Hao Yuan

Mary Theresa

Dipankar Bain

Hussein Fakhouri

Aswani Raveendran

Sabu Thomas

Nandakumar Kalarikkal

Rodolphe Antoine
email: rodolphe.antoine@univ-lyon1.fr

Singlet Oxygen Generation Efficiency and Antimicrobial Ability in Glutathione Protected Ag 31 Nanoclusters

Keywords: silver nanoclusters, antimicrobial agent, quantum clusters, reactive oxygen species Highlights

Ligand protected silver nanoclusters (Ag NCs) have recently been reported to have significant applications as antimicrobial agents. In the current study, glutathione (SG) protected silver nanocluster was synthesized leading to Ag31SG19 by cyclic reduction and oxidative condition and was evaluated for its singlet oxygen generation and antibacterial activity. From the results of the study, Ag31SG19 was found to have inhibitory activity against Escherichia coli and Staphylococcus aureus with a diameter of 15.3 ± 0.58 mm and 16.7 ± 0.58 mm respectively. The MIC was also calculated to have 0.3125 mg/ml and 0.1562 mg/ml respectively. In addition, the Ag31SG19 was found to have high efficiency for the generation of singlet oxygen. Based on the results of the study, the antimicrobial mechanisms of Ag31SG19 could be predicted to be mainly because of the generation of singlet oxygen species in addition to the activity due to the release of Ag + ions from the nanoclusters. These multimechanistic antimicrobial effects make the synthesized Ag31SG19 to be a promising candidate for a broad range of antimicrobial applications.

Introduction

Bacterial infections have caused serious threat to human health with massive global economic loss. [START_REF] Privalsky | Prospects for Antibacterial Discovery and Development[END_REF] Even though antibiotics are being considered as the most effective antibacterial medicine, their widespread use has already resulted in the development of antimicrobial resistance (AMR) among microbial pathogens. [START_REF] Dhingra | Microbial Resistance Movements: An Overview of Global Public Health Threats Posed by Antimicrobial Resistance, and How Best to Counter[END_REF] Here is the relevance of exploring nanotechnological advances for the discovery of new nanoscale materials with high antimicrobial efficiency and safety. [START_REF] Doolan | Advancements in antimicrobial nanoscale materials and selfassembling systems[END_REF][START_REF] Thomas | Enhanced antimicrobial efficacy of biosynthesized silver nanoparticle based antibiotic conjugates[END_REF] Nanomaterials generally cause the generation of reactive oxygen species (ROS) and hence exhibit potential antibacterial activity. The targeted killing action of nanomaterials against pathogenic bacteria hence could be due to the ROS which act as strong oxidants leading to oxidative damage of cellular constituents. [START_REF] Slade | Oxidative Stress Resistance in Deinococcus radiodurans[END_REF][START_REF] Dwyer | Role of reactive oxygen species in antibiotic action and resistance[END_REF] Among the various nanomaterials, ligand protected noble metal nanoclusters (NCs) are novel entities having a well-defined number of metal atoms and ligands. [START_REF] Jin | Atomically precise metal nanoclusters: stable sizes and optical properties[END_REF][START_REF] Bonacic-Koutecky | Enhanced two-photon absorption of ligated silver and gold nanoclusters: theoretical and experimental assessments[END_REF][START_REF] Qian | Quantum Sized Gold Nanoclusters with Atomic Precision[END_REF] When protected by the biomolecular templates, the nanoclusters can exhibit strong luminescence with good biocompatibility, which make them highly appealing as theranostic probes for biomedical applications. [START_REF] Yu | Rational Design of Biomolecular Templates for Synthesizing Multifunctional Noble Metal Nanoclusters toward Personalized Theranostic Applications[END_REF][START_REF] Kulesza | Doubly Charged Silver Clusters Stabilized by Tryptophan: Ag-4(2+) as an Optical Marker for Monitoring Particle Growth[END_REF][START_REF] Yang | Ligand engineering of Au44 nanoclusters for NIR-II luminescent and photoacoustic imaging-guided cancer photothermal therapy[END_REF][START_REF] Yang | Engineering Au44 Nanoclusters for NIR-II Luminescence Imaging-Guided Photoactivatable Cancer Immunotherapy[END_REF][START_REF] Yang | Ultrasmall Coinage Metal Nanoclusters as Promising Theranostic Probes for Biomedical Applications[END_REF] More importantly, NCs can exhibit their long triplet-state lifetime and the singlet oxygen ( 1 O2), the most damaging ROS species against microorganisms, has been reported to be generated by the energy transfer from NCs to molecular oxygen upon light irradiation. [START_REF] Kawasaki | Generation of Singlet Oxygen by Photoexcited Au25(SR)18 Clusters[END_REF][START_REF] Ho-Wu | Efficient Singlet Oxygen Generation in Metal Nanoclusters for Two-Photon Photodynamic Therapy Applications[END_REF] Indeed, as illustrated in Scheme S1 (in supporting materials), an efficient photosensitizer for 1 O2 production requires a high triplet-state yield with a triplet-state energy larger than the energy of 1 O2 (0.98 eV) for efficient energy transfer to 3 O2.

From this viewpoint, the optical gap of thiolate protected silver nanoclusters is usually higher than 1 eV and since presenting long-lifetime, the triplet-state efficiency is high. [START_REF] Bertorelle | Synthesis, characterization and optical properties of low nuclearity liganded silver clusters: Ag31(SG)19 and Ag15(SG)[END_REF] Consequently the nanoclusters have also been observed to display antibacterial capability after being metabolized by the bacterial cells. [START_REF] Zheng | Antimicrobial silver nanomaterials[END_REF][START_REF] Yuan | Highly luminescent silver nanoclusters with tunable emissions: cyclic reductiondecomposition synthesis and antimicrobial properties[END_REF][START_REF] Chen | Amphiphilic silver nanoclusters show active nano-bio interaction with compelling antibacterial activity against multidrug-resistant bacteria[END_REF] In addition to their high capability to generate ROS, the antimicrobial mechanisms of metal NCs may also include the cell wall and membrane disruption, damage to intracellular components. In the case of silver-based NCs, due to their moderate stability the antimicrobial mechanisms also involve release of metal ions. [START_REF] Zheng | Antibacterial metal nanoclusters[END_REF][START_REF] Vijayakumar | Chitosan with pendant (E)-5-((4-acetylphenyl)diazenyl)-6-aminouracil groups as synergetic antimicrobial agents[END_REF][START_REF] Thomas | Inhibitory effect of silver nanoparticle fabricated urinary catheter on colonization efficiency of Coagulase Negative Staphylococci[END_REF] Thus for the ultrasmall Ag + -rich nanoclusters, it has been proposed to have both efficient ROS generation and abundant Ag + ions on its surface resulting in enhanced antibacterial activity. [START_REF] Yuan | Ultrasmall Ag+-rich nanoclusters as highly efficient nanoreservoirs for bacterial killing[END_REF] Compared with the AgNPs, ultrasmall AgNCs can have higher surface to volume ratio which make them to be more susceptible to the oxidative dissolution resulting in a fast release of Ag + and thereby enhanced antibacterial performance. In a previous study, glutathione (GSH)-protected silver nanoclusters which have been produced mainly as mixture (Agn(SG)m) [START_REF] Yuan | Highly luminescent silver nanoclusters with tunable emissions: cyclic reductiondecomposition synthesis and antimicrobial properties[END_REF][START_REF] Wang | Synthesis, characterization and antimicrobial activity of silver nanoparticles: Agn(NALC)m and Agn(GSHR)m[END_REF] were reported to have strong activity against Pseudomonas aeruginosa due to the easily oxidized and released Ag + on the surface of AgNCs. It has been reported by Yuan et al. that the ultrasmall Ag + -rich nanoclusters (GSH-capped Ag + -rich Ag NCs) can have higher antibacterial activity when compared with the neutral Ag-rich counterpart. [START_REF] Yuan | Ultrasmall Ag+-rich nanoclusters as highly efficient nanoreservoirs for bacterial killing[END_REF] Even though preliminary mass spectrometry data were reported in previous studies, insights into atomic precision for the silver cluster synthesis was not enough to explore its structureactivity relationship. Based on a facile aqueous-based synthesis, [START_REF] Antoine | Supramolecular Gold Chemistry: From Atomically Precise Thiolate-Protected Gold Nanoclusters to Gold-Thiolate Nanostructures[END_REF] the current study demonstrated the synthesis of Ag nanoclusters (Ag31SG19) using glutathione as thiolate ligand. Stability and photo-induced ROS efficiency of Ag NCs are explored and compared with ultrasmall Ag nanoparticles. Also their antimicrobial capabilities are reported.

Instrumentation

Electrospray ionization mass spectra (ESI-MS) of silver nanoclusters were recorded on a Thermo Fisher LTQ-Velos linear ion trap mass spectrometer (mass reolution ~7500). UV-Vis absorption spectra were analysed using Avantes AvaSpec-2048 spectrophotometer with an AvaLight DH-S deuterium lamp. Transimission electron microscopy (TEM) images were obtained using a JEOL JEM 2100 microscope. Fluorescence emission spectra were studied using a Horiba FluoroMax-4 spectrophotometer and fluorescence lifetimes were measured using custom-built setup [START_REF] Soleilhac | Correlating Droplet Size with Temperature Changes in Electrospray Source by Optical Methods[END_REF].

Synthesis of atomically precise silver nanoclusters

Two atomically precise silver nanoclusters were sythesized as our previous reports with some modification. For Ag31SG19 synthesis, the procedures of cyclic reduction under oxidative conditions were the same as previous synthesis [START_REF] Bertorelle | Synthesis, characterization and optical properties of low nuclearity liganded silver clusters: Ag31(SG)19 and Ag15(SG)[END_REF]. Here, in a dark environment, 1 mL of AgNO3 solution (55 mg/mL) was dropwise added to an solution of L-glutathione (200 mg in 35 mL of H2O) under ice-water bath. Further to this, 800 mL of 1 M NaOH solution was added to adjust the final pH to 6~7. Then, under constant stiring, 36 mg of NaBH4 in 1 mL of ice-cold water was quickly added.

After 1.5 h stiring, 300 mL of hydrogen peroxide (33%) was further added to the dark brown solution. Then, 140 mL AgNO3 solution (100 mg/mL) and 50 mg GSH were added after 30 min. This was followed by the addition of 500 mL of ice-cold solution of NaBH4 (20 mg/mL) after 5 min. The reaction mixture was kept stirring vigorously for 1.5 h till to a brown-red solution was obtained. For purification, the solution was evaporated to ~5 mL using a rotary evaporator at 30℃. This was further precipitated with the addition of 15 mL methanol followed by centrifugation. The precipitate was further dissolved in 20 mL H2O followed by the addition of 50 ml of glacial acetic acid. This was kept still for 2 h before centrifugation at 13 000 rpm for 30 min. The precipitate was further removed and the supernatant was further concentrated by evaporator and precipitated with methanol. Then the powder was washed with methanol/diethyl ether for 3 times and dried.

The Ag29DHLA12 was synthesized as per the method described previously [START_REF] Russier-Antoine | Tuning Ag-29 nanocluster light emission from red to blue with one and two-photon excitation[END_REF] with some modification. Here, 100 mg of (±)-α-lipoic acid was dissolved in a mixture of 5 ml methanol and 10 ml water. Then, 85 mg of sodium borohydride was added and the solution was stirred for 30 min to reduce the lipoic acid. Then, excess reducer was eliminated by sonication for 1 hour. Further to this, 41 mg of silver nitrate (dissolved in 15 mL of water) was added rapidly, followed by the addition of 1 ml of 10% NH4OH solution. The solution was further heated at 50 °C and 85 mg of sodium borohydride was added again. After 1 hour, 42.5 mg of sodium borohydride was added further when the color became bright orange. After 1 hour, the solution was concentrated using a rotary evaporator. Then, 5 mL of ethanol and 20 mL of diethyl ether were added and was followed with the addition of 50 μL of glacial acetic acid.

This was then centrifuged for 5 minutes at 9000 rpm, the precipitate was collected, washed first with ethanol/diethyl ether and then with diethyl ether. The brown powder obtained was dried by natural evaporation.

Synthesis of ultra-small 5-nm silver plasmonic nanoparticles.

The silver plasmonic nanoparticles were synthesized as per the methods reported by Agnihotri et al. [START_REF] Agnihotri | Size-controlled silver nanoparticles synthesized over the range 5-100 nm using the same protocol and their antibacterial efficacy[END_REF] Here, freshly prepared NaBH4 aqueous solution (2 mM, 24 mL) was added to trisodium citrate (TSC) aqueous solution (4.28 mM, 24 mL). The mixture was heated at 60 °C for 30 mins under vigorous stirring in dark. Then, 2mL of AgNO3 solution (1 mM) was added drop-wise following with increase of temperature to 90 °C. When temperature was raised to 90 °C, the pH was also adjusted to 10.5 with the addition of 0.1 M NaOH. After heating for 30 minutes, the light-yellow solution was cooled down at room temperature. This silver nanoparticle solution was characterized by absorption spectroscopy, and by transimission electron microscopy (TEM). The concentration of AgNPs was verified using the absorption coefficient of 1.9 × 10 3 M -1 cm -1 (for 6 nm AgNPs has reported by Mulvaney et al. [START_REF] Mulvaney | Electrochemistry of multilayer colloids: preparation and absorption spectrum of gold-coated silver particles[END_REF]) at plasmonic peak of ~394 nm according to Lambert-Beer law.

Reactive oxygen species (ROS) generation measurements 2.5.1 Protocol for 1 O2 generation measurements

The singlet oxygen generation efficiency was measured by an indirect method with the 1 O2 sensor of 1,3-diphenylisobenzofuran (DPBF). [START_REF] Ho-Wu | Efficient Singlet Oxygen Generation in Metal Nanoclusters for Two-Photon Photodynamic Therapy Applications[END_REF] Singlet oxygen generation was monitored under the excitation at 473 nm using a continuous wavelength laser (Changchun New Industries Optoelectronics Tech. Co., Ltd, China)

with an output power of 250 mW and a beam diameter of 3 mm. A 532 nm continuous wavelength laser (Spectra-Physics) with an output power set to 282 mW (beam diameter: 3 mm) was also used for the comparison. A typical solution used in the experiment contained nanoclusters/nanoparticles/dyes and DPBF with a concentration of 1.37 × 10 -6 M and 6.15 × 10 -5 M, respectively. All solutions were prepared in ethanol. For Ag NPs, the mixure solution with ethanol-water solvent (50/50 in volume ratio) was used. The samples were loaded in quartz cuvettes (1cm light path length) and the absorbance was were recorded with irradation for different time.

Statistical analysis for 1 O2 generation rate

The concentration of DPBF was calculated from the intensity of absorption peak

Protocol for oxidants generation indication

An excellent oxidants fluorescent indicator, 2′,7′-dichlorodihydrofluorescein (H2DCF) was used for marking the generation of reactive oxygen species, including superoxide (•O2 -), hydroxyl radical (•OH), and hydrogen peroxide (H2O2). [START_REF] Chen | 2',7'-Dichlorodihydrofluorescein as a fluorescent probe for reactive oxygen species measurement: Forty years of application and controversy[END_REF][START_REF] Zhu | Marrying luminescent Au nanoclusters to TiO2 for visible-light-driven antibacterial application[END_REF] H2DCF can be oxidized to green fluorescent 2′,7′-dichlorofluorescein (DCF) with an emission peak at ~525 nm. The ROS generation can be indicated by the fluorescence change on fluorescence spectra.

The commercial dye 2,7-dichlorodihydrofluorescein diacetate (H2DCF-DA) was firstly prepared based on reported procedure [START_REF] Jakubowski | 2,7-dichlorofluorescin oxidation and reactive oxygen species: what does it measure?[END_REF]: 0.5 mL of 2 mM H2DCF-DA in ethanol was added to 2.0 mL of 10 mM NaOH and stilled for 30 mins at room temperature under darkness. Then Dulbecco's phosphate-buffered saline (DPBS, pH7.2, without calcium, magnesium, Thermo Fisher Scientific) was added to a finial concentration at 123 μM. Freshly prepared H2DCF solution was used for further.

For oxidants generation indication, 2 mL mixture aqueous solution containing prepared H2DCF (6.15 × 10 -5 M) and Ag31SG19 (1.37 × 10 -6 M) was irridiated for 10 mins under the excitation of 473 nm continuous wavelength laser. Before and after laser irradiation, fluorescence emssion spectra were recorded with excitation wavelength at 488 nm.

Antibacterial activity analysis of Ag31SG19 and plasmonic AgNPs

The antibacterial activity of Ag31SG19 and plasmonic AgNPs was studied against Gram-positive Staphylococcus aureus (MTCC 96) and Gram-negative Escherichia coli (ATCC 35218) by both agar well diffusion and minimum inhibitory concentration.

Agar well diffusion analysis

The antibacterial activity of Ag31SG19 and plasmonic AgNPs was evaluated against E. coli and S. aureus by the standard well diffusion method in Mueller Hinton Agar (MHA) plates by following the Clinical and Laboratory Standards Institute (CLSI) guidelines (CLSI M02-A13). For this, a 20 mg/ml suspension of Ag31SG19 was prepared. Here, the bacterial inoculum was prepared by adjusting the cell concentration of respective cultures to 1.5 x 10 8 CFU/ml (0.5 McFarland) and the organisms were inoculated onto the MHA plates as lawn culture. Then, wells were cut on the media and 60 µl of Ag31SG19 suspension was added to each well. Plates were further incubated at 37℃ for 24 hours and the diameter of the zone of inhibition was measured using a zone measuring scale and the experiment was carried out in triplicates. The antimicrobial experiment was conducted under visible light exposure (400-700 nm) in Biosafety cabinet level 2.

The Minimum Inhibitory Concentration analysis

The Minimum Inhibitory Concentration (MIC) is defined as the lowest concentration of the antimicrobial agent that can inhibit the growth of bacteria. MIC of Ag31SG19 against S. aureus and E. coli was evaluated by using the standard broth microdilution method as per the CLSI guidelines (CLSI M07-A10). For this, the test organisms were incubated overnight at 37°C in Muller Hinton Broth (MHB) and inoculum was prepared as described above. Further, serial dilution was carried out to obtain different concentrations of Ag31SG19 (5, 2.5, 1.25, 0.625, 0.3125, 0.1563, 0.0781, 0.0391, 0.0195, 0.0098, 0.0049, and 0.0024 mg/ml) in a 96-well microtiter plate and each well was further inoculated with 100 µl of the selected test organisms.

The plate was then incubated under shaking conditions at 100 rpm and 37℃ for 24 hours. After this, 0.01% Resazurin dye (30µl) was added to each of the well followed by incubation in dark for 30-45 mins. MIC value of Ag31SG19 against each organism was identified based on the colour change observed with the addition of Resazurin.

Both sterility and growth controls were also maintained and the experiment was carried out in triplicates. [START_REF] Saied | Aspergillus terreus-Mediated Selenium Nanoparticles and Their Antimicrobial and Photocatalytic Activities[END_REF] 3 Results

Characterization of atomically precise Ag31SG19 silver nanoclusters and their

(photo)stability. The purity of synthesized silver nanoclusters was examined by mass spectrometry (Figure 1 a). [START_REF] Comby-Zerbino | The emergence of mass spectrometry for characterizing nanomaterials. Atomically precise nanoclusters and beyond[END_REF][START_REF] Soleilhac | Size Characterization of Glutathione-Protected Gold Nanoclusters in the Solid, Liquid and Gas Phases[END_REF] A charge state distribution was observed from [M-5H] 5-through [M-7H] 7-. Deconvolution of charge states 5 through 7 provided a mass of 9156 Da for the intact Ag cluster and was consistent with the calculated mass of Ag31SG19 (9164 Da) (see Figure S1). Experimental and simulated isotope pattern of [M-6H] 6-were shown in Figure S2. The pattern shape and mass position match well, though the resolution of our mass spectrometer is too low to resolve isotope pattern.

The absorption spectra (Figure 1b) showed the presence of an intense visible band centred at ~490 nm, in agreement with previous findings for which a structure for Ag31 was proposed (see Fig. S3). [START_REF] Bertorelle | Synthesis, characterization and optical properties of low nuclearity liganded silver clusters: Ag31(SG)19 and Ag15(SG)[END_REF] Excitation and emission spectra for Ag31SG19 nanoclusters dispersed in water and ethanol were also analyzed (Figure 1c and1d). As already observed for Ag29 nanoclusters [START_REF] Russier-Antoine | Tuning Ag-29 nanocluster light emission from red to blue with one and two-photon excitation[END_REF], the solvent-dependent emission spectra were also evidenced for Ag31SG19 NCs. Such a shift might be an indication of a stronger stabilization of the ground state than those of the excited state due to the difference in dielectric constants between H2O and ethanol. Stability and photostability are generally important issues, that limit the applications of thiolate protected silver nanoclusters. As the, Ag29DHLA12 nanocluster was known to have poor stability due to degradation in few hours upon light exposure, [START_REF] Van Der Linden | Characterisation, degradation and regeneration of luminescent Ag29 clusters in solution[END_REF][START_REF] Fakhouri | Effects of Single Platinum Atom Doping on Stability and Nonlinear Optical Properties of Ag29 Nanoclusters[END_REF] it was not possible to study its antimicrobial activity. The emission spectra for fresh and 1-day old solutions clearly demonstrated the complete degradation of Ag29 in less than 24 hours (Figure 2b). On the other hand, the Ag31SG19 was found to be very stable under normal conditions and upon constant irradiation by visible laser light as confirmed by the emission spectra of fresh and 1-day old solution respectively (Figure 2a) and for absorption spectra under several dozen minutes of visible laser irradiation (Figure 2c and S4). In the current study, silver nanoparticles (AgNPs) were synthesized by using a co-reduction approach, where the nucleation and growth kinetics during the synthesis process was precisely controlled and the AgNPs of average size ~6 nm were synthesized with good yield and monodispersity (Figure 3). Here, the silver nanoparticles were found to exhibit a sharp extinction peak at ~394 nm wavelength which confirmed the surface plasmon behavior of such particles as opposed to the multi-band "molecular-like" behavior of silver nanoclusters.

ROS efficiency of Ag31SG19 in solution.

The indirect method was used to quantify the singlet oxygen generation by the photoexcited nanoclusters in solution. Here, 1,3-diphenylisobenzofuran (DPBF) has already been known to be highly reactive toward singlet oxygen resulting in the formation of endoperoxides (with different absorption properties in particular at 412 nm) and hence can be used as an optical probe. [START_REF] Entradas | The detection sensitivity of commonly used singlet oxygen probes in aqueous environments[END_REF][START_REF] Fakhouri | Ligand impact on reactive oxygen species generation of Au10 and Au25 nanoclusters upon one-and two-photon excitation[END_REF] Both the nanoclusters and DPBF can be excited to generate and detect singlet oxygen simultaneously. Here, the generation of singlet oxygen was triggered by the excitation of the nanoclusters with a cw laser emitting at 473 nm (and also a 532 nm) with different time of exposure. Of divided by the concentration of the nanoclusters. To demonstrate the efficiency of Ag31 nanoclusters for 1 O2 production, we compared 1 O2 production by Ag31SG19 to that of the conventional dye photosensitizer new methylene blue (NMB). Under 473 nm irradiation, a superior 1 O2 production efficiency was observed for Ag31SG19 when compared with NMB (Table 1). The photosensitizer ability of Ag31SG19 is mainly due to its high light absorption coefficients, and long lifetime to allow high intersystem crossing efficiencies as demonstrated (Table 1). Under 532 nm irradiation (see Figure 4), a lower 1 O2 production efficiency ( 1 O2 generation rate ( 1 O2/cluster/min) = 0.980) was observed for Ag31SG19 (as compared to 473 nm irradiation). Of note, the 1 O2 generation rate (presented in Table 1) also depends on the absorbance of used laser light. Quantum yield of the singlet oxygen generation allows for a normalization by the absorbance at 473 nm of the cw laser. Using the protocol described in the pioneering paper [START_REF] Marin | Efficient intersystem crossing using singly halogenated carbomethoxyphenyl porphyrins measured using delayed fluorescence, chemical quenching, and singlet oxygen emission[END_REF], we were able to evaluate the quantum yield of the singlet oxygen generation (as described in supplementary materials and Figure S5). The quantum yield of 1 O2 generation was estimated to ~0.01-0.02.

@532 nm

Agar well diffusion analysis

The antibacterial activity analysis of Ag31SG19 was carried out against S. aureus and E. coli. Here, the zone of inhibition could be observed against both S. aureus and E. coli which indicated the Ag31SG19 to have antibacterial activity. The zone of inhibition observed against E. coli and S. aureus was 15.3 ± 0.58 mm and 16.7 ± 0.58 mm respectively It was also observed that a zone of inhibition against E. coli was 12.3 ± 0.58 mm, whereas no zone was observed against S. aureus (Figure 5).

Minimum Inhibitory Concentration analysis

The MIC of Ag31SG19 against S. aureus and E. coli was determined using the standard microtiter plate assay. Here, the MIC value of Ag31SG19 against E. coli and S. aureus was found to be 0.3125 mg/ml and 0.1562 mg/ml respectively (Figure S6).

Discussion

In the study, the antibacterial activity of Ag31SG19 was evaluated against E. coli and S. aureus as test organisms. From the agar well diffusion analysis, Ag31SG19 (20 mg/ml) was found to have significant antibacterial activity against both bacteria with a zone of inhibition measuring 15.3 ± 0.58 and 16.7 ± 0.58 mm respectively against E. coli and S. aureus. The size of the zone of inhibition indicates the degree of antibacterial effectiveness of Ag31SG19. However, the zone of inhibition for plasmonic AgNPs was observed to be 12.3 ± 0.58 mm against E. coli and no zone was observed against S. aureus. MIC analysis was carried out to find out the minimum concentration of Ag31SG19 required to inhibit the growth of E. coli and S. aureus.

Here, the MIC of Ag31SG19 was found to be 0.1562 mg/ml against S. aureus and 0.3125 mg/ml against E. coli. Overall, the results of both the experiments suggest the Ag31SG19 to have significant antibacterial activity and hence it can have the potential to be used as an antimicrobial agent. Here, S. aureus was found to be more susceptible towards Ag31SG19 when compared with E. coli.

In the study, Ag31SG19 nanoclusters were produced with high purity for the quantitative measurements of ROS efficiencies upon visible laser irradiation. Here, the nanoclusters synthesized were found to be highly stable under daily conditions and also under continuous visible laser irradiation. This behavior is quite unique in particular when compared with the Ag29 nanoclusters protected by DHLA ligands.

Here, Ag31SG19 presents high rates of singlet Oxygen generation in the visible range in particular upon 473 nm laser irradiation. Hence these Ag nanoclusters protected by glutathione were further used to investigate its antimicrobial efficacy against Grampositive S. aureus and Gram-negative E. coli. Based on our findings on quantitative measurements of ROS efficiencies and photostability measurements in the solution, the primary mechanism ( 1 O2 generation) leading to the antimicrobial activity could be a possible mechanism. Other mechanism which arise from the degradation of nanoclusters is certainly minimal here due to the high (photo)stability of Ag31SG19.

However, we cannot exclude that the generation of oxidizing species (H2O2, radicals) upon irradiation may play a role on the antimicrobial activity of Ag31SG19. To better address this point, we used a DCF-dye, as already studied by Yuan and coworkers, [START_REF] Liu | Conjugating AIE-featured AuAg nanoclusters with highly luminescent carbon dots for improved visible-light-driven antibacterial activity[END_REF] for probing the oxidizing agents produced following photoexcitation.

We adapted this method to probe type I ROS species generated following 473 nm laser irradiation, and indeed, we confirmed that Ag31SG19 is proned to produce such oxidizing agents in addition to 1 O2 species.

Conclusion

In conclusion, the antibacterial performance indicated the Ag31SG19 with superior stability to have enhanced antibacterial activity when compared to previous works using undefined glutathione protected silver nanoclusters (Table 2). The present study confirms that singlet oxygen generation produced by photo-excited Ag31SG19 nanoclusters is the key process involved in its antibacterial mechanism, while generation of oxidizing species is also involved. Singlet oxygen generation rate as well as quantum yield for 1 O2 generation for Ag31SG19 nanoclusters are compared to NMB dye photosensitizer under cw 473 nm and 532 nm irradiations. As singlet oxygen is considered to have rapid fatal activity and the most damaging ROS species against microorganisms, the currently studied Ag31SG19 nanoclusters can have significant antimicrobial therapeutic applications. Ag n (GSHR) m 1687.5 1687.5

Ref. [START_REF] Wang | Synthesis, characterization and antimicrobial activity of silver nanoparticles: Agn(NALC)m and Agn(GSHR)m[END_REF]

  at 410 nm according to Beer-Lambert law. The rate of 1 O2 generation was obtained by the initial DPBF concentration change over time by a function: and represent the concentration of DPBF at 0 min and 5 min under irradiation, respectively. represents the time under irradiation (5 mins), and =1.37 × 10 -6 M. The photoluminescence lifetime was calculated by two-exponential fitting in the decay curve. The uncertainty was evaluated by measuring 3 times repeatly.

Figure 1

 1 Figure 1 Mass spectrometry (a) and optical (b-d) characterization of Ag31SG19. (a) ESI-MS spectrum recorded in the negative mode of ionization. Inset : Deconvoluted ESI-MS spectra. (b) Absorption spectrum of Ag31SG19 in water. Emission (at excitation 450 nm) and Excitation spectra (at emission 705 nm (c) and 825 nm (d)) of Ag31SG19 (c) in water and (d) in ethanol (EtOH) (d).

Figure 2 :

 2 Figure 2 :Emission spectra of Ag31SG19 and Ag29DHLA12 for fresh and 24 h old solutions under normal conditions ( (a) and (b)) and (c) absorption spectra of Ag31SG19 under 473 nm laser irradition for 0 to 30 minutes).

Figure 3 :

 3 Figure 3: Characterisation of AgNPs. (a) TEM analysis, (b) HRTEM analysis, (c) size histogram, and (d) UV-VIS spectrum of plasmonic AgNPs.

Figure 4 .

 4 Figure 4. The rate of 1 O2 generation was obtained by the initial DPBF concentration

Figure 4 :

 4 Figure 4:Absorption spectra evolution of DPBF-Ag31SG19 mixture solution under cw laser (a) 473 and (b) 532 nm irradiation.

Figure 5 :

 5 Figure 5: Antibacterial activity of (a) Ag31SG19 against E. coli, (b) Ag31SG19 against S. aureus, (c) plasmonic AgNPs against E. coli and (d) plasmonic AgNPs against S. aureus.

  Figure S7 displays the fluorescence spectra of DCF dye without and with 10 min laser irradiation. The strong increase in fluorescence intensity is a signature of generation of type I species.

Table 1

 1 Singlet Oxygen generation rate by Ag31SG19 nanoclusters compared to NMB dye photosensitizer and AgNPs under cw 473 nm irradiation.

	Photosensitizer	Emission	Long lifetime	1 O2 generation rate
			component (ns)	( 1 O2/cluster/min) @473 nm

Table 2 :

 2 Minimum Inhibitory Concentration of silver nanoclusters against E. coli and S. aureus.

	Glutathione	MIC (µg/mL)	Ref.
	protected silver NCs	E. coli	S. aureus
	Ag 31 SG 19	312.5	156.2	Our work
	Glutathione-protected Ag+-Rich NCs	450.0	358.0	Ref.[24]

Acknowledgments

We acknowledge CNRS for funding through International Emerging Actions between Institut Lumière Matière, CNRS, France and Mahatma Gandhi University, India. HY is grateful for PhD fellowships donated by the China Scholarship Council (CSC). DB is grateful for post-doc fellowship donated by Agence Nationale de la Recherche (project MANBAMM, ANR-21-CE29-0020). R.A. and H.Y. acknowledge Shanghai Science and Technology Innovation Program (22520712500) for support. We gratefully acknowledge Marion Girod (Institut des Sciences Analytiques, Villeurbanne, France) for lending us the cw lasers. MT, SK, AR and REK acknowledge KSCSTE-SRS project (307/2022/KSCSTE, Kerala) for the facility and support.

Declaration of Competing Interest

The authors declare no conflict of interest.

Appendix A. Supplementary data

All the data are presented in this article and the Supplementary Materials.


Alexandre Vidal

Jean-Pierre Françoise

Jean-Pierre Franc ¸oise

Canard cycles in global dynamics

Keywords: 1991 Mathematics Subject Classification. Primary 34C29, 34C25, 58F22 Fast-Slow Systems, Canards, Delay to Bifurcation, Transcritical Bifurcation, Relaxation Oscillations

come   L'archive ouverte pluridisciplinaire

Introduction

Systems are often complex because their evolution involves different time scales. Purpose of this article is to present several phenomena which can be observed numerically and analyzed mathematically via bifurcation theory.

A first approximation for the time evolution of fast-slow dynamics is often seen as follows. A generic orbit quickly reaches the vicinity of an attractive invariant set of the fast dynamics. It evolves then slowly close to this attractive part until, under the influence of the slow dynamics, this attractive part bifurcates into a repulsive one. Then, the generic orbit quickly reaches the vicinity of another attractive invariant set until it also loses its stability. This approach is a quite meaningful approximation because it explains many phenomena like hysteresis cycles, relaxation oscillations, bursting oscillations [START_REF] Françoise | Oscillations en Biologie : Analyse qualitative et modèles[END_REF][START_REF] Vidal | Stable periodic orbits associated with bursting oscillations in population dynamics[END_REF][START_REF] Vidal | Periodic orbits of tritrophic slow-fast systems and double homoclinic bifurcations[END_REF] and more complicated alternation of pulsatile and surge patterns of coupled GnRH neurons [START_REF] Clement | Mathematical modeling of the GnRH-pulse and surge generator[END_REF][START_REF] Clement | Foliation-based parameter tuning in a model of the GnRH pulse and surge generator[END_REF].

Discovered by E. Benoit, J.-L. Callot, M. and F. Diener (see [START_REF] Benoit | Chasse au canard[END_REF]), canards were first observed in the van der Pol system: (0.1)

ε ẋ = y -f (x) = y -( x 3 3 + x 2 ) ẏ = x -c(ε)
where c(ε) ranges between some bounds: (0.2)

c 0 + exp - α ε 2 < c(ε) < c 0 + - β ε 2
A canard is an orbit which remains for a while in a small neighborhood of a repulsive branch of the critical manifold y = f (x) i.e. a connected set of repulsive points for the fast dynamics. In the following, we consider the canard phenomenon in its broader sense of delay to the bifurcation of the underlying fast dynamics under the influence of the slow dynamics. More recently, F. Dumortier and R. Roussarie [START_REF] Dumortier | Canard cycles and center manifolds[END_REF] contributed to the analysis of such orbits by blowing-up techniques and introduced the notion of canard cycle. There are now several evidences showing the relevance of this notion to explain experimental facts observed in physiology (see [START_REF] Desroches | Mixed-mode oscillations and slow manifolds in the self-coupled FitzHugh-Nagumo system[END_REF][START_REF] Toporikova | A-type K + current can act as a trigger for bursting in the absence of a slow variable[END_REF]).

The systems presented here display anomalous long delay to ejection from the repulsive part of the fast dynamics. crosses λ = 0. Equation (1.1) displays two equilibria, x = 0 and x = λ. For λ > 0, x = 0 is stable and x = λ is unstable. After the bifurcation, λ < 0, x = 0 is stable and x = λ is unstable. The two axis have "exchanged" their stability. The terminology "Dynamical Bifurcation" (due to R. Thom) refers to the situation where the bifurcation parameter is replaced by a slowly varying variable. In the case of the transcritical bifurcation, this yields:

(1.2) ẋ = -yx + x 2 ẏ = -ε
where ε is assumed to be small. This yields:

ẋ = -(-εt + y 0 )x + x 2 , (y 0 = y(0))
which is an integrable equation of Bernoulli type. Its solution is:

x(t) = x 0 exp[-Y (t)] 1 -x 0 t 0 exp[-Y (u)]du , (x 0 = x(0))
where:

Y (t) = t 0 y(s)ds = t 0 (-εs + y 0 )ds = -ε t 2 2 + y 0 t Figure 1. Orbits of system (1.
2) starting from (0.05i, 1), for i = 1, .., 6, with ε = 0.1. The critical set -formed by the singular points of the fast dynamics -is shown in green and red. On this set, the red points are repulsive for the fast dynamics and the green points are attractive. Each orbit reaches a neighborhood of the attractive manifold x = 0, y > 0, goes down along x = 0. When y becomes negative, despite the repulsiveness of x = 0, y < 0, each orbit remains for a very long time close to x = 0 before drifting away.

If we fix an initial data (x 0 , y 0 ), y 0 > 0, 0 < x 0 < y 0 /2, and we consider the solution starting from this initial data, we find easily that it takes time t = y 0 /ε to reach the axis y = 0. If x 0 is quite small, that means the orbit stays closer and closer of the attractive part of the critical manifold until it reaches the axis x = y and then coordinate x starts increasing. But now consider time cy 0 /ε, 1 ≤ c ≤ 2. Then, a straightforward computation shows that:

Y (t) = c 1 - c 2
y 2 0 ε = k ε and: x(t) = O   x 0 e -k ε 1 -2x 0 y 0  
This shows that, despite the repulsiveness of the half-line x = 0, y < 0 for the fast dynamics, the orbit of (1.2) remains for a very long time close to x = 0, indeed x(cy 0 /ε) < 2x 0 for ε small enough (see Figure 1). Note that, afterwards for larger time, the orbit blows away from this repulsive axis. This phenomenon, although quite simply explained, is of the same nature as the delay to bifurcation discovered for the dynamical Hopf bifurcation, see for instance [START_REF] Baer | The slow passage through a Hopf bifurcation: Delay memory effects and resonance[END_REF][START_REF] Candelpergher | Retard à la bifurcation : du local au global[END_REF][START_REF] Erneux | Slow passage through bifurcation and limit points. Asymptotic theory and applications[END_REF][START_REF] Zoladek | Remarks on the delay of the loss of stability of systems with changing parameters[END_REF]. This well-known effect is instrumental in the systems we study in this article. Some related work has been done in computing entry-exit relation for the passage near single turning points (see [START_REF] Benoit | Equations différentielles: relation entrée-sortie[END_REF][START_REF] De Maesschalk | Time analysis and entry-exit relation near planar turning points[END_REF]).

Enhanced delay to bifurcation.

Consider the system:

(1.3) ẋ = (1 -x 2 )(x -y) ẏ = εx
The critical set -defined as the set of singular points of the fast dynamicsis the union of the straight lines x = -1, x = 1, and y = x. In the following, we call "slow manifolds" the two straight lines x = -1 and x = 1, as they are invariant for the critical system {(1 -x 2 )(x -y) = 0, ẏ = x}. A quick analysis shows that, as the slow variable y (considered as a parameter) evolves, the fast system undergoes two transcritical bifurcations: at x = -1 for y = -1, at x = 1 for y = 1.

As we recalled in Subsection 1.1, a typical orbit starting from an initial data (x 0 , y 0 ), |x 0 | < 1, close to (x = -1, y > -1), first goes down along x = -1 and displays a "delay" along the repulsive part of the slow manifold (x = -1, y < -1). Then, under the influence of the fast dynamics, it quickly reaches the attractive part (x = 1, y < 1) and moves upward to the other transcritical bifurcation point. There, it again displays a delay along the repulsive part (x = 1, y > 1). Then, it quickly reaches (x = 1, y > -1) and starts again. Hence, there is a mechanism of successive enhancements of the delay to bifurcation after each oscillation generated by the hysteresis. We proved in [START_REF] Françoise | Enhanced Delay to Bifurcation[END_REF] the: Theorem 1. For all initial data inside the strip -1 < x < 1, for all δ and for all T , the corresponding orbit spends a time larger than T within a distance less than δ to the repulsive part of the slow manifolds.

We also proved in [START_REF] Françoise | Enhanced Delay to Bifurcation[END_REF] the: Theorem 2. Given any initial data (x 0 , y 0 ) outside the strip |x| ≤ 1, the corresponding orbit is asymptotic to y = x.

Structural stability of the enhanced delay.

The theory of the structural stability of fast-slow systems remains to be found. In this subsection, we actually adopt a very pragmatic approach and restrict Figure 2. Typical orbits of (1.3) with ε = 0.1 starting from a point near the origin in panel 1) and from various initial datas outside the strip |x| < 1 in panel 2). Double arrows are added on the fast parts of the orbits and single arrows on the slow parts. Panel 1): The delay to the transcritical bifurcation undergone by the orbit is enhanced at each passage. Consequently, the time taken to escape a given neighborhood of x = -1, y < -1 on one hand and a given neighborhood of x = 1, y > 1 on the other hand is longer for each oscillation than for the preceding one, although these half-lines are repulsive for the fast dynamics. Panel 2): Each orbit is asymptotic to y = x. These numerical simulations were performed with order 4 Runge-Kutta integration scheme (absolute error value: 10 -9 ; relative error value: 10 -12 ; mean integration step: 10 -5 ).

ourselves to numerical simulations. We perturb the system by adding an arbitrary small perturbation generated by a chaotic system, more precisely the Rössler system [START_REF] Rössler | An equation for continuous chaos[END_REF].

(1.4)

ε ẋ = (1 -x 2 )(x -y) + αn(t) ẏ = x
where n(t) is the first variable of a Rössler system:

(1.5) ṅ = -u -v u = n + au v = b + (n -c)v
with a = 0.1, b = 0.1, and c = 14 with some fixed initial data. We choose α quite small. The perturbation does not change the behavior of the typical orbit for a while. It starts to oscillate between x = 1 and x = -1: it remains for a long time alternatively near each of these lines and undergoes fast motions after ejection. But, as the orbit approaches closely these axes (within the distance of α), the small chaotic perturbation becomes operating and moves slightly the orbit outside of the strip. After crossing one of the straight lines x = -1 or x = 1, the orbit moves quickly to the axis y = x as previously shown. . Despite the very small magnitude of the perturbation -less than 3.10 -3 -the orbit exits the strip |x| ≤ 1 before the third oscillation around the origin. This simulation shows the loss of the oscillatory behavior as, under the influence of a perturbation of very small magnitude, the orbit crosses one of the slow manifolds as soon as x is sufficiently close to -1 or 1. We have thus obtained numerical evidence that the system is not structurally stable.

It is interesting to point out the fact that the numerical simulation of (1.3) highlights also this structural unstability. In fact, the integration scheme itself -whatever the integration step and the tolerances -provides errors. They give rise in the long run to intrinsic approximation: after many oscillations, the orbit starting from |x| < 1 passes so close to the slow manifolds that the numerical integration leads, sooner or later, to the approximation x = 1 or x = -1 or can eventually cross one of these lines.

Canard cycle for a double transcritical system.

In this subsection, we introduce a new system inspired by the preceding (1.3):

(1.6) ẋ = (1 -x 2 )(x -y) ẏ = εx(y + b)(a -y)
where a, b > 1 are parameters. The critical set is the same as the critical set of (1.3), formed by the three straight lines of equation x = -1, x = 1 and y = x. The dynamics displays again two transcritical bifurcations: at x = -1 for y = -1, at x = 1 for y = 1. But, the novelty is in the factors (a -y) and (y + b) in ẏ which yield bounded orbits. We restrict the phase space to the compact set:

K = {(x, y)| -1 ≤ x ≤ 1, -b ≤ y ≤ a} As for (1.
3), the origin is a repulsive focus of (1.6). Except the origin, the singular points of (1.6) in K are the summits of the rectangular boundary of K, (-1, -b), (1, -b), (1, a), (-1, a), and are all of saddle type. As the straight lines x = -1, x = 1, y = a, y = -b are invariant under the flow of (1.6), we deduce the stable and unstable manifolds of each saddle:

saddle stable manifold unstable manifold (-1, -b) x = -1, y < a y = -b, x < 1 (1, -b) y = -b, x > -1 x = 1, y < a (1, a) x = 1, y > -b y = a, x > -1 (-1, a) y = a, x < 1 x = 1, y > -b
Consequently, the interior of K is invariant under the flow and contains only a repulsive focus. The boundary of K is a graphic (ω-limit set formed by the union of the saddles with their separatrices). Following the study on the dynamical transcritical bifurcation, one expects that the value of y along an orbit oscillates between -b and min(a, b+2) if a ≥ b or betweenmin(b, a + 2) and a if a < b. However, the exponential attraction of x = -1, -1 < y < a (resp. x = 1, -b < y < 1), formed by attractive points of the fast dynamics, produces a delay to bifurcation and keeps the orbit near x = -1, -b < y < -1 (resp. x = 1, 1 < y < a), formed by repulsive points of the fast dynamics. The delay could be great enough so that the orbit reaches a small neighborhood of ẏ = 0. Hence, after the fast motion, the orbit tracks the other manifold x = 1, -b < y < 1 (resp. x = -1, -1 < y < a) even closer than during the preceding passage. The delay obtained afterwards is then again enhanced, as the orbit approaches even closer the slow manifolds. Hence, an orbit starting near the origin, after several oscillations, reaches alternatively a small neighborhood of both y = a and y = -b, whatever the values of a and b.

Figure 4 illustrates this global behavior with fine step numerical simulations for various sets of parameter values. As expected, the smaller ε, the faster the orbit reaches a given neighborhood of x = -1 (resp. x = 1). Starting near the origin, if ε is small (for instance, equal to 0.5 for the simulations presented in Figure 4), a few oscillations suffice to obtain a part close to the graphic.

Thus, the orbits of system (1.6) generate a new type of canards. On the contrary of the canards discovered in the solutions of the van der Pol system, all the orbits of system (1.6) starting from K\{(0, 0)} display delays to bifurcation. Moreover, we have a very simple way to modulate these delays by choosing the ) and a = 30 in panel 2). In both cases, the initial data is (10 -3 , 0). Comparison of panels 1) and 2) shows that, whatever the values of a and b, the orbit enters alternatively small neighborhood of x = -1, x = 1, y = -b and y = a (blue straight lines). These numerical simulations were performed with order 4 Runge-Kutta integration scheme (absolute error value: 10 -9 ; relative error value: 10 -12 ; mean integration step: 10 -6 ).

parameter values a and b. It is worth noticing that, as in system (1.3), small perturbations may provoke dramatic changes in the orbits. Now, we explore the ejection mechanism -leading to this type of oscillationsusing the conjugacy of (1.6) with an appropriate semi-local form and the analysis of the transition near the saddle (x, y) = (-1, -b) (the case of (x, y) = (1, a) is similar). First, we translate the origin to the saddle via the change of variables {X = x + 1, Y = y + b} to obtain the new form of the system:

(1.7) Ẋ = X(X -Y + b -1)(2 -X) Ẏ = εY (X -1)(a + b -Y )
Note that the left point of transcritical bifurcation of the fast dynamics (x, y) = (-1, -1) reads now (X, Y ) = (0, b -1). Consider an initial data ( X, Ȳ ) with X ∈]0, 1], b-1 < Ȳ < a+b. As explained above, for ε small, the corresponding orbit of (1.7) reaches quickly the vicinity of X = 0, y > b -1, X decreases very quickly while Y remains almost constant. Hence, along this fast branch of the dynamics, variable Y is given by Ȳ + O(ε) until X = O(ε). Afterwards, approximation of the dynamics near X = 0 by the slow motion and direct integration yield an approximation for the time needed to reach Y = b -1:

T Ȳ →b-1 = ε→0 1 ε(a + b) ln (a + 1) Ȳ (b -1)(a + b -Ȳ ) + O(1)
The leading term of the x-component C exp(-k/ε) displays:

C = (b -1) 2 b-1 a+b (a + 1) 2 a+1 a+b (1.8) k = 2 a + b (b -1) ln( Ȳ ) + (a + 1) ln(a + b -Ȳ ) > 0 (1.9)
We now study the delay to bifurcation and consider the entry of the orbits coming from above Y = b -1. Hence, we consider initial values of X which are exponentially small with respect to ε: C exp(-k/ε).

To specify the transition induced by the flow near the saddle, we consider the two sections:

Σ in = {(X, δ)|0 < X ≤ η} (1.10) Σ out = {(η, Y )|0 < Y ≤ δ} (1.11)
with η > 0 a small fixed parameter and 0 < δ ≤ b -1. We note:

U = {(X, Y )|0 < X ≤ η, 0 < Y ≤ δ}
the rectangle delimited by Σ in , Σ out and the stable and unstable manifolds of the saddle (see Figure 5). As X is small in U (smaller than η), system (1.7) is conjugated to:

(1.12) Ẋ = 2(b -1)X -2XY Ẏ = εY Note that (X, Y ) = (0, b -1)
is also a point of transcritical bifurcation for this system. Direct integration provides the orbit from (X 0 , δ):

X(t) = X 0 exp 2(b -1)t + 2δ ε e -εt - 1
Y (t) = δe -εt
As explained previously, we consider initial data of type

X 0 = C exp(-k/ε), where k, C > 0. If η < C, for ε small enough, i.e.: ε < - k ln η C (X 0 , δ) lies in Σ in . If η ≥ C, as ε, k > 0, all values of ε fulfill this property.
The time T needed to go from (X 0 , δ) to (η, Y (T )) ∈ Σ out along the flow is the solution of:

2(b -1)T + 2δ ε e -εT = k ε + ln η C + 2δ ε
The solution T can be expressed via the "Lambert function" W L -inverse function of w → we w (see [START_REF] Corless | On the LambertW function[END_REF]). This yields:

(1.13) T = 1 ε W L - δ b -1 exp - ǫ ln(η/C) + k + 2δ 2(b -1) + 1 2(b -1) ln η C + k + 2δ ε
As expected, the transition time is O(1/ε) and the leading term of T is:

(1.14) T = ε→0 1 ε W L - δ b -1 e -2δ+k
2(b-1)

+ 2δ + k 2(b -1) + O(1)
This displays the transition function (X 0 , δ) → (η, Y out (X 0 )) with:

(1.15) Y out (Ce -k ε ) = δ exp -W L - δ b -1 exp - ǫ ln(η/C) + k + 2δ 2(b -1) + 1 b -1 ln η C + k + 2δ ε
This shows that, even if the transition time tends to +∞, the exit function displays as O(1)-leading term:

(1.16) Y out (Ce -k ε ) = ε→0 Y 0 out (Ce -k ε ) + O(ε) = ε→0 δ exp -W L - δ b -1 e -2δ+k 2(b-1) - 2δ + k 2(b -1) + O(ε)
Hence, the longer the orbit has remained near the slow manifold, the stronger the ejection (to reach the neighborhood of (η, Y 0 out ) given by (1.16)) is. This property of the orbits -staying a very long time near the repulsive part of the slow manifold without squashing on the ẏ-nullcline -together with the increasing strength of ejection is what we call the "exceptionally fast recovery". Similar study can be done for the other transition of (1.6) and this leads to similar results (where a and b are exchanged).

Finally, note that, for given values of parameter a, b, ε, the values of parameters k and C in (1.14) and (1.16) are approximated for the global dynamics using (1.8) and (1.9).

Saddle-node transcritical ejection in a

prey-predator-superpredator model 2.1. Tritrophic food chain dynamics.

In the late seventies, interest in the mathematics of tritrophic food chain models (composed of prey, predator and superpredator) appear (see, for instance, [START_REF] Freedman | Mathematical analysis of some three-species foodchain models[END_REF][START_REF] Gard | Persistence in food chains with general interactions[END_REF]). Related predator-prey models with parasitic infections were studied later [START_REF] Hadeler | Predator-prey populations with parasitic infection[END_REF]. In the nineties, in [START_REF] Hastings | Chaos in a three-species food chain[END_REF][START_REF] Muratori | Low-and high-frequency oscillations in threedimensional food chain systems[END_REF] and [START_REF] Klebanoff | Chaos in three species food chains[END_REF], the existence of chaotic attractors was discussed. There are many more recent contributions, that we can not refer in more details (see, for instance, [START_REF] Kooi | Consequences of population models for the dynamics of food chains[END_REF][START_REF] Vidal | Periodic orbits of tritrophic slow-fast systems and double homoclinic bifurcations[END_REF]).

We investigate here the following:

dU dT = U R 1 - U K - A 1 V B 1 + U dV dT = V E 1 A 1 U B 1 + U -D 1 - A 2 W B 2 + V (2.1) dW dT = εW E 2 A 2 V B 2 + V -D 2
which represents the interactions between three populations U, V and W . The variable U stands for the prey, V for its predator and W for a superpredator of V . The threshold constant K > 0 and the intrinsic growth rate of the prey R > 0 characterize the logistic evolution of U.

The predator-prey interactions are described by two Holling type II factors defined by the positive parameters:

A j : the maximum predation rates B j : the half-saturation constants D j : the death rates E j : the efficiencies of predation j = 1 relates to the predator V , j = 2 relates to the superpredator W . It is assumed that the evolution of the superpredator is slower than those of the predator and the prey. Then we introduce different time scales by means of the constant 0 < ε ≪ 1.

Bifurcations of the fast dynamics.

The global behavior of this system (existence of global periodic orbit, bifurcations of limit cycles, early crisis in the predator membership) has been studied in [START_REF] Vidal | Stable periodic orbits associated with bursting oscillations in population dynamics[END_REF] and [START_REF] Vidal | Periodic orbits of tritrophic slow-fast systems and double homoclinic bifurcations[END_REF]. We recall briefly the classical situation of interest for us for which the system is bistable.

In order to obtain a simpler and more useful analytic form, Klebanoff and Hastings proposed in [START_REF] Klebanoff | Chaos in three species food chains[END_REF] the following rescalings:

(2.2) x = U K , y = V KE 1 , z = W KE 1 E 2 , t = RT which yields: ẋ = x 1 -x - a 1 y 1 + b 1 x = f (x, y, z) ẏ = y a 1 x 1 + b 1 x -d 1 - a 2 z 1 + b 2 y = g(x, y, z) (2.3) ż = εz a 2 y 1 + b 2 y -d 2 = h(x, y, z)
where a j , b j , d j , j = 1, 2 are positive parameters (see [START_REF] Klebanoff | Chaos in three species food chains[END_REF] or [START_REF] Vidal | Stable periodic orbits associated with bursting oscillations in population dynamics[END_REF] for their expressions in function of A j , B j , D j , E j , R and K. All axes and faces of the positive octant R 3 + are invariant sets of (2.3). Thus, we limit the phase space to this positive octant.

Considering the slow variable z as a parameter, we describe the sequence of bifurcations undergone by the two-dimensional fast dynamics. To this purpose, it is convenient to introduce the critical set:

(2.4) C = {(x, y, z) ∈ R 3 + |f (x, y, z) = 0, g(x, y, z) = 0} formed by the singular points of the so-called Boundary-Layer System (BLS), obtained from (2.3) with ε = 0. For any point (x, ỹ, z) ∈ C, (x, ỹ) is a singular point of the fast dynamics for z = z. In the following, we note π the projection from R 3 + into R 2 + : π(x, y, z) = (x, y).

First, we assume:

(2.5) G = a 1 -d 1 (1 + b 1 ) > 0
to ensure that the singular point:

(2.6)

d 1 G + d 1 , G (G + d 1 ) 2 , 0
lays in the phase space R 3 + . The critical set writes C = ∆ ∪ L where:

∆ = {(1, 0, z)|z ∈ R + } (2.7) L = (x, y L (x), z L (x)) ∈ R 3 + |x ∈ [0, 1] (2.8)
and:

y L (x) = 1 a 1 (1 -x) (1 + b 1 x) (2.9) z L (x) = (a 1 x -d 1 (1 + b 1 x)) (a 1 + b 2 (1 -x)(1 + b 1 x)) a 1 a 2 (1 + b 1 x) (2.10)
Note that ∆ always intersects L at the point T = (1, 0, z T ) where:

(2.11) z T = G a 2 (1 + b 1 )
> 0

In the following, we assume that d 1 is small enough so that:

(2.12) there is a point x P > 0 so that z ′ L (x P ) = 0. We note P = (x P , y P , z P ) = (x P , y L (x P ), z L (x P )). Let us remark that, under assumption (2.12), z T < z P . Consequently, L is ∩-shaped in R 3 + (cf. Figure 6) and we note:

• for each z ∈ ]z T , z P [, S z the unique point (x, y L (x), z) ∈ L such that x > x P ; L S = ∪

z T <z<z P {S z } ; • for each z ∈ [0, z P [, R z the unique point (x, y L (x), z) ∈ L such that x < x P ; L ± = ∪ 0<z<z P {R z }.
Thus, L = T ∪ L S ∪ P ∪ L ± (see Figure 6). The points π(S z ) are saddle type singular points of the fast dynamics, (1, 0) is a saddle for z < z T and an attractive node for z > z T . Additionally, we assume that d 1 is small enough such that there exists a point z H ∈]z T , z P [ so that:

for 0 ≤ z < z H , π(R z ) is a repulsive focus, for z H ≤ z < z P , π(R z ) is an attractive focus.
Figure 6 displays C and its splitting according to the nature of the singular points for the fast dynamics. It can be seen as a bifurcation diagram of the fast dynamics as the bifurcation parameter z varies. Hence, as z decreases, the following sequence of bifurcations occurs:

• for z > z P , the attractive node (1, 0) is the unique singular point. • as z = z P , an inverse saddle-node bifurcation occurs at (x P , y P ).

• for z H < z < z P , there are three singular points: the attractive node (1, 0), the saddle π(S z ), the attractive focus π(R z ). • as z = z H , π(R z ) undergoes a supercritical Hopf bifurcation.

• for z T < z < z H , there are three singular points: the attractive node (1, 0), the saddle π(S z ), the repulsive focus π(R z ) surrounded by an attractive limit cycle, born from the Hopf bifurcation. • as z = z T , a saddle-node transcritical bifurcation occurs at (1, 0) (π(S z ) and (0, 1) exchange their stability). • for 0 ≤ z < z T , there are two singular points in the positive octant: the saddle (0, 1) and the repulsive focus surrounded by an attractive limit cycle.

We note M the union in R 3 + of the planar attractive limit cycles surrounding the points R z for all z H < z < z P . M is thus an attractor for the (BLS).

A precise study of this sequence of bifurcations is available in [START_REF] Vidal | Stable periodic orbits associated with bursting oscillations in population dynamics[END_REF][START_REF] Vidal | Periodic orbits of tritrophic slow-fast systems and double homoclinic bifurcations[END_REF], including the double homoclinic bifurcation that may occur in a certain range of the parameter space. However, here, we focus on the oscillatory behavior of the system in the phase space and control the two types of ejections giving rise to the hysteresis loop: the saddle-node bifurcation and the saddle-node transcritical bifurcation.

2.3.

Global behavior of the system and canard cycles.

The separatrix ż = 0 is the plane of equation:

y = d 2 a 2 -b 2 d 2
For 0 < d 2 < a 2 /b 2 small enough, this plane separates ∆ near which ż < 0 and M near which ż > 0. Hence, any orbit of system (2.3) starting near ∆ -goes down along ∆ -. Once z < z T , it undergoes a delay to bifurcation and keeps tracking ∆ S for a while, although this branch is formed by saddles of the fast dynamics. After this delay, the orbit quickly reaches the vicinity of M. It goes up while spiraling around M, then around L -. As z becomes larger than z P , the orbit reaches the vicinity of ∆ -and repeats the same sequence of motions. Figure 7 displays the geometric invariants of the fast dynamics and a schematic orbit of (2.3). In this setting, the delay to the transcritical bifurcation, that the orbit undergoes, qualifies the terminology of canard cycle. The most surprising behavior of these orbits is the long time needed to escape a neighbordhood of the point (1, 0, 0) and the "exceptionally fast recovery" of the variable y. In this subsection, we perform successively two changes of variables to obtain an appropriate local form near the saddle.

The Jacobian matrix associated with the system (2.1) at the singular point (x, y, z) = (1, 0, 0) reads:

   -1 -a 1 1+b 1 0 0 a 1 1+b 1 -d 1 0 0 0 -εd 2    It admits -1, a 1 1+b 1 -d 1 = G 1+b
1 and -εd 2 as eigenvalues. Under the assumption (2.5), the singular point (1, 0, 0) is then a saddle with a two-dimensional stable manifold and a one-dimensional unstable manifold.

Eigenvectors associated with the eigenvalues are: eigenvalue eigenvector -1

(1, 0, 0)

G 1+b 1 (-1, 1 a 1 (G + 1 + b 1 ), 0) -εd 2
(0, 0, 1) Set:

α = 1 a 1 (G + 1 + b 1 ) = 1 - (1 + b 1 )(d 1 -1) a 1
The parameter α is positive under the assumption (2.5) (G > 0).

Hence, via the change of variables:

x = 1 -X -Y y = αY z = Z
the singular point (x, y, z) = (1, 0, 0) translates to (X, Y, Z) = (0, 0, 0). Now, the eigendirections of the saddle coincide with the (X, Y, Z)-axis. In this new system of coordinates, (2.3) reads:

Ẋ = -(1 -X -Y )(X + Y ) +Y -1 + (X -Y )(d 1 -1) 1 + b 1 (1 + X -Y ) - a 2 Z 1 + b 2 αY Ẏ = Y a 1 (1 + X -Y ) 1 + b 1 (1 + X -Y ) -d 1 - a 2 Z 1 + b 2 αY (2.13) Ż = Zε a 2 αY 1 + b 2 αY -d 2
This allows to consider X > 0 in the following.

Actually, an eigenvector associated with each saddle (0, 1) of the fast dynamics writes:

(2.14) (β(Z), 1) = a 2 Z(1 + b 1 ) 1 + b 1 + G -a 2 Z(1 + b 1 ) , 1
This vector is well-defined in the positive octant as long as:

Z < 1 + b 1 + G a 2 (1 + b 1 ) = 1 a 2 1 + G 1 + b 1
Thus, under the assumption (2.5), it is well-defined at least for Z ∈ [0, z T ] and β(Z) > 0. Note that, for Z = z T , this vector gives the central direction associated with the 0 eigenvalues of the non hyperbolic point (0, 0) of the fast dynamics.

For each Z ∈ [0, z T ], the vector (2.14) gives the tangent line to the unstable manifold of the saddle (0, 0) of the fast dynamics with Z = Z. Thus, we have an approximation for X and Y small of the invariant manifold of (2.13) with ε = 0. As this manifold is normally attractive, it persists for ε > 0 small enough into a normally attractive manifold of (2.13). We proceed with another change of variables:

u = X -β(Z)Y y = Y z = Z locally near X = Y = 0, 0 ≤ Z ≤ z T , system (2.13) reads: u = -u + O(εv, uv, u 2 ) v = G ′ v -a 2 vw + vO(u, v) (2.15) ẇ = -εd 2 w + wO(v 2 )
where:

G ′ = G 1 + b 1 = a 1 1 + b 1 -d 1
The perturbed attractive manifold in this new set of parameters is approximated by u = 0, 0 ≤ Z ≤ z T . The flow on the invariant perturbed manifold is conjugated -locally near u = v = 0 -to the system:

v = v a 1 1 + b 1 -d 1 -a 2 vw (2.16) ẇ = -εd 2 w
After setting w = 2Y /a 2 and replacing εd 2 by ε, we recognize the normal form (1.12) introduced in Subsection 1.4 with parameter values G ′ = (b -1) > 0.

Transition function: recovery of the predator.

Consider the two sections:

Σ in = {(u, v, δ)|0 < v ≤ η} (2.17) Σ out = {(u, η, Y )|0 < u ≤ ξ, 0 < w ≤ δ} (2.18)
where η, ξ > 0 are small but fixed parameters and 0 < δ < z T (see (2.11)) is fixed. We note:

U = {(u, v, w)|0 < u ≤ ξ, 0 < v ≤ η, 0 < w ≤ δ}
The preceding analysis shows that, for ε small enough, an orbit of (2.16) starting from Σ in enters U and exits from U. Hence, the transition function induced by the flow of (2.16) is well-defined from Σ in into Σ out .

As the typical orbits of system (2.3) enter an exponentially small neighborhood of ∆ S (see Figures 7,8 and Subsection 2.3), we restrict initial data of (2.16) to be exponentially close to X = Y = 0 in Σ in . Thus, consider:

u 0 = C 1 e -k 1 ε (2.19) v 0 = C 2 e -k 1 ε (2.20)
By direct application of the local analysis made in Subsection 1.4, we obtain the leading term of the transition time from (u 0 , v 0 , δ) ∈ Σ in to Σ out along the flow of (2.16):

T = ε→0 1 εd 2 W L - a 2 δ G ′ e -a 2 δ+k 2 d 2 G + a 2 δ + k 2 d 2 G ′ + O(1)
The transition function (u 0 , v 0 , δ) → (u out (u 0 , v 0 ), η, w out (u 0 , v 0 )) can be specified as well:

(2.21)

u out (C 1 e -k 1 ε , C 2 e -k 1 ε ) ∼ ε→0 C 1 e -k 1 ε -W L - a 2 δ G ′ e -a 2 δ+k 2 d 2 G ′ - a 2 δ + k 2 d 2 d 2 G ′ (2.22) w out (C 1 e -k 1 ε , C 2 e -k 1 ε ) = ε→0 δ exp -W L - a 2 δ G ′ e -a 2 δ+k 2 d 2 G ′ - a 2 δ + k 2 d 2 d 2 G ′ + O(ε)
It is worth noticing that the u-component of the transition function tends actually to 0 as ε → 0, but that the w-component experiments a basal threshold, even if this threshold is very low. 2.6. Interpretation in terms of ecological parameters: loss of resilience.

In contrast with system (1.6), the two ejection mechanisms in (2.3) are provided by different types of bifurcation of the fast dynamics: a saddle-node transcritical bifurcation at T and a saddle-node bifurcation at P . Near P , canards could eventually appear for which the orbit tracks the manifold L S formed by saddle of the fast dynamics. However, this type of canards can occur only if P is near the plane ż = 0 (within a O( √ ε) distance). But the assumption that this plane separates properly ∆ and the manifold M (d 2 small enough) generally forbids such behavior.

Hence, any orbit in the strictly positive octant displays, sooner or later, the motion described in Subsection 2.3 as there exists a globally attractive periodic orbit of this type (see [START_REF] Vidal | Stable periodic orbits associated with bursting oscillations in population dynamics[END_REF][START_REF] Vidal | Periodic orbits of tritrophic slow-fast systems and double homoclinic bifurcations[END_REF]). Thus, the entry of this periodic orbit -as the entry of any orbit after spiraling around M -in a given neighborhood of ∆ occurs for z = z P + O(ε 2/3 ). The delay to bifurcation after the passage in the vicinity of T is then mostly characterized by z P -z T . In fact, as calculated for (1.6) in (1.9), the contraction exponent between ∆ and the orbit is chiefly determined by z P -z T .

The "exceptionally fast recovery" that we have pointed out therein may have some connection with the notion of "loss of resilience" which has been discussed in several dynamical models related to Ecology [START_REF] Holling | Resilience and stability of ecological systems[END_REF][START_REF] Ludwig | Sustainability, stability and resilience[END_REF][START_REF] Martin | The cost of restoration as a way of defining resilience: a viability approach applied to a model of lake eutrophication[END_REF]. The long return times associated with a loss of resilience are caused by slow dynamics near the unstable equilibrium. We expect that new developments of this subject could possibly benefit of the notion of canards and canard cycles.

1 .

 1 Enhanced delay and canard cycles of planar (1+1)-dynamics 1.1. Dynamical Transcritical Bifurcation. The classical transcritical bifurcation occurs when the parameter λ in the equation: (1.1) ẋ = -λx + x 2

Figure 3 .

 3 Figure 3. Example of an orbit of (1.4), with ε = 0.5 starting from (x, y) = (10 -3 , 0) for α = 10 -4 . Double arrows are added on the fast parts of the orbit and single arrows on the slow parts.

Figure 3

 3 Figure 3 displays an example of such an orbit starting from (0.001, 0) for α = 10 -4 . Despite the very small magnitude of the perturbation -less than 3.10 -3 -the orbit exits the strip |x| ≤ 1 before the third oscillation around the origin. This simulation shows the loss of the oscillatory behavior as, under the influence of a perturbation of very small magnitude, the orbit crosses one of the slow manifolds as soon as x is sufficiently close to -1 or 1. We have thus obtained numerical evidence that the system is not structurally stable.It is interesting to point out the fact that the numerical simulation of (1.3) highlights also this structural unstability. In fact, the integration scheme itself -whatever the integration step and the tolerances -provides errors. They give rise in the long run to intrinsic approximation: after many oscillations, the orbit starting from |x| < 1 passes so close to the slow manifolds that the numerical integration leads, sooner or later, to the approximation x = 1 or x = -1 or can eventually cross one of these lines.

Figure 4 .

 4 Figure 4. Orbits (red) of (1.6) for ε = 0.5, b = 2 and respectively a = 5 in panel 1) and a = 30 in panel 2). In both cases, the initial data is (10 -3 , 0). Comparison of panels 1) and 2) shows that, whatever the values of a and b, the orbit enters alternatively small neighborhood of x = -1, x = 1, y = -b and y = a (blue straight lines). These numerical simulations were performed with order 4 Runge-Kutta integration scheme (absolute error value: 10 -9 ; relative error value: 10 -12 ; mean integration step: 10 -6 ).

Figure 5 .

 5 Figure 5. The transition function induced by the flow of (1.7) is well-defined from Σ in into Σ out . Hence, as Ẏ < 0 in {(X, Y )|0 < X < 1, 0 < Y < X + b -1} and δ ≤ b -1, any orbit starting from Σ in enters U and escapes from U through Σ out . Thus, the transition function induced by the flow is well-defined from Σ in into Σ out .

Figure 6 .

 6 Figure 6. Critical set of (2.3) (in red) -set of singular points of the Boundary-Layer System (BLS) obtained by setting ε = 0. It is splitted according to the nature of the singular points for the fast dynamics with the corresponding value of z: ∆ -and L -formed by attractive nodes, ∆ S and L S by saddles, L + by repulsive foci. The points P , H, T are respectively the points of saddle-node, Hopf and saddle-node transcritical bifurcation of the fast dynamics. For z ∈ [0, z H [, the repulsive focus π(R z ) ∈ L + is surrounded by an attractive limit cycle of the fast dynamics. The union M of these limit cycles (in green) is an invariant attractive manifold for the (BLS).

Figure 7 .

 7 Figure 7. Outline of a typical orbit of system (2.3) generating bursting oscillations. Double arrows are added on the fast parts of the orbit and single arrows on the slow parts.

Figure 8

 8 displays a numerical simulation of a typical orbit of (2.3) with:a 1 = 0.8, b 1 = 4, d 1 = 0.1, ε = 0.1, a 2 = 42, b 2 = 40, d 2 = 1.

Figure 8 .

 8 Figure 8. Orbit of system (2.3) with the parameter values given in the table of Subsection 2.3. Double arrows are added on the fast parts of the orbit and single arrows on the slow parts.

Acknowledgements: The numerical simulations were performed using XPP-AUT: http://www.math.pitt.edu/~bard/xpp/xpp.html

This research is supported by the Agence Nationale de la Recherche with the project ANAR. 1


domain: chim phys spi

text:
Oleg Sidletskiy
email: sidletskiy@isma.kharkov.ua

Pavlo Arhipov

Sergii Tkachenko

Olga Zelenskaya

Sergey Vasyukov

Federico Moretti

Christophe Dujardin

Drastic scintillation yield enhancement of YAG:Ce with carbon doping

Keywords: Scintillators, YAG, YAG:Ce, light yield Scintillators, YAG:Ce, light yield

Scintillation performances of carbon-doped YAG and YAG:Ce crystals obtained by the Czochralski method under novel conditions of Ar+CO reducing atmosphere are compared to their counterparts obtained by conventional technologies. While light yield and energy resolution in YAG crystals grown under different conditions are similar, a fast luminescence decay component of 4-6 ns was observed in YAG:C grown by the new procedure. Optimization of post-growth thermal annealing procedure of YAG:Ce,C scintillator provides the very high light yield of 28200 phot MeV -1 , and the energy resolution of 7.8 -8.5 % at 662 KeV. YAG:Ce [10] HDC b) Мо

Introduction

New high-energy physics experiments at colliders (see, for ex., [START_REF] Coupard | LHC Detectors Upgrade[END_REF] ), as well as future generation of medical tomography equipment [START_REF] Lecoq | [END_REF] need bright scintillators with faster luminescence decay compared to currently widely used Ce 3+ -doped scintillators. Y3Al5O12 (YAG) garnet crystals are considered as an efficient host for laser, phosphor, and scintillation materials. Scintillation properties of undoped YAG were barely explored in detail due to low light yield and slow luminescence decay. Meanwhile, Ce-doped garnets did not attract much attention until the development of extremely efficient Al/Ga-substituted multicomponent garnets with the light yields of up to 58000 phot MeV -1 . [3,4] This value is near the theoretical limit considering the energy gap of these materials. [5] The improvements in the light yield were attributed to the suppression of electron capture on shallow traps due to the electronic structure modification by cationic substitution. [6] In this sense, it is worth to explore whether it is possible to reduce carrier trapping and improve the scintillation performance in "simple" garnets, such as YAG:Ce, without considering solid solutions with other cations. This is important, because Gd3Al5-xGaxO12:Ce (GAGG:Ce) and related Ga-containing crystals have to be grown in inert gases enriched with oxygen to reduce Ga evaporation. Only Ir crucibles are capable to withstand such atmosphere at temperatures near 2000 °C, if oxygen content does not exceed 1-2 vol.%. Nevertheless, Ga evaporation from the melt under these conditions is still rather strong. This, evidently, causes the tendency of crystals to cracking. As a result, the use of Ir crucibles and expensive thermal insulation material, as well as the high energy consumption during the crystal growth of rareearth garnets, makes the crystals expensive for some applications. The economic factor is a problem also for other high-melting point complex oxide scintillators, such as Lu2SiO5:Ce (LSO:Ce) and Lu2-xYxSiO5:Ce (LYSO:Ce). All these factors promote the development of low-cost crystal growth technologies.

Procedures for aluminum garnets and perovskites growth in Mo and W crucibles have been developed since 1970-80s [7][8][9] until much more recently. [10][11][12] Meanwhile, a new procedure of growth [13,14] and post-growth heat-treatment [15] for garnet-based crystals was developed by some of authors of this work, and a series of undoped and Ce-doped YAG crystals were produced. Among the basic features of the novel procedure there is the use of cheap graphite heat insulation in crystallizer and reducing Ar+CO growth atmosphere. The latter is a cause of few hundreds wt. ppm (up to 1 at.%) carbon incorporation in crystals [15] .

Since carbon impacts on YAG optical properties, [15] the crystals obtained in these conditions were denoted as carbon-doped crystals (YAG:C and YAG:Ce,C).

In this paper the scintillation parameters of YAG:C and YAG:Ce,C crystals grown by the developed procedure are compared with YAG and YAG:Ce grown by the conventional technologies.

Methods

Crystal growth: YAG:C and YAG:Ce,C crystals were grown by the developed procedure in Mo and W crucibles using graphite heat insulation under Ar+CO reducing atmosphere. The growth procedure details are described elsewhere. [13,14] The carbon concentration was up to 1 at.% in YAG:C crystals. The Ce concentration in YAG:Ce,C was around 0.1 at.%. Carbon concentration was not measured in YAG:Ce,C, but it is presumed to be the same as in YAG:C, because the crystal growth procedure was the same. Cerium was introduced in the form of CeAlO3 single crystals, whose production technology was developed by us in. [16] Asgrown YAG:C were greenish, and YAG:Ce,C were yellow-greenish as the result of interaction with the reducing atmosphere (Figure 1). The post-growth heat treatment of crystals at 1200 °C under air and at 1500 -1850 °C under Ar+CO atmosphere provided the bleaching of YAG:C and less deep yellow coloration of YAG:Ce,C. The reference YAG and YAG:Ce crystals were grown by the Czochralski method from Ir crucibles in Ar gas atmosphere. Samples with the dimensions of 10x10x2 mm 3 were fabricated for optical and scintillation measurements. Measurement of spectral properties: The absorption spectra were measured using a Specord 40 spectrophotometer (Analytik Jena AG). X-ray luminescence (RL) spectra were measured using a home-made apparatus featuring a CCD detector (Jobin-Yvon Spectrum One 3000)

Experimental results and discussion

YAG:C

The as-grown crystals are colored, while the annealed crystals are highly transparent both in UV-and in the visible. In annealed YAG:C the color centers are not re-formed under any subsequent reducing or oxidizing annealing (see [15] for details). The irreversible bleaching was observed for YAG:C crystals grown in W, Mo, and Ir crucibles (if raw material was prepared under Ar+CO atmosphere). This means that the possible introduction of crucible material atoms in the crystals is not responsible for this phenomenon. The lack of coloration after subsequent annealing was attributed [15] to the existence of competing centers of electron capture linked to complex C-related defects. Thus, a high transparency of annealed YAG:C crystals and their scintillation parameters sustain under any further annealing.

Light yield (10000 -14000 phot MeV -1 ), energy resolution (17 % at 662 keV) and scintillation decay (τ = 400 ns + slow component) of annealed crystals are similar to those reported in, [11,16,17] while light yield and energy resolution of as-grown YAG:C are significantly worse due to the formation of color centers (Table 1). Light yield, phot/MeV <10000 50000/5.5 MeV [11] up to 14000 14300 [16] 60000/5.5MeV [11] Luminescence decay times, ns ~5 + slow 3-4 + slow [17] 35, 179, 974 [11] ~ 400 + slow 460 [16] 3-4, 750-1000 [17] 51, 373 [11] Energy resolution at 662 keV, % Not identified 16-18 [17] 17 11-14 [17] Meanwhile The fast nanosecond component under selective UV-excitation and synchrotron excitation was already reported before in the literature. Probably, the most detailed analysis of emission centers in undoped garnets was presented in [18] where the 2-3 ns decay components with emission band peaked around 400 nm (3.095 eV) and excitation at 3.33; 5.37; 6.56;7.3 eV were attributed to F + -centers localized near YAl antisite defects. The same luminescence band peaked at 397 nm and excited at 370 and 236 nm was registered in our YAG:C crystals [19] (in that work they are denoted as "Type III" crystals). Similar data on fast (<10 ns) luminescence decay in garnets by γ-ray excitation were reported previously in, [17] but it is impossible to evaluate its contribution based on the there presented data. demonstrates just one dominant band peaked at 310 nm, which is evidently a superposition of emission of excitons trapped near antisite defects [20] and Gd 3+ impurity emission, the spectrum of transparent YAG:C crystals contains much wider luminescence band in the same range with the main peak near 300 nm of AD-related exciton emission, and shoulder at ~245 nm attributed to self-trapped exciton emission. The wide 300 nm peaked band is a superposition of peaks of exciton emission related to different types of AD-related defects.

Unlike transparent crystals, in colored YAG:C the complex luminescence band arises at >395 nm, which is evidently related to the registered fast F + -center emission. Similar luminescence band was observed in YAG grown under Ar+H2 atmosphere [11] , and the persistent blue luminescence was revealed in carbon-doped YAG [21,22] , which was proposed for dosimetry applications. However, we have not noticed a clear correlation between the contribution of the fast luminescence decay component and intensity of the >370 nm luminescence bands on Xray luminescence spectra.

YAG:Ce,C

While the light yield of as-grown YAG:Ce,C is similar to the light yields of YAG:Ce reported in most of the papers, [23][24][25] the light yield of YAG:Ce,C after thermal annealing under air or Ar+CO atmosphere increases by 1.5 -2.0 times compared to the as-grown crystal (Figure 4, Table 2). The highest light yield registered after annealing in air is 330 % relatively to BGO taking into consideration the quantum efficiency of these scintillators with a R1307 PMT.

Assuming the averaged light yield of BGO reference sample of 8600 phot MeV -1 according to 8000 -9000 phot/MeV values prevailing in literature, [23,[26][27][28] the light yield of air-annealed YAG:Ce,C is 28200 phot MeV -1 (Table 2). The LSO:Ce reference light yield of 28000 phot/MeV determined by our procedure lies within the standard values for this scintillator, see for example, [29] and confirms the correctness of our data. To the best of our knowledge, just one paper reports the similar light yield around 30000 phot/MeV in YAG:Ce, though without the pulse height spectrum [30] (Table 2). The energy resolution in air-annealed sample presented in Figure 4 is 8.1 %, though it ranges from 7.8 to 8.5 % in different samples. The growth and post-growth annealing under the reducing conditions should promote the reduction of cerium into the optically active trivalent state. We presume that in as-grown YAG:Ce,C a high Ce 3+ concentration does not lead to light yield increase because of high concentration of defects in crystals. The light yield improvement after the reducing annealing coincides with the suppression of the absorption band peaked at 371 nm and related to F + centers, [18,19] as shown in The air annealing significantly improves both light yield and energy resolution. This is evidently attributed to the elimination of oxygen vacancies and related defects. According to the intensity of Ce 3+ absorption bands at 458 and 340 nm, and absorption band around 300 nm attributed to Ce 4+ -O 2-charge transfer complexes, [START_REF] Blasse | Luminescent Materials[END_REF] the weakening of Ce 3+ bands and reinforcement of Ce 4+ -O 2-band in the air-annealed sample compared to CO-annealed sample (see the difference spectrum at the inset in Figure 5) certify a partial oxidation of Ce 3+ into Ce 4+ . According to [START_REF] Nikl | [END_REF] , Ce 4+ competes for electron capture with shallow electron traps in garnets, and the coexistence of Ce 3+ and Ce 4+ reduces electron trapping and promotes fast and bright scintillation.

Comparison of X-ray luminescence spectra (Figure 6 2) are similar to those published in literature (see, for ex., [24] ) with the fast (τ1 = 73 ns) and slow (τ2 = 254 ns) components. Although the calculated decay constants are somewhat longer in the case of the annealed samples (98 ns, and 100 ns), the overall decay is faster, as one may see in Figure 7, due to the decrease the slow component contribution down to 21 -23 %. The luminescence decay almost does not depend on the annealing atmosphere (air or Ar+CO) suggesting that the annealing temperature is the main factor calling the reduction of carrier trapping responsible for the delayed luminescence. The phosphorescence level (background) after 3 µs decreases by more than twice after the thermal annealing (see Table 2). Despite the very high measured light yield, we suppose that there is still room for further improvement of scintillation parameters by the optimization of Ce concentration and other parameters of the preparation process.

Figure 1 .

 1 Figure 1. The view of as-grown YAG (a), YAG:C (b), and YAG:Ce,C (c) crystals, and airannealed YAG:Ce,C scintillation element (d).

  monochromator (Jobin-Yvon Triax 180). RL irradiation was performed by a Philips 2274 X-ray tube. Determination of light yield and energy resolution: Light yield (LY) and energy resolution (R) were measured under irradiation with a 662 keV 137 Cs gamma source. A R1307 Hamamatsu PMT ran at 800 V and HV with linear dynode voltage divider were used for registration of YAG:Ce,C parameters. Meanwhile, a R2059 Hamamatsu PMT run at 1400 V with better sensitivity in UV-range was used for measurements with YAG:C. PMT output was connected to the charge-sensitive preamplifier BUS 2-94 and BUI-3K shaping amplifier. Signal from preamplifier was shaped by a shaping amplifier with 2 µs (for YAG:C) and 8 μs (for Ce-doped YAG) shaping times. In order to collect the whole scintillation light the samples together with open part of PMT photocathode were covered with Teflon reflector. No optical contact between samples and PMT window was provided. For absolute light yield determination the yields were compared to Bi4Ge3O12 (BGO) produced at ISMA, and Lu2SiO5 (LSO:Ce) sample obtained from Proteus Inc., USA. The quantum efficiencies were 0.1867 (BGO), 0.1158 (YAG:Ce,C), 0.2578 (LSO:Ce) for measurements with R1307 PMT, and 0.1472 (BGO) and 0,2108 (YAG:C) for measurements with R2059 PMT based on the luminescence spectra of the crystals and the PMT quantum efficiency. Energy resolutions of 137 Cs 662 keV peaks are determined by approximation of the obtained pulse-height spectra with Gaussian function. Determination of decay times: Scintillation decay times were determined at X-ray excitation (50 kHz excitation repetition rate, FLE 400 filter λ> 400nm). Pulsed X-ray decay measurements were performed using a Hamamatsu N5084 light-excited X-ray tube set at 30 kV as irradiation source. The optical excitation of the tube was performed with a Hamamatsu PLP-10 picosecond light pulser. The scintillation decay was registered by a PMA 165-C PMT from PicoQuant operating in single photon counting regime. According to the data sheet, the transit time is below 180 ps and the rise and fall time are 750 ps providing a time resolution of 180ps.

  , colored YAG:C, apart of few hundred ns component ascribed to the selftrapped exciton (STE) emission, and long microsecond components, possesses a very fast luminescence decay component with τ = 4-6 ns under X-ray excitation. The decay curves are well fit by the three-exponential decay. The contribution W1 of the fast component with τ1 = 4.8 ns at fitting (Figure 2) to the light sum within the 30 microsecond gate is ~2 %, according to the equation 1.

Figure 2 .

 2 Figure 2. Scintillation decay after X-ray irradiation of colored YAG:C crystal fit by the function y = 0.003+0.888*exp(-t/4.8)+0.064*exp(-t/296.4)+0.058*exp(-t/3137.9); inset: the same data within the time range of first 60 ns.

Figure 3 .

 3 Figure 3. Typical X-ray luminescence spectra of YAG and YAG:C crystals

Figure 4 .

 4 Figure 4. Pulse height spectra of YAG:Ce,C crystals compared to the reference BGO and LSO:Ce samples.

Figure 5 .Figure 5 .

 55 Figure 5. Absorption spectra of YAG:Ce,C crystals; on the insetdifference spectra between air-and Ar+CO-annealed samples.

  ) indicates that both position and shape of Ce 3+ luminescence bands at 530 nm are the same in YAG:Ce and YAG:Ce,C, and they are not affected by the thermal treatments of YAG:Ce,C. Note that the most intense X-ray excited Ce 3+ luminescence is registered in CO-annealed samples, while air-annealed YAG:Ce,C possesses the highest light yield. The 310 nm complex band in YAG:Ce, by the analogy to YAG, is related to YAl antisite luminescence and Gd 3+ emission. The excitation peak at 273 nm and emission peak at 312 nm clearly related to Gd 3+ luminescence were detected in YAG:Ce spectra (not shown in this paper). Meanwhile, the weakening of the band peaked 310 nm suggests the decrease of concentration of antisites in Ce,C-codoped crystals. The absence of Gd 3+ -related peak in carbon-codoped samples can be attributed to the enhanced energy transfer from Gd 3+ to Ce 3+ . Therefore, the improved energy transfer to Ce 3+ luminescence centers is an additional reason of the light yield improvement in YAG:Ce,C scintillator.

Figure 6 .

 6 Figure 6. X-ray luminescence spectra of YAG:Ce,C.

Figure 7 .

 7 Figure 7. Luminescence decay curves after X-ray irradiation of YAG:Ce,C crystals (blue scattersas-grown YAG:Ce, red scatters -CO-annealed YAG, black scattersair-annealed YAG.

Table 1 .

 1 Scintillation parameters of YAG:C and YAG crystals.

Table 2 .

 2 Scintillation parameters of YAG:Ce,C crystals compared to their counterparts obtained by the conventional technologies.

	Crystal	Growth method	Crucible material	Light yield [phot MeV -1 ]	Energy [%] resolution	Scintillation decay [ns] time	Background level [%] after 3 µs
	YAG:С,Ce	Cz a) , as-grown	Mo/W	14200	9.7	254 (70%) 73 (30%),	2.94
	YAG:С,Ce	Cz a) , CO-anneal.	Mo/W	21700	12.9	(23%) 100 (77%), 331	1.77
	YAG:С,Ce	Cz a) , Air-anneal.	Mo/W	28200	7.8 -8.5	98 (79%), 349 (21%)	1.30
	a) Czochralski method; b) Horizontal Directional Freezing method

Acknowledgements

Authors are thankful to Prof. N. Shiran (ISMA NAS of Ukraine), Dr. S. Gridin (Department of Physics, Wake Forest University, Winston-Salem, NC, USA), and Prof. A. Vedda (Department of Materials Science, University of Milano-Bicocca) for fruitful discussions. The work is partially supported by the Marie Skłodowska-Curie Research, Innovation Staff Exchange Project H2020-MSCA-RISE-2014 no.644260 "INTELUM". Authors are grateful to COST Action TD1401 "Fast Advanced Scintillator Timing (FAST)" for support of collaboration.

Received: ((will be filled in by the editorial staff))

Revised: ((will be filled in by the editorial staff)) Published online: ((will be filled in by the editorial staff))

Ce [23] Cz a) No data 16700 No data No data No data YAG:Ce [24] Cz a) No data No data No data 119+slow No data 30260 No data No data No data

4. Conclusions Scintillation performance of carbon-doped YAG and YAG:Ce crystals is explored. The YAG:C parameters are similar to those in YAG obtained by other methods. Meanwhile the fast luminescence with τ ~ 5 ns attributed to F + -centers under extrinsic excitation was revealed in as-grown YAG:C. High-quality YAG:Ce,C crystals were obtained by the proposed Czochralski process using Mo/W crucibles and optimized post-growth annealing with a very high light yield of up to 28200 phot MeV -1 and a good energy resolution of 8 % at 662 keV. The contribution of fast ~100 ns luminescence decay component in air-annealed YAG:Ce,C reaches 79 %.

The obtained scintillation parameters are the result of complex interactions between electrons and holes with cerium in different valence states, carbon, as well as various types of intrinsic defects in the garnet structure. Thus, scintillation mechanisms in YAG:Ce,C still need for detailed study.