Aleksander Aleksenskii

Marcus Bleuel
email: markusbleuel@gmail.com

Alexei Bosak
email: alexei.bossak@esrf.fr

Alexandra Chumakova
email: aleksandra.chumakova@esrf.fr

Artur Dideikin
email: dideikin@mail.ioffe.ru

Marc Dubois
email: marc.dubois@uca.fr

Ekaterina Korobkina

Egor Lychagin

Alexei Muzychka

Grigory Nekhaev
email: grigorijnekhaev@yandex.ru

Aleksander Aleksensk

Marcus Bleu

Egor Lychag
email: lychag@nf.jinr.ru

Grigory Nekha

Valery Nesvizhevsky
email: nesvizhevsky@ill.eu

Alexander Nezvanov
email: nezvanov@jinr.ru

Ralf Schweins
email: schweins@ill.eu

Alexander Shvidchenko

Alexander Strelkov

Kylyshbek Turlybekuly

Alexander Vul'
email: alexandervul@mail.ioffe.ru

Kirill Zhernenkov
email: k.zhernenkov@fz-juelich.de

Effect of Particle Sizes on the Efficiency of Fluorinated Nanodiamond Neutron Reflectors

Keywords: detonation nanodiamonds, nanopowder, reflectors of slow neutrons, albedo, size separation of nanodiamonds, fluorination, Monte Carlo

come

Introduction

Slow neutrons are usually subdivided into three ranges: ultracold neutrons (UCNs) [START_REF] Lushchikov | Observation of ultracold neutrons[END_REF][START_REF] Steyerl | Measurements of total cross sections for very slow neutrons with velocities from 100 m/s to 5 m/s[END_REF][START_REF] Ignatovich | The Physics of Ultracold Neutrons[END_REF][START_REF] Golub | Ultracold Neutrons[END_REF][START_REF] Steyerl | Ultracold Neutrons[END_REF], very cold neutrons (VCNs), and cold neutrons (CNs). The characteristic feature of UCNs is their (nearly) total reflection from a material surface provided the neutron velocity is smaller than the critical velocity of the surface material; a typical value of the critical velocity of materials used to build UCN traps is ~5 m/s. The available UCN fluxes are extremely low; however, this property of total reflection makes them an invaluable tool in fundamental neutron physics. CNs are widely used in neutron scattering and particle neutron physics due to their much higher fluxes than those of UCNs. Their wavelengths are slightly larger than interatomic distances; thus, Bragg scattering in solids starts disappearing, and matter starts becoming more transparent. Most of the neutrons are thermalized in cryogenic CN sources [START_REF] Ageron | Cold neutron sources[END_REF] in research nuclear reactors and spallation neutron sources to this energy range. A typical velocity of CNs is ~500 m/s. The intermediate range of VCNs with a typical velocity of ~50 m/s is rarely used because of two reasons: they have much lower fluxes than those of CNs, and there is no storage in traps unlike in the case of UCNs. We explore the possibility of solving both of these problems by means of developing VCN reflectors, which are efficient in the lower half of the so-called "reflectivity gap".

Over a decade ago, it was confirmed that detonation nanodiamond (DND) powders reflect VCNs diffusively at any incidence angle, and they reflect CNs quasi-specularly at small incidence angles [START_REF] Nesvizhevsky | Interaction of neutrons with nanoparticles[END_REF][START_REF] Artem'ev | Estimation of neutron reflection from nanodispersed materials[END_REF][START_REF] Nesvizhevsky | The reflection of very cold neutrons from diamond powder nanoparticles[END_REF][START_REF] Lychagin | Storage of very cold neutrons in a trap with nano-structured walls[END_REF][START_REF] Nesvizhevsky | Application of diamond nanoparticles in low-energy neutron physics[END_REF][START_REF] Cubitt | Quasispecular reflection of cold neutrons from nano-dispersed media at above critical angles[END_REF]. In analogy to UCNs, the phenomenon of efficient reflection of neutrons from DND powder could be used for the definition of the range of velocities/energies of VCNs. Potential applications of these phenomena in neutron technology, as well as the beauty and complexity of the phenomena themselves, motivate active experimental and theoretical research in this area [START_REF] Krylov | Study of bound hydrogen in powders of diamond nanoparticles[END_REF][START_REF] Nesvizhevsky | Fluorinated nanodiamonds as unique neutron reflector[END_REF][START_REF] Nesvizhevsky | Effect of nanodiamond fluorination on the efficiency of quasi-specular reflection of cold neutrons[END_REF][START_REF] Herraiz | A multitechnique study of fluorinated nanodiamonds for low-energy neutron physics applications[END_REF][START_REF] Bosak | Fluorination of diamond nanoparticles in slow neutron reflectors does not destroy their crystalline cores and clustering while decreasing neutron losses[END_REF][START_REF] Nesvizhevsky | Reflectors for VCN and applications of VCN[END_REF][START_REF] Ignatovich | Reflection of slow neutrons from nanorod powder[END_REF][START_REF] Batsanov | Novel synthesis and properties of hydrogen-free detonation nanodiamond[END_REF].

Not only can neutrons be reflected from DNDs, but they can also be lost due to the interaction with DNDs in the course of diffusive motion. As neutron losses in raw DNDs are dominated by a small admixture of hydrogen on the surface of DNDs, the removal of hydrogen by fluorination [START_REF] Ersez | Small angle and inelastic scattering investigations of nanodiamonds[END_REF][START_REF] Teshigawara | Measurement of neutron scattering cross section of nano-diamond with particle diameter of approximately 5 nm in energy range of 0.2 meV to 100 meV[END_REF][START_REF] Grammer | The small-angle neutron scattering extension in MCNPX and the SANS cross section for nanodiamonds[END_REF][START_REF] Granada | Studies of reflector materials for cold neutrons[END_REF][START_REF] Jamalipour | Directional reflection of cold neutrons using nanodiamond particles for compact neutron sources[END_REF] is essential for reducing neutron losses.

The properties of DNDs have been studied over the decades of their research and use, in particular those relevant to neutron reflectors [START_REF] Badziag | Nanometre-sized diamonds are more stable then graphite[END_REF][START_REF] Greiner | Diamonds in detonation soot[END_REF][START_REF] Aleksenskii | The structure of diamond nanoclusters[END_REF][START_REF] Palosz | Analysis of short and long range order in nanocrystalline diamonds with application of powder diffractometer[END_REF][START_REF] Kruger | Usually tight aggregation in detonation nanodiamond: Identification and disintegration[END_REF][START_REF] Baidakova | New prospects and frontiers of nanodiamond clusters[END_REF][START_REF] Avdeev | Pressure induced charges in fractal structure of detonation nanodiamond powder by small-angle neutron scattering[END_REF][START_REF] Enyashin | Atomic and electronic structures and stability of icosahedral nanodiamonds and onions[END_REF][START_REF] Osawa | Monodisperse single nanodiamond particles[END_REF][START_REF] Williams | Size-dependent reactivity of diamond nanoparticles[END_REF][START_REF] Aleksenskii | Deagglomeration of detonation nanodiamonds[END_REF][START_REF] Chang | Confirmation of the electrostatic self-assembly of nanodiamonds[END_REF][START_REF] Mochalin | The properties and applications of nanodiamonds[END_REF][START_REF] Osawa | Deagglomeration of Detonation Nanodiamond[END_REF][START_REF] Avdeev | The spatial diamond-graphite transition in detonation nanodiamond as revealed by small-angle neutron scattering[END_REF][START_REF] Vul | Detonation Nanodiamonds: Science and Applications[END_REF][START_REF] Williams | [END_REF][START_REF] Zousman | Pure nanodiamonds produced by laser-assisted technique[END_REF][START_REF] Krueger | Current issues and challenges in surface chemistry in nanodiamonds[END_REF][START_REF] Dideikin | Rehybridization of carbon on facets of detonation diamond nanocrystals and forming hydrosols of individual particles[END_REF]. In relation to neutron reflectors, DNDs combine the large cross-section of coherent scattering and the low cross-section of neutron losses. As this paper shares the scientific motivations and some experimental methods with another article published in the same issue of this journal [START_REF] Aleksenskii | Clustering of Diamond Nanoparticles, Fluorination and Efficiency of Slow Neutron Reflectors[END_REF], we limit our description here to only the specific results related to the subject of our present research. We refer the reader to this previous publication for a detailed description of the principle of operation and the general properties of such neutron reflectors. In the present publication, we report results of our study to optimize the sizes of DNDs in order to increase the neutron albedo in the most interesting VCN range; the neutron albedo is neutron reflectivity for the isotropic angular distribution of incident neutrons. From below, the VCN velocity range starts directly from UCNs. The most probable VCN velocity at the only existing VCN user facility, PF2 at ILL, is ~50 m/s. The maximum available VCN velocity is ~200 m/s. Better VCN reflectors might allow for an increase in this value.

For every neutron wavelength, there is an optimum nanoparticle diameter that corresponds to the maximum neutron transport cross-section. In the model of diamond nanospheres, the optimal particle diameter D opt is related to the neutron wavelength λ n and the neutron velocity V n as follows [START_REF] Artem'ev | Estimation of neutron reflection from nanodispersed materials[END_REF]:

D opt [nm] ≈ 0.54λ n [nm] ≈ 215/V n [m/s]. (1)
In particular, for a typical VCN velocity of ~50 m/s, the optimal particle diameter is ~4.3 nm, precisely in line with a typical size of commercially available DNDs.

Figure 1 illustrates the calculated optimal particle sizes and the corresponding maximum VCN albedo for a range of neutron velocities from 20 to 200 m/s. This and all other calculations presented below were performed using Monte Carlo method and the model of discrete-sized diamond nanospheres (MDDNS) [START_REF] Bosak | Fluorination of diamond nanoparticles in slow neutron reflectors does not destroy their crystalline cores and clustering while decreasing neutron losses[END_REF][START_REF] Aleksenskii | Clustering of Diamond Nanoparticles, Fluorination and Efficiency of Slow Neutron Reflectors[END_REF]. In the particular case shown in Figure 1, the powder layer thickness is 1 cm and the density is 0.56 g/cm 3 . All particles are assumed to have a spherical shape and monodisperse optimal diameter, at which the transport cross-section for each wavelength reaches its maximum. It is clear that for the broad range of velocities above ~50 m/s, the mean size of particles should be smaller than ~4.3 nm. For comparison with the ideal case, we added a dashed curve corresponding to the albedo calculated for a model of the real powder of deagglomerated fluorinated DNDs (DF-DNDs) with the same density of 0.56 g/cm 3 described in ref. [START_REF] Aleksenskii | Clustering of Diamond Nanoparticles, Fluorination and Efficiency of Slow Neutron Reflectors[END_REF] (the sample with the best performance achieved). The difference between DF-DNDs and optimal diamond nanospheres is due to the deviation from the optimum size and the spread of sizes.

Evidently, a maximum neutron albedo for all wavelengths at once is unattainable for any particular DND powder. However, the albedo can be increased in a broad wavelength range by properly selecting the mean size of DNDs. Therefore, we found a way to separate, by centrifugation, the fraction of finer DNDs from a broad size distribution in the initial powder and experimentally and theoretically studied the performance of a neutron reflector made of such particles. To underline the separated DNDs as well as the small particle size in this powder, we refer to these as S-DNDs to distinguish them from the DF-DNDs, which are used for comparison.

When choosing methods of characterization of DNDs, we took into account that our main goal was to describe the diffusion of neutrons in the DND powder. Therefore, SANS (small-angle neutron scattering) is the main characterization method used in this paper, as it allowed us to describe DNDs as neutron scatterers. Other methods are complementary and were used to confirm the effect of different procedures that we applied to modify the samples (fluorination, size separation, etc.). When applying MDDNS to SANS data, we approximated a real medium with a model one, the main goal of which was to precisely describe all of the SANS data and to extrapolate the experimental data to velocity and angle ranges that are not accessible for direct SANS measurements. Such a model allowed us to describe neutron diffusion in DND powders in all velocity and angular ranges of interest.

This approach allowed us to take into account all relevant properties of the interaction of neutrons with nanostructured media described in this work (diamond cores and This and all other calculations presented below were performed using Monte Carlo method and the model of discrete-sized diamond nanospheres (MDDNS) [START_REF] Bosak | Fluorination of diamond nanoparticles in slow neutron reflectors does not destroy their crystalline cores and clustering while decreasing neutron losses[END_REF][START_REF] Aleksenskii | Clustering of Diamond Nanoparticles, Fluorination and Efficiency of Slow Neutron Reflectors[END_REF]. In the particular case shown in Figure 1, the powder layer thickness is 1 cm and the density is 0.56 g/cm 3 . All particles are assumed to have a spherical shape and monodisperse optimal diameter, at which the transport cross-section for each wavelength reaches its maximum. It is clear that for the broad range of velocities above ~50 m/s, the mean size of particles should be smaller than ~4.3 nm. For comparison with the ideal case, we added a dashed curve corresponding to the albedo calculated for a model of the real powder of deagglomerated fluorinated DNDs (DF-DNDs) with the same density of 0.56 g/cm 3 described in ref. [START_REF] Aleksenskii | Clustering of Diamond Nanoparticles, Fluorination and Efficiency of Slow Neutron Reflectors[END_REF] (the sample with the best performance achieved). The difference between DF-DNDs and optimal diamond nanospheres is due to the deviation from the optimum size and the spread of sizes.

Evidently, a maximum neutron albedo for all wavelengths at once is unattainable for any particular DND powder. However, the albedo can be increased in a broad wavelength range by properly selecting the mean size of DNDs. Therefore, we found a way to separate, by centrifugation, the fraction of finer DNDs from a broad size distribution in the initial powder and experimentally and theoretically studied the performance of a neutron reflector made of such particles. To underline the separated DNDs as well as the small particle size in this powder, we refer to these as S-DNDs to distinguish them from the DF-DNDs, which are used for comparison.

When choosing methods of characterization of DNDs, we took into account that our main goal was to describe the diffusion of neutrons in the DND powder. Therefore, SANS (small-angle neutron scattering) is the main characterization method used in this paper, as it allowed us to describe DNDs as neutron scatterers. Other methods are complementary and were used to confirm the effect of different procedures that we applied to modify the samples (fluorination, size separation, etc.). When applying MDDNS to SANS data, we approximated a real medium with a model one, the main goal of which was to precisely describe all of the SANS data and to extrapolate the experimental data to velocity and angle ranges that are not accessible for direct SANS measurements. Such a model allowed us to describe neutron diffusion in DND powders in all velocity and angular ranges of interest.

This approach allowed us to take into account all relevant properties of the interaction of neutrons with nanostructured media described in this work (diamond cores and non-diamond carbons, nano-pores, interference on neighbor scatterers, etc.). The structures of significantly larger sizes, originating in particular from microporosity and microstructure, have no direct effect on neutron transport characteristics, as they result in the scattering of too-small angles. However, they are also included in our analysis through SANS measurements. The absence of neutron-absorbing impurities must be verified by other methods (neutron activation, neutron prompt-γ analysis, etc.) for each DND powder considered to be used for a real reflector.

There are several options for building neutron reflectors based on DND powders, and these include placing a DND powder in a thin-wall envelope made of materials with low neutron losses as used, for instance, for the first demonstration of VCN storage in a closed trap [START_REF] Lychagin | Storage of very cold neutrons in a trap with nano-structured walls[END_REF], as well as sintering and cold compaction, as is being investigated in the ANR-20-CE08-0034 project (ANR-Agence Nationale de la Recherche, France). The choice between these options depends on the particular applications of such reflectors. However, the results of the present study are valid for all these cases.

The details of the sample preparation process are described in Section 2. In Section 3, we present the results of experimental studies of the particle size distribution. We investigated DF-DND and S-DND samples with complementary techniques as follows: Section 3.1 illustrates the size distribution of DNDs in these samples using transmission electron microscopy (TEM); in Section 3.2, the size distribution of DND cores using X-ray diffraction (XDR) is discussed; SANS results are described in Section 3.3; in Section 4, we discuss the calculated performance of DF-DND and S-DND reflectors and the effect of increased albedo on the yield of neutrons.

Samples

For compatibility with previous results [START_REF] Teshigawara | Measurement of neutron scattering cross section of nano-diamond with particle diameter of approximately 5 nm in energy range of 0.2 meV to 100 meV[END_REF][START_REF] Grammer | The small-angle neutron scattering extension in MCNPX and the SANS cross section for nanodiamonds[END_REF][START_REF] Granada | Studies of reflector materials for cold neutrons[END_REF][START_REF] Jamalipour | Directional reflection of cold neutrons using nanodiamond particles for compact neutron sources[END_REF][START_REF] Aleksenskii | Clustering of Diamond Nanoparticles, Fluorination and Efficiency of Slow Neutron Reflectors[END_REF], we used raw DNDs produced at the Federal State Unitary Enterprise, "Russian Federal Nuclear Center-Academician E.I. Zababakhin All-Russian Research Institute of Technical Physics" (FSUE "RFNC-VNIIF"), Snezhinsk, Russia, Technical Regulations TY 2-037-677-94. Their fluorination was carried out at the Institute of Chemistry of Clermont-Ferrand (Université Clermont Auvergne and CNRS), Aubière, France, in accordance with the procedure described in ref. [START_REF] Teshigawara | Measurement of neutron scattering cross section of nano-diamond with particle diameter of approximately 5 nm in energy range of 0.2 meV to 100 meV[END_REF], to obtain the fluorinated DND (F-DND) samples, which were then deagglomerated as described in ref. [START_REF] Aleksenskii | Clustering of Diamond Nanoparticles, Fluorination and Efficiency of Slow Neutron Reflectors[END_REF] to obtain the DF-DND samples.

While molecular fluorine does not react with sp 3 carbons of the diamond core, it decomposes amorphous carbons when the reaction is carried out at a high temperature such as 450 • C in our case. The lower the crystalline order, the higher reactivity of sp 2 -type carbon. At this temperature, graphitized samples form, in pure F 2 gas, a mixture of (C 2 F) n and (CF) n structural types, whereas amorphous carbons are decomposed into C x F y gases (x = 1, 2, 3, . . . and y = 4, 5, 6, . . . ). As a gas/solid reaction, the F 2 molecules react with the available sp 2 C and functional groups (mainly C-OH, C-H, C=O, and COOH) on the diamond surface. If those groups are located on sp 2 C shells, they are removed together with C x F y gases. When located onto the diamond surface, they are converted into C-F bonds. It is important to note that no diffusion of fluorine occurs inside the diamond core. At boundaries, two opposite cases may occur: F 2 molecules open channels for their diffusion through the decomposition of sp 2 carbons. If this phenomenon does not occur, the impurities located at boundaries are inaccessible for molecular fluorine, and their removal fails.

We separated an S-DND fraction of finer DNDs from a broad size distribution. For this purpose, we used raw DNDs produced at the Federal State Unitary Enterprise Special Design-Technology Bureau (FSUE SDTB) "Technolog", Saint-Petersburg, Russia. These DNDs underwent the deagglomeration process described in ref. [START_REF] Aleksenskii | Deagglomeration of detonation nanodiamonds[END_REF]. The obtained hydrosol of deagglomerated particles was centrifuged (in a Sigma 6-16 centrifuge (SIGMA Laborzentrifugen GmbH, Germany) with the acceleration a max = 1.8 × 10 4 g for 100 min) in order to separate particles by size in water. The supernatant containing S-DND particles with diameters of ~3 nm was carefully separated from the sediment. Note that such diameters are well within the range of stability of DNDs, whose lower bound is ~1.2 nm [START_REF] Enyashin | Atomic and electronic structures and stability of icosahedral nanodiamonds and onions[END_REF].

Results

Transmission Electron Microscopy

Figure 2a,c show examples of TEM images of DF-DND and S-DND samples, respectively, obtained using FEI Tecnai G2 30 S-TWIN, NRC "Kurchatov Institute"-CRISM "Prometey", Russia. For these measurements, a 2-3 mm 3 sample of the powder was added to 1 mL of distilled water, and the container with the mixture was placed in an ultrasonic bath filled with water. It was then sonicated for 30 min. The resulting suspension (2-3 drops) was applied to a carbon replica placed on the plain grid manufactured by Pacific Grid-Tech. After drying, the replica was examined via TEM.

Nanomaterials 2021, 11, x FOR PEER REVIEW 5 of 17

Results

Transmission Electron Microscopy

Figure 2a,c show examples of TEM images of DF-DND and S-DND samples, respectively, obtained using FEI Tecnai G2 30 S-TWIN, NRC "Kurchatov Institute"-CRISM "Prometey", Russia. For these measurements, a 2-3 mm 3 sample of the powder was added to 1 mL of distilled water, and the container with the mixture was placed in an ultrasonic bath filled with water. It was then sonicated for 30 min. The resulting suspension (2-3 drops) was applied to a carbon replica placed on the plain grid manufactured by Pacific Grid-Tech. After drying, the replica was examined via TEM.

The For the presented histograms, we used 10 TEM images for S-DNDs and 1 TEM image for DF-DNDs. The visible projection of an individual particle was described by an ellipse. The particle was assigned a diameter equal to the diameter of a circle of equal area. The total number of particles in TEM images of S-DNDs was 2078, and the total number of DF-DND particles was 264.

The mean particle diameters in DF-DND and S-DND samples are ~4.7(2) nm and ~3.8 [START_REF] Lushchikov | Observation of ultracold neutrons[END_REF] nm respectively. This difference in mean diameters is large enough for the purpose of this study, i.e., for a comparison of the two DND reflectors as a function of the particle size.

X-ray Diffraction

Another non-neutron method of evaluating DND sizes is the broadening of peaks in XRD patterns.

The data were collected from DF-DND and S-DND powders by combining a powder diffractometer Rigaku SmartLab III (Rigaku Corporation, Tokyo, Japan) equipped with a CuKα source (λ = 1.541 Å) and the ID28 diffractometer at ESRF (Grenoble, France) [START_REF] Girard | A new diffractometer for diffuse scattering studies on the ID28 beamline at the ESRF[END_REF] with a PILATUS3 1M area detector (λ = 0.784 Å).

With the Rigaku diffractometer, a planar powder stage was used as the sample holder. With the ID28 diffractometer, the samples were packed in quartz capillaries with the diameter of 200 µm; the data were evaluated using SNBL Toolbox [START_REF] Dyadkin | A new multipurpose diffractometer PILATUS@SNBL[END_REF] and Dioptas [START_REF] Prescher | A program for reduction of two-dimensional X-ray diffraction data and data exploration[END_REF] software.

The results are shown in Figure 3. The positions of the diffraction peaks correspond to the diamond lattice. Broadening of the peaks contains information about particle sizes. A characteristic size of the coherent scattering region was determined by analyzing the full width of the peak at half maximum using the Sedyakov-Scherrer equation and the shape of the peak through the comparison of full width at one-fifth and four-fifths of the maximum [START_REF] Pielaszek | FW1/5/4/5M method for determination of the grain size distribution from powder diffraction line profile[END_REF]. For the latter purpose, the 311 peak was the most convenient, and its width change by ~10% is shown in Figure 3c with a parabolic local background subtracted and the peak height scaled. The results, as expected, are only slightly dependent on the procedure. Rigaku data provide the value of the mean particle size equal to ~3.4 nm for S-DND, while the mean size and dispersion of sizes from the synchrotron data are, respectively, ~4.1 nm and ~2.0 nm for DF-DND, and ~3.7 nm and 1.7 nm for S-DND. The mean value D 4 /D 3 estimated from XRD peak broadening thus decreases by ~10%.

Nanomaterials 2021, 11, x FOR PEER REVIEW 6 of 17

The mean particle diameters in DF-DND and S-DND samples are ~4.7(2) nm and ~3.8 [START_REF] Lushchikov | Observation of ultracold neutrons[END_REF] nm respectively. This difference in mean diameters is large enough for the purpose of this study, i.e., for a comparison of the two DND reflectors as a function of the particle size.

X-Ray Diffraction

Another non-neutron method of evaluating DND sizes is the broadening of peaks in XRD patterns.

The data were collected from DF-DND and S-DND powders by combining a powder diffractometer Rigaku SmartLab III (Rigaku Corporation, Tokyo, Japan) equipped with a CuKα source (𝜆 = 1.541 Å) and the ID28 diffractometer at ESRF (Grenoble, France) [START_REF] Girard | A new diffractometer for diffuse scattering studies on the ID28 beamline at the ESRF[END_REF] with a PILATUS3 1M area detector (𝜆 = 0.784 Å).

With the Rigaku diffractometer, a planar powder stage was used as the sample holder. With the ID28 diffractometer, the samples were packed in quartz capillaries with the diameter of 200 μm; the data were evaluated using SNBL Toolbox [START_REF] Dyadkin | A new multipurpose diffractometer PILATUS@SNBL[END_REF] and Dioptas [START_REF] Prescher | A program for reduction of two-dimensional X-ray diffraction data and data exploration[END_REF] software.

The results are shown in Figure 3. The positions of the diffraction peaks correspond to the diamond lattice. Broadening of the peaks contains information about particle sizes. A characteristic size of the coherent scattering region was determined by analyzing the full width of the peak at half maximum using the Sedyakov-Scherrer equation and the shape of the peak through the comparison of full width at one-fifth and four-fifths of the maximum [START_REF] Pielaszek | FW1/5/4/5M method for determination of the grain size distribution from powder diffraction line profile[END_REF]. For the latter purpose, the 311 peak was the most convenient, and its width change by ~10% is shown in Figure 3c with a parabolic local background subtracted and the peak height scaled. The results, as expected, are only slightly dependent on the procedure. Rigaku data provide the value of the mean particle size equal to ~3.4 nm for S-DND, while the mean size and dispersion of sizes from the synchrotron data are, respectively, ~4.1 nm and ~2.0 nm for DF-DND, and ~3.7 nm and 1.7 nm for S-DND. The mean value 〈𝐷 〉/〈𝐷 〉 estimated from XRD peak broadening thus decreases by ~10%.

Small-Angle Neutron Scattering

SANS provides the most direct and unambitious information for the analysis of scattering and transport of slow neutrons in DND powders. SANS characterization [START_REF] Linder | X-ray and Light: Scattering Methods Applied to Soft Condensed Matter[END_REF] of DF-DND and S-DND samples was performed using three instruments: a time-of-flight spectrometer YuMO in the two-detector mode (FLNP, JINR, Dubna, Russia [START_REF] Kuklin | Neutronographic investigations of supramolecular structures on upgraded small-angle spectrometer YuMO[END_REF]), a diffractometer D11 (ILL, Grenoble, France) [START_REF] Linder | The D11 small-angle scattering instrument: A new benchmark for SANS[END_REF][START_REF] Richard | Analysis and visualization of neutron-scattering data[END_REF], and an NGB30 at the NIST Center for Neutron Research (Gaithersburg,MD, USA) [START_REF] Glinka | The 30m small angle neutron scattering instruments at the National Institute of Standard and Technology[END_REF], and their neutron wavelengths and ranges of transferred momenta (Q) were equal to 0.7-5.0 Å and 7×10 -2 nm -1 < Q < 10 1 nm -1 ; 6 Å and

Small-Angle Neutron Scattering

SANS provides the most direct and unambitious information for the analysis of scattering and transport of slow neutrons in DND powders. SANS characterization [START_REF] Linder | X-ray and Light: Scattering Methods Applied to Soft Condensed Matter[END_REF] of DF-DND and S-DND samples was performed using three instruments: a time-of-flight spectrometer YuMO in the two-detector mode (FLNP, JINR, Dubna, Russia [START_REF] Kuklin | Neutronographic investigations of supramolecular structures on upgraded small-angle spectrometer YuMO[END_REF]), a diffractometer D11 (ILL, Grenoble, France) [START_REF] Linder | The D11 small-angle scattering instrument: A new benchmark for SANS[END_REF][START_REF] Richard | Analysis and visualization of neutron-scattering data[END_REF], and an NGB30 at the NIST Center for Neutron Research (Gaithersburg, MD, USA) [START_REF] Glinka | The 30m small angle neutron scattering instruments at the National Institute of Standard and Technology[END_REF], and their neutron wavelengths and ranges of transferred momenta (Q) were equal to 0.7-5.0 Å and 7 × 10 -2 nm -1 < Q < 10 1 nm -1 ; 6 Å and 10 -2 nm -1 < Q < 10 0 nm -1 ; and 6 Å and 3.4 × 10 -2 nm -1 < Q < 1.2 × 10 0 nm -1 , respectively. The three SANS instruments were used to increase the reliability and precision of the results and to select Q-ranges with maximum statistical accuracy and minimum backgrounds. To match the SANS data measured at different Q-ranges (corresponding to different distances from the sample to the detector) as well as SANS data measured with the various instruments (YuMO, D11, and NGB30), the following procedures were applied:

(1) With each instrument, a few Q-ranges were measured with a significant overlapping and high enough statistics in the overlapping ranges. The data from the overlapping Q-ranges were used to normalize the intensity for the entire Q-range for each instrument.

(2) The absolute intensity of neutron scattering was normalized using the data measured with NGB30, where both the transmitted beam and the scattered intensity were recorded for each measurement. As the same DF-DNDs were used for all three instruments, we matched the absolute intensities taking into account the density of the DF-DND samples used with each instrument. As both DF-DND and S-DND samples were measured with two of the three instruments (D11 and YuMO), we could match the intensities of neutron scattering for the two samples. The Q-ranges used for the summary plot presented in Figure 4 for the S-DNDs are as follows: 1.55 × -2 nm -1 -8.53 × 10 -1 nm -1 from D11 and 8.53 × 10 -1 nm -1 -1.10 × 10 0 nm -1 from YuMO; for the DF-DNDs, they are as follows: 9.03 × 10 -3 nm -1 -1.55 × 10 -2 nm -1 and 7.57 × 10 -2 nm -1 -9.85 × 10 -2 nm -1 from NGB30, 1.55 × 10 -2 nm -1 -7.57 × 10 -2 nm -1 from D11, and 9.85 × 10 -1 nm -1 -1.47 × 10 0 nm -1 from YuMO. The absolute calibration of scattering probabilities is an important input for the simulation of neutron propagation in powders. ) of neutron scattering versus transferred momentum Q (nm -1 ) for DF-DNDs (black squares) and S-DNDs (green circles). To compare only the effect of particle sizes, and not powder density, both curves are normalized to the sample density of 1 g/cm 3 . SANS characterization was performed with YuMO, D11, and NGB30 instruments, and their neutron wavelengths and ranges of transferred momenta (Q) were equal to 0.7-5.0 Å and 7×10 -2 nm -1 < Q < 10 1 nm -1 ; 6 Å and 10 -2 nm -1 < Q < 10 0 nm -1 ; and 6 Å and 3.4×10 -2 nm -1 < Q < 1.2×10 0 nm -1 , respectively. The Q-ranges used for the summary plot for the S-DNDs are as follows: 1.55×10 -2 nm -1 -8.53×10 -1 nm -1 from D11 and 8.53×10 -1 nm -1 -1.10×10 0 nm -1 from YuMO; for the DF-DNDs, they are as follows: 9.03×10 -3 nm -1 -1.55×10 -2 nm -1 and 7.57×10 -2 nm -1 -9.85×10 -2 nm -1 from NGB30, 1.55×10 -2 nm -1 -7.57×10 -2 nm -1 from D11, and 9.85×10 -1 nm -1 -1.47×10 0 nm -1 from YuMO.

Discussion

Approximation of the Size Distribution of DNDs using MDDNS

As previously mentioned, our main goal was not to precisely extract the size distribution of real powders but to find an approximate model with a size distribution that allows us to precisely reproduce neutron scattering. Figure 5 presents the diameter distribution of ideal DNDs obtained from MDDNS in the case of DF-DND and S-DND samples evaluated using the SANS data shown in Figure 4 after the incoherent scattering on hydrogen in S-DNDs was subtracted. The physical basis for the procedure of size evaluation is the model of discrete-sized diamond nanospheres (MDDNS) [START_REF] Jamalipour | Directional reflection of cold neutrons using nanodiamond particles for compact neutron sources[END_REF][START_REF] Aleksenskii | Clustering of Diamond Nanoparticles, Fluorination and Efficiency of Slow Neutron Reflectors[END_REF], which represents ) of neutron scattering versus transferred momentum Q (nm -1 ) for DF-DNDs (black squares) and S-DNDs (green circles). To compare only the effect of particle sizes, and not powder density, both curves are normalized to the sample density of 1 g/cm 3 . SANS characterization was performed with YuMO, D11, and NGB30 instruments, and their neutron wavelengths and ranges of transferred momenta (Q) were equal to 0.7-5.0 Å and 7 × 10 -2 nm -1 < Q < 10 1 nm -1 ; 6 Å and 10 -2 nm -1 < Q < 10 0 nm -1 ; and 6 Å and 3.4 × 10 -2 nm -1 < Q < 1.2 × 10 0 nm -1 , respectively. The Q-ranges used for the summary plot for the S-DNDs are as follows: 1.55 × 10 -2 nm -1 -8.53 × 10 -1 nm -1 from D11 and 8.53 × 10 -1 nm -1 -1.10 × 10 0 nm -1 from YuMO; for the DF-DNDs, they are as follows: 9.03 × 10 -3 nm -1 -1.55 × 10 -2 nm -1 and 7.57 × 10 -2 nm -1 -9.85 × 10 -2 nm -1 from NGB30, 1.55 × 10 -2 nm -1 -7.57 × 10 -2 nm -1 from D11, and 9.85 × 10 -1 nm -1 -1.47 × 10 0 nm -1 from YuMO.

We used Igor macros [START_REF] Kline | Reduction and analysis of SANS and USANS data using Igor Pro[END_REF] to evaluate the SANS data. The samples were placed inside aluminum 1 mm cells with 1 mm windows for measurements with YuMO and in Quartz SUPRASILL 1 mm cells with D11 and NGB30. The DF-DNDs bulk density was equal to 0.56 g/cm 3 . The bulk density of S-DNDs was equal to 0.67 g/cm 3 .

Figure 4 compares the merged results of SANS measurements made with all these instruments. The shape of the scattering intensities for the two samples differs significantly at small values Q < 10 0 nm -1 due to the lower concentration of agglomerates in S-DNDs resulting from the DND size separation procedure. Another natural consequence of this procedure is that the number of individual DNDs (corresponding to large values of Q > 10 0 nm -1 ) increases by 36%. These observations agree with our expectations based on the knowledge of the DND separation method. Incoherent scattering at Q > 7 × 10 0 nm -1 is defined mainly by the presence of hydrogen. It is larger for S-DND powder because it is not fluorinated; however, this fact is not relevant to our calculations, as we do not consider the effect of impurities.

Discussion

Approximation of the Size Distribution of DNDs Using MDDNS

As previously mentioned, our main goal was not to precisely extract the size distribution of real powders but to find an approximate model with a size distribution that allows us to precisely reproduce neutron scattering. Figure 5 presents the diameter distribution of ideal DNDs obtained from MDDNS in the case of DF-DND and S-DND samples evaluated using the SANS data shown in Figure 4 after the incoherent scattering on hydrogen in S-DNDs was subtracted. The physical basis for the procedure of size evaluation is the model of discrete-sized diamond nanospheres (MDDNS) [START_REF] Jamalipour | Directional reflection of cold neutrons using nanodiamond particles for compact neutron sources[END_REF][START_REF] Aleksenskii | Clustering of Diamond Nanoparticles, Fluorination and Efficiency of Slow Neutron Reflectors[END_REF], which represents the medium as ideal independent diamond nanospheres with a discrete set of diameters. Such a model medium scatters neutrons the same way as a real medium but also allows model extrapolation of the measured experimental data to other wavelength and angular ranges not accessible directly with standard SANS devices. The mathematical algorithm of this procedure will be described in detail in a forthcoming publication and is the subject of a patent (RU 2020662675). Although both SANS curves in Figure 4 are of equal statistical quality, the size distribution for MDDNS DF-DNDs is smooth in shape, whereas that for S-DNDs shows considerable fluctuations. This difference might be because fewer nanodiamond diameters effectively contribute to the scattering of neutrons on S-DNDs. Such fluctuations have no effect on the precision of model calculations of neutron transport in S-DNDs on the condition that our model precisely describes the SANS data. In the range of 1.2-10.0 nm, the mean size of ideal scatterers in the S-DND model is ~2.9 nm, while it is ~3.8 nm for the DF-DND model.

Figure 5 confirms the conclusion drawn from the other methods of DND characterization; that is, in S-DNDs, the fraction of smaller particles is larger, while the larger particles disappear completely. Moreover, it can be clearly observed that the size distributions obtained from MDDNS using SANS data have a fraction of noticeably smaller particles Although both SANS curves in Figure 4 are of equal statistical quality, the size distribution for MDDNS DF-DNDs is smooth in shape, whereas that for S-DNDs shows considerable fluctuations. This difference might be because fewer nanodiamond diameters effectively contribute to the scattering of neutrons on S-DNDs. Such fluctuations have no effect on the precision of model calculations of neutron transport in S-DNDs on the condition that our model precisely describes the SANS data. In the range of 1.2-10.0 nm, the mean size of ideal scatterers in the S-DND model is ~2.9 nm, while it is ~3.8 nm for the DF-DND model.

Figure 5 confirms the conclusion drawn from the other methods of DND characterization; that is, in S-DNDs, the fraction of smaller particles is larger, while the larger particles disappear completely. Moreover, it can be clearly observed that the size distributions obtained from MDDNS using SANS data have a fraction of noticeably smaller particles than those obtained with other methods. The reason for this difference is that neutrons are not scattered only on individual DNDs but also on fluctuations in the density of the medium. They are sensitive to the entire structure of the powder. The scattering on pores between DNDs also contributes to this. In MDDNS, scattering on small pores, on non-diamond carbon, on specific types of agglomeration, on specific shapes of DNDs, etc. corresponds to scattering on small nanospheres. Note that to calculate the transport of neutrons in the powder, we only used SANS data. Other methods serve only for a better understanding of the peculiarities of powder modifications and the effect of these modifications.

Albedo Calculations

To calculate the albedo for different cases, as discussed below, we used size distributions shown in Figure 5 in Section 3.3 obtained from SANS data for DF-DNDs and S-DNDs. First, we simulated the neutron albedo for a theoretical case of semi-infinite flat media; second we considered cases of final thickness (flat and spherical layers) with the same or different density. In the latter case, we used the densities of real samples. Note that when preparing samples for SANS, we aimed at minimum but stable density; for DNDs of different types, the density was notably different. For a powder of infinite thickness and a flat geometry, the albedo does not depend on density, and it is defined only by the absorption and scattering cross-sections. For a powder of a finite thickness, the fraction of neutrons passing through the layer depends on the powder density; thus, the albedo depends on density. For a spherical trap, the albedo also changes for the infinite powder thickness, since it depends on the ratio of the radius of curvature of the surface to the depth of the penetration of neutrons into the powder. The calculations were performed within MDDNS, using the Monte Carlo method [START_REF] Spanier | Monte Carlo Principles and Neutron Transport. Problems[END_REF] and our original software. Incident neutrons are isotropic.

Figure 6 shows the calculated neutron albedo from the semi-infinite media for DF-DND and S-DND powders.

Nanomaterials 2021, 11, x FOR PEER REVIEW 10 of 17 within MDDNS, using the Monte Carlo method [START_REF] Spanier | Monte Carlo Principles and Neutron Transport. Problems[END_REF] and our original software. Incident neutrons are isotropic. Figure 6 shows the calculated neutron albedo from the semi-infinite media for DF-DND and S-DND powders. From Figure 6, it can be seen that for VCN velocities <70 m/s, the albedo is higher for DF-DNDs, whereas, for VCN velocities >70 m/s, it is higher for S-DNDs. This is explained by the presence of larger DNDs in DF-DNDs, which are closer to the optimal values for smaller velocities, and, vice versa, the presence of smaller DNDs in S-DNDs, which are closer to the optimal values for larger velocities. Both conclusions are in line with Formula (1) and the calculation presented in Figure 1.

The relatively small difference between these absolute albedo values is translated into a larger difference between loss factors (η = 1 -albedo), especially for albedo values close to 100%. As reciprocal loss factors are proportional to VCN densities, which can be accumulated in a trap with DND walls, we used the results shown in Figure 6 for the calculation of the loss factors ratio shown in Figure 7.

From Figure 6, it can be seen that for VCN velocities <70 m/s, the albedo is higher for DF-DNDs, whereas, for VCN velocities >70 m/s, it is higher for S-DNDs. This is explained by the presence of larger DNDs in DF-DNDs, which are closer to the optimal values for smaller velocities, and, vice versa, the presence of smaller DNDs in S-DNDs, which are closer to the optimal values for larger velocities. Both conclusions are in line with Formula (1) and the calculation presented in Figure 1.

The relatively small difference between these absolute albedo values is translated into a larger difference between loss factors (η = 1 -albedo), especially for albedo values close to 100%. As reciprocal loss factors are proportional to VCN densities, which can be accumulated in a trap with DND walls, we used the results shown in Figure 6 for the calculation of the loss factors ratio shown in Figure 7. From Figure 7, it can be observed that DF-DNDs are more advantageous than S-DNDs for the reflection of neutrons with typical VCN velocities of ~50 m/s from semi-infinite flat media. However, when the goal is to increase the upper boundary of the VCN velocity range, S-DNDs are slightly more appropriate than DF-DNDs. It must be borne in mind, however, that the advantage of S-DND will become more evident for realistic 3D geometries of DND reflectors, which we consider below.

In contrast to the case of semi-infinite media, for a finite-thickness flat layer, powder density is important. Figure 8 shows the calculated neutron albedo from flat 3 cm-thick layers for DF-DNDs and S-DNDs for two cases: an equal density of the samples as well as for the real densities of SANS samples. This comparison allows one to separate the effects of powder density and particle sizes. The probability of neutron absorption is below 1.43% for S-DNDs, and it is below 1.30% for DF-DNDs, at any velocity.

As Figure 8 clearly shows, for a finite thickness geometry, the albedo increases with an increase in powder density. This can be explained by the fact that increasing density is effectively equivalent to increasing thickness (for the same density). The effect of density is more pronounced in a 3D reflector geometry, which is of interest for a VCN source reflector or a VCN trap. It assumes that there is a cavity inside the DND reflector. This effect can be explained by the fact that the probability of return of a VCN to the cavity decreases with an increase in the depth of penetration into the reflector.

Figure 9 shows the results of calculations of the neutron albedo from the wall of a spherical cavity depending on the cavity radius. Neutron velocities are 50 m/s, 100 m/s, and 150 m/s; powder thickness is infinite and 3 cm; and powder densities vary. geometries of DND reflectors, which we consider below.

In contrast to the case of semi-infinite media, for a finite-thickness flat layer, powder density is important. Figure 8 shows the calculated neutron albedo from flat 3 cm-thick layers for DF-DNDs and S-DNDs for two cases: an equal density of the samples as well as for the real densities of SANS samples. This comparison allows one to separate the effects of powder density and particle sizes. The probability of neutron absorption is below 1.43% for S-DNDs, and it is below 1.30% for DF-DNDs, at any velocity. As Figure 8 clearly shows, for a finite thickness geometry, the albedo increases with an increase in powder density. This can be explained by the fact that increasing density is effectively equivalent to increasing thickness (for the same density). The effect of density is more pronounced in a 3D reflector geometry, which is of interest for a VCN source reflector or a VCN trap. It assumes that there is a cavity inside the DND reflector. This effect can be explained by the fact that the probability of return of a VCN to the cavity decreases with an increase in the depth of penetration into the reflector.

Figure 9 shows the results of calculations of the neutron albedo from the wall of a spherical cavity depending on the cavity radius. Neutron velocities are 50 m/s, 100 m/s, and 150 m/s; powder thickness is infinite and 3 cm; and powder densities vary. In contrast to the flat geometry, in a 3D geometry, the effect of density is significant provided the cavity radius is comparable with the depth of penetration of VCNs into the reflector. As shown in Figure 9, this condition is always met for any cavity radius and powder type when the goal is to increase the upper boundary of the velocity range of efficient reflection of VCNs. In this case, VCNs with a maximum velocity from the range penetrate into the DND wall to a depth comparable to the cavity radius.

One could try to compress powders to a higher density. However, there are certain constraints. With the decrease in the DND density, at some point, the density fluctuations would also decrease, thus resulting in a decrease in neutron scattering. The powder becomes more transparent to neutrons, and as it approaches the diamond density, it becomes completely transparent. This phenomenon can develop unevenly. In some areas of the compressed powder, the density may become too high, and in others, it may not. A 3 for DF-DNDs, and it is 0.67 g/cm 3 for S-DNDs. (c) A powder layer with the thickness of 3 cm; the density is 0.56 g/cm 3 for both samples. (d) A powder layer with the thickness of 3 cm; the density is 0.56 g/cm 3 for DF-DNDs, and it is 0.67 g/cm 3 for S-DNDs.

In contrast to the flat geometry, in a 3D geometry, the effect of density is significant provided the cavity radius is comparable with the depth of penetration of VCNs into the reflector. As shown in Figure 9, this condition is always met for any cavity radius and powder type when the goal is to increase the upper boundary of the velocity range of efficient reflection of VCNs. In this case, VCNs with a maximum velocity from the range penetrate into the DND wall to a depth comparable to the cavity radius.

One could try to compress powders to a higher density. However, there are certain constraints. With the decrease in the DND density, at some point, the density fluctuations would also decrease, thus resulting in a decrease in neutron scattering. The powder becomes more transparent to neutrons, and as it approaches the diamond density, it becomes completely transparent. This phenomenon can develop unevenly. In some areas of the compressed powder, the density may become too high, and in others, it may not. A technical realization of this compactification and the compressibility of DNDs of different types are the topic of a separate study.

To summarize the results of the present study, we compare the performance of reflectors based on DF-DNDs and S-DNDs for a realistic geometry of a spherical cavity with a radius of 5 cm and a wall thickness of 3 cm for realistic powder densities (see Figure 10). A comparison of Figures 7 and10 shows that (1) S-DNDs are more advantageous for the reflection of VCNs starting from a smaller velocity equal to ~60 (2) the decrease in the loss factor for S-DNDs is also more important, and it is equal to ~20-25% in a broad range of VCN velocities. The first point is important when estimating the spectrum of VCNs to be reflected with a DND reflector. The second factor is directly translated into the gain in the intensity of VCNs, which can be accumulated in DND traps with S-DND and DF-DND walls, respectively. The relative decrease in the gain factor in Figure 10 is a result of the partial transmission of faster neutrons through the wall of a finite thickness.

As Formula (1) shows, the optimum diameter of DNDs for VCNs with a velocity of ~50 m/s is ~4.3 nm. If the goal is to increase the upper boundary of the velocity range of effective reflection of VCNs, the mean diameter of DNDs must be reduced. However, it cannot be reduced by more than a factor of approximately three because even smaller DNDs do not exist, and because the cross-section of coherent scattering of neutrons decreases dramatically as a function of the DND diameter (as the sixth power of the diameter [START_REF] Nesvizhevsky | Interaction of neutrons with nanoparticles[END_REF]). If the goal is to improve storage times for a softer part of the VCN spectrum, the mean diameter of DNDs must be increased. However, it should not be increased by more than a factor of approximately three, because there are other mechanisms of reflection of slower VCNs that are efficient for such velocities (optical Fermi potential and supermirrors). These arguments roughly define the range of effective reflection of VCNs by DNDs and guide the choice of parameters of DNDs for the design of DND reflectors optimized for the diffusive reflection of VCNs.

Conclusions

In the current study, we investigated the effect of particle sizes on the efficiency of DND neutron reflectors. If typical DNDs with a size of ~4.3 nm are efficient for the reflection of typical VCNs with a velocity of ~50 m/s, other mean sizes are more efficient for other VCN velocities. Thus, up to a factor of approximately three, reduction in the mean A comparison of Figures 7 and10 shows that (1) S-DNDs are more advantageous for the reflection of VCNs starting from a smaller velocity equal to ~60 m/s; (2) the decrease in the loss factor for S-DNDs is also more important, and it is equal to ~20-25% in a broad range of VCN velocities. The first point is important when estimating the spectrum of VCNs to be reflected with a DND reflector. The second factor is directly translated into the gain in the intensity of VCNs, which can be accumulated in DND traps with S-DND and DF-DND walls, respectively. The relative decrease in the gain factor in Figure 10 is a result of the partial transmission of faster neutrons through the wall of a finite thickness.

As Formula (1) shows, the optimum diameter of DNDs for VCNs with a velocity of ~50 m/s is ~4.3 nm. If the goal is to increase the upper boundary of the velocity range of effective reflection of VCNs, the mean diameter of DNDs must be reduced. However, it cannot be reduced by more than a factor of approximately three because even smaller DNDs do not exist, and because the cross-section of coherent scattering of neutrons decreases dramatically as a function of the DND diameter (as the sixth power of the diameter [START_REF] Nesvizhevsky | Interaction of neutrons with nanoparticles[END_REF]). If the goal is to improve storage times for a softer part of the VCN spectrum, the mean diameter of DNDs must be increased. However, it should not be increased by more than a factor of approximately three, because there are other mechanisms of reflection of slower VCNs that are efficient for such velocities (optical Fermi potential and supermirrors). These arguments roughly define the range of effective reflection of VCNs by DNDs and guide the choice of parameters of DNDs for the design of DND reflectors optimized for the diffusive reflection of VCNs.

Conclusions

In the current study, we investigated the effect of particle sizes on the efficiency of DND neutron reflectors. If typical DNDs with a size of ~4.3 nm are efficient for the reflection of typical VCNs with a velocity of ~50 m/s, other mean sizes are more efficient for other VCN velocities. Thus, up to a factor of approximately three, reduction in the mean size allows more efficient reflection of faster neutrons. An increase up to a factor of approximately three in the mean size allows the storage times of slower VCNs to be increased. Simulations based on SANS data and MDDNS (model of discrete-size diamond nanospheres) for a realistic reflector geometry show that the particle size reduction corresponding to the replacement of DF-DNDs (DNDs with the best achieved performance) by S-DNDs (DNDs of smaller sizes) increases neutron albedo in the broad velocity range above ~60 m/s. This increase in albedo results in an increase in the density of faster VCNs in such a reflector cavity of up to ~25%, as well as an increase in the upper boundary of velocities of efficient VCN reflection. As the density of DND reflectors is of importance for their efficiency, an investigation of the practical feasibility of its increase and corresponding feedback is of interest for future research.

Figure 1 .

 1 Figure 1. Optimal diameters of diamond nanospheres (black line) as a function of VCN velocity/wavelength. Solid red line shows the corresponding calculated VCN albedo for the optimal diameters of diamond nanospheres. Dashed red line stands for the VCN albedo from real DF-DNDs (see ref. [46]). Neutrons are only absorbed by carbon atoms.

Figure 1 .

 1 Figure 1. Optimal diameters of diamond nanospheres (black line) as a function of VCN velocity/wavelength. Solid red line shows the corresponding calculated VCN albedo for the optimal diameters of diamond nanospheres. Dashed red line stands for the VCN albedo from real DF-DNDs (see ref. [46]). Neutrons are only absorbed by carbon atoms.

  Figure2a,cshow examples of TEM images of DF-DND and S-DND samples, respectively, obtained using FEI Tecnai G2 30 S-TWIN, NRC "Kurchatov Institute"-CRISM "Prometey", Russia. For these measurements, a 2-3 mm 3 sample of the powder was added to 1 mL of distilled water, and the container with the mixture was placed in an ultrasonic bath filled with water. It was then sonicated for 30 min. The resulting suspension (2-3 drops) was applied to a carbon replica placed on the plain grid manufactured by Pacific Grid-Tech. After drying, the replica was examined via TEM.The size distribution of DF-DNDs and S-DNDs shown in Figure2b,d, respectively, was evaluated using all TEM images available, examples of which are shown in Figure2a,b. For the presented histograms, we used 10 TEM images for S-DNDs and 1 TEM image for DF-DNDs. The visible projection of an individual particle was described by an ellipse. The particle was assigned a diameter equal to the diameter of a circle of equal area. The total number of particles in TEM images of S-DNDs was 2078, and the total number of DF-DND particles was 264.

Figure 2 .

 2 Figure 2. (a,c) Examples of TEM images of the DF-DND and S-DND samples; (b,d) size distributions for DF-DNDs and S-DNDs evaluated using all TEM images available.Figure 2. (a,c) Examples of TEM images of the DF-DND and S-DND samples; (b,d) size distributions for DF-DNDs and S-DNDs evaluated using all TEM images available.

Figure 2 .

 2 Figure 2. (a,c) Examples of TEM images of the DF-DND and S-DND samples; (b,d) size distributions for DF-DNDs and S-DNDs evaluated using all TEM images available.Figure 2. (a,c) Examples of TEM images of the DF-DND and S-DND samples; (b,d) size distributions for DF-DNDs and S-DNDs evaluated using all TEM images available. The size distribution of DF-DNDs and S-DNDs shown in Figure 2b,d, respectively, was evaluated using all TEM images available, examples of which are shown in Figure 2a,b.For the presented histograms, we used 10 TEM images for S-DNDs and 1 TEM image for DF-DNDs. The visible projection of an individual particle was described by an ellipse. The particle was assigned a diameter equal to the diameter of a circle of equal area. The total

Figure 3 .

 3 Figure 3. X-ray diffraction patterns: (a) S-DNDs at Rigaku SmartLab III; background is subtracted. (b) ESRF ID28 data for DF-DNDs and S-DNDs composed of three sets corresponding to three positions of the 2D detector; empty capillary scattering is shown. (c) Close-up of the 311 peak with the subtracted background and peak rescaling.

Figure 3 .

 3 Figure 3. X-ray diffraction patterns: (a) S-DNDs at Rigaku SmartLab III; background is subtracted. (b) ESRF ID28 data for DF-DNDs and S-DNDs composed of three sets corresponding to three positions of the 2D detector; empty capillary scattering is shown. (c) Close-up of the 311 peak with the subtracted background and peak rescaling.

  Nanomaterials 2021, 11, x FOR PEER REVIEW 8 of 17

Figure 4 .

 4 Figure 4. Intensity I (nm -1) of neutron scattering versus transferred momentum Q (nm -1 ) for DF-DNDs (black squares) and S-DNDs (green circles). To compare only the effect of particle sizes, and not powder density, both curves are normalized to the sample density of 1 g/cm3 . SANS characterization was performed with YuMO, D11, and NGB30 instruments, and their neutron wavelengths and ranges of transferred momenta (Q) were equal to 0.7-5.0 Å and 7×10 -2 nm -1 < Q < 10 1 nm -1 ; 6 Å and 10 -2 nm -1 < Q < 10 0 nm -1 ; and 6 Å and 3.4×10 -2 nm -1 < Q < 1.2×10 0 nm -1 , respectively. The Q-ranges used for the summary plot for the S-DNDs are as follows: 1.55×10 -2 nm -1 -8.53×10 -1 nm -1 from D11 and 8.53×10 -1 nm -1 -1.10×10 0 nm -1 from YuMO; for the DF-DNDs, they are as follows: 9.03×10 -3 nm -1 -1.55×10 -2 nm -1 and 7.57×10 -2 nm -1 -9.85×10 -2 nm -1 from NGB30, 1.55×10 -2 nm -1 -7.57×10 -2 nm -1 from D11, and 9.85×10 -1 nm -1 -1.47×10 0 nm -1 from YuMO.

Figure 4 .

 4 Figure 4. Intensity I (nm -1) of neutron scattering versus transferred momentum Q (nm -1 ) for DF-DNDs (black squares) and S-DNDs (green circles). To compare only the effect of particle sizes, and not powder density, both curves are normalized to the sample density of 1 g/cm3 . SANS characterization was performed with YuMO, D11, and NGB30 instruments, and their neutron wavelengths and ranges of transferred momenta (Q) were equal to 0.7-5.0 Å and 7 × 10 -2 nm -1 < Q < 10 1 nm -1 ; 6 Å and 10 -2 nm -1 < Q < 10 0 nm -1 ; and 6 Å and 3.4 × 10 -2 nm -1 < Q < 1.2 × 10 0 nm -1 , respectively. The Q-ranges used for the summary plot for the S-DNDs are as follows: 1.55 × 10 -2 nm -1 -8.53 × 10 -1 nm -1 from D11 and 8.53 × 10 -1 nm -1 -1.10 × 10 0 nm -1 from YuMO; for the DF-DNDs, they are as follows: 9.03 × 10 -3 nm -1 -1.55 × 10 -2 nm -1 and 7.57 × 10 -2 nm -1 -9.85 × 10 -2 nm -1 from NGB30, 1.55 × 10 -2 nm -1 -7.57 × 10 -2 nm -1 from D11, and 9.85 × 10 -1 nm -1 -1.47 × 10 0 nm -1 from YuMO.

Nanomaterials 2021 ,

 2021 11, x FOR PEER REVIEW 9 of 17

Figure 5 .

 5 Figure 5. Size distribution of scatterers evaluated using MDDNS for S-DNDs (green circles) and DF-DNDs (black squares). Points correspond to the results of calculation.

Figure 5 .

 5 Figure 5. Size distribution of scatterers evaluated using MDDNS for S-DNDs (green circles) and DF-DNDs (black squares). Points correspond to the results of calculation.

Figure 6 .

 6 Figure 6. Neutron albedo for the flat semi-infinite media of DF-DNDs (dashed lines) and S-DNDs (solid lines) versus neutron velocity and wavelength.Figure 6. Neutron albedo for the flat semi-infinite media of DF-DNDs (dashed lines) and S-DNDs (solid lines) versus neutron velocity and wavelength.

Figure 6 .

 6 Figure 6. Neutron albedo for the flat semi-infinite media of DF-DNDs (dashed lines) and S-DNDs (solid lines) versus neutron velocity and wavelength.Figure 6. Neutron albedo for the flat semi-infinite media of DF-DNDs (dashed lines) and S-DNDs (solid lines) versus neutron velocity and wavelength.

Figure 7 .

 7 Figure 7. Ratio of loss factors η_(DF-DNDs)/η_(S-DNDs) at reflection from flat semi-infinite media of DF-DNDs and S-DNDs as a function of neutron velocity. The respective neutron albedo is shown in Figure 6.

Figure 7 .

 7 Figure 7. Ratio of loss factors η_(DF-DNDs)/η_(S-DNDs) at reflection from flat semi-infinite media of DF-DNDs and S-DNDs as a function of neutron velocity. The respective neutron albedo is shown in Figure 6.

Figure 8 .

 8 Figure 8. Neutron albedo (black lines) and transmission (red lines) for a flat layer of DF-DNDs (dashed lines) and S-DNDs (solid lines) versus neutron velocity [m/s]. (a)The layer thickness is 3 cm and the powder density is 0.56 g/cm 3 for both samples; (b) the layer thickness is 3 cm and the powder density is 0.56 g/cm 3 for DF-DNDs, and it is 0.67 g/cm 3 for S-DNDs.

Figure 8 .Figure 9 .

 89 Figure 8. Neutron albedo (black lines) and transmission (red lines) for a flat layer of DF-DNDs (dashed lines) and S-DNDs (solid lines) versus neutron velocity [m/s]. (a) The layer thickness is 3 cm and the powder density is 0.56 g/cm 3 for both samples; (b) the layer thickness is 3 cm and the powder density is 0.56 g/cm 3 for DF-DNDs, and it is 0.67 g/cm 3 for S-DNDs. Nanomaterials 2021, 11, x FOR PEER REVIEW 13 of 17

Figure 9 .

 9 Figure 9. Neutron albedo for the velocities of 50 m/s (black lines), 100 m/s (red lines), and 150 m/s (blue lines) for S-DNDs (solid lines) and DF-DNDs (dashed lines) versus the cavity radius. (a) A powder layer of infinite thickness; the density is 0.56 g/cm 3 for both samples. (b) A powder layer of infinite thickness; the density is 0.56 g/cm 3 for DF-DNDs, and it is 0.67 g/cm 3 for S-DNDs. (c) A powder layer with the thickness of 3 cm; the density is 0.56 g/cm 3 for both samples. (d) A powder layer with the thickness of 3 cm; the density is 0.56 g/cm 3 for DF-DNDs, and it is 0.67 g/cm 3 for S-DNDs.

Nanomaterials 2021 , 17 Figure 10 .

 20211710 Figure 10. Ratio of loss factors η/η at reflection from the wall of a spherical cavity with a radius of 5 cm and a wall thickness of 3 cm for realistic powder densities of 0.56 g/cm3 for DF-DNDs and 0.67 g/cm 3 for S-DNDs.

Figure 10 .

 10 Figure 10. Ratio of loss factors η DF-DNDs /η S-DNDs at reflection from the wall of a spherical cavity with a radius of 5 cm and a wall thickness of 3 cm for realistic powder densities of 0.56 g/cm 3 for DF-DNDs and 0.67 g/cm 3 for S-DNDs.

Patents

The algorithm for extracting a model-independent size distribution of scatterers from small-angle scattering data was developed to obtain the results reported in this manuscript. It is protected by the author's certificate of state registration of the software "Structural Nanopowders Analyzer Based on Small-Angle Scattering Data (SNASAS)" RU 2020662675, issued by the Federal Service for Intellectual Property.

Data Availability Statement: Neutron data were obtained in experiments at ILL, Grenoble, France: doi:10.5291/ILL-DATA.3-07-386; doi:10.5291/ILL-DATA.3-07-361.

Acknowledgments:

The authors are grateful to M. Mikhaylov (NRC "Kurchatov Institute"-CRISM "Prometey", St. Petersburg) for producing the TEM images of S-DND and DF-DND samples; to Oleksandr Ivankov for his help in measuring and processing the YuMO data; to C. Gagnon and CHRNS at NIST, E. Yudina (Ioffe Institute), for X-ray characterization; and to V.A. Artem'ev for useful discussions. This work benefited from the use of the SASView (http://www.sasview.org/download/, accessed on 1 October 2021), originally developed under NSF award DMR-0520547. SasView contains code developed with funding from European Union's Horizon 2020 research and innovation program under the SINE2020 project, grant agreement No. 654000. This work was primarily supported by NSF PREM in collaboration with the Center for High Resolution Neutron Scattering (CHRNS) through the award DMR 1827731.

Conflicts of Interest:

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.


Patrice Abry
email: patrice.abry@ens-lyon.fr

Yannick Malevergne
email: yannick.malevergne@univ-paris1.fr

Herwig Wendt
email: herwig.wendt@irit.fr

Stéphane Jaffard
email: stephane.jaffard@u-pec.fr

Marc Senneret
email: msenneret@vivienne-im.com

Laurent Jaffrès
email: ljaffres@vivienne-im.com

Foreign Exchange Multivariate Multifractal Analysis

Keywords: Financial times series, Foreign exchange, basket properties, multivariate multifractal analysis, wavelet leaders

After Mandelbrot's seminal work, scale-free and multifractal temporal dynamics have been recognized as classical stylized facts for financial time series and massively documented. Multifractal analysis in finance has however mainly remained univariate (one time series at a time) when multivariate (or basket) properties are critical for financial applications. This is mostly due to a lack of theoretical foundations and practical tools for multivariate multifractal analysis. Expanding on a theoretically-grounded recently proposed multivariate multifractal formalism, the present work performs an original multivariate analysis for a basket of six Foreign Exchange rate time series. Beyond confirming multifractality for each component independently, the definition of cross-multifractalities amongst components is introduced, assessing cross-dependencies in temporal dynamics not already accounted for by cross-correlations.

The key practical outcome is to show that, essentially, one same multifractal time governs jointly the temporal dynamics of all the Foreign Exchange time series studied here.

I. INTRODUCTION

Context. It has been well-documented that financial time series are characterized by a set of stylized facts [START_REF] Cont | Empirical properties of asset returns: stylized facts and statistical issues[END_REF]. Two notable such facts consist of the absence of temporal autocorrelation (a consequence of the efficient market hypothesis) and the scale-free nature of temporal dynamics (all time scales, or frequencies, drive temporal dynamics). This lead Mandelbrot to propose multifractal fluctuations to model the temporal dynamics of asset prices [START_REF] Mandelbrot | Fractals and scaling in finance, ser. Selected Works of Benoit B. Mandelbrot[END_REF], [START_REF] Calvet | The multifractal model of asset returns[END_REF]. This triggered massive research works on financial time series [START_REF] Arneodo | Direct causal cascade in the stock market[END_REF]- [START_REF] Segnon | Multifractal models in finance: Their origin, properties, and applications[END_REF]. However, multifractal analysis remained essentially univariate, one time series studied at a time, while the multivariate, or joint, or basket, properties of financial time series are critical for financial applications [START_REF] Bacry | Continuous cascade models for asset returns[END_REF], [START_REF] Sornette | Volatility fingerprints of large shocks: endogenous versus exogenous[END_REF]. Assessing cross-multifractalities in a basket of assets is thus the crucial issue addressed here. Related work. In [START_REF] Mandelbrot | A multifractal walk down Wall Street[END_REF], Mandelbrot proposed to modify the Brownian motion B(t) classically used to model financial time series as random walks, via a time warping B(A(t)), where A denotes a multifractal cascading process. This added a key intuition to multifractality: that of an internal clock A(t) that fluctuates around an average smooth evolution A(t) t to model time accelerations (and thus shocks) or decelerations in financial markets. Variations of multifractal models for asset prices where further proposed, such as the Markovswitching multifractal model [START_REF] Lux | The markov-switching multifractal model of asset returns: Gmm estimation and linear forecasting of volatility[END_REF] or the multifractal random walk [START_REF] Bacry | Multifractal random walk[END_REF]. Empirically, multifractality has been evidenced in numerous financial time series different in nature, e.g., [START_REF] Fisher | Multifractality of Deutsche Mark / US dollar exchange rate[END_REF] using the MultiFractal-Detrended Flutuation Analysis [START_REF] Kantelhardt | Multifractal detrended fluctuation analysis of nonstationary time series[END_REF] or the Wavelet Transform Modulus Maxima [START_REF] Muzy | Multifractal formalism for fractal signals: The structure-function approach versus the wavelet-transform modulus-maxima method[END_REF] frameworks. Often, multifractality has been connected with long memory in volatility, while it was shown to be unrelated to the signs of the returns [START_REF] Sornette | Volatility fingerprints of large shocks: endogenous versus exogenous[END_REF], [START_REF] Lux | The markov-switching multifractal model of asset returns: Gmm estimation and linear forecasting of volatility[END_REF], [START_REF] Calvet | Multifractal volatility: theory, forecasting, and pricing[END_REF], [START_REF] Abry | Shuffling for understanding multifractality, application to asset price time series[END_REF]. Except for rare cases where multifractal models were modified to handle several time series jointly [START_REF] Lux | Higher dimensional multifractal processes: A GMM approach[END_REF], multifractal analysis was performed in univariate settings only, in finance as well as in most applications. This is mostly due to a lack of theoretical foundations and practical tools for multivariate multifractal analysis. However, recently a novel formalism based on wavelet leaders has been devised and shown to reach state of the art univariate analysis performance [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF]- [START_REF] Leonarduzzi | p-exponent and p-leaders, Part II: Multifractal analysis. Relations to Detrended Fluctuation Analysis[END_REF]. Moreover, in contradistinction with earlier multifractal formalisms, it permits a natural theoretically wellgrounded formulation of multivariate multifractal analysis [START_REF] Leonarduzzi | Multifractal characterization for bivariate data[END_REF]- [START_REF] Jaffard | Multifractal formalisms for multivariate analysis[END_REF]. This thus permits to envisage the assessments of cross-multifractalities amongst financial time series which, to the best of our knowledge, has not yet been performed. Goals, contributions and outline. The present contribution aims to perform and show the interest of the joint or multivariate multifractal analysis for the exchange rates of six couples of major currencies. Elaborating on [START_REF] Jaffard | Multivariate multifractal analysis[END_REF], [START_REF] Jaffard | Multifractal formalisms for multivariate analysis[END_REF], Section II introduces the theory and practice of multivariate multifractal analysis based on p-leader multiscale quantities, a recent extension of the wavelet leader formalism introduced in [START_REF] Jaffard | p-exponent and p-leaders, Part I: Negative pointwise regularity[END_REF], [START_REF] Leonarduzzi | p-exponent and p-leaders, Part II: Multifractal analysis. Relations to Detrended Fluctuation Analysis[END_REF] in univariate settings. Section II further provides understanding of the meaning and importance of the cross-multifractalities parameter and of multifractal crosscorrelation. The Foreign exchange rates data used in this work are described in Section III. Their multifractalities and cross-multifractalities are assessed, quantified and discussed in Section IV with the major findings that, essentially, one same and unique clock governs jointly the temporal dynamics of all the currencies under consideration and possibly of the entire Foreign Exchange market.

A. Multivariate multifractal Analysis: Theory

Local regularity. Multifractal analysis aims to quantify the fluctuations in time of the local regularity in a time series X(t) [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF]. Local regularity has classically been measured using the Hölder exponent [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF]. It has however been recently proposed that novel measures based on the notion of pexponents (p > 0) could be used, offering a wider versatility to analyze real world data and benefiting from more practical robustness. These p-exponents, referred to as h(t) ≥ 0, are used here. Their precise definition can be found in [START_REF] Jaffard | p-exponent and p-leaders, Part I: Negative pointwise regularity[END_REF], [START_REF] Leonarduzzi | p-exponent and p-leaders, Part II: Multifractal analysis. Relations to Detrended Fluctuation Analysis[END_REF], where they are thoroughly studied. As for the Hölder exponent, the smaller h(t), the more irregular X is around t. Multivariate multifractal spectrum. Though based on a local regularity measurement, multifractal analysis provides a global and geometrical information on how local regularities vary along time and, in a multivariate setting, across components. Let h(t) {h 1 (t), . . . , h M (t)} denote the values of the pexponents taken at time t by each of the M-variate times series X = (X 1 , . . . , X M ). Inspired from [START_REF] Meneveau | Joint multifractal measures -theory and applications to turbulence[END_REF] in a bivariate setting, it has been proposed in [START_REF] Leonarduzzi | Multifractal characterization for bivariate data[END_REF]- [START_REF] Jaffard | Multifractal formalisms for multivariate analysis[END_REF] to define the multivariate multifractal spectrum D of X as the collection of Hausdorff dimensions dim H of the sets of points t ∈ R at which h(t) takes on the values h:

D(h) dim H {t ∈ R + : {h(t)} = {h}} . (1)

B. Wavelet p-leader multivariate multifractal formalism

The estimation of the multivariate multifractal spectrum D(h) is performed via the so-called multifractal formalisms. They are often based on multiscale quantities, constructed from wavelet analysis [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF], [START_REF] Wendt | Bootstrap for empirical multifractal analysis[END_REF]. Let ψ denote an oscillating reference pattern, the mother wavelet, characterized by its number of vanishing moments N ψ , a positive integer defined as ψ ∈ C N ψ -1 and ∀n = 0, . . . , N ψ -1, R t k ψ(t)dt ≡ 0 and R t N ψ ψ(t)dt = 0. The mother wavelet is designed such that dilated and translated templates {ψ j,k (t) = 2 -j/2 ψ(2 -j tk)} (j,k)∈Z 2 form an orthonormal basis of L 2 (R) [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]. The discrete wavelet transform coefficients d X (j, k) of X are defined as d X (j, k) = 2 -j/2 ψ j,k |X .

The wavelet p-leaders of X, X (j, k) are defined as a local L p -norm (p > 0), taken across all d X (j , k ) located in a neighbourhood of t = 2 j k at finer scales j ≤ j. Interested readers are referred to [START_REF] Jaffard | p-exponent and p-leaders, Part I: Negative pointwise regularity[END_REF], [START_REF] Leonarduzzi | p-exponent and p-leaders, Part II: Multifractal analysis. Relations to Detrended Fluctuation Analysis[END_REF] for detailed definitions.

It can be shown that wavelet p-leaders reproduce the pexponents in the limit of fine scales, L X (j, k) ∼ C2 jh(t) as 2 j → 0 for t = 2 j k. Consequently, with q {q 1 , . . . , q M },

1 n j nj k=1 Π M m=1 L Xm (j, k) qm ∼ c q 2 jζ(q) , 2 j → 0. (2)
The so-called scaling exponents ζ(q) in ( 2) are tightly related to D(h) via their Legendre transform, defining the multivariate Legendre spectrum as: which provides an estimate for D(h) for large classes of processes, see [START_REF] Jaffard | Multivariate multifractal analysis[END_REF], [START_REF] Jaffard | Multifractal formalisms for multivariate analysis[END_REF].

L(h) = inf q (1 + q, h -ζ(q)), (3)
Its precise shape thus conveys information regarding the joint or cross fluctuations of local regularities amongst the components of X. For example, in a bivariate setting, a multifractal spectrum L(h 1 , h 2 ) that would stem directly from the univariate spectra of each time series as L 1 (h 1 ) + L 2 (h 2 ) -1 would indicate that multifractalities amongst components are not related. To the converse, a bivariate spectrum whose support would collapse on a line would indicate a total dependence between the multifractalities of each component.

C. Multivariate multifractal analysis: Practice

To measure L(h), elaborating on [START_REF] Castaing | Log-similarity for turbulent flows[END_REF], one can use the multivariate cumulants of order s > 0 of the log-leaders [START_REF] Jaffard | Multifractal formalisms for multivariate analysis[END_REF]. For large classes of multivariate multifractal processes, C s (j) behave as:

(ln L X1 (j, k), . . . , ln L X M (j, k)) at scale 2 j , denoted C s (j) with s = {s 1 , . . . s M }, ∀m, s m ∈ {0, . . . , M } and s = m s m [23]-
C s (j) = c 0 s + j c s ln 2, s ≥ 1 (4)
where the multifractal coefficients c s can be related to the ζ(q)

ζ(q) = s c s Π M m=1 q sm m /s m ! , (5)
and hence to the multivariate Legendre spectrum L(h).

The c s are estimated by means of linear regressions of C s (j) versus log 2 2 j = j.

III. FOREIGN EXCHANGE RATES

The data used here consist of Foreign Exchange rates for six couples of major currencies against the US Dollar (USD): AUD/USD (Australian Dollar), CAD/USD (canadian Dollar), JPY/USD (Japanese Yen), GBP/USD (British Pound), CHF/USD (Swiss Franc), EUR/USD (Euro), from 1999 to 2011. As the years after birth of the Euro shows remarkably stable statistical properties along time, a period of around 18 months, from December 2004 to May 2006 is studied here. Similar conclusions are drawn for the analysis of other 18 month-period, between 2002 and 2008. Data are sampled at an intraday high frequency, of T s = 5min, thus resulting in around 1e5 samples per time series. Rates and returns (increments) for these six couples of currencies and for the chosen period are shown in Fig. 1 For ease of notations, the couple XXX/USD will hereafter be referred as XXX, thus omitting /USD. Because we are interested in cross-dependencies, the bivariate analysis corresponding to the pair of couples XXX/USD and YYY/USD will thus simply be referred to as the pair XXX-YYY.

IV. MULTIVARIATE MULTIFRACTAL FOREIGN EXCHANGE

A. Second-order cross-statistics analysis: cross-selfsimilarity Prior to any practical multifractal analysis, the inspection of the joint second order statistics of the temporal dynamics of X are mandatory. To that end, the multivariate wavelet spectrum S(j) must be computed, defined as the collection of M × M matrix S(j) as functions of the analysis scales 2 j , whose entries consists of the cross correlations of the wavelet coefficients at scale

2 j : S m,m (j) = 1/n j nj k d Xm (j, k)d X m (j, k).
In the present work, the wavelet coefficients are computed using least asymmetric orthornormal wavelets, with N ψ = 3 [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]. For processes with scale-free temporal dynamics, such as multifractal processes, each entry of S(j) behave as a powerlaw with respect to the analysis scale:

S m,m (j) K m,m 2 2jH m,m (6)
The scaling exponents H m,m quantify the auto-and crosscorrelations. Notably, H m,m corresponds to the Hurst (or selfsimilarity) exponent of component m.

To further quantify cross-temporal dynamics, one usually use the wavelet-coherence function

ρ m,m (j) = S m,m (j)/ S m,m (j)S m ,m (j) (7)
that consists of scale-dependent correlation coefficients [START_REF] Whitcher | Wavelet analysis of covariance with application to atmospheric time series[END_REF]. Fig. 2(left) displays the log wavelet-spectrum log 2 S m,m (j) for each component as a function of the log of the analysis scales j = log 2 2 j . Linear behaviours in these log-log plots indicate, for all couples of currencies, scale-free dynamics ranging from j = 2 to j = 9, hence across 7 octaves. These scale-free dynamics are also observed for all pairs log 2 S m,m (j) across the same range of scales (cf. Fig. 2 (middle)). Given the sampling period of T s = 5min, this indicates that scale-free dynamics range remarkably from 10min to 100hours, which, in terms of trading time, means around 2 weeks, i.e., for more than two decades of time scales. Interestingly, these scale-free dynamics also imply that time scales of, e.g., one working day or one working week, that could be a priori considered as typical given the human nature of financial trading, can not be identified in the data. However, when estimated from linear regressions of log 2 S m,m (j) vs. j, none of the estimated scaling exponents H m,m are found significantly different1 from H = 0.5, see Table I (upper triangle). This clearly indicates the absence of any temporal autocorrelation for all components and of any temporal crosscorrelation amongst components, in clear agreement with the efficient market hypothesis. Fig. 2(right) shows that the wavelet coherence functions ρ m,m (j) for all pairs of components are essentially independent of scales, in accordance with the efficient market hypothesis. It moreover indicates that cross-correlations are significantly non-zero for all but the last analysis scale. This is quantified by computing the average across scales of ρ m,m (j), highly reminiscent of the pairwise Pearson correlation coefficient. These pairwise cross-correlations (Table I

B. Beyond second-order cross-statistics: cross-multifractality

Multifractal analysis is here performed with 2-leaders (p = 2), shown to yield the best estimation performance [START_REF] Jaffard | p-exponent and p-leaders, Part I: Negative pointwise regularity[END_REF], [START_REF] Leonarduzzi | p-exponent and p-leaders, Part II: Multifractal analysis. Relations to Detrended Fluctuation Analysis[END_REF].

Often, in practice, to perform multivariate multifractal analysis, use is made only of the cumulants of orders s = 1 and s = 2, thus restricting the M -variate analysis to a collection of bivariate (or pairwise) analyses. For ease of notations, the M cumulants of order 1 (hence univariate) are hereafter labelled C m (j) and the corresponding multifractal coefficients c s in Eq. 4 as c 1 (m). The M × (M + 1)/2 cross-second order cumulants (hence bivariate) and corresponding multifractal coefficients are written C m,m (j) and c 2 (m, m ).

Implicitly, this amounts to perform a parabolic approximation of the pairwise scaling exponents ζ m,m (q), which by Eq. 3 also implies a pairwise parabolic approximation of L m,m around its maximum [START_REF] Leonarduzzi | Multifractal characterization for bivariate data[END_REF]: Inspired from ρ m,m (j), this leads to define,

ζ m,m (q 1 , q 2 ) ≈ c 1 (m)q 1 + c 1 (m )q 2 + c 2 (m, m) 2 q 2 1 + c 2 (m , m ) 2 q 2 + c 2 (m, m )q 1 q 2 (8) L m,m (h 1 , h 2 ) ≈ 1 + c 2 (m, m)b 2 h 1 -c 1 (m) b 2 + c 2 (m , m )b 2 h 2 -c 1 (m ) b 2 - c 2 (m, m )b h 1 -c 1 (m) b h 2 -c 1 (m ) b , (9
ρ M F m,m (j) = C m,m (j) C m,m (j)C m ,m (j) , (10)
that quantifies a scale-free based higher order statistics dependence coefficient, and is hence hereafter referred to as multifractal coherence function, slightly abusively yet with an intuitive meaning. This further prompts for the use of the multifractal correlation coefficient defined as:

ρ M F = -c m,m / √ c m,m c m ,m .
Univariate multifractality. Let us first study the multifractal property of each couple of currencies independently (in a univariate setting). Fig. 3(left column) reports the M first order cumulants C m (j) (top left) and the M second order cumulants C m,m (j) (bottom left) computed independently for each component. These univariate cumulants show remarkable scaling across a broad range of octaves, consistent with that observed in Fig. 2. The C m (j) are superimposing for all components, and their corresponding slopes c 1 (m) are all found not to depart from 0.5 (see Table II), thus further confirming the absence of autocorrelation in the temporal dynamics of each component independently and the consistency with the efficient market hypothesis. However, the C m,m (j) also display remarkable scaling, with decreasing amplitude along scales, and corresponding slopes c 2 (m, m) all significantly negatively departing from 0 (see the diagonal entries of Table III). This is a clear evidence that each of these Foreign Exchange time series displays a clear and robust multifractality, over a broad range of time scales from 10min to above 2 weeks. This indicates that each of these time series possesses independently subtle temporal dynamics in the form of local or transient structures and burstiness, that cannot be captured by the classical correlation and second order statistics. Beyond being significant, the multifractality parameters c 2 (m, m) of all components are interestingly found to be of close values: -0. (see Table III Discussion. As mentioned in the introduction, after Mandelbrot, financial asset prices are often modelled as compound Brownian motions B(A(t)) where the time warping function A(t) entails the multifractal properties. The present analysis suggest that the collection of Foreign exchange rates studied here can be modelled by a multivariate extension of that model {B m (A m (t)), m = 1, . . . , M } [START_REF] Seuret | On multifractality and time subordination for continuous functions[END_REF]. The B m are correlated Brownian motions, with a cross correlation matrix showing significant non-zero entries, both positive or negative. They however do not possess any auto-or cross-correlation in their auto-and cross-temporal dynamics. The A m are multifractal processes, e.g., multiplicative cascades as proposed by Man-delbrot, each with multifractality controlled by c 2 (m, m). In addition, they also possess a cross sectional correlation, with correlation matrix ρ M F , characterized by positive only and significantly departing from 0 entries. This is the correlation amongst those time warping functions A m that induces cross multifractalities and hence transient dependencies in the crosstemporal dynamics of the B m (A m (t)) and thus amongst Foreign exchange times series.

Finally, returning to the original intuition of multifractality understood as a time warping, the fact that the entries of ρ M F are all positive and large, indicates that all time warping functions A m are very similar for all components. In financial terms, this can be interpreted as the fact that the currency rates studied here are governed by one unique and same clock across the world. This provides a different understanding of the Foreign Exchange market compared to the one suggested by classical second-order statistics with two groups of currencies.

To the best of our knowledge, this unique clock timing the the dynamics of the majors currencies had never been evidenced before, because a theoretically-grounded and practically efficient multivariate multifractal formalism was lacking.

In

Figure 1 .

 1 Figure 1. Foreign Exchange data. Left: Rates ; Right: Returns

Figure 2 .

 2 Figure 2. Scale-free dynamics. Log wavelet-spectrum log 2 Sm,m(j) (left), log 2 S m,m (j) (center), and ρ m,m (j) (right) as functions of the log of the analysis scales j = log 2 2 j and for the 6 time series in Fig. 1.

) where b c 2

 2 (m, m)c 2 (m , m ) -c 2 (m, m ) 2 ≥ 0. This shows that the position of the maximum of L m,m is given by (c m , c m ), and that c m,m and c m ,m quantify the widths of the fluctuations independently for each component. Finally, c m,m quantify the departure from an isotropic parabola, thus providing a characterization across components of the joint fluctuations of the regularities of each component.

Figure 3 .

 3 Figure 3. Foreign Exchange multivariate multifractal analysis. Bivariate multifractal spectrum for the (GBP, EUR) pair of currencies (top right). Cm(j) (top left), C m,m (j) (bottom left) and ρ M F m,m (j) (bottom right) as functions of the log of the analysis scales j and for the 6 time series in Fig. 1.

  lower triangle). In addition, Fig.3(central bottom plots) shows the M (M -1)/2 multifractal coherence functions ρ M F m,m , observed to be weakly dependent on scales (except at coarse scales where estimation is of poor quality because of observation finite duration) and all significantly positive. Further, the multifractal correlation coefficients ρ M F (m, m ) are found to all positively depart from 0, with values ranging from 0.47 ≤ ρ M F ≤ 0.95 and often large (see TableIII

  upper triangle). Fig. 3(right columns) finally illustrates for two pairs of couples of currencies (GBP-EUR) and (CHF-EUR) the estimated bivariate multifractal spectra L(h m , h m ). The fact that they both consist of a parabola stretched along the first diagonal, hence with large aspect ratios clearly departing from 1, provides intuition to the meaning of large values for ρ M F . These multifractal spectra clearly differ from a spectrum that would correspond to unrelated multifractalities L m (h m ) + L m (h m ) -1, where the univariate spectra of each component L m (h m ) and L m (h m ) are also shown. These plots and results clearly indicate that the crosstemporal dynamics of the exchange rates for all pairs of couples of currencies studied here show related scale-free dynamics, with dependencies that cannot be measured by crosscorrelation or cross-Fourier spectrum, but are evidenced and quantified by the cross-multifractality parameters c 2 (m, m ) and ρ M F (m, m ). Positive ρ M F (m, m ) indicate that the occurrences of burstiness or of local transient structures are colocalized in time between components.

  upper triangle) identify two groups of currencies, with positive intragroup correlations, and negative inter-group correlations. The first group gathers EUR, GBP and AUD, with strong intragroup correlations ; the second group (JPY, CHF, CAD) shows less intra-group correlations. Correlation and selfsimilarity analysis. ESTIMATED SCALING EXPONENTS H m,m (DIAGONAL AND LOWER TRIANGLE, BLUE). AVERAGE ACROSS SCALES COHERENCE FUNCTION (UPPER TRIANGLE, RED).

		AUD	CAD	JPY	GBP	CHF	EUR
	AUD	0.48	-0.45	-0.50	0.61	-0.61	0.62
	CAD	0.49	0.47	0.26	-0.36	0.40	-0.44
	JPY	0.49	0.48	0.48	-0.56	0.59	-0.59
	GBP	0.50	0.49	0.49	0.49	-0.76	0.76
	CHF	0.50	0.49	0.49	0.50	0.49	-0.93
	EUR	0.50	0.49	0.48	0.50	0.50	0.50
				Table I

  an effort toward reproducible research and open science, Matlab codes implementing multivariate multifractal analysis are made publicly available 2 . Location of the maximum of the multifractal spectrum: c 1 (m). Multivariate multifractal analysis and multifractal parameters. MULTIFRACTALITY PARAMETERS c 2 (m, m ) (DIAGONAL AND LOWER TRIANGLE, BLUE). MULTIFRACTAL CORRELATION COEFFICIENTS ρ M F (m, m ) (UPPER TRIANGLE, RED).

			AUD	CAD	JPY	GBP	CHF	EUR
	c 1 (m)	0.48	0.49	0.49	0.51	0.52	0.52
					Table II
	ρ M F		AUD	CAD	JPY	GBP	CHF	EUR
	AUD	-0.024	0.47	0.70	0.61	0.64	0.68
	CAD	-0.014 -0.039	0.59	0.66	0.67	0.68
	JPY	-0.017 -0.018 -0.024	0.71	0.84	0.95
	GBP	-0.018 -0.025 -0.021 -0.038	0.78	0.86
	CHF	-0.020 -0.026 -0.025 -0.030 -0.038	0.92
	EUR	-0.020 -0.026 -0.028 -0.032 -0.035 -0.037
					Table III

Throughout this work, statistical significance has been assessed by means of wavelet domain block bootstrap procedures, as devised in[START_REF] Wendt | Bootstrap for empirical multifractal analysis[END_REF] and here extended to multivariate analysis; p-values are not reported for space reasons.