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

Freiman, Yu.A.

Goncharov, A.F.

Hemley, R.J.

Mao, H.

Tretyak, S.M.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

© 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 2rot 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/rotSp(...)e(...) =SpeH TH 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]  21= ; = 6 ;2cdBB B B RdR⎛ ⎞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( ) ,nc nnf 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 LLc 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 20 2 4 0 2= 6 20 ( / 2 );cE c c U+ − η + ε η  (8)  2 20 2 2 2 4 42 2 12 20= .7 775 7 5c 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 is0E  does not depend from η  and respective minimum conditions take the form:   rot0 / = 0;E∂ ∂η  (10a)  is rot0 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   rot0 / = 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 20 0= (1 ),LLE 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 LLL L c U L cc∂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 kL Lk c Lkk kc q q B∞ ∞ε δ∑ ∑   (14) Let us expand the combinations L Lc c ′  in power series in 2 :cε   2=0= .n nL L LL cnc 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= ,nn cnp∞η ε∑  (16) where np  can be expressed in terms of the coefficients of the expansion Eq. (15):   ( ) ( ) ( ) ( )02 22 24 442 2 12 20= ...7 775 7 5n n n nnp 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 in i n c ci ky r g n∞ ∞⎛ ⎞− ε ε⎜ ⎟⎜ ⎟⎝ ⎠∑ ∑  (18) Here functions ( ) ;kng  ,ir  and ( )iny  were introduced:  ( ) ( ) ( ) ( ) ( )1 00 02 22 042 4 2 12= ...,75 5 7 5k k k k kg c c c c+ − + +    ( ) ( ) ( ) ( ) ( )2 02 04 24 0412 40 2 30= ...,777 5 5 11 13k k k k kg c c c c+ − + +   ( ) ( ) ( ) ( )3 04 06 2630 28 2= ...,5511 13 5k k k kg c c c+ − +  (19a) for = 1,2,3,...k  and  0 0 01 2 32= ; = 0; = 0;...5g g g  (19b)  0 1= ,i i ir U p + δ  (20) ( ikδ  is the Kroneker symbol).  ( )( ) 0= 2 ( 1) ; ( = 2 , =1,2,3...).iin 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 43 2 3 32 2 215 15= (15 316) ;14 2 49c c cκη κε + κ ε + κ − ε⋅  (23)  rot 2 3 30 2 21 75= .2 14c cE − κε − κ ε  (24) where   01= .15 Uκ −  (25) In Eqs. (23)–(25) rot2 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   20 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   tr1=2= ,3 R cdUb adR⎛ ⎞⎜ ⎟⎝ ⎠  (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 2tr2 2= =3= .8 8R a R aa dU a d UbdR dR⎛ ⎞⎛ ⎞ + ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ (29) The contribution of the rotational degrees of freedom to 2b  as follows from Eq. (24) is  rot 22 rot= ( / ).b B B−κ   (30) The total tot2b  is a sum of the respective contributions   tot tr rot2 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 51 = 9.658 10 ;b−⋅  tr 32 = 21.386 10 ;b −⋅  rot 32 = 6.55 10 .b −− ⋅  Finally we have  3= 3.26 10 ;−δ − ⋅  (32)  6 12 = 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 31 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( ) ( ) np c nnU R R R f R C R−α −β − γ − ∑  (A.1) (R  is the interparticle distance). The damping function  2exp[ ( / 1) ], <( ) =1, >cR R R Rf RR R∗ ∗∗⎧ − −⎪⎨⎪⎩ (A.2) where = ,mR DR∗  mR  is the position of the potential min-imum. The three-body potential:  [ ]{ }3 3 3tr 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 ;    H2tr = 49.49815C ,    D2tr = 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   1. Yu.A. Freiman, S.M. Tretyak, A. Grechnev, A.F. Goncharov, J.S. Tse, D. Errandonea, H.-k. Mao, and R.J. Hemley, Phys. Rev. B80, 094112 (2009).  2. A. Grechnev, S.M. Tretyak, and Yu.A. Freiman, Fiz. Nizk. 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Molecular rotation in p-H₂ and o-D₂ in phase I under pressure

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Molecular rotation in p-H₂ and o-D₂ in phase I under pressure

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)