Structural Properties of Solid Oxygen:
density functional study
Tatsuki ODA
Graduate School of Natural Science and Technology, Kanazawa University
Kanazawa, 920-1192, Japan
Abstract
We have studied structural properties of the
high-pressured solid oxygens with using the
density functional theory. For the ε-phase, we
employed the lattice constants of available experimental data and obtained the stable structure of double chain. The structure factor extracted from the electron density shows a good
agreement with experimental results. The local structure of chain was found to have O 2
pairs, which is consistent with the results of optical measurements. The persistent magnetic
polarization has never been supported.
1
Introduction
Oxygen molecule has a spin-triplet ground
state. The magnetism on molecules is the
main origin of paramagnetic or antiferromagnetic properties of condensed oxygens. It could
make complex on the structural property of
oxygen. High pressures on oxygen have promoted the collapse of magnetism and the induction of metallic feature. In the metallic
phase, the superconducting property was also
reported [1]. In this report, we present a short
review for condensed phases and our recent
work on the structural property in ε-phase [2].
This phase appears in a wide pressure range
of solid oxygen. The atomic positions in the
phase has not been determined yet. The magnetic collapse and the metalization has been
studied at boundaries of the phase[3, 4].
Oxygen forms gas phase at the room temperatures and normal pressures. At the low
temperature of 90 K, the gas changes to a paramagnetic liquid. At further low temperatures,
the liquid transforms to the solid phases; γphase at 54 K, β-phase at 44 K, and α-phases
at 24 K [5]. These are categorized to the molecular crystal that there are covalent bonds in
intramolecule and weak interactions between
intermolecules. The magnetism of molecules is
preserved in these phases [6, 7] and the α-phase
has an anti-ferromagnetic ordering of molecular magnets [8]. In these phases, the rectangular shape of O4 is a typical local structure
of multi-molecules. The isolated O 4 has the
ground state of spin singlet in gaseous phase
[9] and could be found in liquid phase [10, 11].
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 rotational fluctuation of molecules. The crystal
structure of β-phase has the two-dimensional
array of molecules, in which the triangular
lattice causes an magnetic frustration among
anti-ferromagnetic interactions [8].
Imposing the hydrostatic pressure, the αphase transforms to novel states of structure.
The optical measurements implied δ-phase in
a wide range of pressure at low temperatures
[13]. Akahama et al. concluded the direct
transition to ε-phase at 7.6 GPa from the results of X-ray diffraction measurements [14].
From the analysis of spin-polarized neutron
diffraction measurements, Goncharenko et al.
found the phase boundary of different magnetic orders [15]. These recent studies for
phase boundary among α-, δ-, and ε- seem to
be still controversy. The phase diagram for low
pressures is presented in Fig. 1. The boundary
between liquid- and solid-phases could be expected (not shown) in Fig. 1. For ε-phase,
based on the speculation related with mag-
netic interactions and triangular local structures, the noncollinear magnetism has been
studied [16].
For the high pressures, no phase transition
has been observed up to 96 GPa [3], above
which the system of ε-phase transforms to
metallic ζ-phase. The early work on electronic
structure calculation for solid oxygens was devoted to the study on insulator-metal transition [17]. At the transition pressure, where
the discontinuity in the X-ray diffraction profiles was found, the vibron frequency has been
still observed [18]. This is the signature of
O2 molecule. The structure of ζ-phase is not
definitively determined, however the theoretical approach for the high-pressured crystal
proposed the prototype of crystal structure
which has a good agreement in X-ray diffraction profiles [4].
The feature of ε-phase is an occupation over
a wide range of pressures in the phase diagram.
This means a stability of atomic and electronic
structures. So far, however, the atomic position of ε-phase has never been determined yet.
It could be interesting to know a form of molecular arrangement, because solid oxygen have
the peculiar properties which the other solid
of diatomic molecules never has. The determination of structure in ε-phase will prompt the
study on solid oxygen considerably.
Figure 1: Phase diagram of oxygen. As
a boundary between α- and δ- phases, the
boundaries specified by the curve 1 and curve 2
were proposed on the base of the X-ray diffraction analysis [14] and the spin-polarized neutron diffraction [18], respectively.
The structural analysis with X-ray diffraction measurement at ε-phase has been performed first by Johnson et al. and successively
by some groups [3, 18, 19]. The magnetic αphase is transformed to ε-phase around 8 GPa
[14]. The α- and ε-phases are analyzed to
monoclinic phases classified by the same space
group (C2/m), as shown in Fig. 1 [20]. The
molecular axes point in the direction normal to
ab-plane. This is consistent with both of available experimental and theoretical results. The
unit cell of ε-phase is dimerized along both aand b-axes at the transition from that of αphase, resulting to eight molecules in the cell.
The most impressive feature is an appearance
of the new (31̄1̄) line in the diffraction profile
[14]. This is accompanied by the (31̄0) line,
whose d-value (distance between neighboring
diffraction planes) crosses around 50 GPa with
that of the (22̄0) diffraction line [18].
The infrared and Raman spectra of optical measurements have provided the local
structure information. Angnew et al. proposed a simple chain model of O4 unit to explain their data and Gorelli et al. found the
new peak in the far infrared region, confirming the existence of dimerized unit of oxygen
molecules [21, 22, 23]. This O4 unit seems to
be non-magnetic unlike the unit in the liquid
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 firstprinciples. 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 ε-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
priority of study, we used only total energies
and atomic forces for structural optimization.
Our proposed structure, which has a double
chain consisting O2 pairs, shows lower energies than the herringbone-type chain structure
at high pressures and consistencies with exper-
Table 1: Lattice constants at the pressures for
calculations.
(GPa)
Lattice Constants
a (Å) b (Å) c (Å) β (deg.)
9.6a
8.248
5.768 3.814
117.66
b
19.7
7.705
5.491 3.642
116.2
33c
7.39
5.23
3.53
115.6
c
54.5
7.13
4.99
3.43
115.3
71c
7.00
4.83
3.36
115.1
90c
6.89
4.70
3.31
114.8
a Ref. [14], b Ref. [19],
c read from Fig 2 in Ref. [18]
imental results.
2
P(2,2)
1
P(1,1)
P(1,3)
Pressure (GPa)
0
-1
-2
-3
P(3,3)
-4
-5
-6
-7
40
60
80
100
120
Energy Cut-off (Ry)
Figure 2: Energy cut-off dependences of pressure tensor. Four non-vanishing elements are
presented only. The experimental lattice constants of 0.96 GPa were used [14].
Method
For the density functional approach, we used
planewave basis to represent wavefunctions
and electron densities and ultrasoft-type pseudopotentails [26, 27] to include core-valence interactions. The 1s states were included in the
core, while the other states were described explicitly. In the construction of pseudopotentials, 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,
respectively. This level of cut-off shows the
convergence of internal pressures as well as the
total energy of systems. Figure 2 represents
the energy cut-off dependence of pressure tensor which was calculated with the lattice constants at 0.96 GPa [14]. The k-point sampling
of 4 × 4 × 4 meshes for the eight-molecular unit
cell was used. It is essentially enough to compare the total energies for two types of atomic
configuration at fixed volumes for semiconductors. The exchange and correlation energy was
treated in the generalized gradient approximation (GGA) proposed by Perdew and Wang
(PW91) [29].
The ground state of molecule corresponds
to the spin-polarized state with a magnetization of 2 µB and shows a bond length of 1.22
Å(see Fig. 3), a vibrational frequency of 46.7
THz, and a binding energy of 5.99 eV. The two
former are in good agreement with the corresponding experimental values (1.21 Å, 47.39
THz) and the latter is larger than the experimental value (5.12 eV) by 17 % [30]. These
properties are similar to the previous results in
density functional approaches [4, 25, 28]. The
lattice constants used in this work were extracted in the X-ray diffraction measurements
at five points of pressure (9.6, 19.7, 33, 54.5,
71, 90 GPa) [14, 18, 19]. and listed in Table 1.
-31.981
-31.982
-31.983
Energy (Ha)
2
3
-31.984
-31.985
-31.986
-31.987
1.16
1.18
1.2
1.22
1.24
1.26
1.28
Bond Length (Angstrom)
Figure 3: Total energies with respect to bond
length.
3
Result and Discussion
We consider the molecular arrangement of
monoclinic α-phase [8, 20], as an ideal structure, in which molecules form two rectangular
sublattices in the ab-plane and the molecular
u
1
u
∆v
(a) type 1
2
∆x
(b) type 2
7
1'
Figure 4: Distortion of molecules in ab-plane
for (a) type 1 and (b) type 2.
axis is perpendicular to the plane. The c-axis
forms the angle of about 116 degree with the aaxis [3]. This ideal structure shows the extinction of (31̄0) diffraction for the structure factor. To examine this diffraction line in structure factor, we performed the simple simulation in which we assumed the charge density
of Gaussian type on atomic positions. As a
result, it was found that to induce the experimental features we could make the distortion,
alternate chain slidings along [110] direction,
represented in Fig. 4(a) (calls as type 1), in
which the molecules are taken off from the
(31̄0) planes alternatively. In contrast to this,
the atomic configuration of previous work presented as type 2 in Fig. 4(b), shows the extinction of (31̄0) and forms two strong diffraction
lines, (200) and (201̄), around the (001) line.
As shown Figs. 4(a) and (b), the internal coordinates, ∆v and ∆x, for these types of distortion determine the individual atomic positions
and we could use initial atomic configurations
for optimization.
At the pressure of 9.6 GPa, the internal coordinates of ∆v and ∆x are separately optimized
to be 0.037 and 0.067. The successive optimizations revealed that the distortion of type
1 induced an instability to dimerized chains of
the [110] direction. This dimerization of chain
essentially provides a new picture of internal molecular arrangement for ε-phase, which
would correspond to a simple chain model previously proposed [21, 22], but has never been
associated with the crystal structure.
To search a lower energy geometry, we constructed the initial configuration with mixed
distortions, for example, ∆v = 0.04 and ∆x =
0.03. Starting optimizations from these kinds
of configuration, we found that the low energy
configuration presented in Fig. 5 (a) at 9.6
6
8'
b
8
3
5
2
4
1
a
(a) 9.6 GPa
(b) 54.5 GPa
(c) 90 GPa
Figure 5: Molecular arrangements obtained at
(a) 9.6 GPa, (b) 55 GPa, and (c) 90 GPa. The
thin and thick lines are drawn for guide of eyes.
The couple of parameters, u1 and u2 , specifies
the distances of neighboring chains along the
[110] direction.
GPa. We call the optimized structures type
3 configuration. In the dimerized chain, the
individual molecule has two nearest neighbors
for the bond angle with being nearly normal in
ab-plane and there exist the rectangular O 4 ’s
(pairs of O2 ). These structural features completely coincide with results of the optical experiments [21, 22].
The electronic structure is common to the
crystals of type 2 and 3 in forming the energy gap at the Fermi level. The dimerization of molecules in rectangular shape forms
the bonding and anti-bonding orbitals which
consist of the couple of π ∗ molecular orbitals.
Resulting to a non-bonding orbital on each
molecule, in crystal, this orbital connects to
the non-bonding orbital in the other neighboring molecule. On the connection there is a
difference between type 2 and 3, however almost the same size of energy gap (bondingantibonding splitting) is formed. The total en-
0.04
Total Energy (eV/mole.)
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
0
20
40
60
Pressure (GPa)
80
100
Figure 6: Total energies of the type 3 configuration with respect to that of type 2 at calculated pressures.
At higher pressures, the double chain structure is preserved. The structures of 54.5 and
90 GPa are presented in Fig. 5 (b) and (c),
respectively. In the double chain, however, a
novel change in arrangement of molecules occurs, namely, some angles between neighboring bonds are decreased to an angle less than
90 degree. At high pressures near the transition point to ζ-phase, the molecular arrangement within ab-plane becomes similar to that
of α-phase. The distances between neighboring
molecules are listed in Table 2. The distances
connecting nearest neighbors (d12 , d23 , and d34
in the table) are shorter than the others. The
largeness of d36 indicates a novel arrangement
of molecules within the double chain. The
pressure dependence of d23 is not monotonic,
showing a complex feature of molecular relaxation. The bond length in each molecule linearly decreases between 1.212 and 1.187 Å with
respect to pressure, as presented in Fig. 7.
Our proposed crystal structure has the pair
of chains, which are stacked in the direction
normal to the [110] direction. This dimeriza-
Table 2: Distances of neighboring molecules.
The value of dmn specifies the distance of m’th
and n’th molecules numbering in Fig. 5 (a).
pressure
distance (Å)
(GPa)
d12
d23
d34
d36 d280
9.6
2.10 2.14 2.13 2.89 2.76
54.5
1.99 2.04 2.01 2.31 2.28
90
1.99 2.08 1.99 2.10 2.20
tion results in two distances between neighboring chains. The relaxation of these distances
against the increasing pressure could be characterized by an equal rate of decrease for the
lattice constants, a and b. In Fig. 8, we present
pressure dependence of the two distances, u 1
and u2 (see Fig. 5(a)). The shorter distance
shows the gradual decrease as increasing pressure, while the longer one decreases rapidly at
low pressures. This rapid decrease, which is an
indirect evidence of chain structure along [110]
direction, is in good agreement with the previous analysis of X-ray diffraction measurement
at low pressures, which resulted in the similar
decrease of a and b [18].
1.215
1.210
Bond lenght (Angstrom)
ergies of type 3 are as low as those of type 2
(the herringbone-type chain structure). Figure
6 shows the total energy differences between
type 3 and type 2 at five pressures calculated.
The new structure of type 3 is slightly lower in
energy than the type 2 at the high pressures.
This shows a contrast with the previous result
in which the distortion of type 2 is stable only
at lower pressures [25].
1.205
1.200
1.195
1.190
1.185
0
20
40
60
Pressure (GPa)
80
100
Figure 7: Pressure dependence of intramolecular bond length.
The profile of X-ray diffraction measurement
is related with the structure factor contributed
from the electron density. To see the comparison with experimental data, the structure factors, estimated from the calculations, are presented in Fig. 9. The extinction law in the
profiles is in good agreement with the available experimental data [3, 14, 18]. The profiles
2.6
1.2
2.5
1
Intensity (arb. unit)
distance (Angstrom)
2.7
2.4
2.3
2.2
(001)
(2-20)
(40-1)
(3-10)(2-2-1)
(002)
(400)
(3-1-1)
90 GPa
71 GPa
0.8
54.5 GPa
0.6
(5-1-1)
33 GPa
0.4
2.1
19.7 GPa
0.2
2
(200)
(020)
0
1.9
5
0
20
40
60
Pressure (GPa)
80
100
Figure 8: Pressure dependence of two distances between neighboring chains.
closely reproduce the features of experimental
data at the relatively high pressures. As pressure increases, the intensity of (3 1̄1̄) and (31̄0)
reduces and the intensity of (401̄) grows. The
crossings of (31̄0) with (22̄0), and (13̄0) with
(400) are clearly shown in the calculation, as
observed in the work of Weck et al.. In the
present work, some weak lines are expected in
the range of low angles at the low pressures,
where no indication exits in the corresponding experimental diffraction profiles. This fact
and some differences in relative intensity between observed and calculated profiles at the
low pressures (strong intensity of (3 1̄1̄) and
weak of (401̄) at 9.6 GPa in Fig. 9, compared
with experimental results [14]) imply that our
proposed molecular arrangement presented in
Fig. 5 (a) are not completely enough to describe details of the structural property.
At low pressures, due to the property of
molecular crystal, the intermolecular interaction is important for determining detailed geometry of molecules. The discrepancy between
calculation and experiment will be attributed
to the shortcoming that the DFT do not contain the many-electron effect on weak molecular pair interaction like a Van Der Waals interaction [31]. The DFT is also incomplete in
electron correlation effects on electronic structure. The band gap is considerably underestimated; direct gap(Γ point) of 0.88 eV and
indirect gap (Γv − Ac ) of 0.75 eV at 9.6 GPa
(experiment: 2.4 eV around 10 GPa) [32]. The
overlap between valence and conduction bands
10
(1-30)
(400)
15
2 θ (degree)
(040)
20
9.6 GPa
25
Figure 9: Structure factors at the pressures
calculated as a function of the diffraction angle
of measurement. The wave length of X-ray,
λ = 0.4817 Å was used.
appears at the range of 33∼55 GPa.
Due to the constraint of periodic boundary
condition, the infinite chain exits in the crystal
structure of our model. This recalls a possibility of sliding along the chain. The alternate
distortion of sliding, resulting in a 4 × 4 superlattice of ideal unit cell (structure of α-phase),
did not improve the profile of structure factors at low pressure. Another candidate of improvement is a variation of long period. The
existence of modulation is not denied by the restricted information deduced from our present
work.
The ε-phase is bound on the magnetic
phases at the low pressures. The recent neutron diffraction measurement revealed that the
magnetic alignment changed at the boundary
between α- and δ-phases [8, 15]. In both of
phases, the antiferromagnetic alignment along
a [110] direction exists. In general, there is
no relation between two phases on first order transition except for their energetics. It
is interesting, however, to speculate the role
of magnetic interaction. The chain of herringbone type (type 2) is nonmagnetic, while the
single chain of [110] before the transition can
preserve their magnetic state. This magnetic
interaction might stabilize the linear structure
along [110] direction. Successively, the dimerized chain becomes nonmagnetic due to the
hybridization of non-bonding orbitals between
chains. The noncollinear magnetic scheme
[33, 34], which efficiently searches the lowest energy configuration of magnetic system,
found the ground state to be the same nonmagnetic configuration described here. The
proposal of noncollinear magnet for ε-phase
has never been supported in our calculations
[4, 16].
4
Summary
We gave a short review on the structural properties for condensed phases. We presented
calculated results for crystal structures of εphase, with use of the DFT and assumption
of the experimental lattice constants. We proposed the new structure which has the dimerized chain of O2 molecules. The local structure
(rectangular O2 pair) is consistent with results
of the optical measurements and structure factors of crystal almost agree with the profiles
of X-ray diffractions. The double chain along
[110] direction would be used to refine the analyses in high-pressured oxygens.
[C,D class; 29000K (A), 3000K (B)]
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