Poisson’s ratio in cryocrystals under pressure
We present results of lattice dynamics calculations of Poisson’s ratio (PR) for solid hydrogen and rare gas solids (He, Ne, Ar, Kr and Xe) under pressure. Using two complementary approaches — the semi-empirical many-body calculations and the first-principle density-functional theory calculations w...
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irk-123456789-1278312017-12-29T03:03:27Z Poisson’s ratio in cryocrystals under pressure Freiman, Yu.A. Grechnev, A. Tretyak, S.M. Goncharov, Alexander F. Gregoryanz, Eugene 10th International Conference on Cryocrystals and Quantum Crystals We present results of lattice dynamics calculations of Poisson’s ratio (PR) for solid hydrogen and rare gas solids (He, Ne, Ar, Kr and Xe) under pressure. Using two complementary approaches — the semi-empirical many-body calculations and the first-principle density-functional theory calculations we found three different types of pressure dependencies of PR. While for solid helium PR monotonically decreases with rising pressure, for Ar, Kr, and Xe it monotonically increases with pressure. For solid hydrogen and Ne the pressure dependencies of PR are nonmonotonic displaying rather deep minimums. The role of the intermolecular potentials in this diversity of patterns is discussed. 2015 Article Poisson’s ratio in cryocrystals under pressure / Yu.A. Freiman, Alexei Grechnev, S.M. Tretyak, Alexander F. Goncharov, and Eugene Gregoryanz // Физика низких температур. — 2015. — Т. 41, № 6. — С. 571-575. — Бібліогр.: 25 назв. — англ. 0132-6414 PACS: 67.80.F–, 67.80.B–, 62.20.dj http://dspace.nbuv.gov.ua/handle/123456789/127831 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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10th International Conference on Cryocrystals and Quantum Crystals 10th International Conference on Cryocrystals and Quantum Crystals |
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10th International Conference on Cryocrystals and Quantum Crystals 10th International Conference on Cryocrystals and Quantum Crystals Freiman, Yu.A. Grechnev, A. Tretyak, S.M. Goncharov, Alexander F. Gregoryanz, Eugene Poisson’s ratio in cryocrystals under pressure Физика низких температур |
| description |
We present results of lattice dynamics calculations of Poisson’s ratio (PR) for solid hydrogen and rare gas
solids (He, Ne, Ar, Kr and Xe) under pressure. Using two complementary approaches — the semi-empirical
many-body calculations and the first-principle density-functional theory calculations we found three different
types of pressure dependencies of PR. While for solid helium PR monotonically decreases with rising pressure,
for Ar, Kr, and Xe it monotonically increases with pressure. For solid hydrogen and Ne the pressure dependencies
of PR are nonmonotonic displaying rather deep minimums. The role of the intermolecular potentials in this
diversity of patterns is discussed. |
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Article |
| author |
Freiman, Yu.A. Grechnev, A. Tretyak, S.M. Goncharov, Alexander F. Gregoryanz, Eugene |
| author_facet |
Freiman, Yu.A. Grechnev, A. Tretyak, S.M. Goncharov, Alexander F. Gregoryanz, Eugene |
| author_sort |
Freiman, Yu.A. |
| title |
Poisson’s ratio in cryocrystals under pressure |
| title_short |
Poisson’s ratio in cryocrystals under pressure |
| title_full |
Poisson’s ratio in cryocrystals under pressure |
| title_fullStr |
Poisson’s ratio in cryocrystals under pressure |
| title_full_unstemmed |
Poisson’s ratio in cryocrystals under pressure |
| title_sort |
poisson’s ratio in cryocrystals under pressure |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2015 |
| topic_facet |
10th International Conference on Cryocrystals and Quantum Crystals |
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http://dspace.nbuv.gov.ua/handle/123456789/127831 |
| citation_txt |
Poisson’s ratio in cryocrystals under pressure / Yu.A. Freiman, Alexei Grechnev, S.M. Tretyak, Alexander F. Goncharov, and Eugene Gregoryanz // Физика низких температур. — 2015. — Т. 41, № 6. — С. 571-575. — Бібліогр.: 25 назв. — англ. |
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Физика низких температур |
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2025-07-09T07:49:31Z |
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6, pp. 571–575
Poisson’s ratio in cryocrystals under pressure
Yu.A. Freiman, Alexei Grechnev, and S.M. Tretyak
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: freiman@ilt.kharkov.ua
Alexander F. Goncharov
Geophysical Laboratory, Carnegie Institution of Washington, 5251 Broad Branch Road NW, Washington DC 20015, USA
Center for Energy Matter in Extreme Environments and Key Laboratory of Materials Physics, Institute of Solid State
Physics, Chinese Academy of Sciences, 350 Shushanghu Road, Hefei, Anhui 230031, China
Eugene Gregoryanz
School of Physics and Centre for Science at Extreme Conditions, University of Edinburgh, Edinburgh EH9 3JZ, UK
Received February 18, 2015, published online April 23, 2015
We present results of lattice dynamics calculations of Poisson’s ratio (PR) for solid hydrogen and rare gas
solids (He, Ne, Ar, Kr and Xe) under pressure. Using two complementary approaches — the semi-empirical
many-body calculations and the first-principle density-functional theory calculations we found three different
types of pressure dependencies of PR. While for solid helium PR monotonically decreases with rising pressure,
for Ar, Kr, and Xe it monotonically increases with pressure. For solid hydrogen and Ne the pressure dependen-
cies of PR are nonmonotonic displaying rather deep minimums. The role of the intermolecular potentials in this
diversity of patterns is discussed.
PACS: 67.80.F– Solids of hydrogen and isotopes;
67.80.B– Solid 4He;
62.20.dj Poisson’s ratio.
Keywords: Poisson’s ratio, rare gas solids, solid hydrogen, intermolecular potential.
At low temperatures and pressures solid helium is an ul-
timate quantum solid displaying such phenomena as zero-
temperature quantum melting and quantum diffusion. As
atomic masses and interatomic forces increase in the se-
quence Ne, Ar, Kr, and Xe quantum effects in their proper-
ties become progressively less pronounced. Solid hydrogen
is the only molecular quantum crystal where both transla-
tional and rotational motions of the molecules are quan-
tum. Translational quantum effects decrease with increas-
ing pressure.
Quantum and classical solids respond to the applied
pressure differently. When pressure is applied to a classical
solid the atoms are “pushed into” the hard cores of the po-
tential; as a result of this core, the compressibility is usual-
ly quite small. Typically, the pressure of 1 GPa results in a
few percent change in molar volume. At the same time,
quantum solids hydrogen and helium are highly compress-
ible. For hydrogen the pressure of 1 GPa results in a 100%
change in volume. The physical reason for this is that the
lattice is highly blown up due to the zero-point kinetic en-
ergy. The initial compression works against the weaker
“kinetic pressure” rather than the harder “core pressure”.
One of fundamental thermodynamic characteristics de-
scribing behavior of a material under mechanical load is
Poisson’s ratio [1,2]. For isotropic elastic materials the
Poisson's ratio is uniquely determined by the ratio of the
bulk modulus B to the shear modulus G, which relate to
the change in size and shape respectively [3]:
1 3 / 2= .
2 3 / 1
B G
B G
−
σ
+
(1)
As can be seen from this equation, PR can take values be-
tween –1 ( / 0)B G → and 1/2 ( / ).B G →∞ The lower limit
corresponds to the case where the material does not change
© Yu.A. Freiman, Alexei Grechnev, S.M. Tretyak, Alexander F. Goncharov, and Eugene Gregoryanz, 2015
Yu.A. Freiman, Alexei Grechnev, S.M. Tretyak, Alexander F. Goncharov, and Eugene Gregoryanz
its shape and upper limit corresponds to the case when the
volume remains unchanged. Materials with small PR
(small / ),B G such as cork, are more easily compressed
than sheared, whereas those with PR approaching 1/2
(large / )B G are rubber-like: they strongly resist compres-
sion in favor of shear.
For most isotropic materials PR lies in the range
0.2 < 0.5≤ σ [4]. Materials with 0 < < 0.2σ are rare —
beryllium ( = 0.03),σ diamond ( = 0.1σ ) — and are very
hard [1]. Typically, PR increases with pressure near
linearly with the rate 3/ 10P −∂σ ∂ (GPa)–1 indicating a
continuous loss of shear strength [5–7].
An unusual pressure dependence of PR decreasing with
rising pressure in solid hydrogen in the pressure range up
to 24 GPa [8] and solid helium up to 32 GPa [9] was found
by Zha et al. With the aim to investigate the distinctions in
the response of quantum and classical solids to the applied
pressure we calculated pressure dependencies of PR in the
quantum (He, H2, Ne) and classical (Ar, Kr, Xe) cryo-
crystals under pressure. The calculations were performed
using complementary semi-empirical (SE) and density
functional theory (DFT) with generalized gradient appro-
ximation (GGA) approaches. The DFT calculations were
performed using the FP–LMTO code RSPt, while the SE
calculations were done using our own code. The calcu-
lation details have been published previously [10]. It is
important to notice that the two approaches treat solid
hydrogen in fundamentally different ways. SE approach
deals with interaction between H2 molecules, which are
treated as nearly spherically symmetrical quantum rotators,
while the DFT can only treat fully oriented (classical) H2
molecules, ignoring the zero-point rotations. The 12Pca
oriented structure has been used for our calculations.
One of the signatures of a quantum crystal is that it melts
at temperature mT much lower the Debye temperature [11]:
/ 1.D mTΘ >> (2)
Figure 1 shows pressure dependencies of DΘ and mT for
solid hydrogen, helium, neon, and argon. At zero pressure
and temperature the ratio /D mTΘ is infinitely large for
helium and 8.5 for parahydrogen. The ratio rather slowly
decreases with rising pressure. For example, at 1 GPa it is
still as high as 3.75 for helium and 3.0 for parahydrogen.
For solid Ne at zero pressure /D mTΘ = 2.7 thus making
solid Ne a candidate for the manifestation of quantum
effects. Other RGS, with /D mTΘ = 1, 0.55, and 0.35 for
Ar, Kr and Xe respectively, can be regarded as essentially
classical solids.
Rewriting Eq. (1) in terms of the ratio of the bulk
(hydrodynamic) Bv to the transverse (shear) sound
velocity Sv we have [2]:
2
2
( / ) 21= .
2 ( / ) 1
B S
B S
−
σ
−
v v
v v
(3)
The hydrodynamic or bulk sound velocity Bv can be
found from Equation of State:
1/22
1/2= [ / ] = ,B
V PP
V
∂
∂ ∂ρ −
µ ∂
v (4)
where P is pressure, µ is molar mass, and V is molar
volume. In the calculations of Bv for H 2 we used our SE
and DFT–GGA EOS from Ref. 17, for He, Ar and Xe from
Refs. 16, 18, and for Kr from Ref. 19. We have also
included zero-point vibrations in the Debye approximation
in our calculations of ( )P V and ( )B Vv [10].
Generally, to find sound velocities Pv and Sv one has
to find a complete set of elastic moduli .ijC In the case of
hcp lattice there is a simplified scheme based on lattice
dynamics [20,21], which makes it possible to circumvent
the problem of calculations of elastic moduli. In particular,
in this approach it is possible to relate frequency ν of the
Fig. 1. Debye temperature and melting temperature vs pressure.
He and H2 (insert) (a); Ne and Ar (insert) (b). Experimental
melting curves: He and Ne [12], Ar [13]. H2 melting curve
corresponds to Kechin Eq. [14]. Debye temperatures were
calculated using the many-body potentials: for He and H2
[10,15], for Ne and Ar [16].
572 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6
Poisson’s ratio in cryocrystals under pressure
Raman-active 2gE phonon mode of hcp lattice and the
shear elastic constant 44:C
2
44 2
1= ( ),
4 3 g
c mC E
a a
ν (5)
where a, c are the lattice parameters and m is the mole-
cular mass. The pressure dependencies of 2( )gEν and
44C were found for H2 [17] and hcp RGS (hcp He, Ar, Kr,
and Xe) [18,22] using both ab initio DFT and SE lattice
dynamics approaches. The shear velocity Sv was obtained
using the relation
44= / ,S C ρv (6)
where ρ is the density, disregarding the elastic anisotropy
of the crystal. A special case is solid Ne which preserves
the fcc structure up to at least 208 GPa [23] which makes
the outlined procedure impossible. For this reason for solid
Ne we used results of lattice dynamics calculations by
Gupta and Goyal [19].
The sound velocities for H2 and He are given in Ref. 15;
the data for Ne by Gupta and Goyal were published in Ref.
19; the data for Ar, Kr, and Xe will be published elsewhere.
Pressure dependencies of Poisson’s ratios for helium,
hydrogen, neon, argon, krypton, and xenon calculated from
sound velocities using Eq. (3) are shown in Figs. 2–5.
Figure 2 shows the pressure dependence of Poisson’s
ratio in solid He obtained in the framework of SE and
DFT–GGA approaches in comparison with experimental
results from Refs. 9, 24. Results which account for zero-
point vibrations (ZPV) and those obtained disregarding
ZPV are presented. Both SE and DFT–GGA calculations
agree with the somewhat surprising experimental result of
Poisson’s ratio decreasing with pressure. There is a
reasonable fair agreement between the SE theoretical curve
(comprising ZPV) and experimental data. Usually SE results
are preferable at smaller pressures while at higher pressures
the DFT approach works better. Comparing the SE and DFT
theoretical curves it is hard to say in which way the low-
pressure SE results could continuously go over to the high-
pressure DFT ones. It should be noted that the experimental
points may show that around 30 GPa there is a minimum
point at the pressure dependence of PR.
As was said above, typically [5–7] Poisson’s ratio
increases with pressure and tends to 1/2 (the limit of zero
compressibility) when pressure goes to infinity. It would
appear reasonable to consider anomalous behavior of PR in
such quantum solids as He and H2 as a manifestation of
quantum effects. Reasons for such understanding is the
following. It is known that the He and H2 lattices are
swelled due to large zero-point vibrations (ZPV). If ZPV
were not present, “classical” solid He and H2 would have
much smaller zero molar volumes cl
0( (He)V ≈ 11.2 cm3/mol;
cl
0 2(H )V ≈ 7.4 cm3/mol), i.e., the swelling effect is huge [18].
Until the volume reaches about cl
0 ,V the main effect of the
external pressure is the suppression of the zero-point
vibrations and not the compression of the electron shells.
To check whether this explanation is correct we
calculated PR of He disregarding ZPV, that is, for
“classical” He both in the SE and DFT approaches (dot-
dash and dotted curves, respectively, Fig. 2). As can be seen,
the pressure dependence of PR with and without ZPV is
qualitatively the same. Thus, the anomalous (descending
with rising pressure) behavior of PR is not a quantum effect.
As can be seen from Fig. 2, the contribution of ZPV into PR
is positive. This fact is easily understood if we take into
account that the introduction of ZPV is a step to liquation
but PR of liquid is an upper bound for PR of any substance.
Naturally, the relative value of this contribution increases
with decreasing pressure and as pressure goes to zero it
increases up to 15%. The effect of ZPV is much higher in
the case of 3He. Nieto et al. [24] showed that the mixture
3He–4He has higher PR than pure 4He. For pure 3He they
gave value of PR 0.473 rather close to the liquid limit.
The theoretical and experimental pressure dependencies
of PR for solid H2 are shown in Fig. 3. As can be seen,
the SE and DFT–GGA approaches give the opposite signs
of the pressure effect on PR: PR decreases with rising
pressure for SE and increases for DFT–GGA. Since in the
experimentally studied pressure range (up to 24 GPa) the SE
result agrees qualitatively with experiment [8], we conclude
that at low pressures PR decreases with rising pressure for
solid H2. It is known that while the SE approach works well
for molecular solids at low pressures, for higher pressures
the DFT–GGA approach is preferable. Thus the PR(P) curve
can be subdivided into three regions: At the low-pressure
region SE is expected to work well, while at high pressures
we can use the DFT–GGA approach. In the intermediate
pressure range both approaches fail. The dot-dot dash curve
shows schematically a possible continuous transition from
the low-pressure asymptote to high-pressure one. Resulting
pressure dependence of PR for H2 is nonmonotonic dis-
Fig. 2. (Color online). Poisson’s ratio of solid He as a function
of pressure. Theory: this work. Experiment: Zha et al. [9];
Nieto et al. [24].
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6 573
Yu.A. Freiman, Alexei Grechnev, S.M. Tretyak, Alexander F. Goncharov, and Eugene Gregoryanz
playing rather deep minimum. It should be noted that the
transient region from the descending to the ascending
curves falls on phase II of the hydrogen phase diagram. As
mentioned above, the SE and DFT approaches treat the
orientational degrees of freedom in H2 in completely
different ways: the former regards H2 molecules as nearly
spherically symmetric quantum rotators (as in phase I),
while the latter considers classically oriented H2 molecules
(as in phase III), completely ignoring any quantum
rotations or librations. It seems likely that this is the reason
why SE and DFT give such drastically different PR(P)
curves for H2, while results for helium are qualitatively
similar. It would mean that the PR minimum in hydrogen
is related to the orientational transition at around 110 GPa,
however more detailed study of this question is beyond the
scope of the present work.
A similar curve with a deep minimum was obtained for
PR in solid neon (Fig. 4). The pressure dependence of PR
was obtained from the SE theoretical results on sound
velocities obtained by Gupta and Goyal [19]. Unfor-
tunately, experimental data on sound velocities in solid Ne
exist for very narrow pressure range 5–7 GPa [25]. In this
region PR ≈ 0.37.
Figure 5 shows the pressure dependencies of PR
obtained in the SE approach for Ar, Kr, and Xe. In contrast
with He, H2, and Ne, we obtained that PR for the heavy
RGS increases with rising pressure.
In conclusion, we present results of lattice dynamics
calculations of Poisson’s ratio for solid hydrogen and rare
gas solids (He, Ne, Ar, Kr and Xe) under pressure. Using
two complementary approaches: lattice dynamics based on
the semi-empirical many-body potentials and ab initio
DFT–GGA we found three different types of the behavior
of PR with pressure. While for solid He PR monotonically
decreases with rising pressure, for Ar, Kr, and Xe it mono-
tonically increases with pressure. For solid H2 and Ne PR
are nonmonotonic with pressure displaying rather deep
minimums. To investigate the role of quantum effects we
performed the calculations of PR disregarding zero-point
vibrations and found qualitatively similar results, that is, we
proved that the effects have a nonquantum origin. We may
rather say that the anomalies, discovered for H2, He and Ne,
and quantum effects in these cryocrystals have common
origins: weak intermolecular interactions and small masses
of constituent atoms and molecules.
We thank J. Peter Toennies for valuable discussions.
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574 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6
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