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Theory of the mechanisms of pressure-induced phase transitions in oxygen
Hannelore Katzke1,* and Pierre Tolédano2
1Institute
of Geosciences, Crystallography, University of Kiel, Olshausenstraße 40, 24098 Kiel, Germany
2Laboratory of Physics of Complex Systems, University of Picardie, 33 rue Saint-Leu, 80000 Amiens, France
共Received 2 February 2009; published 15 April 2009兲
A theoretical description is proposed for the transition mechanisms relating the different phases of oxygen.
In contrast to the current view, the antiferromagnetic order existing in the ␣ and ␦ phases is shown to be
structurally driven by the spontaneous deformations taking place at the  → ␣ and  → ␦ transitions. The
structural mechanism giving rise to 共O2兲4 clusters in the high-pressure ⑀ phase is described and shown to be
correlated with the magnetic collapse occurring at the ␦ → ⑀ transition. The large intermediate region of
coexistence between the ⑀ and yet unknown higher-pressure structures is suggested to correspond to a
low-symmetry ferroelastic phase.
DOI: 10.1103/PhysRevB.79.140101
PACS number共s兲: 64.70.kt, 62.50.⫺p
Different types of structural transition mechanisms are involved in the phase diagram of solid oxygen. Below 8 GPa,
one finds the reconstructive ␥ →  transition, which exhibits
large displacements and orientational ordering of the molecules, and the magnetostructural  → ␣ and  → ␦
transitions,1,2 where small deformations are assumed to be
induced by the magnetic ordering.3–5 Further compression
leads to the ⑀ phase across a remarkable type of transition
characterized by a magnetic collapse and the formation of
molecular clusters containing four O2 molecules.6,7 At 96
GPa, the ⑀ phase transforms progressively into the
phase,8–10 corresponding to a profound modification of the
electronic structure, giving rise to metallization in the absence of molecular-to-atomic dissociation. Because of the
diversity of transition mechanisms between the phases of
oxygen, there has been no attempt to give a unifying theoretical description of its phase diagram. Here, we propose a
comprehensive picture of the transition mechanisms occurring in oxygen. In contrast with the current view, the antiferromagnetic order observed in ␣ and ␦-O2 is interpreted as
structurally driven. The magnetic collapse occurring at the
␦ → ⑀ transition and the correlated structural mechanism inducing the formation of 共O2兲4 clusters in the ⑀ structure are
described.
The relationship between oxygen structures assumed in
our description and the phase diagram of oxygen are shown
in Fig. 1. The structures of five solid phases, namely ␣, , ␥,
␦, and ⑀, are presently known. The cubic ␥ phase occupies a
narrow range of temperature confined between the melting
curve and the  phase. Figure 2共a兲 shows the large molecular
displacements transforming the ordered  structure into the
disordered ␥ structure 关Fig. 2共b兲兴. The corresponding reconstructive transition mechanism can be described as proceeding via a rhombohedral shear-deformed ␥ structure shown in
Fig. 2, displaying an eightfold  unit cell, induced by an
instability at the L-point 共0 , 2 / a , / 3c兲 of the rhombohedral Brillouin zone.11
The structural mechanism transforming -O2 into ␣-O2 is
shown in Fig. 3. It consists of antiparallel displacements of
about 0.1 Å along the 关22̄1兴 hexagonal direction, corresponding to a shear strain exz which reduces the monoclinic
 angle from 133.96° in -O2 to 132.53° in ␣-O2. The
1098-0121/2009/79共14兲/140101共4兲
 → ␦ transition mechanism involves displacements of the
same amount along 关210兴, reorienting the O2 molecules
along the 关212兴 direction 共Fig. 3兲. Since the ␦ structure is
not group subgroup related to -O2, the transition can be
described as occurring via an intermediate common monoclinic substructure 共Fig. 3兲, having the same structure as
␣-O2 but a different orientation with respect to -O2; the ␦
structure resulting from a shearing of the ␣ monoclinic structure.
Theoretical12,13 and experimental studies4,5,14 suggest that
the  → ␣ and  → ␦ structural transitions are magnetically
driven. This interpretation can be tested by considering the
effective free energy associated with the magnetostructural
FIG. 1. 共a兲 Symmetry relationship between oxygen phases assumed in our description. White boxes represent hypothetical intermediate phases in the transition mechanisms. The irreducible representations associated with the symmetry-breaking order
parameters are indicated along the lines relating the structures, following the notation of Stokes and Hatch 共Ref. 11兲. 共b兲 Temperaturepressure phase diagram of oxygen from Refs. 1, 6, 14, 17, and 20.
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HANNELORE KATZKE AND PIERRE TOLÉDANO
FIG. 2. 共Color online兲 Displacive mechanism associated with
the -O2 → ␥-O2 phase transition. 共a兲 Trigonal structure of -O2
projected along 关001兴. The -to-␥ phase transition is described as
proceeding via a common trigonal substructure 共R3c, Z = 48兲 with
the O2 centers occupying the Wyckoff position 6a: 共0 0 0兲 in -O2
and 共0 0 0.0625兲 in ␥-O2 and 18b: 共0 1/2 1/2兲 in -O2 and 共1/12
5/12 23/48兲 in ␥-O2. The arrows indicate displacements of the molecular centers. 共b兲 Cubic structure of ␥-O2 projected along 关111兴.
The cubic unit cell contains eight O2 molecules centered at the
Wyckoff positions 2a and 6c of space group Pm3̄n. The molecules
centered at the 2a positions are statically disordered with their axes
lying with equal probability along one of the four body diagonals.
The other six molecules are disordered about the 6c positions. Atoms on different heights along 关001兴 in 共a兲 and 关111兴 in 共b兲 are
shown in different gray shades 共colors兲. The trigonal and cubic unit
cells of -O2 and ␥-O2 are shown as gray polyhedra. The trigonal
unit cell of the common R3c substructure is represented by solid
lines in 共a兲 and 共b兲.
 → ␣ transition,15 in which only the spin-density component
M y and the shear strain exz have nonzero equilibrium values.
It reads as
F1共M y,exz兲 =
a1 2
2 My
+
b1 4
4 My
− d1M 2y exz −
+
b2 6
6 My
c
c
c
2
3
4
+ 21 exz
+ 32 exz
+ 43 exz
d2 2 2
2 M y exz .
共1兲
Minimizing F1 with respect to M y and exz yield
2
M y共a1 + b1M 2y + b2M 4y − 2d1exz − d2exz
兲 = 0,
exz共c1 + c2exz +
2
c3exz
−
d2M 2y 兲
−
d1M 2y
= 0,
共2兲
共3兲
which show the following: 共1兲 in the absence of magnetic
ordering 共M y = 0兲, a nonzero equilibrium shear strain exz
= 2c1 3 共−c2 ⫾ 冑c22 − 4c1c3兲 takes place, whereas exz = 0 implies
M y = 0. It indicates that in contrast with the current interpretation, the  → ␣ transition is structurally driven, i.e., the
primary structural mechanism induces the onset of the magnetic ordering, imposing a first-order character to the  → ␣
3
transition, due to the existence of a cubic invariant exz
in F1.
Therefore, clamping the ␣ phase with the suitable conjugated
strain xz should cancel the antiferromagnetic order. Let us
emphasize that the asymmetry of the free energy F1 and the
resulting quadratic-linear coupling d1M 2y exz between the
magnetic and structural order parameters are related to the
existence in the paramagnetic -symmetry R3̄m1⬘ of the
time-reversal operation which acts on M y but not on exz.
Such asymmetry was overlooked in previous theoretical
FIG. 3. 共Color online兲 Displacive mechanisms associated with
the  → ␣, ␣ → ␦, and  → ␦ phase transitions in oxygen. 共a兲 Monoclinic structure of ␣-O2 projected along 关010兴. 共b兲 Trigonal structure
of -O2 projected along 关1̄1̄0兴. The -to-␣ phase transition is associated with atomic displacements of the order of 0.1 Å within the
共a / c兲 plane with the oxygen atoms occupying the Wyckoff position
4i: 共0.1154 0 0.1731兲 in -O2 and 共0.0890 0 0.1530兲 in ␣-O2 of the
common C2 / m unit cell. 共c兲 Orthorhombic structure of ␦-O2 projected along 关01̄0兴. The ␣-to-␦ phase transition is associated with
displacements of the oxygen atoms occupying the Wyckoff position
4i: 共0.0890 0 0.1530兲 in ␣-O2 and 共0.0871 0 0.1742兲 in ␦-O2. 共d兲
Trigonal structure of -O2 projected along 关010兴. The -to-␦ transition proceeds via a common monoclinic substructure 共C2 / m, Z
= 4兲 with the oxygen atoms occupying the Wyckoff position 4i:
共0.0577 0 0.9423兲 in -O2 and 共0.0871 0 0兲 in ␦-O2. 共e兲 Orthorhombic structure of ␦-O2 projected along 关010兴. The -to-␦ phase transition involves a rotation of the O2 molecular axes by 13.6°, resulting in a parallel alignment of the molecular axes along the 关001兴
direction in ␦-O2. Atoms on different heights along 关010兴 in 共a兲,
关1̄1̄0兴 in 共b兲, 关01̄0兴 in 共c兲, and 关010兴 in 共d兲, and 共e兲 are shown in
different gray shades 共colors兲. The conventional unit cells are
shown as gray polyhedra. The unit cells of the common substructures are represented by solid lines.
studies12,13 in which the magnetic interactions were expressed, via the Heisenberg Hamiltonian, independently from
the interaction potential between O2 molecules.
共2兲 The phase diagram associated with F1 共Fig. 4兲 shows
that three phases denoted I to III can be stabilized below the
paramagnetic  phase 共M y = 0, exz = 0兲. 共i兲 Two isostructural
variants 共phases I and II兲 of the ␣ phase 共M y ⫽ 0, exz ⫽ 0兲
displaying the same C2 / m structural symmetry and C p2 / m
magnetic symmetry. It supports the observation by Jodl et
al.16 of an intermediate phase between the ␣ and ␦ phases,
although infrared and x-ray measurements do not confirm
this result.17
共ii兲 A nonmagnetic monoclinic structure 共M y = 0, exz ⫽ 0兲
of symmetry C2 / m 共phase III兲, coinciding with the intermediate structure assumed in our proposed  → ␦ transition
mechanism, may correspond to the nonmagnetic ␦II phase
reported by Goncharenko,6 whereas infrared spectroscopy
measurements by Gorelli et al.14 found no evidence of a
distinct ␦II region.
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FIG. 4. Theoretical phase diagram associated with the  → ␣
and  → ␦ transitions in the 共c1 , a1兲 plane deduced from the minimization of the free energy F1 given by Eq. 共1兲. Full- and hatcheddotted lines represent, respectively, first-order transition and limit of
stability lines. T1, T2, and T3 are triple points. K is a critical point.
␦-O2 → ⑀-O2 transition. 共a兲 Orthorhombic structure of ␦-O2 pro-
共3兲 Stability of the ␦ phase requires a shear strain exz
transforming the C2 / m symmetry of phase III into Fmmm
and triggering an antiferromagnetic order 共M y ⫽ 0兲 with
magnetic symmetry Am⬘mm⬘. The triggering mechanism can
be foreseen from Eqs. 共2兲 and 共3兲. A nonzero strain exz ⫽ 0
induces the magnetic component M y ⫽ 0, which is preserved
when exz vanishes.
The property of the magnetic ordering in oxygen to be
structurally driven allows the interpretation of the magnetic
collapse occurring at the ␦ → ⑀ transition. The transition is
associated with a four-component order parameter denoted
共1, 2, 3, and 4兲, corresponding to the L1 instability of the
orthorhombic F Brillouin zone. The ⑀ structure is stabilized
for 1 = ⫾ 4 and 2 = 3 = 0. The effective transition free energy is18
F2共1, 4,exz,Ly兲 =
␣1
2
2 共1
+ 24兲 +
1
2
4 共1
+ 24兲2 +
2
4
4 共1
+ 44兲
+ ␥1共21 + 24兲exz + ␥2共21 + 24兲L2y
2
+ 2 L2y .
+ 21 C44exz
共4兲
The ␥1 coupling gives rise to an induced shear strain
␥
exz = − C441 共21 + 24兲 in the ⑀ phase. The ␥2 term represents
the magnetostructural coupling between the displacive
order parameter and the induced magnetic ordering. Ly is the
projection along y of the antiferromagnetic vector
L = s1 + s2 − s3 − s4, where the si are atomic spins related by
face centering in the ␦ magnetic structure. Minimizing F2
with respect to Ly gives Ly = 0, i.e., the magnetic ordering
cancels in the ⑀ phase as the result of the ␦ → ⑀ structural
transition.
Our proposed structural mechanism transforming
␦-O2 into ⑀-O2 is shown in Fig. 5. It consists of
critical displacements by about 0.4 Å, within the 共a / c兲
plane, of oxygen atoms in positions 4i in the ⑀
structure and of correlated displacements by the same
amount of atoms in position 8j in general directions. The
order-parameter components associated with the displacements read as 关1共k1兲 , 2共k2兲 , 3共k3兲 , 4共k4兲兴, where
FIG. 5. 共Color online兲 Structural mechanism associated with the
jected along the wave vector k1 corresponding to the direction
关001兴. The orthorhombic unit cell is shown as a gray polyhedron.
The monoclinic unit cell with its origin shifted to p = 共0 , 1 / 2 , 0兲 is
represented by solid lines. The arrows indicate the directions of the
magnetic spins. The ␦-to-⑀ phase transition involves periodic displacements of the atoms occupying two Wyckoff positions 4i:
共0.79355 0 0.17420兲, 共0.29355 0 0.17420兲 in ␦-O2 and 共0.82600 0
0.17500兲, 共0.24700 0 0.19200兲 in ⑀-O2 and one Wyckoff position
8j: 共0.04355 0.25000 0.17420兲 in ␦-O2 and 共0.03790 0.26660
0.18500兲 in ⑀-O2 shown by modulation waves in different colors
corresponding to atoms in different heights along 关001兴. 共b兲 Resulting structure of ⑀-O2 projected along the wave vector k1 corresponding to the direction 关104兴. Atoms on different heights along
关001兴 in 共a兲 and 关104兴 in 共b兲 are shown in different gray shades
共colors兲.
k1 = 共− / a , − / b , / c兲,
k2 = 共 / a , / b , / c兲,
k3
= 共− / a , / b , − / c兲, and k4 = 共 / a , − / b , − / c兲 represent
the four branches of the star of the wave vector corresponding to the L1 instability. In ␦-O2 k1, k2, k3, and k4 are perpendicular, respectively, to the 共001兲, 共111兲, 共010兲, and 共100兲
planes, whereas in ⑀-O2 they are perpendicular to the 共001兲,
共22̄1兲, 共010兲, and 共401兲 planes. From the calculated displacements occurring at the ␦ → ⑀ transition given in Fig. 5, one
can deduce that the transition mechanism consists of
opposite rotations by ⫾2.756° of k1 and k4 which bring into
coincidence the 共001兲 and 共100兲 ␦ planes onto the 共001兲 and
共401兲 ⑀ planes, respectively, the directions of k2 and k3 remaining unchanged. Figure 5 shows that the amplitude of the
critical displacements is modulated periodically along the
关010兴 ␦ direction in a way that it lowers the distances between molecules displaying opposite magnetic spins in ␦-O2.
The displacement field transforms each two pairs of neighboring molecules in the ␦ structure into almost regular rhombohedra forming the 共O2兲4 clusters in the ⑀ structure. The
volume drop of about 7%, occurring at the first-order ␦ → ⑀
transition,17 coincides with a collapse of the spin-density amplitude and the correlated formation of an equilibrium configuration of 共O2兲4 clusters. Our proposed ␦ → ⑀ transition
mechanism reflects at the structural level the underlying
changes in the electronic structure leading to the formation
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HANNELORE KATZKE AND PIERRE TOLÉDANO
of intermolecular bonding, which have been deduced from
inelastic x-ray scattering measurements.19
Above 96 GPa, the ⑀ structure transforms progressively to
a metallic molecular state .8–10 The transformation proceeds
in two stages. 共1兲 An intermediate stage extending from 96
GPa to at least 110 GPa in which one observes a disappearance of the ⑀-O2 lattice reflections, a continuous evolution of
the lattice parameters, and changes in the mosaicity of the
crystal.9 共2兲 A stabilization of the structure, with a new
position of the reflections above a slightly discontinuous
transition. The almost continuous character of the transformation is against an interpretation of the intermediate stage
as representing an extended region of coexistence between
the ⑀ and phases.10 A more likely interpretation is that it
constitutes an intermediate phase between ⑀-O2 and -O2,
into which a reorganization of the intermolecular bonding
occurs. The indexing of the measured diffraction pattern in
the intermediate region with the structural model of the ⑀
phase9 indicates a close relationship between the two phases.
Therefore, one can assume that the intermediate phase is
either incommensurately modulated or possesses a
pseudomonoclinic ⑀ structure with a lower triclinic P1̄ sym-
ACKNOWLEDGMENT
This work has been supported by the German Science
Foundation under Grant No. DE 412/33-1.
12
*hanne@min.uni-kiel.de
1
metry giving rise to ferroelastic domains. Along this line, the
discontinuous reformation of a single domain crystal above
110 GPa 共Ref. 9兲 reflects the recovery of a higher-symmetry
structure.
In summary, the structural phase-transition mechanisms
occurring in oxygen have been described and analyzed theoretically. The antiferromagnetic order existing in the ␣ and ␦
phases has been interpreted as structurally driven. Two isostructural ␣ variants have been suggested to be stable,
whereas an intermediate nonmagnetic monoclinic phase was
shown to be possibly stable in the  → ␦ transformation process. The magnetic collapse and formation of 共O2兲4 clusters
observed at the ␦ → ⑀ transition have been correlated by the
same displacement-field mechanism, corresponding to opposite rotations of the 共001兲 and 共100兲 ␦-atomic planes. The
existence of an intermediate low-symmetry phase is suggested between the ⑀ and phases.
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F1 expresses the coupling between the structural 共R3̄m
→ C2 / m兲 and magnetic 共R3̄m1⬘ → C p2 / m兲 orderings occurring
at the  → ␣ transition. The effective form of F1 is deduced from
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