanon | Testing three-body forces in light nuclei: Lifetime measurements in 20 O in the femtosecond range[END_REF]. The correlation between the excitation energy of 20 O and the emitted γ-rays is shown in Fig. 1. The transitions used in the line-shape analysis for the lifetime extraction are highlighted in red in the figure.

The prompt Doppler-corrected tracked γ-ray spectrum measured in AGATA in triple coincidence using the CD 2 target is shown in Fig. 4 + 1 , 2 + 2 and 3 + 1 excited states are clearly visible. The reconstruction of the level scheme obtained via particleγ spectroscopy combining MUGAST and AGATA is reported in detail in [START_REF] Zanon | Testing three-body forces in light nuclei: Lifetime measurements in 20 O in the femtosecond range[END_REF][START_REF] Zanon | [END_REF][39]. The complete spectroscopic information is summarized in the Supplemental Material [35]. The branching ratios of the 2 + 2 → 2 + 1 and 2 + 2 → 0 + 1 transitions were measured to be 0.88(1) and 0.12(1), respectively. For the 3 + 1 → 2 + 2 and 3

+ 1 → 2 + 1
transitions, the measured branching ratios were 0.28(1) and 0.72(1), respectively.

Lifetime measurements.-Previous experiments provided the lifetime measurements of the 2 + 1 (τ =10.5(4) ps [40]) and 2 + 2 (τ =150 +80 -30 fs [28]) states. In the present work, lifetimes were extracted by fitting the line-shape of the transitions in the γ-ray spectra of the CD 2 +Au dataset with realistic Monte Carlo simulations. The simulations have been performed using the AGATA Geant4 simulation code [41], that includes the geometry of the array and the reaction event with the emission of the beamand target-like particles as well as γ rays. Line-shape analysis based on Monte Carlo simulations for AGATA was already performed in the fs range [42][43][START_REF] Michelagnoli | The lifetime of the 6.79 MeV state in 15 O as a challenge for nuclear astrophysics and gammaray spectroscopy: a new DSAM measurement with the AGATA Demonstrator array[END_REF], proving the capabilities of the apparatus in this range of lifetimes.

The simulation has been optimized by adjusting the parameters corresponding to the response function of the detectors at the time of the experiment to reduce the sources of systematic errors. The velocity distributions at reaction point were measured and used as an input of the simulation of the decay for each of the investigated states. The reproduction of the energy loss in the target and degrader was tested for both CD 2 and CD 2 +Au datasets on the 2 + 1 → 0 + 1 transition. The energy and Full-Width Half Maximum of the transition were in agreement within the detector resolution. The comparison is provided in the Supplemental Material [35] (see also references [45-47] therein).

The lifetime of the 2 + 2 state was extracted by fitting the line-shape of the simulated 2 + 2 → 2 + 1 transition to the experimental one, obtained by requiring the coincidence with the 2 + 2 state in the excitation energy and thus removing the influence of feeders. The simulations were performed by varying two parameters: the energy of the transition at rest and the lifetime of the 2 + 2 state. The minimum was attested at 70 (10) fs (70( 14) fs for 90% confidence). In Fig. 3, the red hatched area represents the simulation output within the 1σ limit. The two dimensional χ 2 surface is shown in the Supplemental Material [35] (see also references [48,49] therein). To extract the lifetime of the 3 + 1 state, the 3 + 1 → 2 + 1 transition at 3552.6 keV was chosen. Similarly to the procedure for the 2 + 2 state, the simulated spectrum was fitted to the experimental one after gating on the excitation energy of the 3 + 1 state. Using the least-χ 2 procedure, a lifetime of τ = 54(9) fs was obtained (54 (12) fs for 90% confidence).

Discussion.-The present measurement confirms the short lifetime of the 2 + 2 state and the conclusion drawn in [28] in spite of a tension between the results of the two experiments. The origin of the discrepancy is not completely understood. The computation of a longer lifetime can arise from systematical effects in the initial velocity distribution model or unobserved side-feeding contribution. Such systematic errors are cancelled in our improved experimental approach. The use of the (d, p) reaction combined to a thin target layer where 20 O is produced, followed by an optimized gold thick foil to develop the DSAM profile, allows one to determine, on an event-by-event basis, the entry point in the recoil from the measured excitation energy and the initial velocity distribution for each state used to extract the lifetimes. The reduced transition probabilities have been extracted from the measured transition energies, branching ratios and lifetimes, and are reported in Table I. A measured value of the mixing ratio δ(E2/M 1) for the 2 + 2 → 2 + 1 transition reported earlier is -0.18(8) [50]. The mixing ratios of the 3 + 1 → 2 + 1,2 transitions are experimentally unknown, preventing us to extract model independent transition probabilities.

The present experimental results have been compared to ab-initio calculations using the valencespace in-medium similarity renormalization group (VS-IMSRG) [51]. The calculations were performed at the VS-IMSRG(2) level, building an effective shell-model Hamiltonian for 20 O in the 0d 5/2 , 1s 1/2 , 0d 3/2 configuration space for protons and neutrons. The E2 and M 1 transition operators were evolved consistently with the VS-IMSRG keeping up to two-body operators, but meson-exchange currents, explored for M 1 transitions in very light nuclei [52][53][54], were not included. As a starting point, three well established nuclear Hamiltonians based on chiral EFT with three-nucleon forces were used: (i) 1.8/2.0(EM) [55,56], which reproduces very well groundstate energies up to heavy nuclei [3,57] and was used for 20 O in Ref. [28]; (ii) the more recent N 3 LO lnl [58] and (iii) N 2 LO GO [59], which includes explicit ∆-isobar degrees of freedom. In addition, standard shell-model calculations using the USDB interaction [60] in the same configuration space were performed. Unlike in the VS-IMSRG calculations, the bare M 1 and E2 operators and therefore neutron effective charges e n = 0.4 [61], were used along with the USDB interaction. For the VS-IMSRG and configuration-interaction calculations the codes im-srg++ [62] and KSHELL [63] were used, respectively. Fig. 4 compares the experimental low-lying excitation spectra of 20 O with the results of the theoretical calculations. Additionally, the experimental and calculated level scheme of 19 O are also shown in the Supplemental Material [35]. The excitation energies obtained in the ab-initio approaches and the shell model are in general in good agreement with experiment, within hundreds of keV. The 1.8/2.0(EM) and USDB results are in the best agreement with the data. We emphasize the good agreement for the 1/2 + excited state corresponding to the (d 5/2 ) 2 (s 1/2 ) 1 configuration in 19 O. Consistently, the evolution of its excitation energy, for different chiral EFT Hamiltonians, is correlated to the 2 + 1 → 2 + 2 energy difference in 20 O (Supplemental Material [35]). It should be noted that there was no nuclear structure information on excitation energies etc. in oxygen or similar systems used for the derivation of the chiral EFT Hamiltonians in ab-initio approaches, while the shell model USDB interaction resulted from the fit to the selected nuclear structure data. The calculated wave functions of the states observe two main structures: the 0 + 1 , 2 + 1 and 4 + 1 yrast states are mainly due to the neutron (0d 5/2 ) 4 configuration, while the 2 + 2 and 3 + 1 states are dominated by the (0d 5/2 ) 3 (1s 1/2 ) 1 configuration. The color code on the level shows the amplitude of the main configurations (≥ 10%). The ab-initio and shell-model calculations are in good agreement, but the shell model suggests more fragmented wave functions.

The mixing ratio of transitions from the 3 + 1 state are experimentally unknown. For the following discussion, the theoretical values obtained using the USDB [60] interaction were used to obtain the corresponding B(E2) and B(M 1). The experimental and theoretical reduced transition probabilities are presented in Table I and reported in Fig. 4. The B(E2) between the 2 + 2 state and the 0 + 1 and 2 + 1 states, and from the 3 + 1 state, were found experimentally small, consistently with their single-particle character, interpreted as (0d 5/2 ) 4 → (0d 5/2 ) 3 (1s 1/2 ) 1 single-particle transition.

The B(E2) values are systematically underestimated in the ab-initio calculations, as already observed in Ref. [61] for 21 O and discussed first in Ref. [65] and recently in much detail in Ref. [66,67]. The likely reason is the restriction to the VS-IMSRG(2) level, which leaves many-particle-many-hole correlations out of the evolved VS-IMSRG operator. With a neutron effective charge e n = 0.4, USDB B(E2) results present a good agreement with experimental values, in particular, those involving the 3 + 1 state. An even better agreement for some transitions is observed in USDB calculations using an effective charge e n = 0.5. However, high-precision spectroscopy data reveal that simple effective charges in the shell model seem not to reproduce all transitions simultaneously (see also the discussion in Ref. [61]) and a more sophisticated treatment would be desirable. In addition, the B(E2; 2 + 2 → 2 + 1 ) is underestimated by orders of magnitude in all the models, which suggests some deficiency of the wave functions. It could be related to the limited configuration space or to the insufficient configuration mixing. The latter is consistent with the results of the measurements of the spectroscopic factor (S) by Hoffman et al. [36]. The cross sections for 0 + 1 and 4 + 1 states were computed with L = 2 transfer and large S-factor obtained are compatible with the occupancy of the d-orbitals only. Similarly, the cross section for 2 + 2 state is dominated by L = 0 transfer consistent with the USDB N

→ 2 + 1 (3 + 1 → 2 +
2 ) have been calculated using a theoretical mixing ratio of δ = 0.13 (δ = 0.019), obtained from USDB. The binding energy with respect to 16 O is presented in MeV.

single-particle excitation into the 1s 1/2 neutron orbital. In contrast, the cross section for 2 + 1 state was obtained with large contributions of L = 0 (S = 0.19) and L = 2 transfers (S = 0.43) corresponding to a more fragmented wave function of the 2 + 1 state.

The theoretical B(M 1) reduced transition probabilities for the 2 + 2 → 2 + 1 transition are about a factor 3 smaller than those obtained in the experiment, with a small difference between USDB and 1.8/2.0(EM) and lower values for the other two VS-IMSRG calculations. The B(M 1) for the 3 + 1 state are in reasonably good agreement with the experiment, especially for USDB, where there is agreement within (1 -2)σ. The three chiral EFT Hamiltonians well reproduce the 3 + 1 → 2 + 1 B(M 1) value, but they overestimate the 3 + 1 → 2 + 2 reduced transition probability by about a factor 2. Overall, for the M 1 tran-sitions there is a better agreement for the phenomenological USDB interaction. The ab-initio results are of similar quality, with a slight preference for 1.8/2.0(EM) over the other two chiral EFT Hamiltonians. The agreement may improve when including meson exchange currents. Hence, the measurements of the B(M 1) reduced transition probabilities appear to be very pertinent for testing ab-initio calculations based on chiral EFT Hamiltonians. It should be noted that the short lifetime measured in this work (lower than 100 fs) for the 2 + 2 state is incompatible with having at the same time a low B(E2) and a low B(M1), in the range of the theoretical predictions, for the 2 + 2 → 2 + 1 transition.

Conclusion.

-The lifetimes of the 2 + 2 and 3 + 1 excited states in 20 O were measured by means of the DSAM technique via the direct (d, p) reaction in inverse kine-matics using a radioactive post-accelerated beam of 19 O. A feeding-free lifetime for the 2 + 2 and the 3 + 1 states was extracted. For the first time in the key isotopic chain of oxygen, all spectroscopic observable obtained for yrast and non-yrast excited states in the neutron rich 20 O were compared simultaneously to the results of ab-initio calculations using chiral EFT forces and provide the results in reasonable agreement with the experimental data. The reduced transition probabilities, B(M 1) and B(E2) in particular, provide a very constraining test of the performance of the ab-initio models. Many improvements in ab-initio calculations are still to be envisaged like including meson exchange currents or many-particlemany-hole correlations by releasing the restriction to the VS-IMSRG(2) level, so that the predictive power can reach and exceed that of the conventional phenomenological shell-model approaches. The comparison between data and predictions will be strengthened by systematic estimation of the theoretical uncertainties. This observation serves as motivation for future endeavors aimed at quantifying uncertainties in VS-IMSRG calculations. This work paves the way for lifetime measurements in exotic nuclei using next-generation radioactive beam facilities under construction worldwide to be compared with state of the art ab-initio calculations.

Data Availability Statements. The supporting data for this article are from the e775s experiment and are registered as https://doi.org/10.26143/GANIL-2020-E775S following the GANIL Data Policy.

2 FIG. 1 .

 21 FIG. 1. (Color online) Two dimensional correlation between the 20 O excitation energy on the y-axis and the corresponding γ-ray decays on the x-axis. The transitions from which the lifetimes are extracted are highlighted in red. The weakest transitions depopulating the 2 + 2,3,4 states are marked with black boxes.

+ 3 FIG. 2 .

 32 FIG. 2. Prompt γ-ray spectrum following the 19 O(d, p) 20 O reaction measured in AGATA in coincidence with MUGAST and VAMOS++.

FIG. 3 .

 3 FIG. 3. (Color online) Comparison between experimental data and simulation for the lifetime of the 2 + 2 state in 20 O.

  2. The decays of the 2 + 1 ,

	Excitation energy (MeV)	1 2 3 4 5 6	(keV) 1500 2000 2500 3000 3500 4000 γ Energy h1 Entries 19926 Mean x 1900 Mean y 4.102 RMS x 713 h1 Entries 19926 Mean x 1900 Mean y 4.102 RMS x 713 Ex:gammaEDC*1000 {PromptAu && ELab>0.} 1 + 2 1 + 4 2 + 2 1 RMS y 1.129 RMS y 1.129 +

TABLE I .

 I FIG. 4. (Color online) Experimental 20 O excited states compared to theoretical USDB shell-model calculations and VS-IMSRG results obtained with three different Hamiltonians. For theoretical states, the leading configurations are reported with a color code for each level: (d 5/2 ) 4 (black), (d 5/2 ) 3 (s 1/2 ) 1 (green) and (d 5/2 ) 2 (s 1/2 ) 2 (violet). Other configurations are shown in gray. The colored bars length is proportional to each contribution in the wave-function. The measured and calculated B(E2)s (blue) and B(M 1)s (red) are also reported proportionally to the arrows thickness. Comparison between experimental, shell-model (USDB) and ab-initio (N 3 LO lnl , 1.8/2.0(EM), N 2 LOGO) transition probabilities. The B(E2)s are given in e 2 fm 4 and the B(M 1)s in µ 2 N . The experimental transition probabilities of the 3 + 1

					(d 5/2 ) 4	(d 5/2 ) 3 (s 1/2 ) 1
					(d 5/2 ) 2 (s 1/2 ) 2	other conf.
	Energy [MeV]	2 + 4 + 2 + 3 +	2 + 4 + 2 + 3 +	2 + 4 + 2 + 3 +	2 + 4 + 2 + + 3	2 + 3 + 4 + 2 +
		0 +	0 +	0 +	0 +	0 +
				3 LO lnl	1.8/2.0(EM)	N 2 LO GO
			Exp.	USDB	N 3 LO lnl	1.8/2.0(EM)	N 2 LOGO
	B(E2; 2 + 1 → 0 + 1 )	5.9(2)	3.25	0.79	0.89	0.80
	B(E2; 2 + 2 → 0 + 1 )	1.3(2)	0.77	0.21	0.20	0.26
	B(E2; 2 + 2 → 2 + 1 )	4(2)	0.0005	0.089	0.070	0.18
	B(M 1; 2 + 2 → 2 + 1 )	0.05(2)	0.019	0.014	0.017	0.012
	B(E2; 3 + 1 → 2 + 1 )	0.32(7)	0.57	0.16	0.17	0.17
	B(M 1; 3 + 1 → 2 + 1 )	0.016(4)	0.029	0.023	0.028	0.0089
	B(E2; 3 + 1 → 2 + 2 )	0.7(2)	1.24	0.14	0.15	0.11
	B(M 1; 3 + 1 → 2 + 2 )	0.19(4)	0.32	0.53	0.55	0.56
	Binding energy	-23.74 [64]	-23.63	-19.67	-20.51	-22.71

We acknowledge the GANIL facility for provision of heavy-ion beams and we would like to thank J. Goupil, G. Fremont, L. Ménager, A. Giret for assistance in using the G1 beam line and its instrumentation. This work was supported by STFC(UK). This research was also supported by the OASIS Project No. ANR-17-CE31-0026, by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract number DE-AC02-06CH11357, by the European Regional Development Fund with Contract No. GINOP-2.3.3-15-2016-00034 and by the MCIN/AEI /10.13039/501100011033, Spain, grants PID2020-118265GB-C41, PID2020-118265GB-C42, by Generalitat Valenciana, Spain, grants PROME-TEO/2019/005, CIAPOS/2021/114 and by the FEDER EU funds, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project-ID 279384907 -SFB 1245, and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 101020842

). Cloud Veneto is acknowledged for the use of computing and storage facilities [68]. The nucleon-nucleon and three-nucleon interaction matrix elements used in the VS-IMSRG calculations are generated by NuHamil [69]. We acknowledge the AGATA collaboration for the use of the spectrometer.


Romane Berthelin
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Peniel Adounkpe

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Nanée Chahinian

Alain Dezetter

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Labcom Hydr.IA: a joint Company-Academic laboratory to develop new flood forecasting services Case Study on the Nîmes Flash-Floods (Southeaster France) Authors

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Labcom Hydr.IA: a joint Company-Academic laboratory to develop new flood forecasting services. Case Study on the Nîmes Flash-Floods (Southeastern France)

Mines Alès is certified by

In hydrology, the studied systems are dynamic and non-linear.

Neural Networks are particularly appropriate for representing these complex natural systems. Ongoing studies

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  have in solutions that are often perceived as opaque, and whose operational implementation is rare Despite the growing development of AI-related solutions in many complex fields, and the demonstration of their relevance to several hydro(geo)logical issues, there is a deficit in the transfer of these technologies to end-users. /www.follow.solutions/fr/ G, Crippa J, Djerboua A, Dobricean O, Dongar F, Eugene A, Raymond M. 2019. ESPADA : un outil pour la gestion en temps réel des crues éclairs urbaines en pleine modernisation. La Houille Blanche 3-4: 57-66.

  The flood events observed in Nîmes are used to train and test the Neural Network in order to improve flood forecasting in this catchment. References: Hornik K, Stinchombe M, White H (1989) Multilayer feedforward networks are universal approximators. Neural Netw 2:359-366 Barron AR (1993) Approximation bounds for superpositions of a sigmoidal function. Proceedings, San Antonio, TX, USA, 17-22 January 1993. 930-945.