Shift of close-packed basal planes as an order parameter of transitions between antiferromangetic phases in solid oxygen: II. Temperature/pressure dependence of sound velocities and lattice parameters
In the present paper we generalize a phenomenological model developed by Gomonay and Loktev (Fiz. Nizk. Temp. 31, 1002 (2005)) for the description of magnetostructural phase transitions and related peculiarities of elastic properties in solid oxygen under high pressure and/or temperature below 40 K....
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| Zitieren: | Shift of close-packed basal planes as an order parameter of transitions between antiferromangetic phases in solid oxygen: II. Temperature/pressure dependence of sound velocities and lattice parameters / E.V. Gomonay, V.M. Loktev // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 711-718. — Бібліогр.: 16 назв. — англ. |
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Gomonay, E.V. Loktev, V.M. 2017-06-16T07:13:37Z 2017-06-16T07:13:37Z 2007 Shift of close-packed basal planes as an order parameter of transitions between antiferromangetic phases in solid oxygen: II. Temperature/pressure dependence of sound velocities and lattice parameters / E.V. Gomonay, V.M. Loktev // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 711-718. — Бібліогр.: 16 назв. — англ. 0132-6414 PACS: 75.50.Ee; 61.50.Ks; 81.40.Vw https://nasplib.isofts.kiev.ua/handle/123456789/121772 In the present paper we generalize a phenomenological model developed by Gomonay and Loktev (Fiz. Nizk. Temp. 31, 1002 (2005)) for the description of magnetostructural phase transitions and related peculiarities of elastic properties in solid oxygen under high pressure and/or temperature below 40 K. We show that variation of all the lattice parameters in the vicinity of a–b phase transition is due to both the shift of basal closed-packed planes and appearance of the long-range magnetic order. Competition between these two factors from one side and lattice compression below Tab from another produces nonmonotonic temperature dependence of lattice parameter b (along monoclinic axis). Steep decrease of the sound velocities in the vicinity of Tab can be explained by the softening of the lattice with respect to shift of the close-packed planes (described by the constant K₂) prior to phase transition point. We anticipate an analogous softening of sound velocities in the vicinity of a–d phase transition and nonmonotonic pressure dependence of sound velocities in a phase. We would like to acknowledge all the participants of CC-2006 Conference for keen interest to our presentations and fruitful discussions. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Classical Cryocrystals Shift of close-packed basal planes as an order parameter of transitions between antiferromangetic phases in solid oxygen: II. Temperature/pressure dependence of sound velocities and lattice parameters Article published earlier |
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| title |
Shift of close-packed basal planes as an order parameter of transitions between antiferromangetic phases in solid oxygen: II. Temperature/pressure dependence of sound velocities and lattice parameters |
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Shift of close-packed basal planes as an order parameter of transitions between antiferromangetic phases in solid oxygen: II. Temperature/pressure dependence of sound velocities and lattice parameters Gomonay, E.V. Loktev, V.M. Classical Cryocrystals |
| title_short |
Shift of close-packed basal planes as an order parameter of transitions between antiferromangetic phases in solid oxygen: II. Temperature/pressure dependence of sound velocities and lattice parameters |
| title_full |
Shift of close-packed basal planes as an order parameter of transitions between antiferromangetic phases in solid oxygen: II. Temperature/pressure dependence of sound velocities and lattice parameters |
| title_fullStr |
Shift of close-packed basal planes as an order parameter of transitions between antiferromangetic phases in solid oxygen: II. Temperature/pressure dependence of sound velocities and lattice parameters |
| title_full_unstemmed |
Shift of close-packed basal planes as an order parameter of transitions between antiferromangetic phases in solid oxygen: II. Temperature/pressure dependence of sound velocities and lattice parameters |
| title_sort |
shift of close-packed basal planes as an order parameter of transitions between antiferromangetic phases in solid oxygen: ii. temperature/pressure dependence of sound velocities and lattice parameters |
| author |
Gomonay, E.V. Loktev, V.M. |
| author_facet |
Gomonay, E.V. Loktev, V.M. |
| topic |
Classical Cryocrystals |
| topic_facet |
Classical Cryocrystals |
| publishDate |
2007 |
| language |
English |
| container_title |
Физика низких температур |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
In the present paper we generalize a phenomenological model developed by Gomonay and Loktev (Fiz. Nizk. Temp. 31, 1002 (2005)) for the description of magnetostructural phase transitions and related peculiarities of elastic properties in solid oxygen under high pressure and/or temperature below 40 K. We show that variation of all the lattice parameters in the vicinity of a–b phase transition is due to both the shift of basal closed-packed planes and appearance of the long-range magnetic order. Competition between these two factors from one side and lattice compression below Tab from another produces nonmonotonic temperature dependence of lattice parameter b (along monoclinic axis). Steep decrease of the sound velocities in the vicinity of Tab can be explained by the softening of the lattice with respect to shift of the close-packed planes (described by the constant K₂) prior to phase transition point. We anticipate an analogous softening of sound velocities in the vicinity of a–d phase transition and nonmonotonic pressure dependence of sound velocities in a phase.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121772 |
| citation_txt |
Shift of close-packed basal planes as an order parameter of transitions between antiferromangetic phases in solid oxygen: II. Temperature/pressure dependence of sound velocities and lattice parameters / E.V. Gomonay, V.M. Loktev // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 711-718. — Бібліогр.: 16 назв. — англ. |
| work_keys_str_mv |
AT gomonayev shiftofclosepackedbasalplanesasanorderparameteroftransitionsbetweenantiferromangeticphasesinsolidoxygeniitemperaturepressuredependenceofsoundvelocitiesandlatticeparameters AT loktevvm shiftofclosepackedbasalplanesasanorderparameteroftransitionsbetweenantiferromangeticphasesinsolidoxygeniitemperaturepressuredependenceofsoundvelocitiesandlatticeparameters |
| first_indexed |
2025-11-25T20:35:29Z |
| last_indexed |
2025-11-25T20:35:29Z |
| _version_ |
1850526274465824768 |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7, p. 711–718
Shift of close-packed basal planes as an order parameter
of transitions between antiferromangetic phases in solid
oxygen: II. Temperature/pressure dependence of sound
velocities and lattice parameters
E.V. Gomonay1,2 and V.M. Loktev1
1
Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine
14-b Metrologichna Str., Kyiv 03680, Ukraine
E-mail: vloktev@bitp.kiev.ua
2
National Technical University «KPI», 37 Peremogy Ave., Kyiv 03056, Ukraine
E-mail: malyshen@ukrpack.net
Received September 25, 2006
In the present paper we generalize a phenomenological model developed by Gomonay and Loktev (Fiz. Nizk.
Temp. 31, 1002 (2005)) for the description of magnetostructural phase transitions and related peculiarities of
elastic properties in solid oxygen under high pressure and/or temperature below 40 K. We show that variation of
all the lattice parameters in the vicinity of �–� phase transition is due to both the shift of basal closed-packed
planes and appearance of the long-range magnetic order. Competition between these two factors from one side
and lattice compression below T�� from another produces nonmonotonic temperature dependence of lattice pa-
rameter b (along monoclinic axis). Steep decrease of the sound velocities in the vicinity of T�� can be explained
by the softening of the lattice with respect to shift of the close-packed planes (described by the constant K2) prior
to phase transition point. We anticipate an analogous softening of sound velocities in the vicinity of �–� phase
transition and nonmonotonic pressure dependence of sound velocities in � phase.
PACS: 75.50.Ee Antiferromagnetics;
61.50.Ks Crystallographic aspects of phase transformations; pressure effects;
81.40.Vw Pressure treatment.
Keywords: magnetostructural phase transition, order parameter, lattice parameters.
1. Introduction
Solid oxygen that belongs to a family of cryocrystals is
being studied for more than 100 years and still attracts at-
tention of researches. This simple molecular crystal has
very complicated temperature–pressure T–P phase dia-
gram consisting of magnetic and nonmagnetic, metallic
and dielectric, studied and nondiscovered phases (see,
e.g., review [2] and references therein). Phase transitions
and drastic change of the magnetic, electronic, elastic
properties of solid O2 can be triggered by variation of
temperature, pressure, external magnetic field, etc. The
thorough analysis of corresponding regularities is rather
nontrivial problem because the O2 crystal lattice is main-
ly hold by the weak van der Waals forces and so is very
soft in comparison with ordinary solids. Besides, in con-
trast to other crystals, exchange magnetic interactions at
low temperature prove to be of the same order as lattice
energy. As a result, application of external fields (tempe-
rature, stress, magnetic) gives rise to a pronounced vari-
ation of all the lattice and magnetic parameters. In this
situation precise microscopic calculations should be com-
bined with general thermodynamic treatment.
In the 1st part of this paper [1] we made an attempt to
develop a phenomenologic (Landau-type) model aimed at
the description of the lattice and magnetic properties of
solid oxygen in a sequence of temperature/pressure in-
duced � � �� � transitions.
The proposed model was based on the following as-
sumptions:
1. Magnetoelastic coupling is so strong that abrupt
change of the magnetic structure leads to the noticeable
variation of the crystal lattice.
© E.V. Gomonay and V.M. Loktev, 2007
2. The primary order parameter in the series of
� � �� � phase transitions is a homogeneous shift of
closed-packed planes. The order parameter is defined
with respect to the virtual hexagonal, D 6
1
h pra-phase.
3. The only macroscopic parameter that controls these
transitions is the specific volume. Temperature depend-
ence of the specific volume cannot be calculated within
the model and should be taken from the experiment.
As it was shown, this model gives good agreement
with experimental data. However, some questions have
been left beyond its scope. In particular, we have consid-
ered only two factors that define crystal structure of dif-
ferent phases — shift of the closed packed planes and spe-
cific volume of the crystal. More thorough analysis
should account for rather strong rhombic deformations
within the plane and separate contribution of interplane
distance and isotropic in-plane strain into the change of
specific volume.
In the present paper we try to refine the model by tak-
ing into account all the parameters that define the crystal
structure of �, �, and � phases and explain observed pe-
culiarities of temperature/pressure dependence of lattice
constants and sound velocities.
We would also like to mention with our great pleasure
that the preliminary results of this paper were presented at
International Conference CC-2006 (in Kharkov) devoted
to the prominent Ukrainian scientist and wonderful wo-
man Antonina Fedorovna Prihot’ko who is well known
for her brilliant experiments in physics of cryocrystals.
She is also famous for fundamental results in optical in-
vestigation of � and � phases in solid oxygen. With the
present paper we try to pay our tribute of respect to her
memory.
2. Model
It was already mentioned that the crystal structure of
�, �, and � phases can be considered as a homogeneously
deformed hexagonal lattice. In general, such a deforma-
tion can be consistently described with the use of four in-
dependent variables: lattice parameters a, b, c, and angle
� of monoclinic cell. Corresponding combinations that
form representation of the D 6
1
h space symmetry group
(parent pra-phase) are:
i) relative shift of closed-packed planes� �� ( / ) cosc a ;
i i ) isotropic strain in the basal plane �s s/ 0 �
�
( ) ( )ab a b a b0 0 0 0 ;
iii) relative extension/contraction u h h hzz � ( ) /0 0 in
the direction perpendicular to closed-packed planes (Z
axis), where h c� sin�;
iv) shear strain in the basal plane u a b� �( )3
�
( )3 0 0
1 2a b .
The quantities with subscript «0» are attributed to a
certain reference state (at zero T or P). In assumption of
small variation of interplane distance (formally expressed
by inequality u zz
1) relative change of the specific vol-
ume �v v/ can be readily expressed in a form of a simple
sum � �v v s s u zz/ /� �0 .
Equilibrium values of �, u zz , u, and �s s/ 0 at a given
temperature T and hydrostatic pressure P are calculated
from standard conditions for minimum of Gibbs’ poten-
tial� that is supposed to be invariant with respect to oper-
ations of the symmetry group D 6
1
h .
Using results of symmetry analysis given in [1] (see
Table 1 therein) general expression for � can be repre-
sented as a sum of structural �str , magnetic �mag , elastic
�elast contributions, and interaction term � int :
� � � � �� � � �str mag elast int . (1)
The structure of the first term in (1)
�str � � � �
�
K s h
K
2 1 2 1 2
4
2 2 2
1
4
( , )[cos cos cos ( )]
[
�� �� � � �
cos cos cos ( )]4 4 41 2 1 2�� �� � � �� � � ��
�
�
�
�
�P
s
s
uzz
�
(2)
reflects the translational invariance of the crystal with re-
spect to relative shift u of neighboring close-packed
planes of hexagonal lattice, namely: 2 1 2 1 2�� , ,� b u, where
b1 2, are vectors of the reciprocal pra-phase lattice.
Phenomenological constants K 2 and K 4 can be consid-
ered as coefficients in Fourrier series of lattice potential.
The coefficient K 2 is supposed to be a linear function
of the isotropic in-plane strain and interplane distance
with corresponding phenomenological constants
� � �s h, �
v
(where �
v
�10 GPa was estimated in [1]):
K K
s
s
us h zz2 0( )v � � �
�
.
Both constants � �s h, � 0 are supposed to be positive sign
basing on general considerations (confirmed with further
calculations, see below). Namely, the effective constant
K 2 describes the strength of intermolecular forces that
keep relative arrangement of basal planes. Increase of the
average intermolecular distance should give rise to weak-
ening of intermolecular bonds and hence to a decrease of
K 2.
Magnetic contribution in (1)
�mag �
� �
� �J J
J
j
j
j
j( ) ( ) ( ) ( )
(
( ) ( )
k l k l
k
13
1
2
2
12
1
3
2� �
14
1
3
2) ( )
( )
j
j
�
� l
� (3)
accounts for the exchange interaction only, J j( )k � 0 are
Fourier components of exchange integrals labeled ac-
cording to stars k j ( j � 12, 13, 14) of irreducible repre-
sentations of D 6
1
h space group, antiferromagnetic (AFM)
vectors l
( )� , l
( )� , l
( )� unambiguously describe the mag-
712 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
E.V. Gomonay and V.M. Loktev
netic ordering in �, �, and � phases*. It should be also
stressed that the magnetic ordering in � phase has
short-range nature (so-called correlation ordering with
three sublattice 120° Loktev structure [2,6,7]). Collinear
long-range magnetic ordering characterized by one of
l
( )� vectors, is established in � phase. An effective Neel
temperature TN is close to the temperature T�� of �–�
transition (T�� = 23.5 K and TN � 40 K at ambient pres-
sure) [2], so, saturation magnetization M 0 and, corre-
spondingly, | |( )
l
� � M 0 noticeably depend on tempera-
ture in the vicinity of T��.
The elastic contribution in (1)
�elas ( � ) [( ) ]u c u u u c uxx yy xy zz� � ��1
2
4
1
2
2 2
33
2 (4)
does not include strain tensor components u xz , u yz , that
can be considered as small excitations over finite defor-
mations given by the order parameter �1 2, . An expression
(4) is valid in the limit of infinitely small strains
u uxx yy , u zz . In such an approximation operations of
plane shift and deformation commute, hence, all the pa-
rameters of the model can be referred to the same «zero»
state. Otherwise, one should specify the succession of lat-
tice transformations, use different reference frames at
each stage and work within nonlinear elasticity approach.
Interaction energy includes terms that describe differ-
ent crossover effects. Here we are concentrated on several
of them. First, variation of interatomic distances gives
rise to the change of intra- (characterized with interaction
constants � intra , �� ), and inter- (constants � inter , � ||)
plane exchange integrals**. This effect has magnetoelas-
tic origin and can be formally described by two expres-
sions. The first one is responsible for isotropic
� �
�
�
mag-el
iso)
intra inter
( ( ) ( )
(� ��
�
�
�
�
�
�
�� �s
s
u zz
j 1
2
l lj zz
j
j
s
s
u
( ) ( ) ( ) ( )
) (
� � � �2
1
3
� ��
�
�
�
�
�
�
��
�
�
intra inter
) ( ) ,
( ) ( ) ( )2
1
3
2� ��
�
�
�
�
�
�
��
�
�
intra inter
� � �s
s
u zz
j
jl
(5)
and another one for anisotropic effects
� �
mag-el
an)(
||
( ) ( ) ( )
[( ) cos ( ) cos� �� � ��� ��l l
1
2
1 2
22 2 2 3
2
1 2
1
2
1
2
2
�
�
( ) cos ( )]
[( ) cos (
( )
( ) ( )
||
l
l l
�
� �
� � �
���
2
2
2 3
2
1 22 2
( ) ( )
( )
) cos ( ) cos ( )]
{(
� �
�
�� � � ��
�
l
� u uxx yy xy xx yyu u u)( ) [( ) ( ) ]} {(
( ) ( ) ( ) ( )
l l l
1
2
2
2
3
22
� � � �� �� )( ) [( ) ( ) ]} .
( ) ( ) ( )
l l l
1
2
2
2
3
22
� � �� u xy (6)
Another coupling effect originates from strong anhar-
monicity which is a peculiar feature of molecular crystals.
Corresponding contribution into thermodynamic poten-
tial has a form
�anhar � �
�
� ��
�� � �
e xx yy
xy
u u
u
{( ) cos
[cos cos (
2
2 2 2
1
2 1 �� �2 )]} ,
�s
s
u zz (7)
where the coupling constants � e , �� are of the same na-
ture as� s ,� h but from the general point of view should be
much smaller in value (see Table 1). Really, � s , � h de-
scribe variation of average interatomic distances that
result from reconstruction of crystal lattice in the course
of phase transition. These constants are proportional to
Grünaisen parameter which is very high. At the same time
� e and �� are responsible for a «differential» effect,
namely, variation of anisotropy of the crystal lattice in the
course of phase transition.
Table 1. Phenomenological coefficients
Coefficient Value Comments
�s
0 31 10 12. � cm2/dyne
3 2 10 12. � cm2/dyne
at amb. P,
at T = 19 K, P � 1–10 GPa
�P T( )
1 6 10 4. � 1 / K
1 7 10 3. � 1 / K
T < 23.5 K
T > 23.5 K
Keff 3 GPa calc. at amb. P
K4 5 GPa calc. at amb. P
�s
13 GPa
0.2 GPa
at amb. P,
at T = 19 K, P � 1–10 GPa
�h 10 GPa in assumption that c33 = 10 GPa
�� � c13 1 GPa in assumption that c33 = 10 GPa
�e 0.06 GPa in assumption that c� = 1 GPa
�
||
( )�
M
0
2 –0.04 GPa
��
( )�
M
0
2 –0.02 GPa
�interM
0
2 �0.06 GPa in all phases
Shift of close-packed basal planes as an order parameter of transitions in solid oxygen: II
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 713
* We assume that the series (3) may also include a wave vector corresponding to 4-sublattice structure of recently decoded
phase [3–5]. Detailed analysis of this phase is out of scope of that paper.
** We keep superscripts ( )� , ( )� , and ( )� in the constants of magnetic nature only, in order to emphasize a step-like change of mag-
netic structure in the phase transition point. Variation of other phenomenologic coefficients is not so crucial.
Standard minimization procedure enables to obtain the
value of lattice parameters and stability conditions of �,�
and � phases.
3. Temperature/pressure dependence of lattice
parameters
Equations for order parameter and stability regions
were derived and discussed in details in [1]. The refine-
ment of the model (splitting of in- and out-of-plane con-
tribution into specific volume) gives rise to no qualitative
changes. In this section we discuss only some additional
effects that can be explained and predicted in the frame-
work of the model. As in [1] we consider a homoge-
neous (from crystallographic point of view) phase with
� � �� �
1 2 2. An effective macroscopic order parameter
that vanishes in � phase is introduced as ! ��� �1 2 2cos .
Magnetic ordering in � and � phases is described by a sin-
gle AFM vector l
1
( )�
and l
1
( )�
, correspondingly.
3.1. Order parameter and isomorphic in-plane strain
In the «more symmetrical»� and � phases an order pa-
rameter takes the limiting values ! � 0 (� phase) and ! � 1
(� phase). Saturation magnetization M 0 (or correlation
parameter in� phase) are constant. Pressure and tempera-
ture dependence of the isomorphic in-plane strain �s s/ in
these cases cannot be calculated from general thermody-
namic considerations and is determined experimentally as
�
�
� �
s
s
T P dT T Ps
T
s� � �"
� �
� �
�
0
( ' ; ) ' ( ) [
( )
intra intra
ph p
�
h
( )
] ,
�
M 0
2
(8)
where the in-plane thermal expansion coefficient� s T P( ; )
and isothermal� s T( ) compressibility vary in a wide range
with pressure and temperature (for example, Abramson et
al. observed 10% change of compressibility per 1 GPa in�
phase at room temperature [8]).
In � phase the temperature/pressure dependence of !
and �s s/ can be calculated from the system
K K M
s
s
u
K
s h zz0 4 0
2
4
2 31
2
3
� #
$%
&
'(
�
�
�
�
�
||
( )
( )
� � � !
! ! ||
( )
,
( ' ; ) ' ( )
�
� �
� �
! �
M
s
s
T P dT Ps
T
s
s s
s
0
2
0
2
0
2
3
�
� � "�
�intra
( )
( ) .
�
M T0
2
(9)
(The behavior of interplane deformation u zz will be dis-
cussed in the next section.) The values of phenome-
nological constants derived from fitting of functions
!( , )T P , �s s T P
( , ) to experimental data [9,10] with due
to account of M T0( ) temperature dependence [11,12] are
given in Table 1. It can be seen from the Table 1 that
K K K Meff � �0 4 0
2�
||
( )�
and K 4 are comparable in or-
der of value with the shear modulus of the material [13].
The value of � s which we associate with the «seed» com-
pressibility is much smaller compared with experimetally
observed values [2]. This means that the main contri-
bution into compressibility arises from shifts of close-
packed planes and magnetic interactions.
3.2. Interplane distance
Distance h between the close-packed planes can be cal-
culated from equations
u
c
s
s c
M
c
s
s
zz �
�
�
�
�
�
�
�
33 33
0
2
33
� �
�
inter for phase
( )
,
�
! �
� �
�
h
h
c c
M T
c
s
s
2 33
2
33
0
2
33
� �
�
�
inter for phase
( )
( ) ,
2 33 33
0
2
c c
M
�
inter for phase
( )
,
�
�
(10)
once the dependencies !( , )T P and �s s T P
( , ) are known.
Analysis of equations (10) shows that in the � phase
temperature/pressure dependence of u zz (and hence inter-
plane distance) is related with lattice compression within
the close-packed plane. Thus, constant �� coincides with
the elastic modulus c13. Estimated ratios c c13 33 01
� .
at ambient pressure and c c13 33
� 0.24 at T �19 K,
P � 1–7 GPa and also the positive sign of c13 0� are in
agreement with observations made by Abramson et al.
[8].
Figure 1 shows temperature dependence of interplane
distance experimentally measured (squares) and calcu-
lated from (10) (fitting parameters calculated in assump-
tion that c33 10� GPa are given in Table 1). Below T�� an
interplane distance abruptly diminishes and then its tem-
perature derivative changes sign. This fact may be ex-
714 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
E.V. Gomonay and V.M. Loktev
10 20 30 40
3.752
3.754
3.756
3.758
3.760
3.762
3.764
3.766
T��
� 2-O�-O2
In
te
rp
la
n
e
d
is
ta
n
ce
,
�
T, K
Fig. 1. Temperature dependence of interplane distance:
(squares) calculated according to experimental data [9]. Theo-
retical curve (solid line) is calculated from Eq. (10). Arrow in-
dicates the point of �–� transition.
plained by the combined influence of the magnetic order-
ing (characterized by difference � �
inter inter
( ) ( )� � ) and shift
of basal planes (corresponding interaction constant � h ).
In� phase each O2 molecule is situated over the center of
underlying triangle. In-plane compression (that arises due
to cooling or application of pressure) pushes out the over-
laying molecules in vertical direction, so interplane dis-
tance increases. In � phase hexagonal planes are not only
shifted from an ideal «close-packed» (with respect to un-
derlying layer) position but can also «slip» under com-
pression. Appearance of an additional degree of freedom
allows lattice compression not only in-plane but also in
perpendicular direction. Formally this means that con-
stant � h is positive, in agreement with monotonic
decrease of interplane distance under pressure-induced
compression (see Fig. 2).
Contribution of magnetic interactions into interplane
distance is important not only at �–�, but also at �–� tran-
sition as can be seen from Fig. 2. If we assume that� and �
phases have the same magnetic structure, then pressure
dependence of interplane distance according to (10)
should be continuous up to (hypothetical) 2nd order tran-
sition point at P2 75� . GPa (dashed line in Fig. 2). Change
of magnetic structure that originates from interplane ex-
change interactions induces 1st order phase transition at
P1 6� GPa
P2 followed by abrupt change of the order
parameter ! and all the related parameters including u zz .
So, difference between the observed and hypothetical u zz
value above P1 (see Fig. 2 and Eq. (10)) is proportional to
� �
inter inter
( ) ( )� � and is of magnetic nature.
3.3. In-plane deformation: parameters a, b
It is quite obvious from symmetry considerations that
establishment of long-range magnetic order in � phase is
fol lowed by in-plane deformat ion descr ibed by
u u uxx yy � . Analysis of the expressions (6), (7) shows
that the same effect can be produced by the shift of basal
planes (term with � e) as seen from the following
equation:
u
c
M T
c
e�
�
�
�
��
! !
�
( )
( )
( )
2 0
2
3
�
. (11)
Pressure dependence of u uxx yy calculated from
Eq. (11) along with experimental data is given in Fig. 3. It
is clearly seen that in �-phase lattice anisotropy u uxx yy
monotonically increases in absolute value with pressure.
This effect can be explained from very simple consider-
ations. Pressure produces compression in all directions in
the basal plane but relative shift of planes can induce an
effective tension in b direction as seen from Fig. 4 (O2
molecules move apart in b direction when overlaying
molecule moves toward the edge). As a result, deforma-
tion in basal plane is essentially anisotropic and aniso-
Shift of close-packed basal planes as an order parameter of transitions in solid oxygen: II
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 715
1 2 3 4 5 6 7 8
3.25
3.30
3.35
3.40
3.45
3.50
3.55
3.60
P2P1
T = 19 K
~��inter
P, GPa
In
te
rp
la
n
e
d
is
ta
n
ce
,
�
� 2-O�-O2
Fig. 2. Pressure dependence of interplane distance at T = 19 K:
(squares) calculated according to experimental data [10]. Theo-
retical curve (solid line) is calculated from Eq. (10). Arrow indi-
cates the point of the real, P1, and hypothetical, P2, �–� transition.
1 2 3 4 5 6 7 8
–0.19
–0.18
–0.17
–0.16
–0.15
–0.14
–0.13
–0.12
–0.11
P, GPa
u
–
u
x
x
y
y
T = 19 K
� 2-O�-O2
Fig. 3. Pressure dependence of in-plane rhombic strain
u uxx yy (solid line) calculated from Eq. (11). Squares: experi-
mental data [10].
b
Fig. 4. In-plane displacements (arrows) of O2 molecules in-
duced by relative shift of the overlaying (open circle) plane.
tropy is more pronounced at smaller interatomic distances
(i.e., at higher pressure).
The same effect takes place during cooling � phase
at ambient pressure. Figure 5 shows temperature de-
pendence of a and b lattice parameters below T��.
Intermolecular lattice distance a decreases monotonically
because of cooling-induced compression. In contrast,
temperature dependence of the parameter b is nonmono-
tonic. Increase of b during cooling from 23.5 to 18 K
means that tension in this direction is stronger than cool-
ing-induced compression. In particular, Eq. (11) demon-
strates that two mechanisms may be responsible for this
behavior: repulsion of ferromagnetically ordered neigh-
bors [2] (described by the term ��
( )
( )
�
M T0
2 , M 0
2 increases
with temperature decrease) and already mentioned shift
of basal planes.
Comparison of the above theoretical dependences with
experimental data obtained by different groups using dif-
ferent technique makes it possible to estimate the values
and range of phenomenological constants (see Table 1).
All the coefficients could be grouped in three categories:
i) large ( � 10 GPa) constants responsible for anharmo-
nicity effects (� s h, ); ii) intermediate (� 1 GPa) constants
(K eff , K 4 , c13) that characterize elastic properties of crys-
tal and iii) small (below 0.1 GPa) constants that describe
magnetic interactions and anisotropy effects (rest of the
constants). Though small in value the magnetic interac-
tions reveal themselves in the case of competition be-
tween different structural interactions like pressure in-
duced compression and repulsion that arises due the shift
of basal planes. Strong anharmonicity is quite natural
phenomena for molecular crystals. It is remarkable that at
ambient pressure both coefficients � s and � h are of the
same order and the main contribution into effective
structural constant K 2 arises from isomorphic in-plane
strain* �s s
.
4. Peculiarities of sound velocities in the vicinity
of phase transition points
Experimental study of sound velocity in solid oxygen
provides information about interactions that play the
leading role in phase transitions. High-pressure measure-
ments in � phase [8] show monotonic increase with pres-
sure of all the elastic constants except shear modulus c44 ,
the value of which abruptly decreases in the vicinity of
�–�-transition point (approximately 8 GPa at 295 K).
Low-temperature curves [13,14] have a pronounced min-
ima at T��.
Critical behavior of sound velocities can be quite natu-
rally explained in the framework of the developed model
using the concept of Goldstone mode. Really, the 2nd (or
weak 1st**) order phase transitions are usually accompa-
nied with softening of a certain bond responsible for ap-
pearance of the order parameter. So, characteristic fre-
quency of the corresponding excitations (symmetry
related to the order parameter) should vanish or at least
noticeably diminish in the vicinity of the phase transition
point (Goldstone mode). In the case of � � �� � transi-
tions an order parameter (shift of basal planes) is symme-
try related with strain tensor component u zx (see Table 2
in [1]) and, as a result, with the transverse acoustic modes
propagating in [0001] and [1000] directions. Thus, these
modes should have peculiarity in phase transition points.
Sound velocities of the «soft» transverse, vt , and lon-
gitudinal, vl , acoustic waves propagating in [1000] direc-
tion are expressed through the elastic moduli in a standard
way
v
v
t
l
c c c c c
c c
2
11 55 11 55
2
15
2
2
11 55
1
2
4
1
2
� � �
� � �
)
)
[ ( ) ] ,
[ ( ) ] ,c c c11 55
2
15
24 �
(12)
where ) is crystal density.
Elastic moduli that equal to the 2nd derivatives of ther-
modynamic potential � (see Eq. (1)) with respect to strain
tensor components are calculated at equilibrium values of
lattice parameters:
716 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
E.V. Gomonay and V.M. Loktev
5 10 15 20 25
5.37
5.38
5.39
5.40
5.41
b
,
�
a,
�
T, K
3.423
3.424
3.425
3.426
3.427
T��
Fig. 5. Temperature dependence of lattice parameters a and b
(solid lines) calculated from Eqs. (9), (11). Squares: experi-
mental data [9].
* We are grateful to Prof. Yu.A. Freiman who has drawn our attention to this fact.
** In other words, 1st order phase transition between the phases that are in subgroup relation, i.e., 1st order close to 2nd one.
c
s s
c
c
a
c
11
2
2
0
55
2 2
2
0
1
�
*
*
+
+
+ �
�
�
�
�
�
�
*
*
+
+
+
�
�
�
( )
,
sin
,
�
�
5
2
0
�
*
* *
+
+
+ �
�
�
�
�
�
�
c
a s s
du
c
a
dxz
sin
( )
,
sin
.
�
�
�
�
�
�
(13)
Factor ( ) sinc a
� in (13) is deduced from geometrical
considerations.
Then, omitting some terms immaterial for further dis-
cussion we obtain that in all three phases c s11
1� � ,
c15 = 0 in� and � phases and
c
c
a
T P Ks s s55
2
4 3
2
�
�
�
�
�
�
� #
$%
&
'(
� �
� � �
sin
( ) eff
(14)
in� phase,
c
c
a
P T K K Ms s s
55
2
4
4
2
�
�
�
�
�
�
� �
� � �
� �
� � � �
sin
[ ( )
||
( )
eff � 0
2] (15)
in � phase, and
c
c
a
P T K Ks s s
55
2
4
4
3 1
2
�
�
�
�
�
�
� �
�
�
�
�
�
� �
� � � !
!
sin
( ) eff
�
�
#
$
%
&
'
(
�
sin ,
sin
sin
2
15
2
4
2
��
� �
� ��c
c
a
s
(16)
in � phase.
It can be easily seen from (14) that in � phase shear
modulus increases with temperature and decreases with
pressure, at least in the vicinity of transition point, in con-
sistency with observations [8].
More interesting is comparison of experimental and
theoretical temperature dependencies* ct l t l, ,� )v
2 shown
in Fig. 6. Anomalous softening of ct agrees well with the-
oretical predictions and thus may be explained by high
compliance of O2 crystal lattice with respect to shift of
the closed-packed molecular planes. Softening of cl is not
so obvious from general point of view but in the case
of molecular crystal may originate from strong anhar-
monicity.
An analogous softening of elastic moduli and corre-
sponding sound velocities is also expected in the vicinity
of�–� transition. Figure 7 shows hypotetical pressure de-
pendence of the moduli product** c ct l calculated on the
basis of equations (12)–(16) with characteristic values of
phenomenologic parameters (see Table 1). It is clear that
this dependence should be nonmonotonic, with notice-
Shift of close-packed basal planes as an order parameter of transitions in solid oxygen: II
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 717
10 15 20 25 30 35 40
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
c
,
G
P
a
t
T, K
� 2-O
� 2-O
�-O2
�-O2
10 15 20 25 30 35 40
4.0
4.2
4.4
4.6
4.8
5.0
5.2
c
,
G
P
a
l
T, K
a
b
Fig. 6. Temperature dependences of shear modulus ct (a) and
compression modulus cl (b) calculated from (12)–(16) (solid
line). Squares: experimental data [13].
1 2 3 4 5 6 7
10
14
18
22
26
30
c
c
,
G
P
a
t
l
(
)2
P, GPa
T = 19 K
� 2-O�-O2
P��
Fig. 7. Pressure dependence of the product of c ct l calculated
from (12)–(16) (solid line).
* Temperature dependence of c s11
1� � is taken from fitting the data in � phase.
** The lack of experimental data makes it impossible to separate ct and cl contributions.
able decrease of moduli while approaching to ��- and
��-phase boundaries.
5. Conclusions
In summary, we have calculated temperature and pres-
sure dependence of all the crystal lattice parameters of
solid oxygen in the magnetic �, �, � phases. The above
phenomenological model ascribes the leading role in
structural changes to the shift of close-packed basal
planes in satisfactory agreement with the experiment.
In particular, shift of the planes is responsible for
nonmonotonic temperature dependence of lattice parame-
ter b in � phase (effective repulsion), change of the effec-
tive thermal coefficient in c direction in �–� transition
(nonmonotonic t dependence of interplane distance),
softening of shear modulus and corresponding velocities
in the vicinity of T��.
We assume that with certain stipulation the developed
model may be applied to interpretation of IR spectra
[15,16] which pressure dependence looks similar to pres-
sure dependence of the order parameter. In particular,
we predict nonmonotonic pressure dependence of elastic
moduli in � phase and critical behavior in �–�-transition
point. At the same time some of the problems are still
being outside the theoretical treatment. First of all it
concerns magnetocrystalline structure of phase, me-
tallization of highly compressed oxygen along with corre-
sponding mechanism of superconductivity, optical pro-
perties of high-pressure phases, etc. These and other
problems are the subject of future investigations.
We would like to acknowledge all the participants of
CC-2006 Conference for keen interest to our presenta-
tions and fruitful discussions.
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