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ON THE PRESSURE INDUCED
ROTATION-LIBRATION TRANSITION IN MOLECULAR SOLID HYDROGEN
I. Aviram, S. Goshen, R. Thieberger
To cite this version:
I. Aviram, S. Goshen, R. Thieberger. ON THE PRESSURE INDUCED ROTATION-LIBRATION TRANSITION IN MOLECULAR SOLID HYDROGEN. Journal de Physique Colloques, 1984, 45 (C8), pp.C8-207-C8-209. �10.1051/jphyscol:1984838�. �jpa-00224339�
JOURNAL DE PHYSIQUE
Colloque C8, supplément au n ° l l , Tome 45, novembre 198* page C8-207
ON THE PRESSURE INDUCED ROTATION-LIBRATION TRANSITION IN MOLECULAR SOLID HYDROGEN
+ + +*
I. Aviram , S. Goshen and R. Thieberger
physics Department, N.R.C.N., P.O. Box 9001, Beer Sheva, Israel Physics Department, Ben Gurion University, Beer Sheva, Israel
Résumé - Nous avons calculé par une méthode varlationnelle la transition rotation-libration induite par pression dans les solides moléculaires para-H2, ortho-D2, para-T2. La fonction d'onde d'essai Comprend deux paramètres et les intégrales variationnelles sont évaluées par une mé- thode Monte-Carlo.
Abstract - The pressure induced rotation-libration transition in solid molecular para-H2, ortho-D2 and para-T2, is calculated by a variational procedure. The trial wave function incorporates two variational parameters and the variational integrals are evaluated by a Monte Carlo procedure.
Solid H2 with a molar fraction greater than 0.55 of ortho-H2 (J=l) has at temperatures below 3K a first order transition from an orientationally disor- dered hep phase to an orientationally ordered fee phase, as a result of electric quadrupole-quadrupole interactions. A similar situation is measured also in solid D2 for a high para-D2 (J=l) mixture. Cullen et al. showed that this transi- tion is of first order, also in the case when one ignores, the hep-fee transition.
The reason that only ortho-H2 molecules or para-D2 molecules take part is that the wave function of para-H2 and ortho-D2 are spherically symmetric CJ=0) an<i have no preferred direction.
For heavier di-atomic molecules, like N2, we know that the even-J type does also give a transition. The reason is that the ground state, wave function is a mixture of the even-J states. Hydrogen is the exception as its interaction is weak. It seems immediately reasonable to expect that at high pressures the inter- action will suffice to mix the J-terms, and therefore para-H2 and ortho-D2 are expected to show the same kind of transition.
Measurements performed by Silvera and Wijngaardent2} gave the value of 278 kbar for the D2 transition, and indicate that the transition for H2 is above 900 kbar.
The first calculations on this problem were done by Raich and Etters (3).
using a mean field approach. These calculated pressure values at the transition point were substantially lower than the measurements.
In the calculations which we wish to describe here, it is assumed, as in the previous calculations, that the translational and orientational coordinates of the molecules may be decoupled. This leads to the consideration only of the rotational part of the Hamiltonian, the molecular centers being fixed at the static lattice
sites.
The procedure is described in detail elsewhereW ; here we wish to outline briefly the main features of the procedure. The Hamiltonian for N molecules is of the form:
N 2 1
H = -B Z V + 4 Z'V. • 1=1 Mi 2 ij ^
where^3) B = 59.34 cm"1 for H2, 29.91 cm"1 for D2 and 20.67 cm"1 for T2 • u.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984838
C8-208 JOURNAL DE PHYSIQUE
i s t h e u n i t v e c t o r s p e c i f y i n g t h e o r i e n t a t i o n of molecule i with r e s p e c t t o t h e l a b o r a t o r y frame of r e f e r e n c e , and V . i s t h e a n i s o t r o p i c p a r t o f t h e two-body i n t e r a c t i o n p o t e n t i a l o f t h e form: I'
V. . = Cm('ij)Y2m($)Y2-m(~j) m
where g i i s t h e o r i e n t a t i o n o f molecule i w i t h r e s p e c t t o t h e i - j i n t e r - molecular a x i s , and r i j = Rij/Ro i s t h e d i s t a n c e between molecular c e n t e r s i n u n i t s of t h e nearest-neighbour d l s t a n c e a t zero p r e s s u r e . The c o e f f i c i e n t s Cm(r) were determined from t h e Ree-Bender A n attempt was a l s o made t o i n c o r p o r a t e i n t o Vi. a Y k O term but no s i g n i f i c a n t changes i n t h e r e s u l t s were observed, so t h i s at$empt was abandoned.
The amount of o r i e n t a t i o n a l o r d e r o f t h e molecules i s measured by t h e o r d e r parameter
1 2
<Y20S = ~f 1 J, (~l?l"..,~)Y20(~i!i)d~l...d~
1
where J, i s t h e N-particle wave f u n c t i o n , and ui s p e c i f i e s t h e o r i e n t a t i o n of molecule i r e l a t i v e t o i t s own symmetry a x i s [one o f t h e f o u r t h r e e - f o l d axes of t h e f c c s t r u c t u r e ) . The unnormalized wave f u n c t i o n i s assumed t o be of Jastrow type,
N N
J , ( g 1 , . - . , ~ I = @(fiil ilf(gj,&)
i=l j<k
where
@ ( 9 = YOO($ + y Y z o ( 9
2 2
exp[hcos S.cos Ek] f o r j , k n e a r e s t neighbour,
f'4.41 =
{
J1 otherwise
i s t h e p o l a r angle of t h e d i r e c t i o n ui with r e s p e c t t o t h e symmetry a x i s of molecule i . The energy i n t e g r a l s a r e evaluated by a Monte Carlo sampling procedure on a system of N = 108 r o d l i k e , non-polar "molecules" whose c e n t e r s a r e l o c a t e d a t t h e s i t e s of t h e f c c l a t t i c e .
The s e a r c h f o r t h e t r a n s i t i o n d e n s i t y was c a r r i e d o u t by mapping t h e t o t a l energy a s a f u n c t i o n o f t h e v a r i a t i o n a l parameters h,y, and of t h e d e n s i t y , and by looking f o r a jump i n t h e l o c a t i o n of t h e minimum.
To determine t h e t r a n s i t i o n p r e s s u r e t h e use of an equation o f s t a t e (EOS) i s c a l l e d f o r . We f i r s t used t h e EOS proposed by S i l v e r a and oldm man(^)(^^) and l a t e r t h e EOS r e c e n t l y discussed by Ross, Ree and (denoted by Y R ) . The r e s u l t s a r e given i n t h e t a b l e ; we s e e t h a t t h e YR equation b r i n g s t h e c a l c u l a t e d r e s u l t s c l o s e r t o experiment. Because o f t h e steepness o f t h e EOS function, t h e r e s u l t s a r e very s e n s i t i v e t o t h e EOS chosen. I t would be u s e f u l t o have more p r e c i s e EOS i n o r d e r t o b e t t e r a s s e s s our c a l c u l a t i o n s . The t a b l e a l s o shows t h e c a l c u l a t e d values of T2. Because o f t h e lower t r a n s i t i o n p r e s s u r e , t h e
d i f f e r e n c e between t h e two EOS i s small, a s expected. I t would t h e r e f o r e be u s e f u l t o have an experimental r e s u l t f o r T2 which would check our c a l c u l a t i o n s
i r r e s p e c t i v e o f t h e EOS chosen.
Table : The t r a n s i t i o n parameters.
Ro T r a n s i t i o n Volume T r a n s i t i o n P r e s s u r e (Kbar)
Isotope (in) cm /mol 3 ~ x ~ e r i m e n t ' ~ ) - - SG YR
References
1. J.R. Cullen, D. Mukamel, S. Shtrikman, L.C. L e v i t t and E . C a l l e n , S o l i d S t a t e S o l i d S t a t e Commun. 2, 195 (1972)
2. I . F . S i l v e r a and R . J . Wijngaarden, Phys. Rev. L e t t . 47, 39 (1981) 3. J . C . Raich and R.D. E t t e r s , J . Low Temp. Phys. 2, 22F(1972)
4. I . Aviyam, S. Goshen and R. Thieberger, J . Low Temp. Phys. - 52, 397 (1983);
55, 349 (1984); J . Chem. Phys. 80, 5337 (1984)
5. F.K Ree and C.F. Bender, J . ~ h e m r ~ h y s , 2, 5362 (1979) 6 . I . F . S i l v e r a and V.V. Goldman, J . Chem. Phys. 69, 4209 (1978) 7. M. Ross, F.H. Ree and D.A. Young, J . Chem. Phys. 2, 1487 (1983)
