$\alpha$-$\beta$ and $\beta$-$\gamma$ phase boundaries of solid oxygen observed by adiabatic magnetocaloric effect

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by  Toshihiro Nomura

α-β and β-γ phase boundaries of solid oxygen
observed by adiabatic magnetocaloric effect
T. Nomura,∗ Y. Kohama, Y. H. Matsuda,† and K. Kindo
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
T. C. Kobayashi
Department of Physics, Okayama University, Okayama 700-8530, Japan
(Dated: January 19, 2017)
7
1 The magnetic-field-temperature phase diagram of solid oxygen is investigated by the adiabatic
0 magnetocaloriceffect(MCE)measurementwithpulsedmagneticfields. Relativelylargetemperature
2 decrease with hysteresis is observed at just below the β-γ and α-β phase transition temperatures
owing to the field-induced transitions. The magnetic field dependences of these phase boundaries
n
a are obtained as Tβγ(H) = 43.8−1.55×10−3H2 K and Tαβ(H) = 23.9−0.73×10−3H2 K. The
J magneticClausius-ClapeyronequationquantitativelyexplainstheH dependenceofTβγ,meanwhile,
doesnotT . TheMCEcurveat T isoftypicalfirst-order,whilethecurveat T seemstohave
3 αβ βγ αβ
both characteristics of first- and second-order transitions. We discuss the order of the α-β phase
1
transition and propose possible reasons for theunusualbehavior.

]
i
c
I. INTRODUCTION temperature (H-T) phase diagram has not been studied
s
- until recently. In 2014, θ phase of solid oxygen was dis-
l
tr Molecularoxygen,O2hasaspinquantumnumberS = covered in ultrahigh magnetic field over 100 T using the
m 1 andbehaves as a magnetic molecule. In the condensed single-turncoil(STC)technique[18,19]. Thisisthefirst
. phases, magnetic interaction between O2 molecules has observation of the field-induced phase transition of oxy-
at animportantroleforthecondensationenergyinaddition gen. At that time, strangely, the field-induced α-β and
m to van der Waals interaction [1–4]. As decreasing tem- β-γphasetransitionswerenotobserved[19],andthefield
- perature, antiferromagnetic (AFM) correlation develops dependences ofthese transitiontemperatures (Tαβ, Tβγ)
d and three phases which have different crystallographic have never been clarified experimentally [4]. The H-T
n and magnetic structures appear as γ, β, and α. The phase diagram is the most fundamental information of
o γ phase (54.4–43.8 K, paramagnetic, cubic) is called as the magnetic material for discussing thermodynamical
c
a plastic phase where molecules are rotating at certain property. Therefore, the clarification of the H-T phase
[
lattice sites [2, 5]. In the β phase (43.8–23.9 K, short- diagramis animportantissue for the oxygen-relatedsci-

1 rangeAFM,rhombohedral),molecularaxisis orderedto ence and technology.
v
one direction with large volume contraction. The geo-
6
metrical frustration due to the triangular lattice of the
9
0 basal plane suppresses the long-range AFM ordering. In Calorimetricmeasurementisstraightforwardtoclarify
5 the α phase (23.9–K,long-rangeAFM, monoclinic), the the thermodynamicalrelationinthe phasediagram. Re-
0 frustration is lifted by the lattice deformation and the
cently, adiabatic magnetocaloric effect (MCE) measure-
. long-range order is realized [6–8].
1 ment is developed for the pulse magnetic field technique
0 The β-γ phase transition is of first-order where larger andappliedforvariouskindsofmaterials[20–22]. Inthe
7 entropythanfusionisreleased[9,10]. Incontrast,theor-
adiabatic MCE measurement, we measure the H depen-
1 der of the α-β phase transitionis notclear despite many
dence of T which changes to conserve the total entropy.
v: calorimetricstudies[4,9–12]. Inrecentstudies,mostau- Inotherwords,eachMCEcurvecorrespondstotheisen-
i thors are inclined toward the opinion of ”first-order but tropic curve in the H-T phase diagram. A contour plot
X
close to second-order” [13, 14]. However, the highest-
of entropy enables the quantitative discussionon the en-
r resolved heat capacity measurement suggests that there
a tropy relation between the phases.
is an intermediate phase between the α and β phases
[11], and the detail of the α-β phase transition is still
controversial.
In the last 50 years, the pressure-temperature (P-T) In this study, we conducted the adiabatic MCE mea-
phasediagramofsolidoxygenhasbeenextensivelystud- surement of condensed oxygen up to 56 T using pulse
iedandfourhigh-pressurephases(δ,ǫ,ζ, η)werediscov- magnetic fields. From the obtained isentropes, the mag-
ered [4, 15–17]. On the other hand, the magnetic-field- netic field dependences of Tβγ and Tαβ were revealedfor
the first time. We comment on the reason why these
phase boundaries had not been observed in the previous
high-field measurements [2, 18, 19]. We also comment
∗ [email protected] on the long lasting problem, the order of the α-β phase
† [email protected] transition.
2
FRP base FRP tube inhomogeneity of temperature during the calibration of
the thermometers. Near the phase boundaries of solid
Stainless tube
oxygen, the absolute error of T can be recalibrated by
Heater Tβγ = 43.8 K and Tαβ = 23.9 K where heat capacity
diverges. These valuesarerecommendedfor the thermo-
Pickup coil
metric fixed points [12].
Cernoxfor calibration
Kapton tube
RuO for MCE
Condensed O 2 III. RESULT
2
Cernoxfor MCE
Vacuum (few Pa)
Summarized results of the MCE measurement of con-
Liquid He
densedoxygenareshowninFig. 2for(a)higherand(b)
lower temperature regions. The γ-liquid, β-γ, and α-β
FIG. 1. Schematic setting of theMCE measurement for con- phase boundaries at zero field (TγL(0), Tβγ(0), Tαβ(0))
densed oxygen.
areshownbydottedlines. Largernoiselevelinhightem-
perature region is due to the mechanical vibration and
relatively lower sensitivity of the thermometer. The ef-
II. EXPERIMENTAL SETUP fectofvibrationis suppressedatlowtemperatureby the
solidification of oxygen.
TheconceptoftheadiabaticMCEmeasurementisde- ∆T foreachinitial temperature (T0) is summarizedin
scribed in Ref. [20]. The MCE data were obtained in Fig. 3 (a). Corresponding amounts of heat (∆Q) and
the pulsed magnetic fields with the duration time of 36 entropy change (∆S) are shown in Figs. 3 (b) and (c),
ms. Two types of resistance thermometers, Cernox bare respectively. The definitions are as follows.
chip(CX-1050orCX-1030,LakeShoreCryotronics)and
RuO2 film(EZ-13,TanakaKikinzokuKogyoK.K.)were ∆Q=−c0∆T, (1)
employed. The sizes of the Cernox and the RuO2 film
were 1×0.8×0.2 mm3 and 2×1.5×0.1 mm3, respec-
∆S =∆Q/T0, (2)
tively. RuO2 wasemployedonlyforthe lowtemperature
region below 10 K where the reliability becomes better wherec0 isheatcapacityatT0 andzerofieldreportedby
than Cernox. Resistance was measured by the standard Fagerstroemand Hollis Hallett [10]. Data plots near the
ac four-probe method using numerical lock-in technique phaseboundariesareremovedsinceheatcapacitygreatly
at the frequency of 100 kHz. dependsonT andH. ∆T at50Tislessthan1Kinmost
Schematic setup near the thermometer is shown in T0 region. Only near the β-γ and α-β phase boundaries,
Fig. 1. Solidoxygenwascondensedfrom high-purity O2 larger temperature decrease is observed with hysteresis.
gas (99.999%) at the bottom of the tube made of fiber- Thisindicatesthattheβ-γ andα-β phasetransitionsare
reinforcedplastic(FRP).The innerdiameterofthe tube induced by magnetic fields.
was 10 mm. Two thermometers were directly buried in- The enlargedMCE curves with different T0 are shown
sidethecondensedoxygenforidealthermalcontact. The in Figs. 4 for the (a) β-γ and (b) α-β boundaries. Even
thermometers located at 10 mm below the FRP base to if T0 is slightly changed, all MCE curves reach to the
reduce the heattransferfrom the constructionparts and same temperature at the top of the field and go back to
to realize the adiabatic condition. The Cernox was fixed T0withhysteresis. Byconnectingthecenterofhysteresis
ontheKaptontube(diameter1mm,thickness0.06mm) withquadraticfunction,themagneticfielddependenceof
by vanish to suppress mechanical vibration. The RuO2 the phase boundaries are obtained as the dashed curves.
film was freely hanged in the condensed oxygen. The Noteworthy,the MCE curvesataroundthe α-β phase
thermometers are rigidly fixed inside solid oxygenat be- boundary depend on the thermal history. When the α
low 43.8 K. phase is prepared with keeping the temperature above
The temperature dependence of the ac resistance is 20 K, the MCE curves become different from Fig. 4 (b)
calibrated for each setup using the calibrated Cernox at although the obtained phase boundary is similar (Sup-
zero field. The effect of magnetoresistance at each tem- plemental Material S-I [24]). Lipinski et al. also pointed
perature is estimated from the same measurement for outthatthe sampleofthe αphasehastobe preparedat
solidargonwheretheintrinsictemperaturechange(∆T) below 20 K to obtain reproducible data of heat capacity
is negligible. The artificial ∆T caused by magnetoresis- [11,12]. Inthispaper,alldataneartheα-βphasebound-
tance of the thermometer is subtracted as background. arywerecollectedwiththesamplewhichexperiencedthe
Inthisanalyticalprocess,non-linearityandangledepen- temperature below 20 K.
dence of the magnetoresistance could cause large error In this study, there was no indication of the field-
of ∆T [23]. The relative error of ∆T at 50 T is esti- induced γ-liquid phase transition near TγL(0). This
mated as ±0.2 K. The absolute error of T is estimated would be because TγL is almost independent on H [19].
as±0.5K.Theabsoluteerrormainlyoriginatesfromthe Since the magnetic susceptibilities of the γ and liquid
3
90 45
1 (a)
(a) Tbg (0)
0
K) -1
T ( D T at 50 T
85 40 D -2 Calculated
-3
a b g liquid
-4
50
(b)
80 35
ol)
m
J/ 0
Q (
D
75 30
-50
0.1 (c)
R
K) 70 25 S/ 0.0
e ( Tba (0) D
ur
at
per -0.1
m
Te 65 20 0 20 40 60 80
T (K)
0
FIG. 3. (a) Temperature changes at 50 T as a function of
T0. The blue curve shows the estimated value by Eq. (4).
60 15 Correspondingamountsof(b)heat,and(c)entropy. Entropy
is normalized bythe gas constant, R=8.31 JK−1mol−1.
IV. DISCUSSION
55 10
TgL (0) At first, we discuss the common results of the MCE
where no phase transition occurs. The magnetocaloric
relation [25, 26]
50 5 ∂T TH ∂χ
( ) =− ( ) , (3)
S H
∂H c ∂T
H
where c is the specific heat at constant magnetic field,
H
(b)
manifests that the sign of ∂χ/∂T determines the sign
45 0 of MCE. ∂χ/∂T is negative for the liquid and γ, and
0 20 40 60 0 20 40 60
positive for the β and α phases [1]. Therefore, ∆T in
Magnetic field (T)
Fig. 3 (a) qualitatively agrees with Eq. (3).
To a first approximation, ∆T is obtained by fixing
FIG.2. MCEcurvesofcondensedoxygenin(a) 90-45Kand
in(b)45-0K.Phaseboundariesatzerofield(TγL(0),Tβγ(0), cH =c0, T =T0, and (∂χ/∂T)H=0 as
T (0)) are shown by dotted lines.
αβ ∆T =−T0(∂χ/∂T)H=0H2. (4)
2c0
The expected ∆T at 50 T is shown by the blue curve in
Fig. 3 (a) (details are givenin Supplemental MaterialS-
II [24]). Itquantitativelyagreesforallphases,but isnot
oxygen are only slightly different [2], external magnetic perfect for the γ and β phases. That means the approx-
fieldcannotbeadrivingforceofthisphasetransition. T imation fails for these phases, at most by the factor of
sweep at fixed H would be necessary to study the phase two. Mostprobably,itisduetotheinaccuracyof∂χ/∂T
boundary. under the external magnetic field.
4
to overcome the entropy barrier ∆S and to transform
44 Tbg (0) the entire β phase into the γ phase.βγ
The fraction of the γ phase at 53 T can be estimated
43
K) by the equation of entropy. We write the entropy as
e ( functionsofH andT fortheβ andγ phasesasS (H,T)
ur 42 β
at and Sγ(H,T), respectively. The initial entropy is equal
per to the average of them as
m 41
e
T Sβ(0 T,T0)=cβSβ(H,T)+cγSγ(H,T), (5)
40
(a) where c and c are the fractions of the β and γ phases,
β γ
39 respectively. Here, the contribution of mixing entropy is
0 20 40 60
neglected. Byintroducingtheentropydifferencebetween
Magnetic field (T)
theβ andγ phases∆S (H,T)andusingtherelationof
βγ
c =1−c ,
β γ
24 Tba (0)
Sβ(0 T,T0)−Sβ(H,T)
c = . (6)
γ
K) ∆Sβγ(H,T)
e (
ur 23 The entropy difference between the β and γ phases at
at
er zero field is reported as 2.04R [10]. Figure 3 (c) shows
mp that ∆Sβγ(H,T) decreases approximately by 0.1R at 50
Te T. Therefore, ∆Sβγ(53 T,39.6 K) = 2.04R − 0.1R =
22 1.94R. By using the isentropic relation (blue curve in
(b) Fig. 4(a)), Sβ(53 T,39.6 K) = Sβ(0 T,40.2 K) = 3.88R
[24]. If we consider the case of T0 = 43.0 K, the initial
0 20 40 60 entropyis S (0 T,43.0 K)=4.22R[24]. c atthe top of
β γ
Magnetic field (T)
the field is obtained as c =0.18. Even at 53 T, the β-γ
γ
phasetransformationoccursonlypartiallyundertheadi-
FIG.4. EnlargedMCEresultsataround(a)T and(b)T .
βγ αβ abatic condition. If T continues to decrease with the
βγ
Blackdashedcurvesarethequadraticfunctionshownforthe dashed curve, 100±10T is necessary for c =1.
phase boundaries. γ
This is the reason why the field-induced β-γ phase
transition had never been observed in the early pulsed-
fieldexperiments[2,18,19]. Theywereconductedatthe
Inthe followingsections,wequantitativelydiscussthe
adiabatic condition because of the short duration of the
β-γ and α-β phase boundaries. For estimating the en-
field. In the optical and magnetization measurements,
tropy at zero field, the specific heat data reported by
averaged results from the coexisting β and γ phases are
Fagerstroem and Hollis Hallett [10] are employed (Sup-
obtained. Ifc graduallyincreases,itisdifficulttodetect
plemental Material S-III [24]). γ
the phase transitionandseparatethe contributions from
coexisting phases. We emphasize that this phase coex-
istence is only the case for adiabatic condition. If the
A. β-γ phase boundary system is isothermal, thermal energy is provided by the
heat bath and the phase transition finishes at a certain
Here,wediscusstheresultsneartheβ-γ phasebound- field.
aryinFig. 4(a). EvenwhenT0 changes,allMCEcurves Next, we compare the obtained phase boundary with
followthe sameboundary(dashedcurve)andgobackto the expected one derived from the thermodynamical re-
T0 with hysteresis. This behavior is due to the phase lation. The slope of the phase boundary is described by
equilibrium between the β and γ. When the magnetic the magnetic Clausius-Clapeyronequation
field reaches to the first-order β-γ phase boundary, the
β starts to transform to the γ. Here, the transition oc- dTc/dHc =−∆M/∆S =−∆χH/∆S. (7)
curs only partially since the total entropy is conserved
Here,∆χ isthe differences ofmagneticsusceptibility be-
in the adiabatic condition. When the fraction of the γ
tween two phases. If the ratio λ = ∆χ/2∆S is indepen-
phaseincreases,temperature hasto decreaseto compen-
dent on H, an integrated form is written as
sate the entropy difference ∆S = 2.04R [10]. Thus,
βγ
the total entropy is conserved by balancing the fraction Tc(H)=Tc(0)−λH2. (8)
andtemperaturealongtheβ-γ phaseboundary. Because
of this balance, all MCE curves reach to the same point The β-γ phase boundary is wellfitted by this formula as
(H = 53 T, T = 39.6 K) regardless of the different T0.
In other words, the magnetic field of 53 T is not enough T (H)=43.8−1.55×10−3H2. (9)
βγ
5
By using the reported values of ∆χ = 51.2 × 10−3 1990, most researches insist first-order by the measure-
βγ
JT−2mol−1 [1] and ∆S = 16.9 JK−1mol−1 [10], λ at ments of heat capacity [11, 12], x-ray diffraction [13],
βγ
zero field is estimated as λ = 1.51×10−3 KT−2. This optical spectroscopy [27, 28]. Our results of the adia-
value is in good agreement with the experimental one. batic MCE measurements, which indicate two-phase co-
The correspondence between the experiment and Eq. existence with hysteresis, agree with these researches.
(8) means that λ is independent on H. However, Fig. However, the reason of why the MCE curves also show
3(c) shows that ∆S decreases as H increases. For the the continuous-transition-like behavior is not clear. In
βγ
compensation, it is indicated that ∆χ also decreases the following discussion, three possible reasons for this
βγ
inmagneticfields. Thisisexplainedbythe Brillouin-like behavior are proposed.
magnetizationcurveoftheparamagneticγ phase. When
Thefirstpossiblereasonisinhomogeneousstressinthe
the magnetization curve of the γ phase shows a trend of
sample. Thesampleusedinthisstudy ispolycrystalline.
saturationinhighfields,∆χ decreases. Forthecaseof
βγ
The local stress at the grain boundaries could slightly
the β-γ phase boundary,these contributionscompensate
change the transition field. Especially, the α-β phase
and λ fortuitously stays constant.
transition is considered to be martensitic [2, 30, 31],
which implies sensitive to stress. However,we confirmed
that the MCE curves in Fig. 4 (b) were reproduced in
B. α-β phase boundary
four different samples with different settings. If it orig-
inates from local stress, it should depend on the sam-
Compared with the β-γ phase boundary, the MCE ple qualityandshowdifferentbehaviorsforeachsample.
curves near the α-β phase boundary are difficult to in- Therefore, the effect of local stress does not seem to be
terpret since they have both characteristics of first- and the dominant reason.
second-order transitions. As a characteristic of the first-
order,allMCEcurvesreachtothesamepoint(H =56T, ThesecondoneisthemagneticanisotropyoftheAFM
T =21.8 K) with slight hysteresis, indicating that the α ordered α phase [7]. Since the magnetic susceptibility of
and β phases are coexisting. However,one unique phase the αphase isanisotropic,the transitionfieldcouldvary
boundary can not obtained by connecting the center of dependingontheorientationofeachdomain. Thediffuse
hysteresis. Thisisbecausethetemperaturedecreasewith α-β phase boundary for the polycrystalline sample was
hysteresis starts before it reaches the α-β phase bound- proposedinRef. [7]eventhoughitdoesnotcoincidewith
ary. Thisbehaviorresembleswiththesecond-order(con- Eq. (10). In this sense, the α-β phase transition should
tinuous) transition which shows continuous temperature take place in broad area of the H-T phase diagram for
change prior to the phase transition. In Fig. 4(b), we polycrystal.
propose a feasible phase boundary by the dashed curve
Thethirdoneisthe intermediatephasebetweenthe α
as
and β phases. The high-resolution measurement of heat
T (H)=23.9−0.73×10−3H2. (10) capacity indicated that the peak of the α-β phase tran-
αβ sitionis composedoftwo sharperpeaks withthe separa-
tion of 0.02 K [11, 12]. The authors argued that the
Wecomparetheobtainedα-βphaseboundarywiththe
double peak is due to the intermediate phase. Inter-
predictedonebyJansenandAvoird[7]. Theirprediction
estingly, the helical-ordered intermediate phase is the-
wasbasedonEq. (8),andtheparameterswereemployed oretically predicted by Slusarev et al. [32, 33] and dis-
as ∆χ =14.1×10−3 JT−2mol−1 [1] and ∆S =3.85
αβ αβ cussed by Gaididei and Loktev [34]. However, the exis-
JK−1mol−1 [29]. λ at zero field was predicted as λ =
tence of the intermediate phase has not been confirmed
1.8 × 10−3 KT−2. This value is more than twice the
in other measurements because of its tiny temperature
value obtained in this study.
range. If the intermediate phase exists, the behavior of
The discrepancyis consideredtobe due to the inaccu-
the MCE curve is not clear; it depends on the orders
racy of ∆χ and ∆S . The Clausius-Clapeyronequa-
αβ αβ of α-intermediate and intermediate-β transitions. More-
tion can be applicable for any points in the phase di-
over, the equilibrium is not guaranteed for the pulsed
agram. However, ∆χ and ∆S have to be estimated for
field measurement. In any cases, the existence of the
theinfinitesimal∆T. Thisestimationisdifficultnearthe
intermediatephasecouldaffecttheMCEcurveinanun-
α-β phase transition since χ and S show tendency of di-
usual way.
vergence. Actually, reported ∆S differs each other by
αβ
around 20% even if we exclude the reports of ”no latent At this stage, we cannot conclude which is the main
heat” [4]. Therefore, the employed values of ∆χαβ and factor to explain the MCE curve of the α-β phase tran-
∆Sαβ couldbe inappropriatefortheestimation. Thera- sition. For further discussions, single crystal of the solid
tio obtained in this study (λ = ∆χ/2∆S = 0.73×10−3 oxygen α phase is necessary. For single crystal, the ef-
KT−2) is considered to be more reliable. fectofstrainwouldbesuppressedandthetransitionfield
Next, wediscussthe long-lastingproblemofsolidoxy- can be discussed for eachaxis. For the third reason,fur-
gen, the order of the α-β phase transition. The history ther investigation is difficult in the pulse field because
of the controversyis well summarized in Ref. [11]. After the precise control of T and H is necessary.
6
V. CONCLUSION equation, while the α-β phase boundary was not. The
discrepancy of the α-β phase boundary was attributed
to the difficulty of estimating ∆χ and ∆S .
αβ αβ
TheMCEcurveattheβ-γphaseboundaryisoftypical
The adiabatic MCE measurement was conducted for first-ordertransition. Ontheotherhand,theMCEcurve
liquidandsolidoxygenupto56T.∆T wasqualitatively at the α-β phase boundary has both characteristics of
discussed for each phase in terms of the magnetocaloric first-andsecond-ordertransitions. We arguedthis could
relation. Relativelylargetemperaturedecreasewithhys- be due to the polycrystalline sample or the intermediate
teresis was observed at just below T (0) and T (0), phase between the α and β phases. In any cases, our
βγ αβ
suggesting the β-γ and α-β phase transitions occurred. resultsagreewiththepreviousstudieswhichsuggestthat
In the adiabatic condition, the entropy of the system is the α-β phase transition is of first-order.
conserved. When the high-entropy phase is induced by
magnetic fields, temperature decreasesfor compensating
the entropy difference at the first-order transition. At ACKNOWLEDGMENTS
that time, the fraction of the high-field phase gradually
increases along the phase boundary with decreasing T. TNwassupportedbyJapanSocietyforthePromotion
This is the reasonfor that the field-induced β-γ and α-β of Science through the Program for Leading Graduate
phase transitions had not been observed in pulsed mag- Schools (MERIT) and a Grant-in-Aid for JSPS Fellows.
neticfields. Theobtainedβ-γ phaseboundarywasquan- This work was partly supported by JSPS KAKENHI,
titatively explained by the magnetic Clausius-Clapeyron Grant-in-Aid for Scientific Research (B) (16H04009).
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