First-Principles Study on Stability and Magnetism of Ni_𝑚Al_𝑛 (𝑚=1--3, 𝑛=1--9) Clusters

Abstract

The investigation on the structures, stabilities, and magnetism of Ni_𝑚Al_𝑛 (𝑚=1--3, 𝑛=1--9) clusters has been made by using first principles. We found some new ground-state structures which had not been found before. These mixed species prefer to adopt three-dimensional (3D) structures starting from four atoms. All the ground-state structures for the Ni-Al clusters are different from those of the corresponding pure Al clusters with the same number of atoms except for three atoms. The Mulliken population analysis shows that some charges transfer from the Al atoms to the Ni atoms. NiAl_𝑛 (𝑛 = odd number) cations, Ni₂Al₆ neutral, Ni₂Al₁ and Ni₃Al cations and anions, and Ni₃Al₅ anion have the magnetic moments of 2 𝜇_B. The magnetic moments of NiAl₄ and NiAl₆ cluster neutrals and cations are 2 𝜇_B and 3 𝜇_B, respectively. All the other cluster neutrals and ions do not have any nontrivial magnetic moments. The 3d electrons in Ni atoms are mainly responsible for the magnetism of the mixed Ni-Al clusters.

1. Introduction

Aluminum is one of the most popular metals used in quantum-effect electronic devices. It attracts scientists’ great attention to its nanoscale clusters. Cohen et al. have explained the stability of small Al_𝑛 clusters by using jellium model because of the valence electrons of Al-like free electron [1, 2]. Khanna et al. have discovered that aluminum clusters have chemical properties similar to single atoms of metallic and nonmetallic elements when they react with iodine [3, 4]. For instance, Al₁₃ cluster behaves like a single iodine atom, while Al₁₄ cluster is similar to an alkaline earth atom. Fournier have obtained the ground-state structures of Al_𝑛,Al_𝑛⁻,andAl_𝑛⁺(4≤𝑛≤15) by performing “Tabu Search” (TS) global optimizations directly on the BPW91/LANL2DZ potential energy surface [5]. They have found that all clusters (4≤𝑛≤15) have the lowest spin state as their ground state except Al₄ (triplet), Al₄⁺ (quartet), Al₇⁻ (triplet), and maybe Al₅⁺ (singlet and triplet are degenerate). Furthermore, Al₇⁺ and Al⁻₁₃ had special stability and larger HOMO-LUMO gaps.

The studies of the Al clusters doped with impurity atoms have been also reported. Ren and Li have investigated the structures, stabilities, and magnetism of zinc-doped Al_𝑛 (𝑛=1--9) clusters in detail by using first-principles density functional theory [6]. Ding and Li have done the calculation on Al_𝑛Si_𝑚−𝑛(𝑚=6,9,10,𝑛≤𝑚) by using the first-principles methods [7]. Li and Wang have performed first-principles calculations on the ground states of both neutral and anionic Al₁₂𝑋(𝑋=C,Si,Ge,Sn,Pb) clusters [8].

Nickel is a transition metal. Aluminum clusters doped Ni atoms have many novel properties and chemical properties. Ni_𝑛Al(𝑛=2--8) neutral clusters have been investigated by Wen et al. using the density functional theory and generalized gradient approximation (GGA) with the exchange-correlation potential (BPW91) [9]. It is found that the atomization energies per atom have the same trend as the binding energies per atom for Ni_𝑛(𝑛=3--9) clusters. It is shown that Ni₅Al is the relatively most stable structure in the series. Besides that,Ni average magnetic moment decreases when alloyed with Al atoms than that in pure Ni clusters. Deshpande et al. have reported the magnetic properties of small Ni_13−𝑛Al_𝑛 clusters with 𝑛=0--13 calculated in the framework of density functional theory [10]. The overall magnetic moment of the Ni_13−𝑛Al_𝑛 cluster decreases with the sequential substitution of Ni by Al atoms. This could be attributed to the antiferromagnetic alignment of small individual atomic moments arising due to the hybridization of Al-p and Ni-3d orbitals. Calleja et al. have reported ab initio molecular dynamics simulations of Ni₂,Al₂,Ni₁₃,Al₁₃,andNi₁₂Al clusters using a fully self-consistent density-functional method [11]. Recently, Sun et al. have performed the study on the growth behavior and electronic properties of NiAl_𝑛(𝑛=1--14) clusters in the framework of density-functional theory (DFT) theoretically [12]. In the ground-state structures of NiAl_𝑛 clusters, the equilibrium site of Ni atom gradually moves from convex and surface to interior site as the number of Al atom varying from 2 to 14. Ferrando et al. have performed very thorough review on alloyed clusters [13]. Bailey et al. have presented their research results about nickel and aluminum clusters and nickel-aluminum nanoalloy clusters with up to 55 atoms [14]. The effect of doping Al atoms into pure Ni clusters and vice versa has been investigated.

Although the investigations about Al, Ni, and Ni_𝑛Al clusters mentioned above have been performed, our knowledge about the properties of the Ni-Al clusters with different Ni/Al atomic ratio is still limited. In this paper, our main goal is to explore the effect of the Ni atoms on the geometric, electronic, and magnetic properties of the pure Al_𝑛(𝑛=2--9) clusters.

The paper is organized as follows: a brief account of the computational methodology is given in Section 2, followed by a detailed presentation and discussion of the structures of different size Ni_𝑚Al_𝑛(𝑛=1--9,𝑚=1--3) clusters in Section 3. A summary of the findings and conclusions is given in Section 4.

2. Computational Method

We optimized the Ni-Al cluster structures by using first-principles density functional theory. Our calculations were performed with the generalized gradient approximation (GGA) by means of the Becke-Perdew functional, which uses Becke’s [15] gradient correction to the local expression for the exchange energy and Perdew’s [16] gradient correction to the local expression of the correlation energy, as implemented in the Amsterdam Density Functional (ADF) codes [17]. The choice of such a gradient-corrected exchange-energy functional gives us satisfactory results, which are in good agreement with the experimental results. For example, the calculated vertical electron affinity and vertical ionization potential for Al₅ cluster are 2.16 eV and 6.61 eV, respectively. The available experimental values are 2.25 eV and 6.45 eV, respectively [18, 19]. In the ADF program, molecular orbitals (MOs) were expanded using a large, uncontracted set of Slater-type orbitals (STOs): TZ2P. The TZ2P basis is an all-electron basis of triple-ζ quality, augmented by two sets of polarization functions. The frozen-core approximation for the inner-core electrons was used. The orbitals up to 3p  for nickel and 2p for aluminum were kept frozen. An auxiliary set of 𝑠,𝑝,𝑑,𝑓,and𝑔 STOs was used to fit the molecular density and to represent the Coulomb and exchange potentials accurately in each self-consistent field (SCF) cycle. The self-consistent field was converged to a value of 10⁻⁵ in Hartree.

Frequency analyses at the stable structures are carried out at the same theoretical level to clarify if they are true minima or transition states on the potential energy surfaces of specific clusters. All of the most stable clusters in the paper are characterized as energy minima without imaginary frequencies.

3. Results and Discussions

Searching for the ground-state structures of the atomic clusters, especially the mixed clusters, is a difficult task. The following approach is applied in an attempt to find the global energy minimum structures. Firstly, about more than ten thousand initial geometrical configurations are produced by random selections of atomic positions within a three-dimensional box, or a cage, or a ball in real space. The distance between two nearest atoms is properly chosen to avoid overlapping or loosely packing. Next, the Amsterdam Density Functional (ADF) package is used for geometrical optimizations with their spins being not unrestricted. After the structural optimization on the initial configurations is performed, it is found that some structures are stable, but others are not convergent. Sometimes, several initial structures can gave out the same final structure after the first-principles optimizations. In the end, the structures with the largest binding energies calculated using generalized gradient approximation (GGA) are considered to be the ground-state geometries. The binding energy (BE) for the Ni_𝑚Al_𝑛 cluster is calculated according to the following atomization reaction: Ni_𝑚A1_𝑛→𝑚Ni+𝑛A1. It is defined by the following: BE=𝑚𝐸_Ni+𝑛𝐸_A1−𝐸_Ni𝑚A1𝑛, where 𝐸_Ni𝑚A1𝑛 is the total energy of the Ni_𝑚Al_𝑛 cluster. 𝐸_Ni and 𝐸_Al are the energy of Ni and Al atoms, respectively.

All the lowest energy structures of Al₃, Al₄, and Al₅ clusters are planar structures. They are an equilateral triangle, a planar quadrangle, a trapezoid-like, respectively. The ground-state structures for Al_𝑛(𝑛=6--10) clusters are three-dimensional. The Al₆ and Al₇ clusters have a distorted tetragonal bipyramid (D_2d) and an antitrigonal prism with a capping atom (C_3v), respectively. But, Fournier found that the ground-state structure for the Al₆ cluster was a distorted trigonal prism by using the BPW91/LANL2DZ method [5]. For the Al₇ cluster, its ground-state structure can be obtained by putting an atom on one triangular face of the prism for the Al₆ cluster. We have also performed calculations on the two structures found by Fournier with the ADF program. It is found that both of them are metastable. Our results are 0.15 and 0.36 eV more stable than his, respectively. For the Al₈ cluster, our result is a transcapped octahedron with C_2h symmetry. But, in Fournier’s report, the ground-state structure is a distorted transcapped octahedron, or an antitrigonal prism with a face-capping atom and an edge-capping atom. We have performed structural optimization on his structure. It is found that his structure finally changes into the C_2h structure. The Al₉ cluster has a C_2v ground-state structure. The most stable structure of pure Al₁₀ cluster can be obtained from the Al₉ ground-state isomer by putting a capping atom. All of the structures obtained by the ADF program are in excellent agreement with those reported by Chuang et al. using a genetic algorithm coupled with a tight-binding interatomic potential [20]. They are shown in Figure 1.

Figure 1 

The ground-state structures of the Al𝑛(𝑛=3--10) clusters.

The ground-state structures of NiAl_𝑛(𝑛=2--9) clusters are presented in Figure 2. The ionic structures similar to the neutral structures are not repeated in Figure 2. But the structure of NiAl₉ cluster cation is presented in Figure 2 because it has a severe distorted ground-state structure compared with its neutral and anion. Their symmetries are listed in Tables 1, 2, and 3. The NiAl₃ and NiAl₄ clusters and their ions have one three-dimensional structure, which are different from the planar structures of the Al₄ and Al₅ clusters. All the most stable structures for the NiAl_𝑛(𝑛=5--9) cluster neutrals, anions and cations are three-dimensional structures, but they are obviously different from the structures of the corresponding Al_𝑛(𝑛=6--10) clusters with the same number of atoms. Positive NiAl₉ cluster ion with C₁ symmetry has a severe distorted ground-state structure compared with its neutral and anion (see the NiAl₉⁺ structure in Figure 2). Mulliken population analysis shows that nonuniform charge loss of the Al atoms in the NiAl₉⁺ structure results in its severe structure distortion.

Table 1 

The symmetries, the total magnetic moments (𝑀, in 𝜇_B), and the binding energies (BE, in eV) for NiAl_𝑛 (𝑛=2--9) clusters and their ions. The energy gaps (𝐸_𝑔, in eV), the vertical electron affinities (EA, in eV), and vertical ionization potentials (IP, in eV) of neutral NiAl_𝑛 (𝑛=2--9) clusters.

Table 2 

The symmetries, the total magnetic moments (𝑀, in 𝜇_B), and the binding energies (BE, in eV) for Ni₂Al_𝑛 (𝑛=2--9) clusters and their ions. The energy gaps (𝐸_𝑔, in eV), the electron affinities (EA, in eV), and ionization potentials (IP, in eV) of neutral Ni₂Al_𝑛 (𝑛=2--9) clusters.

Table 3 

The symmetries, the total magnetic moments (𝑀, in 𝜇_B), and the binding energies (BE, in eV) for Ni₃Al_𝑛 (𝑛=2--9) clusters and their ions. The energy gaps (𝐸_𝑔, in eV) and the electron affinities (EA).

Figure 2 

The ground-state structures of the NiAl𝑛(𝑛=2--9) clusters. The black ball refers to Ni atom in the clusters.

The cluster’s magnetism is usually described by its magnetic moment. We define magnetic moment (in Bohr magnetron unit 𝜇_B) as the difference between the electronic occupation number of spin-up and spin-down. We think the cluster is magnetic if its structure with nonzero spin is more stable than that with zero spin. The investigation on the magnetic moments for the Ni-Al clusters shows that anions, neutrals, and cations for the NiAl₂ and NiAl₈ clusters have the magnetic moments of 1 𝜇_B, 0, 1 𝜇_B, respectively. Negative, neutral, and positive NiAl₄ and NiAl₆ clusters have the magnetic moments of 1 𝜇_B, 2 𝜇_B, and 3 𝜇_B, respectively. The NiAl,NiAl₃,NiAl₅,NiAl₇,NiAl₉ clusters with odd-aluminum atoms have the magnetic moments of 1 𝜇_B. Their negative ions have no magnetic moment, but the positive ions show the magnetic moment of 2 𝜇_B.

The most stable structures of Ni₂Al_𝑛(𝑛=1--9) clusters are presented in Figure 3. It is found from observing Figure 3 that the geometries are also different from those in Figures 1 and 2. Obviously, the second impurity Ni atom further changes the structures of Al host clusters. The lowest energy structure Ni₂Al₅a of Ni₂Al₅ cluster which is obtained by a random way has a C₁ symmetry. The second most stable structure Ni₂Al₅b which produced by substitution from Al₈ has a C_s symmetry and lies 0.07 eV above Ni₂Al₅a structure. All the ground state structures of the Ni₂Al₆, Ni₂Al₇,  and  Ni₂Al₉ clusters have the same C_2v symmetry. The Ni₂Al₇ cluster can be obtained by substitution, by a random way, or by putting a capping atom on the Ni₂Al₆ cluster. The Ni₂Al₈ cluster has a low C₁ symmetry.

Figure 3 

The ground-state structures of the Ni2Al𝑛(𝑛=2--9) clusters. The black ball refers to Ni atoms in the clusters.

By performing calculation on the magnetic moments for the Ni₂Al_𝑛(𝑛=1--9) clusters, we obtain the following results. All the Ni₂Al_𝑛(𝑛=odd-number) clusters have the total magnetic moment of 1.0 𝜇_B due to one unpaired electron. This suggests that the two impurity Ni atoms do not change the magnetism of the Al clusters. On the other hand, the Ni₂Al₂, Ni₂Al₄, and Ni₂Al₈ clusters with even number of electrons have no magnetic moment because all the electrons are paired together in their respective molecular orbitals. But, unexpectedly, the Ni₂Al₆ cluster shows magnetic moment of 2 𝜇_B. Each Ni atom in the Ni₂Al₆ cluster has the local magnetic moment 0.40 𝜇_B and the total charge transferring from the Al atoms to them is −0.74 e. However, the average magnetic moment of each Al atom is 0.20 𝜇_B, which is only half of that of the Ni atoms. The Mulliken population analyses indicate that the local magnetic moment 0.40 𝜇_B of Ni atoms is mainly from the d orbital. On the other hand, if we add up the magnetic moments for the Al atoms and the Ni atoms, respectively, the total magnetic moments for Al atoms are 1.2 𝜇_B, which is larger than 0.8 𝜇_B for Ni atoms. Hence, Al atoms contribute more to total spin than Ni atoms.

The Ni₂Al⁻andNi₂Al⁺ cluster ions are expected to produce zero magnetic moment. But our calculations show that the ions have the magnetic moment of 2 𝜇_B. It is found that the Ni₂Al₃,Ni₂Al₅,Ni₂Al₇,andNi₂Al₉ clusters ions have no magnetic moment. The magnetic moment for the Ni₂Al₄,Ni₂Al₆,andNi₂Al₈ cluster ions is 1 𝜇_B. Furthermore, it is found that the electron added or removed in the neutral Ni₂Al_𝑛(𝑛=1--9) clusters does not change the basic geometrical structures.

The lowest energy structures of Ni₃Al_𝑛(𝑛=1--9) clusters are presented in Figure 4. All the structures are three-dimensional. They differ from the geometries of the NiAl_𝑛 and Ni₂Al_𝑛(𝑛=1--9) clusters in Figures 2 and 3. The clusters have C_3v,C_2v,C_s,orC₁ symmetry. The ground-state structure of the Ni₃Al₈ cluster is degenerate. The Ni₃Al₈a structure lies only 0.03 eV below the Ni₃Al₈b structure. The Ni₃Al₈b structure can be obtained by putting two Al capping atoms in the Ni₃Al₆ structure. The Ni₃Al₉a structure is 0.26 eV more stable than the Ni₃Al₉b structure.

Figure 4 

The ground-state structures of the Ni3Al𝑛(𝑛=2--9) clusters. The black ball refers to Ni atoms in the clusters.

The Ni₃Al_𝑛(𝑛=odd-number) clusters have the magnetic moment of 1.0 𝜇_B due to one unpaired single electron. The magnetic moment is trivial. For the Ni₃Al_𝑛(𝑛=even-number) clusters show no magnetic moment because the molecular orbitals are doubly occupied. For their ions, some clusters show magnetism. The Ni₃Al⁻,Ni₃Al⁺,andNi₃Al₅⁻ cluster ions are expected to produce zero magnetic moment. But our calculations indicate that they have the magnetic moment of 2 𝜇_B. All the Ni₃Al₃, Ni₃Al₇, and Ni₃Al₉ clusters ions including the Ni₃Al₅⁺ cation have no magnetic moments. The magnetic moment for the Ni₃Al₄, Ni₃Al₆, and Ni₃Al₈ clusters ions is 1 𝜇_B, which is trivial.

To investigate the bonding character of the mixed Ni-Al clusters in relation to molecular orbitals, we have also examined the density of states in terms of the contributions of the different orbital components (s, p, d) and the frontier orbital (HOMO and LUMO) states. The Mulliken population analyses indicate that some of charge transfer from the Al atoms to the Ni atoms. Here, we take the Ni₂Al₆ cluster with the magnetic moment of 2 𝜇_B as an example to illustrate their bonding properties. The orbitals of HOMO and LUMO states of the Ni₂Al₆ cluster are shown in Figure 5. For the orbitals of the HOMO state, the cluster binds in the term of 𝜎 and 𝜋 orbital. Likewise, the binging characteristic varies from 𝜎 orbital to 𝜋 orbital and 𝛿 feature between Ni and Al atoms for the LUMO state.

(a)

(b)

Figure 5 

The HOMO and LUMO orbitals of the Ni2Al6 cluster.

In order to better understand the relative stability of the Ni-Al clusters, we have also investigated the second difference of cluster energies, the energy gaps, and their positive and negative ion clusters. The second difference of cluster energies Δ₂𝐸=𝐸(Ni_𝑚Al_𝑛+1)+𝐸(Ni_𝑚Al_𝑛−1)−2𝐸(Ni_𝑚Al_𝑛)(𝑚=1--3,𝑛=2--9), which is a sensitive quantity that reflects the relative stability of the mixed clusters. Maximum Δ₂𝐸 indicates that the cluster is more stable than its neighboring clusters. The maximums for the Ni₁Al_𝑛(𝑛=1--9)and Ni₂Al_𝑛(𝑛=1--9) clusters are found at 𝑛=3and7 (see Figure 6). The conclusion is a good agreement with that obtained by Sun et al. [12]. For the Ni₃Al_𝑛(𝑛=1--9) clusters, the maximums are at 𝑛=4and6. It is found from observing Figure 6 that the minimum for the mixed clusters is at 𝑛=5. This implies that the Ni_𝑚Al₅(𝑚=1--3) clusters are less stable than their neighboring clusters.

(a)

(b)

(c)

(d)

Figure 6 

The second difference of the cluster energies, the energy gaps 𝐸𝑔 between the highest-occupied molecular orbital (HOMO) and the lowest-unoccupied molecular orbital (LUMO), the electron affinities (EA), and the ionization potentials (IP) for Ni𝑚Al𝑛(𝑚=1--3,𝑛=1--9) clusters.

The maximums of Δ₂E for the pure Al_𝑛(𝑛=2--10) clusters are also at 𝑛=3and7. The stability probably can be explained by jellium model [2, 21]. In this model, the nuclei together with the innermost electrons form a positive charged background, whereupon the valence electrons coming from individual atoms are then subjected to this potential. The clusters containing 8, 20, 40, 58… valence electrons correspond to filled electronic shells and exhibit enhanced stability. But the stability is associated not only with closed electronic shells (𝑁𝑒=20,40) but also, to a lesser extent, with numbers of electrons just above and below the shell closing numbers [5]. The simultaneous stability of the neutral, anion, and cation of a given size can happen only for clusters of elements having more than one itinerant electron per atom. Therefore, the higher stability of Al₃andAl₇ clusters is easily understood. It is found from observing Figure 6 that the mixed Ni_𝑚Al₃andNi_𝑚Al₇(𝑚=1or2) clusters also have high stability. This indicates that the stability of the nickel-doped Al_𝑛 clusters is determined by the host Al clusters. But it should be noted that more heteroatoms would change their relative stability, as we have seen in the Ni₃Al_𝑛(𝑛=2--9) clusters. Al₅ cluster is less stable compared to other Al clusters. This is the reason that the Ni_𝑚Al₅(𝑚=1--3) clusters are less stable than their neighboring clusters.

The energy gaps 𝐸_𝑔s as an important function of the cluster size present a characteristic quantity of metal clusters’ chemical activity. Here we define 𝐸_𝑔s as the gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). For the pure Al_𝑛 clusters, it is found that the Al₂ cluster has a large gap of 3.05 eV. Other gaps change within range from 0.32 to 0.84 eV. For the smaller mixed clusters, their energy gaps change drastically. But they show relatively small fluctuation as the atomic number increases. It is worth noting that the 𝐸_𝑔 curves display an even/odd alternating pattern as a function of cluster size for the Ni₃Al_𝑛(𝑛=1--9) clusters. Even though the Ni₁Al_𝑛andNi₂Al_𝑛(𝑛=1--9) clusters with larger than five atoms do not present the even/odd alternating pattern, their change trend is similar as the atomic number increases.

Finally, we have also investigated the Ni_𝑚Al_𝑛(𝑚=1--3,𝑛=1--9) positive and negative ion clusters. The electron affinities EA=𝐸(Ni_𝑚Al_𝑛)−𝐸(Ni_𝑚Al_𝑛⁻)(𝑚=1--3,𝑛=1--9) are the amount of energy released when the cluster obtains an electron from its neutral state. In our research, all the vertical electron affinities (EA) of the Ni_𝑚Al_𝑛(𝑚=1--3,𝑛=1--9) clusters are positive, suggesting that the clusters have a tendency to gain an electron, as it is energetically favorable to do so. The vertical electron affinities (EA) change with atomic number, shown as Figure 6. The graph of vertical electron affinities versus atomic number shows that the mixed clusters with one, two, and three nickel atoms have a similar change trend. For the positive ion clusters, similar energies called vertical ionization potentials are calculated. The graph of vertical ionization potentials versus atomic number displays different changes. For the Ni₂Al_𝑛(𝑛=2--9) clusters with two nickel atoms, the vertical ionization potentials (IP) show an even/odd alteration with the number of aluminum atoms. For the Ni₃Al_𝑛(𝑛=2--9) clusters with three nickel atoms, a similar change trend (exceptfor𝑛=8) is observed. But, for the mixed clusters including one nickel atom, the property does not exist.

4. Conclusions

We have optimized the geometric structures of the mixed Ni_𝑚Al_𝑛(𝑚=1--3,𝑛=1--9) clusters by using first-principles density functional theory. Their ground-state structures are obtained. Most of them are different from those of the host Al clusters. The impurity Ni atoms result in the local structural distortion of the mixed clusters in comparison with the corresponding pure Al clusters. At the same time, we calculated magnetic moment of the mixed cluster neutrals and ions. Some of them present nontrivial magnetic moments. The NiAl_𝑛(𝑛=oddnumber) positive ions, the Ni₂Al₆neutral, theNi₂Al₁andNi₃Al positive and negative ions, and the Ni₃Al₅ negative ion possess the magnetic moments of 2 𝜇_B. NiAl₄andNiAl₆ cluster neutrals and positive ions have the magnetic moments of 2 𝜇_B and 3 𝜇_B, respectively. But all the other cluster neutrals and ions do not have any nontrivial magnetic moments. Some of charge transfer from the Al atoms to the Ni atoms. The local magnetic moment of Ni atoms is mainly from the d orbital. Both the structural effect and chemical bonding are responsible for the magnetism of the mixed Ni-Al clusters.

Acknowledgments

The Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant No. 200320), the Natural Science Foundation of Zhejiang Province (Grant no. Y6100098), and the National Natural Science Foundation of China (Grant no. 11074062) supported this work.