Ekaterina Mamontova
email: ekaterina.mamontova@etu.umontpellier.fr

Jérôme Long
email: jerome.long@umontpellier.fr

Rute A S Ferreira
email: rferreira@ua.pt

Alexandre M P Botas
email: a.botas@ua.pt

Dominique Luneau
email: dominique.luneau@univ-lyon1.fr

Yannick Guari
email: yannick.guari@umontpellier.fr

Luis D Carlos
email: lcarlos@ua.pt

Joulia Larionova
email: joulia.larionova@umontpellier.fr

Magneto-Luminescence Correlation in the Textbook Dysprosium(III) Nitrate Single-Ion Magnet

Keywords: lanthanides, single-ion magnets, single-molecule magnet, luminescence, magnetic relaxation

Multifunctional Single-Molecule Magnets (SMMs) or Single-Ion Magnets (SIMs) are intriguing molecule-based materials presenting an association of the slow magnetic relaxation with other physical properties. In this article, we present an example of a very simple molecule based on Dy 3+ ion exhibiting a field induced SIM property and a characteristic Dy 3+ based emission. The [Dy(NO 3 ) 3 (H 2 O) 4 ]•2H 2 O (1) complex is characterized by the means of single crystal X-Ray diffraction and their magnetic and photo-luminescent properties are investigated. We demonstrate here that it is possible to correlate the magnetic and luminescent properties and to obtain the Orbach barrier from the low temperature emission spectra, which is often difficult to properly extract from the magnetic measurements, especially in the case of field induced SIMs.

Introduction

Single-Ion Magnets (SIMs) are intriguing mononuclear coordination complexes which represent one of the smallest magnetically bistable units since they are composed of a single paramagnetic transition metal or lanthanide ion [START_REF] Woodruff | Lanthanide single-molecule magnets[END_REF][START_REF] Layfield | Organometallic single-molecule magnets[END_REF][START_REF] Liddle | Improving f-element single molecule magnets[END_REF]. Consequently, such molecular objects appear as promising candidates for high-density data storage or quantum computing [START_REF] Woodruff | Lanthanide single-molecule magnets[END_REF][START_REF] Luzon | Lanthanides in molecular magnetism: So fascinating, so challenging[END_REF]. In the case of lanthanide ions, the magnitude of the anisotropic barrier responsible for the reversal of the magnetization arises from the intrinsic electronic structure and in particular from the sub-levels structure of the Stark components. Such energetic considerations rely on numerous parameters including the nature of the lanthanide ion (Kramers/non-Kramers character, oblate/prolate electronic density), its geometry as well as the overall coordination environment often governed by the donor strength of the ligands [START_REF] Rinehart | Exploiting single-ion anisotropy in the design of f-element single-molecule magnets[END_REF]. In addition, the relaxation of the magnetization may be governed by various mechanisms that are either spin-lattice processes (Orbach, Raman and direct relaxations) or the Quantum Tunneling of the Magnetization (QTM). The latter is known to depend also on numerous parameters such as the occurrence of magnetic exchange interactions, hyperfine coupling or dipolar interactions [START_REF] Pointillart | Magnetic memory in an isotopically enriched and magnetically isolated mononuclear dysprosium complex[END_REF]. Understanding the mechanisms that govern the relaxation of the magnetization and the parameters that affect these relaxation processes is particularly important to ultimately optimize the SIM features.

In this line of scope, luminescent SIMs are particularly appropriate for these investigations since the photo-luminescence measurements permit determining spectroscopically the Stark sub-level structure of the lanthanides ions and comparing it with the magnetic data. However, examples of such magneto-optical correlations are rather scarce [START_REF] Cucinotta | Magnetic anisotropy in a dysprosium/dota single-molecule magnet: Beyond simple magneto-structural correlations[END_REF][START_REF] Long | A bifunctional luminescent single-ion magnet: Towards correlation between luminescence studies and magnetic slow relaxation processes[END_REF][START_REF] Yamashita | A luminescent single-molecule magnet: Observation of magnetic anisotropy using emission as a probe[END_REF][START_REF] Pointillart | A redox-active luminescent ytterbium-based single molecule magnet[END_REF][START_REF] Gavey | Placing a crown on dy(III)-A dual property lniii crown ether complex displaying optical properties and smm behaviour[END_REF][START_REF] Shintoyo | Crystal field splitting of the ground state of terbium(iii) and dysprosium(III) complexes with a triimidazolyl tripod ligand and an acetate determined by magnetic analysis and luminescence[END_REF][START_REF] Ren | A layered erbium phosphonate in pseudo-D 5h symmetry exhibiting field-tunable magnetic relaxation and optical correlation[END_REF][START_REF] Long | A high-temperature molecular ferroelectric Zn/Dy complex exhibiting single-ion-magnet behavior and lanthanide luminescence[END_REF][START_REF] Rechkemmer | Comprehensive spectroscopic determination of the crystal field splitting in an erbium single-ion magnet[END_REF][START_REF] Gregson | A monometallic lanthanide bis(methanediide) single molecule magnet with a large energy barrier and complex spin relaxation behaviour[END_REF][START_REF] Bi | Thermostability and photoluminescence of dy(III) single-molecule magnets under a magnetic field[END_REF][START_REF] Hareri | Encapsulation of a [Dy(OH 2 ) 8 ] 3+ cation: Magneto-optical and theoretical studies of a caged, emissive smm[END_REF] in comparison with the large number of reported lanthanides-based SIMs. This fact is often due to the weak or completely quenched emission in these complexes. In order to overcome this problem, antenna ligands are used to enhance the lanthanide emission. Thus, in all cases of magneto-luminescent SIM, the observed lanthanide luminescence relies on the use of coordinated ligands or complexes as sensitizers to enhance the lanthanide luminescence via an energy transfer mechanism [START_REF] Bunzli | Taking advantage of luminescent lanthanide ions[END_REF]. We report in this article that such magneto-optical correlations can be extended to a textbook dysprosium(III) nitrate coordination complex by direct excitation in the f-f transitions of the lanthanide ion. Surprisingly, this complex behaves as a field induced SIM allowing to perform a magneto-optical correlation.

Results

Structure

X-Ray Single Crystal diffraction indicates that [Dy(NO 3 ) 3 (H 2 O) 4 ]•2H 2 O (1) crystallizes in the triclinic P-1 space group (Table S1) with a unique crystallographic molecule. The coordination sphere of the Dy 3+ ion is constituted by three bidentate nitrate and four water molecules leading to an overall coordination number of 10 (Figure 1). The Dy-O distances are comprised between 2.357(6) and 2.792(5) Å. The analysis of the geometry using the SHAPE software [START_REF] Casanova | The rich stereochemistry of eight-vertex polyhedra: A continuous shape measures study[END_REF] indicates that it is close to a spherocorona (Table S2). Study of the crystal packing reveals a complex hydrogen bonds network involving oxygen nitrates and both coordinated and uncoordinated water molecules giving rise to a three-dimensional supramolecular network. The shortest Dy-Dy distance in the crystal is found to be 6.3282 [START_REF] Cucinotta | Magnetic anisotropy in a dysprosium/dota single-molecule magnet: Beyond simple magneto-structural correlations[END_REF] Å.

Magnetochemistry 2016, 2, 41 2 of 11

In this line of scope, luminescent SIMs are particularly appropriate for these investigations since the photo-luminescence measurements permit determining spectroscopically the Stark sub-level structure of the lanthanides ions and comparing it with the magnetic data. However, examples of such magneto-optical correlations are rather scarce [START_REF] Cucinotta | Magnetic anisotropy in a dysprosium/dota single-molecule magnet: Beyond simple magneto-structural correlations[END_REF][START_REF] Long | A bifunctional luminescent single-ion magnet: Towards correlation between luminescence studies and magnetic slow relaxation processes[END_REF][START_REF] Yamashita | A luminescent single-molecule magnet: Observation of magnetic anisotropy using emission as a probe[END_REF][START_REF] Pointillart | A redox-active luminescent ytterbium-based single molecule magnet[END_REF][START_REF] Gavey | Placing a crown on dy(III)-A dual property lniii crown ether complex displaying optical properties and smm behaviour[END_REF][START_REF] Shintoyo | Crystal field splitting of the ground state of terbium(iii) and dysprosium(III) complexes with a triimidazolyl tripod ligand and an acetate determined by magnetic analysis and luminescence[END_REF][START_REF] Ren | A layered erbium phosphonate in pseudo-D 5h symmetry exhibiting field-tunable magnetic relaxation and optical correlation[END_REF][START_REF] Long | A high-temperature molecular ferroelectric Zn/Dy complex exhibiting single-ion-magnet behavior and lanthanide luminescence[END_REF][START_REF] Rechkemmer | Comprehensive spectroscopic determination of the crystal field splitting in an erbium single-ion magnet[END_REF][START_REF] Gregson | A monometallic lanthanide bis(methanediide) single molecule magnet with a large energy barrier and complex spin relaxation behaviour[END_REF][START_REF] Bi | Thermostability and photoluminescence of dy(III) single-molecule magnets under a magnetic field[END_REF][START_REF] Hareri | Encapsulation of a [Dy(OH 2 ) 8 ] 3+ cation: Magneto-optical and theoretical studies of a caged, emissive smm[END_REF] in comparison with the large number of reported lanthanides-based SIMs. This fact is often due to the weak or completely quenched emission in these complexes. In order to overcome this problem, antenna ligands are used to enhance the lanthanide emission. Thus, in all cases of magneto-luminescent SIM, the observed lanthanide luminescence relies on the use of coordinated ligands or complexes as sensitizers to enhance the lanthanide luminescence via an energy transfer mechanism [START_REF] Bunzli | Taking advantage of luminescent lanthanide ions[END_REF]. We report in this article that such magneto-optical correlations can be extended to a textbook dysprosium(III) nitrate coordination complex by direct excitation in the f-f transitions of the lanthanide ion. Surprisingly, this complex behaves as a field induced SIM allowing to perform a magneto-optical correlation.

Results

Structure

X-Ray Single Crystal diffraction indicates that [Dy(NO3)3(H2O)4]•2H2O (1) crystallizes in the triclinic P-1 space group (Table S1) with a unique crystallographic molecule. The coordination sphere of the Dy 3+ ion is constituted by three bidentate nitrate and four water molecules leading to an overall coordination number of 10 (Figure 1). The Dy-O distances are comprised between 2.357(6) and 2.792(5) Å. The analysis of the geometry using the SHAPE software [START_REF] Casanova | The rich stereochemistry of eight-vertex polyhedra: A continuous shape measures study[END_REF] indicates that it is close to a spherocorona (Table S2). Study of the crystal packing reveals a complex hydrogen bonds network involving oxygen nitrates and both coordinated and uncoordinated water molecules giving rise to a three-dimensional supramolecular network. The shortest Dy-Dy distance in the crystal is found to be 6.3282(7) Å.

Magnetic Properties

The magnetic properties of 1 were investigated using a Superconducting QUantum Interference Device (SQUID) magnetometer (Quantum Design, San Diego, CA, USA) working in the temperature range 1.8-350 K up to 7 T in direct current (DC) and alternating currents (AC) modes.

Magnetic Properties

The magnetic properties of 1 were investigated using a Superconducting QUantum Interference Device (SQUID) magnetometer (Quantum Design, San Diego, CA, USA) working in the temperature range 1.8-350 K up to 7 T in direct current (DC) and alternating currents (AC) modes.

DC Magnetic Properties

The room temperature value of χT is equal to 14.04 cm 3 •K•mol -1 , which is in a good agreement with the value of 14.17 cm 3 •K•mol -1 expected for a Dy 3+ ion using the free-ion approximation. Upon cooling, χT remains constant down to 50 K before decreasing to reach the value of 11.87 cm 3 •K•mol -1 at 1.8 K (Figure 2). Such a phenomenon can be ascribed to the thermal depopulation of the Stark sublevels. The field dependence of the magnetization measured at 1.8 K reaches the value of 5.85 µ B under 70 kOe without evidence of a clear saturation (inset of Figure 2).

DC Magnetic Properties

The room temperature value of χT is equal to 14.04 cm 3 •K•mol -1 , which is in a good agreement with the value of 14.17 cm 3 •K•mol -1 expected for a Dy 3+ ion using the free-ion approximation. Upon cooling, χT remains constant down to 50 K before decreasing to reach the value of 11.87 cm 3 •K•mol -1 at 1.8 K (Figure 2). Such a phenomenon can be ascribed to the thermal depopulation of the Stark sublevels. The field dependence of the magnetization measured at 1.8 K reaches the value of 5.85 μB under 70 kOe without evidence of a clear saturation (inset of Figure 2).

AC Magnetic Properties

AC magnetic properties were performed in order to investigate the occurrence of a slow relaxation of the magnetization. Under a zero DC field, no significant out-of-phase susceptibility, χ′′, can be observed. This directly reflects a strong QTM which may arise from external factors such as dipolar interactions, intrinsic effects due to hyperfine interactions or symmetry deviations [START_REF] Pointillart | Magnetic memory in an isotopically enriched and magnetically isolated mononuclear dysprosium complex[END_REF]. Applying DC fields induce a short-cut of this QTM and allow the observation of an out-of-phase component. The highest relaxation time, τ, is found at 4.0 K around 1500-2000 Oe DC fields (Figure 3). For higher field values, a decrease of τ is observed and can be imputed to the direct process that becomes predominant. Thermally activated and Raman processes are not field dependent but cannot be neglected even at 4.0 K, and the field dependence of the relaxation time can be fitted by using the following equation [START_REF] Zadrozny | Slow magnetization dynamics in a series of two-coordinate iron(II) complexes[END_REF]:

τ -1 = DH 4 T + B1/(1 + B2H 2 ) + τ0 -1 exp(-∆/kT) + CT m . ( 1
)
The first term accounts for the direct process (for Kramers-ion), the second one stands for the QTM while the third and fourth are relative to thermally activated and Raman processes, respectively. The best fit leads to the following values: D = 1.37 × 10 -12 s -1 •K -1 •Oe -4 ; B1 = 293 s -1 , B2 = 2.6 × 10 -6 Oe -2 ; C = 0.0005 s -1 •K -9 ; ∆ = 39 cm -1 ; τ0 = 8 × 10 -10 s. The magnitudes of the B1 and B2 parameters directly reflect the degree of mixing between the ±mJ levels and consequently are a sign of the magnitude of the QTM process.

AC Magnetic Properties

AC magnetic properties were performed in order to investigate the occurrence of a slow relaxation of the magnetization. Under a zero DC field, no significant out-of-phase susceptibility, χ , can be observed. This directly reflects a strong QTM which may arise from external factors such as dipolar interactions, intrinsic effects due to hyperfine interactions or symmetry deviations [START_REF] Pointillart | Magnetic memory in an isotopically enriched and magnetically isolated mononuclear dysprosium complex[END_REF]. Applying DC fields induce a short-cut of this QTM and allow the observation of an out-of-phase component. The highest relaxation time, τ, is found at 4.0 K around 1500-2000 Oe DC fields (Figure 3). For higher field values, a decrease of τ is observed and can be imputed to the direct process that becomes predominant. Thermally activated and Raman processes are not field dependent but cannot be neglected even at 4.0 K, and the field dependence of the relaxation time can be fitted by using the following equation [START_REF] Zadrozny | Slow magnetization dynamics in a series of two-coordinate iron(II) complexes[END_REF]:

τ -1 = DH 4 T + B 1 /(1 + B 2 H 2 ) + τ 0 -1 exp(-∆/kT) + CT m . ( 1
)
The first term accounts for the direct process (for Kramers-ion), the second one stands for the QTM while the third and fourth are relative to thermally activated and Raman processes, respectively. The best fit leads to the following values: D = 1.37 × 10 -12 s -1 •K -1 •Oe -4 ; B 1 = 293 s -1 , B 2 = 2.6 × 10 -6 Oe -2 ; C = 0.0005 s -1 •K -9 ; ∆ = 39 cm -1 ; τ 0 = 8 × 10 -10 s. The magnitudes of the B 1 and B 2 parameters directly reflect the degree of mixing between the ±m J levels and consequently are a sign of the magnitude of the QTM process. The frequency dependence of χʺ for various temperatures under a 2000 Oe DC field reveals a frequency dependent asymmetric peak (Figure 4a). The maximum of the peak shifts to higher temperatures upon increasing frequency, which indicates a field-induced slow relaxation of the magnetization. At low temperature, a plateau at low frequencies can be observed demonstrating the occurrence of a second relaxation process. We note that this process is essentially frequency independent, suggesting that it results from a collective behavior mediated by dipolar interactions and/or hydrogen bond network. As regards the frequency dependent process, it can be noticed that the magnitude of the out-of-phase signals increases with frequency, which is usually ascribed to the presence of intermolecular interactions or spin-glass behavior [START_REF] Bilyachenko | Heterometallic Na 6 Co 3 phenylsilsesquioxane exhibiting slow dynamic behavior in its magnetization[END_REF]. The same feature can also be observed on the temperature dependence of the out-of-phase susceptibility with various frequencies (Figure 4b). The frequency dependence of χ for various temperatures under a 2000 Oe DC field reveals a frequency dependent asymmetric peak (Figure 4a). The maximum of the peak shifts to higher temperatures upon increasing frequency, which indicates a field-induced slow relaxation of the magnetization. At low temperature, a plateau at low frequencies can be observed demonstrating the occurrence of a second relaxation process. We note that this process is essentially frequency independent, suggesting that it results from a collective behavior mediated by dipolar interactions and/or hydrogen bond network. As regards the frequency dependent process, it can be noticed that the magnitude of the out-of-phase signals increases with frequency, which is usually ascribed to the presence of intermolecular interactions or spin-glass behavior [START_REF] Bilyachenko | Heterometallic Na 6 Co 3 phenylsilsesquioxane exhibiting slow dynamic behavior in its magnetization[END_REF]. The same feature can also be observed on the temperature dependence of the out-of-phase susceptibility with various frequencies (Figure 4b). The frequency dependence of χʺ for various temperatures under a 2000 Oe DC field reveals a frequency dependent asymmetric peak (Figure 4a). The maximum of the peak shifts to higher temperatures upon increasing frequency, which indicates a field-induced slow relaxation of the magnetization. At low temperature, a plateau at low frequencies can be observed demonstrating the occurrence of a second relaxation process. We note that this process is essentially frequency independent, suggesting that it results from a collective behavior mediated by dipolar interactions and/or hydrogen bond network. As regards the frequency dependent process, it can be noticed that the magnitude of the out-of-phase signals increases with frequency, which is usually ascribed to the presence of intermolecular interactions or spin-glass behavior [START_REF] Bilyachenko | Heterometallic Na 6 Co 3 phenylsilsesquioxane exhibiting slow dynamic behavior in its magnetization[END_REF]. The same feature can also be observed on the temperature dependence of the out-of-phase susceptibility with various frequencies (Figure 4b). The presence of a second relaxation process at a low temperature becomes evident when looking at the Cole-Cole plots (Figure 5a). Although a second semi-circle is not observed, a plateau is present. Therefore, the Cole-Cole plots were fitted by taking into account only the data of the semi-circle corresponding to the high-temperature process (Table S3). The values of the α parameter are ranging between 0.114 and 0.227, indicating a narrow distribution of the relaxation processes. Further insights into the mechanisms of the slow relaxation of the magnetization can be obtained by studying the temperature dependence of τ (Figure 5b). It turns out that a deviation from the thermally activated behavior can be observed at low temperatures and originates from the occurrence of other relaxation processes, such as Raman and/or direct. Fitting the temperature dependence of the relaxation time was performed using the following model: [START_REF] Meihaus | Influence of pyrazolate vs. N-heterocyclic carbene ligands on the slow magnetic relaxation of homoleptic trischelate lanthanide(III) and uranium(III) complexes[END_REF] τ

-1 = τ 0 -1 exp(-∆/kT) + CT m + AT n , (2)
for which the first term accounts for a thermally activated process such as Orbach or direct spin-phonon transitions involving higher excited states [START_REF] Ungur | Strategies toward high-temperature lanthanide-based single-molecule magnets[END_REF], while the second and third ones stand for two-phonon Raman and direct relaxation processes, respectively. To avoid the over-parameterization, the values of m = 9 and n = 1 were fixed to the values found for two-phonon Raman (for Kramers ions) and direct processes [START_REF] Shrivastava | Theory of spin-lattice relaxation[END_REF][START_REF] Scott | Spin-lattice relaxation in some rare-earth salts at helium temperatures; observation of the phonon bottleneck[END_REF].

Magnetochemistry 2016, 2, 41 5 of 11

The presence of a second relaxation process at a low temperature becomes evident when looking at the Cole-Cole plots (Figure 5a). Although a second semi-circle is not observed, a plateau is present. Therefore, the Cole-Cole plots were fitted by taking into account only the data of the semi-circle corresponding to the high-temperature process (Table S3). The values of the α parameter are ranging between 0.114 and 0.227, indicating a narrow distribution of the relaxation processes. Further insights into the mechanisms of the slow relaxation of the magnetization can be obtained by studying the temperature dependence of τ (Figure 5b). It turns out that a deviation from the thermally activated behavior can be observed at low temperatures and originates from the occurrence of other relaxation processes, such as Raman and/or direct. Fitting the temperature dependence of the relaxation time was performed using the following model: [START_REF] Meihaus | Influence of pyrazolate vs. N-heterocyclic carbene ligands on the slow magnetic relaxation of homoleptic trischelate lanthanide(III) and uranium(III) complexes[END_REF] τ

-1 = τ0 -1 exp(-∆/kT) + CT m + AT n , (2)
for which the first term accounts for a thermally activated process such as Orbach or direct spinphonon transitions involving higher excited states [START_REF] Ungur | Strategies toward high-temperature lanthanide-based single-molecule magnets[END_REF], while the second and third ones stand for two-phonon Raman and direct relaxation processes, respectively. To avoid the overparameterization, the values of m = 9 and n = 1 were fixed to the values found for two-phonon Raman (for Kramers ions) and direct processes. [START_REF] Shrivastava | Theory of spin-lattice relaxation[END_REF][START_REF] Scott | Spin-lattice relaxation in some rare-earth salts at helium temperatures; observation of the phonon bottleneck[END_REF].

(a) (b) The best fit parameters give ∆ = 40 ± 4 cm -1 ; τ0 = (1.38 ± 0.09) × 10 -9 s and C = 0.003 ± 0.001 s -1 •K -9 while the value for A is found negligible. Consequently, the deviation from linearity observed in Figure 5b can be explained by the overlap between thermally activated and Raman processes at low temperature. Fitting the temperature dependence of the relaxation time with only a Raman process (m being free) did not provide meaningful results with an m exponent value larger (10.7) than the expected value for the Kramers ion. Consequently, the thermally activated relaxation process seems to dominate in the high-temperature region.

Luminescence

The photoluminescence properties of 1 were investigated at 14 K and at 300 K, Figure 6a,b. Independently of the temperature, the emission spectra reveal the characteristic luminescence of the Dy 3+ ions ascribed to the 4 F9/2→ 6 H15/2-11/2 transitions. By decreasing the temperature from 300 K to 14 K, a decrease in the full-width-at-half-maximum (fwhm) is observed, as well as the Stark splitting of the intra-4f 9 lines. The excitation spectra monitored within the more intense 4 F9/2→ 6 H13/2 transition reveal a series of intra-4f 9 straight lines attributed to transitions between the 4 K15/2,17/2, 4 I9/2-13/2, 4 G9/2-11/2, The best fit parameters give ∆ = 40 ± 4 cm -1 ; τ 0 = (1.38 ± 0.09) × 10 -9 s and C = 0.003 ± 0.001 s -1 •K -9 while the value for A is found negligible. Consequently, the deviation from linearity observed in Figure 5b can be explained by the overlap between thermally activated and Raman processes at low temperature. Fitting the temperature dependence of the relaxation time with only a Raman process (m being free) did not provide meaningful results with an m exponent value larger (10.7) than the expected value for the Kramers ion. Consequently, the thermally activated relaxation process seems to dominate in the high-temperature region.

Luminescence

The photoluminescence properties of 1 were investigated at 14 K and at 300 K, Figure 6a,b. Independently of the temperature, the emission spectra reveal the characteristic luminescence of the Dy 3+ ions ascribed to the 4 F 9/2 → 6 H 15/2-11/2 transitions. By decreasing the temperature from 300 K to 14 K, a decrease in the full-width-at-half-maximum (fwhm) is observed, as well as the Stark splitting of the intra-4f 9 lines. The excitation spectra monitored within the more intense 4 F 9/2 → 6 H 13/2 transition reveal a series of intra-4f 9 straight lines attributed to transitions between the 4 Similarly to what is found for the emission spectra, apart from a decrease of the fwhm, the intra-4f 9 lines, the excitation spectrum observed at 14 K resembles that measured at 300 K. The 4 F 9/2 emission decay curves were measured under UV excitation at 355 nm (Figure S1) for 1. Both decay curves are well-described by a single-exponential function yielding lifetime values of τ = (7.7 ± 0.2) × 10 -9 s (14 K) and τ = (5.9 ± 0.3) × 10 -9 s (300 K). Similarly to what is found for the emission spectra, apart from a decrease of the fwhm, the intra-4f 9 lines, the excitation spectrum observed at 14 K resembles that measured at 300 K. The 4 F9/2 emission decay curves were measured under UV excitation at 355 nm (Figure S1) for 1. Both decay curves are well-described by a single-exponential function yielding lifetime values of τ = (7.7 ± 0.2) × 10 -9 s (14 K) and τ = (5.9 ± 0.3) × 10 -9 s (300 K). In order to gain additional insight into the correlation between SIM behaviour and luminescence properties, the crystal field splitting of the ground state of the Dy(III) ion in 1 was studied. The low temperature (14 K) high-resolution emission spectrum in the spectral region showing the 4 F9/2→ 6 H15/2 transitions was acquired for several crystals (Figure 7 and Figure S2). Apart from changes in the relative intensity of the Stark components, all the spectra reveal the presence of 11 components, as illustrated in Figure 7 for a selected ensemble of crystals.

Attending to the fact that the Dy(III) ions occupy a low-symmetry group, the splitting of the electronic levels ( 4 F9/2 and 6 H15/2) into the maximum number of allowed components to (2J + 1)/2 Kramers doublets is expected, leading to five and eight components for 4 F9/2 and 6 H15/2, respectively. Thus, at least eight Stark components are expected for the 4 F9/2→ 6 H15/2 transitions assuming that only the lower-energy Stark component of the 4 F9/2 excited state is populated and all the transitions end at the Stark components of the ground multiplet 6 H15/2. Since 11 components are clearly expressed, it readily points out that the second Stark component of the 4 F9/2 excited state must also be populated. This is a feasible situation due to the low-energetic difference between the first Stark component and the second one (35 ± 5 cm -1 ) which enables the population of the second Stark component of the 4 F9/2 at 14 K. The high-resolution emission spectrum was fitted with 11 components using Gaussian functions, whose energy was constrained to the peak position analysis based on the spectrum observation taking into account the experimental uncertainty (±5 cm -1 ); the fwhm and the relative intensity were free to vary. From the emission spectra best fit, an energy diagram of the Stark-sublevels is proposed in Figure 7d. From the data acquired for the three distinct ensemble of crystals (Figure S2), an average energy barrier, ∆Orbach = 34 ± 5 cm -1 is estimated, which is in excellent agreement with that obtained from the AC magnetic susceptibility data. In order to gain additional insight into the correlation between SIM behaviour and luminescence properties, the crystal field splitting of the ground state of the Dy(III) ion in 1 was studied. The low temperature (14 K) high-resolution emission spectrum in the spectral region showing the 4 F 9/2 → 6 H 15/2 transitions was acquired for several crystals (Figure 7 and Figure S2). Apart from changes in the relative intensity of the Stark components, all the spectra reveal the presence of 11 components, as illustrated in Figure 7 for a selected ensemble of crystals.

Attending to the fact that the Dy(III) ions occupy a low-symmetry group, the splitting of the electronic levels ( 4 F 9/2 and 6 H 15/2 ) into the maximum number of allowed components to (2J + 1)/2 Kramers doublets is expected, leading to five and eight components for 4 F 9/2 and 6 H 15/2 , respectively. Thus, at least eight Stark components are expected for the 4 F 9/2 → 6 H 15/2 transitions assuming that only the lower-energy Stark component of the 4 F 9/2 excited state is populated and all the transitions end at the Stark components of the ground multiplet 6 H 15/2 . Since 11 components are clearly expressed, it readily points out that the second Stark component of the 4 F 9/2 excited state must also be populated. This is a feasible situation due to the low-energetic difference between the first Stark component and the second one (35 ± 5 cm -1 ) which enables the population of the second Stark component of the 4 F 9/2 at 14 K. The high-resolution emission spectrum was fitted with 11 components using Gaussian functions, whose energy was constrained to the peak position analysis based on the spectrum observation taking into account the experimental uncertainty (±5 cm -1 ); the fwhm and the relative intensity were free to vary. From the emission spectra best fit, an energy diagram of the Stark-sub-levels is proposed in Figure 7d. From the data acquired for the three distinct ensemble of crystals (Figure S2), an average energy barrier, ∆ Orbach = 34 ± 5 cm -1 is estimated, which is in excellent agreement with that obtained from the AC magnetic susceptibility data.

Discussion

Compound 1 behaves as a field induced SIM showing two relaxation processes. Multiple relaxation processes have been widely observed in lanthanide zero-field or field induced SIMs [START_REF] Liddle | Improving f-element single molecule magnets[END_REF][START_REF] Blagg | Magnetic relaxation pathways in lanthanide single-molecule magnets[END_REF][START_REF] Amjad | Multiple magnetization reversal channels observed in a 3d-4f single molecule magnet[END_REF]. For 1, the independent frequency process observed in the low frequency range most likely arises from a collective behavior resulting from strong dipolar interactions or hydrogen bond network involving water molecules. Such intermolecular interactions may also affect the frequency dependent process associated with a slow relaxation of the magnetization. Indeed, a clear increase of the out-of-phase signal are observed with increasing frequencies. This fact is frequently viewed as the signature for either spin-glass or superparamagnetic systems with strong intermolecular interactions [START_REF] Mydosh | Spin Glasses: An Experimental Introduction[END_REF]. Dilution into a diamagnetic matrix would have been the ultimate proof to confirm the origin of the frequency independent relaxation process. However, attempts to grow crystals of a solid solution of Y 3+ /Dy 3+ ions were unsuccessful.

The field-induced SIM character of 1 appears at first glance rather surprising regarding the coordination environment of the dysprosium site. Based on the electrostatic proposed by Rinehardt and Long [START_REF] Rinehart | Exploiting single-ion anisotropy in the design of f-element single-molecule magnets[END_REF] and later extended by Chilton et al. [START_REF] Chilton | An electrostatic model for the determination of magnetic anisotropy in dysprosium complexes[END_REF], the stabilization of the oblate electronic density of the Dy 3+ ion requires an axial crystal field. Although this model only relies on the charge of the atoms belonging to the ligands (it does not take into account the donor strength), it allows for estimating the orientation of the anisotropic axis by using the MAGELLAN software [START_REF] Chilton | An electrostatic model for the determination of magnetic anisotropy in dysprosium complexes[END_REF]. The orientation of the latter is highly dependent on the magnitude of the interaction between the negatively charged atom of the ligand and the Dy 3+ ion. It turns out that the anisotropic axis (Figure 8) is found nearly collinear (deviation of 23.62° with respect to the central nitrogen atom) to the nitrate anion presenting the shortest Dy-O distance (Dy-O8 = 2.463 Å). The ideal stabilization of oblate density would have been to place two nitrate in trans fashion. However, the three nitrate are located in cis fashion to each others, forming a pseudo-triangular arrangement. We note also that the second shortest Dy-O4 = 2.491 Å distance forms an angle of 32.48° with the anisotropic axis. The deviation of the anisotropic axis with respect to the O4-Dy-O8 bond most likely relies on the effect of the positively charged nitrogen that attracts the electron density of the lanthanide ions. Significant longer Dy-O bonds (Dy-O1 2.529 Å and Dy-O5 = 2.568 Å) are found almost perpendicular to the anisotropic axis in order to limit the repulsion with the radial plane of the oblate electron density from the Dy 3+ ion.

Discussion

Compound 1 behaves as a field induced SIM showing two relaxation processes. Multiple relaxation processes have been widely observed in lanthanide zero-field or field induced SIMs [START_REF] Liddle | Improving f-element single molecule magnets[END_REF][START_REF] Blagg | Magnetic relaxation pathways in lanthanide single-molecule magnets[END_REF][START_REF] Amjad | Multiple magnetization reversal channels observed in a 3d-4f single molecule magnet[END_REF]. For 1, the independent frequency process observed in the low frequency range most likely arises from a collective behavior resulting from strong dipolar interactions or hydrogen bond network involving water molecules. Such intermolecular interactions may also affect the frequency dependent process associated with a slow relaxation of the magnetization. Indeed, a clear increase of the out-of-phase signal are observed with increasing frequencies. This fact is frequently viewed as the signature for either spin-glass or superparamagnetic systems with strong intermolecular interactions [START_REF] Mydosh | Spin Glasses: An Experimental Introduction[END_REF]. Dilution into a diamagnetic matrix would have been the ultimate proof to confirm the origin of the frequency independent relaxation process. However, attempts to grow crystals of a solid solution of Y 3+ /Dy 3+ ions were unsuccessful.

The field-induced SIM character of 1 appears at first glance rather surprising regarding the coordination environment of the dysprosium site. Based on the electrostatic proposed by Rinehardt and Long [START_REF] Rinehart | Exploiting single-ion anisotropy in the design of f-element single-molecule magnets[END_REF] and later extended by Chilton et al. [START_REF] Chilton | An electrostatic model for the determination of magnetic anisotropy in dysprosium complexes[END_REF], the stabilization of the oblate electronic density of the Dy 3+ ion requires an axial crystal field. Although this model only relies on the charge of the atoms belonging to the ligands (it does not take into account the donor strength), it allows for estimating the orientation of the anisotropic axis by using the MAGELLAN software [START_REF] Chilton | An electrostatic model for the determination of magnetic anisotropy in dysprosium complexes[END_REF]. The orientation of the latter is highly dependent on the magnitude of the interaction between the negatively charged atom of the ligand and the Dy 3+ ion. It turns out that the anisotropic axis (Figure 8) is found nearly collinear (deviation of 23.62 • with respect to the central nitrogen atom) to the nitrate anion presenting the shortest Dy-O distance (Dy-O8 = 2.463 Å). The ideal stabilization of oblate density would have been to place two nitrate in trans fashion. However, the three nitrate are located in cis fashion to each others, forming a pseudo-triangular arrangement. We note also that the second shortest Dy-O4 = 2.491 Å distance forms an angle of 32.48 • with the anisotropic axis. The deviation of the anisotropic axis with respect to the O4-Dy-O8 bond most likely relies on the effect of the positively charged nitrogen that attracts the electron density of the lanthanide ions. Significant longer Dy-O bonds (Dy-O1 2.529 Å and Dy-O5 = 2.568 Å) are found almost perpendicular to the anisotropic axis in order to limit the repulsion with the radial plane of the oblate electron density from the Dy 3+ ion. Photoluminescence studies show that the characteristic emission of Dy 3+ ion in 1 can be obtained without any sensitizer ligands by direct excitation into the f-f transitions. To our knowledge, this constitutes the first example of a luminescent SIM showing such an excitation process. The magnetooptical correlation shows that ∆ obtained by magnetism (i.e., 40 ± 4 cm -1 ) is very close to the energy gap between the ground and first-excited Kramers doublet extracted from photoluminescence measurements (i.e., 34 ± 5 cm -1 ), indicating that the dominant relaxation pathways in the high temperature region is of the Orbach type. The observed slight deviation most likely arises from the overlap between Orbach and Raman processes due to the limited frequency of the oscillating field by standard SQUID magnetometry (i.e., 1488 Hz), even in the highest temperature range. The energy gap between the ground and first-excited Kramers doublet obtained from photoluminescence corresponds definitely to the correct value. Consequently, fitting of the temperature dependence of the relaxation time with Equation (2) can be performed by fixing ∆Orbach to the one obtained by luminescence (~34 cm -1 ). The best fit parameters obtained are τ0 = (5.7 ± 0.4) × 10 -9 s and C = 0.0009 ± 0.0003 s -1 •K -9 , while the value for A is still negligible. Attempts to let the m parameter free does not improve significantly the quality of the fit, and the m parameter is found slightly higher than its theoretical value of nine (fit m = 10.7) [START_REF] Orbach | Spin-lattice relaxation in multilevel spin systems[END_REF]. From these results, the C parameter is found lowered with respect to the fit without fixing ∆Orbach. This confirms that the main source of relaxation is clearly an Orbach process (red dashed line, Figure 5b). This constitutes a rare example of a luminescent SIM without an underestimation of ∆ obtained by magnetism as frequently observed in luminescent SIMs [START_REF] Cucinotta | Magnetic anisotropy in a dysprosium/dota single-molecule magnet: Beyond simple magneto-structural correlations[END_REF][START_REF] Long | A bifunctional luminescent single-ion magnet: Towards correlation between luminescence studies and magnetic slow relaxation processes[END_REF][START_REF] Yamashita | A luminescent single-molecule magnet: Observation of magnetic anisotropy using emission as a probe[END_REF][START_REF] Pointillart | A redox-active luminescent ytterbium-based single molecule magnet[END_REF][START_REF] Gavey | Placing a crown on dy(III)-A dual property lniii crown ether complex displaying optical properties and smm behaviour[END_REF][START_REF] Shintoyo | Crystal field splitting of the ground state of terbium(iii) and dysprosium(III) complexes with a triimidazolyl tripod ligand and an acetate determined by magnetic analysis and luminescence[END_REF][START_REF] Ren | A layered erbium phosphonate in pseudo-D 5h symmetry exhibiting field-tunable magnetic relaxation and optical correlation[END_REF][START_REF] Long | A high-temperature molecular ferroelectric Zn/Dy complex exhibiting single-ion-magnet behavior and lanthanide luminescence[END_REF][START_REF] Rechkemmer | Comprehensive spectroscopic determination of the crystal field splitting in an erbium single-ion magnet[END_REF][START_REF] Gregson | A monometallic lanthanide bis(methanediide) single molecule magnet with a large energy barrier and complex spin relaxation behaviour[END_REF][START_REF] Bi | Thermostability and photoluminescence of dy(III) single-molecule magnets under a magnetic field[END_REF][START_REF] Hareri | Encapsulation of a [Dy(OH 2 ) 8 ] 3+ cation: Magneto-optical and theoretical studies of a caged, emissive smm[END_REF].

Materials and Methods

Synthesis and Crystal Structure

The compound was purchased from Alfa Aesar and used as received. Single crystals of 1 were selected and measured on an Xcalibur, Onyx diffractometer (Oxford Diffraction, Oxford, UK). The crystal was kept at 300 K during data collection. Using Olex2 [START_REF] Dolomanov | 2: A complete structure solution, refinement and analysis program[END_REF], the structure was solved with the Superflip [START_REF] Palatinus | A computer program for topological analysis of discrete electron densities[END_REF] structure solution program using Charge Flipping and refined with the olex2.refine [START_REF] Bourhis | The anatomy of a comprehensive constrained, restrained refinement program for the modern computing environment-Olex2 dissected[END_REF] refinement package using Gauss-Newton minimisation.

Magnetic Measurements

Magnetic susceptibility data were collected with a Quantum Design (San Diego, CA, USA) MPMS-XL SQUID magnetometer working between 1.8 and 350 K with the magnetic field up to 7 Tesla. The data were corrected for the sample holder and the diamagnetic contributions calculated Photoluminescence studies show that the characteristic emission of Dy 3+ ion in 1 can be obtained without any sensitizer ligands by direct excitation into the f-f transitions. To our knowledge, this constitutes the first example of a luminescent SIM showing such an excitation process. The magneto-optical correlation shows that ∆ obtained by magnetism (i.e., 40 ± 4 cm -1 ) is very close to the energy gap between the ground and first-excited Kramers doublet extracted from photoluminescence measurements (i.e., 34 ± 5 cm -1 ), indicating that the dominant relaxation pathways in the high temperature region is of the Orbach type. The observed slight deviation most likely arises from the overlap between Orbach and Raman processes due to the limited frequency of the oscillating field by standard SQUID magnetometry (i.e., 1488 Hz), even in the highest temperature range. The energy gap between the ground and first-excited Kramers doublet obtained from photoluminescence corresponds definitely to the correct value. Consequently, fitting of the temperature dependence of the relaxation time with Equation (2) can be performed by fixing ∆ Orbach to the one obtained by luminescence (~34 cm -1 ). The best fit parameters obtained are τ 0 = (5.7 ± 0.4) × 10 -9 s and C = 0.0009 ± 0.0003 s -1 •K -9 , while the value for A is still negligible. Attempts to let the m parameter free does not improve significantly the quality of the fit, and the m parameter is found slightly higher than its theoretical value of nine (fit m = 10.7) [START_REF] Orbach | Spin-lattice relaxation in multilevel spin systems[END_REF]. From these results, the C parameter is found lowered with respect to the fit without fixing ∆ Orbach . This confirms that the main source of relaxation is clearly an Orbach process (red dashed line, Figure 5b). This constitutes a rare example of a luminescent SIM without an underestimation of ∆ obtained by magnetism as frequently observed in luminescent SIMs [START_REF] Cucinotta | Magnetic anisotropy in a dysprosium/dota single-molecule magnet: Beyond simple magneto-structural correlations[END_REF][START_REF] Long | A bifunctional luminescent single-ion magnet: Towards correlation between luminescence studies and magnetic slow relaxation processes[END_REF][START_REF] Yamashita | A luminescent single-molecule magnet: Observation of magnetic anisotropy using emission as a probe[END_REF][START_REF] Pointillart | A redox-active luminescent ytterbium-based single molecule magnet[END_REF][START_REF] Gavey | Placing a crown on dy(III)-A dual property lniii crown ether complex displaying optical properties and smm behaviour[END_REF][START_REF] Shintoyo | Crystal field splitting of the ground state of terbium(iii) and dysprosium(III) complexes with a triimidazolyl tripod ligand and an acetate determined by magnetic analysis and luminescence[END_REF][START_REF] Ren | A layered erbium phosphonate in pseudo-D 5h symmetry exhibiting field-tunable magnetic relaxation and optical correlation[END_REF][START_REF] Long | A high-temperature molecular ferroelectric Zn/Dy complex exhibiting single-ion-magnet behavior and lanthanide luminescence[END_REF][START_REF] Rechkemmer | Comprehensive spectroscopic determination of the crystal field splitting in an erbium single-ion magnet[END_REF][START_REF] Gregson | A monometallic lanthanide bis(methanediide) single molecule magnet with a large energy barrier and complex spin relaxation behaviour[END_REF][START_REF] Bi | Thermostability and photoluminescence of dy(III) single-molecule magnets under a magnetic field[END_REF][START_REF] Hareri | Encapsulation of a [Dy(OH 2 ) 8 ] 3+ cation: Magneto-optical and theoretical studies of a caged, emissive smm[END_REF].

Materials and Methods

Synthesis and Crystal Structure

The compound was purchased from Alfa Aesar and used as received. Single crystals of 1 were selected and measured on an Xcalibur, Onyx diffractometer (Oxford Diffraction, Oxford, UK). The crystal was kept at 300 K during data collection. Using Olex2 [START_REF] Dolomanov | 2: A complete structure solution, refinement and analysis program[END_REF], the structure was solved with the Superflip [START_REF] Palatinus | A computer program for topological analysis of discrete electron densities[END_REF] structure solution program using Charge Flipping and refined with the olex2.refine [START_REF] Bourhis | The anatomy of a comprehensive constrained, restrained refinement program for the modern computing environment-Olex2 dissected[END_REF] refinement package using Gauss-Newton minimisation.

Magnetic Measurements

Magnetic susceptibility data were collected with a Quantum Design (San Diego, CA, USA) MPMS-XL SQUID magnetometer working between 1.8 and 350 K with the magnetic field up to 7 Tesla. The data were corrected for the sample holder and the diamagnetic contributions calculated from Pascal's constants. The AC magnetic susceptibility measurements were carried out in the presence of a 3 Oe oscillating field in zero or applied external DC field.

Photoluminescence Measurements

The photoluminescence spectra were recorded at 14 K and at 300 K with a modular double grating excitation spectrofluorimeter with a TRIAX 320 emission monochromator (Fluorolog-3, Horiba Scientific (Kyoto, Japan)) coupled to a R928 photomultiplier (Hamamatsu) using a front face acquisition mode. The excitation source was a 450 W Xe arc lamp. All the emission spectra were measured under vaccum conditions (10 -7 bar) and corrected for detection and optical spectral response of the spectrofluorimeter, and the excitation spectra were corrected for the spectral distribution of the lamp intensity using a photodiode reference detector. The nominal spectral dispersion response of the spectrofluorimeter is 2.64 nm•cm -1 , yielding a spectral resolution of 0.1 nm (±5 cm -1 ) for the high-resolution emission spectra (slits width of 5 × 10 -2 mm). The emission decay curves were measured at 14 K and at 300 K using a pulsed LED peaking at 355 nm (SpectraLED-355, Horiba Scientific (Kyoto, Japan)) coupled to a TBX-04 photomultiplier tube module. The emission spectrum of the SpectraLED-355 displays a Gaussian profile with a fwmh of ca. 15 nm. The room temperature emission quantum yield was measured using the C9920-02 (Hamamatsu) setup with a 150 W Xe lamp coupled to a monochromator for wavelength discrimination, an integration sphere as sample chamber and a multichannel analyzer for signal detection. The accuracy is within 10% according to the manufacturer. Three measurements were made for each sample, revealing values below the detection limits of our experimental apparatus (0.01).

Conclusions

Although known for decades and thoroughly used for the synthesis of SIMs and SMMs, we have demonstrated that the dysprosium(III) nitrate salt, [Dy(NO 3 ) 3 (H 2 O) 4 ]•2H 2 O (1), behaves as a field-induced SIM. This can be rationalized by the presence of the negatively charged oxygen of the nitrate ions, which helps to stabilize the oblate electronic density of the Dy 3+ ion. In addition, the excitation in the intra 4f transitions induces the characteristic luminescence of the Dy 3+ ion and allows to perform a magneto-structural correlation, giving additional insights into the relaxation processes. In the high-temperature region, it proceeds via an Orbach mechanism, which overlaps with the Raman process upon lowering of the temperature.

Figure 1 .

 1 Figure 1. (a) molecular structure of 1. Color code: orange, Dy; red, O; blue, N; light grey, C. Hydrogen atoms are omitted for clarity; and (b) view of the packing arrangement along the a crystallographic axis.

Figure 1 .

 1 Figure 1. (a) molecular structure of 1. Color code: orange, Dy; red, O; blue, N; light grey, C. Hydrogen atoms are omitted for clarity; and (b) view of the packing arrangement along the a crystallographic axis.

Figure 2 .

 2 Figure 2. Temperature dependence of χT measured under a 1000 Oe DC field. Inset: field dependence of the magnetization measured at 1.8 K.

Figure 2 .

 2 Figure 2. Temperature dependence of χT measured under a 1000 Oe DC field. Inset: field dependence of the magnetization measured at 1.8 K.

Figure 3 .

 3 Figure 3. Field dependence of the relaxation time measured at 4.0 K. The red solid line represents the fit using Equation (1).

Figure 4 . 1 .Figure 3 .

 413 Figure 4. (a) frequency dependence of the in-phase (χ′) and out-of-phase (χ′′) components of the AC susceptibility under an optimal magnetic DC field of 2000 Oe for 1; and (b) temperature dependence of the in-phase (χ′) and out-of-phase (χ′′) components of the AC susceptibility under a 2000 Oe DC field for 1. Solid lines are guides to the eye.

Magnetochemistry 2016, 2 , 41 4 of 11 Figure 3 .

 2113 Figure 3. Field dependence of the relaxation time measured at 4.0 K. The red solid line represents the fit using Equation (1).

Figure 4 .Figure 4 .

 44 Figure 4. (a) frequency dependence of the in-phase (χ′) and out-of-phase (χ′′) components of the AC susceptibility under an optimal magnetic DC field of 2000 Oe for 1; and (b) temperature dependence of the in-phase (χ′) and out-of-phase (χ′′) components of the AC susceptibility under a 2000 Oe DC field for 1. Solid lines are guides to the eye.

Figure 5 .

 5 Figure 5. (a) Cole-Cole plots using the AC data performed under a 2000 Oe DC field. The red solid lines correspond to the fit with a generalized Debye model; (b) temperature dependence of the relaxation time using the AC data under a 2000 Oe DC field. The blue solid line corresponds to the fit with Equation (2), the purple solid line represents the fit with Equation (2) by fixing the ∆Orbach parameter from photoluminescence, while the red dashed line represents the Orbach process and (∆Orbach fixed from photoluminescence and τ0 obtained from Equation (2)).

1 Figure 5 .

 15 Figure 5. (a) Cole-Cole plots using the AC data performed under a 2000 Oe DC field. The red solid lines correspond to the fit with a generalized Debye model; (b) temperature dependence of the relaxation time using the AC data under a 2000 Oe DC field. The blue solid line corresponds to the fit with Equation (2), the purple solid line represents the fit with Equation (2) by fixing the ∆ Orbach parameter from photoluminescence, while the red dashed line represents the Orbach process and (∆ Orbach fixed from photoluminescence and τ 0 obtained from Equation (2)).

Figure 6 .

 6 Figure 6. (a) emission; and (b) excitation spectra (14 K and 300 K) excited at 385 nm and monitored at 573 nm, respectively. In (b), the 6 H 11/2 ground state is omitted and only the excited state is represented for clarity.

Figure 7 .

 7 Figure 7. (a,b) magnification of the 4 F9/2→ 6 H15/2 transition at 14 K and excited at 385 nm. Multi-Gaussian functions envelope fit (circles) and the components arising from the (orange shadow) first and (purple shadow) second 4 F9/2 Stark sublevels to the 6 H15/2 multiplet; (c) fit regular residual plot; and (d) schematic diagram of the radiative transitions between the Stark sublevels of the 4 F9/2 and 6 H15/2 multiplets of the Dy(III) ion.

Figure 7 .

 7 Figure 7. (a,b) magnification of the 4 F 9/2 → 6 H 15/2 transition at 14 K and excited at 385 nm. Multi-Gaussian functions envelope fit (circles) and the components arising from the (orange shadow) first and (purple shadow) second 4 F 9/2 Stark sublevels to the 6 H 15/2 multiplet; (c) fit regular residual plot; and (d) schematic diagram of the radiative transitions between the Stark sublevels of the 4 F 9/2 and 6 H 15/2 multiplets of the Dy(III) ion.

Figure 8 .

 8 Figure 8. Arrangement of the anisotropic axis (purple) obtained with the MAGELLAN software.

Figure 8 .

 8 Figure 8. Arrangement of the anisotropic axis (purple) obtained with the MAGELLAN software.

  K 15/2,17/2 , 4 I 9/2-13/2 , 4 G 9/2-11/2 , 4 M 15/2-21/2 , 6 P 3/2-7/2 , 4 F 7/2-9/2 , and 4 H 15/2 excited levels and the 6 H 11/2 ground multiplet.

Magnetochemistry 2016, 2, 41; doi:10.3390/magnetochemistry2040041 www.mdpi.com/journal/magnetochemistry

Acknowledgments: The authors thank the University of Montpellier, Centre Nationale de la Recherche Scientifique (CNRS) and Plateforme d'Analyse et de Caractérisation (PAC) Balard Institut Charles Gerhardt Montpellier (ICGM). E.M. acknowledges the financial support from the LABoratoire d'EXcellence (LABEX) CheMISyst Agence Nationale de la Recherche ANR-10-LABX-05-01. This work was developed within the scope of the project CICECO-Aveiro Institute of Materials, POCI-01-0145-FEDER-007679 (FCT Ref. UID /CTM /50011/2013) and I3N (UID/CTM/50025/2013) financed by national funds through the FCT/MEC and co-financed by Fonds Européen de Développement Régional (FEDER) under the PT2020 Partnership. A.M.P.B. thanks FCT for a PhD fellowship (SFRH/BD/104789/2014). The Portugal-France bilateral action, PESSOA Program (Hubert Curien) "Multifunctional magneto-luminescent molecular architectures" is also acknowledged.

Supplementary Materials:

The following are available online at www.mdpi.com/2312-7481/2/4/41/s1, Table S1: Crystal and structure refinement data for compound 1; Table S2: SHAPE analysis for compound 1; Table S3: Author Contributions: All authors contributed equally to this work. Rute A. S. Ferreira, Alexandre M. P. Botas and Luis D. Carlos performed, interpreted and wrote the luminescence part; Dominique Luneau resolved the crystal structure; Ekaterina Mamontova and Jérôme Long performed the magnetic characterization; Jérome Long wrote the draft of the article; Yannick Guari and Joulia Larionova discussed the results and implications and commented on the manuscript at all stages.

Conflicts of Interest:

The authors declare no conflict of interest.


Lionel De Boisde¤re
email: lionel.de.boisde¤re@wanadoo.fr

Dropping Rational Expectations

Keywords: sequential equilibrium, temporary equilibrium, perfect foresight, existence, rational expectations, …nancial markets, asymmetric information, arbitrage

, Du¢ e-Sha¤er's (1985) or De Boisde¤re's (2021), which prove the generic existence of equilibrium when agents make perfect forecasts. From an asymptotic argument, our main theorem is derived from De Boisde¤re's (2007), which characterizes the existence of equilibria on purely …nancial markets by a no-arbitrage condition.

Introduction

When agents'information is incomplete or asymmetric, the issue of how markets may reveal information is essential and, yet, debated. Quoting Ross [START_REF] Starr | The new palgrave …nance: a dictionary of economics[END_REF], "the theory with asymmetric information is not well understood at all. In short, the exact mechanism by which prices incorporate information is still a mystery and an attendant theory of volume is simply missing." A traditional response is given by the REE (rational expectations equilibrium) model by assuming, quoting [START_REF] Radner | Rational expectations equilibrium: generic existence and the information revealed by prices[END_REF], that "agents have a 'model' or 'expectations' of how equilibrium prices are determined". Under this assumption, agents know the relationship between private information signals and equilibrium prices, along a so-called 'forecast function'.

In the context of the particular [START_REF] Radner | Rational expectations equilibrium: generic existence and the information revealed by prices[END_REF] model of asymmetric information, equilibrium prices are typically distinct across agents' joint information signals. Thus, with a forecast function, agents could theoretically infer all information initially detained by other agents from observing such 'separating'prices. However, the forecast function is endogeneous, depending on all agents', typically private, characteristics. The eductive process by which agents might infer that forecast function is, therefore, unclear. Moreover, if agents had the 'structural knowledge'of how equilibrium prices were determined, their using it to correctly infer information at a REE equilibrium would require a reckonning skill, which is seen as unrealistic.

The actual process by which markets incorporate information needs, therefore, be studied. Along [START_REF] Cornet | Elimination of arbitrage states in asymmetric information models[END_REF] and De Boisde¤re (2016, 2021), markets reveal that information, with no price model, through arbitrage.

Under asymmetric information, the rational expectation assumption states that agents have a price model a la [START_REF] Radner | Rational expectations equilibrium: generic existence and the information revealed by prices[END_REF], from which they infer information at a 1 Documents de travail du Centre d'Economie de la Sorbonne -2021.09 revealing REE. With symmetric information, the rational expectation assumption, also called the 'perfect foresight hypothesis', states that agents make their trade and consumption plans with the perfect knowledge of future contingent prices, along [START_REF] Radner | Existence of equilibrium plans, prices and price expectations in a sequence of markets[END_REF]. On same grounds, this assumption faces similar criticisms as Kurz and Wu's (1996): "agents need to know the maps from states at future dates to prices in the future and it is entirely unrealistic to assume that agents can …nd out what this sequence of maps is." [START_REF] Radner | Equilibrium under uncertainty[END_REF] himself acknowledges that it "seems to require of the traders a capacity for imagination and computation far beyond what is realistic". Yet, the classical sequential equilibrium relies on perfect foresight.

The current paper drops both rational expectation assumptions and extends the concept of sequential equilibrium. Agents with no price map learn information from markets, whenever possible, along a rational inference behaviour called the 'noarbitrage principle'(see De Boisde¤re, 2016 and 2021). Agents'anticipations of admissible states and prices are exogenous. After being re…ned from the no-arbitrage principle, such anticipations preclude arbitrage on …nancial markets. Accordingly, the extended concept of equilibrium only requires that anticipations be self-ful…lling ex post, instead of perfect. Namely, a sequential equilibrium is reached when agents anticipate the true spot price on every spot market as a possible, instead of certain, outcome and all other conditions of equilibrium hold unchanged. We introduce and refer to such a sequential equilibrium as a 'correct foresight equilibrium (CFE)'.

At a CFE, agents no longer have a price map to refer to and, yet, face no bankruptcy or unexpected price or event across periods. Therefore, cautious rational agents would typically anticipate a continuum of admissible spot prices, making the model in…nite dimensional. This complexity may explain why standard sequential equilibrium models always assumed rational expectations. To our best knowledge, standard models explore no alternative condition than a price map to insure that agents made correct inferences and self-ful…lling anticipations. The current paper attempts to …ll this gap. It recalls what information markets may reveal to agents with no price map. It states a condition, which insures that their anticipations are self-ful…lling and that equilibria exist, whatever the …nancial and information structure. These outcomes of the paper di¤er from the classical models', namely, the models of rational expectations, bounded rationality and temporary equilibrium.

As a consequence of their having no price map or structural knowledge, we show that agents face an incompressible uncertainty over the set of future spot prices, called the 'minimum uncertainty set'. This set is non-empty and consists of all possible equilibrium spot prices at agents'arbitrary beliefs, which are private. This price uncertainty is akin to Kurz andWu's (1994 and[START_REF] Kurz | Endogenous uncertainty in a general equilibrium model with price contingent contracts[END_REF], when described as "primarily endogenous and internally propagated (...) generated by the actions and beliefs of the agents (...) and by their uncertainty about the actions of other agents".

That minimum uncertainty set, or a bigger set, might be inferred, we argue, by a tradehouse or …nancial institution from observing and treating past data on long time series. Consumers themselves would not have the required computational capacities. But they could build personal beliefs on public institutions'forecasts.

No agent or institution would forecast equilibrium prices with certainty, because this would typically require to know every agent's private characteristics and actions ex ante. Only a set of possible clearing-market prices might be inferred, namely, the minimum uncertainty set, possibly endowed with a probability distribution. The precise location of future prices in that set would remain uncertain (see Section 3).

The correct foresight equilibrium (CFE) is thus de…ned as De Boisde¤re's (2007) equilibrium, except for agents'expectations, which need no longer be unique in every state, but form sets containing the true spot prices. The CFE, we argue, reconciles into one concept the sequential and temporary equilibria, which [START_REF] Grandmont | Temporary general equilibrium theory[END_REF] describes as dichotomic. It is sequential, since anticipations are self-ful…lling. It is temporary, since forecasts are exogenously given and not endogenous to the model.

Along our main Theorem, whether …nancial assets be nominal or real and whether agents shared the same beliefs or disagreed about the future, a CFE exists whenever all consumers'anticipations embed the minimum uncertainty set. The current paper proves this theorem in a model, which drops all forms of rational expectations.

The approach to information transmission and equilibrium, proposed hereafter and in our earlier papers, seems to model agents' actual behaviours. Consumers have limited observational and reckonning capacities, hence, no forecast function.

They are unaware of the primitives of the economy, make exogenous anticipations and face uncertainty over future spot prices, as on actual markets. They may infer a unique coarsest arbitrage-free re…nement of prior expectations from observing trade, along De Boisde¤re (2016). However, once reached, this re…nement can no longer be improved and market forces, driven by prices, lead to equilibrium.

The paper is organized as follows: Section 2 presents the model. Section 3 states the existence Theorem and discusses its speci…c anticipation assumption. Section 4 proves the Theorem. An Appendix proves technical Lemmas.

The model

Throughout the paper, we consider a two-period pure exchange economy, where agents face uncertainty, receive information signals and exchange consumption goods and …nancial assets on markets. The sets, I, J, L and S, respectively, of consumers, securities, goods and states of nature are all given and …nite. The …rst period is referred to as t = 0 and the second, as t = 1. At t = 0, agents are uncertain which state of nature will randomly prevail at t = 1. At t = 1, all uncertainty is removed.

The non-random state, at t = 0, is denoted by s = 0 and we let 0 := f0g [ for every subset, , of S. Perishable goods, l 2 L, may be exchanged on spot markets for consumption purposes at both dates. Financial assets, also called securities, are exchanged at t = 0 and pay o¤ in goods and/or in cash at t = 1. We let l = 0 be the unit of account (payo¤ in cash) and L 0 := f0g [ L be the set of all payo¤s at t = 1.

Markets, information and beliefs

At t = 0, each agent, i 2 I, receives a private information signal, S i S, which informs her correctly that no state s 2 SnS i will prevail at t = 1. Hence, the pooled information set, henceforth denoted by S := \ i2I S i , is non-empty. The information structure, (S i ) := (S i ) i2I , is set as given throughout. Agents are unaware of the primitives of the economy and of other agents'beliefs, information and actions. They fail to know how market prices are determined and, therefore, face uncertainty over future spot prices. At t = 0, the generic i th agent elects a private set of anticipations, in each state s 2 S i , out of a set of admissible prices,

P := fp 2 R L ++ : kpk = 1g.
We refer to := S P as the set of forecasts and denote by ! its generic element, and by B( ) its Borel -algebra. A forecast, ! := (s; p) 2 , is thus a pair of a random state, s 2 S, and a conditional spot price, p 2 P , expected (as possible) in that state.

Remark 1 Strictly positive prices in P are related to strictly increasing preferences, as assumed below. For simplicity, but w.l.o.g., the price set, P , normalizes all agents'price expectations to one. In each state, this common value of one could be replaced by any other positive value without changing the model's properties.

Agents may transfer wealth across periods and states by exchanging, at t = 0, …nitely many assets, j 2 J, which pay o¤ at t = 1, conditionally on the realization of the forecast. In each state, …nancial payo¤s may be nominal (i.e., in cash) or real (in goods) or a mix of both. All assets'payo¤s de…ne a payo¤ function, V : ! R J , which is henceforth given and continuous, from the de…nition. Thus, at asset price, q 2 R J , agents may buy or sell unrestrictively portfolios of assets, z = (z j ) 2 R J , for q z units of account at t = 0, against the promise of delivery of a ‡ow, V (!) z, of conditional payo¤s in cash across forecasts, ! 2 . We now de…ne anticipations.

De…nition 1 An anticipation set is a closed subset of := S P . A collection of anticipation sets, i := [ s2Si fsg P i s , de…ned for each i 2 I, is an anticipation structure if the following conditions hold:

(a) 8(i; s) 2 I S i , P i s 6 = ?; (b) 8s 2 S, \ i2I P i s 6 = ?.
We let AS be the set of anticipation structures. A structure, ( e i ) 2 AS, is called a re…nement of ( i ) 2 AS, and denoted by

( e i ) ( i ), if

The agent' s behaviour and the concept of equilibrium

The generic i th agent (for i 2 I) is granted an endowment in form of a vector,

e i := (e is ) 2 R L S 0 i +
, promising the commodity bundles, e i0 2 R L + at t = 0, and e is 2 R L + , in each state s 2 S i , if it prevails. When they elect their strategies, at t = 0, agents have reached a structure of beliefs, ( i ) 2 SB, de…ning their anticipation structure, (supp( i )) 2 AS. The structures ( i ) 2 SB or (supp( i )) 2 AS are exogenous and assumed (costlessly from De Boisde¤re (2016, 2021)) to be arbitrage-free, that is, to meet the following condition of absence of future arbitrage opportunity (AFAO):

@(z i ) 2 R J I : P i2I z i = 0 and V (! i ) z i > 0; 8(i; ! i ) 2 I supp( i ); with one strict inequality.
The generic i th agent's consumption set, which depends on her belief, i , is denoted by X i and de…ned as the set of continuous maps, x : f0g [ supp( i ) ! R L + , which relate state s = 0 to the …rst period consumption, x 0 2 R L + , and every forecast,

! 2 supp( i ), to the conditional consumption, x ! 2 R L + at t = 1, chosen if ! obtains.
The asset and commodity prices at t = 0 are observed by every agent and denoted by ! 0 := (p 0 ; q). As in Remark 1, we restrict w.l.o.g. their admissible values to the set P 0 := fp 2 R L + : kpk 6 1g fq 2 R J : kqk 6 1g. Given her belief, i , and the observed prices, ! 0 := (p 0 ; q) 2 P 0 , at t = 0, the generic i th agent's budget set is

B i (! 0 ) := f (x; z) 2 X i R J : p 0 (x 0 -e i0 ) 6 q z; p s (x ! -e is ) 6 V (!) z; 8! := (s; p s ) 2 supp( i ) g.
The generic agent, i 2 I, has ordered preferences, represented, ex post, by a utility function, u i : R 2 + ! R + , over her consumptions at both periods, and, ex ante, by the V.N.M. utility function:

x 2 X i 7 ! U i i (x) := R !2 u i (x 0 ; x ! )d i (!).
The above economy, denoted by E ( i ) = f(I; S; L; J); V; (S i ); ( i ); (e i ); (u i )g, retains the small consumer price-taker hypothesis, by which no single agent may, alone, have a signi…cant impact on prices. It is standard if it meets the following Assumptions:

A1 (strong survival): 8i 2 I; e i 2 R L S 0 i ++ ;
A2 for each i 2 I, u i is continuous, strictly concave and strictly increasing, namely: [(x; y; x 0 ; y 0 ) 2 R 4L + ; (x; y) 6 (x 0 ; y 0 ); (x; y) 6 = (x 0 ; y 0 )] ) [u i (x 0 ; y 0 ) > u i (x; y)].

Remark 3 Strict concavity is retained in Assumption A2 to alleviate the proof of a correspondence selection amongst optimal strategies (see proof of Lemma 4).

In the economy E ( i ) , each consumer elects an optimal strategy in her budget set. This leads to the following concept of equilibrium: De…nition 2 Given ( i ) 2 SB, a collection of prices, ! 0 := (p 0 ; q) 2 P 0 and p := (p s ) 2 P := P S , and strategies, (x i ; z i ) 2 B i (! 0 ), de…ned for each i 2 I, is a sequential equilibrium of the economy, E ( i ) , or correct foresight equilibrium (CFE), if:

(a) 8i 2 I; (x i ; z i ) 2 arg max (x;z)2B i (!0) U i i (x)

The existence Theorem

This Section introduces a set of incompressible uncertainty and states the paper's main Theorem. A discussion of its speci…c anticipation assumption follows.

Endogenous uncertainty and the existence of equilibrium

Agents' incompressible uncertainty, sketched above, is characterized by a socalled and never empty 'minimum uncertainty set', de…ned hereafter. Theorem 1, below, shows that equilibrium always exists, in a standard economy, when agents' anticipations embed that set of minimum uncertainty. This full existence result di¤ers from the generic ones of the standard models, based on rational expectations.

De…nition 3 Let be the set of prices, p := (p s ) 2 P := P S , which support a CFE, that is, are equilibrium prices of a standard economy, E (e i ) , for some beliefs, (e i ) 2 SB.

The (possibly empty) set of forecasts, := f! 2 : 9p := (p s ) 2 ; 9s 2 S; ! = (s; p s )g, which support a CFE, is called the minimum uncertainty set.

The set is that of forecasts, which may be shared by agents and clear markets tomorrow, for at least one structure of beliefs today. The following Theorem 1 shows that this set is never empty. Lemma 1 and Assumption A3, stated before, rely on the convention that the empty set is included in all other sets.

Lemma 1 Under Assumptions A1-A2, there exists 2 R ++ such that:

S [ ; 1] L .
Proof See the Appendix.

Assumption A3 (correct foresight): beliefs, ( i ) 2 SB, satisfy

\ i2I supp( i ).
Theorem 1 Let ( i ) 2 SB be given. Under Assumptions A1-A2-A3, the economy,

E ( i )
, admits a CFE. The set is, therefore, non-empty.

Proof See Section 4, below.

Endogenous uncertainty and how to reach correct anticipations

Along Theorem 1, as long as agents have correct foresight, a CFE exists whatever their beliefs. That is, markets always clear ex post at some common forecast. We argue why is a set of 'minimum uncertainty'and how it could be assessed.

On the …rst issue, when beliefs are private, possibly changing, no prospective clearing-market price should be ruled out a priori by cautious rational agents. This outcome would remain if agents with no price map were symmetrically informed and knew the structural characteristics of the economy. Any common clearing-market forecast might obtain tomorrow, in relation to some unknown collection of beliefs today. From a theoretical viewpoint, is, indeed, a set of minimum uncertainty.

The second issue questions the possibility that agents inferred the set , or a bigger set. This problem might be solved empirically, from treating past data.

The above model speci…es normalized spot prices, in the sense of Remark 1. It is often possible to observe past prices and reckon their relative (normalized) values, at many dates and in a wide array of situations, which typlically replicate over time, like the realizations of random states. If time series are long enough, it seems sensible to assume, in each type of situations (or 'state s' ), that the true spot price in state s (up to a scale factor) belongs to the convex hull of its empirical values.

Under this assumption, an embedding of the set may easily be constructed.

To be more speci…c, assume that long series of prices between two dates, t which seems realistic: the longer the time series, the larger the array of events and beliefs is likely to be encountered, up to spanning all equilibrium possibilities.

A statistical method of this kind and its iterative veri…cation over long series require no price map and need not be performed by consumers themselves. They could be implemented by a …nancial or public institution, which have greater computational facilities and could also derive useful applications of the method.

On consumer side, agents with no price model should seek to re…ne their information and take advice from specialists to reduce their uncertainty. Yet, disagreements may persist, with agents receiving typically incomplete or asymmetric (yet correct) pieces of information. As a result, prior anticipations need not be consistent with equilibrium. If not, agents observing markets would update their beliefs from the no-arbitrage principle, up to a unique arbitrage-free re…nement. From De Boisdeffre (2016), this re…nement never eliminates forecasts, which agents share ex ante.

Hence, typical agents would keep idiosyncratic beliefs and share enough uncertainty, represented by a set of common forecasts, to reach equilibrium along Theorem 1.

The existence proof

Hereafter, we set as given beliefs, ( i ) 2 SB, which meet Assumption A3 and the AFAO Condition. We de…ne a standard economy, E ( i ) = f(I; S; L; J); V;

(S i ); ( i ); (e i ); (u i )g,
whose anticipation structure will be denoted by ( i ) := (supp( i )) throughout. The proof proceeds in three steps. Sub-Section 4.1 de…nes, via …nite partitions, a nondecreasing sequence, f( n i )g n2N , of …nite re…nements of ( i ), whose limit is dense in

( i ).

Finite partitions of agents'anticipation sets

For every n 2 N (starting from n = 1), let

K n := N L \ [1; 2 n 1 ] L . The collection of sets (n;s;k) := fsg ( l2L ] k l 1 2 n 1 ; k l 2 n 1 ]
), de…ned for every triple

(n; s; k) 2 N S K n , is a partition of S ]0; 1] L . Let K (i;n) := ft := (s; k) 2 S i K n : t (i;n) := i \ (n;s;k) 6 = ?g, for every (i; n) 2 I N. Then, f t (i;n) g t2K (i;n)
is a partition of i , for every (i; n) 2 I N, and we de…ne the following sets and maps, which meet the properties of Lemma 2: For every triple, (i; n; t) 2 I N K (i;n) , we set as given (exactly) one element,

! t (i;n) 2 t (i;n) := i \ (n;s;k) , and let n i := f! t (i;n) g t2K (i;n) be a …nite subset of i .
We de…ne the maps, n i :

n i ! R ++ and n i : i ! n i , by n i (! t (i;n) ) := i ( t (i;n) )
and

n i (!) := ! t (i;n) , for every (i; n; t; !) 2 I N K (i;n) t (i;n) Lemma 2
The following Assertions hold:

(i) for every (i; n) 2 I N, the partition f t

(i;n+1) g t2K n+1 i re…nes f t (i;n) g t2K n i
; therefore, we may assume that n i n+1 i

;

(ii) for each i 2 I, the set [ n2N n i is dense in i ;

(iii) for every i 2 I and every ! 2 i , n i (!) converges uniformly to !;

(iv) 9N 2 N : 8n > N; ( n i ) & (S [ n i
) are arbitrage-free along the AFAO Condition.

Proof Assertions (i) to (iii) are immediate from the de…nitions.

Assertion (iv) For each i 2 I, let Z i := fz 2 R J : V (!) z = 0; 8! 2 i g and Z ? i be its orthogonal complement. We de…ne

Z o := P i2I Z i and Z := f(z i ) 2 i2I Z ? i : k(z i )k = 1g.
Assume that Assertion (iv) fails. Then, from the de…nitions and above, it holds that:

8N 2 N : 9 n > N; 9(z n i ) 2 Z : P i2I z n i 2 Z o and V (! i ) z n i > 0; 8(i; ! i ) 2 I n i :
Since Z is compact, the sequence f(z n i )g may be assumed to converge to (z i ) 2 Z. From Lemma 2-(i)-(ii) and the continuity of V , the above relations yield (for N ! 1):

12 Documents de travail du Centre d'Economie de la Sorbonne -2021.09

P i2I z i 2 Z o and V (! i ) z i > 0; 8(i; ! i ) 2 I i .
Since ( i ) is arbitrage-free, the above relation, (z i ) 2 i2I Z ? i , and the latter yield

(z i ) 2 i2I Z i \Z ?
i = f0g and contradict the above relation (z i ) 2 Z. This contradiction proves that ( n i ), hence (S [ n i ), are arbitrage-free, if n 2 N is large enough.

The auxiliary economies, E n

Given n 2 N, we de…ne a …nite economy, E n =f(I; S n ; L; J); V n ; (S n i ); (e n i ); (u n i )g, akin to Cornet-De Boisde¤re's (2002), with two periods, and with the same sets of agents, goods and assets as in Section 2. The information structure is de…ned as follows:

S n i := S [ n i
is the generic i th agent's information set. Formally, S is de…ned as the set of realizable states, and n i (for each i 2 I) as the i th agent's idiosyncratic set of unrealizable states. 2 We let S 0n i := S 0 [ n i , for each i 2 I, and

S n := [ i2I S n i .
In each realizable state, s 2 S, the generic i th agent has perfect foresight.

In each formal state, (s; p) 2 n i , the generic i th agent solely expects p 2 P .

In the economy E n (for n 2 N), admissible prices are restricted to the sets P 0 := fp 2 R L + : kpk 6 1g fq 2 R J : kqk 6 1g, at t = 0, and P := P S := fp 2 R L ++ : kpk = 1g S , at t = 1. For all prices, ! 0 := (p 0 ; q) 2 P 0 and p := (p s ) 2 P, the generic i th agent's consumption set, budget set, and V.N.M. utility function are, respectively:

X n i := R L S 0n i +
, whose generic element is denoted by

x := [(x s ) s2S 0 ; (x ! ) !2 n i ]; B n i (! 0 ; p) := f (x; z) 2 X n i R J : p 0 (x 0 e i0 ) 6 q z; p s (x s e is ) 6 V (s; p s ) z; 8s 2 S and p i s (x ! e is ) 6 V (!) z; 8! := (s; p i s ) 2 n i g;
and x 2 X n i 7 ! u n i (x) := 1 n#S X s2S u i (x 0 ; x s ) + (1 1 n ) X !2 n i u i (x 0 ; x ! ) n i (!).
The concept of equilibrium in the economy E n is de…ned as follows:

De…nition 4 A collection of prices, ! 0 := (p 0 ; q) 2 P 0 and p := (p s ) 2 P, and decisions,

(x i ; z i ) 2 B n i (! 0 ; p), for each i 2 I
, is an equilibrium of the economy, E n , if the following Conditions hold:

(a) 8i 2 I; (x i ; z i ) 2 arg max (x;z)2B n i (!0;p) u n i (x); (b) P i2I (x is e is ) = 0; 8s 2 S 0 ; (c) P i2I z i = 0.
From De Boisde¤re (2007), for every n > N along Lemma 2, the economy E n admits an equilibrium, say

C n := f! n 0 := (p n 0 ; q n ); p n ; [(x n i ; z n i )]g, henceforth set as given, such that k! n 0 k 2 [1; 2].
The sequence fC n g := fC n g n>N satis…es the following Lemmas:

Lemma 3
The following Assertions hold:

(i) the price sequences, f! n 0 := (p n 0 ; q n )g and fp n s g, may be assumed to converge, say to ! 0 := (p 0 ; q ) 2 P 0 and p := (p s ) 2 P := P S , s.t. ! s := (s; p s ) 2 \ i2I i , for all s 2 S;

(ii) the sequences f(x n is ) s2S 0 g and fz n i g, for each i 2 I, may be assumed to converge, say to (x is ) s2S 0 2 R L S 0 + and z i 2 R J , such that P i2I (x is -e is ) s2S 0 = 0, and

P i2I z i = 0.
Lemma 4 For all (i; ! := (s; p); z) 2

I i R J , let B i (!; z) = fx 2 R L + : p (x-e is ) 6 V (!) zg.
Using the notations of Lemma 3, the following Assertions hold, for each i 2 I:

(i) \ !2 i B i (!; z i ) 6 = ?;
(ii) the correspondence ! 2 i 7 ! arg max u i (x i0 ; x), for x 2 B i (!; z i ), is a continuous map, whose embedding, x i : f0g [ i 7 ! x i! , is a consumption plan, i.e., x i 2 X i ;

(iii) x i! s = x is , for each s 2 S;

(iv) U i i (x i ) = lim n!1 u n i (x n i ).
14 Documents de travail du Centre d'Economie de la Sorbonne -2021.09

Proofs of the Lemmas See the Appendix.

An equilibrium of the initial economy

We now prove Theorem 1, via the following Claim 1.

Claim 1 The collection of prices, ! 0 2 P 0 and p 2 P, and strategies, (x i ; z i ), for each i 2 I, de…ned from the above Lemmas 3 and 4, is a CFE of the economy E ( i ) .

Proof Let the collection C := f! 0 ; p ; [(x i ; z i )]g be de…ned as in Claim 1. We show that C also meets Condition (a) of De…nition 2.

From De…nition 4, the relations p n 0 (x n i0 e i0 ) 6 q n z n i hold, for each i 2 I and each n > N , and imply in the limit, from Lemma 3: p 0 (x i0 e i0 ) 6 q z i for each i 2 I.

From Lemma 4, the relations x i 2 X i and p s (x i! e is ) 6 V (!) z i hold for each i 2 I

and every ! = (s; p s ) 2 i , and imply, from above:

(x i ; z i ) 2 B i (! 0 ) for each i 2 I.
For every i 2 I, every consumption plan, x 2 X i , and every n > N , we let

x n = [(x n s ) s2S 0 ; (x n ! ) !2 n i ] 2 X n
i be de…ned by x n ! := x ! , for every ! 2 n i , x n 0 := x 0 and x n s := x ! s , for every s 2 S, where ! s := (s; p s ) 2 i . From the above de…nitions and Lemmas and the uniform continuity of u i and x on compact sets, there exists N ( ;x) 2 N, for all > 0 and all bounded-valued consumption plan, x 2 X i , such that:

(I) jU i i (x)-u n i (x n )j < R !2 i ju i (x 0 ; x ! )-u i (x 0 ; x n i (!) )jd i (!) + < 2
holds, for every i 2 I and every n > N ( ;x) . Assume, by contraposition, that C fails to meet Condition (a) of De…nition 2, that is, there exist i 2 I, " 2 R ++ and (X; Z) 2 B i (! 0 ), such that:

For each i 2 I, we let Z i := fz 2 R J : V (!) z = 0; 8! 2 i g, Z ? i be its orthogonal complement, and z 0n i be the orthogonal projection of z n i on Z ? i , for each n > N . We also let

Z o := P i2I Z i and Z := f(z i ) 2 i2I Z ? i : k(z i )k = 1g.
We now show that the sequence f(z 0n i )g is bounded. Assume, by contraposition, that there exists an extracted sub-sequence,

f(z 0'(n) i )g n>N , of non-zero portfolios, such that lim n!1 k(z 0'(n) i )k = 1. Then, the portfolio collection, (z n i ) := (z 0'(n) i ) k(z 0'(n) i
)k , belongs to Z, for every n > N , a compact set. Therefore the sequence, f(z n i )g, may be assumed to converge, say to (z i ) 2 Z. We let := 1 + k(e i )k > 0 be given and observe that the following relations hold, from the de…nition of auxiliary equilibria:

P i2I z n i 2 Z o and V (!) z n i > k(z 0'(n) i )k ; 8(i; !) 2 I n i ; 8n > N , and
P i2I z i 2 Z o and V (!) z i > 0; 8(i; !) 2 I
i , when passing to the limit.

Since ( i ) meets the AFAO Condition, the latter relations imply V (!) z i = 0 for every (i; !) 2 I i , hence, z i 2 Z i \ Z ? i = f0g for every i 2 I, which contradicts the fact that (z i ) 2 Z. This contradiction proves that f(z 0n i )g is bounded. Moreover, from the equilibrium relations

P i2I z 0n i 2 Z o (for n > N )
, the sequence f(z n i )g may be assumed to be bounded (since f(z 0n i )g is), hence, to converge, say to f(z i )g. Then, the equilibrium relations, P i2I z n i = 0 (for n > N ), yield, in the limit:

P i2I z i = 0. Lemma 4 For all (i; ! := (s; p); z) 2 I i R J , let B i (!; z) = fx 2 R L + : p (x-e is ) 6 V (!) zg.
Using the notations of Lemma 3, the following Assertions hold, for each i 2 I:

(i) \ !2 i B i (!; z i ) 6 = ?;
(ii) the correspondence ! 2 i 7 ! arg max u i (x i0 ; x), for x 2 B i (!; z i ), is a continuous map, whose embedding, x i : f0g [ i 7 ! x i! , is a consumption plan, i.e., x i 2 X i ; From Lemmas 2 and 3, the relations (x i0 ; !; z i ) = lim n!1 (x n i0 ; n i (!); z n i ) hold for every ! 2 i . From the continuity of (x 0 ; !; z) 2 R L + R 7 ! arg max x2Bi(!;z) u i (x 0 ; x), the relations, fx n i n i (!) g = arg max x2Bi( n i (!);z n i ) u i (x n i0 ; x), for all n > N and ! 2 i , pass to the limit and yield a continuous map, ! 2 i 7 ! x i! := arg max x2Bi(!;z i ) u i (x i0 ; x), whose embedding, x i : ! 2 0 i := f0g [ i 7 ! x i! , is a consumption plan of E ( i ) , i.e., x i 2 X i . Assertion (iv) Let x i 2 X i be de…ned from above. Resuming the proof of Assertion We recall that the relations (x i0 ; !; z i ) = lim n!1 (x n i0 ; n i (!); z n i ), u i (x i0 ; x i! ) = U i (x i0 ; !; z i ) and u i (x n i0 ; x n i n i (!) ) = U i (x n i0 ; n i (!); z n i ) hold, from Lemmas 2 and 3 and above, for every ! 2 i and every n > N . We also recall that B i (R) is bounded. Then, the following relations follow from above, the uniform continuity of u i and U i on compact sets, Lemma 2 and the de…nitions of the utility functions:

(iii) x i! s = x is , for each s 2 S; (iv) U i i (x i ) = lim n!1 u n i (x n i ). 19
(I) Assertion (iv) results immediately from relations (I), (II) and (III), above.

  ; (b) 8(i; s) 2 I S), (s; p s ) 2 supp( i ); (c) 8s 2 S 0 ; P i2I (x i!s e is ) = 0, where ! s := (s; p s ); (d) P i2I z i = 0.

  Sub-Section 4.2 constructs a sequence of …nite auxiliary economies, which all admit an equilibrium along De Boisde¤re (2007). Sub-Section 4.3 derives a CFE of the economy E ( i ) from that sequence of auxiliary equilibria. 11 Documents de travail du Centre d'Economie de la Sorbonne -2021.09

  From the above Lemmas, C meets Conditions (b)-(c)-(d) of De…nition 2 of equilibrium.

+ R 7 !

 7 (iii) Let s 2 S be given. For every n > N along Lemma 2, let ! n s := (s; p n s ) 2 i . Resuming the proof of Assertion (ii), above, the map,(x 0 ; !; z) 2 R L arg max x2Bi(!;z) u i (x 0 ; x), is continuous. From De…nition 4, the relations x n i n i (! n s ) = arg max x2Bi( n i (! n s );z n i ) u i (x n i0 ; x) and x n is = arg max x2Bi(! n s ;z n i ) u i (x n i0 ;x) hold, for every n > N . Passing to the limit, these relations yield, from Lemma 3 and above:x i! s = lim n!1 x n i n i (! n s ) = arg max x2Bi(! s ;z i ) u i (x i0 ; x) = lim n!1 x n is = x is .

(R 7 !

 7 ii), we let ' i : (x 0 ; !; z) 2 R L + arg max x2Bi(!;z) u i (x 0 ; x) be a continuous map. The continuity of U i : (x 0 ; !; z) 2 R L + R 7 ! u i (x 0 ; ' i (x 0 ; !; z)) follows from that of u i and ' i . 20 Documents de travail du Centre d'Economie de la Sorbonne -2021.09

  We let SB be the set of structures of beliefs.

	Remark 2 The support of a belief, , is de…ned as the set of forecasts, ! 2 ,
	whose neighbourhoods,

it is smaller than ( i ) for the inclusion relation. A structure of beliefs is a collection of probabilities, i , over ( ; B( )), called beliefs and de…ned for each i 2 I, whose supports, supp( i ), de…ne an anticipation structure, (supp( i )) 2 AS. U , always satisfy (U ) > 0. Beliefs have closed supports from the de…nition. Thus, anticipation sets, seen as supports of beliefs, are necessarily closed. The condition of De…nition 1 that they be closed does not restrict the model. 6 Documents de travail du Centre d'Economie de la Sorbonne -2021.09

  For each s 2 S, it observes the number of times, N s , that state s occurs between t 1 and t 2 . For each s 2 S, it deduces the time series, , of the empirical spot price in state s, its convex hull, C s , and the set

	fp k s g 16k6Ns P

[START_REF] Arrow | Le rôle des valeurs boursières pour la répartition la meilleure des risques[END_REF]

2 N and t 2 >> t 1 , are observed by a so-called 'tradehouse', and treated as follows. The tradehouse partitions the events occuring between t 1 and t 2 , into a …nite set, S, of realizable states. Thus, at each date, t 2 ft 1 ; t 1 + 1; :::; t 2 g, the tradehouse knows which state, s 2 S, prevails.

P s := fp 2 P : 9p s 2 C s ; p = ps kpsk g. The above assumption states that [ s2S fsg P s ,

  Proof Let i 2 I be given throughout. , and imply the following one, from Lemmas 2 and 3 and the continuity of V : 0 2 \ !2 i B i (!; z i ). (ii) From De…nition 4, Lemma 2 and Assumption A2, the following relations hold for each n > N and every ! 2 i : fx n Let Z := fz 2 R J : kzk 6 2kz i k + 1g and R := f(!; z) 2 i Z : B i (!; z) 6 = ?g be given non-empty sets. The correspondence B i is closed continuous and B i (R) is bounded from the de…nition of i . From Berge Theorem(Debreu, 1959, p. 19) and Assumption A2, (x 0 ; !; z) 2 R L

	Assertion (i) From De…nition 4, the relations, p i s e is 6 p i s (x n !	e is ) 6 V (!) z n i ,
	hold, for every n > N and every ! := (s; p i s ) 2 n
	Assertion i n i (!) g = arg max x2Bi( n i (!);z n i ) u i (x n i0 ; x).

Documents de travail du Centre d'Economie de la Sorbonne -2021.09 i + R 7 ! arg max x2Bi(!;z) u i (x 0 ; x) is a continuous map.

  8" > 0; 9N " 2 N : 8n > N " ; 8! 2 i , j u i (x i0 ; x i! ) u i (x n R !2 i u i (x i0 ; x i! )d i (!);

						i0 ; x n i n i (!) ) j < ".
	(II)	U i i (x i ) :=
	(III)	u n i (x n ) := 1 n#S	X	u i (x n 0 ; x n s ) + (1 1 n )	X	u i (x n 0 ; x n ! ) n
			s2S		!2 n i

i (!), for each n > N .

Documents de travail du Centre d'Economie de la Sorbonne -2021.09

Documents de travail du Centre d'Economie de la Sorbonne -2021.09

From Lemma 3 and the fact that i is closed, X and x i are bounded-valued.

Then, from the above relations, the following ones hold, for every n > N (";x i ) + N (";X) :

From Assumption A1, the de…nitions of i (a closed set) and of the price sets and from the uniform continuity of u i on a compact set, we may assume that there exists > 0, small enough, such that the above strategy, (X; Z) 2 B i (! 0 ), satis…es:

From Lemmas 2 to 4, and the latter constraints, the relations

. From De…nition 4, the latter relations, joined to relations (I) to (III), yield the following:

for every n > N 0 . Relations (IV ) embed a contradiction. This contradiction proves that C also meets Condition (a) of De…nition 2, hence, from above, is a CFE.

Appendix

Lemma 1 Under Assumptions A1-A2, there exists 2 R ++ such that:

Proof We introduce new notations and let, for every (i; s; x := (x 0 ; x s )) 2 I S R L L + : The proof of Lemma 1 relies on the following Lemmata:

The following Assertions hold:

(i) 8s 2 S, P s is a closed, hence, compact set;

(ii) 9 > 0 : 8s 2 S, P s [ ; 1] L .

Proof of Lemmata 1 Notice from de…nitions that p n s 2 P s holds for all n > N , s 2 S.

Assertion (i) Let s 2 S and a converging sequence fe p k g k2N of elements of P s be given. Its limit, p, is in the closure, P , of P . We may assume there exist (a same) j 2 I and a sequence, fx k g k2N := f(x k i )g k2N , of elements of A s , which meet the condition of the de…nition of P s , and which converges to some x := (x i ) 2 A s , a compact set.

The relations e p k (x k js e js ) > 0 hold from the de…nition, for each k 2 N and yield, in the limit, p (x js e js ) > 0. We show that (p; j; x) meets the conditions of the de…nition of P s (hence, p = lim e p k 2 P s and P s is closed). By contraposition, assume there exists

+ , such that u j (x j0 ; y) > u j (x j0 ; x js ) and p (y x js ) < 0. Then, we show:

If not, one has u j (x k j0 ; y) 6 u j (x k j0 ; x k js ), for k big enough, which implies, in the limit (k ! 1), u j (x j0 ; y) 6 u j (x j0 ; x js ), in contradiction with the above opposite relation.

Hence, relations (I) hold. From the de…nition of the sequences fx k g and fe p k g, relations (I) imply: e p k (y x k js ) > 0, whenever u j (x k j0 ; y) > u j (x k j0 ; x k js ). Hence, in the limit (k!1), p (y x js ) > 0, in contradiction with the above opposite inequality. This contradiction proves that p := lim p k 2 P s , hence, P s is a compact set, for each s 2 S.

Assertion (ii) Let (s; l ) 2 S L and p := (p l ) 2 P s be given. Let e := (e l ) 2 R L be such that e l = 1 and e l = 0 for every l 2 Lnfl g. We prove that p l = p e > 0.

Indeed, let (p; j; (x i )) 2 P s I A s meet the conditions of the de…nition of P s . For every n > 1, we let y n 2 R L + be such that y n := (1 1 n )x js . It satis…es p (y n x js ) < 0 (since p x js > p e js > 0, from Assumption A1 and the de…nition of P s ).

From Assumption A2, there exists n 2 N, such that y := y n +(1

satis…es u j (x j0 ; y) > u j (x j0 ; x js ), which implies: p x js 6 p y < p x js + (1 1 n )p e, hence,

The continuous mapping ' (s;l) : p 2 P s 7 ! p e 2 R ++ attains a minimum on the compact set P s , say (s;l) > 0, and Assertion (ii) holds for := min (s;l)2S L (s;l) .

Proof of Lemma 1 It follows from Lemmata 1 and the inclusion [ s2S fsg P s .

Lemma 3

The following Assertions hold:

(i) the price sequences, f! n 0 := (p n 0 ; q n )g and fp n s g, may be assumed to converge, say to ! 0 := (p 0 ; q ) 2 P 0 and p := (p s ) 2 P := P S , s.t. ! s := (s; p s ) 2 \ i2I i , for all s 2 S;

(ii) the sequences f(x n is ) s2S 0 g and fz n i g, for each i 2 I, may be assumed to converge, say to (x is ) s2S 0 2 R L S 0 + and z i 2 R J , such that P i2I (x is -e is ) s2S 0 = 0, and

Proof Assertion (i) From the de…nitions and Lemma 1, the relations p n s 2 [ ; 1] L \ P and (s; p n s ) 2 hold, for every n > N along Lemma 2, and every s 2 S. The sequence fp n s g n>N may, hence, be assumed to converge, say to p s 2 P , such that (s; p s ) 2 , for every s 2 S. The relation (\ i2I i ) holds from Assumption A3 and De…nition 1.

Similarly, the sequence f! n 0 := (p n 0 ; q n )g may be assumed to converge in the compact set P 0 , say to ! 0 := (p 0 ; q ) 2 P 0 . Assertion (i) follows.

Assertion (ii) The non-negativity and market clearance conditions over auxiliary equilibrium allocations imply that f(x n is ) s2S 0 g n>N is bounded, hence, may be assumed to converge, for each i 2 I, say to (x is ) s2S 0 2 R L S 0 + . The market clearance relations, P i2I (x n is e is ) s2S 0 = 0, hold for each n > N , and yield, the limit:

x is e is ) s2S 0 = 0.