domain: chim phys spi

text:
Sofiane Mahtout
email: mahtout_sofiane@yahoo.fr

Chaouki Siouani

Franck Rabilloud
email: franck.rabilloud@univ-lyon1.fr

Growth Behavior and Electronic Structure of Noble Metal-Doped Germanium Clusters

Structures, energetics and electronic properties of noble metal-doped germanium (MGen with M = Cu, Ag, Au; n=1-19) clusters are systematically investigated by using the density-functional theory (DFT) approach. The endohedral structures in which the metal atom is encapsulated inside a germanium cage appear at n = 10 when the dopant is Cu, and n = 12 for M = Ag and Au. While Cu doping enhances the stability of the corresponding germanium frame, the binding energies of AgGen and AuGen are always lower than those of pure germanium clusters. Our results highlight the great stability of the CuGe10 cluster in a D4d structure, and to a lesser extent that of AgGe15 and AuGe15 which exhibits a hollow cage-like geometry. The sphere-type geometries obtained for n = 10-15 present a peculiar electronic structure in which the valence electrons of the noble metal and Ge atoms are delocalized and exhibit a shell structure associated with the quasi-spherical geometry. It is found that the coinage metal is able to give both s-and d-type electrons to be reorganized together with the valence electrons of Ge atoms through a pooling of electrons. The cluster size dependence of the stability, the frontier orbital energy gap, the vertical ionization potentials, and electron affinities are given.

Introduction

Germanium is an important element in the field of semiconductor materials and the microelectronic industry. Pure and metal-doped germanium clusters have been intensively studied during the last decades in both theoretical and experimental fields, as this class of materials which exhibits size and shape dependent properties provides a rich source to discover new nanostructures with new physical and chemical properties. Their structure and specific properties have been the object of intense researches in order to better understand the stability and growth patterns and to explore the possibility of applications in nanotechnology. Doping Gen clusters with a metal atom is expected to significantly change the chemical and physical properties and modify the framework. For example, the dopant atom may stabilize germanium cage structures and induced a large electronic gap.

A large number of investigations on a transition metal encapsulating Gen clusters have been carried out . Zhao and Wang [START_REF] Zhao | Geometries, stabilities, and electronic properties of FeGen (n = 9-16) clusters: Density-functional theory investigations[END_REF] have systematically investigated geometries, stabilities, and electronic properties of FeGen (n = 9-16) clusters. They found that the doping with one Fe atom contributes to enhance the stability of the germanium clusters and the electron charges always transfer from Fe atom to the neighboring Ge atoms. In the case of MnGen (n = 2-16) clusters, the same authors [START_REF] Zhao | Geometries, stabilities, and magnetic properties of MnGen (n = 2-16) clusters: Density-functional theory investigations[END_REF] shown that the dopant Mn atom contributes to increase the stabilities of the germanium clusters while the magnetic moment that mainly originates from the unpaired electrons of the 3d state was not quenched when Mn is embedded in Gen clusters. Lu and Nagase [START_REF] Lu | Metal-doped germanium clusters MGen at the sizes of n = 12 and 10: Divergence of growth patterns from the MSin clusters[END_REF] have showed that MGe12 (M =W, Os, Zn) exhibits a higher chemical stability among MGen with M= Hf, W, Os, Ni, and Zn, while Ni doped Gen clusters have higher stability as compared to Mn and Co doped Gen clusters 4 . For Chromium doped germanium clusters, there are conflicting results. A study has concluded than the Cr atom tends to be located at exohedral positions and do not show tendency to fall into the center of Ge outer frame [START_REF] Kapila | First principle investigation into structural growth and magnetic properties in GenCr clusters for n=1-13[END_REF] , while a more recent investigation suggested than CrGe10 and CrGe14 might be endohedral structures resulting in a quenching of the Cr magnetic moment [START_REF] Dhaka | Study of the electronic structure, stability and magnetic quenching of CrGen (n=1-17) clusters: a density functional investigation[END_REF] . However large magnetic moments are found in small CrGen 0,+ (n= 1-5) clusters [START_REF] Hou | Chromium-doped germanium clusters CrGen (n = 1-5): geometry, electronic structure, and topology of chemical bonding[END_REF] . Deng et al. [START_REF] Deng | Photoelectron spectroscopy and density functional calculations of VGen -(n = 3-12) Clusters[END_REF] have investigated the structural, electronic and magnetic properties of VGen -/0 (n =3-12) clusters. They found an electron transfer from Ge atoms to the V atom and related the minimization of the magnetic moments to the structural pattern. In the case of TM@Gen (TM = Ti, Zr and Hf; n = 1-20) clusters, Kumar et al. [START_REF] Kumar | Architecture, electronic structure and stability of TM@Gen (TM = Ti, Zr and Hf; n = 1-20) clusters: a density functional modeling[END_REF] found that the clusters having 20 valence electrons turn out to be relatively more stable in both the neutral and the anionic series. Also, the TM atom is adsorbed endohedrally in the pure germanium cage for n > 9 and the spin on the TM atom is quenched for clusters with n > 7. Similarly, Bandyopadhyay and Sen [START_REF] Bandyopadhyay | Density functional investigation of structure and stability of Gen and GenNi (n = 1-20) clusters: validity of the electron counting rule[END_REF] have showed that NiGen clusters having 20 valence electrons are relatively more stable in both neutral and cationic species. Wang and Han [START_REF] Wang | A computational investigation of copper-doped germanium and germanium clusters by the density-functional theory[END_REF][START_REF] Wang | A Theoretical Study on Growth Patterns of Ni-Doped Germanium Clusters[END_REF][START_REF] Wang | Geometries and electronic properties of the tungsten-doped germanium clusters: WGen (n = 1-17)[END_REF][START_REF] Wang | stabilities, and vibrational properties of bimetallic Mo2doped Gen (n = 9-15) clusters: a density functional investigation[END_REF] investigated CuGen (n=2-13), NiGen (n=1-13), WGen (n=1-17) and Mo2Gen (n= 9-15) clusters. They show that the doping with Cu atom decreases the binding energies while the doping with Ni or W strengthens the stability of germanium clusters. Studying the properties of neutral and cationic MoGen (n=1-20) clusters, Trivedi et al. [START_REF] Trivedi | Study of electronic properties, stabilities and magnetic quenching of molybdenum-doped germanium clusters: a density functional investigation[END_REF] have showed that clusters containing more than nine germanium atoms are able to absorb a Mo atom endohedrally into a germanium cage and the spin magnetic moment of the Mo atom is quenched in all Mo-doped Gen clusters. The most stable structures of Mo2Gen (n= 9-13) clusters have one Mo atom inside a Gen cage and another Mo atom on the surface at smaller cluster sizes while the Mo2 dimer is completely encapsulated into the germanium clusters when the cluster size increases to 15 [START_REF] Wang | stabilities, and vibrational properties of bimetallic Mo2doped Gen (n = 9-15) clusters: a density functional investigation[END_REF] . For CoGen clusters, the magnetic moment is not quenched, the doping of the Co atom generally enhances the stability of the host Gen clusters and the Co atom is completely falls into the center of the Ge outer frame from n= 9 16- 17 . Singh et al. [START_REF] Kumar Singh | Thorium encapsulated caged clusters of germanium: Th@Gen, n =16, 18, and 20[END_REF] have reported from ab initio calculations that thorium encapsulation can be used to stabilize highly symmetric cages of germanium with 16 and 20 atoms. Bi2Gen-2 (n= 3-8, 12) clusters [START_REF] Zdetsis | Silicon-bismuth and germanium-bismuth clusters of high stability[END_REF] have been found to be characterized by high stability and symmetry and relatively large highest occupied-lowest unoccupied molecular orbital energy gaps (HOMO-LUMO gaps). ZrGen clusters [START_REF] Jaiswal | Growth behavior and electronic structure of neutral and anion ZrGen (n = 1-21) clusters[END_REF] have been predicted to stabilize cage-like structures for n ≥ 13. Structural and electronic properties of Ruthenium doped germanium clusters have been performed using a particle swarm optimization metaheuristic coupled with density functional theory computations by Jin et al [START_REF] Jin | Structural and electronic properties of ruthenium-doped germanium clusters[END_REF] . They found that there is an inherent tendency of the formation of an endohedral cage of RuGen clusters with increasing stability as the size increase. More recently, we found that the vanadium atom in VGen can largely contribute to strengthen the stability of the framework from n = 7, while the cage-like VGe14 cluster with a high symmetry and a high stability exhibits a peculiar electronic structure in which the valence electrons of V and Ge atoms are delocalized and exhibit a shell structure associated with the quasi-spherical geometry. This cluster is proposed to be a good candidate to be used as the building blocks for developing new materials [START_REF] Siouani | Structure, Stability, and electronic and magnetic properties of VGen (n = 1-19) clusters[END_REF] . In experiment, the anion photoelectron spectroscopy has been performed for some germanium clusters containing transition or lanthanide-metal atom [START_REF] Atobe | Anion photoelectron spectroscopy of germanium and tin clusters containing a transition-or lanthanide-metal atom; MGen -(n=8-20) and MSnn -(n=15-17)[END_REF] .

Although the doping of Gen with a transition metal atom have been intensively studied, there is no systematic investigation on the doping by a noble metal atom in literature to the best of our knowledge. However some recent studies have considered small clusters in a limited size range. Li et al. [START_REF] Li | Size-selective effects in the geometry and electronic property of bimetallic Au-Ge nanoclusters[END_REF] have studied the size-selective effects of geometrical structures and electronic properties on neutral and anionic AuGen 0/-(n = 1-13) clusters. Their results indicate that the addition of an extra electron to neutral cluster may enhance its kinetic stability. Recently, the photoelectron spectroscopy of small AuGen -clusters revealed an electronic structure and a very high detachment energy for AuGe12 -compatible with a highsymmetry geometry like Ih structure [START_REF] Lu | Transition from exohedral to endohedral structures of AuGen (n = 2-12) clusters: photoelectron spectroscopy and ab initio calculations[END_REF] . Bandyopadhyay [START_REF] Bandyopadhyay | Architectures, electronic structures, and stabilities of Cu-doped Gen clusters: density functional modeling[END_REF] has reported endohedral structures for CuGen clusters with n > 7, while the doping of Ge9 and Ge10 with several types of metal atoms including noble metals have been also considered [START_REF] Tai | Enhanced stability by three-dimensional aromaticity of endohedrally doped clusters X10M 0/with X = Ge, Sn, Pb and M = Cu, Ag[END_REF][START_REF] Tai | The group 14 cationic clusters by encapsulation of coinage metals X10M + , with X = Ge, Sn, Pb and M = Cu, Ag, Au: Enhanced stability of 40 valence electron systems[END_REF][START_REF] Qin | Structures and stability of metal-doped GenM (n = 9, 10) clusters[END_REF] . Interestingly, the global minimum of CuGe10 + clusters were presented to be a magic cluster with D4d symmetry and 40 valence electrons that satisfy the jellium electron shell model [START_REF] Brack | The physics of simple metal clusters: self-consistent jellium model and semiclassical approaches[END_REF] . The authors also proposed a concept of doubly spherical aromatic based on a counting rule for both pi-type and sigma-type valence electrons [START_REF] Tai | The group 14 cationic clusters by encapsulation of coinage metals X10M + , with X = Ge, Sn, Pb and M = Cu, Ag, Au: Enhanced stability of 40 valence electron systems[END_REF] .

Here we present a systematic investigation of the geometry, relative stabilities, and electronic properties of coinage metal germanium doped clusters MGen (M=Cu, Ag, Au; n = 1 -19) in the framework of the density-functional theory (DFT). We aim to highlight the modifications in the properties of Gen clusters due to the doping atom, and to study the role of the nature of the dopant coinage metal atom. It is hoped that this work would be useful to understand the influence of different noble metal on the properties of germanium clusters and could provide powerful guidelines for future experimental studies. The present paper is organized as follows: In Section 2, we describe our theoretical approach. The structural and energetic properties of MGen clusters (n = 1-19) clusters are presented sections 3, following by a discussion about the electronic properties.

Computational methodology

Spin polarized calculations are carried out by generalized gradient approximation (GGA) with PBE exchange-correlation functional [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] . Self-consistent field procedures are carried out with a convergence criterion of 10 -4 a.u. on the energy and electron density. Calculations have been performed by using the method of norm-conserving Troullier-Martins nonlocal pseudo-potentials [START_REF] Troullier | Efficient pseudopotentials for plane-wave calculations[END_REF] in the Kleinman-Bylander projectors [START_REF] Kleinman | Efficacious form for model pseudopotentials[END_REF] as implemented in SIESTA packages [START_REF] Ordejón | Self-consistent order-N density-functional calculations for very large systems[END_REF] . We used a double zeta basis plus polarization orbitals (DZP) for dopant atoms species (Cu, Ag, Au) and double zeta basis (DZ) for Ge atoms. In order to avoid interaction between neighboring clusters, a large cubic cell of 40 Å edge lengths with a periodic boundary condition is taken. The k grid integration has been carried using the Γ point approximation. The atomic positions have been optimized by the conjugate gradients dynamics algorithm until the residual forces are smaller than 10 -3 eV/Å, and without any symmetry constraints. Further analysis of the electronic properties and molecular orbitals have been performed with the software Gaussian09 38 using PBE and the Gaussian-type basis sets cc-pvtz for Ge and LanL2DZ for coinage metal atoms.

The number of geometric isomers or local minima increases exponentially with the cluster size. For this reason, lots of putative isomers including some high and low symmetries were relaxed for each cluster size. Amongst them, some initial structures of Gen+1 and metaldoped Gen clusters were taken from literature [START_REF] Kapila | First principle investigation into structural growth and magnetic properties in GenCr clusters for n=1-13[END_REF][START_REF] Bandyopadhyay | Density functional investigation of structure and stability of Gen and GenNi (n = 1-20) clusters: validity of the electron counting rule[END_REF][START_REF] Wang | A computational investigation of copper-doped germanium and germanium clusters by the density-functional theory[END_REF][START_REF] Siouani | Structure, Stability, and electronic and magnetic properties of VGen (n = 1-19) clusters[END_REF][START_REF] Shi | A computational investigation of aluminum-doped germanium clusters by density functional theory study[END_REF][START_REF] Deutsch | Binding energies of germanium clusters Gen (n = 2-5)[END_REF] . Also, the putative structures of MGen were obtained by local relaxation after the substitution of one Ge atom by M atom in several isomers of the original pure Gen+1 cluster. The different initial positions of the Ge atom in the Gen+1 clusters lead to different MGen isomers. But the search for the lowest isomer cannot include a global optimization procedure of the potential energy surface, and we cannot be sure that a more stable structure than those found in our calculations does not exist. In the next section, we only show the best calculated structures for each cluster size. The spin state of all clusters is found to be singlet or doublet for even and odd number of electrons respectively.

In order to check the reliability of our computational method, the dimers Ge2, Cu2, Au2 and Ag2 dimers were first considered. The bond lengths of Ge2, Ag2, Au2 and Cu2 were calculated to be 2.450, 2.595, 2.627, and 2.280 Å respectively. These values are in good agreement with the experimental data of 2.57 (Ref [41]) and 2.44 Å (Ref [42]) for Ge2, 2.53 Å for Ag2 (Ref [43]), 2.47 Å for Au2 (Ref [44]), and 2.22 Å for Cu2 (Ref [45]). The binding energy per atom, calculated to be 1.44, 1.07, 1.37 and 1.13 eV for Ge2, Ag2, Au2 and Cu2 respectively, are also in good agreement with both theoretical and experimental data [START_REF] Wang | A computational investigation of copper-doped germanium and germanium clusters by the density-functional theory[END_REF][START_REF] Bandyopadhyay | Architectures, electronic structures, and stabilities of Cu-doped Gen clusters: density functional modeling[END_REF][START_REF] Kingcade | Electronic structure and bonding of the molecule Ge2 from all-electron ab initio calculations and equilibrium measurements[END_REF][START_REF] Rabilloud | Structure and stability of coinage metal fluoride and chloride clusters (MnFn and MnCln, M = Cu, Ag, or Au; n = 1-6)[END_REF] .

Results 3a Structural properties

The lowest energy been structures of pure Gen clusters have already reported in the literature [START_REF] Siouani | Structure, Stability, and electronic and magnetic properties of VGen (n = 1-19) clusters[END_REF] . Shortly, the lowest-energy isomer of the trimer Ge3 presents a D3h triangular geometry, while that of Ge4 is a rhombus structure with D2h symmetry. The triangular bipyramid framework with D3h symmetry is found to be the global minimum structure of the Ge5 pentamer. The hexamer Ge6 has two lowest-energy isomers quasi-degenerate, the first one is a bicapped quadrilateral structure with C2v symmetry and the second one shows a bicapped rectangular geometry with D4h symmetry. The most stable isomer of the heptamer Ge7 adopts a structure having C2 point group symmetry. The ground state structure of Ge8 cluster can be seen as a capped pentagonal bipyramid with Cs symmetry. The Ge9 cluster forms a capped cube like structure with C4v symmetry. For Ge10, the most stable structure is a capped irregular pentagonal prism structure with C1 symmetry. Ge11 is a caped pentagonal structure with C2v point group symmetry. Ge12 displays a compact and nearly spherical structure with C2v symmetry. For the clusters size range between 13 and 16, a prolate geometry with C1 point group symmetry is always favored. The lowest-energy structure of Ge17 presents a nearly spherical shape with Cs symmetry while a prolate-like structure with C2 symmetry is observed in the case of Ge18. The ground state isomer for Ge19 has a prolate shape with C2v point group symmetry, while the best structure of Ge20 adopts a cage like structure with lower symmetry.

We now discuss optimized geometries for the three systems MGen (M = Cu, Ag, Au) in the size range of n = 1 to 19. The lowest energy structures together with some low-lying isomers are presented in Figure 1, 2 and 3 for AgGen, AuGen and CuGen respectively. Meanwhile, the corresponding symmetry group, binding energy per atom, HOMO-LUMO gap, vertical ionization potential, vertical electronic affinity, and average bond lengths aGe-Ge and aGe-M (M=Cu, Ag, Au) are summarized in Table 1, 2 and 3 for AgGen, AuGen and CuGen respectively.

Both AgGe and AuGe dimers have a binding energy per atom of 0.813 eV which is 0.372 eV less stable than the dimer CuGe. The bond lengths are calculated to be 2.591, 2.645 and 2.333 Å for AgGe, AuGe and CuGe respectively. For CuGe, our present data are very near to those predicted by Bandyopadhyay [START_REF] Bandyopadhyay | Architectures, electronic structures, and stabilities of Cu-doped Gen clusters: density functional modeling[END_REF] .

For small clusters, the lowest-energy isomers are similar whatever the nature of the doping atom M = Ag, Au, or Cu. A triangular structure with C2v symmetry is found for MGe2 and a planar rhombus with C2v symmetry is found for MGe3. For CuGen (n=1-3) clusters our results are similar to those reported by Bandyopadhyay [START_REF] Bandyopadhyay | Architectures, electronic structures, and stabilities of Cu-doped Gen clusters: density functional modeling[END_REF] . The best isomer of MGe4 can be seen as a distorted Ge4 rhombus with a capped dopant on one face. For MGe5 clusters, the metal M atom is capping a Ge5 triangular pyramid. Up to size n = 8, the structures of MGen clusters are roughly similar for M = Ag, Au and Cu. The substitution of one germanium atom by one noble metal atom in the capped bi-pyramid pure Ge8 structure gives the optimized ground state structure of MGe7 with M atom capping a pentagonal by-pyramid. The structure of MGe8 can be seen as MGe6 on which two additional Ge atoms are localized on surface, or alternatively it can also be seen as a distorted cube-like MGe7 structure with a capping Ge atom.

Starting from n = 9, the optimized structures of Cu-doped Gen clusters differ from those of Ag-and Au-doped Gen. A prolate-like structure with the M atom located out of the frame of germanium cluster is obtained as the best isomer of MGe9 clusters. Cu atom is highly coordinated to the neighboring Ge atoms in CuGe9 comparatively to Ag and Au atoms in AgGe9 and AuGe9 clusters. The endohedral structures in which the metal atom is encapsulated inside a germanium cage appear at n= 10 when the dopant is Cu, and n = 12 for M = Ag and Au. Indeed, the lowest-energy structure of CuGe10 is a D4d symmetry cage-like structure with Cu atom highly coordinated at the center of the cage. This structure has been already found previously [START_REF] Tai | Enhanced stability by three-dimensional aromaticity of endohedrally doped clusters X10M 0/with X = Ge, Sn, Pb and M = Cu, Ag[END_REF] . CuGe11 has also a cage-like structure with a Cu atom highly coordinated. In contrast, Ag and Au atoms are still in peripheral position in MGe10-11. All ground states isomers of MGe12 are high symmetry cage-like structures with M atom highly coordinated at the center of the germanium cage. The lowest-energy isomer of CuGe12 belongs to the high-symmetry D5d point group while that of AgGe12 and AuGe12 clusters has a D2d symmetry. The latter can be described as a biccapped pentagonal prism and were previously proposed by Li et al. [START_REF] Li | Structure, stability and electronic property of the gold-doped germanium clusters: AuGen (n = 2-13)[END_REF] as a putative structure for AuGe12, lying below the hexagonal prism with Cs symmetry (isomer e) well known to be the ground state for several transition metal Ge doped clusters [START_REF] Siouani | Structure, Stability, and electronic and magnetic properties of VGen (n = 1-19) clusters[END_REF][START_REF] Goicoechea | On the structural landscape in endohedral silicon and germanium clusters, M@Si12 and M@Ge12[END_REF] . The isoelectronic cluster RuGe12 3-is also found to lie in the D2h structure [START_REF] Espinoza-Quintero | Synthesis and characterization of [Ru@Ge12] 3-: An endohedral 3-connected cluster[END_REF] . Previous works on the anion AuGe12 -have furnished conflicting results since Li et al. [START_REF] Li | Size-selective effects in the geometry and electronic property of bimetallic Au-Ge nanoclusters[END_REF] has predicted a D2d structure to be the lowest-isomer with hybrid DFT + MP2 level calculations, while Lu et al. [START_REF] Lu | Transition from exohedral to endohedral structures of AuGen (n = 2-12) clusters: photoelectron spectroscopy and ab initio calculations[END_REF] obtained a Ih-symmetry structure using DFT + CCSD calculations. The latter is in better agreement with the photoelectron spectroscopy results [START_REF] Lu | Transition from exohedral to endohedral structures of AuGen (n = 2-12) clusters: photoelectron spectroscopy and ab initio calculations[END_REF] . In our calculation, we find that the Ih-symmetry structure is the most stable isomer for the anion while the neutral AuGe12 cluster adopts a D2-symmetry structure. For AgGe12, the D5d symmetry structure is found to be stable but lying 0.09 eV above the D2d structure.

From n=13 to n = 19, Cu occupies an endohedral position for each size. Ag and Au are generally located in a cage-like structure with some exceptions for AgGe15,16,18 and AuGe15,18, where the metal atom prefers a peripheral position with a low number of coordination. Interestingly, the most stable isomer of MGe14 is a cage belonging a high symmetry D4h point group for the three metal M = Ag, Au, Cu. A high Oh symmetry have been previously predicted to be highly stable for W-or V-encapsulated Ge14 clusters [START_REF] Wang | Geometries and electronic properties of the tungsten-doped germanium clusters: WGen (n = 1-17)[END_REF][START_REF] Siouani | Structure, Stability, and electronic and magnetic properties of VGen (n = 1-19) clusters[END_REF] but for noble metal atom the Oh symmetry structure was found to relax in a D4h symmetry geometry. As we can see clearly from Tables 1 and2, the average Ge-Ag and Ge-Au bond lengths are larger than the average Ge-Ge bond length. In contrast, the average Ge-Cu bond length is smaller than the average Ge-Ge bond length as we can see from Table 3. These parameters can significantly affect the stability and others properties of each MGen species.

3b Energetics

The binding energy par atom of MGen clusters is defined as where E(Ge) is the total energy of free Ge atom, E(M) the total energy of free M = Ag, Au, Cu atom and E(MGen) is the total energy of the MGen cluster. The calculated binding energies for all clusters are reported in Table 1, 2 and 3, and their evolution with the cluster size n is shown in Figure 4 for the best isomers of each system. For comparison, the evolution of binding energy of their corresponding ground state pure germanium clusters is also plotted. As expected, the binding energy generally increases with increasing size, which means that these clusters can continuously gain energy during the growth process. This is likely associated to the increase in the average number of neighbors per atom. The binding energy par atom of all systems increases rapidly from n=1 to n=6, then a slow and non-monotonic growth is observed. The Cu doping atom enhances the stability of the corresponding Gen frame from n = 10, i.e. when CuGen clusters adopt cage-like structures with Cu atom highly coordinated at central position. In contrast, the binding energies of AgGen and AuGen are always lower than those of pure germanium clusters, AgGen clusters being slightly more stable than gold-doped clusters. This behavior can be due to the nature of the bonding involving Cu, Ag and Au species, since copper presents a stronger covalent character than Ag and Au and a greater ability to make chemical bonds [START_REF] L'hermite | Metastable fragmentation of silver bromide clusters[END_REF] . For copper-doped clusters, a pronounced value of binding energy is observed at n=10 and 12 in which Cu is encapsulated in a quasi-perfect Gen cage. The binding energies per atom for CuGe10 and CuGe12 are calculated to be 3.080 and 3.099 eV respectively. They are somewhat similar to the binding energy of 3.058-3.179 eV calculated for VGe10-15 cage-like structures [START_REF] Siouani | Structure, Stability, and electronic and magnetic properties of VGen (n = 1-19) clusters[END_REF] . For AgGen and AuGen, the largest value is obtained at n = 15, with 3.019 and 2.997 eV/atom respectively, and corresponding to a peculiar hollow cage-like geometry where the metal atom is inserted into the germanium structure without having a particular position.

In cluster physics, the second-order difference in energy (∆2E) can well reflect the relative stabilities of the corresponding clusters. It is generally compared with the relative abundances determined in mass spectroscopy experiments. For the ground-state MGen clusters, it can be calculated as

∆2E = E(MGen+1) + E(MGen-1) -2 E(MGen),
where E is the total energy of the ground state cluster. In Figure 05, we show the variation of ∆2E as a function of the size n. The plots show some pronounced positive peaks at n = 5, 8, 12 and 15 for all MGen (M=Cu, Ag, Au) clusters. A relative high stability is found for CuGe10, CuGe12, AgGe10, AgGe12, AgGe15, AuGe12 and AuGe15. It is mainly due to the metalgermanium bonds since the corresponding initial pure germanium clusters do not show a peculiar stability.

The chemical activity of clusters can be characterized by the measurement of the HOMO-LUMO gap. In general, a large HOMO-LUMO gap implies a low chemical activity and a high chemical stability, while the latter decreases as the HOMO-LUMO gap decreases. The calculated HOMO-LUMO gaps for all AgGen, AuGen and CuGen clusters are reported in Table 1, 2 and 3 respectively. Their evolution as a function of the size for the ground states structures is plotted in Figure 06. The decreasing tendency with increasing size is strong for pure germanium clusters but less significant for metal-doped Gen clusters. As expected, the gap of MGen is lower than that of the pure Gen+1 cluster, except for sizes n= 10, indicating that the metal dopant in the germanium clusters generally increases the chemical activity of clusters. Values oscillate in the 0.15-1.3 eV range but the evolution with the cluster size shows a non-monotonic behavior. Pronounced value of HOMO-LUMO gaps are observed at n = 4, 6, 13, 15 and 18 for AgGen and at n = 4, 6, 10, 13 16 and 18 for AuGen and at n = 5, 10, 13, 16 for CuGen clusters which indicate that these clusters have a higher stability and a low reactivity than their neighbors. Particularly, the high relative value of 1.324 eV for CuGe10 reflects the chemical stability of this cluster which may likely be a good candidate to be used as a building block for developing new materials.

The vertical electroaffinity (VEA) and vertical ionization potential (VIP) are defined as VEA = E(MGen) -E(MGen -) and VIP = E(MGen + ) -E(MGen), where E(MGen -) and E(MGen + ) are the energy of the corresponding anionic and cationic clusters calculated at the geometry of the neutral system. The obtained calculated VEA and VIP are reported in Tables 1, 2 and 3, while their evolutions as a function of the cluster size are plotted in Figure 07 and Figure 08 together with those of Gen+1 clusters. These data could be compared to experiment even if to our knowledge no experimental data are available nowadays. The VEA values present a non-monotonic increasing with the cluster size, while the VIP values present a nonmonotonic decreasing with the cluster size. Both VEA and VIP are relatively few dependent on the nature of the doping metal, even if values for copper-doped clusters are often slightly larger than those for Ag-and Au-doped clusters. CuGe12, with its highly D5d symmetry structure, presents the highest values.

Discussion

Our results highlight a great stability for CuGe10 cluster in an endohedral structure. This is line with the experimental mass spectrometric stability investigation by Lievens et al. [START_REF] Neukermans | Mass spectrometric stability study of binary MSn clusters (S = Si, Ge, Sn, Pn, and M = Cr, Mn, Cu, Zn)[END_REF] that exhibited remarkably high abundance for CuGe10 + and interpreted it in terms of peculiarly stable dopant-encapsulated cage-like structures. In Ref [27], the 42 valence electrons (4 from Ge and 1 for Cu) in CuGe10 -are found to be distributed in the following energy sequence of electronic shell model [1S 2 1P 6 1D 10 1F 2 2S 2 1F 8 2P 2 1F 4 2P 4 1G 2 ]. In our work, the d electrons of Cu are found to participate as well to the delocalized orbitals and then 51 electrons (4 from Ge and 11 from Cu) are in the delocalized electronic shells with the following filling [1S 2 1P 6 1D 10 2D 10 2S 2 1F 14 2P 6 1G 1 ]. Due to a lowering symmetry (D4d instead of Kh for a sphere), some shells are splitting subshells. However one can expect that the number of electrons fit with shell closings numbers for the cation CuGe10 + , and such an organization contributes to the high stability of the cluster through the pooling of electrons. Density of states (DOS) together with representative Kohn-Sham orbitals are given in Figure 9 for CuGe10,12,14. All molecular orbitals can be found in Supporting Information. We can easily distinguish the orbitals associated to S, P, D, F, G supershells. For the endohedral structure corresponding to the lowest-energy isomer of CuGen, with n>14, we were not able to assign a symmetry character of molecular orbitals, and we think the supershell filling is no more valid. The great values of VIP and VEA calculated for CuGe12 is due to the high symmetry (D5d) of its geometrical structure resulting to a large gap between two successive electronic shells (Figure 9). We show in figure 10 the density of states (DOS) for the lowest-energy isomer of AgGe12,14,15, and AuGe12,14,15. These structures are of sphere-type where the germanium cage encapsulates the metal atom resulting in a high symmetry structure, except for n=15, where the metal atom is inserted into the germanium structure without having a particular position, resulting in a hollow cage-like geometry. Our above results highlight a relative high stability for n = 15. As a matter of fact, the electronic configuration of AgGe15 and AuGe15, [1S 2 1P 6 2S 2 1D 10 1F 14 3S 2 2D 10 2P 6 1G 18 1H 1 ] see Table 4, fits well with shell closings numbers with a single excess electron, similarly to the configuration found for CuGe10. The high stability of the three clusters, CuGe10, AgGe15 and AuGe15, can be explained by such organization. The Kohn-Sham molecular orbitals of AgGe15 and AuGe15 are showed in Supporting Information, we can note that the molecular orbitals 1F, 2D, and 1G are mixed. It is very significant that the coinage metal is able to give both s-and d-type electrons to be reorganized together with the valence electrons of Ge atoms through a pooling of electrons. To the best of our knowledge, that has never been reported previously. And it is in line with the recent photoelectron spectroscopy results on small AuGen -which has suggested strong interactions between the 5d orbitals of the Au atom and the 4s4p hybridized orbitals of the Ge atoms [START_REF] Lu | Transition from exohedral to endohedral structures of AuGen (n = 2-12) clusters: photoelectron spectroscopy and ab initio calculations[END_REF] .

The electron shell configuration of endohedral cage-like structure MGen (M = Cu, Ag, Au ; n = 10, 12, 14) and hollow cages of AgGe15 and AuGe15 are given in Table 4. The density of states together with Kohn-Sham molecular orbitals can be found in Supporting information. The orbitals character can be easily identified for all cages, even if some mixing were found for 2D-, and 1G-type orbitals in case of low-symmetry geometries.

Conclusion

We have performed a systematic investigation on the coinage metal-doped germanium (MGen with M = Cu, Ag, Au; n=1-19) clusters by using first-principles DFT calculations. Doping with Cu atom enhances the stability of the corresponding germanium frame when CuGen clusters adopt cage-like structures with Cu atom highly coordinated at central position. In contrast, the binding energies of AgGen and AuGen are always lower than those of pure germanium clusters. The endohedral structures in which the metal atom is encapsulated inside a germanium cage appear at n = 10 when the dopant is Cu, and n = 12 for M = Ag and Au. Our results highlight the great stability of the CuGe10 cluster in D4d structure which exhibits a large binding energy, a large frontier orbital energy gap, and a relative small electron affinity. AgGe15 and AuGe15 which exhibit a hollow cage-like geometry where the metal atom is inserted into the germanium structure without having a particular position present a relative high stability as well. The sphere-type structures obtained for n = 10-15 present a peculiar electronic structure in which the valence electrons of the noble metal and Ge atoms are delocalized and exhibit a shell structure associated with the quasi-spherical geometry. Each Ge gives 4 electrons while each coinage atom gives 11 electrons. It is very significant that the coinage metal is able to give both s-and d-type electrons to be reorganized together with the valence electrons of Ge atoms through a pooling of electrons. We hope that the present study will motivate appropriate experimental investigations probing the electronic properties and the stability of noble metal-doped germanium clusters.

Figure 2 . 4 .

 24 Figure 2. Ground state structures and corresponding isomers for AuGen (n=1-19)clusters. For each size, the lowest-energy isomer is indicated in bold characters.

Figure 5 .

 5 Figure 5. Second energy differences (∆2E) the ground state structures of MGen (M=Cu, Ag, Au) clusters.

6 .

 6 Size dependence of the HOMO-LUMO gap for the ground state structures of MGen (M=Cu, Ag, Au) clusters.

Figure 7 .

 7 Figure 7. Size dependence of the electron affinity for the ground state structures of MGen (M=Cu, Ag, Au) clusters.

8 .

 8 Size dependence of the vertical ionization potential for the ground state structures of MGen (M=Cu, Ag, Au) clusters.

Figure 9 . 10 .

 910 Figure 9. Density of states for most stable isomer of CuGe10,12,14 together with some representative Kohn-Sham molecular orbitals. More molecular orbitals are given in Supporting Information.

Table 1 .

 1 Symmetry group, Binding energy per atom Eb (eV/atom), HOMO-LUMO gap ΔE(eV), Vertical Ionization Potential (VIP) (eV), Vertical Electronic Affinity (VEA) (eV), and average bond length aGe-Ge and aGe-Ag for AgGen (n=1-19) clusters.

	n	Sym	Eb(eV/atom)	ΔE (eV)	VIP (eV)	VEA (eV)	aGe-Ge(Å)	aGe-Ag(Å)
	1	a-C∞v	0.813	0.554	6.897	1.169	-	2.591
	2	a-D∞h	0.988	0.404	6.916	1.653	-	2.567
		b-C2v	1.721	0.206	6.945	1.571	2.339	2.696
	3	a-C2v	2.032	0.885	7.280	1.913	2.446	2.620
		b-Cs	1.999	0.618	6.835	1.372	2.667	2.828
	4	a-Cs	2.324	1.127	6.602	1.651	2.623	2.860
		b-C2v	2.306	0.873	6.787	1.711	2.674	2.907
		c-C2v	2.113	0.486	6.798	1.991	2.421	2.823
	5	a-C1	2.376	0.758	6.595	1.940	2.570	2.811
		b-Cs	2.572	1.060	7.216	2.038	2.702	2.783
		c-C2v	2.256	0.465	6.963	2.492	2.542	2.784
	6	a-Cs	2.557	0.326	6.905	2.566	2.666	2.855
		b-Cs	2.662	1.156	6.783	2.131	2.764	2.849
		c-C2v	2.650	1.017	6.747	1.982	2.765	2.827
		d-C1	2.603	0.936	6.418	1.732	2.673	2.783
	7	a-Cs	2.712	0.688	6.346	1.969	2.756	2.758
		b-C1	2.695	0.708	6.337	1.984	2.772	2.809
		c-C2v	2.656	0.476	6.686	2.278	2.737	2.864
	8	a-Cs	2.778	0.665	6.610	2.378	2.750	2.789
		b-C1	2.743	0.859	6.419	2.116	2.758	2.788
		c-C1	2.762	0.582	6.612	2.328	2.787	2.780
		d-Cs	2.755	0.782	6.843	2.447	2.740	2.914
	9	a-Cs	2.784	0.755	6.512	2.436	2.767	2.724
		b-C1	2.797	0.667	6.267	2.223	2.774	2.789
		c-C1	2.792	0.657	6.296	2.142	2.760	2.752
		d-C1	2.794	0.758	6.403	2.204	2.711	2.802
		e-C1	2.808	0.754	6.132	2.222	2.742	2.765
		f-C1	2.804	0.884	6.442	2.416	2.756	2.851
		g-C1	2.806	0.921	6.254	2.107	2.766	2.813
	10	a-C1	2.849	1.015	6.315	2.278	2.762	2.829
		b-Cs	2.898	0.611	6.667	2.719	2.800	2.768
		c-Cs	2.818	0.422	6.888	2.916	2.804	2.863
		d-Cs	2.835	0.762	6.660	2.725	2.752	2.951
	11	a-C1	2.828	0.380	6.652	2.836	2.778	2.834
		b-Cs	2.848	0.547	5.918	2.175	2.707	2.598
		c-C1	2.869	0.380	6.637	2.802	2.806	2.803
		d-C1	2.847	0.729	6.271	2.550	2.765	2.922
		e-C2v	2.893	0.568	6.663	2.623	2.779	2.764
	12	a-D5d	2.919	0.236	7.120	3.356	2.950	2.805
		b-Cs	2.870	0.555	6.525	2.800	2.795	2.761
		c-C1	2.866	0.416	6.525	2.856	2.719	2.857
		d-Cs	2.879	0.342	6.463	2.794	2.826	2.837
		e-C2h	2.910	0.126	6.383	2.892	2.653	2.878
		f-d2d	2.926	0.744	6.390	2.699	2.661	2.879
	13	a-C2v	2.879	0.566	6.383	2.712	2.794	2.778
		b-C2v	2.890	0.765	6.084	2.462	2.789	2.742
		c-C1	2.906	0.655	6.448	2.854	2.758	2.891
		d-C1	2.903	0.495	6.508	2.996	2.780	2.866
		e-C1	2.914	0.750	6.359	2.715	2.797	2.797
	14	a-D4h	2.963	0.171	6.495	3.029	2.736	2.949
		b-C1	2.911	0.638	6.238	2.769	2.766	2.771
		c-C1	2.917	0.627	6.331	2.880	2.746	2.907
		d-C1	2.952	0.552	6.514	2.861	2.791	2.922
	15	a-C1	2.947	0.786	6.167	2.706	2.788	2.841
		b-C1	2.932	0.586	6.245	2.905	2.771	2.885
		c-C2v	3.019	0.732	6.662	3.024	2.762	2.886
		d-C1	2.946	0.594	6.157	2.772	2.774	2.913
	16	a-C1	2.964	0.511	6.358	2.880	2.806	2.937
		b-C1	2.919	0.476	6.278	2.918	2.770	2.847
		c-C1	2.985	0.336	6.581	3.274	2.784	2.886
		d-Cs	2.915	0.405	6.045	2.841	2.813	2.823
		e-C1	2.943	0.618	6.074	2.808	2.827	2.886
	17	a-C1	2.933	0.583	6.028	2.631	2.853	2.909
		b-C1	2.966	0.601	6.116	2.818	2.819	3.009
		c-C1	2.940	0.455	6.192	2.991	2.750	2.967
		d-C1	2.942	0.589	5.839	2.508	2.755	2.912
	18	a-C1	2.991	0.708	6.076	2.817	2.783	2.876
		b-C1	2.988	0.583	6.172	1.551	2.744	2.841
		c-C1	2.978	0.698	6.059	2.791	2.766	2.935
		d-C1	2.987	0.623	6.034	2.796	2.714	2.948
		e-C1	2.981	0.615	6.082	2.784	2.787	2.682
	19	a-C1	3.017	0.437	6.304	3.086	2.802	2.953
		b-C1	2.973	0.439	6.362	3.119	2.737	2.887
		c-C1	3.001	0.600	6.199	3.038	2.756	2.946
		d-C3v	2.990	0.644	6.223	2.906	2.743	2.942
		e-C1	2.988	0.542	6.253	3.082	2.797	2.967
		f-C2v	2.976	0.445	6.217	3.021	2.720	2.969

Table 2 .

 2 Symmetry group, Binding energy per atom Eb (eV/atom), HOMO-LUMO gap ΔE(eV), Vertical Ionization Potential (VIP) (eV), Vertical Electronic Affinity (VEA) (eV), and average bond length aGe-Ge and aGe-Au for AuGen (n=1-19) clusters. Symmetry group, Binding energy per atom Eb (eV/atom), HOMO-LUMO gap ΔE(eV), Vertical Ionization Potential (VIP) (eV), Vertical Electronic Affinity (VEA) (eV), and average bond length aGe-Ge and aGe-Cu for CuGen (n=1-19) clusters.

	n	Sym	Eb(eV/atom)	ΔE (eV)	VIP (eV)	VEA (eV)	aGe-Ge(Å)	aGe-Au (Å)
	1	a-C∞v	0.813	1.354	6.931	1.227	-	2.645
	2	a-D∞h	0.997	0.469	6.924	1.660	-	2.619
		b-C2v	1.697	0.177	6.913	1.608	2.347	2.762
	3	a-C2v	2.015	0.910	7.208	1.893	2.451	2.680
		b-Cs	1.985	0.627	6.806	1.385	2.672	2.892
		c-C2v	1.981	0.388	6.441	1.400	2.434	2.849
	4	a-Cs	2.300	0.849	6.902	1.853	2.665	2.848
		b-C2v	2.093	0.468	6.781	1.983	2.426	2.882
		c-Cs	2.310	1.103	6.596	1.698	2.623	2.936
	5	a-C2v	2.550	1.127	7.167	2.017	2.698	2.736
		b-Cs	2.553	1.059	7.194	2.050	2.705	2.854
		c-C2v	2.494	1.273	6.767	2.090	2.656	2.780
	6	a-Cs	2.538	0.570	6.731	2.362	2.661	2.927
		b-Cs	2.655	1.144	6.677	1.976	2.703	2.846
		c-C2v	2.627	0.951	6.678	1.953	2.769	2.904
	7	a-C1	2.464	0.643	6.559	2.521	2.525	2.833
		b-Cs	2.696	1.141	6.421	2.053	2.723	2.838
		c-C1	2.681	0.949	6.492	2.091	2.739	2.813
		d-C2v	2.648	0.479	6.670	2.276	2.737	2.927
	8	a-Cs	2.764	0.654	6.585	2.372	2.752	2.854
		b-C2	2.740	0.645	6.500	2.220	2.789	2.879
		c-C1	2.709	0.526	6.989	2.679	2.758	2.915
		d-Cs	2.742	0.746	6.821	2.453	2.746	2.972
	9	a-Cs	2.772	0.785	6.476	2.424	2.767	2.793
		b-C1	2.785	0.616	6.201	2.200	2.755	2.853
		c-C2v	2.765	0.902	6.238	2.216	2.791	3.070
		d-C1	2.782	0.693	6.340	2.176	2.713	2.873
		e-C1	2.797	0.793	6.149	2.257	2.743	2.833
		f-C1	2.795	0.973	6.283	2.141	2.764	2.723
	10	a-C1	2.838	0.975	6.320	2.298	2.763	2.803
		b-Cs	2.794	0.484	6.816	2.871	2.822	2.915
		c-Cs	2.797	0.912	6.088	2.207	2.743	2.974
	11	a-C1	2.817	0.517	6.535	2.757	2.752	2.833
		b-Cs	2.838	0.507	5.899	2.168	2.708	2.662
		c-C1	2.831	0.743	6.117	2.367	2.789	2.780
		d-C1	2.834	0.722	6.256	2.541	2.767	2.887
		e-C2v	2.882	0.535	6.687	2.659	2.781	2.832
	12	a-C2	2.875	0.586	6.718	2.999	2.824	2.937
		b-Cs	2.863	0.553	6.511	2.799	2.795	2.818
		c-C1	2.858	0.450	6.516	2.856	2.717	2.822
		d-Cs	2.866	0.348	6.396	2.749	2.828	2.915
		e-Cs	2.885	0.374	6.422	2.746	2.715	2.904
		f-D2d	2.912	0.733	6.401	2.726	2.680	2.901
	13	a-C2v	2.871	0.575	6.370	2.701	2.793	2.841
		b-Cs	2.865	0.617	5.854	2.374	2.804	2.875
		c-C1	2.919	0.831	6.297	2.736	2.759	2.904
		d-C1	2.889	0.444	6.487	2.997	2.778	2.942
		e-C1	2.904	0.751	6.350	2.717	2.799	2.864
		f-C1	2.870	0.503	6.543	2.922	2.744	2.941
	14	a-C1	2.904	0.646	6.229	2.769	2.767	2.848
		b-D4h	2.934	0.165	6.423	2.972	2.755	2.972
		c-C1	2.933	0.594	6.486	2.815	2.807	2.969
	15	a-C1	2.946	0.841	6.016	2.682	2.771	2.918
		b-Cs	2.930	0.758	6.366	2.889	2.811	2.886
		c-Cs	2.997	0.495	6.609	3.201	2.791	2.789
		d-C1	2.923	0.715	6.234	2.752	2.749	2.872
	16	a-C1	2.967	0.660	6.140	2.728	2.786	2.995
		b-C1	2.912	0.474	6.311	2.927	2.771	2.908
		c-C1	2.925	0.630	6.125	2.868	2.803	2.945
	17	a-C1	2.925	0.589	6.033	2.628	2.839	2.927
		b-C1	2.974	0.499	6.132	2.831	2.734	2.957
		c-C1	2.933	0.446	6.208	3.004	2.753	2.995
		d-C1	2.948	0.494	6.063	2.754	2.769	2.872
	18	a-C1	3.004	0.656	6.194	2.780	2.775	2.932
		b-C1	2.981	0.465	6.328	3.054	2.728	2.996
		c-C1	2.973	0.643	6.115	2.903	2.780	2.965
		d-C1	2.974	0.579	6.013	2.774	2.708	2.988
	19	a-C1	3.006	0.425	6.305	3.086	2.805	2.979
		b-C1	2.999	0.600	6.029	2.906	2.770	2.977
		c-C1	2.972	0.551	6.273	3.055	2.768	2.973
		d-C2v	2.967	0.438	6.203	3.008	2.727	2.991

Acknowledgement

FR thanks the Pôle Scientifique de Modélisation Numérique (PSMN) at Lyon, France, and the GENCI-IDRIS (Grant i2016086864) center for generous allocation of computational time.

Supporting Information. Density of states for the most stable and low-lying isomers of CuGe10,12,14, AgGe12,14,15, and AuGe12,14,15, together with Kohn-Sham molecular orbitals.


Juan Pablo Scotta

Ignacio Marquez

Gilles Noguere

David Bernard

Study on neutron scattering in light water

It is presented a method to produce covariance matrices of the light water total cross section from thermal scattering laws of the JEFF-3.1.1 nuclear data library and CAB model. The generalized least square method was used to fit the LEAPR module parameters of the processing tool NJOY with light water experimental transmission measurements at 293.6K with CONRAD code. The marginalization technique was applied to account for systematic uncertainties.

Introduction

The probability of an incident neutron with energy and solid angle (E;Ω) of scattering a light water molecule and being emitted at a different energy and different solid angle (E';Ω') can be expressed with the double differential scattering cross section:

( ) β α π σ σ β , ' 4 ) ' , ' ( 2 2 S e E E kT E E dE d d b - = Ω → Ω → Ω (1)
where σ b is the characteristic bound cross section of the material, T is the temperature and k is the Boltzmann constant.

The double differential cross section depends on the thermal scattering law function S(α,β), where α and β are the dimensionless momentum and energy transfer respectively:

AkT E E E E θ α cos ' 2 ' - + = ; kT E E - = ' β (2)
where cos θ is the cosine of the scattering angle in the laboratory system and A is the ratio between the mass of the scattering molecule and the neutron mass.

a Corresponding author: juan-pablo.scotta@cea.fr

In order to evaluate the function S(α,β) some approximations are often used: the incoherent and the Gaussian approximation. The processing of the thermal scattering law is done with LEAPR module of NJOY code [1], and is computed with these approximations.

The main parameter that defines the dynamics of the function S(α,β) is the frequency spectrum of the material, which defines its excitation states. When using LEAPR module, the frequency spectrum is a resultant of three different components: a solid type spectrum, a translational spectrum and discrete oscillators.

Models for H into H2O

It will be presented two different models that describe the frequency distribution of H into H 2 O.

The first one is IKE model, developed by M. Mattes and J. Keinert [START_REF] Mattes | Thermal Neutron Scattering Data for the moderator Materials H2O, D2O and ZrHx in ENDF-6 Format and as ACE Library for MCNP(x) Codes[END_REF]. It is based on experimental frequency spectrum measured by B. C. Haywood and D. I. Page [START_REF] Page | The Harwell scattering law program: frequency distributions of moderators[END_REF]. The rotational mode of hydrogen bound to the water molecule is represented by a solid type spectrum with a weight w R . Two discrete oscillators, with energies E 1 and E 2 , define the intra-molecular modes of vibration (bending and stretching modes), with corresponding weight w 1 and w 2 . Finally, an effective mass describes the molecular clustering effect of the water molecule with a weight w t . The sum of all the weights must be equal to unity.

The nuclear data library JEFF-3.1.1 adopted this model in its database.

Covariance matrices of differential and total cross section 3.1 Governing equations

In this work, the full covariance matrix between the model parameters is defined as follow:

        Σ Σ Σ Σ = Σ 22 21
12 11 [START_REF] Page | The Harwell scattering law program: frequency distributions of moderators[END_REF] where each element of the matrix is obtained by: ( ) ( )

1 1 11 - - + = Σ x T x x T T x x T x x G G G G M G G G G M θ θ θ (4) θ θ M G G G G T x x T x 1 12 ) ( - - = Σ (5)
θ M = Σ 22 [START_REF] Russell | Total cross section measurements of H2O[END_REF] in which M x stands for the covariance matrix between the best-fit values of the model parameter and M θ represents the covariance matrix between the nuisance parameters that account systematic uncertainties. The nuisance parameters included in the present study were the normalization and the background. In our context, the model parameter vector x and the nuisance parameter vector θ are:

) , , , , , , , ( x 2 1 2 1 ω ω ω ω E E c R t ∆ = → ; ) , ( B N = → θ (7)
The matrices G x and G θ correspond to the derivatives of the calculated quantity z to the model and the nuisance parameters respectively:

              = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2 2 2 2 2 2 1 1 1 ... ... ... ... ω ω ω ω ω ω k t k k t t z z c z z z c z z z c z x G M M M ;               = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ B z N z B z N z B z N z k k G M M 2 2 1 1 θ (8)
To calculate the covariance matrices of the total cross sections CONRAD code [START_REF] Saint | CONRAD, a new software for nuclear data analysis[END_REF] was used. The Generalized Least Square method (GLS) was used in association with the marginalization technique in which the zero variance penalty condition accounts for systematic uncertainties [START_REF] Noguere | [END_REF].

The "best estimates values" of the LEAPR module parameters were obtained from water transmission experimental measures.

Sensitivity analysis of the model parameters

A sensitivity analysis of the LEAPR module parameters to the H 2 O transmission at 293.6 K was done. Their sensitivity as a function of the incident neutron energy is plotted in figure 3 for the H 2 O transmission calculated with CAB model.

The parameters ω R and ∆ are one order of magnitude more sensitive to the transmission than the rest. This highlights the relevance of the continuous part of the frequency spectrum, described by these two parameters. Their anticorrelation suggests that only one of them should be taken into account in the marginalization technique.

The parameters c and ω t follow an equivalent trend because they are physically correlated. The parameters not included in figure 3 (E 1 , E 2 , ω 1 and ω 2 ) have a negligible sensitivity. Their uncertainties are not taken into account in the present study.

Results

According to our sensitivity analysis, the uncertainty on the total cross section is mainly driven by the parameters ∆ and ω t . The relative uncertainties and the correlation matrices of the total cross section for JEFF-3.1.1 and CAB model at 293.6 K are shown in figure 4.

In the cold neutron energy range, below 10 -3 eV, the uncertainty associated to JEFF-3.1.1 is larger than the uncertainty associated to CAB. This is explained by the better agreement of the CAB model with the low energy total cross section measurements.

Around 5 meV, the uncertainty found for CAB model is low. Additional work is needed for improving this underestimated magnitude.

Above the thermal energy, the relative uncertainties remain lower than 2%. The main contribution is related to the 1 H free atom cross section whose uncertainty is lower than 1% [START_REF] Mughabghab | Atlas of Neutron Resonance[END_REF].

Fig. 1 .

 1 Fig. 1. Frequency spectrum for JEFF-3.1.1 and CAB model at 293.6 K.

Fig. 2 .

 2 Fig. 2. Total cross section for JEFF-3.1.1 and CAB model, at 293.6 K.

Fig. 3 .

 3 Fig. 3. Sensitivity (in %/%) of the LEAPR model parameters to the H2O transmission as a function of the energy calculated with CAB model.

Fig. 4 .

 4 Fig. 4. Relative uncertainties and correlation matrices of the total cross section for JEFF-3.1.1 (left) and CAB model (right) at 293.6 K.

  Table 1 resumes the LEAPR parameters of this model at 293.6K.The second model, CAB model, was calculated by I. Marquez Damian[START_REF] Marquez Damian | [END_REF] and is based on molecular dynamic simulations with GROMACS code. It was obtained an equivalent model to IKE model but the translational mode was replaced by the diffusional model of Egelstaff-Schofield[5] to better describe low energy dynamics. The key parameter in this model is the diffusion constant c. Table1resumes the LEAPR parameters of CAB model at 293.6K.

	WONDER-2015
	Frequency spectrum parameters	JEFF -3.1.1 CAB Model
	Translational weight ω ω ω ω t Solid-type spectrum weight ω ω ω ω R	0.021739 0.489130	0.007918 0.522080
	Scissoring mode oscillator energy E 2 (meV)	205	205
	Scissoring mode oscillator weight ω ω ω ω 2	0.16304	0.15667
	Stretching modes oscillator energies E 1,3 (meV)	436	430
	Stretching modes oscillator weights ω ω ω ω 1,3	0.3261	0.31333
	Dimensionless diffusion constant c	0	3.969

Figure 1 compares the frequency spectrum for IKE model (used in the JEFF-3.1.1 nuclear data library) and CAB model at 293.6K.

The total cross section is obtained integrating equation (

1

) in all emitted neutron energies and solid angles. It was computed for JEFF-3.1.1 library and CAB model at 293.6 K. Figure

2

compares them with experimental data

[START_REF] Russell | Total cross section measurements of H2O[END_REF][START_REF] Zaitsev | [END_REF][8]

.

Tab. 1. IKE and CAB model parameters of H into H2O at 293.6K for the LEAPR module of NJOY.

Conclusions

The covariance matrices of the light water total cross sections from thermal scattering laws of JEFF-3.1.1 nuclear data library and CAB model were obtained by using CONRAD code. In the cold neutron energy range, differences in the relative uncertainties between both models are important. The uncertainties associated to JEFF-3.1.1 are larger because the agreement with the low energy total cross section is poor.

Present work is being done to propagate uncertainties due to the thermal scattering laws to integral calculations. Preliminary calculations performed on MOX fuel provide uncertainties on the multiplication factor ranging from 100 pcm with the CAB model to 200 pcm with JEFF-3.1.1.

The next step would be to determine uncertainties on the water potential parameters used in the molecular dynamic simulation code GROMACS, in order to obtained directly uncertainties on the frequency spectrum introduced in LEAPR module.


Tiago Da

Silva Nascimento
email: tiagodasnascimento@gmail.com

Emanuel Coutinho
email: emanuel@virtual.ufc.br

Carla Ilane Bezerra

José Neuman De Souza

A Metric for Resource Usage Evaluation in Cloud Computing Environments

Cloud Computing (CC) has become a subject of considerable research in the scientific and industrial communities, as well as other technologies. Although there are several studies and recent developments stating CC is still in the process of evolution and increasingly new technologies are emerging that can be added to it, such as Software Defined Networks (SDN) and Network Functions Virtualization (NFV). The quality evaluation of these technologies tends to become increasingly important, and this directly impacts the users. Due to the emergence of CC, SDN and NFV technologies, and the fact that they are highly related, it was necessary to evaluate the quality of these environments in an integrated way. This work proposes a metric for the quality evaluation of integrated environments.

Introduction

Over the years, Cloud Computing (CC) has become a subject of considerable research in the scientific and industrial communities, as it represents a new era for the availability and use of Information Technology services through the Internet [START_REF] Singh | A systematic review on cloud computing[END_REF]. The cloud computing paradigm is based on a usage-based payment model, and suggests the integration of various technology models to provide hardware infrastructure, development platforms, and applications such as on-demand services [START_REF] Ferreira Coutinho | Elasticity in cloud computing: a survey[END_REF].

Although there are several studies and recent developments, CC is still in the process of evolution and increasingly new technologies are emerging that can be added to it [START_REF] Azodolmolky | Sdn-based cloud computing networking[END_REF]. This is the case of Software Defined Networks and Network Functions Virtualization.

Software Defined Networking (SDN) represents a paradigm that promises to change the state of traditional IP (Internet Protocol) networks, which are considered complex and difficult to manage, because they have the control plane and the data plane grouped into the network devices. SDN breaks vertical integration with the separation of these planes (control and data). In this way, the control functionality is removed from devices, which become simple packet forwarding elements [START_REF] Kreutz | Software-defined networking: A comprehensive survey[END_REF].

Network Functions Virtualization (NFV) emerged as a solution to the challenges faced by telecom service providers, as it leads to virtualization technology to provide a new way to design, deploy, and manage network services [START_REF] Rao | Network security function virtualization (nsfv) towards cloud computing with nfv over openflow infrastructure: Challenges and novel approaches[END_REF]. NFV disassociates the Network Functions of the physical equipment where they are executed, which can lead to significant reductions in operating expenses (OPEX) and capital Expenses (CAPEX), in addition to promoting greater return and agility in the implementation of new services [START_REF] Mijumbi | Network function virtualization: State-of-the-art and research challenges[END_REF].

Cloud computing network infrastructure is shared with a number of independent entities. Therefore, network management is a challenge. In this context, SDN has highly valuable features for cloud computing systems, and also plays an important role in NFV technology infrastructure by making networks more flexible, dynamic and cost-effective, while simplifying the operational complexities [START_REF] Rao | Network security function virtualization (nsfv) towards cloud computing with nfv over openflow infrastructure: Challenges and novel approaches[END_REF].

With the great use of Information and Communication Technologies at all levels of human activity, the evaluation of the quality of these technologies tends to become increasingly important [START_REF] Da | Especialista em Sistemas de Informação, com Pós-Graduação em Gestão Estratégica da Informação e Consultora da UCP/PNAFM/MF[END_REF]. Thus, due to the emergence of CC, SDN and NFV technologies, and the fact that they are highly related, it was necessary to evaluate the quality of these environments in an integrated way. A quality evaluation can be performed by measuring the quality attributes of the technology in question [START_REF]Systems and software engineering-systems and software quality requirements and evaluation (square)-system and software quality models[END_REF].

In this context of cloud quality assessment, this work proposes a metric for the evaluation of the quality of integrated Cloud Computing, Network Defined by Software and Network Function Virtualization environments, with the main target audience being the researchers areas. This work is still in progress, and therefore the experiments only used the environment of a computational hybrid cloud. This paper presents an initial idea of a metric to analyze the quality of the environment. It is also a way of knowing the used tool, to analyze its flexibility in the inclusion of new metrics. However, the environment did not use SDN or NFV resources, remaining for future work and application of the tool in these environments.

Related Work

[12] presented a proposal for a comprehensive quality model for assessing the quality of Software as a Service (SaaS) cloud services. Based on ISO/IEC 9126 [9], these authors identified characteristics, quality attributes and defined metrics to measure the quality of these attributes. This work does not make any practical application in a real environment of defined metrics and its quality assessment model is directed to a specific model of cloud computing (SaaS).

[16] proposed a quality model for cloud services called CLOUDQUAL, which specifies six quality assessment metrics that serve cloud services in general. In this work, the authors designed a case study involving three real-world storage clouds: Amazon S3, Microsoft Windows Azure Blob Storage (Azure Blob) and Aliyun Open Storage Service (Aliyun OSS). By the use of experimental results, the effectiveness of CLOUDQUAL was analyzed and its use for evaluating the quality of cloud services is feasible.

[14] performed a systematic mapping on the current state of the art of the proposed Web service quality models, using ISO/IEC 25010 [START_REF]Systems and software engineering-systems and software quality requirements and evaluation (square)-system and software quality models[END_REF] for articulation of the analysis. This work presented characteristics and attributes of quality, with the definitions most used in Web service quality models for these dimensions. [START_REF] Emanuel | Physics and microeconomics-based metrics for evaluating cloud computing elasticity[END_REF] proposed same metrics for elasticity evaluation in cloud environments, based on concepts from Physics and Microeconomics. This metrics could be used in some future work in SDN and NFV environments for evaluating the elasticity effects under different workloads.

The idea of these works was to discuss quality assessment for cloud computing environments, as well as aspects, characteristics and quality attributes. This is one of the ideas of our work.

3 Description of the Hybrid Cloud Environment

Testbed

The experiments used two different cloud computing environments: a private cloud and a public cloud. For the private cloud we used OpenNebula 3.8. All physical machines had 5 or 7 cores, 24 GB RAM, Linux Ubuntu Server 12.04 64-bit operating system and KVM hypervisor. Each virtual machine was created with 1 VCPU, 1 GB of RAM and Linux operating system Ubuntu Server 12.04 64 bit. For the public cloud, the Microsoft Azure platform was used, with instances created with standard type A1 (1 core and 1.75 GB of RAM) and Linux operating system Ubuntu Server 14.04 64 bit.

The experiments used a maximum of four virtual machines, varying the number of instances used in the private and public cloud, according to the design of the experiment. Apache Tomcat was used as web server, NGINX as load balancer and HTTPERF as workload generator. Figure 1 displays the used testbed, based on the architecture proposed in [START_REF] Emanuel | An architecture for providing elasticity based on autonomic computing concepts[END_REF].

Workload

The workloads generated for the experiments were given by: requests forwarded directly to the load balancer virtual machine, generated by HTTPERF (with load of a multiplication of matrices) and by web browsers, distributed among the others virtual machines allocated; and requests executed directly on the virtual machines used by the infrastructure. Thus, competition for resources in a CC environment can be emulated [START_REF] Emanuel | An architecture for providing elasticity based on autonomic computing concepts[END_REF]. Figure 2 represents the applied workload.

Metrics Analysis Support Tool

To assist in the visualization and analysis of the collected data, a tool was used that allows the generation of graphs for the various metrics collected from the environment resources [START_REF] Coutinho | A tool for resource monitoring in computational clouds[END_REF] (Figure 3). This application was developed with Java programming language. It is based on the reading of text files, which contains information collected from the environment, such as CPU and memory usage, as well as application data such as response time for requests. The log files have their own structure read by the application. Examples of log files are: virtual machine allocation and average CPU consumption. These files are illustrated in Table 1 and Figure 4 File

Application of Metrics

For experiments performing, a metric of resource utilization was used. In addition to the resource utilization metric, the average CPU usage metric was used to aid in the evaluation of resource utilization. The following formula was used for the resource utilization metric:

R u = R a R p (1)
Where R u , R a and R p are equivalent to the used resources, allocated resources, and preallocated resources, respectively. In our experiments, we used virtual machines as resources.

Figure 3: General screen of the application for metrics analysis [START_REF] Coutinho | A tool for resource monitoring in computational clouds[END_REF] Figure 4: Excerpts from the log files for virtual machines allocation and average CPU utilization

The implementation of the resource utilization metric was basically due to the relationship of resource utilization data with the total sampling time. The CPU Average metric shows an average CPU utilization of the hybrid system completely. This metric was already implemented in the tool, as previously mentioned. In the same way, as the allocation of virtual machines.

Experiments

For the metrics application, we used the database of the experiments described in the following subsections (4.1 e 4.2), available in [START_REF] Ferreira | FOLE: Um Framework Conceitual para Avaliao de Desempenho da Elasticidade em Ambientes de Computação em Nuvem. Doutorado[END_REF].

Experiment 1 -One Virtual Machine in the Private Cloud and One Virtual Machine in the Public Cloud

This experiment lasted 36min10s. Only two virtual machines were involved in this experiment: one in the private cloud and one in the public cloud. In this way, it was possible to verify if the designed infrastructure would provide resources of the two clouds, constituting a hybrid cloud, according to the necessity generated by the workload.

Figure 5 shows the graphs generated from the application of the Resource Utilization and CPU Average metrics, with the allocation of the virtual machines for complementary analysis.

From the analysis of the graph illustrated by Figure 5, it can be seen that the CPU utilization average varied between practically all points of the ordinate axis (% CPU), reaching points of use of 100%. From this information, it can be seen that many requests were occurring and there were few virtual machines to meet these requests. Relating the CPU Average graph to the Resource Usage chart, it can be noted that when the CPU reaches an average usage of approximately 80% the load balancer starts to allocate more resources, in which case it allocates another machine in the public cloud to aid in the processing of excess load. This experiment lasted 11min14s. Four virtual machines were involved in this experiment: three in the private cloud and one in the public cloud. Thus, it was possible to verify if the infrastructure performance would be impacted by the addition of a virtual machine from a public cloud, and whether the SLA (Service Level Agreement) would be maintained. Figure 6 displays the graphs generated from the Resource Utilization and Average CPU metrics application. It also presents the allocation of virtual machines for further analysis.

From the analysis of the graph illustrated by Figure 6, it can be seen that the average CPU utilization in this experiment remained more constant than that of the previous experiment. This is due to the fact that there were more resources, that is, more virtual machines for the distribution of workloads. By relating the CPU Average graph to the Resource Utilization graph, it can be noted that when the CPU reaches an average usage of approximately 80% the load balancer starts to allocate more resources. When the CPU Utilization Average reaches 100%, all resources are allocated to assist in processing the excess load, which includes the allocation of the public cloud machine.

Final Remarks

Managing resource utilization in CC environments is an essential task. With the emergence of new technologies such as SDN and NFV, and their integrations with CC, this management becomes even more important. Therefore, the need for metrics that assess the quality of service of such environments is real. In the two presented experiments, the Resource Utilization metric shown results that match what was expected of the resource utilization of the analyzed environment. With the aid of the Average CPU metric it was possible to note that whenever the clouds were overloaded, more resources were allocated (virtual machines), which were available in the environment and the resources were being deallocated according to the load reduction.

In an integrated CC, SDN, and NFV environment, there is likely to be a need for resource allocation and deallocation (scaling up and scaling down)). However, the Resource Utilization metric would identify this allocation, but could have different behavior than it did in the experiments performed on the testing environment. This is future work to be planned and executed. Other intended future work is to design new metrics for quality evaluation for integrated environments, and to perform new experiments. Also, to use and scaling containers instead only virtual machines.

Figure 1 :

 1 Figure1: Autonomic architecture for elasticity in cloud computing using private and public clouds[START_REF] Emanuel | An architecture for providing elasticity based on autonomic computing concepts[END_REF]

Figure 2 :

 2 Figure 2: Workload applied to the infrastructure [6]

Figure 5 :

 5 Figure 5: Experiment 1: One virtual machine in the private cloud and one virtual machine in the public cloud. CPU Average Usage, Virtual Machine Allocation and Resource Utilization graphics

Figure 6 :

 6 Figure 6: Experiment 2: Three virtual machine in the private cloud and one virtual machine in the public cloud. CPU Average Usage, Virtual Machine Allocation and Resource Utilization graphics

Table 1 :

 1 Structure of the log files

		Structure
	Virtual Machine Allocation	Date of collection; Maintain (m), Increase (a) or Decrease (d); Number
		of allocated virtual machines
	Average CPU consumption	Date of collection; Average percentage of CPU value in all virtual
		machines

Acknowledgments

This work was partially supported by Universal MCTI/CNPq 01/2016 program (process 422342/2016-5).