Molecular rotation in p-H₂ and o-D₂ in phase I under pressure
The orientational order parameter, rotational ground-state energy, and lattice distortion parameter (the deviation of the c/a ratio from the ideal hcp value 1.633) in hcp lattice of phase I of p-H₂ and o-D₂ are calculated using a semi-empirical approach. It is shown that the lattice distortion in th...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Цитувати: | Molecular rotation in p-H₂ and o-D₂ in phase I under pressure / Yu.A. Freiman, S.M. Tretyak, A.F. Goncharov, H. Mao, R.J. Hemley // Физика низких температур. — 2011. — Т. 37, № 12. — С. 1302–1306. — Бібліогр.: 24 назв. — англ. |
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irk-123456789-1188002017-06-01T03:04:54Z Molecular rotation in p-H₂ and o-D₂ in phase I under pressure Freiman, Yu.A. Tretyak, S.M. Goncharov, A.F. Mao, H. Hemley, R.J. Физические свойства криокристаллов The orientational order parameter, rotational ground-state energy, and lattice distortion parameter (the deviation of the c/a ratio from the ideal hcp value 1.633) in hcp lattice of phase I of p-H₂ and o-D₂ are calculated using a semi-empirical approach. It is shown that the lattice distortion in these J-even species is small compared with that found in n-H₂, and n-D₂. The difference presumably is caused by the J-odd species. 2011 Article Molecular rotation in p-H₂ and o-D₂ in phase I under pressure / Yu.A. Freiman, S.M. Tretyak, A.F. Goncharov, H. Mao, R.J. Hemley // Физика низких температур. — 2011. — Т. 37, № 12. — С. 1302–1306. — Бібліогр.: 24 назв. — англ. 0132-6414 PACS: 62.50.–p, 64.70.kt, 67.80.F– http://dspace.nbuv.gov.ua/handle/123456789/118800 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Физические свойства криокристаллов Физические свойства криокристаллов |
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Физические свойства криокристаллов Физические свойства криокристаллов Freiman, Yu.A. Tretyak, S.M. Goncharov, A.F. Mao, H. Hemley, R.J. Molecular rotation in p-H₂ and o-D₂ in phase I under pressure Физика низких температур |
| description |
The orientational order parameter, rotational ground-state energy, and lattice distortion parameter (the deviation of the c/a ratio from the ideal hcp value 1.633) in hcp lattice of phase I of p-H₂ and o-D₂ are calculated using a semi-empirical approach. It is shown that the lattice distortion in these J-even species is small compared with that found in n-H₂, and n-D₂. The difference presumably is caused by the J-odd species. |
| format |
Article |
| author |
Freiman, Yu.A. Tretyak, S.M. Goncharov, A.F. Mao, H. Hemley, R.J. |
| author_facet |
Freiman, Yu.A. Tretyak, S.M. Goncharov, A.F. Mao, H. Hemley, R.J. |
| author_sort |
Freiman, Yu.A. |
| title |
Molecular rotation in p-H₂ and o-D₂ in phase I under pressure |
| title_short |
Molecular rotation in p-H₂ and o-D₂ in phase I under pressure |
| title_full |
Molecular rotation in p-H₂ and o-D₂ in phase I under pressure |
| title_fullStr |
Molecular rotation in p-H₂ and o-D₂ in phase I under pressure |
| title_full_unstemmed |
Molecular rotation in p-H₂ and o-D₂ in phase I under pressure |
| title_sort |
molecular rotation in p-h₂ and o-d₂ in phase i under pressure |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2011 |
| topic_facet |
Физические свойства криокристаллов |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118800 |
| citation_txt |
Molecular rotation in p-H₂ and o-D₂ in phase I under pressure / Yu.A. Freiman, S.M. Tretyak, A.F. Goncharov, H. Mao, R.J. Hemley // Физика низких температур. — 2011. — Т. 37, № 12. — С. 1302–1306. — Бібліогр.: 24 назв. — англ. |
| series |
Физика низких температур |
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AT freimanyua molecularrotationinph2andod2inphaseiunderpressure AT tretyaksm molecularrotationinph2andod2inphaseiunderpressure AT goncharovaf molecularrotationinph2andod2inphaseiunderpressure AT maoh molecularrotationinph2andod2inphaseiunderpressure AT hemleyrj molecularrotationinph2andod2inphaseiunderpressure |
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2025-07-08T14:40:09Z |
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2025-07-08T14:40:09Z |
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1837090063887892480 |
| fulltext |
© Yu.A. Freiman, S.M. Tretyak, Alexander F. Goncharov, Ho-kwang Mao, and Russell J. Hemley, 2011
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 12, p. 1302–1306
Molecular rotation in p-H2 and o-D2 in phase I
under pressure
Yu.A. Freiman and S.M. Tretyak
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: freiman@ilt.kharkov.ua
Alexander F. Goncharov, Ho-kwang Mao, and Russell J. Hemley
Geophysical Laboratory, Carnegie Institution of Washington
5251 Broad Branch Road NW, Washington, DC 20015, USA
Received July 7, 2011, revised August 9, 2011
The orientational order parameter, rotational ground-state energy, and lattice distortion parameter (the devia-
tion of the c / a ratio from the ideal hcp value 1.633) in hcp lattice of phase I of p-H2 and o-D2 are calculated us-
ing a semi-empirical approach. It is shown that the lattice distortion in these J-even species is small compared
with that found in n-H2, and n-D2. The difference presumably is caused by the J-odd species.
PACS: 62.50.–p High-pressure effects in solids and liquids;
64.70.kt Molecular crystals;
67.80.F– Solids of hydrogen and isotopes.
Keywords: solid hydrogen, lattice distortion, axial ratio, molecular rotation, orientational order parameter.
At zero pressure the molecules in J-even solid hydro-
gens (p-H2 and o-D2) are virtually spherical. Such spheres
crystallize into a perfect hcp structure with the ideal axial
ratio / = 8 / 3.c a A competition between the gain in the
anisotropic energy and a loss in that of the isotropic com-
ponent determines the deviation = / 8 / 3c aδ − from the
ideal hcp value for given pressure and temperature. The
orientational state of molecules in the distorted lattice is
characterized by the orientational order parameter .η
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.
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 smaller
[1,2]. In the case of hydrogens the situation under discus-
sion is rather controversial. There have been numerous ex-
perimental attempts to determine /c a in solid hydrogens
starting from the first measurements by Keesom et al. [3]
who found that at zero pressure p-H2 has an ideal hcp lat-
tice. Precision x-ray zero-pressure measurements by Krups-
kii et al. [4] confirmed this result and extended it for o-D2.
Systematic high-pressure neutron and x-ray structural
studies of different isotopes and spin-nuclear species of
solid hydrogen were started in the early of 1980’s. The first
structural studies 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 [5,6]. It was
found that the ratio /c a is practically constant and is
slightly less than the ideal hcp value (1.631±0.002). A
large number of x-ray [7,9,10] and neutron-diffraction
[8,11,12] studies of the axial ratio of solid hydrogen and
deuterium were performed in the late 1980’s and early
1990's. These measurements on normal ortho-para samples
established that the /c a ratio is very close to the ideal hcp
value and weakly depends on pressure.
Synchrotron single-crystal x-ray diffraction measure-
ments of n-H2 and n-D2 up to megabar pressures at room
temperature [13] revealed an approximately linear decrease
of the /c a ratio with increasing pressure. The observed
lattice distortion has been interpreted as a gradual effect of
orientation of the H2 molecular axis within phase I.
No isotope effect in the pressure dependence of the /c a
ratio was found.
Molecular rotation in p-H2 and o-D2 in phase I under pressure
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 12 1303
A somewhat weaker pressure dependence of /c a was
displayed in the x-ray powder diffraction study of solid deu-
terium [14]. The lattice distortion parameter δ = –0.03 was
obtained for 94 GPa at 83 K, ∼25% smaller than the room
temperature single-crystal value by Loubeyre et al. [13].
When these results for the lattice distortion are compared
with that for rare-gas solids ( 310−∼ at 100 GPa [1,2]) it is
apparent that the difference in more than the order of magni-
tude is due to the rotational degrees of freedom.
There have been many attempts to calculate the pres-
sure behavior of the axial ratio using different theoretical
approaches: Hartree–Fock [15,16], local density approxi-
mation [17], Path Integral Monte Carlo [18], ab initio mo-
lecular dynamics [19]. In all proposed theoretical ap-
proaches the rotation-lattice coupling gives rise to positive
δ which is in contradiction with experiment. In this paper
we developed a self-consistent approach based on the
many-body semi-empirical intermolecular potential pro-
posed originally in Ref. 20. It is shown that the pressure
behavior of the c / a-ratio in solid p-H2 and o-D2 in phase I
is determined mainly by the translational degrees of free-
dom and is comparable in magnitude with that for rare-gas
solids. The behavior of the axial ratio found in Ref. 13
is due to the presence of the J-odd component in normal
ortho-para mixtures of solid hydrogens [21].
The molecular-field rotational Hamiltonian has the
form [20,22,23]
( )2 2
rot rot 0 2 20 0= 4 / 5 / 2,cH B L U Y U− η+ε π + η (1)
where the first term in the right side is the rotor kinetic
energy rot(B is the characteristic rotational constant, L is
the angular momentum operator), 0U is the molecular
field constant, and LMY are spherical harmonics.
The orientational order parameter
20= 4 / 5 ,Yη π 〈 〉 (2)
where
/rot
/rot
Sp(...)e(...) =
Spe
H T
H T
−
−
〈 〉
denotes thermodynamic averaging with the Hamiltonian
Eq. (1). The rotational Hamiltonian Eq. (1) includes the
lattice-rotation coupling term
int 2 20= 4 / 5 ,cH Y−ε π (3)
where 2cε is the crystal field parameter [23]
2
1= ; = 6 ;
2c
dBB B B R
dR
⎛ ⎞ε δ − +⎜ ⎟
⎝ ⎠
(4)
and ( )B R is the radial function of the single-molecular
term in the anisotropic intermolecular potential.
1( ) = exp( )B B BB R R R−α +β + γ −
=6,8,10
( ) ,n
c n
n
f R b R−− ∑ (5)
The ground-state wave function can be written as a series
in spherical harmonics LMY with L even and = 0:M
0 0
=0
= , ( = 2 ).L L
L
c Y L n
∞
Ψ ∑ (6)
The expansion coefficients Lc obey the normalization
condition
2 =1.Lc∑ (7)
It can be shown that in order to account for terms up to the
fourth order in ,δ only three first harmonics should be
retained in the expansion Eq. (6). Then the rotational part
of the ground-state energy and the rotational order parame-
ter take the form
rot 2 2 2
0 2 4 0 2= 6 20 ( / 2 );cE c c U+ − η + ε η (8)
2 2
0 2 2 2 4 4
2 2 12 20= .
7 775 7 5
c c c c c cη + + + (9)
The order parameters η and δ can be found from minima
conditions of the total zero-point energy with respect to
these parameters. The isotropic part of the zero-point ener-
gy is
0E does not depend from η and respective minimum
conditions take the form:
rot
0 / = 0;E∂ ∂η (10a)
is rot
0 0( ) / = 0.E E∂ + ∂δ (10b)
Thus, the complete minimization can be carried out in two
stages, first, with the help of Eq. (10a) we find η as a
function of 0U and δ and then by minimizing the total
ground-state energy with respect to δ (Eq. (10b)) we find
δ and η as a function of 0U (pressure).
The minimum condition (10a) can be replaced by the
set of conditions
rot
0 / = 0, ( = 2 , = 0, 1, 2,...).LE c L n n∂ ∂ (11)
Equations (11) are necessary and sufficient conditions of
the satisfiability of Eq. (10a). Together with the normaliza-
tion condition Eq. (7) they are a complete set of equations
defining the ground-state rotational wave function.
We will seek the minimum condition Eq. (11) with the
constraint Eq. (7) using the Lagrange method, that is, we
will seek the minimum of the function
rot 2
0 0= (1 ),L
L
E E c+ λ −∑ (12)
Yu.A. Freiman, S.M. Tretyak, Alexander F. Goncharov, Ho-kwang Mao, and Russell J. Hemley
1304 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 12
where λ is the Lagrange factor to be found. The resulting
set of equations for the coefficients Lc ( = 2 ,L n
= 0,1,2,...)n has the form
( )0 2( 1) 2 = 0.L c L
L
L L c U L c
c
∂η
+ − η+ ε − λ
∂
(13)
For small 0U we will seek solutions of the set Eq. (13) in the
form of series in powers of the crystal field parameter 2cε or
in fact in powers of the lattice distortion parameter δ
2
=0 =0
= = .k k k
L Lk c Lk
k k
c q q B
∞ ∞
ε δ∑ ∑ (14)
Let us expand the combinations L Lc c ′ in power series
in 2 :cε
2
=0
= .n n
L L LL c
n
c c c
∞
′ ′ε∑ (15)
Then the orientational order parameter Eq. (9) can be also
presented in the form of the expansion in 2 :cε
2
=0
= ,n
n c
n
p
∞
η ε∑ (16)
where np can be expressed in terms of the coefficients of
the expansion Eq. (15):
( ) ( ) ( ) ( )
02 22 24 44
2 2 12 20= ...
7 775 7 5
n n n n
np c c c c+ + + + (17)
Expressing the Lagrange factor λ from Eq. (13) for 0c
and substituting it to the remaining equations we arrive at
the following set of algebraic equations
( ) ( )
2 2
=1 =0
= 0, = 1, 2,3...i k k i
n i n c c
i k
y r g n
∞ ∞⎛ ⎞
− ε ε⎜ ⎟⎜ ⎟
⎝ ⎠
∑ ∑ (18)
Here functions ( ) ;k
ng ,ir and ( )i
ny were introduced:
( ) ( ) ( ) ( ) ( )
1 00 02 22 04
2 4 2 12= ...,
75 5 7 5
k k k k kg c c c c+ − + +
( ) ( ) ( ) ( ) ( )
2 02 04 24 04
12 40 2 30= ...,
777 5 5 11 13
k k k k kg c c c c+ − + +
( ) ( ) ( ) ( )
3 04 06 26
30 28 2= ...,
5511 13 5
k k k kg c c c+ − + (19a)
for = 1,2,3,...k and
0 0 0
1 2 3
2= ; = 0; = 0;...
5
g g g (19b)
0 1= ,i i ir U p + δ (20)
( ikδ is the Kroneker symbol).
( )( )
0= 2 ( 1) ; ( = 2 , =1,2,3...).ii
n Ly L L c L n n+ (21)
Equations (18) are supplemented by the normalization
condition (7), which can be written as a set of equalities
( ) ( ) ( ) ( )
00 22 44 66 ..., =1,2,3,...i i i ic c c c i+ + + + (22)
Using the successive approximation method we can find
solutions of Eq. (18) in any necessary approximation. Up
to the third order in the crystal-field parameter the orienta-
tional order parameter and orientational ground-state ener-
gy have the following form:
2 4
3 2 3 3
2 2 2
15 15= (15 316) ;
14 2 49c c c
κ
η κε + κ ε + κ − ε
⋅
(23)
rot 2 3 3
0 2 2
1 75= .
2 14c cE − κε − κ ε (24)
where
0
1= .
15 U
κ
−
(25)
In Eqs. (23)–(25) rot
2 0, ,c Eε 0U are given in units of rot .B
The expansion parameter 2cε (Eqs. (4), (5)) is negative
for all pressures, so the expansions Eqs. (23), (24) are os-
cillating and converge if the terms of the expansions are
decreasing.
To find the total ground state energy we must include
into consideration the contribution of the translational de-
grees of freedom. As known, pure pair potentials are too
stiff to describe properly EOS at high pressures and many-
body corrections should be taken into account. In the cal-
culations we used a many-body potential having pair and
triple forces (Appendix, Eqs. (A.1), (A.2)).
Using the smallness of δ and expanding the total
ground state energy in powers of δ we may restrict our-
selves to terms of the second order in δ
2
0 0 1 2( ) = (0) ,E E b bδ + δ + δ (26)
where 0E is the ground-state energy of the ideal lattice,
( = 1,2)ib i are the coefficients depending on the parame-
ters of the intermolecular potential and molar volume. Mi-
nimizing 0 ( )E δ over ,δ we obtain
1 2= / (2 ).b bδ − (27)
As was shown in Ref. 1, the contributions of the first two
shells of neighbors to 1b are exactly equal to zero. The first
non-vanishing contribution comes from the two molecules
of the third shell. The contribution from the pair intermole-
cular potential ( )U R is
tr
1
=
2= ,
3 R c
dUb a
dR
⎛ ⎞
⎜ ⎟
⎝ ⎠
(28)
where a and c are the lattice parameters of hcp lattice.
The contribution of the rotational degrees of freedom to 1b
Molecular rotation in p-H2 and o-D2 in phase I under pressure
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 12 1305
as can be seen from Eq. (24) is also exactly zero. The con-
tribution of the first shell from the pair potential is
2 2
tr
2 2
= =
3= .
8 8R a R a
a dU a d Ub
dR dR
⎛ ⎞⎛ ⎞ + ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
(29)
The contribution of the rotational degrees of freedom to 2b
as follows from Eq. (24) is
rot 2
2 rot= ( / ).b B B−κ (30)
The total tot
2b is a sum of the respective contributions
tot tr rot
2 2 2= .b b b+ (31)
Due to the presence in the terms of the expansions Eqs.
(23), (24) of a singular factor κ (Eq. (25)), a validity of
the developed analytical solution is limited by the condi-
tion 0 < 15,U which corresponds to pressure of 30∼ GPa
for the case of solid p-H2. To extend the solution into the
higher pressure region a numerical analysis should be used.
Let us estimate obtained crystal field 2 .cε For
30 GPaP ≈ ( 4.6V ≈ cm3/mol) using parameters given in
Appendix we obtained
( )B R = 1.44 310−⋅ a.u.; B = 5.80 310−⋅ a.u.;
tr 5
1 = 9.658 10 ;b −⋅ tr 3
2 = 21.386 10 ;b −⋅ rot 3
2 = 6.55 10 .b −− ⋅
Finally we have
3= 3.26 10 ;−δ − ⋅ (32)
6 1
2 = 18.9 10 a.u. = 4.1 cm .c
− −ε − ⋅ − (33)
The two third of the total lattice distortion is due to the
translational degrees of freedom. Without regard for the
rotational degrees of freedom we would have tr =δ
tr tr 3
1 2= / (2 ) = 2.26 10 .b b −− − ⋅
Comparing with the results obtained by Loubeyre et al.
[13] we see that the lattice distortion parameter for J-even
species (p-H2, o-D2) is more than the order of magnitude
smaller than that for normal ortho-para mixtures. These
results are in accord with conclusions made by Goncharov
et al. in Ref. 24. The authors have measured low-frequency
Raman spectra at low temperature for the pressure range
up to the I–II phase transition. These spectra were em-
ployed to estimate the crystal-field parameters. The authors
estimated 2cε parameter 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| | 10 ,−δ ∼ more than the order of magnitude smaller
than experimental data by Loubeyre et al. [13].
In conclusion, the rotational order parameter, rotational
ground-state energy, and the lattice distortion parameter
for hcp lattice of phase I of p-H2, o-D2 are calculated using
the semi-empirical many-body intermolecular potential. It
is shown that the lattice distortion in these J-even species
is small compared with that found in n-H2, and n-D2 where
the main contribution to the lattice distortion is introduced
by the J-odd species.
Appendix
The pair potential:
2
=6,8,10
( ) = exp( ) ( ) n
p c n
n
U R R R f R C R−α −β − γ − ∑ (A.1)
(R is the interparticle distance). The damping function
2exp[ ( / 1) ], <
( ) =
1, >
c
R R R R
f R
R R
∗ ∗
∗
⎧ − −⎪
⎨
⎪⎩
(A.2)
where = ,mR DR∗ mR is the position of the potential min-
imum.
The three-body potential:
[ ]{ }3 3 3
tr 1 2 3 t 1 2 3= exp ( ) rU A a r r r C r r r− − + + + ×
( )1 2 31 3cos cos cos ,× + φ φ φ (A.3)
Potential parameters (in atomic units)
Parameters of the pair potential:
= 1.713α , = 1.5671β , = 0.00993.γ
Dispersion coefficients:
6 = 12.14C , 8 = 215.2C , 10 = 4813.9.C
Parameters of the damping function:
= 1.28D , = 6.44.mR
Parameters of the three-body potential:
tr = 6.085A , tr = 0.737858a ;
H2
tr = 49.49815C , D2
tr = 48.84650.C
Parameters of the function ( )B R (Eq. 5):
= 1.3252Bα − , = 1.9292Bβ − , = 3.8103.Bγ
6 = 1.368b , 8 = 50.53b , 10 = 1185.b
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