Isotopic and spin-nuclear effects in solid hydrogens (Review Article)
The multiple isotopic family of hydrogens (H₂, HD, D₂, HT, DT, T₂) due to large differences in the de Boer quantum parameter and inertia moments displays a diversity of pronounced quantum isotopic solid-state effects. The homonuclear members of this family (H₂, D₂, T₂) due to the permutation symmetr...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1753602021-02-01T01:29:18Z Isotopic and spin-nuclear effects in solid hydrogens (Review Article) Freiman, Y.A. Crespo, Y. The multiple isotopic family of hydrogens (H₂, HD, D₂, HT, DT, T₂) due to large differences in the de Boer quantum parameter and inertia moments displays a diversity of pronounced quantum isotopic solid-state effects. The homonuclear members of this family (H₂, D₂, T₂) due to the permutation symmetry are subjects of the constraints of quantum mechanics which link the possible rotational states of these molecules to their total nuclear spin giving rise to the existence of two spin-nuclear modifications, ortho- and parahydrogens, possessing substantially different properties. Consequently, hydrogen solids present an unique opportunity for studying both isotope and spin-nuclear effects. The rotational spectra of heteronuclear hydrogens (HD, HT, DT) are free from limitations imposed by the permutation symmetry. As a result, the ground state of these species in solid state is virtually degenerate. The most dramatic consequence of this fact is an effect similar to the Pomeranchuk effect in ³He which in the case of the solid heteronuclear hydrogens manifests itself as the reentrant broken symmetry phase transitions. In this review article we discuss thermodynamic and kinetic effects pertaining to different isotopic and spin-nuclear species, as well as problems that still remain to be solved. 2017 Article Isotopic and spin-nuclear effects in solid hydrogens (Review Article) / Y.A. Freiman, Y. Crespo // Физика низких температур. — 2017. — Т. 43, № 12. — С. 1687-1706. — Бібліогр.: 97 назв. — англ. 0132-6414 PACS: 67.80.ff http://dspace.nbuv.gov.ua/handle/123456789/175360 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The multiple isotopic family of hydrogens (H₂, HD, D₂, HT, DT, T₂) due to large differences in the de Boer quantum parameter and inertia moments displays a diversity of pronounced quantum isotopic solid-state effects. The homonuclear members of this family (H₂, D₂, T₂) due to the permutation symmetry are subjects of the constraints of quantum mechanics which link the possible rotational states of these molecules to their total nuclear spin giving rise to the existence of two spin-nuclear modifications, ortho- and parahydrogens, possessing substantially different properties. Consequently, hydrogen solids present an unique opportunity for studying both isotope and spin-nuclear effects. The rotational spectra of heteronuclear hydrogens (HD, HT, DT) are free from limitations imposed by the permutation symmetry. As a result, the ground state of these species in solid state is virtually degenerate. The most dramatic consequence of this fact is an effect similar to the Pomeranchuk effect in ³He which in the case of the solid heteronuclear hydrogens manifests itself as the reentrant broken symmetry phase transitions. In this review article we discuss thermodynamic and kinetic effects pertaining to different isotopic and spin-nuclear species, as well as problems that still remain to be solved. |
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Freiman, Y.A. Crespo, Y. |
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Freiman, Y.A. Crespo, Y. Isotopic and spin-nuclear effects in solid hydrogens (Review Article) Физика низких температур |
| author_facet |
Freiman, Y.A. Crespo, Y. |
| author_sort |
Freiman, Y.A. |
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Isotopic and spin-nuclear effects in solid hydrogens (Review Article) |
| title_short |
Isotopic and spin-nuclear effects in solid hydrogens (Review Article) |
| title_full |
Isotopic and spin-nuclear effects in solid hydrogens (Review Article) |
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Isotopic and spin-nuclear effects in solid hydrogens (Review Article) |
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Isotopic and spin-nuclear effects in solid hydrogens (Review Article) |
| title_sort |
isotopic and spin-nuclear effects in solid hydrogens (review article) |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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http://dspace.nbuv.gov.ua/handle/123456789/175360 |
| citation_txt |
Isotopic and spin-nuclear effects in solid hydrogens (Review Article) / Y.A. Freiman, Y. Crespo // Физика низких температур. — 2017. — Т. 43, № 12. — С. 1687-1706. — Бібліогр.: 97 назв. — англ. |
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Физика низких температур |
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AT freimanya isotopicandspinnucleareffectsinsolidhydrogensreviewarticle AT crespoy isotopicandspinnucleareffectsinsolidhydrogensreviewarticle |
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2025-07-15T12:36:51Z |
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2025-07-15T12:36:51Z |
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1837716484692901888 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12, pp. 1687–1706
Isotopic and spin-nuclear effects in solid hydrogens
(Review Article)
Yuri A. Freiman
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Nauky Ave., Kharkiv 61103, Ukraine
yuri.afreiman@gmail.com
Yanier Crespo
International Center for Theoretical Physics (ICTP), IT-34014 Trieste, Italy
E-mail: yanier@hotmail.com
Received April 2, 2017, published online October 25, 2017
The multiple isotopic family of hydrogens (H2, HD, D2, HT, DT, T2) due to large differences in the de Boer
quantum parameter and inertia moments displays a diversity of pronounced quantum isotopic solid-state effects.
The homonuclear members of this family (H2, D2, T2) due to the permutation symmetry are subjects of the con-
straints of quantum mechanics which link the possible rotational states of these molecules to their total nuclear
spin giving rise to the existence of two spin-nuclear modifications, ortho- and parahydrogens, possessing sub-
stantially different properties. Consequently, hydrogen solids present an unique opportunity for studying both
isotope and spin-nuclear effects. The rotational spectra of heteronuclear hydrogens (HD, HT, DT) are free from
limitations imposed by the permutation symmetry. As a result, the ground state of these species in solid state is
virtually degenerate. The most dramatic consequence of this fact is an effect similar to the Pomeranchuk effect in
3He which in the case of the solid heteronuclear hydrogens manifests itself as the reentrant broken symmetry
phase transitions. In this review article we discuss thermodynamic and kinetic effects pertaining to different iso-
topic and spin-nuclear species, as well as problems that still remain to be solved.
PACS: 67.80.ff Molecular hydrogen and isotopes.
Keywords: isotopic family of hydrogens, homo- and heteronuclear species, ortho- and parahydrogens, isotope
and spin-nuclear effects, ortho- and para conversion, Pomeranchuk effect, reentrant broken symmetry phase
transition.
Contents
1. Introduction ............................................................................................................................................1688
2. Ortho-para conversion ............................................................................................................................1689
2.1. Experiment .....................................................................................................................................1689
2.2. Theoretical considerations ..............................................................................................................1690
2.3. Conversion in solid D2 ...................................................................................................................1690
2.4. Validity of the concept of ortho-para states at high pressure ..........................................................1691
3. Thermodynamic properties of J = 0 – J = 1 mixtures .............................................................................1692
3.1. Molar volume .................................................................................................................................1692
3.2. HCP lattice distortion in solid H2 as a function of o-p composition ...............................................1692
3.3. Experimental determination of the crystal field splitting of isolated = 1J impurities in solid
parahydrogen .................................................................................................................................1694
3.4. HCP lattice distortion in solid HD ..................................................................................................1695
4. Broken symmetry phase transition in solid HD and o-p mixtures ...........................................................1695
4.1. Introduction .....................................................................................................................................1695
4.2. Experimental results ........................................................................................................................1697
© Yuri A. Freiman and Yanier Crespo, 2017
Yuri A. Freiman and Yanier Crespo
4.3. Theoretical studies .......................................................................................................................... 1698
4.3.1. Quantum orientational melting in all-J system ........................................................................ 1698
4.3.2. The reentrant behavior from numerical simulations .................................................................. 1700
4.4. Reentrant phase transitions in ortho-para mixtures ......................................................................... 1701
5. Conclusion .............................................................................................................................................. 1704
References ................................................................................................................................................... 1705
1. Introduction
The large isotopic family of hydrogens (H2, HD, D2,
HT, DT, T2) presents an unique possibility for studying the
diversity of quantum isotopic effects [1–3]. The differ-
ences in properties cannot be, as a rule, related solely to the
de Boer quantum parameter, since symmetry-related nu-
clear spin effects turn out to be far more essential.
The requirements of quantum mechanics on a homo-
nuclear hydrogen species (H2, D2, T2) rigidly link the rota-
tional momentum J and the total nuclear spin molI of the
molecule. According to the principle of the indistinguish-
ability of identical particles, the molecular wave function
of any homonuclear diatomic molecule must be symmetric
or antisymmetric under nuclear exchange. The nucleus of
hydrogen atom is fermion with nuclear spin = 1/ 2NI and
the nucleus of deuterium atom is boson with nuclear spin
= 1NI . So the total molecular wave function, which is a
product of vibrational, spin, and rotational wave functions,
mol vib spin rot=ψ ψ ψ ψ , must be antisymmetric for the hyd-
rogen molecule and symmetric for the deuterium molecule.
The vibrational part, vibψ , is always symmetric. The spin
part, spinψ , for the hydrogen molecule is antisymmetric for
the singlet state mol = 0I , and symmetric for the triplet
state mol = 1I . Hence the rotational part rotψ is symmetric
(rotational quantum number = 0, 2, 4, ...J ) for the singlet
state and antisymmetric ( = 1, 3, ...J ) for the triplet state.
For the deuterium molecule the spin part is symmetric for
mol = 2;0I and antisymmetric for mol = 1I . The rotational
part rotψ respectively is symmetric ( = 0J or even) for the
symmetric spin state mol = 2;0I state and antisymmetric
(Jodd) for the mol = 1I spin state. The states with the parity
of the largest possible molI value are called ortho while the
states with the other parity are para. Transitions between
states of different parity (J or molI ), an ultimately quantum
process, called ortho-para conversion, are strictly forbidden
in a single molecule and is a low-probability process in a con-
densed state. This enables treating the ortho- and para-
species as rather stable different substances (orthohydro-
gen, o-H2, and parahydrogen, p-H2, and their isotopic
counterparts p-D2 and o-D2) with their own properties.
The equilibrium ortho-para composition in the mixtures
can be calculated by using the rotational energy of a dia-
tomic rotator rot rot= ( 1)E B J J + , where rotB is the rota-
tional constant (85.25 K for H2 and 42.97 K for D2 [3]),
and assuming a Boltzmann distribution. At high tempera-
ture the equilibrium ratio of ortho- to para species is de-
termined by their statistical weights and is therefore 3:1 for
hydrogen and 2:1 for deuterium. Such mixtures are called
normal (n-H2 and n-D2). The thermodynamic equilibrium
ortho-para concentration for H2 and D2 is shown in Fig. 1.
Thus the constraints of quantum mechanics link the possi-
ble rotational states of the hydrogen molecule to its total
nuclear spins. This symmetry-related link imparts large
differences in the solid-state properties of ortho- and para
modifications and may be considered as one of the most
striking macroscopic manifestations of the microscopic
laws of quantum mechanics. Due to this unique relation
between the rotational and spin quantum numbers of
homonuclear molecules the transitions between J -even
and J -odd rotational states with = 1, 3, ...J∆ ± ± require
the simultaneous change mol = 1I∆ ± of the total nuclear
spin of the molecule. Resulting transformation of the two
species into each other is called ortho-para conversion. In
isolated molecules the probability of such transitions is
negligible and they may be considered as rigorously for-
bidden. The conversion in solid hydrogen and deuterium
will be considered in Sec. 2.
In contrast to H2 and D2, the nuclei in the heteronuclear
HD molecule are distinguishable, and no symmetry re-
quirements exist on the nuclear wave functions of HD. As
a result, the heteronuclear molecules do not have ortho-
para species, and for both possible total nuclear spin
mol = 1/ 2, 3 / 2I all angular momentum states = 0, 1, 2, ...J
and transition between them are allowed.
Fig. 1. Equilibrium ortho-para concentration of noninteracting
hydrogen and deuterium as a function of temperature from Ref. 1.
1688 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12
Isotopic and spin-nuclear effects in solid hydrogens
2. Ortho-para conversion
2.1. Experiment
While ortho-para conversion is strictly forbidden for
isolated molecules, conversion does occur in solid. The
ortho-para conversion in solid H2 was first studied experi-
mentally as far back as in the beginning of the 1930s by
Cremer and Polany [4,5] soon after the discovery of the
two hydrogen modifications. At ambient pressure ortho-para
conversion in the solid is very slow, taking weeks for
a sample to equilibrate [1,2]. It was found that the conversion
reaction is autocatalytic, that is, the rate of the reaction is
determined by the average number of nearest o-H2 neigh-
bors M surrounding each o-H2 molecule: / =dx dt Mx− ,
where x is the o-H2 concentration. As a result, for the
normal equilibrium distribution the rate equation describ-
ing the time dependence of the ortho concentration is
2/ =dx dt kx− , (1)
where k is the macroscopic conversion constant. The ana-
lytic solution of this equation is 1 1
0 =x x kt− −− , which can
be used for comparison with experimental results.
There have been numerous experimental studies of the
ortho-para conversion rate at pressures below 1 GPa by
using various methods [6–12]. Ahlers [7] determined the
conversion rate by measurement of the heat conduction
of a gaseous sample before and after solidification. Pedroni
et al. [8] and Schmidt [9] used nuclear magnetic resonance
(NMR): the ortho molecules with mol 1= = 1I I contribute
to the nuclear magnetization and thus to the NMR signal,
while the para molecules with mol 0= = 0I I do not.
Silvera’s group [6,11,12] used Raman measurements. The
ortho concentration was determined by comparing the rela-
tive intensities 0I and 1I of the respective Raman-active
rotational transitions = 0 2J → and = 1 3J → . At low
temperatures << 6Bk T B , the respective equation takes the
form [13]
1
0 1
5 5= / .
3 3
x I I
−
+
(2)
The experimental data on the conversion rate as a func-
tion of density at small compressions up to 0/ 1.7ρ ρ ≈
from Refs. 6–12 are shown in Fig. 2. Here 0ρ is zero-
pressure density and ρ is the density at pressure P . The
experimental results are seen to be mutually consistent
over the whole pressure range.
The most extensive measurements at zero pressure were
made by Schmidt by NMR at three temperatures 12, 4.2,
and 1.57 K in long-term experiment up to 900 h. He gets a
conversion rate of = (1.90 0.03) % /k h± which is in agree-
ment with most of the previous experiments. Measure-
ments at small pressures showed that the conversion rate
monotonously increases with pressure. Driessen et al. [6]
found a maximum conversion rate at 0.5 GPa followed by
a drop up to 0.7 GPa (Fig. 2).
Phase diagram studies at high pressures have inspired a
renewed interest in the ortho-para conversion in solid H2.
Hemley et al. [13] using Raman method raised the pressure
limit in the conversion measurements up to 58 GPa. Several
characteristic low-temperature (10 K) spectra at 21.6 GPa
are given in Fig. 3. They show the decrease in the integrat-
ed intensity of the 0 (1)S peak with respect to the 0 (0)S
peak with time (here the roton bands 0 (0)S and 0 (1)S in-
volve transitions = 0 = 2J J→ , = 1 = 3J J→ , respec-
tively). A rather intricate non-monotonous pressure de-
pendence of the conversion rate emerged from these
studies (Fig. 4).
Following maximum at about 0.5 GPa the conversion
slows to a minimum at about 3 GPa nearly a factor of two
below the ambient pressure rate constant. At higher pres-
sures the conversion rate increases rapidly to about 260%/h
at 58 GPa. The results are compared with NMR measure-
ments to 12.8 GPa [15] ( 0/ 3.7ρ ρ ≈ ). On the whole, there
is an excellent agreement between results obtained by the
Fig. 2. The conversion rate in solid H2 as a function of the re-
duced density at small compressions (after Ref. 6).
Fig. 3. Changes in the Raman spectra of solid H2 at 21.6 GPa and
10 K with time elapsed from cool-down completion (after Eggert
et al. [14]).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12 1689
Yuri A. Freiman and Yanier Crespo
two methods with possible discrepancies arising from tem-
perature differences in the two studies.
2.2. Theoretical considerations
A theory of the ortho-para conversion in solid hydrogen
was first developed by Motizuki and Nagamiya [17]. They
showed that the conversion rate is controlled by the three
mechanisms: (1) the magnetic dipole-dipole interaction of
the nuclear spins of ortho molcules; (2) magnetic-dipole in-
teraction between the nuclear spins and the rotational mag-
netic moments; (3) magnetic interaction with paramagnetic
impurities.
The conversion transition ratio W can be calculated
from the golden rule:
2
int
2= | | | | ( ),i i f
if
W P f i E Eπ
〈 〉 δ −∑
(3)
where iP is the probability that the system is initially in
one of the states i , and f is the final state; iE and fE are
the initial and final energies of the system. int is the in-
teraction Hamiltoman and has three additive parts which
can change the nuclear spin states: ss , the nuclear spin-
spin interaction; rs , the rotation-spin interaction; and qq
, the electric quadrupole interaction, (explicit equations can
be found in Refs. 3, 18). ss is the dominant term with rs
yielding a 2–3% correction to ss . The delta-function in
Eq. (3) requires conservation of energy. The rotational ener-
gy change in a conversion is rot ( 1)B J J + . According to the
theoretical description of ortho-para conversion in solid
hydrogen as developed by Motizuki and Nagamiya [17], an
ortho-para transition requires a large magnetic field gradi-
ent acting on a molecule to flip a proton spin. This field
gradient is supplied by the rotational magnetic moment of
neighboring ortho molecules. Thus two ortho-neighbors
are needed for one of them to convert. As a result, if the
ortho distribution is random, the rate equation Eq. (1) de-
scribes the time dependence of the ortho concentration.
The rotational energy difference between a = 1J ortho
molecule and a = 0J para molecule, 0 1 rot= 2 =E B←∆
= 171 K, released by ortho-para conversion must be ab-
sorbed by the lattice. The Debye temperature DΘ gives an
estimate for the limiting phonon energy. Since at ambient
pressure DΘ = 118.5 K [3] the energy conservation re-
quires the simultaneous creation at least two phonons. The
two-phonon theory of Motizuki and Nagamiya predicts a
drop in the conversion rate with increasing pressure over
about 1 kbar where the two-phonon conversion rate is neg-
ligible. Thus, neither the two-phonon theory nor the one-
phonon theory in the harmonic approximation can explain
experimental results [7,8,10] which showed the conversion
rate to be an increasing function of density. Therefore Ber-
linsky [19] suggested that the effect of the large anhar-
monisity in solid hydrogen is to broaden the high-energy
features of the phonon spectrum into a high-energy tail
which extends far above B Dk Θ . On the basis of this con-
jecture Berlinsky [19] developed a detailed theory of one-
phonon processes and found that the one-phonon con-
version rate should initially increase with pressure before
falling above about 0.5 GPa because of density of states
factors and the rapid increase in phonon energy.
On the base of the conversion rate data [14,15] a new
conversion mechanism was proposed by Strzhemechny and
Hemley in Ref. 16 and by Strzhemechny et al. in Ref. 20
that differs from that employed to explain low-pressure
data. The steep increase in conversion rates at high pres-
sures was explained by a conversion channel that involves
an intermediate state in which new excitations are created
due to the electric quadrupole-quadrupole (EQQ) coupling
between rotational momenta. The enhancement was explain-
ed by a gap closing that arises when the EQQ interaction
becomes sufficient to substantially diminish the conversion
energy released because of the lowering of the ground-
state level of the converted ortho molecule. It was shown
that this concentration-sensitive channel comes into play at
compressions 0= /ξ ρ ρ between 3 and 4 and ceases to ope-
rate at higher values ( 6–7ξ ≈ ), depending on the ortho
concentration. Thus, this theory predicts a significant re-
duction in the conversion rate at about 80 GPa; The exten-
sion of the measurements to pressures over 50 GPa would
provide a critical test of the conversion mechanisms pro-
posed in Refs. 16, 20.
2.3. Conversion in solid D2
Theory of conversion in solid D2 was developed first by
Motizuki [21]. The conversion in D2 differs from that in
H2 in a number of essential aspects. First of all, the deuter-
ium nucleus, deuteron, possesses a nonzero quadrupole
Fig. 4. The conversion rate vs. reduced density (the crystal densi-
ty ratio reduced to the = 0P value) at high compression. Solid
circles are from Refs. 13, 14, empty squares are from Ref. 15.
The solid curve is the theoretical prediction for hcp n-H2 from
Ref. 16. The open circles are earlier data (see Fig. 2). Inset: Low
density region (see Ref. 14). After Eggert et al. [14].
1690 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12
Isotopic and spin-nuclear effects in solid hydrogens
moment. The interaction of this quadrupole moment with
electric field gradients due to neighboring molecules pro-
vides another mechanism for conversion in addition to
dipolar interactions. Also, the deuteron has = 1DI , so the
final state of the = 0J D2 molecule may be any of the six
nuclear states corresponding to = 0I or 2. If the final nu-
clear state is = 2I , then this molecule will cause magnetic
field gradients at neighboring molecules, and thus it may
still contribute to the conversion process. This is in con-
trast to the case of solid H2 where the para molecules have
= 0J , mol = 0I and thus do not contribute to further con-
version.
One more significant difference between the conversion
in D2 and H2 lies in the fact that the rotational constant of D2,
D2
rot = 42.97 KB [3], about a half of that for H2 (85.25 K)
with the respective change of the rotational energy in going
from the = 1J to the = 0J state, while the Debye tem-
perature ( D2
DΘ ) = 114.1 K [3] is nearly the same ( H2
DΘ ) =
= 118.5 K [3]. Thus the one-phonon process is allowed for
conversion in solid D2.
The presence of the conversion mechanism that in-
volves = 0J molecules with = 2I changes the rate equa-
tion describing the time dependence of the concentration of
= 1J molecules:
2/ = (1 ).dx dt kx k x x′− − − (4)
The general solution of this equation with the initial
condition 0( = 0) =x t x has the form
1 1 1
0 0= ( 1 / )[1 exp ( )].x x x k k k t− − − ′ ′− − + − (5)
A characteristic distinction of the conversion in deuteri-
um is that at low = 1J fraction the conversion rate is linear
in the concentration x of the = 1J molecules. Experi-
mental results obtained by different methods (see Table 1)
gave rather close results:
Table 1. The conversion constants in solid deuterium at = 0P
Author, Ref., Method 4 1(10 h )k − − 4 1(10 h )k − −′
Hardy and Berlinsky [22] (NMR) 5.6 =k k′
Milenko and Sibileva [23] 5.6 0.5± 5.3 0.3±
Berkhout, Minneboo, and Silvera
[24] (Raman)
6.3 0.1± =k k′
Calkins, Banke, Li, and Meyer [25]
(NMR)
5.5 0.5± =k k′
Bagatskii, Krivchikov, Manzhelii
et al. [26]
6.4 0.2±
Strzhemechny and Tokar [27]
(Theory)
6.1 14.2
Comparison with the conversion rate in H2 ( 2 11.9 10 h− −⋅ )
shows that in D2 it proceeds about 30 times slower. The
efficiency of various conversion channels in solid deuterium
at high pressures have been considered by Strzhemechny
and Hemley [28]. They found that the standard phonon-
assisted channels are inefficient at high pressures in D2 as
they are in H2, and the idea of the intermediate state with
subsequent participation of the EQQ interaction remains
productive for D2.
2.4. Validity of the concept of ortho-para states
at high pressure
At small pressures when the wave function of the crys-
tal may be represented as a product of molecular wave
functions there are no any doubts that the notion of ortho-
para states is a well justified quantum-mechanical concept.
To what extent this concept remains valid at high pressures
when the rotational quantum number J is no longer a good
quantum number? How does an essential increase in the
conversion rate with pressure affect the ortho-para con-
cept? These important questions were analyzed in detail by
Silvera [29] and Silvera and Pravica [30]. Their main con-
clusion is that the only important quantum number for the
o-p states is parity under exchange and that the nuclear
spin states preserve up to pressures when the dissociative
Wigner-Huntington transition occurs.
In solid molecular hydrogen or deuterium the many-
body wave function must be symmetrized with respect to
exchange of nucleons. Nucleon exchange can take place
within a molecule or between molecules. If exchange of
nucleons between neighboring molecules took place, then
the structure of the wave functions under nucleon permuta-
tion would be very complicated and the ortho-para concept
associated with isolated molecules would not be applicable.
The total molecular wave function can be written as
a superposition of pure ortho and pure para states:
ortho para=Ψ Ψ +αΨ . (6)
For the validity of the concept of ortho-para species, the
mixing parameter α should be very small.
The perturbation which causes mixing is the same that
causes conversion. Taking into consideration the mecha-
nism of Motizuki and Nagamiya Silvera obtained the fol-
lowing density dependence of the resultant mixing parame-
ter [29]:
10 4/3
1 0| | = 8 10 ( / ) .c−α ⋅ ρ ρ (7)
where 1c is the ortho concentration. From Eq. (7) we see
that at zero pressure the mixing parameter is exceptionally
small. The highest compression 0= /ξ ρ ρ reached in the
equation of state experiments at 180 GPa is 10.4 [31]. The
mixing parameter increases by a factor 22.7 but remains
very small of order 810− , and the concept of ortho-para
species remains valid to very high pressure.
There is an interesting question concerning a possible
implication of ortho-para species for metallization [30]. At
a finite temperature we have a mixture of ortho- and
parastates which violates the translational invariance. This
can possibly give rise to Anderson localization and sup-
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12 1691
Yuri A. Freiman and Yanier Crespo
press an insulator–metal transition that would occur in a
pure lattice [32]. It might be the case that even pure ortho-
or para lattices suffer from Anderson localization. Since
the electron motions are very much faster than the nuclear
motions, an electron moving through a pure lattice sees a
snapshot of molecules with random orientations, although
on the average (in the quantum mechanical sense), the
molecules have well defined distributions. At this time it is
not clear from theoretical considerations whether the elec-
tron feels a translationally invariant or a disordered lattice.
It is possible that the failure to observe metallization in the
phase III of hydrogen, above 150 GPa, is due to this type
of disorder.
3. Thermodynamic properties of J = 0 – J = 1 mixtures
3.1. Molar volume
In 1972 Silvera et al. [33] studying Raman-active 2gE
phonon mode in hcp hydrogen and deuterium found unex-
pectedly large downward shift ( 5% ) in the measured
frequency when going from pure = 1J to pure = 0J solid.
This effect was explained by Silvera et al. [12] by the dif-
ference in molar volume of = 1J and = 0J species, the
former having a smaller molar volume. The free energy
has translational (lattice) contributions and orientational
contributions related with isotropic and anisotropic interac-
tions, respectively. When compressions are not too high,
we may restrict ourselves to the contribution from EQQ
(electric quadrupole-quadrupole) interactions. The quadru-
pole pressure difference between = 1J and = 0J H2 and
D2 in the orientationally ordered Pa 3 state is shown
in Fig. 5 [11]. Hydrogen is a quantum solid and is therefore
highly compressible. Due to the large compressibility, the
weak anisotropic forces related with = 1J species have a
non-negligible effect on the molar volume. The zero-
pressure molar volume V of hydrogen and deuterium as a
function of = 1J concentration is shown in Fig. 6. In the
orientationally disordered state of hexagonal close packed
(hcp) solids at = 4.2 KT we have the following dependen-
cies [1]: 2 3
2( ) = 23.16 0.091 0.217 cm / mol (H )V x x x− − ,
and 2 3
2( ) = 19.95 0.16 0.04 cm / mol (D )V x x x− − .
3.2. HCP lattice distortion in solid H2
as a function of o-p composition
At zero pressure the molecules in J -even solid
hydrogens (p-H2, o-D2) are virtually spherical. Rigid
spheres in the undistorted lattice crystallize into fcc (face
center cubic) or hcp lattice. As compared with fcc, the hcp
lattice has an additional degree of freedom associated with
the /c a ratio. A lattice of closed packed hard spheres has
/ = 8 / 3 1.633c a ≈ (an ideal hcp structure). The quantity
= / 8 / 3c aδ − , the lattice distortion parameter, describes
the deviation of the axial ratio from the ideal value. In the
case of < 0δ , this distortion involves extension within
close-packed planes, and contraction along the c-axis di-
rection, and vice versa, for >0δ the lattice is expanded along
the c axis and contracted within close-packed planes.
Calculations with pair isotropic potentials have shown
that the ideal hcp lattice does not correspond to minimal
lattice energy [35–37]. Calculations with a many-body
potential and DFT calculations performed for solid He
showed [38] that the pressure dependencies of the lattice
distortion parameter ( )Pδ for a many-body (two- plus
three-body) and for pair intermolecular potentials are
qualitatively different. The three-body forces flatten the
lattice ( < 0δ ) while the pair forces at large compressions
tend to elongate it ( > 0δ ). Thus, it was shown that the lat-
tice distortion parameter is a thermodynamic characteristic
which is very sensitive to the many-body component of the
intermolecular potential and can therefore be used as a
probe of the many-body forces [39].
The deviation of the axial ratio from the ideal value in
the hcp solids can be attributed ultimately to a lowering of
the band-structure energy through lattice distortion. In the
molecular hcp solids we can explicitly ascribe this effect to
the reduction in the ground-state energy due to lattice dis-
tortion. In the case of solid hydrogens the effect of lattice
distortion both on the isotropic and rotational components
of the zero-point energy is essential.
The orientational state of molecules in the distorted lat-
tice is characterized by the orientational order parameter
20= 4 / 5 Yη π 〈 〉, where (...)〈 〉 means thermodynamic av-
eraging with the rotational Hamiltonian. The lattice-
rotation coupling is described by the the Hamiltonian
int 2 20= 4 / 5c Y−ε π , where 2cε is the crystal-field pa-
rameter [2] which is linear with respect to δ :
2
1= ; = 6 ,
2c
dBB B B R
dR
ε δ − +
(8) Fig. 5. The quadrupole pressure in p-H2 and o-D2 in 3Pa ordered
state at = 0T (after Jochemsen et al. [11]).
1692 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12
Isotopic and spin-nuclear effects in solid hydrogens
where ( )B R is the radial function of the single-molecular
term in the anisotropic intermolecular potential [2]. Thus,
the state of the lattice can be described by two coupled
order parameters, ( , )P Tη and ( , )P Tδ , which can be found
by the minimization of free energy with respect to these
parameters. Figure 6 shows the volume dependence of the
lattice distortion parameter δ and orientational order pa-
rameter η in parahydrogen and orthodeuterium calculated
using the many-body intermolecular potential from Ref. 40.
The negative sign of δ means that the lattice is flattened
compared with the ideal one.
For all hcp elemental solids except helium, hydrogen,
and high-pressure Ar, Kr, and Xe the behavior of δ with
pressure and temperature is well established from both
theory and experiment. Typical values are of the order
of 10 2− . For solid helium δ is an order of magnitude small-
er [38,41]. The first measurements of /c a ratio in solid
hydrogens were done by Keesom et al. [42] who found that
at zero pressure hcp lattice of p-H2 is close to ideal. X-ray
zero-pressure study by Krupskii et al. [43] confirmed this
result ( / = 1.633 0.001c a ± ) and extended it for o-D2. In
fact, the only structural study of p-H2 and o-D2 at elevated
pressures up to 2.5 GPa and low temperatures were made by
Ishmaev et al. using the neutron diffraction method [44,45].
It was found that the ratio /c a is practically constant and is
slightly less than the ideal hcp value (1.631 0.002± ).
In the absence of direct experimental data some qualita-
tive conclusions on the lattice distortion parameter of p-H2
were obtained by Goncharov et al. [46] from low-fre-
quency Raman spectra at low temperature. The authors
have measured low-frequency Raman spectra at low tem-
perature for the pressure range up to the I–II phase transi-
tion and used these spectra to estimate the crystal-field
parameter 2cε . Assuming that only the second-order crystal
field is responsible for the splitting of the roton triplet band
0 (0)S and get that 2| |cε 1 cm–1 and thus 3 4| | 10 10− −δ −
in accord with theoretical data shown in Fig. 7.
There were numerous structural studies of n-H2 and n-D2
[3,31,48–51]. Synchrotron single-crystal x-ray diffraction
measurements of n-H2 [31,48] and n-D2 [48] up to mega-
bar pressures at room temperature revealed an approxi-
mately linear decrease of the /c a ratio with increasing
pressure (Fig. 8). No isotope effect in the pressure depend-
ence of the /c a ratio was found.
Fig. 6. The molar volume at zero pressure as a function of con-
centration of = 1J modification (after Driessen et al. [34]).
Fig. 7. (Color online) Lattice distortion parameter δ and orien-
tational order parameter η in parahydrogen and orthodeuterium
as a function of molar volume [47].
Fig. 8. (Color online) Lattice distortion parameter in normal or-
tho-para mixture of solid hydrogens. Theory: solid line [47]; ex-
periment: red squares — data from Ref. 48, blue triangles — data
from Ref. 31.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12 1693
Yuri A. Freiman and Yanier Crespo
For a single = 1J molecule in a lattice of = 0J mole-
cules there is an additional contribution to the ground-state
energy arising from the polarization of the surrounding
= 0J molecules by the the electric quadrupole field of the
= 1J molecule. The polarization energy due to the interac-
tion of the quadrupole moment of the = 1J molecule with
the induced dipole moments of the surrounding nearest
neighboring = 0J molecules equal to
2 8
1 = 18 ,Q R−ε − α (9)
where α is the polarizability of the = 0J molecules and Q
is the quadrupole moment of the = 1J molecule, and R is
the intermolecular distance.
If the crystal shows a homogeneous deviation from the
ideal hcp structure, the polarization energy contains a crys-
tal-field term [2]
2 20
4= ( ),
5c cV Yπε Ω (10)
where Ω specifies the orientation of the = 1J molecule
with
2 1
24= .
7c cε − ε δ (11)
The splitting of the = 1J level in the crystal field cV is
given by
2
3= ( ) (0) = | |,
5c cE E∆ ± − ε (12)
where ( )E M is the energy of the state = 1J , =zJ M with z
along the c axis. The positive sign of c∆ implies that the
singlet state = 0zJ is the ground state of the triplet. Due to
this splitting the ground-state energy is brought down by
22 | | /5cε . Thus, the gain in the anisotropic interaction
contribution to the zero-point energy is linear in the lattice
distortion parameter δ . At the same time, this distortion
increases the contribution to the ground-state energy from
the isotropic part of the intermolecular interaction. This
contribution is quadratic in the lattice distortion parameter
δ [52]. The loss in the isotropic part and the gain in the
anisotropic part of the ground-state energy determines the
lattice distortion parameter at the given molar volume. One
cannot obtain a reliable value of δ from Eqs. (9)–(12) be-
cause there are other contributions to the splitting Eq. (12)
[2]. Assuming that the volume dependence of the polariza-
tion energy holds the same volume dependence as in Eq.
(9) in spite of these contributions, we will get the pressure
dependence of the lattice distortion parameter shown in
Fig. 8. When this result is compared with that for p-H2
(Fig. 7) it is seen that the presence of = 1J molecules in-
creases the hcp lattice distortion by two order of magni-
tude.
3.3. Experimental determination of the crystal field
splitting of isolated = 1J impurities in solid parahydrogen
An isolated = 1J orthohydrogen substitutional mole-
cule in the pure = 0J hcp crystal lattice is an ideal system
which has been studied extensively both theoretically and
experimentally [2]. The heat capacity anomaly in the = 0J
solid hydrogen samples containing small concentrations of
= 1J impurity molecules has been found by Mendesson,
Ruhemann and Simon as early as in 1931 [53] and ex-
plained by Schaefer [54] in 1939 by the splitting c∆ of the
triplet level of the = 1J impurity molecules (Eq. (12)). In
succeeding years this splitting has been studying in numer-
ous heat capacity [57,58] and NMR [22,63–65] measure-
ments, and theoretical works [55,56,59–61,66,67].
The first direct measurements of the crystal field split-
ting of the = 1J sublevels were performed by Dickson,
Bayers and Oka [68] who observed the fine structure of the
infrared 3(0)Q second overtone transition ( = 3 0,ν ←
= 0 0)J ← at 11758 cm–1 in parahydrogen crystals con-
taining approximately 0.1% orthohydrogen. The transition
is caused by the dipole moments induced in = 0J H2 due
to the quadrupolar electric field of the = 1J H2.
Three main sets of features were observed: the stronger
sets separated by nearly 0.5 cm–1 which arise from nearest
neighbor pairs of = 1J H2 and = 0J H2, while the weaker
set near the center is due to next nearest neighbor pairs.
The large splitting of 0.5 cm–1 corresponds to that of the
= 1J H2 next to the vibrationally excited = 0J H2: the
vibrational excitation breaks the 3hD symmetry of the crys-
tal and the splitting is greatly enhanced since the lattice
sum for nearest neighbors no longer vanishes. The break-
down of symmetry also removes the degeneracy of the
= 1M ± levels; this together with the slight difference in
energy for in plane and out of plane pairs leads to the intri-
cate structure of each set.
The small = 1J H2 splitting without vibrational excita-
tion for = 0J H2 is shown in Fig. 9 where the set of transi-
tions near 11758.39 cm–1 is given with an expanded fre-
quency scale for the polarization of the laser radiation both
parallel and perpendicular to the crystalline c axis. Alto-
Fig. 9. The low-frequency set of transitions for both parallel (up-
per) and perpendicular (lower) polarizations. The first and third
lowest frequency transitions are due to out-of-plane nearest neigh-
bor = 1 / = 0J J hydrogen molecules while the second and fourth
are due to in-plane neighbors [68].
1694 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12
Isotopic and spin-nuclear effects in solid hydrogens
gether eight such pairs of transitions are observed and led
to the value = (0.0102 0.0003) Kc∆ + and to the observa-
tion that the = 1M ± levels are higher in energy than is the
= 0M level. This value agrees with the results of
Schweizer et al. [64] within their quoted uncertainty and is
close to the theoretical estimate of Raich and Kanney [66].
3.4. HCP lattice distortion in solid HD
The principal difference between the HD molecule and
its homonuclear isotopic analog is that the center of mass
is considerably shifted relative to the center of electron
density distribution. A free HD molecule rotates around its
center-of-mass, which is separated from the midpoint be-
tween the two nuclei, the center-of-charge, by a distance
1
6 er , where er is the internuclear distance ( = 0.74116 Åer ).
Due to such mismatch the heteronuclear solid isotopes HD,
HT, and DT have some properties that distinguish them
from the homonuclear species. The most remarkable case —
the reentrant BSP phase transition in solid HD will be dis-
cussed in Sec. 4. In this section we discuss the effect of the
molecular asymmetry on the axial ratio of the hcp lattice
parameters /c a .
Structure data obtained using x-ray diffraction [69]
(Fig. 10) gave an unexpected result: c/a = 1.618 ± 0.003 at
the zero-temperature limit while for p-H2 x-ray diffraction
measurements by Krupskii et al. [43] gave the ideal value
1.633 ± 0.001. Surprisingly, the /c a ratio showed different
temperature behavior for different species: whereas in both
normal and para H2 the c/a ratio slightly decreased [69]
upon warming, the c/a ratio in HD grew with temperature,
reaching the value 1.623 ± 0.005 at 17 K. The authors qua-
litatively explained the found effect by the different virtual
contributions of the excited rotational states into the mole-
cular ground rotational state of the molecule. Theory of the
effect developed recently by Strzhemechny and Hemley
[74] explains how differences in the rotational dynamics
affect the value of the /c a ratio at low temperatures. Their
approach is based on the following equation:
el rot[ ( ) ( )] = 0,E E∂
∆ δ + ∆ δ
∂δ
(13)
where elE and rotE are the densities of the elastic and
ground-state rotational energies, respectively, as functions
of the deviation δ . While the loss in the elastic energy
el ( )E∆ δ is proportional to 2δ , the gain in the rotational
energy rot ( )E∆ δ due to the quantum self-polarization effect
calculated in the second perturbation order in the aniso-
tropic interactions is linear in δ . Taking both contributions
into consideration one gets a nonzero deviation δ . The
final estimate for δ in solid HD is
3(HD) = (0.61 0.08) 10 ,−δ − ± ⋅ (14)
the deviation δ in HD is also negative and approximately
50 times larger in magnitude than in H2 or D2 at zero pres-
sure (see Fig. 7). Though the above value is by a factor of
approximately 20 smaller compared with what was deter-
mined experimentally [69], the trend is correct.
4. Broken symmetry phase transition in solid HD
and o-p mixtures
4.1. Introduction
To a very good approximation, the electron density dis-
tribution in the H2 and HD molecules are the same, but in
the HD molecule the center-of-charge does not coincide
with the center-of-mass. Since the molecule rotates around
its center-of-mass but the intermolecular interaction is re-
lated to the center-of-charge molecule, rotations of the mo-
lecules are accompanied by translational displacements of
the center-of-mass. Thus, rotation and translation of the
molecule are coupled dynamically. As a result of such off-
center rotation, an additional Heisenberg-like terms appear
in the anisotropic part of the intermolecular potential [2,71].
Evidence for differences in properties of symmetric and
asymmetric hydrogens resulted from the rotational-trans-
lational coupling in the condensed state has been reported
since the sixties of the last century [72].
The heteronuclear hydrogen molecules have nonzero
static electric dipole moments. The experimental value of
the HD dipole moment, as derived from pure rotational
spectra of gaseous HD, is 4= 5.85 10p −⋅ D [73]. Because
of the nonequivalency of the ends of the HD molecule,
there is an additional dynamics in solid HD in comparison
with homonuclear isotopes H2 and D2, namely, end-to-end
reorientational dynamics [2]. In principle, by ordering the
dipole electric moments it is possible to minimize the di-
pole-dipole interaction energy. Thermodynamics and ki-
netics of the dipole ordering have been discussed in litera-
ture in connection with the problem of the dipolar ordering
and zero-temperature entropy in solid CO, a diatomic pos-
sessing in addition to a large quadrupole moment a small
permanent dipole moment 0.1 D, two orders of magni-
tude larger than HD [75]. In the case of solid CO the tem-
Fig. 10. Temperature dependencies of the axial ratio of the hcp
lattice parameters c/a for deuterohydrogen [69], parahydrogen [43],
and normal hydrogen [70]. After Ref. 69.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12 1695
Yuri A. Freiman and Yanier Crespo
perature of the hypotetical dipole ordering phase transition
is about 5 K [3], but the transition is frozen because the
molecules are ordered by much stronger quadrupole-
quadrupole interaction at T 40 K. The qualitative dis-
cussion of the possible dipole ordering in solid HD was
done by Chijoke and Silvera [76]. The characteristic dipo-
lar energy of an HD molecule in the field of the nearest
neighbors 2 3/dE zp R (where z is the number of nearest
neighbors and nnR is the nearest neighbor distance) gives
for the transition temperature an estimate 1 mK. Such
transition cannot take place in the quantum phase I since
all the molecules are in the = 0J state and the expectation
value of the dipole moment in this state is zero. It has been
speculated by the authors of Ref. 76 that the conditions to
observe the dipole ordering transition in HD are much
more favorable at high pressure. They estimated that at
P 150 GPa the dipole moment induced in the molecule
by the quadrupole moment of neighboring molecule in
= 1J state is 24 times larger than the permanent dipole
moment and this could increase the interaction energy to
tens of K. Summarizing results of the IR study Ref. 76 the
authors stated that in their low-temperature spectra they
“found no indication of a transition of electric dipole order-
ing”. We think that even though their considerations about
the strong enhancement of the dipolar interaction at high
pressure are correct, the dipolar ordering in HD is inobserv-
able due to the same reason why it cannot be realized in
solid CO: dipole end-to-end reorientations are frozen in
a strong quadrupolar field.
The BSP transitions in the even-J species (parahydrogen
and orthodeuterium) and all-J system (HD) provide an ex-
ample of quantum phase transitions where the orientational
ordering (OO) is realized by a competition between the
potential and kinetic energies of the system as opposed to
the classical phase transitions in the J -odd species (ortho-
hydrogen and paradeuterium) where the OO is realized
by a competition between the potential energy and the en-
tropy contribution to the free energy of the system. As a re-
sult, strong differences are observed in the OO properties
of odd-J, even-J, and all-J (e.g., HD) systems. The odd-J
systems order at any pressure at low temperature, while the
molecules in the even- and all-J systems at low pressure
are in the = 0J quantum state and thus are spherical enti-
ties. The OO in the even- and all-J systems is realized by
increasing the pressure, that is by increasing the potential
energy of the molecules. By analogy with quantum melting
of solid helium the quantum orientational disordering in
even- and all-J solid hydrogens was taken the title Quan-
tum Orientational Melting (QOM) [80–83].
The phase diagram of all-J systems possesses a surprising
anomalous feature: there is a pressure range 0< <mP P P
0(P is the ordering pressure at = 0T , mP is a minimal or-
dering pressure at the transition line) where the disordered
phase is reentrant (see Fig. 11). At = 0T in this pressure
range the system is in the disordered state but on heating
we cross the phase transition line and get to the ordered
phase. By increasing the temperature further at constant P
we get once again to the disordered phase. And conversely,
if we start from the ordered state inside the pressure range
0< <mP P P , we can get from the ordered to the disordered
state by two ways: either by heating or by cooling! In the
former case the order is destroyed by thermal fluctuations,
in the latter case the order is destroyed by quantum fluctua-
tions. This reentrant behavior was first predicted for the
all-J system by mean-field calculations [80,81] and found
experimentally in HD by Silvera’s group [78].
As was shown in Refs. 80, 81, there is a very close
analogy between the behavior of the melting line in solid
3He and the BSP transition in HD. As known, at the low-
temperature section of the melting line of 3He, the entropy
of liquid is smaller than that of the solid (Pomeranchuk
effect). As a result, there is a minimum in the pressure-
temperature dependence of the melting line in 3He — the
melting transition in 3He is reentrant. It was shown that the
reentrance in the BSP transition line in all-J rotor systems
has also entropy-driven origin [80,81].
Path integral Monte Carlo (PIMC) technique for studying
the reentrant behavior in solid HD was applied in Ref. 84.
The system of asymmetric rotors was considered with cen-
ters fixed on fcc lattice with the EQQ interaction as the
anisotropic interaction potential. The results were only of
qualitative value. Recently a more accurate approach was
implemented by Crespo et al. [85] where the authors ap-
plied to solid HD at high densities PIMC within the con-
stant pressure ensemble. They considered both the transla-
tional and rotational degrees of freedom and different
sources for the anisotropic interaction potential. The HD
phase line was calculated for both fcc and hcp lattices, dis-
playing the reentrant behavior in both cases. Good agree-
ment with the experimental data was obtained in the case
of the hcp lattice, including the minimum pressure ( mP )
where the BSP transition occurs ( mP = 56 GPa) and tem-
perature at the minimum point ( mT = 25 K) to be compared
with experiment: mP = 53 GPa, mT = 30 K [78].
Fig. 11. Schematic experimental phase diagram for H2, ortho-D2
and HD from data reported in Refs. 77–79.
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Isotopic and spin-nuclear effects in solid hydrogens
4.2. Experimental results
Experimentally, “Remarkable high pressure phase line
of orientational order in solid hydrogen deuteride” (as was
the title of the first publication), was first found by means
of Raman scattering by Silvera’s group [78] and then studied
to higher pressure by using near infrared spectroscopy [76].
It was observed that in HD the BSP transition pressure first
decreases and then increases with increasing temperature, a
very unusual reentrant behavior, different from the homo-
nuclear isotopes H2 and D2. In this Raman study the authors
measured both the Raman active rotons and vibrons in HD
to pressures of 120 GPa and temperatures as low as 3.5 K
(Fig. 12(a) [78]). The HD vibron frequency shifts substan-
tially with changing pressure at the phase line and due to
the sharpness of the vibron the detection of this discontinu-
ity was used as a method for determining the phase line. In
Fig. 12(b) the discontinuity is seen on the 64 GPa curve.
The intensity of the peak shifts from one vibron to the other
when the phase line is crossed. At the lowest temperature
of this experiment (3 K) a critical pressure = 68. Pa3 GcP
GPa was obtained. The critical pressure at = 0T was esti-
mated by extrapolation to be 0 = (69.0 2) GPaP ± , lying
between the values for H2 and D2 solids.
As the Raman line broadened with increasing pressure
and the shift of the line became too small to detect the
BSP, the work Ref. 78 was limited in pressure by 74 GPa.
Another technique that can be used to study this phase
transition at low temperatures is the infrared (IR) absorp-
tion. In Ref. 76 the authors reported the first infrared ab-
sorption measurements on HD to pressures of 159 GPa. In
Fig. 13 the IR spectra are shown in the region of the first-
order band for pressures between 11 and 159 GPa at 5 K.
These spectra correspond to the (0 1)v → vibron transition.
The 0 1→ vibrons of HD in hcp solid are symmetry for-
bidden for IR absorption at low temperature when the
thermal occupation of the = 1J level is close to zero. The
transition to the BSP phase was determined by the appear-
ance of two strong and relatively sharp absorption peaks
with linewidths of 18 and 40 cm–1 (see the curve at 77 GPa
in Fig. 13), as well as very weak third peak, indicated by
arrows in Fig. 13. These lines were interpreted as IR-
allowed v (0 1→ ) vibrons in the BSP phase. The reentrant
behavior of the BSP phase was observed by the appearance
and disappearance of these lines as the temperature was
raised at a fixed pressure of 56 GPa. In the ultra-high-
pressure region (159 GPa) it was found that the system
undergoes a transition from the BSP to the phase III. At
pressures of (157 ± 3) GPa it was observed the appearance
of a new spectral feature in the IR spectrum, a new peak at
3890 cm–1 (see the arrow in 159 GPa in Fig. 13). This is
Fig. 13. Absorption spectra for several pressures, collected at
temperatures of 4–8 K (after Ref. 78).
Fig. 12. Raman spectra of HD for several pressures with tempera-
ture 5–7 K. (a) Rotational spectra, (b) vibron spectra (after Ref. 78).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12 1697
Yuri A. Freiman and Yanier Crespo
the same pressure range and spectral response for the onset
of the phase III in H2 [86] and D2 [87]. As a result, these
data were were interpreted as a first observation of phase
III in HD [76]. In Fig. 14 the absorption spectra are shown
collected at 159 GPa for several temperatures. As can be
seen from Fig. 14, the intensity of the peaks corresponding
to both the phase III and BSP decreases as temperature is
raised becoming at 120 K the characteristic spectrum of the
hcp disordered phase. The phase transition line determined
from these methods are shown in Fig. 15.
4.3. Theoretical studies
4.3.1. Quantum orientational melting in all-J system
The BSP transition in solid hydrogen, is a realistic ex-
ample of Quantum Orientational Melting (QOM) that can
occur at zero temperature. This quantum phase transition is
driven by a competition between the potential energy, which
tends to order the system, and the kinetic energy, which
tends to delocalize the system. Early mean field theore-
tical models have shown that this type of transition can be
observed in a system of rotors under appropriate condi-
tions [80–82]. The mean field approximation considers a
system of N linear quantum rotors described by the Hamil-
tonian [80–82]
2 2
rot 20
ˆ= ( ) / 2,
N N
i
i i
B L U Y UN− η Ω + η∑ ∑ (15)
where L̂ is the angular momentum operator, U is a mo-
lecular field constant (U P ), rotB is the rotational con-
stant = 4 / 5,η π 20 ( )iY〈 Ω 〉 is the order parameter, and
< > denotes thermodynamic averaging with the Hamil-
tonian (15).
The computational scheme used to find the spectrum
of the Hamiltonian (15) is given in Refs. 80–82, 90. Figure 16
shows the energy spectrum for a linear rotor in the field
2 (cos ( ))VP− ϑ where =V Uη. As can be seen from this fi-
gure, in the disordered phase ( = 0η ) the gap ∆ between
the ground state and the first exited triplet state is
rot= 2B∆ . In the ordered phase of an all-J system, instead,
this degeneracy is splitted into a singlet and a doublet so
that ∆ decreases as increasing η (for constant U ). In the
the strong interaction limit the ground state becomes a
doublet. This reduction of the gap allows the occupation of
this level even at low temperatures and as a result there is
an additional contribution to the entropy, of the order of
ln 2Bk per a molecule where Bk is the Boltzmann con-
stant, that stabilizes the ordered phase at low temperatures.
The previous reasoning was confirmed by the calcula-
tions of the entropy and order parameter with temperature.
Figure 17 shows the temperature dependence of the entro-
py of both the one corresponding to the free rotor and
that of the OO state for different values of /U B
(11.5 < / < 15.0U B ). As can be seen, in the low-T region
the entropy of the OO phase is larger than that of the dis-
ordered one. The inset shows the temperature dependence
of the difference in entropy ( 0=S S S∆ − ) between the or-
dered, S , and the disordered, 0S , phases for / 12.9U B ≈ .
The free energy of the system as a function of the order
parameter ( )η was calculated using the obtained spec-
Fig. 14. Absorption spectra collected at 159 GPa for several tem-
peratures in the range between 5–160 K (after Ref. 76).
Fig. 15. Phase diagram of HD, the phase lines determined in
Ref. 76 are shown as solid lines and the phase line determined
earlier by Raman measurements as dashed lines [78]. The phase
lines of D2 (dotted lines) [88] are also shown for comparison
(after Ref. 76).
1698 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12
Isotopic and spin-nuclear effects in solid hydrogens
trum (Fig. 16), and from the condition / = 0∂ ∂η the or-
der parameter was found as a function of temperature.
Let us consider the situation for 0< <mP P P where or-
dering at zero temperature is absent. As one can see from
Fig. 18, there is a range of pressures where the ordered
phase emerges with increasing temperature. Within this
range of pressures the system undergoes two first-order phase
transitions with increasing temperature, viz., the low-tem-
perature one from disordered to ordered state and the high-
temperature one from ordered to disordered state. The for-
mer is of a quantum nature, QOM, the latter is the common
orientational disordering. This behavior has the following
explanation. At = 0T the only ordering factor is the poten-
tial energy; as temperature increases the singlet of the
splitted triplet start to get occupied increasing the contribu-
tion of the entropy term (T S∆ ) of the free energy, when the
sum of the contribution coming from the potential energy
and the term T S∆ became bigger than the contribution
from the kinetic energy a phase transition to the ordered
Fig. 16. Energy spectrum of a hindered linear rotor in the field
2(cos ( ))VP ϑ . E and V are given in units of the rotational con-
stant rotB . The numbers at the curves indicate the degeneracy of
the even- and odd-J energy levels for H2, D2, and HD, respec-
tively, taking into account nuclear spin degeneracy (after
Refs. 80, 81, 89).
Fig. 17. Temperature dependence of the entropy for free rotor
and for the orientational ordered state of an all-J system in the
field 2(cos( ))U P− η ϑ at different values of U (dot-dashed lines).
The solid line in the inset shows the entropy difference between
the ordered and disordered phases for / 12.9U B ≈ (after Ref. 81).
Fig. 18. Temperature dependence of the order parameter η of
a system of all-J rotors. The dash curve is the line of equilibrium
phase transitions, the dotted curve is the loci of absolute instabi-
lity of the orientationally ordered state (after Ref. 81).
Fig. 19. The –U T phase diagram for a system of interacting rotors
under the field 2(cos( ))VP− θ where =V Uη (after Refs. 80, 81).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12 1699
Yuri A. Freiman and Yanier Crespo
phase takes place. When temperature grows further higher
levels start to get occupied, as a consequence S∆ arrive to
a maximum and begin to decrease till the entropy of the
disordered state became grater than the ordered one and a
second phase transition to the disordered phase occurs. As
a result the phase diagram of this system became reentrant
as shown in Fig. 19 where the U T− phase diagram for
systems of interacting rotors is plotted [80,81].
4.3.2. The reentrant behavior from numerical simulations
The first attempt to go beyond mean field approach and
study the reentrant behavior in solid HD by using standard
numerical simulations was made in Ref. 84. In this work
the Path Integral Monte Carlo technique [91] (PIMC) was
employed using two main approximations: (i) the authors
consider asymmetric rotors with centers fixed at sites of
fcc lattice instead of hcp, and (ii) their anisotropic interac-
tion potential was limited by the pure EQQ interaction:
50 5
0
224 3/2
( ) 1( ) = ,
(4 )
nn
nn
nn
R
B R
R
Γ
π
(16)
where 0
nnR is the nearest neighbor distance at zero tempera-
ture. Even when the EQQ interaction is the leading term of
the pair interaction potential its selection is essential for
getting the right transition pressure [85]. It results that the
anisotropic interaction potential used in this work [84] was
too soft [85] and as a consequence the minimum transition
pressure mP obtained was small 10mP ≈ GPa when com-
pared with the experimental value = 53mP GPa (see
Fig. 20). Recently a more accurate approach was imple-
mented by Crespo et al. in Ref. 85. In this paper, the au-
thors model the HD solid, using the constant pressure
PIMC (CP–PIMC) formalism. The HD solid was simulated
as an assembly of rigid molecules, positioned on both fcc
or hcp lattices and having rotational and translational de-
grees of freedom. The pair interaction potential used was
essentially that of Cui et al. [92]
pair
Cui iso ani
< <
= ( ) ( , , ) ,ij i j ij
i j i j
U U R U+ κ Ω Ω∑ ∑ R (17)
where iso ( )ijU R is the isotropic component, ijR is the vec-
tor connecting the center points of two molecules,
= ( , )i i iΩ θ φ , is a vector containing the inclination and
azimuthal angles ( , )i iθ φ in spherical coordinates and κ is
a rescaling factor. Two sources for the anisotropic coeffi-
cients pair
aniU were considered: first Schaefer et al. [93],
denoted as Schaefer
CuiU ; second, Burton et al. [94], denoted
by Burton
CuiU . In order to determine the BSP line of HD it is
necessary to define an appropriate order parameter to
measure and monitor the angular order. Suppose we know
the structure of the ordered phase; then it is possible to
define N unit vectors iu corresponding to the orientation
of molecule on all sites = 1, ,i N and define the order
parameter:
( )
2
2
,
=1 =1
1= 3 cos ( ) 1 ,
2
M N
p i m i
m i
O
MN
⋅ −
∑∑ n u (18)
where M is the total number of Trotter slices [91], ,i mn
labels the orientation of the molecule i in the Trotter slice
m for a given configuration visited with the PIMC algo-
rithm. This order parameter measures the extent of order-
ing (and by difference, its deviation) relative to a given
orientational structure defined by the set { }iu and have
been extensively used to study the BSP of solid hydrogens
[84,85,92,95,96]. Nevertheless the order parameter in Eq. (18)
suffers from an important limitation, namely: if the system
reaches an ordered structure different from the reference
one associated to the chosen set { }iu , still 0pO ≈ . Thus,
the condition = 0pO is not sufficient to ensure that the
system is in an orientationally disordered phase. To address
this issue further, a second order parameter defined as
3 2Total
=1 , =1
1 1= ,
6
N
Q jk
i j k
O
N
∑ ∑ (19)
can be monitored, where
( )
2
Total 2
, , 0
=1
1= 3
T MMC
i i
jk j a k a jk
MC a
r r r
T M
− δ∑ (20)
with MCT is the MC time. Total
jk is the quadrupolar moment
of a system of MCT M molecules with a charge per atom of
1( )MCT M − [85]. If the molecules rotate in MC or Trotter
time [91] (showing spherical symmetry), then Total = 0jk ;
if, on the contrary, the molecules are frozen in a quadru-
polar configuration both in MC and Trotter time, then
23 Total
, =1
1 = 1
6 jkj k
∑ . Thus QO〈 〉 signals quadrupolar
Fig. 20. The temperature-pressure phase diagram of solid HD.
The open circles are the experimental data [78]. The up open
triangles are the constant volume PIMC results reported by Shin
et al. [84]. All other symbols are constant pressure PIMC results
obtained by Crespo et al. [85].
1700 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12
Isotopic and spin-nuclear effects in solid hydrogens
order in a more general way than pO〈 〉 , which is restricted
to a particular choice of { }iu .
The inset of Fig. 20 shows the order parameter pO as
a function of MC time at four pressures: 52 GPa (open
circles), 54 GPa (open squares), 56 GPa (solid diamond)
and 58 GPa (solid up triangles). The abrupt change seen in
the order parameter indicates the presence of a phase tran-
sition from a disordered phase to an ordered one, suggesting
that HD, at a temperature of = 30 KT , orders at 58≈ GPa.
Similar results were reported for the second order parame-
ter Total
jk [85]. Figure 20 shows the calculated tempera-
ture-pressure phase diagram of solid HD, along with the
experimental transition line (open circles, from Ref. 78),
also compared to theoretical phase diagrams obtained in
previous calculations. The up open triangles are the con-
stant volume PIMC results reported by Shin et al. [84].
The rest of the symbols are CP-PIMC results considering
different lattices (fcc, hcp) and different interaction poten-
tial ( Schaefer
CuiU and Burton
CuiU ) obtained by Crespo et al. [85].
As can be seen the reentrant behavior for the HD phase
line is obtained on both the realistic hcp and the fictitious
fcc lattices, and for both choices of the anisotropic poten-
tials. The minimum pressure at which the transition occurs
is 56mP ≈ GPa for the hcp lattice with the Burton
CuiU poten-
tial, close to the experimental value of = 53mP GPa. Using
alternatively Schaefer
CuiU yields instead 70mP ≈ GPa. The
edge point temperature = 25 KmT is also in good agree-
ment with the experimental value of = 30 KmT . In gen-
eral, the obtained BSP line with the Burton
CuiU potential is in
good agreement with the experiment. These results show
the crucial role played by the choice of the EQQ interac-
tion in determining the BSP transition pressure. Interest-
ingly, the BSP transition line for the fcc lattice is strongly
shifted downwards by about 30 GPa, regardless of the po-
tential used. In that case the BSP structure is Pa3, and the
stronger tendency to order is evidently due to the lack of
frustration in this structure. In other words, it is the angular
frustration present in the hcp lattice, and the connected
poorer relative stability of the 2 /C c structure [85], that
renders the BSP angular structure much more prone to
melting than the 3Pa .
To estimate the jump in entropy during this transition
the slope of the BSP line can be used. In fact according to
the Clapeyron equation / = /dP dT S V∆ ∆ , therefore di-
rectly connected to the entropy jump S∆ between the low
temperature quantum rotationally melted phase, and the
higher temperature BSP solid phase. During this first order
phase transition, the entropy jump vanish both at = 0T
(because of the Nernst theorem), and at the reentrant edge
point, where the entropies of both phases have the same
value. In between, the entropy jump is finite. From the
slope of the calculated phase line a maximum value of
0.4 0.2BS k∆ ± near 60 GPa at 15 K can be obtained, to
be compared with 0.5 Bk of the experimental slope at same
pressure and temperature (see Fig. 20) and the contribution
to the entropy, predicted by mean field theory, in the
strong interaction limit [81], equal to ln 2 = 0.693B Bk k ,
confirming that the entropy is an additional factor that sta-
bilizes the ordered phase.
4.4. Reentrant phase transitions in ortho-para mixtures
As was shown in the preceding section, depending on
the parity of the rotational quantum number J, solid hyd-
rogens exhibit either pressure-driven BSP quantum phase
transitions (even-J species p-H2 and o-D2) or usual order-
disorder classical phase transitions (odd-J species o-H2 and
p-D2). Solid HD was found to possess an additional anom-
aly: its BSP phase transition line displays a minimum, indi-
cating that the disordered phase is reentrant. Such a strong
difference in the pressure behavior of even-, odd-, and all-J
species raises an intriguing question of the possible phase
diagrams of their mixtures.
Two theoretical papers were devoted to this problem.
We will start from a mean-field theory developed by
Freiman, Tretyak, Mao and Hemley [89]. To circumvent
the main difficulty which one encounters considering mix-
tures, the systems without a translational invariance, the
authors used the following trick. It was supposed that each
site of the lattice is occupied by a superposition of even-J
and odd-J rotors. Two limiting cases are considered: mix-
tures at thermodynamic equilibrium, where the conversion
time is small or comparable with the thermalization time,
and the opposite case, frozen mixtures, when the conver-
sion time is large compared with all other relevant times.
In the former case, the fractions of even and odd species in
the mixture are temperature dependent and equal to ther-
modynamically average concentrations; for the latter, the
fraction of the species are fixed. The molecular field Ham-
iltonian (15) was used to describe dynamics of orient-
ational degrees of freedom.
In the case of the frozen systems the partition function
of the system is a product of the partition functions of the
even and odd systems:
even odd=Z Z Z . (21)
In this case the free energy of the system is a sum of
free energies of the even and odd systems, and there is a
simple relation between the order parameter of the system
20= Yη 〈 〉 and order parameters of even even 20 even= Yη 〈 〉
and odd odd 20 odd= Yη 〈 〉 systems:
20 20 even 20 odd= (1 ) .Y c Y c Y〈 〉 − 〈 〉 + 〈 〉 (22)
In the case of the equilibrium systems, the partition
function of the system is the sum of the partition functions
of even and odd systems:
even odd= .Z Z Z+ (23)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12 1701
Yuri A. Freiman and Yanier Crespo
In this case a simple expression relating 20= Yη 〈 〉 with
even 20 even= Yη 〈 〉 and odd odd 20 odd= Yη 〈 〉 does not exist.
The resulting phase diagrams are shown in Figs. 21 and 22
for the thermodynamically equilibrium and frozen even-J
odd-J mixtures, respectively. In Fig. 21, phase diagrams
for the pure even-J and pure odd-J systems, as well as that
for the all-J system, are also shown for comparison.
As one can see, there is a striking difference between
these diagrams. The equilibrium even-J odd-J mixtures
exhibit the same type of reentrant behavior as known for
the all-J systems (see Subsecs. 4.2, 4.3). The strongest
reentrance is displayed by the even-J odd-J linear rotor
mixture (equilibrium o-p-H2 mixture), the least one by the
equilibrium o-p mixture of D2 with HD intermediate be-
tween them. The frozen systems display qualitatively dif-
ferent behavior of their phase transition lines (Fig. 21). For
any even-J odd-J compositions excluding the case of the
pure even-J system the phase transition lines go monoton-
ically to the point = 0P with decreasing temperature. This
means that the passage to the limit of the zero fraction of
the odd modification is discontinuous.
An analysis of the entropy contribution to the free ener-
gy (Figs. 23, 24) furnishes an understanding of the nature
of the reentrant behavior and of the distinctions between
Fig. 23. Entropy of the equilibrium o-p mixture of solid hydro-
gens (after Freiman et al. [89]).
Fig. 22. Frozen o-p phase diagram (after Freiman et al. [89]).
Fig. 24. Entropy of the frozen o-p mixture of solid hydrogens
(after Freiman et al. [89]).
Fig. 21. Equilibrium o-p phase diagram (after Freiman et al. [89]).
1702 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12
Isotopic and spin-nuclear effects in solid hydrogens
the equilibrium and frozen ortho-para mixtures. As can be
seen from Fig. 23, the entropy of the equilibrium and all-J
systems is anomalous: in the low-temperature region the
entropy of the ordered phase is higher than that of the dis-
ordered phase. Thus, the entropy contribution is an addi-
tional ordering factor; its value increases with temperature
and, as a result, the ordering pressure goes down as tem-
perature increases. At a certain temperature, the entropy of
the disordered phase becomes higher than that of the or-
dered phase and the entropy contribution turns into a dis-
ordering factor and the ordering pressure turns upward
after passing the minimum point. In contrast with that, for
the frozen systems no such anomaly exists for the frozen
systems (see Fig. 25).
In its turn, the entropy anomaly can be traced at the mi-
croscopic level to characteristic features of the spectrum
of the ordered phase (Fig. 16). For the all-J system (HD)
the ground state is formed by two closely spaced levels
(a doublet in the strong interaction limit) as opposite to the
singlet ground state of the disordered phase. As a result,
there is an additional contribution ln 2R to the ordered
phase entropy. At the same time, both even- and odd-J
pure systems have singlet ground states and their transition
lines do not exhibit reentrance. The same entropy-based
considerations can be applied to the case of the BSP transi-
tions in the even-J odd-J mixtures of quantum linear ro-
tors. In the equilibrium case the partition function of the
homonuclear molecules is the sum of the partition func-
tions of the even-J and odd-J states taken with their nu-
clear spin- symmetry weights. As a result, for equilibrium
mixtures the ground state phase of the ordered phase is a
quadruplet for H2 (in a large V limit) and a octuplet for D2.
Taking into account that the disordered phase entropies
(at zero temperature) are zero for H2 and ln 4R for D2,
additional contributions to the ordered phase entropies for
H2 and D2 are ln 4R and ln(4 / 3)R , respectively. This
explains why the reentrance decreases in the sequence H2,
HD, D2. In the case of the frozen mixtures, the partition
function of the mixture is a product of the partition func-
tions of the components, each having a singlet ground
state, with no reentrance resulting.
Hetényi et al. [97], using a multi-parametrized mean-
field (MF) formalism, also found that the equilibrium
even-J odd-J mixtures have reentrant phase diagrams.
The main advantage of their formalism is that contrary to
the previous work where it was supposed that each site of
the lattice is occupied by a superposition of even-J and
odd-J rotors the new formalism allows for onsite distinc-
tion of different spin-nuclear species that can be distributed
randomly inside the lattice. This is a more realistic approx-
imation that allows also for the inclusion of local order
parameter correlation functions and therefore it is possible
to differentiate short-range order from long-range order a
feature that is absent in the standard MF theory where only
phases of complete order or disorder are possible.
The multiorder parameter (MOP) mean-field theory is
based on the trial Hamiltonian
5
2 0
0 rot 20
=1 <
1ˆ= ( )
2
N N
i i i
iji i j
R
H B L K Y
R
+ Ω − γ ×
∑ ∑
*
40(224;00) ( ),j ijC Y× γ Ω (24)
where K is the coupling strength, (224;00)C is a Clebsch–
Gordan coefficient, ijΩ denote the direction of the vector
connecting rotors i and j and iγ are parameters. Variation
of the free energy leads to the self-consistent expression
20 0= ( )i iYγ 〈 Ω 〉 . (25)
Fig. 25. Phase diagrams of the pure systems H2 and D2 at thermal
equilibrium distribution calculated using multiorder parameter
mean-field theory and the standard mean-field theory (MF). After
Ref. 97.
Fig. 26. Multiorder parameter MF phase diagrams for coupled
quadrupolar rotor models corresponding to solid molecular hyd-
rogen frozen at various ortho concentrations. After Ref. 97.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12 1703
Yuri A. Freiman and Yanier Crespo
Since the trial Hamiltonian in Eq. (24) is a sum of single-
rotor Hamiltonians, it follows that for each i in Eq. (25)
the average needs to be performed over the corresponding
coordinate i only. Ortho-para distinction can be imple-
mented by restricting a particular average to be over odd-J
or even-J states. This formalism was applied to a system
of coupled quadrupolar rotors whose centers of mass form
a fcc lattice. The methodology for the self-consistent calcu-
lation of the parameters { iγ } is described in Ref. 97. Order
is then signaled by nonzero solutions for iγ , disorder by
solutions in which all iγ are identically zero.
The orientational ordering of the system with ortho-para
concentrations at thermal equilibrium is shown in Fig. 26
along with the phase diagrams of the pure systems. For H2
and D2 the calculated phase diagrams via the MOP mean-
field theory and via the standard MF theory are very simi-
lar (see Fig. 21) and the reentrance phase line for H2 and
D2 is also obtained within this approximation. In Fig. 26
the results of the formalism presented above are shown for
solid H2 at a frozen concentration of ortho species. As in
the case of standard MF as the ortho concentration is de-
creased the system tends towards disorder entering the
ordered state at higher coupling constants for a given tem-
perature (see also Fig. 22). Nevertheless a noteworthy re-
sult is obtained with this new formalism. As one can see
from Fig. 26 even at an ortho concentration of 1% the sys-
tem enters an ordered state at coupling constants quite dif-
ferent from that of pure parahydrogen, and that for any
ortho concentration the ground state is always ordered.
This result is not observed in the case of standard MF theo-
ry (see the 0.01 curve in Fig. 22). The reason for this dif-
ference is because the orientational order is short ranged at
low temperatures. This feature can be just capture with the
MOP mean-field approximation where it is possible to dif-
ferentiate short-range order from long-range order while
only long-range order could be observed in standard MF
theory. In order to assess the nature of the ordering the cor-
relation functions ( )G r of the local order parameters { }iγ
was calculated using the expression.
2( ) = [ (0) ( )]G r rγ γ . (26)
The correlation functions for different temperatures
along the reentrant phase diagram (Fig. 25) are shown
for H2 in Fig. 27. As the temperature increases correlation
increases along the phase boundary. At high temperatures
( / > 1T B ) the order is definitely long range. The short-range
order is expected to be present at low-T up to / < 0.75T B .
The increases of the correlation along the phase boundary
with increasing temperature indicates the onset of long-
range order. The onset of long-range order is due to the
fact that ortho-para distinction ceases as temperature and
coupling constant (pressure) are increased.
5. Conclusion
In this Section we summarize the results discussed in
this review article with a stress on open experimental and
theoretical problems that still remain to be solved. These
problems span the whole region of the phase diagram.
Naturally, the most “hot” are problems at high-pressure
frontiers.
In the Introduction (Sec. 1) we have discussed the con-
straints which quantum mechanics impose on the possible
link between rotational states of the hydrogen molecules
and their total nuclear spin. This symmetry-related link
gives rise the existence of ortho- and paramodifications dis-
playing large differences in the solid-state properties.
In Sec. 2 we reviewed experimental and theoretical
works devoted to studies of the ortho-para conversion in
solid hydrogen and deuterium with the emphasis on the
high-pressure studies. The highest pressures for which the
experimental conversion rate data exist are at present
58 GPa. The extension of these studies to higher pressures
would provide a critical test of the conversion mechanisms
proposed by theory. It will need further theoretical studies
to validate the conception of ortho-para states at ultra-high
pressures.
In Sec. 3 we discuss two of numerous effects displaying
by the = 0 = 1J J− mixtures of solid hydrogen and deuter-
ium: the effect of = 1J admixtures on the molar volume,
and the hcp lattice distortion.
Hydrogen is a quantum solid and is therefore highly
compressible. Due to the large compressibility, the weak
quadrupole-quadrupole forces related with = 1J species
have a non-negligible effect on the molar volume.
As compared with fcc, the hcp lattice has an additional
degree of freedom associated with the /c a ratio. In this
Section we discuss the deviation of the axial ratio for the
= 0 = 1J J− mixtures from the ideal value 8 / 3 1.633≈ .
It was shown that the presence of = 1J molecules increas-
es the hcp lattice distortion by two order of magnitude.
Fig. 27. Correlation functions along the phase boundary of the so-
lid H2 at thermal equilibrium. After Ref. 97.
1704 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12
Isotopic and spin-nuclear effects in solid hydrogens
Section 4 is devoted to the peculiarities of the phase di-
agram of solid HD. Near 50 GPa it exhibits a reentrant
phase transition where a rotationally ordered («broken sym-
metry») crystalline phase surprisingly transforms into a
orientationally disordered high-symmetry phase upon cool-
ing. The molecular-field theory gives the qualitative reason
for reentrance: the higher entropy of the broken symmetry
phase, due to the inequivalence of H and D, as opposed to
the low entropy of the high-symmetry phase where the
rotational melting is quantum mechanical — a Pome-
ranchuk-like mechanism. The entropy jump across the tran-
sition is found to be comparable with ln 2, the value expect-
ed for the Pomeranchuk mechanism. Path Integral Monte
Carlo calculations give a reentrant phase boundary in good
agreement with experiment.
On the ground of the similar entropy-based considera-
tions the molecular-field theory forecasts that the broken-
phase transitions in thermodynamically equilibrium ortho-
para mixtures of homonuclear species of H2 and D2 should
exhibit reentrant phase transition lines. This prediction
makes the respective experimental studies and the devel-
opment of theory of the BSP transition in such systems
beyond the MF approximation important actual problem
In Sec. 5 we listed a number of problems both in exper-
iment and in theory that still remain to be solved. These
problems span the whole region of the phase diagram.
Naturally, the most “hot” are problems at the high-pressure
frontiers.
The authors are deeply indebted to Erio Tosatti and Mi-
khail Strzhemechny for numerous fruitful discussions.
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1706 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 12
1. Introduction
2. Ortho-para conversion
2.1. Experiment
2.2. Theoretical considerations
2.3. Conversion in solid D2
2.4. Validity of the concept of ortho-para states at high pressure
3. Thermodynamic properties of J = 0 – J = 1 mixtures
3.1. Molar volume
3.2. HCP lattice distortion in solid H2 as a function of o-p composition
3.3. Experimental determination of the crystal field splitting of isolated impurities in solid parahydrogen
3.4. HCP lattice distortion in solid HD
4. Broken symmetry phase transition in solid HD and o-p mixtures
4.1. Introduction
4.2. Experimental results
4.3. Theoretical studies
4.3.1. Quantum orientational melting in all-J system
4.3.2. The reentrant behavior from numerical simulations
4.4. Reentrant phase transitions in ortho-para mixtures
5. Conclusion
eferences
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