Structural Properties of Solid Oxygen: density functional study Tatsuki ODA Graduate School of Natural Science and Technology, Kanazawa University Kanazawa, 920-1192, Japan Abstract the liquid transforms to the solid phases; γ- phase at 54 K, β-phase at 44 K, and α-phases We have studied structural properties of the at 24 K [5]. These are categorized to the molec- high-pressured solid oxygens with using the ular crystal that there are covalent bonds in density functional theory. For the "-phase, we intramolecule and weak interactions between employed the lattice constants of available ex- intermolecules. The magnetism of molecules is perimental data and obtained the stable struc- preserved in these phases [6, 7] and the α-phase ture of double chain. The structure factor ex- has an anti-ferromagnetic ordering of molecu- tracted from the electron density shows a good lar magnets [8]. In these phases, the rectan- agreement with experimental results. The lo- gular shape of O4 is a typical local structure cal structure of chain was found to have O2 of multi-molecules. The isolated O4 has the pairs, which is consistent with the results of op- ground state of spin singlet in gaseous phase tical measurements. The persistent magnetic [9] and could be found in liquid phase [10, 11]. polarization has never been supported. In the latter phase, the chain of O2 has been implied [12]. This was because the γ-phase had the molecular chains accompanied with the ro- 1 Introduction tational fluctuation of molecules. The crystal structure of β-phase has the two-dimensional Oxygen molecule has a spin-triplet ground array of molecules, in which the triangular state. The magnetism on molecules is the lattice causes an magnetic frustration among main origin of paramagnetic or antiferromag- anti-ferromagnetic interactions [8]. netic properties of condensed oxygens. It could make complex on the structural property of Imposing the hydrostatic pressure, the α- oxygen. High pressures on oxygen have pro- phase transforms to novel states of structure. moted the collapse of magnetism and the in- The optical measurements implied δ-phase in duction of metallic feature. In the metallic a wide range of pressure at low temperatures phase, the superconducting property was also [13]. Akahama et al. concluded the direct reported [1]. In this report, we present a short transition to "-phase at 7.6 GPa from the re- review for condensed phases and our recent sults of X-ray diffraction measurements [14]. work on the structural property in "-phase [2]. From the analysis of spin-polarized neutron This phase appears in a wide pressure range diffraction measurements, Goncharenko et al. of solid oxygen. The atomic positions in the found the phase boundary of different mag- phase has not been determined yet. The mag- netic orders [15]. These recent studies for netic collapse and the metalization has been phase boundary among α-, δ-, and "- seem to studied at boundaries of the phase[3, 4]. be still controversy. The phase diagram for low Oxygen forms gas phase at the room tem- pressures is presented in Fig. 1. The boundary peratures and normal pressures. At the low between liquid- and solid-phases could be ex- temperature of 90 K, the gas changes to a para- pected (not shown) in Fig. 1. For "-phase, magnetic liquid. At further low temperatures, based on the speculation related with mag- netic interactions and triangular local struc- The structural analysis with X-ray diffrac- tures, the noncollinear magnetism has been tion measurement at "-phase has been per- studied [16]. formed first by Johnson et al. and successively For the high pressures, no phase transition by some groups [3, 18, 19]. The magnetic α- has been observed up to 96 GPa [3], above phase is transformed to "-phase around 8 GPa which the system of "-phase transforms to [14]. The α- and "-phases are analyzed to metallic ζ-phase. The early work on electronic monoclinic phases classified by the same space structure calculation for solid oxygens was de- group (C2=m), as shown in Fig. 1 [20]. The voted to the study on insulator-metal transi- molecular axes point in the direction normal to tion [17]. At the transition pressure, where ab-plane. This is consistent with both of avail- the discontinuity in the X-ray diffraction pro- able experimental and theoretical results. The files was found, the vibron frequency has been unit cell of "-phase is dimerized along both a- still observed [18]. This is the signature of and b-axes at the transition from that of α- O2 molecule. The structure of ζ-phase is not phase, resulting to eight molecules in the cell. definitively determined, however the theoret- The most impressive feature is an appearance ical approach for the high-pressured crystal of the new (31¯1¯) line in the diffraction profile proposed the prototype of crystal structure [14]. This is accompanied by the (31¯0) line, which has a good agreement in X-ray diffrac- whose d-value (distance between neighboring tion profiles [4]. diffraction planes) crosses around 50 GPa with The feature of "-phase is an occupation over that of the (22¯0) diffraction line [18]. a wide range of pressures in the phase diagram. The infrared and Raman spectra of op- This means a stability of atomic and electronic tical measurements have provided the local structures. So far, however, the atomic posi- structure information. Angnew et al. pro- tion of "-phase has never been determined yet. posed a simple chain model of O4 unit to ex- It could be interesting to know a form of molec- plain their data and Gorelli et al. found the ular arrangement, because solid oxygen have new peak in the far infrared region, confirm- the peculiar properties which the other solid ing the existence of dimerized unit of oxygen of diatomic molecules never has. The determi- molecules [21, 22, 23]. This O4 unit seems to nation of structure in "-phase will prompt the be non-magnetic unlike the unit in the liquid study on solid oxygen considerably. or gaseous phases [9, 11]. The recent study based on the density functional theory (DFT)[24] predicted the herringbone-type chain structure(space group Cmcm) in dense oxygen [25]. In this study, stability of the phase was investigated by estimating enthalpies accurately from first- principles. The assumption for size of unit cell, however, may not be commensurate with the results of X-ray diffraction. We proposed a new crystal structure which models the "-phase. This study was based also on the density functional approach [24], but performed with referring the result of X-ray diffraction closely. Taking into account this Figure 1: Phase diagram of oxygen. As priority of study, we used only total energies a boundary between α- and δ- phases, the and atomic forces for structural optimization. boundaries specified by the curve 1 and curve 2 Our proposed structure, which has a double were proposed on the base of the X-ray diffrac- chain consisting O2 pairs, shows lower ener- tion analysis [14] and the spin-polarized neu- gies than the herringbone-type chain structure tron diffraction [18], respectively. at high pressures and consistencies with exper- Table 1: Lattice constants at the pressures for 3 calculations. 2 P(2,2) 1 (GPa) Lattice Constants P(1,1) a (A)˚ b (A)˚ c (A)˚ β (deg.) 0 P(1,3) 9.6a 8.248 5.768 3.814 117.66 -1 b -2 19.7 7.705 5.491 3.642 116.2 -3 c Pressure (GPa) 33 7.39 5.23 3.53 115.6 -4 P(3,3) c 54.5 7.13 4.99 3.43 115.3 -5 c 71 7.00 4.83 3.36 115.1 -6 c -7 90 6.89 4.70 3.31 114.8 40 60 80 100 120 Energy Cut-off (Ry) a Ref. [14], b Ref. [19], c read from Fig 2 in Ref. [18] Figure 2: Energy cut-off dependences of pres- imental results. sure tensor. Four non-vanishing elements are presented only. The experimental lattice con- stants of 0.96 GPa were used [14]. 2 Method THz) and the latter is larger than the exper- For the density functional approach, we used imental value (5.12 eV) by 17 % [30]. These planewave basis to represent wavefunctions properties are similar to the previous results in and electron densities and ultrasoft-type pseu- density functional approaches [4, 25, 28]. The dopotentails [26, 27] to include core-valence in- lattice constants used in this work were ex- teractions. The 1s states were included in the tracted in the X-ray diffraction measurements core, while the other states were described ex- at five points of pressure (9.6, 19.7, 33, 54.5, plicitly. In the construction of pseudopoten- 71, 90 GPa) [14, 18, 19]. and listed in Table 1. tials, we took the cut-off radius of 1.15 a.u. with including a d-symmetry local orbital [28]. The energy cut-offs of 100 and 400 Ry were taken for wavefunction and electron density, -31.981 respectively. This level of cut-off shows the -31.982 convergence of internal pressures as well as the total energy of systems. Figure 2 represents -31.983 the energy cut-off dependence of pressure ten- -31.984 sor which was calculated with the lattice con- Energy (Ha) -31.985 stants at 0.96 GPa [14]. The k-point sampling of 4×4×4 meshes for the eight-molecular unit -31.986 cell was used. It is essentially enough to com- -31.987 1.16 1.18 1.2 1.22 1.24 1.26 1.28 pare the total energies for two types of atomic Bond Length (Angstrom) configuration at fixed volumes for semiconduc- tors.