日本原子力研究開発機構機関リポジトリ
Japan Atomic Energy Agency Institutional Repository

Title
Author(s)Citation
Text Version

Bulk moduli and equations of state of ice VII and ice VIII
Klotz S., Komatsu Kazuki, Kagi Hiroyuki, Kunc K., Sano Asami,
Machida Shinichi, Hattori Takanori
Physical Review B, 95(17), p.174111_1-174111_7
Publisher's Version

URL

https://jopss.jaea.go.jp/search/servlet/search?5059302

DOI

https://doi.org/10.1103/PhysRevB.95.174111

Right©2017 American Physical Society

Japan Atomic Energy Agency

PHYSICAL REVIEW B 95, 174111 (2017)

Bulk moduli and equations of state of ice VII and ice VIII
S. Klotz,1,* K. Komatsu,2 H. Kagi,2 K. Kunc,1 A. Sano-Furukawa,3 S. Machida,4 and T. Hattori3
1

2

IMPMC, UMR 7590, Université P&M Curie, 4 Place Jussieu, F-75252 Paris, France
Geochemical Research Center, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
3
J-PARC Center, Japan Atomic Energy Agency, 2-4 Shirakata, Tokai, Naka, Ibaraki 319-1195, Japan
4
CROSS-Tokai, Research Centre for Neutron Science and Technology, 162-1 Shirakata, Tokai, Naka, Ibaraki 319-1106, Japan(Received 5 January 2017; revised manuscript received 7 April 2017; published 30 May 2017)
The compression behavior of deuterated ice VII and VIII was investigated by high pressure neutron scatteringin the pressure range 2–13.7 GPa between 93 and 300 K. We establish equations of state which contain accurate
values for the bulk moduli B0 , their pressure derivatives B0 , as well as the ambient pressure volume V0 . Theseequations of state hold over a large part of the stability domain of ice VII, by comparison with available x-ray
data, and to at least ≈13 GPa for ice VIII. They are indistinguishable at low pressures, but beyond ≈7 GPa and at
low temperatures ice VIII appears to become stiffer than expected. This might be related to an anomalous phonon
hardening observed previously in ice VIII in this P /T range [D. D. Klug et al., Phys. Rev. B 70, 144113 (2004)].
DOI: 10.1103/PhysRevB.95.174111

Ice VII and VIII are the dominant high pressure phases ofsolid water. At 300 K, ice VII is stable between 2 and ≈70 GPa
where it transforms continuously into hydrogen-centered ice
X. Upon cooling to below ≈270 K it converts to its hydrogenordered form ice VIII which is stable down to 0 K. The
compressional behavior of these two phases has been studied
by several groups [1–15] up to the megabar range. These
were mainly carried out by x-ray diffraction techniques ina diamond anvil cell (DAC) providing density as a function ofpressure, and focused almost exclusively on ice VII which does
not require cryogenic techniques. Fits to the V (P ) data using
various types of equation of state (EoS) relations are expected
to give the bulk modulus B0 , its pressure derivative B0 , andthe ambient-pressure volume (density) V0 . But, although theseequations of state all give acceptable fits to the data over theconsidered pressure range, the extracted parameters scatterstrongly between different measurements. The reported bulk
moduli of ice VII, for example, scatter between 4–5 GPa[11–13] and 24 GPa [3], and even in the most recent work
vary between 13 GPa [14] and 21 GPa [15]. It hence appears
that most of the reported equations of state are merely fits to
V (P ) data with little physical significance concerning B0 , B0 ,and V0 . It should be noted that ice VII can be decompressed toambient pressure (recovered) [16] and that therefore B0 , B0 ,and V0 are well-defined thermodynamic material properties.
Equations of state with unphysical parameters are not very
useful for drawing any conclusions on binding properties at a
microscopic level. They can hardly be compared with theory,and they cannot be extended significantly beyond the pressure
range they were established for.
Here we present equations of state of ice VII at 298 K and
ice VIII at low temperatures to 93 K which give physicallymeaningful values for B0 , B0 , and V0 which are consistent withmeasured values on recovered samples. Our EoS is based onhigh pressure neutron diffraction data between 2 and 13.7 GPa,using lead as pressure marker, similar to previous experiments [14] but with a significantly extended pressure range.
Compared to DAC measurements, this approach has various*

Corresponding author: stefan.klotz@impmc.upmc.fr

2469-9950/2017/95(17)/174111(7)advantages. The spheroidal pressure chamber used in such a
setup is known to produce an almost homogeneous pressure
distribution [17], contrary to the situation in DAC work wherepressure differences of typically 10% are observed [7]. Also,
the bulk modulus of Pb is similar to that of the sample which
eliminates the possibility of false pressure reading throughthe “Lamé effect” [18]. The pressure determined by Pb wasrecently calibrated to the NaCl scale and gives an accuracy onpressure which is comparable to that of the ruby scale, if notbetter [19].
Neutron diffraction measurements were carried out on two
sample loadings at the high pressure beamline PLANET [20]at MLF, the Japan Proton Accelerator Research Complex (JPARC), Tokai, Ibaraki, Japan. Both runs used double-toroidal
sintered diamond anvils [21], encapsulating TiZr gaskets,
deuterated water (99.6% D) from Eurisotop (France), andhighly pure lead from New Metals and Chemicals Ltd. The
first loading was entirely dedicated to ice VII at 298 K
and used a VX4-type Paris-Edinburgh load frame [22] withthe position of the sample (which is critical for accurate
determination of lattice parameters) maintained to within±0.1 mm relative to the laboratory frame using a previously
determined calibration of the position as a function of load.
The second loading applied a variable-temperature “Mito”
system [23] and sintered diamond anvils with reduced sample
volume compared to the first loading. In these measurements
the sample position was determined by a scan along the beam
direction and monitoring the scattered intensity of the sample,
again to within ±0.1 mm. In both loadings about 1 mm3 lead
was cut into several pieces and distributed across the sample
chamber. Lattice parameters (hence unit cell volumes) of ice
VII/VIII and Pb were determined from Rietveld refinements asshown in Figs. 1 and 2. Due to the small sample volume in thesecond loading the diffraction patterns in these measurements
contain reflections from the anvil material, i.e., diamond. Thiswas considered as an additional phase and included in thefits. The refinements were carried out using Fullprof [24]based on the known structures of ice VII (space group P n3m)with O at (0.25,0.25,0.25) and D at (0.41,0.41,0.41) with
50% occupancy, ice VIII (space group I 41 /amd) with O at
(0,1/4,0.10) and D at (0,0.43,0.21), Pb (space group F m3m),

174111-1©2017 American Physical Society

S. KLOTZ et al.

PHYSICAL REVIEW B 95, 174111 (2017)
TABLE I. Measured lattice parameters of ice VII and Pb at 298 K
and corresponding pressures.

FIG. 1. Neutron diffraction patterns of ice VII collected at 298 K,
at the lowest and highest pressures of loading 2. The lines throughthe data (dots) are results of Rietveld fits. Tick marks are at positionsof Bragg reflections of the sample. Asterisks indicate strongest
reflections of the lead pressure marker and the arrows point to thestrongest reflections of diamond, the anvil material. Accumulation
times are 30 min.

FIG. 2. Neutron diffraction patterns of ice VIII collected at 93 K,
at the lowest and highest pressures. The lines through the data (dots)
are results of Rietveld fits. Tick marks are at positions of Braggreflections of the sample. Asterisks indicate strongest reflectionsof the lead pressure marker and the arrows point to the strongest
reflections of diamond, the anvil material. Accumulation times are
30 min.

Run

aVII (Å)

VVII (Å3 )aPb (Å)P (GPa)

24331
24332
24333
24334
24335
24336
24337
24338
24339
24340
24342
24343
30598
30608
30615
30627
30639
30648
30660

3.3581(1)3.3578(1)3.3373(1)3.3160(1)3.2900(1)3.2634(1)3.2405(1)3.2233(1)3.2044(1)3.1805(1)3.1315(2)3.1318(2)3.3096(1)3.3081(1)3.2503(1)3.2042(1)3.1456(2)3.1276(1)3.0962(1)37.868(3)37.860(3)37.169(2)36.463(2)35.611(2)34.753(2)34.014(2)33.489(2)32.904(2)32.174(2)30.707(4)30.717(4)36.252(2)36.203(2)34.337(4)32.896(4)31.126(5)30.593(3)29.683(2)4.878(1)4.879(1)4.865(1)4.846(1)4.828(1)4.806(1)4.785(1)4.770(1)4.752(1)4.730(1)4.682(3)4.681(3)4.844(1)4.843(1)4.794(2)4.751(2)4.696(2)4.677(1)4.643(1)2.10(3)2.09(3)
2.57(3)3.21(3)3.88(4)4.78(4)5.66(4)6.38(4)7.23(5)8.36(5)11.09(16)
11.16(21)3.30(2)
3.34(3)5.33(8)7.27(9)10.28(14)11.44(8)13.69(5)and diamond (space group F d3m), as well as a minimumof refineable parameters. Apart from lattice parameters, these
are the fractional atomic coordinate of O (phase VIII) and D(phases VII and VIII), isotropic thermal displacement factors,
and profile coefficients. The instrumental parameters which
determine the conversion from time-of-flight to d spacing
were determined by refining a pattern of a SRM 640d silicon
powder sample from NIST, collected in the respective cycles,
and imposing the lattice parameter to 5.431625 Å as recommended by the sample certificate. This procedure ensuresthat all lattice parameters cited in this paper are calibrated
against the Si standard for which the lattice parameters areaccurate to 2 × 10−4 Å. The pressure values were obtained
from the refined lattice parameters of Pb using its equationof state published in Ref. [19] and an ambient pressurelattice parameter of 4.95216 Å as determined in a separate
measurement on the same instrument. An important detailconcerns pressure/temperature changes: to minimize potential
nonhydrostatic strains all pressure changes were carried out atroom temperature, followed by cooling at constant load which
results in essentially isochoric temperature variation. Temperatures were measured with an accuracy better than ±1 K by twothermocouples attached to each anvil at a distance of 10 mmfrom the sample. The results of the refinements are summarizedin Table I for ice VII and Table II for ice VIII at 93 and 196 K.
The V (P ) data (Tables I and II) were then fitted tothree equation of state relations. With the definitions X =(V /V0 )(1/3) , the bulk modulus at ambient pressure B0 =−(∂P /∂lnV )0 , and its pressure derivative B0 = (∂B0 /∂P ),these are:
(1) A third-order Birch equation [25] (“Birch-Murnaghan”
EoS, in the following abbreviated as “BM”):

174111-2


P (V ) = 32 B0 [X−7 − X−5 ] 1 − 34 (4 − B0 )(X−2 − 1) . (1)BULK MODULI AND EQUATIONS OF STATE OF ICE VII . . .

PHYSICAL REVIEW B 95, 174111 (2017)

TABLE II. Measured lattice parameters of ice VIII and Pb at 93
and 196 K and corresponding pressures.
Run

T (K)

VVIII (Å3 )aPb (Å)P (GPa)

30608
30625
30636
30659
30608
30625
30636
30659

93
93
93
93
196
196
196
196

143.97(2)136.26(3)129.49(2)121.42(2)144.17(2)136.37(3)129.57(3)121.53(3)4.841(1)4.791(1)4.738(1)4.669(1)4.843(1)4.793(1)4.740(1)4.670(1)2.75(2)
4.83(4)7.39(5)
11.48(7)3.00(2)5.05(4)7.60(5)
11.63(7)For B0 = 4, this relation reduces to the “second-order BirchMurnaghan” EoS, often simply called “Birch-EoS”.
(2) Rydberg-Vinet EoS [26], in the literature mostly called“Vinet” EoS and in the following abbreviated as “RV”:


3
3B0
(2)
P (V ) = 2 (1 − X) exp (B0 − 1)(1 − X) .
X
2(3) Holzapfel’s AP1 EoS [27], formerly denoted as
H11 [28], and in the following abbreviated as “AP1”:


3 
3B0
(3)
P (V ) = 5 (1 − X) exp (B0 − 3)(1 − X) .
X
2
In this analysis we deliberately exclude the simple Murnaghan
equation [29] due to its well-known shortcomings. This
relation implies a volume-independent B  which leads to an
unrealistic V (P ) at compression ratios V /V0 smaller than≈0.9, see Ref. [30] for a detailed discussion on this issue.
Since compression ranges in our experiments are as small
as 0.7%, the Murnaghan equation is clearly an inappropriate
choice.
Fits to the data were carried out with the program DatLab [31], and checks were made using two other commercial
softwares. These gave slightly different results, though consistent within the errors, probably due to details of the numerical
implementation.
I. ICE VII

Results for ice VII are plotted Fig. 3 in which we include forcomparison two other recently published measurements on ice
VII. These are neutron powder diffraction data [14] measured
between 2 and 6.7 GPa, and x-ray powder data collected ina DAC between 2.7 and 9.8 GPa [15]. We regard these twodata sets as the most trustworthy V (P ) in the 2–10 GPa range,
compared to previous measurements which mostly focused onpressures in the Mbar range [4,5,7,8,11].
Figure 3 reveals a remarkable agreement between ourneutron data and those of Fortes et al. [14], despite the factthat they were obtained on different instruments, at different
neutron sources. The reported EoS by Bezacier et al. [15]gives systematically larger unit cell volumes, up to ≈0.7% at≈5 GPa, equivalent to a difference in pressure of 0.4 GPa.
In an initial analysis, the parameter V0 was included in
the fits which gave (B0 , B0 , V0 ) values of (14.1 ± 1.1 GPa,3

5.8 ± 0.3, 42.2 ± 0.3 Å ) for BM, (13.3 ± 1.0 GPa, 6.3 ± 0.3,

FIG. 3. Pressure dependence of unit cell volume of ice VII. Dotsand circles are measured data, lines are fits to our data [Eqs. (1), (2),and (3), not distinguishable over this pressure range] and previously
published EoS relations [14,15]. Inset: Enlarged 0–2.5 GPa range
with extrapolations of fits to 0 GPa.
3

42.3 ± 0.2 Å ) for RV, and (13.7 ± 1.0, 6.1 ± 0.3, 42.3 ±3
0.2 Å ) for AP1, with all errors corresponding to 68%
confidence limit. From this it is immediately evident that the3value of V0 is close to 42.2 Å and that the bulk modulus of
ice VII is approximately 13–14 GPa, irrespective of the EoS
relation used. Given the remarkable agreement of the V0 value
3found by all three EoS forms V0 = 42.25 Å was imposed in
the subsequent analysis. This gives final results on B0 , B0 , andV0 as shown in Table III.
In order to gauge how far this EoS is extendable inpressure we show in Fig. 4 P (V ) curves extrapolated to
100 GPa (1 Mbar) and compare it with three data setsobtained in diamond anvil cells [4,7,11]. We first recognize theconsiderable scattering of the data, both within a given data set
as well as between different experiments. In the low pressure
range up to 10 GPa the data by Wolanin et al. [7], which werecorrected for nonhydrostatic stresses, agree rather well with
our neutron results and the agreement with the extrapolated
EoS holds to at least ≈20 GPa. Above ≈25 GPa, the scatteringof the data is considerable and no clear conclusion can be
TABLE III. Final results of fits to V (P ) data of ice VII (Table I)
3using three different equations of state. A value V0 = 42.25 Å at
298 K was imposed in the fit as explained in the text.

Third-order Birch
Rydberg-Vinet
Holzapfel AP1

174111-3

3

B0 (GPa)

B0

V0 (Å )13.8(2)
13.6(2)13.7(1)5.9(1)
6.2(1)
6.0(1)

42.25
42.25
42.25

S. KLOTZ et al.

PHYSICAL REVIEW B 95, 174111 (2017)

FIG. 4. Pressure dependence of unit cell volume of ice VII withextrapolations to 100 GPa [upper: Eq. (1), middle: Eq. (3), and lower:
Eq. (2)], compared to three data sets from x-ray diffraction in diamondanvil cells [4,7,11].

drawn on which result is the most accurate. Above ≈30 GPa
the x-ray volumes appear to be on average below our EoS,a fact which hardly can be ascribed to the ice VII-ice X
transition [12,13] which occurs between 60 and 70 GPa [8].
What is clear, on the contrary, it that for pressures >85 GPa, theagreement with the x-ray data becomes better and this holds
to 160 GPa (not shown in Fig. 4), the highest pressure attainedby Wolanin et al. [7] and Loubeyre et al. [11]. In overall it
can be concluded that, despite being established on data in the
0–13.7 GPa range, our EoS extrapolates rather well up to atleast ≈30 GPa, possibly even higher, and therefore covers alarge part of the stability range of ice VII.
It is clear from these measurements that, whatever the
choice of the EoS relation, the zero pressure bulk modulus
B0 of ice VII is approximately 14 GPa or slightly smaller,
and its derivative B0 between 5.9 and 6.2. This places ourB0 on the lower edge compared to most previous diffraction
experiments; consequently, our B0 is larger than reported in the
majority of these investigations. However, our predicted bulk
moduli fit almost perfectly the B(P ) dependence obtained
from Shimizu’s et al. Brillouin scattering measurements [6]to at least 7 GPa. We note that these measurements determine
bulk moduli (adiabatic, which can be converted to isothermalvalues) without a fit to EoS relations, i.e., a comparison withresults from such model-independent methods is particularly
relevant.
II. ICE VIII

Results of the refinements of ice VIII patterns at 93 and196 K (Table II) are plotted in Fig. 5. This plot includes forcomparison the only two published measurements on ice VIII,
i.e., x-ray powder diffraction data by Pruzan et al. [8] andYamawaki et al. [10]. The former were obtained on coldcompressed samples which might explain the larger deviationfrom our data compared to the results of Yamawaki et al.

FIG. 5. Pressure dependence of unit cell volume of ice VIIIat 93 and 196 K. High pressure data (dots and circles) are from
Table II. The two 0 GPa data points correspond to a measured
value from recovered samples (93 K) as well an estimated value(196 K) from an interpolation scheme, see Fig. 6 and text. The solidlines are fits to a Rydberg-Vinet EoS [Eq. (2)] and the dotted lines
correspond to expected V (P ) based on the EoS of ice VII. Published
x-ray data at approximately the same temperatures are shown forcomparison [8,10].

Similar to the analysis of ice VII, the data were fitted
for each temperature to three EoS forms which gave results
listed in Table IV. In the 93 K data set we included an
3ambient pressure value of 160.35 Å determined from neutron
diffraction measurements at ISIS on a recovered sample atthis temperature [32], see the discussion further below. Sincefits to this data set gave V0 values which deviate by only
3
3
0.01 Å , V0 = 160.35 Å was imposed in the final analysis.
For the 196 K isotherm, only the high pressure data were
included in the fits. These gave fitted V0 values which are less
TABLE IV. Final results of fits to V (P ) data of ice VIII (Table II)3using three different equations of state. Values of V0 = 160.35 Å
3
at 93 K and V0 = 164.05 Å at 196 K were imposed in the fits as
explained in the text.

Third-order Birch
93 K
196 K
Rydberg-Vinet
93 K
196 K
Holzapfel AP1
93 K
196 K

174111-4

3

B0 (GPa)

B0

V0 (Å )18.7(2)15.6(3)5.7(1)
6.2(2)

160.35164.0518.5(2)15.4(2)6.0(1)
6.4(2)

160.35164.0518.6(2)15.6(3)5.9(1)
6.2(2)

160.35164.05BULK MODULI AND EQUATIONS OF STATE OF ICE VII . . .

PHYSICAL REVIEW B 95, 174111 (2017)

Vinet formalism [26] by
B0[2 + (η − 1)X − ηX2 ]exp[η(1 − X)],
X2
4 + (3η − 1)X + η(η − 1)X2 − η2 X3,
B0 (T ) =3[2 + (η − 1)X − ηX2 ]

B0 (T ) =FIG. 6. Thermal expansion of ice VII at ambient pressure. Dots
are measured values on recovered samples [16,32], the square is theV0 value of ice VII at 300 K. The line is an extrapolation as explainedin the text.
3

constrained, but nevertheless differed by only 0.1 Å . A value
3of V0 = 164.05 Å was therefore imposed in the final analysis.
Due to the limited number of pressure points the error
bars on V0 , B0 , and B0 are larger than in the case of ice
VII, in particular for the 196 K isotherm where no measured
V0 value is available. However, it is beyond uncertainty
that the bulk modulus substantially increases with decreasing
temperature—as expected—to reach approximately 18.5 GPa
at 93 K. The situation is less clear for B0 which appears tohave little temperature dependence, if at all.
It was pointed out by Besson et al. [33] that an EoS
established for ice VII must also hold for ice VIII, at least closeto ≈270 K. The reason is that the VII-VIII transition line has aslope dT /dP = 0 with a relative volume difference between
the two phases smaller than 10−3 . Even beyond ≈15 GPa, theexperimental evidence of a significant volume difference is
weak. This means that, above (in ice VII) and below (in ice
VIII) the transition line, the equations of state must be strictly
identical. If the thermal expansion of ice VII is known, its EoS
can therefore be extended to low temperatures using the Vinet
formalism [26]. This allows a direct comparison between ice
VII and ice VIII and to gauge to which extent an EoS common
to both phases can be established.
For this purpose we use V (T ) neutron diffraction data ofrecovered ice VII collected in the 6–100 K range at ISIS [16,32]
and parametrized them by an empirical ∝T 3 law as shown
Fig. 6. Above 100 K, where recovered ice VII is unstable, we
assume a linear behavior of the thermal expansion coefficientα(T) = 1.36 × 10−4 K−1 + 1.21 × 10−6 K−2 × (T − 98 K).
When integrated, the resulting V (T ) curve connects to3
V0 = 42.25 Å (12.72 cm3 /mol) at 298 K as required byour high pressure measurements discussed above. It gives
an expansivity of 3.8 × 10−4 K−1 at ambient temperature
which agrees within ≈50% to values found in ice phases Ih
and VI [14] but which is significantly larger than values citedin Refs. [5,9] for ice VII. With this thermal expansion, thetemperature dependence of B0 (T ) and B0 is then given in the

(4)
(5)

where η = 3/2(B0 − 1), X = [V (T )/V0 ]1/3 , and B0 , B0 , andV0 are the usual ambient condition values (0 GPa, 298 K)
as defined above. For T = 93 K (V0 = 12.07 cm3 /mol) thisanalysis predicts for ice VII B0 = 18.5 GPa and B0 = 5.7, andfor T = 196 K (V0 = 12.32 cm3 /mol) B0 = 16.5 GPa and
B0 = 5.9. The corresponding V (P ) curves are shown in Fig. 5
as dotted lines. In the low pressure range up to ≈5 GPa theagreement with the measured data on ice VIII is remarkable,in particular for the 93 K isotherm where the predicted andmeasured bulk moduli coincide within experimental error, i.e.,
B0 = 18.5 GPa. Note that this isotherm makes no assumption
on the thermal expansion coefficient since it uses only themeasured V0 (93 K) value to determine B0 and B0 . Above≈7 GPa the deviations seem to become significant, at least
for the 93 K isotherm. This is obviously related to the factthat at low temperatures the measurements find a larger B0 for
ice VIII than expected from an extrapolation of the EoS of
ice VII. In other words, the ambient pressure bulk moduli are
undistinguishable, but under pressure ice VIII becomes stiffer,at sufficiently low temperatures. The limited number of data
points does not allow a firm conclusion but—if confirmed—
this potential anomaly in the EoS could well be related tothe anomalous stiffening of phonon modes observed in thesame P /T range in infrared measurements and first-principles
calculations [34]. Its microscopic origin is believed to be
in a subtle pressure-induced phonon instability which wasreported [34] to have also an effect on the equation of state ofice VIII. The neutron data also observe an anomaly in the c/a
ratio (not shown) which might be related to this phenomenon.
III. DISCUSSION AND CONCLUSIONS

The values of B0 we find for ice phases VII and VIII,
i.e., 13–18 GPa, are consistent with the fact that ice Ih has
a bulk modulus B0 of 8.5 GPa [35] and general observations
on how bulk moduli scale with density. Apart from very few
exceptions which involve solids with electronic transitions,the bulk modulus of a high pressure phase is always larger
than of its low pressure phase, see Anderson and Nafe [36]for a systematic study an numerous oxides, halides, andsemiconductors. Although hydrogen-bonded systems were
not investigated there, it is unquestionable that a similar
relation holds for ice phases, i.e., one would expect B0 (Ih) <B0 (VI) < B0 (VII). A B0 value for ice VII smaller than thatof ice Ih [11–13] despite a 56% larger density (12.62 vs
19.64 cm3 /mol for ice Ih) is highly unlikely, if not impossible.
Similar arguments might be applied to B0 : Strong covalentbonds tend to show values close to 4 or smaller, weaker metallic
and van der Waals bonds values between 4 and 7. Again, our
B0 = 5.7–6.2 is physically reasonable, given the value for
ice Ih which ranges between 5.4 and 6.6, determined fromultrasonic and neutron diffraction measurement, see Ref. [37]and references therein.

174111-5

S. KLOTZ et al.

PHYSICAL REVIEW B 95, 174111 (2017)

A comparison of the elastic properties of the two phases
at a given temperature shows no significant difference below≈7 GPa (Fig. 5). This demonstrates that hydrogen ordering
has a very small effect on the bulk modulus, and a similar
situation is expected for other ice phases which are related toeach other by hydrogen-ordering phenomena, such as ices II
and IX. This observation seems to confirm the general belief
that the bulk modulus of a solid is determined essentially bydensity, i.e., that structural details have only a minor effect on
B0 (Ref. [38] and references therein).
The reason for the large scattering among previous
diffraction results with a tendency toward overestimating B0
are multiple: potential presence of strongly nonhydrostaticpressure conditions in DAC experiments [4,7], poor samplingof the low-pressure region [4,7,9,11,12], imposing particularvalues for B0 [7,39], for B0 [15] or for V0 [12], or imposing
particular EoS relations [14].
Our equation of state of ice VIII differs from the one
presented previously by Besson et al. [33]. The reason forthe significant difference is that Besson’s et al. EoS [33] wasbased on Hemley’s et al. [4] 300 K EoS of ice VII whichstrongly overestimates the pressure in the 2–25 GPa range, seeFig. 4, hence overestimates significantly the bulk modulus.
Strictly speaking, the equations of state presented here arevalid only for deuterated ices VII and VIII. However, the x-ray
data of Munro et al. [2] demonstrate that there is no detectabledifference between the EoS of D2 O and H2 O to at least 30 GPa,and Shimizu’s et al. Brillouin results to 7.5 GPa come to thesame conclusion [6]. We note as well that a potential differencein V0 between recovered D2 O and H2 O samples is at most 0.04
cm3 /mol for ice VII and 0.02 cm3 /mol for ice VIII [40], i.e.,
smaller than 0.3%.

In conclusion, we have carried out neutron diffraction
experiments to determine the equation of state of ice VII at
298 K between 2 and 13.7 GPa, as well as the EoS of ice VIII to
11.6 GPa and down to 93 K. Fits to these data sets using various
EoS relations can reproduce the elastic behavior of ice VII toat least 20 GPa, and possibly even much higher, by comparison
with accurate low-pressure neutron data to 6.7 GPa [14]and x-ray data obtained in DACs up to 100 GPa [8,11]. In
the 0–7 GPa range, the equations of state of the two phases
are indistinguishable, using a Rydberg-Vinet description withthe thermal expansion determined from recovered ice VII
samples. For higher pressures we find evidence of a small
but significant deviation of the two equations of state at low
temperatures which we attribute to an anomalous stiffeningof phonon modes in ice VIII reported previously [34]. We
believe that these equations of state provide the most accurate
description of the thermoelastic behavior of the dominant highpressure ice phases currently available.

[1] B. Olinger and P. M. Halleck, J. Chem. Phys. 62, 94 (1975).
[2] R. G. Munro, S. Block, and F. A. Mauer, and G. Piermarini J.
Appl. Phys. 53, 6174 (1982).
[3] L.-G. Liu, Earth Planet. Sci. Lett. 61, 359 (1982).
[4] R. J. Hemley, A. P. Jephcoat, H.-K. Mao, C. S. Zha, L. W. Finger,
and D. E. Cox, Nature (London) 330, 737 (1987).
[5] Y. W. Fei, H.-K. Mao, and R. J. Hemley, J. Chem. Phys. 99,
5369 (1993).
[6] H. Shimizu, T. Nabetani, T. Nishiba, and S. Sasaki, Phys. Rev.
B 53, 6107 (1996).
[7] E. Wolanin, Ph. Pruzan, J. C. Chervin, B. Canny, M. Gauthier,D. Häusermann, and M. Hanfland, Phys. Rev. B. 56, 5781
(1997).
[8] Ph. Pruzan, J. C. Chervin, E. Wolanin, B. Canny, M. Gauthier, and M. Hanfland, J. Raman Spectroscop. 34, 591(2003).
[9] M. R. Frank, Y. Fei, and J. Hu, Geochim. Cosmochim Acta 68,
2781 (2004).
[10] H. Yamawaki, H. Fujihasa, M. Sakashita, A. Nakayama, and K.
Aoki, Physica B 344, 260 (2004).
[11] P. Loubeyre, R. LeToullec, E. Wolanin, M. Hanfland, and D.
Häusermann, Nature (London) 397, 503 (1999).
[12] E. Sugimura, T. Iitaka, K. Hirose, K. Kawamura, N. Sata, and
Y. Ohishi, Phys. Rev. B. 77, 214103 (2008).

[13] Y. Asahara, K. Hirose, Y. Ohishi, N. Hirao, and M. Murakama,
Earth Planet. Sci. Lett. 299, 474 (2010).
[14] A. D. Fortes, I. G. Wood, M. G. Tucker, and W. G. Marshall,
J. Appl. Crystallogr. 45, 523 (2012).
[15] L. Bezacier, B. Journaux, J.-P. Perrillat, H. Cardon, M. Hanfland,
and I. Daniel, J. Chem. Phys. 141, 104505 (2014).
[16] S. Klotz, J. M. Besson, G. Hamel, R. J. Nelmes, J. S.
Loveday, and W. G. Marshall, Nature (London) 398, 681(1999).
[17] S. Klotz, J. M. Besson, and G. Hamel, High Press. Res. 26, 277(2006).
[18] Y. Wang, D. J. Weidner, and Y. Meng, Properties of the
Earth and Planetary Materials at High Pressure and High
Temperature, Geophysical Monograph Vol. 101 (American
Geophysical Union, 1998), p. 365.
[19] Th. Strässle, S. Klotz, K. Kunc, V. Pomjakushin, and J. S. White,
Phys. Rev. B. 90, 014101 (2014).
[20] T. Hattori et al., Nucl. Instrum. Meth. Phys. Res. A 780, 55(2015).
[21] S. Klotz, J. M. Besson, G. Hamel, R. J. Nelmes, J. S. Loveday,
W. G. Marshall, and R. M. Wilson, Appl. Phys. Lett. 66, 1735
(1995).
[22] S. Klotz, G. Hamel, and J. Frelat, High Press. Res. 24, 219(2004).

ACKNOWLEDGMENTS

This work is based on experiments performed at theJapanese neutron spallation source MLF under proposal
number 2016I0011 and was supported by JSPS KAKENHI
Grants No. 26246039 and No. 15K21712. S.K. is grateful for
beamtime obtained at ISIS (Rutherford Appleton Laboratory,
Chilton, Didcot, U.K.) in 1999 to collect the low temperature
data shown in Fig. 6 and to J. S. Loveday for help during thesemeasurements. We acknowledge helpful discussions with K.
Syassen (MPI Stuttgart) who also made the DatLab [31]software available to the Paris group.

174111-6

BULK MODULI AND EQUATIONS OF STATE OF ICE VII . . .
[23] K. Komatsu, M. Moriyama, T. Koizumi, K. Nakayama, H. Kagi,
J. Abe, and S. Hario, High Press. Res. 33, 208 (2013).
[24] J. Rodríguez-Carvajal, Physica B 192, 55 (1993); http://wwwllb.cea.fr/fullweb/powder.htm.
[25] F. Birch, J. Appl. Phys. 9, 279 (1938).
[26] P. Vinet, J. R. Smith, J. Ferrante, and J. H. Rose, Phys. Rev. B
35, 1945 (1987).
[27] W. B. Holzapfel, High Press. Res. 16, 81 (1998).
[28] O. Schulte and W. B. Holzapfel, Phys. Rev. B 52, 12636(1995).
[29] F. D. Murnaghan, Am. J. Math. 59, 235 (1937).
[30] K. Syassen, Isothermal EOS Functions for Solids (Max-PlanckInstitut, Stuttgart, 2012), unpublished.
[31] K. Syassen, DatLab Computer Code (Max-Planck-Institut,
Stuttgart, 2005).
[32] S. Klotz, ISIS Experimental Report 1999, RB 10412,
Rutherford-Appleton Laboratory, Chilton, Didcot, UK.

PHYSICAL REVIEW B 95, 174111 (2017)[33] J. M. Besson, Ph. Pruzan, S. Klotz, G. Hamel, B. Silvi, R. J.
Nelmes, J. S. Loveday, R. M. Wilson, and S. Hull, Phys. Rev. B
49, 12540 (1994).
[34] D. D. Klug, J. S. Tse, Z. Liu, X. Gonze, and R. J. Hemley,
Phys. Rev. B 70, 144113 (2004).
[35] R. Feistel and W. Wagner, J. Phys. Chem. Rev. Data 35, 1021
(2006).
[36] O. L. Anderson and J. E. Nafe, J. Geophys. Res. 70, 3951 (1965).
[37] Th. Strässle, S. Klotz, J. S. Loveday, and M. Braden, J. Phys.:
Condens. Matter 17, S3033 (2005).
[38] L. Glasser, Inorg. Chem. 49, 3424 (2010).
[39] Note that B0 = 14.9 GPa was imposed in the data analysis ofRef. [7]. A three-parameter fit in the 0–13 GPa range gives
B0 = 27.8 GPa [7]. The data of this paper are identical to theone of a later work by Pruzan et al. [8].
[40] S. Klotz, Th. Strässle, C. G. Salzmann, J. Philippe, and S. F.
Parker, Europhys. Lett. 72, 576 (2004).

174111-7