F Browne

S Chen

P Doornenbal

A Obertelli

K Ogata

Y Utsuno

K Yoshida

N L Achouri

H Baba

D Calvet

F Château

N Chiga

A Corsi

M L Cortés

A Delbart

J-M Gheller

A Giganon

A Gillibert

C Hilaire

T Isobe

T Kobayashi

Y Kubota

V Lapoux

H N Liu

T Motobayashi

I Murray

H Otsu

V Panin

N Paul

W Rodriguez

H Sakurai

M Sasano

D Steppenbeck

L Stuhl

Y L Sun

Y Togano

T Uesaka

K Wimmer

† K Yoneda

O Aktas

T Aumann

L X Chung

F Flavigny

S Franchoo

I Gasparic

R.-B Gerst

J Gibelin

K I Hahn

D Kim

T Koiwai

Y Kondo

P Koseoglou

J Lee

C Lehr

B D Linh

T Lokotko

M Maccormick

K Moschner

T Nakamura

S Y Park

D Rossi

E Sahin

P-A Söderström

D Sohler

S Takeuchi

H Toernqvist

V Vaquero

V Wagner

S Wang

V Werner

X Xu

H Yamada

D Yan

Z Yang

M Yasuda

L Zanetti

Pairing forces govern population of doubly magic 54 Ca from direct reactions

(Dated: January 11, 2021) Direct proton-knockout reactions of 55 Sc at ∼200 MeV/nucleon were studied at the RIKEN Radioactive Isotope Beam Factory. Populated states of 54 Ca were investigated through γ-ray and invariant-mass spectroscopy. State energies were calculated from the shell model, employing the GXPF1Br interaction, and their exclusive cross sections from distorted-wave impulse approximation single-particle estimates factored with the shell model spectroscopic factors. Despite the calculations showing a significant amplitude of excited neutron configurations in the ground-state of 55 Sc, valence proton removals populated predominantly the ground-state of 54 Ca. This counter-intuitive result is attributed to pairing effects leading to a dominance of the ground-state spectroscopic factor.

Owing to the ubiquity of the pairing interaction, this argument should be generally applicable to direct knockout reactions from odd-even to even-even nuclei.

Perhaps the most profound of the interactions within the atomic nucleus is that of pairing. It is well described by a strongly attractive contact force between two nucleons (protons and neutrons). Notably, it is why, without exception, even-even nuclei have zero total angular momentum and positive parity, J π = 0 + , ground states. Moreover, it is the origin of odd-even staggering of neutron and proton separation energies (S n and S p , re-57 spectively). Recently, pairing has been shown to have a 58 prominent role in single-nucleon removal cross sections in 59 so-called neutron-rich exotic nuclei which have an excess 60 of neutrons compared to their stable counterparts [1]. In 61 this Letter, a further implication of pairing to direct re-62 actions is demonstrated.

Along with pairing, the interaction of a nucleon's intrinsic (s) and orbital ( ) angular momenta has farreaching consequences. In particular, it is necessary for the theoretical reproduction of the canonical nuclear "magic numbers" [2,3], 2, 8, 20, 28, 50, 82..., which relate to large energy gaps between shells of protons and neutrons. Nuclei with both proton and neutron numbers equalling magic numbers are known as doubly magic. A consequence of the pairing interaction, most obvious in doubly magic nuclei, is the existence of excited states based on the promotion of identical nucleon pairs to higher-lying shells into 2-particle-2-hole (2p-2h) states.

A feature of the tensor force, which describes meson exchanges between nucleons, is the attraction of unlike nucleons of the same (= s, p, d, f...) with different intrinsic spin directions, j > = + s and j < = -s, where E(j > ) < E(j < ) [4]. It implies an evolution of the energies of the spin-orbit generated shells as a function of nucleon number in exotic nuclei [5,6]. Stable counterparts of neutron-rich nuclei have occupied proton j > orbits, lowering the corresponding neutron j < orbital energy. Low occupancy of the proton j > in the neutron-rich case results in less interaction with the neutron j < orbital allowing E(j < ) to increase. In 54 Ca, this effect, owing to a lack of π(proton)0f 7/2 occupation, allows for a high-lying ν(neutron)0f 5/2 orbital which creates the non-canonical N (neutron number) = 34 magic number [7][8][9]. A direct analogue exists in 24 O where an unoccupied π0d 5/2 orbital allows a large, compared even to the nearby 27 Ne [10], gap between ν0d 3/2 and ν1s 1/2 , to form, leading to the non-canonical N = 16 magic number [11,12].

There is some recent evidence that the removal of the valence proton from the π0d 5/2 orbital in 25 F leads to ∼ 60% population of excited states of 24 O [13]. The explanation for this in Ref. [13] was the induced configuration mixing of the 24 O core neutron states by the lowering of the ν0d 3/2 energy through an artificially strengthened tensor interaction with the π0d 5/2 proton. However, from the point of view of the pairing interaction, the spectroscopic factor to the ground-state would expected to be large. This can be illustrated in a two-level model of 0p-0h (ground state) and 2p-2h neutron states for 25 F and 24 O. The spectroscopic factor to the ground-state of 24 O following a π0d 3/2 proton removal is dependent on the product of amplitudes of the 2p-2h states in 25 F and 24 O which is positive owing to the definite signs of the off-diagonal two-body matrix elements of the pairing interaction [14].

Whether the population patterns from direct proton removal reactions 55 Sc(p, 2p) adhere to the pairing interaction expectation or tensor-driven effects is addressed in this letter. As with 25 F, 55 Sc has a non-canonical doubly magic core that results from the absence of tensor-driven attraction and a single proton in the orbital that acts to lower the valence neutron energies. In the present work and Ref. [13], projectiles were produced under the same surrounded by a time projection chamber (TPC) situated at the focal plane designated "F13" which is ∼40 m downstream of the 7 th focal plane. Surrounding MINOS was the DALI2 + array of 226 NaI(Tl) crystals [21,22] for the high-efficiency detection (∼23% for a 2 MeV γ ray emitted at β ion ≈ 0.6) of γ rays emitted by excited states populated by secondary reactions. Behind MINOS, reaction residues were identified, as shown in Fig. 1(b), with the SAMURAI spectrometer by their Bρ through a single dipole magnet, TOF from the target to a hodoscope, and energy loss in the hodoscope [23].

Reaction vertices in the LH 2 target were identified with a precision of ∼2 mm (σ) by tracking incoming fragments with beam-line detectors and tracking protons ejected from the target with the surrounding TPC [24].

Fragment velocities at the reaction vertex were reconstructed using the vertex position and velocity measured in SAMURAI. Beam-velocity neutron-detection was realised with NEBULA [23,25], comprising two assemblies of 60 vertically-aligned 12×12×180 cm 3 detectors arranged in a 2×30 configuration, and NeuLAND [START_REF] Aumann | Technical Report for the design, construction 451 and commissioning of NeuLAND: The high-resolution 452 neutron time-of-flight spectrometer for R3B[END_REF], one assembly of 400 5×5×250 cm 3 detectors arranged in

0 + 1 (2 + 1 ) (3 + 1 ) (3 - 1 ) (2,3,4,5) - S n

10 FIG. 1 .

 101 FIG. 1. Identified ions in front of (a) and behind (b) the reaction target. In (a), the ellipse indicates the software gate placed on 55 Sc. In (b), gates applied to select 53 Ca and 54 Ca are shown as dashed and solid ellipses, respectively. The scale is counts per bin and applies to (a) and (b).

120thick 9 Energy

 9 conditions and the same reaction is employed at an en-121 ergy of ∼ 200 MeV/nucleon. 122 The first study of 54 Ca populated through direct re-123 actions is presented. Exclusive cross sections of ground 124 and excited states have been measured as well as values 125 of removed nucleons that populated them. Only through 126 application of invariant-mass spectroscopy to the heavi-127 est exotic nucleus to date were such measurements pos-128 sible for the unbound states. Comparisons of the mea-129 surements were made to distorted-wave impulse approx-130 imation (DWIA) [15, 16] and conventional shell model 131 calculations, the latter employing the GXPF1Br interac-132 tion [7] in the full sd-pf -gds model space. From the com-133 parisons, values of removed nucleons and tentative J π 134 assignments to excited states of 54 Ca were determined. 135 The results show predominant ground-and excited-state 136 population of 54 Ca through valence and core proton re-137 movals, respectively. 138 Experimental investigations were conducted at the Ra-139 dioactive Isotope Beam Factory, operated by the RIKEN 140 Nishina Center and the Center for Nuclear Study, Uni-141 versity of Tokyo. A 240 pnA beam of 70 Zn 30+ was ac-142 celerated to 345 MeV/nucleon and secondary beams of 143 isotopes produced from its fragmentation on a 10-mm-144 Be target situated at the entrance of BigRIPS [18], 145 a two-stage fragment separator. Constituents of the 146 secondary beams were selected and separated up until 147 the 3 rd focal plane of BigRIPS by their magnetic rigid-148 ity (Bρ) through two dipole magnets and energy loss 149 (∆E) through an Al wedge-shaped degrader situated be-150 tween the dipole magnets. The time-of-flight (TOF), 151 Bρ, and ∆E were recorded for each ion between the 152 3 rd and 7 th focal planes. These measurements were 153 combined to provide unambiguous particle identification 154 (PID) of the ions' mass-to-charge ratio and atomic num-155 ber [19], shown in Fig. 1(a). Following identification, 156 the secondary beams were transported to the MINOS de-157 vice [20], a 151(1)-mm-long liquid hydrogen (LH 2 ) target 158

FIG. 2 .

 2 FIG. 2. (a) Doppler-corrected γ-ray spectrum of the 55 Sc(p, 2p) reaction populating bound states. The total fit function (red line) comprises simulated response functions of DALI2 + (blue short-dashed lines) and a double-exponential background (black long-dashed line). The inset spectrum is that in coincidence with the 2021-keV transition, where the blue dashed lines are intensity-fixed responses assuming the shown level scheme. Arrow widths on the level scheme represent relative γ-ray intensities. (b) Relative energy spectrum of 55 Sc(p, 2p) 54 Ca*→ 53 Ca+1n. The total fit function (red line) is the sum of the simulated responses of NeuLAND+NEBULA and beam-line detectors (blue short-dashed lines), and a non-resonant background (black long-dashed line). Inset is the decay scheme to 53 Ca, level energies are with respect to the 53 Ca ground-state energy and arrow widths are relative decay strengths. (c-f) PMDs of residual 54 Ca (c,d) and 53Ca+1n (e,f) following 55 Sc(p, 2p) reactions in coincidence with various states. "Inc. unbound" infers the inclusive PMD for unbound states, otherwise labels of (c-f) correspond to (g), which shows the energies, cross sections, and J π assignments of observed states. Data points in (c-f) are observed and solid curves are predicted by the DWIA for the d-, f -, and s-wave proton removals normalized to experimental cross sections. Where appropriate, Sn and S2n[17] values are indicated.

180Fig. 2 (

 2 Fig. 2(a), with intensities which imply the level scheme 192

2 FIG. 3 . 256 Figure 2

 232562 FIG. 3. (Top panel) Below Sn, the measured cross sections to states from γ-ray spectroscopy are shown. Above Sn, the population cross sections of the unbound states from invariant mass spectroscopy are shown at the energy centroids of fitted values and likely represent contributions from several states. States with conclusive PMDs are colored accordingly to the -value of their removed nucleon, otherwise are black. (Bottom panel) Theoretical predictions of state energies and their population cross sections are shown. Contributions to cross sections from proton removals from the different orbitals are indicated. Unlabelled levels have J π = 2, 3, 4, 5 -.

  The inclusive cross section to bound states was measured to be 1.69(3) mb, close to the predicted 1.83 mb, suggesting little quenching compared to what is expected from such reactions[33].Above S n , it is stressed that whilst the cross sections are indicated at the level energies shown in Fig.2(g), they are likely distributed across many unresolvable states. The over-prediction in the range S n < E level < S 2n may result from some levels predicted to lie below S 2n in-fact existing beyond it and decaying via 2n emission, since the bound-state decays are well reproduced by the prediction, quenching of spectroscopic factors is unlikely. In addition, it is noted that exclusion of the deeply bound π1s 1/2 contribution lowers predicted cross section towards the measured value. As for cross sections of the strengths beyond S 2n , direct comparison to theory is difficult owing to the lack of sensitivity to the 2n decay channel. If the bold assumption is made that the levels observed in the 53 Ca+1n E rel spectra only decay via 1n emission, excluding the π1s 1/2 contribution brings the predicted value to agreement with observation. Proton removal from π0d 3/2 dominates the excited states, reflected by the inclusive PMD. Whilst a strong population of the 4 - 9 state following π1s 1/2 proton removal is predicted it is not observed clearly in any PMD, possibly due to the influence of states populated by π0d 3/2 proton removal.Negligible population of excited states of 54 Ca originates from the proton removal from π0f 7/2 in 55 Sc. All excited states are consistent with population from coreproton removals, predominantly from π0d 3/2 , and potentially π1s 1/2 . These findings are contrary to the analogous case of 25 F(p, 2p)[13], in which valence proton removals preferentially populated excited states of24 O.The shell model calculations result in 0p-0h (closedN =34shell) contributions of the 55 Sc and 54 Ca groundstate wavefunctions as 64.2% and 89.4%, respectively. The remainder of the contributions are dominated by 2p-2h configurations. Despite this erosion of the N = 34 closure in 55 Sc, which is supported by experiment [34], the spectroscopic factors (C 2 S) show removal from π0f 7/2 mostly populates the ground-state of 54 Ca, as observed experimentally. This can be understood through a simple two-level system of both 55 Sc and 54 Ca. Assuming each have wavefunctions that only have contributions from 0p-0h (φ 0p0h ) and 2p-2h (φ 2p2h ) configurations, 337 Ψ 55 Sc = α 0p0h φ 0p0h 55 Sc + α 2p2h φ 2p2h 55 Sc , Ψ 54 Ca = β 0p0h φ 0p0h 54 Ca + β 2p2h φ 2p2h 54 Ca , where α and β are real numbers satisfying α 2 0p0h +α 2 2p2h = 338 1 and β 2 0p0h +β 2 2p2h = 1. With vanishing overlaps of φ 0p0h 339 and φ 2p2h , the spectroscopic factor to the ground state 340 of 54 Ca can be expressed as,

	Broadly speaking, the location of level strengths are in
	good agreement with prediction. Between S n and S 2n
	a total level strength of 3.57(30) mb is observed, com-
	pared with predicted 5.30, 5.19, 4.87, and 4.75 mb con-
	sidering the df s, ds, df , and d contributions, respectively,
	and subtracting the 3 -1 contribution. Beyond S 2n , within
	the experimentally sensitive range, 2.12, 2.09, 0.96, and
	0.93 mb population strengths are predicted for df s, ds,
	df , and d contributions, respectively, compared to a lower
	limit of >1.01 mb deduced from a 1.14(13) mb strength
	in the 1n channel.
	267	+ 1 ), 0.121(84) mb, states are repro-
	268	duced by the calculated, 1.65 mb and 0.086 mb, respec-
	269 270	tively. Moreover, the measured (2 + 1 ) energy is in agree-ment with theory. Some population of the 0 + 2 state is
	271	predicted with a cross section consistent with that of the
	272	low-significance 561-keV γ-ray candidate, not shown on
	273	Fig. 3. To a very good approximation, the predicted
	274 275 276	3 + 1 at 2.99 MV has zero cross section, consistent with observation of the (3 + 1 ) state at the lower 2.47 MeV. The predicted 3 -1 state is unbound, contrary to observa-
	277	tion, however, the measured cross section, 0.316(43) mb,
	278	agrees well with the predicted 0.35 mb. It is notewor-
	279	thy that similar contributions to the 3 -1 state from the
		π1s 1/2 , 0.19 mb, and π0d 3/2 , 0.16 mb, orbitals are pre-

280

dicted. The state's experimental PMD shown in Fig.

2(d

)

281

does not reflect such a contribution.

ter accelerator staff for the stable and high-intensity transport of the Zn primary beam, and the BigRIPS team for their preparation of the magnetic settings of the secondary beam. F. B. is supported by the RIKEN Special Postdoctoral Researcher Program. S. C. acknowledges support from the IPA program at the RIKEN Nishina Center. K. O. and K. Y. acknowledge the support from Grants-in-Aid of the Japan Society for the Promotion of Science under Grants No. JP16K05352. This work was supported by JSPS KAKENHI Grants No. JP16H02179 and JP18H05404, the DFG under Grant No. BL 1513/1-1 HGS-HIRe, the BMBF (grant No. 05P19RDFN1), Swedish Research Council under Grant Nos. 621-2014-5558 and 2019-04880. L.X.C and B.D.L are supported by the Vietnam MOST via the Physics Development Program Grant No. ÐTÐLCN.25/18. D. So. was supported by the the European Regional Development Fund contract No. GINOP-2.3.3-15-2016-00034 and the National Research, Development and Innovation Fund of Hungary via Project No. K128947.

C 2 S g.s. = (α 0p0h β 0p0h + α 2p2h β 2p2h ) 2 .

(1)

Since pairing forces have a definite sign in their off-342 diagonal two-body matrix elements, the amplitudes α 2p2h 343 and β 2p2h take on the same sign [14]. Therefore,


L Lalanne
email: louis.lalanne@ijclab.in2p3.fr

O Sorlin

M Assié

F Hammache

N De Séréville

S Koyama

D Suzuki

F Flavigny

D Beaumel

Y Blumenfeld

B A Brown

F De

Oliveira Santos

F Delaunay

S Franchoo

J Gibelin

V Girard-Alcindor

J Guillot

O Kamalou

N Kitamura

V Lapoux

A Lemasson

A Matta

B Mauss

P Morfouace

M Niikura

J Pancin

A Poves

T Roger

T Saito

C Stodel

J-C Thomas

Evaluation of the 35 K(p,γ) 36 Ca reaction rate using the 37 Ca(p,d) 36 Ca transfer reaction

HAL is

I. INTRODUCTION

Type-I X-ray bursts are among the most energetic events known, which occur in binary systems consisting of a neutron star accreting H/He-rich material from its companion star [1]. As the accreted material builds up on the surface of the neutron star, high temperatures and densities (T peak ≥ 0.9×10 9 K and ρ ≈ 10 6 g cm -3 ) are reached. A thermonuclear runaway occurs, leading to a sharp increase of X-ray emission from the star that lasts approximately 10-100 s. One of the most important challenges in studying X-ray bursts is understanding the observed luminosity profile, which is directly related to the energy released by the nuclear reactions occurring during the thermonuclear explosion. The comparison of the observed light curves to the X-ray burst model predictions may be used to constrain the composition of the neutron star's crust as well as its properties (mass, radius) [START_REF] Zamfir | The Astrophysical[END_REF]. X-ray burst models are sensitive to the nuclear reaction rate inputs and recent sensitivity studies [3,4] have shown that among the thousands of reactions involved, only the ones participating in the breakout of the hot-CNO cycle and a few tens of (α, p) and (p, γ) reactions have a strong impact on the energy generation of the burst and the final abundances.

The most important (α, p) reactions to be studied are usually those involving waiting point nuclei [3], where arXiv:2107.04317v1 [nucl-ex] 9 Jul 2021 the nuclear reaction flow stalls due to a (p, γ) -(γ, p) equilibrium. This implies to await the β + decay, unless the (α,p) reaction is fast enough to bypass the waiting point and reach higher Z nuclei. The 34 Ar nucleus is such a waiting point and the reaction flow is expected to escape it through an (α, p) reaction, unless a series of two proton captures leading to 36 Ca can compete. The 35 K(p, γ) 36 Ca reaction rate, studied in the present work, has been found to have a significant influence on this reaction pathway, as well as the predicted X-ray burst light curve, when its nominal value is increased by a factor of one hundred [3].

At the typical temperature of an X-ray burst, T = 0.5-2 GK [START_REF] Iliadis | Nuclear Physics of Stars[END_REF], the Gamow window for the 35 K(p,γ) 36 Ca reaction lies between E c.m. =0.37 MeV and E c.m. =1.93 MeV (S p =2599.6(61) keV). It corresponds to excitation energies in the 36 Ca compound nucleus between 2.97 and 4.53 MeV. Considering the 3/2 + ground state (g.s.) spin value of 35 K and the fact that the most relevant proton captures will mostly occur through an s-wave ( = 0), the resonances of interest in 36 Ca have J π =1 + ,2 + .

Historically, owing to the lack of experimental spectroscopic information on 36 Ca, several studies [START_REF] Herndl | [END_REF][7][8][9] have estimated the 35 K(p, γ) 36 Ca reaction rate using theoretical predictions for partial widths and a 2 + energy either calculated or adopted from that of the mirror nucleus 36 S (3.291 MeV). The contribution from the 1 + resonance, identified in the mirror nucleus to be at 4.523 MeV, was not considered in these works. The reaction Q-value was derived from the experimental atomic masses of 36 Ca and 35 K, known at that time from the 40 Ca( 4 He, 8 He) 36 Ca [10] and 40 Ca( 3 He, 8 Li) 35 K [11] transfer reactions, respectively.

Since then, the excitation energy of the 2 + first-excited state in 36 Ca was measured at GANIL [12], GSI [13] and NSCL [START_REF] Amthor | Experimental and theoretical study of nuclear reaction rates in the rp-proces[END_REF] by means of one-neutron knockout reactions from a 37 Ca secondary beam. Taking the most precise measurement, its energy is found to be 3045.0 ± 2.4 keV [START_REF] Amthor | Experimental and theoretical study of nuclear reaction rates in the rp-proces[END_REF]. Moreover, the mass of 35 K was precisely measured using a Penning trap mass spectrometer at ISOLDE [START_REF] Yazidjian | [END_REF]. Therefore, the major uncertainty on the reaction Q-value, and thereby the 2 + resonance energy E c.m. r (2 + ), came from the uncertainty on the atomic mass of 36 Ca (∆M ( 36 Ca)= -6440 ± 40 keV [10]). The most recent estimate of the reaction rate was made by Iliadis et al. [16] by taking into account a 2 + state located at 3015 ± 16 keV [13] corresponding to a resonance energy of E c.m. r (2 + )=459 ± 43 keV. The presence of this 2 + state at relatively low energy induces an enhanced resonant capture component in the reaction rate, as compared to what was estimated in Fig. 7 of Ref. [START_REF] Herndl | [END_REF], using E c.m. r (2 + )=700 keV. While writing the present paper, a more precise mass excess of ∆M ( 36 Ca)= -6483.6 (56) keV has been obtained using a Time of Flight -Ion Cyclotron Resonance measurement in a Penning trap [START_REF] Surbrook | First penning trap mass measurement of 36 Ca[END_REF]. This leads to a reaction Q-value of 2599.6(61) keV and E c.m. r (2 + )=445 ± 7 keV, combining the precise mass measurements of 36 Ca [START_REF] Surbrook | First penning trap mass measurement of 36 Ca[END_REF] and 35 K [START_REF] Yazidjian | [END_REF].

In this paper, we report the spectroscopy of 36 Ca through the one neutron pick-up reaction 37 Ca(p, d) 36 Ca in inverse kinematics. Excitation energies and proton branching ratios of the excited states in or near the Gamow window were obtained. Moreover, the mass excess ∆M ( 36 Ca), presently obtained using the reaction Q-value of the (p,d) transfer reaction and well-known atomic mass of 37 Ca [START_REF] Ringle | [END_REF] was compared to the values of Ref. [10,[START_REF] Surbrook | First penning trap mass measurement of 36 Ca[END_REF]. These relevant pieces of information are used to better constrain the 35 K(p,γ) 36 Ca reaction rate at X-ray burst temperatures and above.

II. DESCRIPTION OF THE EXPERIMENT

A. Secondary beam production

The 37 Ca nuclei were produced at GANIL in fragmentation reactions of a 95 MeV nucleon -1 40 Ca 20+ beam, with an average intensity of ≈2 µAe, on a 2 mm 9 Be target. They were separated from other reaction products by the LISE3 spectrometer [19]. A 546 µm wedgeshaped Be degrader was inserted at the intermediate focal plane to induce a Bρ -∆E -Bρ selection among the nuclei transmitted after the first magnetic rigidity selection. The Wien filter, located at the end of the spectrometer, was operated at 2400 kV m -1 in order to induce an additional velocity selection among the transmitted nuclei. The 37 Ca nuclei were produced at 48 MeV nucleon -1 with a mean rate of 3500 pps and a purity of 20%. Two low-pressure multi-wire devices, CATS [20], were placed 67.8 and 118.8 cm upstream of the liquid Hydrogen target CRYPTA [21], in order to track the incident ions and determine their interaction point on the target. A schematic layout of the set-up is shown in Fig. 1. As shown in the left part of Fig. 2, the incoming nuclei were identified through their time-of-flight (TOF) measurement, between the CATS detectors and the cyclotron radio-frequency, and their energy loss in an ionization chamber placed at the LISE spectrometer's image focal plane. The main nuclei transmitted along with 37 Ca were 36 K, 35 Ar and 32 S. Another setting of the spectrometer (not shown here) was used to select a cocktail of secondary beam nuclei, shifted by one unit of mass closer to stability, among which the 38 Ca and 33 S nuclei were produced. These nuclei, along with 35 Ar and 32 S, were used to calibrate the detectors with the deuterons emitted in the (p,d) reaction, as described in Sect. II C. As the 37 Ca nucleus is separated in TOF from the other nuclei, the focal plane ionization chamber was only inserted at the beginning and end of each run to control the incoming beam content.

B. Experimental setup

The liquid Hydrogen (T ≈ 18 K) of the CRYPTA target was contained, at a pressure of 0.9 bar, in a 7.6 cm diameter Al cell with circular apertures of 20 mm at both ends, closed by 6.47 µm thick Havar foils. To reach the liquefaction point of the H 2 gas (20 K at 1 bar), the vacuum inside the reaction chamber had to be maintained below 2×10 -6 mbar. Due to the important difference in pressure between the target cell and the reaction chamber, the filling of the target with liquid H 2 introduced a significant deformation of the Havar foils. This deformation has been parametrized, using a 10 µm precision position measurement from a laser system, in order to correct from the energy loss of the particles inside the target event by event. The target thickness spanned from 0.5 mm (at the edges) to 1.5 mm (at the center), the latter corresponding to an effective thickness of 9.7 mg cm -2 . The target cell was surrounded by a heat shield made of 0.2 µm aluminized Mylar foils to protect the target cell from radiations. During the experiment, gas contaminants were condensing on the heat shield, forming a layer of ice of unknown thickness and composition (estimated to be less than 10 µm equivalent H 2 O) crossed by the light ejectiles. To minimize this effect and keep the H 2 in a liquid phase, the target was warmed-up and cooled down three times during the 12 days of experiment in order to evaporate the ice layer.

After interaction with the target nuclei, the trajectories of the transfer-like nuclei, their atomic number Z and their time-of-flight (referenced to the CATS detectors) were determined by means of a Zero Degree Detection (ZDD) setup, composed of an ionization chamber, a set of two XY drift chambers located at 85 cm and 103 cm from the target, followed by a 1 cm thick plastic scintillator. The angular acceptance of the ZDD does not induce kinematical cuts on the detection of the recoil nuclei. Their identification in Z was performed through the measurement of their energy losses in the ionization chamber, as shown in the right part of Fig. 2.

The energy and angle of the light ejectiles were measured by a set of 6 MUST2 telescopes [START_REF] Pollacco | The 4th International Conference on Exotic Nuclei and Atomic Masses[END_REF] arranged in the forward direction to cover angles from 3 to 37 • in the laboratory frame. Four of them, placed at 22 cm from the target, were covering angles from 5 to 37 • and two more were placed 57 cm behind them to cover smaller angles from 3 to 5 • . For the 37 Ca(p,d) 36 Ca reaction, this corresponds to center-of-mass angles between 2 • and 160 • .

Each telescope consisted of a 300 µm thick Doublesided Silicon Stripped Detector (DSSD) with 128 strips on each side, backed by sixteen 4 cm thick CsI detectors, read out by photodiodes which provide energy-loss (∆E) and residual energy (E) measurements, respectively. Light particles identification was obtained from a ∆E -E matrix for punching through particles. Their total kinetic energy was obtained from the sum of their energy loss in the DSSD and their residual energy in the CsI crystals, after being corrected by the calculated energy losses in the cryogenic target, its windows and heat shields. The emission angle of the light ejectiles is deduced from the information on the impact point of the incident beam on target reconstructed from CATS detector information, and the position measurement of the ejectile in a given strip of the DSSD, with a precision better than 1 • .

C. Energy Calibrations of the MUST2 detectors

Even if the atomic mass of 36 Ca has now been measured with a good accuracy [START_REF] Surbrook | First penning trap mass measurement of 36 Ca[END_REF], it is interesting to determine its value with another method, based on transfer reactions. Even though less precise, this method is more generic and can also be applied to the determination of masses of unbound nuclei. In the present work, the atomic mass and the energy of the excited states of 36 Ca have been determined through the measurement of the energies and angles of the deuterons produced in the 37 Ca(p, d) 36 Ca transfer reaction. Moreover, when populating unbound states in 36 Ca, protons are also emitted and their kinematics can be used as well to determine the energy of the resonant states. Thus a precise energy calibration of both deuterons and protons is required in the DSSD as well as in the CsI crystals, in which they are stopped.

The DSSDs were calibrated strip by strip using a mixed alpha source ( 239 Pu, 241 Am, 244 Cm) placed at the target position, leading to a global energy resolution of about 40 keV (FWHM) at 5.5 MeV for each telescope.

The first step for the energy calibration of the CsI crystals relies on the E -∆E correlation of the light particles. The incident energy of each light particle is deduced from its measured energy loss ∆E in the DSSD. The residual energy in the CsI crystals is subsequently calculated from the measured energy loss in the silicon detector and used to calibrate the CsI crystals. Dead layers are accounted for in this procedure. The correlation between the calculated residual energy (in MeV) and its experimental channel number in the ADC was fitted with a second order polynomial function to determine the calibrated CsI energy E ∆E CsI . Given that the CsI crystals show surface and volume inhomogeneities in the light collection (of about ± 5%) caused by their aging, each CsI crystal was virtually divided into 64 pixels using the position information from the DSSD. The energy calibration of the CsI crystals was then done pixel by pixel, eventually cover- ing their full surface. Since the amount of scintillation light produced in a CsI crystal depends on the type of particle, this calibration procedure was performed for the deuterons and the protons independently.

The second step of the energy calibration of the CsI crystals aimed at correcting the effects of the deformation uncertainties of the target's windows and the nonhomogeneity of the dead layers. For this purpose, reference transfer reactions with well known Q-values (with less than 2 keV uncertainty) such as 38 Ca(p, d) 37 Ca g.s. and 35 Ar(p, d) 34 Ar g.s. were measured during the experiment and used for a refined energy calibration in the energy range of the deuterons resulting from the 37 Ca(p, d) 36 Ca transfer reactions to the ground and excited states up to 6 MeV (see blue, magenta and green crosses in Fig. 3). The reference energy E ref CsI , calculated event by event, is the expected energy, at the angle where the deuteron was measured, to reconstruct the known mass of the reference nuclei. The error on E ref CsI arises from the propagation of the uncertainties on the measured angle. Due to the lack of statistics this second step calibration could only be applied to the CsI crystal and not to each pixel as in the first step. The calibrated values of Fig. 3

display a linear relationship between E ref

CsI and E ∆E

CsI over a large range of deuteron energy. Data points corresponding to the population of the 36 Ca g.s.

(cyan) are enclosed between the three reference (p, d) reactions. Due to the lack of reference reactions giving rise to a precise determination of the proton energy, the above procedure could only be applied to deuterons. This second step calibration allows to improve the resolution on the excitation energy by 20% and to reduce the uncertainty on the mass measurement by a factor 3.

III. EXPERIMENTAL RESULTS

A. Mass excess of 36 Ca

The mass excesses of 37 Ca, 34 Ar, and 36 Ca, given in Table I, have been determined from the invariant mass value of their ground state population through (p,d) reactions. The error bars obtained for reference nuclei show the best reachable precision on mass excess measurement with this method, since they are the nuclei used for the calibration. The mass excess of 36 Ca , ∆M = -6480(40) keV, measured in this work, is in good agreement with the recent measurement, ∆M = -6483.6(56) keV of Ref. [START_REF] Surbrook | First penning trap mass measurement of 36 Ca[END_REF]. As expected, our uncertainty on the 36 Ca mass is larger than the one obtained from the penning trap measurement [START_REF] Surbrook | First penning trap mass measurement of 36 Ca[END_REF], but similar to that obtained in another transfer reaction [10]. This uncertainty is dominated by systematic errors arising from multiple effects such as the propagation of errors on the measured angle and energy of the deuteron and on the energy calibration of the CsI. They have been estimated combining the standard deviation of independent measurements performed using the 4 MUST2 telescopes, located at the closest distance from the target. Taking the most precise atomic mass values of 36 Ca and 35 K, the proton separation energy of 36 Ca is deduced to be S p = 2599.6(61) keV. The missing mass method has been applied in inverse kinematics to determine the excitation energy (E x ) of the states produced in 36 Ca. After gating on an incoming 37 Ca the excitation energy is reconstructed from the total kinetic energy and the angle of the deuterons produced in the (p, d) reaction. Figures 4a) and4b) display the E x spectra gated on the outgoing Ca or K nuclei in the ZDD (as shown in the right part of Fig. 2), respectively. The fit of the excitation energy spectrum has been performed using multiple Gaussian function, assuming that the natural width of the states is much smaller than the experimental resolution. The red lines in Fig- ures 4a) and 4b) show the best total fits obtained and the colored dashed lines are the individual state contributions used for each fit. All the parameters of the fit are free except the resolution. The energy-dependence of the peak widths was estimated using the nptool package [25], in which the whole set-up was simulated. The resolution was then strongly constrained in the fit, using the reference width of the known and isolated ground state and the simulated energy-dependence. The number of contributions used in the fit was guided by the number of levels populated in the mirror reaction [26] and by the statistical test of the p-value.

The peaks corresponding to the feeding of the ground and first 2 + excited states in 36 Ca are well identified in Fig. 4a). As expected, the peak corresponding to the g.s. disappears when gated on K nuclei. The energy of the 2 + state is measured at 3059 [START_REF] Bohr | Nuclear Structure, Nuclear Structure[END_REF] keV in Fig. 4a) and 2982 (120) keV in Fig. 4b) (blue curve), in agreement with the value of 3045 (2.4) keV [START_REF] Amthor | Experimental and theoretical study of nuclear reaction rates in the rp-proces[END_REF], within the error bars. The relatively large uncertainties arise from a nearby resonance, as will be discussed below. As the 2 + state is unbound with respect to one and two proton emissions, a certain fraction of its decay occurs to the ground state of 35 K, bound by only 83.6 (5) keV [START_REF] Wang | [END_REF], with the emission of a proton. This is discussed in the following.

In Fig. 4c), the one-proton energy spectrum E c.m. p has been reconstructed in the 36 Ca center-of-mass from the measured energy and angle of the proton in coincidence with the deuteron of the (p, d) reaction and the outgoing K nuclei. For convenience, the one-proton separation energy (S p ( 36 Ca ) = 2599.6(61) keV) has been added in Fig. 4c) to the proton energies to ease the comparison with the excitation energy spectra of Figs. 4a,b). The resulting excitation energy resolution is 2 to 4 times better when reconstructed with the protons than with the deuterons: 130 keV at E x = 3 MeV and 300 keV at 5 MeV with the protons and an almost constant value around 550 keV with the deuterons. This effect arises from the more forward focused kinematics of the protons, as compared to deuterons. In addition, as the proton energy spectrum is constructed with less than half of the CsI crystals, the systematic uncertainty caused by their inhomogeneities is smaller in the E c.m. p spectrum. Thus, the peak corresponding to the 2 + 1 state is better separated from the others in the E c.m. p spectrum of Fig. 4c), as compared to the excitation energy peak shown in Fig. 4b). Note also that the triple coincidence (deuteron, proton and K nucleus) cleans the E c.m. p spectrum from all type of background. The fit of the E c.m. p spectrum has been performed using multiple Gaussian functions, whose energy-dependent widths have been constrained from simulations, assuming again that their natural width is much smaller than the experimental resolution. The energy of the 2 + 1 state is found at 3057 (20) keV. Its uncertainty comes from the moderate statistics. The presently determined 2 + 1 energy agrees well with the ones of 3036 (11) [12] and 3045(2.4) [START_REF] Amthor | Experimental and theoretical study of nuclear reaction rates in the rp-proces[END_REF] keV, determined by γ-decay, as well as the value of 3059 [START_REF] Bohr | Nuclear Structure, Nuclear Structure[END_REF] keV derived from our fit of the excitation energy spectrum of Fig. 4a). As shown in Fig. 4c), several states are identified up to 6 MeV. One of them at E x = 4.243(40) MeV lies inside the Gamow window of the 35 K(p, γ) 36 Ca reaction. This state is also visible at a compatible energy in the excitation energy spectrum of Fig. 4b). According to the selection rules of the (p, d) transfer reaction, J π =1 + and 2 + states are populated with a (s 1/2 ) 1 (d 3/2 ) 1 configuration in the removal of an =0 neutron from the 2s 1/2 orbital. This assumption is confirmed by the fact that one J π =1 + and two 2 + states are populated in the same energy range in 36 S by means of the mirror reaction 37 Cl(d, 3 He) 36 S [26]. The isobaric analogue 1 + 1 state was measured at E x (1 + )= 4523.0 (6) keV. Therefore we tentatively assign a spin-parity of 1 + for the excited state of 36 Ca at 4.243(40) MeV. Two shell model calculation was performed, one in the sd valence space with USDB interaction and the other using the full sdpf valence space with sdpfu-mix plus Coulomb interaction. Calculation in sd valence space predicts the position of this 1 + 1 state in 36 Ca at E x = 4161 keV and while calculations in sdpf valence space predicts E x = 4000 keV.

Given that the energy resolution of the proton spectrum is accurately determined from simulations, two states are needed between 4 and 5 MeV to achieve the best fit of the spectrum. Besides the 1 + state at 4.24 MeV discussed above, a ( 2 + 2 ) state has been identified at E x = 4.706 (100) MeV in 36 Ca , close to the upper border of the Gamow window. The identification of two states (tentatively 1 + and 2 + 2 ) in this energy range is reinforced by a similar observation in the mirror nucleus 36 S, with a (1 + , 2 +

2 ) doublet at 4.523 and 4.572 MeV, fed within similar relative intensities in the 37 Cl(d, 3 He) 36 S reaction. The energy and feeding of these states, obtained from Fig. 4c), are compatible with those obtained independently in the excitation energy spectrum of Fig. 4b) from the detection of deuterons.

Other states are observed between 5 and 8 MeV, but since they are well outside of the Gamow window, they will have no impact on the reaction rate and will not be discussed here. As a last remark, despite the fact that all states discussed here ( 2 + 1 ,1 + ,2 + 2 ) are unbound by twoproton emission (S 2p ≈ 2683 keV), no peak is observed at the corresponding resonance energy in the Ar-gated E x spectrum (not shown here). Therefore, we shall neglect their 2p decay contribution to determine their proton to γ decay branches in the following section.

C. Experimental proton branching ratios

The first 2 + excited state of 36 Ca at E x = 3.045 MeV has been identified both in the excitation energy spectra gated by Ca and K, meaning that it decays through γray and proton emission with respective partial widths Γ γ and Γ p . We propose here to determine its experimental proton branching ratio B p =Γ p /Γ tot , with Γ tot =Γ γ +Γ p , using two methods. As for the 1 + and 2 + 2 states, no counts are observed in the γ decay branch, we shall determine minimum values of B p , based on the fit of the K-gated and Ca-gated excitation energy spectra. First method-The experimental proton-deuteron angular correlation is shown in Fig. 5 as a function of the proton center-of-mass emission angle in the 36 Ca frame. This correlation function is obtained after normalisation of the number of decaying protons per angle by the total number of counts observed in the excitation energy peak and Tentative spins and parities J π , measured excitation energies Ex (in keV) and proton branching ratios are listed for the three states identified in 36 Ca . Results of Shell Model calculations for partial γ-width (meV), proton spectroscopic factors C 2 S and proton width (in meV) as well as their corresponding proton branching ratios are presented. Two different shell model calculations have been performed, one restricted to the sd orbitals and USDB interaction, the other using the full sdpf valence space with sdpfu-mix plus Coulomb interaction. The results are compared to the shell model results of Ref. [START_REF] Herndl | [END_REF]. Predicted widths, obtained for a given calculated excitation energy, have been renormalized to the experimental values given in the second column. The proton spectroscopic factors are given for the orbital which contributes the most to the proton width (i.e. s 1/2 orbital for all states) Present work

Herndl [START_REF] Herndl | [END_REF] Exp correction of the relative geometrical efficiency between protons and deuterons (which have significantly different angular distributions). The geometrical efficiency was computed using the nptool simulation package where the 37 Ca(p,d) 36 Ca transfer reaction and the subsequent proton emission were both simulated with an isotropic angular distribution. It has been restricted to events corresponding to proton center-of-mass energies ranging from 2.5 to 3.5 MeV to focus on the study of the 2 + 1 decay. Errors for each point of the angular correlation are dominated by statistical uncertainties.

This correlation function W (θ) can be described by a sum of even Legendre polynomials, P k (cos(θ)) [28]:

W (θ) = kmax k=0 A k P k (cos(θ)), (1)
where A k are coefficients obtained from a fit to the experimental angular correlation. The sum is truncated at a maximum value of k max = min( + , 2J), where and are the possible proton orbital angular momenta, and J is the spin of the decaying state. The value of k max =2, which results from the best fit shown in Fig. 5, can then be used to constrain the spin assignment of the decaying 36 Ca state. Given the fact that the ground state of 35 K has J π = 3/2 + , this implies that the first excited state in 36 Ca has either J = 1 or J = 2 assignment. This is in agreement with the J π = 2 + value expected from the mirror nucleus and shell model calculations.

By integrating the angular correlation function over the full 4π solid angle, a proton branching ratio of B p = Γ p /Γ tot = 0.16 (2) is determined. The uncertainty results from the error propagation of the fitted parameters. Second method-By counting the number of events associated to the feeding of the first 2 + excited state in the K-gated and Ca-gated excitation energy spectra, a Γ p /Γ γ = 0.21 (3) ratio is found. Using the fact that:

B p = 1/(1 + Γ γ /Γ p ), (2)
a branching ratio of B p = 0.17 (2) is obtained. The uncertainty is due to the fitting of the peaks in the excitation energy spectrum, as well as in the choice of the gates in Ca and K in the ZDD spectrum of Fig. 2. The proton branching ratio values for the 2 + state derived with the two methods are in excellent agreement.

The second method was applied to compute the following B p lower limits: > 0.96 for the 1 + state and > 0.97 for the 2 + 2 at one σ confidence level. The fit of the Cagated excitation energy spectrum of Fig. 4a) was used to estimate their maximum contributions to the γ-decay channel, such as the small one to the 1 + state, displayed in brown color.

As shown in Table II, the weighted average value B p = 0.165 (10) for the 2 + state is found to be significantly weaker than all theoretical predictions, while the deduced B p lower limits for the higher 1 + and 2 + 2 states are consistent with the predictions.

D. Calculated gamma widths

The predicted Γ γ partial widths of Table II, expressed in meV, were computed using the following relation [START_REF] Iliadis | Nuclear Physics of Stars[END_REF]:

Γ γ (ωL) = 8π(L + 1) L[(2L + 1)!!] 2 E γ c 2L+1 B(ωL), (3)
where ω names the nature of the transition (Electric or Magnetic) and L its multipolarity. B(ωL) is the reduced transition probability for the γ decay of the resonant state (in unit of e 2 f m 4 for E2 transitions and µ 2 N for M1 transitions) and E γ the energy of the γ-ray transition. Shell model calculations usually predict B(ωL) and E γ values, from which Γ γ is calculated. However, as the experimental excitation energy of the 2 + 1 state is known and differs from the calculated ones, the predicted partial widths Γ γ listed in Table II are obtained from Eq. 3 using experimental energies and calculated B(E2) values.

Two different shell model calculations have been performed in the present work, one restricted to the sd orbitals and USDB interaction, the other using the full sdpf valence space with sdpfu-mix plus Coulomb interaction. The reduced transition probabilities, which contribute the most to the γ-ray width of each state, obtained in sd valence space are: B(E2;

2 + 1 → 0 + g.s. ) = 2.4 e 2 fm 4 , B(M 1; 1 + → 0 + g.s. ) = 0.01 µ 2 N , B(M 1; 1 + → 2 + 1 ) = 1.4 µ 2 N , B(M 1; 2 + 2 → 2 + 1 ) = 0.002 µ 2 N , B(E2; 2 + 2 → 0 + g.s. ) = 0.02 e 2 fm 4 . Values obtained in sdpf shell are B(E2; 2 + 1 → 0 + g.s. ) = 4.7 e 2 fm 4 , B(M 1; 1 + → 0 + g.s. ) = 0.04 µ 2 N , B(M 1; 1 + → 2 + 1 ) = 1.5 µ 2 N , B(M 1; 2 + 2 → 2 + 1 ) = 0.06 µ 2 N , B(E2; 2 + 2 → 0 + g.s. ) = 2.2 e 2 fm 4 .
The major difference between the shell model calculations presented here or in Ref. [START_REF] Herndl | [END_REF], resides in the size of their valence spaces: when restricted to sd shells the proton core is closed, while the use of a broader sdpf valence space allows proton excitations. When using the sd valence space, the 2 + 1 state in 36 Ca ( 36 S) is of pure neutron (proton) origin. It follows that the B(E2) values of the two mirror nuclei can simply be derived from their squared neutron to proton effective charges ratio, B(E2)( 36 Ca)= e 2 n /e 2 p B(E2)( 36 S), where e n (e p ) are the neutron (proton) effectives charges usually adopted to be 0.5 (1.5).

As the 2 + 1 state in 36 Ca is certainly not totally of pure neutron origin, the calculated Γ γ using a sd valence space (≈ 0.4 meV) represents a lower limit. At the other extreme, a maximum Γ γ of about 3.7 meV is obtained for 36 Ca when assuming the same B(E2) value as in the mirror nucleus 36 S, after correcting from their different 2 + 1 energies. This latter assumption would imply that the 2 + state has a very mixed (and similar) structure in both nuclei. This is very unlikely for two reasons. First, the two nuclei are likely doubly magic, at least based on the high excitation energy of their first excited states. Second, the 2 + 1 state in 36 S is very well populated by the 37 Cl(d, 3 He) 36 S proton removal reaction, with spectroscopic factors values for the 2 + 1 and 1 + 1 states (0.86 and 0.75 respectively [26]) that are close to the single particle values, meaning it has a strong proton component rather than a mixed proton and neutron one.

E. Calculated proton widths

The proton widths Γ p of the states listed in Table II are obtained by multiplying their single-particle width Γ sp with the spectroscopic factor C 2 S:

Γ p = Γ sp × C 2 S. (4)
The C 2 S values are obtained from shell model calculations, while Γ sp are calculated by scattering phase shifts in a Woods-Saxon potential [29] whose depths are adjusted to match the experimental resonance energies. The Wood-Saxon potential parameters used for calculation can be found in page 239 of Ref. [START_REF] Bohr | Nuclear Structure, Nuclear Structure[END_REF]. In the present work, the widths of the 2 + 1 state obtained in the sd and sdpf shell model calculations agree very well with each other, while those for the 1 + state differ by more than a factor two.

It is important to note that the Γ p values are obtained by multiplying a very small C 2 S number (of the order of 10 -3 ) by large barrier penetrability factors for the protons, especially for those having > 0. Despite this, the Γ p values obtained with the two calculations agree reasonably well. The C 2 S values are small as they correspond to the emission of a proton from an almost pure 1p1h neutron state, selectively populated here by the (p,d) transfer reaction.

IV. THE 35 K(p,γ) 36 CA REACTION RATE
The thermonuclear reaction rate per particle pair is given by [START_REF] Longland | the 2010 Evaluation of Monte Carlo based Thermonuclear Reaction Rates[END_REF]:

< σν >= 8 πµ 1/2 1 (kT ) 3/2 ∞ 0 Eσ(E)e -E/kT dE, ( 5
)
where µ is the reduced mass of the two interacting nuclei, k is the Maxwell-Boltzmann constant, T is the temperature in Kelvin, E is the center-of-mass energy in MeV and σ(E) is the nuclear reaction cross section in barn.

The 35 K(p, γ) 36 Ca reaction rate depends on resonant capture (RC) and direct capture (DC) contributions, that we shall discuss in the following.

A. Resonant capture

In the presence of narrow resonances, the reaction rate can be expressed as:

< σν >= 1.5399 × 10 11 N A µ T 9 3/2 i (ωγ) i e -11.605Ei/T9 , (6) where (ωγ
) i = 2J i + 1 (2J p + 1)(2J35 K + 1) Γ γ,i Γ p,i Γ i , (7)
is the resonance strength of the i th resonance with Γ γ,i , Γ p,i and Γ i its partial γ-ray, proton and total width in MeV, respectively, E i the resonance energy in MeV, J i the spin of the resonance, J p and J35 K are the proton spin (1/2) and the g.s. spin of 35 K (3/2), respectively. T 9 is the temperature in GK and µ is the reduced mass. This assumption of narrow resonance is valid as the resonant states considered here have a total width far below their resonance energies. As shown in Eq. 6, the resonance energy, the spin, as well as the total and partial widths of all resonances are needed to compute the reaction rate. The resonance energy E r for the 2 + 1 state has been determined from the excitation energy of Ref. [START_REF] Amthor | Experimental and theoretical study of nuclear reaction rates in the rp-proces[END_REF] (being the most precise measurement performed by γ-ray spectroscopy) and the recent mass measurement of Ref. [START_REF] Surbrook | First penning trap mass measurement of 36 Ca[END_REF]. For the 1 + and 2 + 2 states, excitation energies are the one determined in the present work. The spin values used for the computation are the ones proposed in Sect. III B.

As we could only determine precisely the proton branching ratio in the present work (and only a lower limit for the 1 + and 2 + 2 states), we choose to fix the Γ γ partial widths using the sdpf shell model calculation which makes use of the broadest valence space and correctly reproduces the energy of the first 2 + state. Once Γ γ is fixed, Γ p and Γ tot can be derived for the 2 + 1 state using the experimental B p value as the proton and γ decays are the only open channels.

As for the 1 + and the 2 + 2 resonances, the proton partial width dominates the total width. It follows that the resonance strength of the Eq. 7 can be approximated by ωγ Γγ 8 (2J r + 1), with J r the spin of the resonance. All the resonance parameters needed to compute the reaction rate are listed in Table . III

The reaction rate has been computed using the Monte-Carlo code RatesMC [START_REF] Longland | the 2010 Evaluation of Monte Carlo based Thermonuclear Reaction Rates[END_REF], allowing a statistically meaningful evaluation of the reaction rate based on experimental and/or theoretical uncertainties. A Gaussian probability density function is assumed for the resonance ener-gies and a log-normal distribution is used as a probability density function for γ-width and proton-width values.

A central value Γ γ of 0.99 meV was used for the 2 + 1 state with an uncertainty factor of 1.7, which corresponds to values between 0.58 and 1.7 meV at one sigma. This way, we accept the lower (0.4 meV) and upper limit (3.7 meV) of Γ γ , discussed in Sect. III D, at about 2σ. The same uncertainty factor is assumed for the Γ γ widths of the 1 + and the 2 + 2 states. The uncertainty on Γ p of the 2 + 1 is deduced from that on Γ γ and on the experimental B p value, following Eq. 2. a with a uncertainty factor 1.7

B. Direct capture

The DC component mostly occurs through an =2 proton capture between the ground state of 35 K and that of 36 Ca, which have J π = 3/2 + and 0 + , respectively. In a simplified picture, 3 protons are occupying the 1d 3/2 orbital in the g.s. of 35 K and one more can occupy it by the direct capture process to the ground state of 36 Ca . The calculations of Ref. [START_REF] Herndl | [END_REF] predict a proton spectroscopic factor C 2 S p for 36 Ca of 3.649, while the sdpf shell model calculation predicts a slightly smaller value of 3.37. We propose the use of the mean value between the two (3.5) and an uncertainty of 0.15 to agree, within one sigma, with the two calculations.

C. Discussion

The evolution of the calculated rates with temperature for the 35 K(p,γ) 36 Ca reaction is shown in Fig 6 . The top part displays the results obtained when using the presently discussed resonances and corresponding widths, while the bottom part shows the results obtained from the work of Iliadis et al. [16], in which only the 2 + resonance was considered (with the following parameters: E r = 459(43) keV, Γ γ = 0.38 meV and Γ p = 1.4 meV) in addition to the DC part. In the X-ray burst temperature range (0.5-2 GK), the reaction rate is dominated by this FIG. 6. The 35 K(p,γ) 36 Ca reaction rate calculated (in units of cm 3 mol -1 s -1 ) in this work (top) and in [16] (bottom). The red curve represent the total reaction rate, which includes the contributions of resonant (RC) and direct (DC) captures. The thickness of the curves represents a coverage probability of 68%. The indicated range of 0.5 -2 GK is typical of X-ray bursts temperatures.

resonance. Therefore the major differences between our work and the one of Ref. [16] reside in the values used for the partial γ-width (to be discussed in the next paragraph), the partial proton-width, the resonance energy of the 2 + state and their associated uncertainties.

For the partial Γ γ width, an arbitrary value of 50% was assumed in [16] with a central value of 0.4 meV for the 2 + state. It corresponds to a log-normal confidence interval for the widths of [0.25-0.64] meV at one sigma, which partly overlaps with our current determination of the γ-ray partial width. The uncertainty on the 2 + res-FIG. 7. Ratio of rates normalized to our recommended reaction rate. The area delimited by the thick/thin black lines and centered around 1 represent the recommended value at the 68% and 95% confidence levels, respectively. The thick and dashed blue lines correspond to the reaction rate given by Iliadis et al. [16] and at the 68% confidence level, normalized to our recommended value.

onance energy considered in the present work was reduced using the very recent precise measurement of the mass of 36 Ca from [START_REF] Surbrook | First penning trap mass measurement of 36 Ca[END_REF] and the excitation energy from [START_REF] Amthor | Experimental and theoretical study of nuclear reaction rates in the rp-proces[END_REF]. As shown in Fig. 6, the contributions of the 1 + and 2 + 2 resonances to the total reaction rate, not taken into account in [16], start to be significant at temperatures above T = 2 GK.

The ratio of the calculated reaction rate by Iliadis et al. [16] to our recommended value (given numerically in Tab. IV) is shown in Fig. 7. The colored areas outlined by the thick/thin black lines show the uncertainty on the recommended reaction rate calculated in this work with a coverage probability of 68% and 95% respectively. The thick and dashed blue lines correspond to the reaction rate given in [16] with the associated 68% uncertainties respectively, normalized to our recommended reaction rate. For the temperature range of interest, the results are similar. We have also estimated that the contributions to the 0 + 2 ( = 2 proton capture) and 3 -( = 1) states, not identified here but present in the mirror nucleus, are negligible in the Gamow window. At temperatures higher than 2 GK, our recommended reaction rate is systematically higher due to the contributions of the 1 + and 2 + 2 resonances, not included in [16]. This reaction rate should, however, be considered as a lower limit, as higher-energy resonances may additionally contribute to the reaction rate beyond 2 GK.

The sensitivity study of Cyburt et al. [3] concluded that the 35 K(p,γ) 36 Ca reaction would lead to a significant modification of the X-ray burst light curve if the reaction rate was a factor of one hundred higher than that of Iliadis et al. [16]. Such an increase is absolutely ruled out by our study for which a factor of 3 difference is found at maximum for the reaction rate between 0.5 and 2 GK.

V. CONCLUSION

The spectroscopy of 36 Ca was investigated via the one neutron pick-up reaction 37 Ca(p, d) 36 Ca in inverse kinematics, in view of determining useful information for the 35 K(p,γ) 36 Ca reaction rate and compare it to earlier works such as [16]. The 36 Ca atomic mass was measured and matches very well with previous values [10,[START_REF] Surbrook | First penning trap mass measurement of 36 Ca[END_REF]. The energy of the first 2 + excited state was confirmed and new resonances have been reported in the vicinity of the Gamow window, at excitation energies E x = 4.243(40) and 4.706(100) MeV. Based on shell model calculations in the sdpf valence space and the comparison to the mirror nucleus ( 36 S), spins and parities 1 + and 2 + were proposed for these two new observed states, respectively. The proton branching ratio B p = 0.165 (10) of the first 2 + state was measured with two independent methods and lower limits, B p (1 + ) > 0.96 and B p (2 +

2 ) > 0.97 were estimated for the two other resonant states.

A Monte Carlo procedure [START_REF] Longland | the 2010 Evaluation of Monte Carlo based Thermonuclear Reaction Rates[END_REF], which consistently takes into account the uncertainties on the energy, spin parity, partial and total widths of the 36 Ca states, was then used to calculate the 35 K(p,γ) 36 Ca reaction rate between 0.01 and 10 GK with its corresponding uncertainty. Shell model predictions of B(ωL) were used to compute the only non experimentally-constrained parameter for the resonant states: Γ γ . The factor 1.7 uncertainty associated to this prediction dominates the total uncertainty of the reaction rate in the X-ray burst temperature range of interest. Therefore, the determination of the gamma width (or lifetime) of the 2 + 1 state is still desirable, as it would provide an almost fully experimentallyconstrained reaction rate.

The present work provides constrains in a broad range of temperatures for the 35 K(p,γ) 36 Ca reaction rate. It should be noted, however, that some additional contributions (not studied in this work) may further increase the reaction rate above 2 GK. Up to 4 GK, our recommended value is consistent, within one sigma, with the one of [16], previously used in X-ray burst models. Based on the reaction sensitivity tests of Cyburt et al. [3], our measured reaction rate is not sufficiently different from previous estimation to modify significantly the luminosity profile of X-ray burst. Therefore, the 35 K(p,γ) 36 Ca reaction can be removed from the list of the proton radiative captures reactions having a strong impact on the light curve.

FIG. 1 .FIG. 2 .

 12 FIG. 1. Schematic layout (not on scale) of the experimental setup. The MUST2 telescopes are represented together with the two CATS beam tracker detector, the CRYPTA liquid Hydrogen target, the ionization chamber (IC), the drift chambers (DC), and the plastic scintillator.

FIG. 4 .

 4 FIG. 4. Excitation energy spectrum of 36 Ca reconstructed from the measurement of energy and angle of the deuteron and gated on outgoing Ca a) or K b) nuclei in the ZDD system. The center-of-mass energy of the protons emitted from unbound states in 36 Ca has been added to the one-proton separation energy in the excitation energy spectrum E c.m. p of c). In all spectra, the red lines display the best fits (p-value = 0.67 for Ex and 0.82 for E c.m. p ) and colored dashed-lines represent the different resonances in 36 Ca used for the fit. Those especially relevant for X-ray bursts are shown in the level scheme on the left with the same color codes. Energy of exited states are taken from Ref. [14] for the 2 + 1 state and from our work otherwise (highlighted in red).

FIG. 5 .

 5 FIG. 5. Proton-deuteron angular correlation corresponding to the 2 +1 state as a function of the center-of-mass angle of the proton in the 36 Ca frame. The red line shows the best fit obtained with kmax=2 (p-value = 0.52).

Used for the calibration g.s. Ca 37 Ca(p,d) 38 g.s. Ar 34 Ar(p,d) 35 1.6 Ca 37 Ca(p,d) 38 g.s. Ca 36 Ca(p,d) 37

  FIG. 3. E ref CsI vs. E ∆E CsI scatter plot for one CsI of one MUST2 telescope. The data points from the 38 Ca(p,d) 37 Cag.s. (blue), 35 Ar(p,d) 34 Arg.s. (green), 37 Ca(p,d) 36 Cag.s. (cyan) and 38 Ca(p,d) 37 Ca1.6MeV (magenta) reactions are shown. A linear fit (red curve) was done using the 38 Ca(p,d) 37 Cag.s. and 35 Ar(p,d) 34 Arg.s. reactions. The linear fit describes well the data points from 0 to 6 MeV in excitation energy (written in red) in 36 Ca (R 2 =0.992).

	<
	[MeV]	16				6 MeV
	ref	14 CsI			3 MeV
	E
			12	0 MeV
			10
			8
			8	10	12	E	14 CsI E ∆	16 [MeV]

TABLE I .

 I Mass excesses (∆M ), obtained in the present work for 37 Ca, 34 Ar, and 36 Ca using the (p,d) reaction are compared to other experimental works. As derived from Q-values, our results use the precise experimental atomic masses of 38 Ca[START_REF] Ringle | [END_REF], 35 Ar[START_REF] Wang | [END_REF], and 37 Ca[START_REF] Ringle | [END_REF], respectively.

	Nucleus	∆M (keV)	∆M (keV)
		this work	literature
	36 Ca	-6480(40)	-6450(40)[10]; -6483.6(56)[17]
	37 Ca	-13141(13)	-13136.1(6) [18]
	34 Ar	-18403(25)	-18378.29(8) [24]
	B. Excited states in 36 Ca

TABLE II .

 II The experimental and theoretical results for the resonant states in 36 Ca obtained in this work are presented.

TABLE III .

 III Resonances parameters used in this work to compute the 35 K(p,γ) 36 Ca reaction rates. Resonance spin-parity value, experimental energy and calculated γ-width (using the sdpf valence space) are given in the three first columns. The proton-width Γp, derived from the calculated γ-width and the experimental proton branching ratio for the first 2 + state, is given in the fourth column. The resonant strength of each state is listed in column five.

	J π	Er [keV]	Γγ [meV]	Γp [meV]	ωγ [meV]
	(2 + )	445 (7)	0.99 a	0.20	0.102(50)
	(1 + )	1643 (41)	65.4 a		25 (14)
	(2 + 2 )	2106 (100)	7.4 a		4.6 (25)

TABLE IV .

 IV Low, recommended and high thermonuclear rates of the 35 K(p,γ) 36 Ca reaction (in units of cm 3 mol -1 s -1 ) as a function of temperature. Interval between low and high rates represents a confidence level of 68% (1 σ).

	T [GK]	Low	Recommended	High
	0.010	3.191×10 -51	3.350×10 -51	3.516×10 -51
	0.011	2.345×10 -49	2.461×10 -49	2.591×10 -49
	0.012	1.051×10 -47	1.104×10 -47	1.160×10 -47
	0.013	3.151×10 -46	3.316×10 -46	3.487×10 -46
	0.014	6.785×10 -45	7.123×10 -45	7.487×10 -45
	0.015	1.100×10 -43	1.157×10 -43	1.218×10 -43
	0.016	1.407×10 -42	1.478×10 -42	1.555×10 -42
	0.018	1.281×10 -40	1.346×10 -40	1.414×10 -40
	0.020	6.238×10 -39	6.547×10 -39	6.886×10 -39
	0.025	1.498×10 -35	1.573×10 -35	1.654×10 -35
	0.030	5.656×10 -33	5.950×10 -33	6.257×10 -33
	0.040	3.229×10 -29	3.393×10 -29	3.565×10 -29
	0.050	1.507×10 -26	1.585×10 -26	1.665×10 -26
	0.060	1.633×10 -24	1.719×10 -24	1.808×10 -24
	0.070	6.881×10 -23	7.243×10 -23	7.612×10 -23
	0.080	1.503×10 -21	1.581×10 -21	1.663×10 -21
	0.090	2.050×10 -20	2.152×10 -20	2.262×10 -20
	0.100	2.112×10 -19	2.274×10 -19	2.501×10 -19
	0.110	2.632×10 -18	3.680×10 -18	5.509×10 -18
	0.120	6.334×10 -17	1.066×10 -16	1.794×10 -16
	0.130	1.416×10 -15	2.425×10 -15	4.077×10 -15
	0.140	2.174×10 -14	3.653×10 -14	6.068×10 -14
	0.150	2.322×10 -13	3.847×10 -13	6.272×10 -13
	0.160	1.837×10 -12	3.007×10 -12	4.807×10 -12
	0.180	5.706×10 -11	9.077×10 -11	1.426×10 -10
	0.200	8.717×10 -10	1.370×10 -09	2.113×10 -09
	0.250	1.116×10 -07	1.720×10 -07	2.577×10 -07
	0.300	2.683×10 -06	4.073×10 -06	6.069×10 -06
	0.350	2.502×10 -05	3.786×10 -05	5.595×10 -05
	0.400	1.298×10 -04	1.955×10 -04	2.894×10 -04
	0.450	4.570×10 -04	6.891×10 -04	1.018×10 -03
	0.500	1.227×10 -03	1.856×10 -03	2.738×10 -03
	0.600	5.214×10 -03	7.901×10 -03	1.163×10 -02
	0.700	1.413×10 -02	2.145×10 -02	3.173×10 -02
	0.800	2.911×10 -02	4.418×10 -02	6.535×10 -02
	0.900	4.983×10 -02	7.581×10 -02	1.125×10 -01
	1.000	7.534×10 -02	1.150×10 -01	1.712×10 -01
	1.250	1.518×10 -01	2.313×10 -01	3.445×10 -01
	1.500	2.356×10 -01	3.564×10 -01	5.288×10 -01
	1.750	3.295×10 -01	4.886×10 -01	7.116×10 -01
	2.000	4.551×10 -01	6.538×10 -01	9.253×10 -01
	2.500	8.745×10 -01	1.197×10 +00	1.653×10 +00
	3.000	1.618×10 +00	2.196×10 +00	3.118×10 +00
	3.500	2.790×10 +00	3.810×10 +00	5.535×10 +00
	4.000	4.507×10 +00	6.124×10 +00	8.876×10 +00
	5.000	9.892×10 +00	1.291×10 +01	1.811×10 +01
	6.000	1.872×10 +01	2.316×10 +01	3.048×10 +01
	7.000	3.204×10 +01	3.773×10 +01	4.696×10 +01
	8.000	5.114×10 +01	5.785×10 +01	6.893×10 +01
	9.000	7.719×10 +01	8.471×10 +01	9.723×10 +01
	10.000	1.104×10 +02	1.196×10 +02	1.330×10 +02

ACKNOWLEDGMENTS

The continued support of the staff of the GANIL facility is gratefully acknowledged. We thank Richard Long-land and Phil Adsley for their help with the use of the RatesMC code. We acknowledge significant support from NFS grant PHY-1811855.


Loïc France

Florent Bruguier

Maria Mushtaq

David Novo

Pascal Benoit

Implementation of Rowhammer Effect in gem5

Modern computer memories have been shown to have reliability issues. The main memory is the target of a security attack called Rowhammer, which causes bit flips in adjacent victim cells of aggressor rows. Existing architectures simulator don't provide any implementation of unintended memory modifications like bit-flips. In this paper, we propose an implementation of the Rowhammer effect in the gem5 architecture simulator.

Introduction

Memory is a key component to modern computing architectures. Dynamic Random Access Memory (DRAM) is used in computers to store data during runtime, with intermediate cache memories between the processor and the DRAM to speed up access to frequently used data. As computer technologies became more efficient and smaller, manufacturers have been able to put more memory in much smaller spaces, resulting in better performances and lower cost [START_REF] Hennessy | Computer architecture: a quantitative approach[END_REF]. However, making DRAM smaller resulted in higher vulnerability to what Kim et al. [START_REF] Kim | Flipping bits in memory without accessing them: An experimental study of DRAM disturbance errors[END_REF] depicted as disturbance errors: activating a row slightly disturbs adjacent rows due to electromagnetic coupling between adjacent wordlines, and repeated activation of neighbors of a victim row can make the capacitors lose their charge, effectively deleting the data. This error rapidly became a major threat as attacks exploiting this error, named Rowhammer (RH) attacks, appeared to precisely flip bits to gain kernel privileges, even from sandboxes or without a malicious program running on the victim system [START_REF] Kim | Revisiting rowhammer: An experimental analysis of modern dram devices and mitigation techniques[END_REF].

To counter the RH attack, multiple mitigation techniques have been proposed, with a lot of them implying a modification of the memory controller and/or a new hardware component in the architecture. However, the most efficient solutions require architectural modifications, e.g on the memory controller. Therefore, most of these solutions only propose a concept and simulate its behavior mathematically.

For those solutions, a simulation environment would help to prove their efficiency against existing and future attacks.

gem5 [START_REF] Binkert | The gem5 simulator[END_REF] is a modular computer architecture simulator used for research. It is used to define a custom architecture with cores and memories, and simulate this architecture running programs and operating systems. However, its memory simulator is not accurate regarding timings and does not simulate memory corruption due to RH attacks. Ramulator [START_REF] Kim | Ramulator: A fast and extensible dram simulator[END_REF] was introduced to bring timingaccurate main memory simulation to gem5 but is not capable of simulating RH effects.

In this paper, we propose a modification of gem5 to introduce the Rowhammer effect to architectural simulations. This modification will allow us to develop new attack methods and associated counter-measures.

Integration of Rowhammer effect in gem5

In order to simulate the memory corruption due to the RH effect, we created a Memory-Corruption (M-C) module, and integrated it inside gem5 by modifying some existing components.

The M-C module stores an associative map where the key is the DRAM row and the value is a counter. When a DRAM row is activated, the concerned row is removed from the map; for each of its neighbor rows, if it is not yet referenced in the map, it is added and its counter initialized to 1, otherwise its counter is incremented. When a counter is incremented, if the value is above the RH threshold, every bit of the line has a low probability of flipping from 1 to 0.

In order to integrate the M-C module, we had to modify existing gem5 and Ramulator components to let the M-C module know when rows are activated: we added the row activation information to the request callback when accessing the main memory, and made periodic refresh events, originally generated and handled by Ramulator alone, notify gem5 when they happen.

Usage

The M-C module is used to generate bit-flip in the host memory when the simulated memory is the target of a Rowhammer attack. This module can be used to test if attacks achieve memory corruptions and if defense mechanisms can prevent the attack from doing any harm to the memory.

Configuration

The M-C module can be configured to simulate the RH effect as precisely as possible, and to add information to the log in order to compare and improve the attacks' efficiency. Multiple parameters can be passed to the M-C module:

• The threshold at which the rows start to have bit-flips; • Enable or disable the modification of the host memory (leaving only a log when the threshold is exceeded); • The address row to physical placement mapping, used to check which row is adjacent to the activated one; • A set of target addresses, to limit the possible corruptions to those addresses.

Limitations

There is some limitations regarding the fidelity of the simulated RH effect.

• The refresh events used by Ramulator are fired with a period corresponding to the refresh interval, which is usually 7.8µs = 64ms/8196. This refresh interval is used by the DRAM controller to refresh a small subset of all DRAM rows. We considered that the activate counters are reset once every 8196 refresh events • The mapping has to be the same across all DRAM banks. • The data pattern of the victim and aggressor rows has no impact on the corruption speed • All bit-flips happen in the same direction (1 to 0).

Perspectives

We recently published an article where we used traces generated with gem5 to train machine learning (ML) models into detecting RH attacks by classifying hardware event traces [START_REF] France | Vulnerability assessment of the rowhammer attack using machine learning and the gem5 simulator-work in progress[END_REF]. At this time, we had to consider that some programs are harmful even if the simulated memory was not corrupted. Using this new contribution, we are able to create programs that perform memory corruption on the simulated memory, generate datasets from the simulations and train ML models correctly. This contribution will also allow designing RH mitigation mechanisms with proof of work, and providing working simulations of architecture integrating state of the art mitigation techniques.

Conclusion

In this paper, we presented the Memory-Corruption gem5 module which brings the Rowhammer effect simulation to gem5. By generating DRAM bit-flips during Rowhammer attacks, this gem5 module allows researchers and designers to simulate a system under attack and securely create attacks targeting not-yet-available architectures, and test defense mechanisms that cannot be tested on existing platforms.

Figure 1 .

 1 Figure 1. cell-to-cell disturbance in DRAM: hammering row (a) disturbs adjacent rows (b) and delete data stored in victim cells (c)

Figure 2 .

 2 Figure 2. Integration of the M-C module in gem5

Acknowledgements

The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR), under grant ANR-19-CE39-0008 (project ARCHI-SEC). They also acknowledge the French Ministère des Armées -Agence de l'innovation de défense (AID) under grant ID-UM-2019 65 0036.


Alfonso Garcia-Marquez

Antonio Frontera

Thierry Roisnel

Rafael Gramage-Doria

Ultrashort Hδ+Hδ-intermolecular distance in a supramolecular system in the solid state

The understanding of chemical bonding is of fundamental importance for the advancement of chemical sciences. 1 In this regard, the origin and consequences of chemical interactions play a central role in fields as diverse as inorganics, biochemistry, solid matter as well as physical and organic chemistry. 2 The last years has witnessed a renewed interest to explore the limits of a certain type of interactions less common than classical hydrogen bonding or ion pairing. 3 In particular, recent reports dealing with ultrashort H•••H non-covalent interactions (i.e. dH…H < 3.08 Å) have shown their relevance from an structural and theoretical point of view. [START_REF] Name | [END_REF] By means of unique, purely-organic molecular designs, Pascal, Jr. and co-workers reported a number of small cage-shaped molecules featuring extremely short intramolecular H•••H distances ranging from 1.89 to 1.76 Å for physicochemically identical hydrogen atoms (A and B, Fig. 1). 5 Further computational studies point out that even shorter distances up to 1.56 Å might be feasible. 5,[START_REF] Mandal | [END_REF] Bullock and co-workers reported the shortest intramolecular H•••H distance found for an organometallic complex. 7,8 The quest for ultrashort intermolecular H•••H distances is more challenging because the systems are prone to display higher degrees of freedom, resulting in longer distances compared to the intramolecular versions. In this respect, the seminal work from Schreiner and co-workers is of paramount importance as they reported the shortest intermolecular H + •••H + distance of 1.57 Å (C, Fig. 1). 9 This was found between two electronically-identical hydrogen atoms in a dimeric species comprising extremely bulky trityl moieties with tert-butyl groups behaving as dispersion energy donors, thus featuring attractive London dispersion interactions. On the other hand, electronically-different, short intermolecular H + •••H  distances (up to 1.66 Å) have been observed for purely ionic, highly charged molecular entities containing BH4 -anions. 10 Following our interest on weak interactions, 11 we herein report the discovery of the shortest H  •••H  intermolecular distance found for two electronically-different hydrogen atoms (D, Fig. 1). In the course of our studies on ligand design for supramolecular catalysis, 12 we synthesized the phosphinephenol 1 and the corresponding borane adduct 2 (Scheme 1) following a modified procedure from the literature (see ESI). 13 Both compounds delivered single crystals suitable for analysis by X-ray diffraction studies. 14 In the solid state, compound 1 was organized in a supramolecular polymeric chain stabilized by classical hydrogen bonding involving the hydroxyl groups (see

[dB-H(51C)•••H(2)-O(2)]
was unambiguously detected (Fig. 2). Importantly, the O(2) atom linked to the hydrogen atom involved in this short distance is the most populated one by far (61.8%). To the best of our knowledge, this is the shortest intermolecular H + •••H  distance between electronically-different hydrogen atoms ever reported for non-ionic species. [START_REF]Extensive searches in the Cambridge Structural Database (CSD) failed to find any molecular system with such a short H-H distance besides some highly charged ionic species[END_REF] Unfortunately, attempts to obtain large crystals of a size suitable to collect lowtemperature neutron diffraction data, which is well capable of determining the exact hydrogen positions (as it was the case for C, Fig. 1), failed so IR spectroscopy studies were performed on 1 and 2 and compared to pure phenol and Ph3P-BH3 (Fig. S5 and Table S1 in the ESI). The main OH vibration (3221 cm -1 ) of compound 2 is shifted ca. 118 cm -1 to lower wavenumbers compared to 1 (3339 cm -1 ). This observation points out to a sort of interaction in the solid state between the O-H group and the borane group of 2 similar to what has been found in the gas phase of a phenol-borane system elsewhere. 18 The appearance of a shoulder at 3401 cm -1 in 2 also suggests a loss of symmetry of the vibration mode. As expected, the B-H vibration signal appears at 2378 cm -1 and additional peaks with maxima at 2346 and 2260 cm -1 . Multinuclei NMR was not conclusive, hereby suggesting that the observed phenomenon is not likely in solution. A theoretical density-functional theory (PBE0 19 -D3 20 /def2-TZVP 21 ) study was performed in order to rationalize the H + •••H  short distance in 2 (see ESI for computational details and Fig. S6). First, the molecular electrostatic potential (MEP) minima and maxima surfaces of species 2 was calculated (Fig. S6 in the ESI). As expected, the MEP maximum (+57.1 kcal/mol) is located at the phenolic hydrogen atom (H + ). On the other hand, the MEP minimum is not located at the extension of the B-H bond, but rather along the P-B bond (-35.1 kcal/mol). The MEP value at the borane-connected hydrogen atom (H -) is slightly more positive (-31.4 kcal/mol).

Next, we combined the quantum theory of atoms in molecules (QTAIM) 22 with the non-covalent interaction (NCI) plot index analysis to further study the H + •••H  interaction in 2 (Fig. 3). 23 In the dimer of 2, the H + •••H  interaction is characterized by a bond critical point (red sphere) and bond path interconnecting the hydrogen atoms. Importantly, a blue NCI isosurface is located between both H + and H  atoms in 2, thus revealing a moderately strong, attractive interaction. In fact, the BSSE-corrected dimerization energy was calculated to be -6.0 kcal/mol (BSSE correction: 0.3 kcal/mol). The value of the Laplacian and total energy density at the bond critical point are both positive (∇ 2 ρ = 0.0814 a.u. and Hr = 0.0024 a.u.), thus confirming the non-covalent nature of the interaction. 24 The NCI plot index analysis reveals the existence of additional green isosurfaces between both monomers, thus suggesting the existence of ancillary long range van der Waals forces (vdW in Fig. 3). We have estimated the contribution of the H + •••H  interaction by using the potential energy density (Vr) predictor, using the expression developed by Espinosa and co-workers (E = ½Vr). 25 The resulting energy for the H + •••H  contact is -4.9 kcal/mol and, consequently, the contribution of the long range van der Waals forces can be estimated by difference, i.e. -1.1 kcal/mol. The NCI plot isosurface is also represented using the RDG = 0.5 a.u. isosurface and ρ = 0.04 cut-off for noncovalent interactions. The colour range is -0.035 a.u. ≤ sign(λ2)ρ ≤ 0.035 a.u. This dimer corresponds to the X-ray geometry. The value of ρ at the bond CP is given in italics (a.u.). The BSSE-corrected interaction energy is also indicated.

To further analyze the energetic features of the dimer of compound 2, the interaction energy was decomposed by means of the symmetry adapted perturbation theory (SAPT) 26 calculations (see ESI for details). The SAPT terms are grouped into four chemically relevant categories: electrostatics, exchange-repulsion, induction and dispersion. The total SAPT energy (-5.7 kcal/mol) is similar to the BSSE-corrected dimerization energy (-6.0 kcal/mol), thus giving reliability to the partition method. The examination of the individual contributions to the interaction energy (Fig. 4) reveals that 2013,00,123Please do not adjust margins Please do not adjust margins electrostatic and dispersion terms are similar, being the latter the largest attractive term (in green). The large contribution of the electrostatic term is quite intuitive, due to the opposite electronic nature of the hydrogen atoms involved in the interaction. The large contribution of dispersion is lesser intuitive, though it agrees well with previous investigations that have demonstrated the decisive role of dispersion interactions to explain the attractive nature of very short C-H•••H-C interactions in hydrocarbons. 9 The SAPT results also show that the contribution of induction is important (28.7 % of the attractive terms). As usual, the largest magnitude is the exchange-repulsion (+16.7 kcal/mol). Charge transfer effects have been also investigated using two different methods (NPA and Merz-Kollman) to derive atomic charges (see ESI for details). Both methods suggest a similar net charge transfer from the BH3 group to the phenolic H-atom (i.e., 0.021 e and 0.019 e for NPA and M-K charges, respectively. To investigate the role of packing effects, the dimer of 2 was computed in the gas phase at the same level of theory (see ESI for details). The corresponding optimized dimer (PBE0-D3/def2-TZVP level of theory) elongates the H + •••H  distance contact to 1.789 Å with additional - stacking (see ESI for details), a feature that is not found in the solid state (Fig. 2). The dependence of the O-H•••H-B distance upon the choice of functional and basis set gives minimal differences (see ESI for details). Further theoretical studies (MARIJ-PB86-D3/def2-SVP level of theory, see ESI for details) focused on a pentamer of 2 (Fig. 5) that structurally resembles the supramolecular polymeric chain observed in the solid state of 2. In contrast to the computed dimeric species, no - stacking was observed which is line with the findings in the solid state of 2 (vide supra). Interestingly, besides the presence of several common O-H•••H-B distances, [START_REF] Such | H-B distances are common, for example[END_REF] one very short O-H•••H-B distance of 1.66 Å was found (Fig. 5). Although this value is again longer than the experimental value of 1.61 Å, it is much shorter than the value computed in the gas phase for the dimeric species (vide supra). Therefore, crystal packing is, in contrast to compound C (Fig. 1), decisive for stabilizing the observed short H + •••H  distance of 1.61 Å in the supramolecular polymeric chain of 2 (Fig. 2). In summary, we have demonstrated that the shortest H + •••H  intermolecular distance for two electronically-different hydrogen atoms can reach values as low as 1.61 Å, which was so far unknown to date for non-ionic species. In silico thorough calculations evidenced the non-covalent nature of the interaction and that electrostatic and dispersion forces are equally important as attraction forces between both electronically different hydrogen atoms. Overall, trivial molecules such as 2 can display unexpected physical properties as it is here discussed. These findings provide a fundamental understanding for ultrashort, electronically-different H•••H noncovalent interactions and they are expected to pave the way towards further studies in view to disclose the limits for uncommon chemical bonding phenomena.

Financial support from Agence Nationale de la Recherche (ANR-19-CE07-0039), CNRS and University of Rennes 1 is acknowledged. A.F. thanks the MICIU/AEI from Spain for financial support (Project CTQ2017-85821-R, FEDER funds)

Fig. 1

 1 Fig. 1 Non-ionic chemical systems featuring the shortest intramolecular H -•••H -(A) and H + •••H + (B) distances as well as intermolecular H + •••H + (C) and H + •••H -(D, current contribution) distances. NRD = neutron diffraction data.

  Fig. S1 in the ESI). 2 | J. Name., 2012, 00, 1-3 Please do not adjust margins Please do not adjust margins In stark contrast, compound 2 led to a supramolecular polymeric chain featuring O-H•••H-B very close contacts (Fig. S2 in the ESI). 15 In the unit cell, two crystallographically independent molecules were present, each one being connected with each other via different type of O-H•••H-B weak interactions (see ESI for atom labelling details). In one case, two borane protons lie at 1.78 Å [dB-H(1B)•••H(52)-O(52)] and 1.85 Å [dB-H(1C)•••H(1)-O(1)], respectively, from the hydroxyl protons from the other molecule (Fig. S3-S4 in the ESI). 16 Due to a slight disorder in the hydroxyl groups around the three phenyl groups (i.e. the population of O(52) and O(1) is 33% and 22%, respectively), such O-H•••H-B distances might not be fully reliable nor representative of the system. Interestingly, in the other case, a single O-H•••H-B interaction as short as 1.61 Å

Fig. 2

 2 Fig. 2 Molecular structure of phosphine-borane adduct 2 displayed as ORTEP drawing with thermal ellipsoids at 50% probability. Non-relevant hydrogen atoms have been removed for the sake of clarity.

Fig. 3

 3 Fig. 3 Distribution of bond and ring critical points CPs and bond paths for the dimer of 2.The NCI plot isosurface is also represented using the RDG = 0.5 a.u. isosurface and ρ = 0.04 cut-off for noncovalent interactions. The colour range is -0.035 a.u. ≤ sign(λ2)ρ ≤ 0.035 a.u. This dimer corresponds to the X-ray geometry. The value of ρ at the bond CP is given in italics (a.u.). The BSSE-corrected interaction energy is also indicated.

Fig. 4

 4 Fig.4SAPT decomposition of the interaction energy (in kcal/mol) for the dimer of 2 (Xray geometry). The contribution of exchange (repulsion), electrostatic, induction and dispersion terms are indicated.

Fig. 5

 5 Fig. 5 Optimized geometry of the pentameric assembly 25 at the MARIJ-PB86-D3/def2-SVP level of theory, distances (blue color) are indicated in Å. Only the B/O-bonded Hatoms are shown for the sake of clarity.

Scheme 1 Synthesis of phosphine 1 and borane adduct 2.

Conflicts of interest

There are no conflicts to declare.

Notes and references