Ionic model for highly compressed solid hydrogen
We propose a simple ionic model for high-pressure conducting phase IV of solid hydrogen observed recently at room temperature. It is based on an assumption of dissociative ionization of hydrogen molecules 3H₂= 2H₂⁽⁺⁾2H⁽⁻⁾ induced by high compression. The model proposed predicts the first order tra...
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irk-123456789-1184482017-05-31T03:08:13Z Ionic model for highly compressed solid hydrogen Yakub, E.S. 9th International Conference on Cryocrystals and Quantum Crystals We propose a simple ionic model for high-pressure conducting phase IV of solid hydrogen observed recently at room temperature. It is based on an assumption of dissociative ionization of hydrogen molecules 3H₂= 2H₂⁽⁺⁾2H⁽⁻⁾ induced by high compression. The model proposed predicts the first order transition of molecular hydrogen solid into partly ionic conducting phase at megabar pressures and describes the temperature dependence of resistivity at room temperature. Its predictions are consistent with high temperature shockcompression experiments which exhibit conductivity of multiply shocked hydrogen. Location of phase transition line, change of volume, and ionization degree in solid phase IV are estimated. 2013 Article Ionic model for highly compressed solid hydrogen / E.S. Yakub // Физика низких температур. — 2013. — Т. 39, № 5. — С. 541–547. — Бібліогр.: 19 назв. — англ. 0132-6414 PACS: 05.70.Ce, 67.80.F–, 67.63.Cd, 64.60.Ej http://dspace.nbuv.gov.ua/handle/123456789/118448 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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9th International Conference on Cryocrystals and Quantum Crystals 9th International Conference on Cryocrystals and Quantum Crystals Yakub, E.S. Ionic model for highly compressed solid hydrogen Физика низких температур |
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We propose a simple ionic model for high-pressure conducting phase IV of solid hydrogen observed recently
at room temperature. It is based on an assumption of dissociative ionization of hydrogen molecules 3H₂= 2H₂⁽⁺⁾2H⁽⁻⁾ induced by high compression. The model proposed predicts the first order transition of
molecular hydrogen solid into partly ionic conducting phase at megabar pressures and describes the temperature
dependence of resistivity at room temperature. Its predictions are consistent with high temperature shockcompression
experiments which exhibit conductivity of multiply shocked hydrogen. Location of phase transition
line, change of volume, and ionization degree in solid phase IV are estimated. |
| format |
Article |
| author |
Yakub, E.S. |
| author_facet |
Yakub, E.S. |
| author_sort |
Yakub, E.S. |
| title |
Ionic model for highly compressed solid hydrogen |
| title_short |
Ionic model for highly compressed solid hydrogen |
| title_full |
Ionic model for highly compressed solid hydrogen |
| title_fullStr |
Ionic model for highly compressed solid hydrogen |
| title_full_unstemmed |
Ionic model for highly compressed solid hydrogen |
| title_sort |
ionic model for highly compressed solid hydrogen |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2013 |
| topic_facet |
9th International Conference on Cryocrystals and Quantum Crystals |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118448 |
| citation_txt |
Ionic model for highly compressed solid hydrogen / E.S. Yakub // Физика низких температур. — 2013. — Т. 39, № 5. — С. 541–547. — Бібліогр.: 19 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT yakubes ionicmodelforhighlycompressedsolidhydrogen |
| first_indexed |
2025-07-08T14:00:16Z |
| last_indexed |
2025-07-08T14:00:16Z |
| _version_ |
1837087557516525568 |
| fulltext |
© E.S. Yakub, 2013
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5, pp. 541–547
Ionic model for highly compressed solid hydrogen
E.S. Yakub
Cybernetics Dept., Odessa National Economic University
8 Preobrazhenskaya Str., Odessa 65082, Ukraine
E-mail: yakub@oneu.edu.ua
Received October 29, 2012
We propose a simple ionic model for high-pressure conducting phase IV of solid hydrogen observed recently
at room temperature. It is based on an assumption of dissociative ionization of hydrogen molecules
( ) ( )
2 23H 2H 2H induced by high compression. The model proposed predicts the first order transition of
molecular hydrogen solid into partly ionic conducting phase at megabar pressures and describes the temperature
dependence of resistivity at room temperature. Its predictions are consistent with high temperature shock-
compression experiments which exhibit conductivity of multiply shocked hydrogen. Location of phase transition
line, change of volume, and ionization degree in solid phase IV are estimated.
PACS: 05.70.Ce Thermodynamic functions and equations of state;
67.80.F– Solids of hydrogen and isotopes;
67.63.Cd Molecular hydrogen and isotopes;
64.60.Ej Studies/theory of phase transitions of specific substances.
Keywords: solid hydrogen, ionization, phase transition, conductivity, molecular dynamics.
1. Introduction
The quest for metallic hydrogen has begun many years
ago just after publication of well-known theoretical work
of Wigner and Huntington [1] and a lot of progress in ex-
perimental, theoretical and simulation work has been
achieved. Electrical conductivity was observed both in
static (see [2] for references) and dynamic [3] experiments
but the rigorous proof of metallization still has not been
found.
Recent diamond anvil cell (DAC) experiments [4] at
room temperature and megabar pressures reveal unusual
behavior of highly compressed solid hydrogen. According
to Eremets and Troyan [4], solid hydrogen transforms first
at 230 GPa into a nonmetallic conducting phase, which
exists up to 270 GPa. This phase is characterized by a low
and rising with temperature conductivity, which is not typ-
ical for metals. Metallic hydrogen occur presumably (see
discussion if Ref. 2) at higher pressures after the second
phase transition [4].
The aim of this work is an attempt to understand the na-
ture of this intermediate conducting nonmetallic phase
(phase IV) of highly compressed hydrogen, discovered by
Eremets and Troyan [4] at room temperature. We propose
a simple model which explains anomalous properties of
this nonmetallic phase by pressure induced ionization of
H2 molecules. According to our estimations, first order
phase transition into partly ionized state may occur at cer-
tain density and temperature when the energy needed for
ionization of H2 molecule is compensated by the sum of
the Coulomb attraction and polarization interaction of
emerging
( )
2H and H
(–)
ions.
Parameters of ionic model, pressure, energy, and con-
ductivity of partly ionic solid hydrogen are calculated di-
rectly using molecular dynamics technique and results are
compared with the recent DAC experiments.
The ionic model is introduced in the next section. In
Sec. 3 we describe the details of the underlying potential
model, determine parameters of atom–atom, ion–ion and
ion–atom interactions, and analyze the dependence of the
Helmholtz free energy of highly compressed hydrogen
solid on ionization degree. Properties of conducting phase
estimated in molecular dynamics simulation are presented
in Sec. 4. The results obtained and problems remaining are
discussed in the last section.
2. Partly ionic model for solid hydrogen
Relatively low and slowly rising with temperature con-
ductivity of solid hydrogen, observed in DAC experiments
of Eremets and Troyan [4] is not typical for a metal but is
quite usual in electrolytes. This rather trivial statement
prompted us to check to what degree the simplest version
mailto:yakub@oneu.edu.ua
E.S. Yakub
542 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5
of an ionic model is able to explain the observed behavior
of highly compressed hydrogen solid.
Ionization of molecular hydrogen may correspond to
the following reaction:
( ) (–)
2 23H 2H + 2H . (1)
Energetics of this reaction in ideal-gas state can be easi-
ly estimated using well-known constants [5]: electron af-
finity (0.75 eV) of hydrogen atom, ionization potential
(15.61 eV) and dissociation energy (4.75 eV) of H2 mole-
cule. The reaction enthalpy ( I = 11.48 eV per H2 mole-
cule) is rather high and equilibrium ionization degree is
negligible at room temperatures and atmospheric pressure.
Nevertheless it can become noticeable at high temperatures
or extreme compressions.
Energy needed for ionization of H2 molecule in con-
densed phase is essentially reduced by the Coulombic at-
traction of emerging
( )
2H and H
(–)
ions and their polariza-
tion interaction with surrounding H2 molecules. At high
compression this effect may compensate the energy loss
due to ionization and transition to the partly ionized state
may occur.
To check this possibility quantitatively, we apply the
following equation for Helmholtz free energy of partly
ionic solid [6]:
( )
(id) ( )
lat
( , , ) ( , , ) ln
k
fk
k
v
F T F T kT N
v
. (2)
Here
(id)
lat
( , , )F T is the free energy of ideal lattice gas,
T is temperature, (0) ( ) ( ){ , , },
k
N N N N
0
N is num-
ber of atoms in neutral H2 molecules,
( )N is number of
positive and
( )N is number of negative atomic ions in vo-
lume V (here and below we treat the
(+)
2H molecular ion as a
diatomic composed from two positive ions having formal
charge 0.5 | |);e )
( ) ( ) (0)N N N N is the total num-
ber of protons, /v V N is specific volume (per one proton),
/2N V is molar density, is ionization degree, and
(0) ( ) ( )
, ,
f f f
v v v are, respectively, atomic free volume of a
neutral atom in H2 molecule, of a positive ion in
(+)
2H mole-
cular ion, and of a negative atomic ion H
(–)
.
The degree of ionization , when expressed in terms of
k
N is as follows:
( ) ( ) (0) ( ) ( )( )/( )N N N N N . (3)
Within harmonic approximation, free volumes
(0)
,
f
v
( ) ( )
, ,
f f
v v in turn, could be expressed through elastic con-
stants
(0) ( ) ( )
, , of atomic (ionic) vibrations [6]:
3/2
( )
( )
k
f k
kT
v . (4)
Finally, substituting the expression for ideal lattice gas
contribution
(id)
lat
( , , )F T [6], one can write the final ex-
pression for Helmholtz free energy
(id)
( , , )F T as
(st) 3
0
( ) ( ) 2
( , , ) ( , ) ln
4
(1 ) ln (1 ) ln .
3
NF T U N I NkT
NkT
(5)
Here
(st)
( , )NU is the static potential energy of partly
ionized solid at T = 0.
Ionic model defined by Eq. (5) corresponds to the first
(Einstein) approximation for the Helmholtz free energy of
a solid. It neglects not only the anharmonicity of atomic
vibrations but also all correlations between displacements
of pairs, triplets, etc. of atoms.
Nevertheless, as it was shown in Ref. 6, in a wide range
of temperatures this approximation provides a reasonable
estimation for Helmholtz free energy of a real solid. Kno-
wing parameters of this model one may determine the
equilibrium ionization degree at given density and tem-
perature by minimizing free energy with respect to .
3. Interparticle interaction in highly compressed
hydrogen solid
To apply the model described above, one must know at
least four functions of density and ionization degree :
the static potential energy
(st)
NU and three elastic constants
(0) ( ) ( )
, , . The simplest way to evaluate these quan-
tities and find the equilibrium degree of ionization is
their direct computation on the basis of a potential model
for atom–atom, ion–atom and ion–ion interactions.
Such potential model is described below. It includes two
different types of interaction: intramolecular (inside 2H and
(+)
2H diatomics) and intermolecular (interparticle). The last
one include short range (repulsion + ion–atom polarization),
and long range Coulomb ion–ion interactions:
(st) (intra) (short)
( , ) ( , ) ( , ) ( , )
C
N N N NU U U U .
(6)
N-particle potential energy in this work for all types of
interaction was represented within atom–atom approxima-
tion (AAA) [7], i.e.:
( ) ( )
( )
N
k k
ijijN
i j
U r , (7)
where
( )
( )
k
ij r are atom–atom, atom–ion or ion–ion central
interaction potentials.
Below we describe specific types of interaction in-
cluded in our potential model in details.
Ionic model for highly compressed solid hydrogen
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5 543
3.1 Intramolecular interaction
Potential energy
(intra)
( , )NU includes interaction of
hydrogen atoms inside neutral H2 molecules and positive
ions inside
(+)
2H molecular ions. H–H interaction inside
neutral H2 molecule was supposed to be the same
1
g -curve as in ideal diatomic gas.
It was represented by the modified Hulburt–Hirshfelder
potential [11]:
1 3( ) [exp( 2 ) 2exp( ) (1 )exp( )],eU r D x x ax bx cx
(8)
where ( / 1).ex r r We adopted the following set of po-
tential parameters for H2: 1.4403; er = 0.74126 Å,
eD 4.767 eV, a = 0.1156, b = 1.0215, c = 1.72, which
give an excellent approximation of the
1
g -curve within
a wide range of distances (0.3–5 Å) [11].
Potential energy of
(+)
2H molecular ion was described by
the same analytical expression Eq. (8). The exact
1
g-inte-
raction energy curve of
(+)
2H ion [12] was fitted by Eq. (8)
and the following set of parameters obtained:
1.3558 ; er = 1.0584 Å, eD = 2.793 eV, a = 0.2803,
b = 0.000407, c = 5.37.
3.2. Short-range repulsion
Potential energy of short-range repulsion
(short)
( , )NU
includes contributions from all pairs of atoms and ions,
both free or bonded in different diatomics. H2–H2 interac-
tion energy within AAA [7] is the sum of four central
atom–atom potentials H–H ( )r defined as
1 3
H–H
1 3
( ) ( ) ( )
4 4
r U r U r . (9)
Here 1( )U r and 3( )U r are potential energies of two
hydrogen atoms in their singlet
1
g and triplet 3
u states.
We used the analytical representation of the atom–atom po-
tential Eq. (9) proposed by Saumon and Shabrier [8]:
* *
H–H 1 2( ) { exp[ 2 ( )] (1 )exp[ ( )]}.r s r r s r r
(10)
Five parameters of this potential: *r 3.2809 Å, =
= 1.74·10
–3
eV, γ = 0.4615, s1 = 1.6367 Å
–1
, 2s
11.2041 Å have been determined in Ref. 8.
At moderate compressions this set of parameters de-
scribes the equation of state [7], as well as the melting and
orientation phase transitions [9]. At higher compressions
the interatomic distances become too short and differences
between predicted and measured pressures (see Ref. 10 and
references therein) become essential. Potential Eq. (10)
with above parameters overestimates repulsion of hydro-
gen atoms at short distances.
We performed re-calibration of the atom–atom potential
for H–H interaction by fitting two of its parameters: *r
and s1. The resulting values 1*
14.527Å; 0.9År s
provide an excellent fit of the pressure–volume relation for
molecular hydrogen calculated in Ref. 10 in volumes range
from 2.0 to 8.0 cm
3
/mol.
For all other short-range repulsive interactions we
adopted the following simple analytical form, which re-
flects the extremely soft repulsion in hydrogen:
(rep)
5
( )
ij
ij
A
r
r
. (11)
Parameter Aij, in general, must be different for different
types of interactions (neutral atom–ion and ion–ion). Be-
low we will discuss the problem of determination of this
parameter for different types of interaction.
3.3. Ion–atom polarization potential
An important contribution to the short-range ion–atom
interaction is polarization of surrounding hydrogen mole-
cules. Its inclusion makes the ionic model more realistic in
prediction of stability conditions in partly ionized hydro-
gen solid. Contrariwise, inclusion of mutual polarization of
ions is less important, because it is only a small portion of
their strong Coulombic interaction. Therefore in this work
we treat ions as nonpolarizable particles.
Additionally, we ignore within current potential model
the difference between polarizability of free H atom and
atom in H2 molecule and estimate the potential energy of
ion–atom polarization interaction at long distances by us-
ing the standard asymptotic form:
2
(pol)
4
| |
( )
2
ij
ij
e Z
r
r
, (12)
where is polarizability of free hydrogen atom
( = 4.5 a.u.), ijr is the distance between ith atom and jth
ion and Z is its formal charge (Z = –1 for the negative ion
and Z = 0.5 for the positive one) [13].
3.4. Short-range ion–atom interaction potential
At high densities the distances between ions and sur-
rounding neutral atoms become too short to apply Eq. (12)
without any correction. Therefore we introduce a more
general form of the ion–atom interaction potential, which
includes both short-range repulsion Eq. (11) and polariza-
tion contributions Eq. (12):
5 42
(short)
4
| |
( )
2
ij ij
ij
i
R Re Z
r
r rR
. (13)
Here ijR is the radius (characteristic size) of ion–atomic
interaction (at ijr R the short-range repulsion compen-
sates the polarization attraction, i.e.,
(short)
( ) 0).ijij R At
the long distances Eq. (13) tends to Eq. (12) and at small
distances approaches to Eq. (11).
E.S. Yakub
544 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5
3.5. Long-range interaction
The last (but not least) contribution to the ionic interac-
tion is the intensive and long-range Coulombic part
( )
( , )
C
NU . It is the most important type of interionic inte-
raction. This contribution was taken into account according
to the method of effective (pre-averaged) Ewald interionic
potentials proposed earlier [14]
2 2
( )
01 1 1,
3 1
( )
16 2
N N N
C i
ijN
mi i j j i
e Z
U r
r
. (14)
Here ijr is the effective ion–ion potential defined as
22
0
1
1 3 , ,
( ) 4 2
0, .
i j
m
ij m m
m
e Z Z r r
r r
r r r r
r r
(15)
iQ are charges of ions, and mr is the radius of sphere hav-
ing the same volume as the computer simulation cell.
This method was proved to be an effective tool for si-
mulation of disordered ionic solids and was used in our
simulations before [15,16] (see Refs. 14–16 for details).
4. Properties of conducting phase
from computer simulation
Molecular dynamics technique, analogous to that used
earlier in our study of pre-melting and melting phase tran-
sitions in ionic solids [16], when combined with the poten-
tial model described in the previous Section was applied to
calculate pressure, energy, and electrical conductivity of
partly ionic solid hydrogen directly.
4.1. General scheme of computation procedure
The adopted computation procedure includes four stages.
1. Minimization of the static potential energy with re-
spect to positions of all atoms and ions in simulation cell at
given density and ionization degree (at 0T ), i.e.,
computation of
(st)
( , );NU and estimation of elastic con-
stants
(0) ( ) ( )
, , by sequential displacing of all
atoms and ions in the cell.
2. Determination of equilibrium ionization degree by
minimizing Helmholtz free energy Eq. (5) at given density
and temperature in partly ionic phase.
3. Estimation of the phase transition line (i.e., transition
pressure and densities of co-existing phases at given tem-
perature) using Eq. (5).
4. Evaluation of electrical conductivity in molecular
dynamics simulation at given density, temperature and
equilibrium ionization degree .
4.2. Calibration of the model
Of course, any assessment based on the ionic model de-
pends on the adopted parameters of the potential model
and of the size of simulation cell. In general, radii ijR in
Eq. (13) for positive and negative ions as well as repulsion
parameters Aij in Eq. (11) for ion–atom and ion–ion inte-
ractions are different. However, in this work we adopted
the simplest possible scheme, which has only one free pa-
rameter for all ions and all interactions.
We used the standard definition of ionic radius IR as a
distance, where Coulomb forces compensate the short-
range attraction of positive H
(+)
and negative H
(–)
ions.
Parameters ijR for H–H
(–)
and H–H
(+)
curves in Eq. (13)
were set equal to .IR
Repeating the above stages 1–3 with different values of
this effective radius, we adopted finally the value Rj =
= 1.68 Å which allow reproducing the observed pressure of
transition to the conducting phase at room temperature
(230 GPa) [4]. The corresponding interaction potentials of
atoms and ions are shown in Fig. 1. Below we discuss the
details of above calculations and some results obtained
during simulations.
4.3. Equilibrium ionization degree
The dependence of the excess Helmholtz free energy,
i.e., the difference between absolute Helmholtz free energy
of partly ionized and molecular hydrogen: ( , , )F T
( , , ) ( , ,0),F T F T from ionization degree at room
temperature and different densities is presented in Fig. 2.
As one can see, at the relatively low density
( < 1.0 g/cm
3
) this dependence is monotonous. The ex-
cess free energy increases with increase of number of
ions because the sum of polarization energy and Coulomb
attraction cannot compensate the ionization energy. Mo-
lecular solid remains the only stable state here. At higher
Fig. 1. Interaction potentials of atoms and ions of hydrogen. Min-
imum of H
(+)
–H
(–)
curve and zeroes of H–H
(–)
and H–H
(+)
curves
correspond to the adopted ionic radius Ri = 1.68 Å.
Ionic model for highly compressed solid hydrogen
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5 545
density the initial (positive) slope of this curve (which is
determined by the difference between ionization and po-
larization energies) decreases. At a certain degree of ioni-
zation the Coulombic contribution overcomes this ten-
dency: the excess free energy reaches maximum and
began decrease with the increase of ionization degree.
This decrease then slows down; free energy reaches a
minimum and then again increases due to increasing role
of the short-range repulsion forces.
As a result, at the F( )-curve appear two extreme: a
maximum at lower ionization degree and a minimum at
higher ionization degree. The equilibrium degree of ioniza-
tion at given temperature and density was determined as
the abscissa of this minimum. At certain conditions these
minima may correspond to thermodynamically stable states
of partly ionized hydrogen solid. Locations of these minima
are plotted in Fig. 3 as functions of pressure at two tempera-
tures (T = 300 K and T = 3000 K). Open symbols correspond
to metastable states of partly ionized hydrogen solid and full
symbols to stable states (details of the phase equilibria calcu-
lation are explained in the next subsection). The maximum
degree of ionization, which was estimated in this way, is
slightly increasing with pressure but remains small: about 8%
at room temperature and about 10% at T = 3000 K.
4.4. Phase transition line
After determination of the equilibrium ionization de-
gree ( , ),T the transition pressure tP and densities of
coexisting phases were estimated using a standard method
of double tangent. Helmholtz free energy of molecular
hydrogen ( , ,0)F T and of the partly ionic state
( , , ( , ))F T T were plotted at fixed temperature against
molar volume 1( ),V and the transition pressure tP
was determined as a slope of their common tangent. At
tP 230 GPa and room temperature the volume change
V was found to be relatively small (0.065 cm
3
/mol).
With increasing temperature the transition pressure de-
creases and the volume change increases ( tP =110 GPa and
V 0.3 cm
3
/mol at 3000 K). This is illustrated in Fig. 4.
The estimated location of the transition line from molecu-
lar (phase III) into conducting phase (IV) is shown by
dashed line in Fig. 5.
4.5. Estimation of electrical conductivity
Molecular dynamics technique provides an easy way to
determine conductivity of ionic systems by applying an
Fig. 2. Excess Helmholtz free energy (per atom) of partly ionized
solid hydrogen as a function of ionization degree at different
densities (shown in legend in g·cm
–3
).
Fig. 3. Equilibrium ionization degree as a function of pressure.
Open symbols correspond to metastable and full symbols to sta-
ble states at T = 300 K, T = 3000 K.
Fig. 4. Pressure dependence of density in compressed solid hy-
drogen at two temperatures (300 and 3000 K). Vertical line seg-
ments correspond to locations of phase transition. Cross indicates
the state in which electrical conductivity was observed in shock-
compression experiments in liquid phase [3].
E.S. Yakub
546 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5
external electric field and monitoring numbers of positive
and negative charges entering and leaving the cell.
According to estimations made by Nellis, Ruoff and
Silvera [2], conductivity in phase IV discovered by Ere-
mets and Troyan [4] is very low (about 10
–2
–1
m
–1
).
Unfortunately, in our computer simulation (we used a ra-
ther small cell containing only 216 hydrogen atoms) one
cannot determine the corresponding charge flux in accept-
able simulation time using the same field strength as in
experiments of Eremets and Troyan [4]. Therefore we were
forced to apply external field two or three orders of magni-
tude higher to get results of reasonable significance level in
acceptable computer time.
Figure 6 illustrates the results of such simulations at
room temperature and two pressures. In both cases the esti-
mated average slope of the logarithm of electrical conductiv-
ity in the Arrhenius plot was small and negative. The pre-
dicted temperature dependence of electrical conductivity is
in a good agreement with experimental data. According to
measurements of Eremets and Troyan [4], the activation
energy in conducting phase of solid hydrogen is about
8 meV while our estimations are 7.4 meV at 250 GPa and
8.0 meV at 270 GPa. The predicted pressure dependence of
resistivity was also negative like in DAC experiments [4]
but is much less pronounced. At the same time the predicted
absolute values of conductivity are much higher then in ex-
periment [4]. By decreasing external field we found a strong
decrease of estimated conductivity. This gives some hope
for the agreement between the results of modeling and expe-
riment in weaker fields that are not available now for us to
simulate.
5. Conclusions
Recent DAC experiments of Eremets and Troyan at
room temperature and megabar pressures [4] reveal transi-
tion of solid hydrogen to a nonmetallic phase IV with a
low and rising with temperature conductivity. To explain
such behavior of compressed hydrogen solid we propose in
this work a simple ionic model.
This model accounts for dissociative ionization of hy-
drogen molecules into stable positive molecular
(+)
2H and
negative atomic H
(–)
ions. The model proposed has only
one parameter – ionic radius, which was fitted to reproduce
the pressure of transition to the conducting phase at room
temperature [4]. We estimated the equilibrium ionization
degree in partly ionic conducting state by minimizing
Helmholtz free energy, transition pressure and volume
change and evaluated electrical conductivity by molecular
dynamics technique.
The main conclusion, which can be made on the basis
of our calculations, is that the ionic migration mechanism
can explain some characteristics of the compressed con-
ducting hydrogen. Ionic model reproduces the negative
temperature dependence of resistivity, observed in static
DAC experiment [4] at room temperature. Our estimations
are also in line with results of dynamic experiments on
multiply shocked hydrogen at T = 3000 K [3]. We found a
reasonable agreement between the predicted pressure of
transition at 3000 K (110 GPa), and parameters of conduct-
ing state observed in the fluid phase (140 GPa and
0.6 g/cm
3
) in shock compression experiments [3].
Correctness of assumptions used in this work can be ve-
rified experimentally. First of all, the presence of negative
Fig. 5. Phase diagram of compressed hydrogen. Open circle
shows the transition from phase III to phase IV and open square
the next transition discovered by Eremets and Troyan [4] at T =
= 300 K. Vertical bars indicate probable location of melting line
predicted ab initio [19]. Dashed line shows the location of the
transition line from molecular solid phase III to the partly ionic
solid phase IV estimated in this work.
Fig. 6. Temperature dependence of electrical conductivity (in
–1
·m
–1
) of partly ionic hydrogen solid estimated at two pres-
sures.
Ionic model for highly compressed solid hydrogen
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5 547
hydrogen ions, even in a small amount, can be detected in
optical spectra of compressed hydrogen. The estimated
vibronic frequency of
(+)
2H is about half as much as vi-
bronic frequency of H2 and may be obscured by the well-
known lattice absorption band of diamond [18]. This is a
challenge for experimentalists.
Secondly, the fact of the volume change V increase
with increasing temperature can be verified. Such beha-
vior, if confirmed by experiment, may be related to exis-
tence of a lower critical point on the transition line. This, in
turn, may answer the question why so many sophisticated
DAC experiments (see discussion in Ref. 2 and references
therein) do not reveal any conductivity in solid hydrogen at
cryogenic temperatures and megabar pressures.
Of course, the model proposed in this work is actually
only a basic one, developed to explain only the principal
features of solid hydrogen in conducting phase. It does not
take into account many aspects which may be important in
a wider context. It ignores the difference in effective radii
of positive and negative ions, as well as contribution of
quantum effects, anharmonicity and correlations, as well as
contribution of other mechanisms of electric charge trans-
port, like polaron hopping etc. In this regard it should be
noted that very high electric fields applied in our simula-
tions make partly ionic hydrogen solid close to the electric
breakdown and the mechanism of conductivity observed in
experiments [4] may differ from that under simulation
conditions. We must note also other attempts to explain
unusual behavior of phase IV [17].
An unanswered question remains also the possibility to
reconcile ionic model with ab initio simulations [19]. It
seems not clear yet how to interpret the results of ab initio
calculations of in terms of the ionic model.
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