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COHERENT POTENTIAL APPROXIMATION AND STRONGLY ANHARMONIC SYSTEMS : CRYSTALS OF PARA-HYDROGEN AND NEON RESP. HYDROGEN ATOMS

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COHERENT POTENTIAL APPROXIMATION AND

STRONGLY ANHARMONIC SYSTEMS : CRYSTALS

OF PARA-HYDROGEN AND NEON RESP.

HYDROGEN ATOMS

K. Menn, W. Biem

To cite this version:

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JOURNAL DE PHYSIQUE

CoZZoque 66, suppZ6ment au n '1 2, Tome 42, de'cembre 1981 page C6-87

COHERENT POTENTIAL APPROXIMATION AND STRONGLY ANHARMONIC SYSTEMS : CRYSTALS OF PARA-HYDROGEN AND NEON RESP, HYDROGEN ATOMS

K. Menn and W. Biem

I n s t i t u t e of TkeoreticaZ Physics, University of Giessen, 0-6300 Giessen, F . R. G.

Abstract.- For mixtures of p-hydrogen and neon in the solid state phonon calculations have been performed by combining the CPA and the self consistent phonon approximation ( S C P ) to include the strong anharmonicity of the system. The structure changes and the phonon spectrum have been calculated as function of the con- centration. The method has also been applied to mixtures of p-Hz and hydrogen atoms.

Under the solids which are known to be quantum crystals there are several systems of two or more kinds of nolecules statistically mixed on the lattice positions, e.g. o-HZ, p-HZ mixtures, HZ-D2, Hz-Ne mix- tures. These systems are strcngly anharmonic because of changing force constants as function of the nean distance of the molecular pairs and of the large amplitudes of the zero-point motion. To calculate their phonon resp. roton spectra the theories of statistically disordered crystals which are harmonic have to be generalized to take into ac- count strong anharmonicities.

The single site coherent potential approximation (CPA) is well known to be the best working method to calculate phonon spectra of dis- ordered crystals abstracting from its weakness in describing pair and clustering effects. We have combined the CPA and the self-consistent phonon approximation (SCP) to get a first insight into the dynamics of anharnonic ana disturbed crystals of two compounds.

One of the simplest disturbed systems seems to us to be the p-hy- drogen neon system since the pair potentials are roughly equal and the mass ratio is 10 ("isotopical disorder"). The large nass ratio produces a big aifference in the zero point motion. Therefore, the structure in the neighbour'nood of one guest molecule in a host crystal is strongly changeu and with it the force constants. iie have sinplified the prob- lem to a certain extent to recluce the numerical difficulties: the stracture has been face centered cubic for all concentrations though high hyarogen concentrations are stable in the hexagonal closed packed structure. We have use6 always the regular cubic point symnetry for every lattice site occupied by a guest molecule. We have not considered

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C6-88 JOURNAL DE PHYSIQUE

11 r i o d e l vllh 4 rclaxinp aclghbovr shclls 6 1 1 : s y s t e m ,

2 1 m o d e l w i t h nearest nelgllbour rclaxrtlon only 2: concontration,

3 1 E I x l = CX~~-I~,~~~V~~I; 3: ncan lairlcc c o n s t a n t as function of C ,

4 1 f < x l = ~ ~ [ ~ ~ -air ~+ ~a*rz>"2 ~ 4 ~ 1 ~4 L R l c l ~ ~ ~ 1 ~ ~ ~ ~ + 5 1 p r e l i m i n a r y r e a v 1 t s onlyl

refined quantum effects (no quantum aiffusion). This may turn out to be too much simplification: There nay occur deviations from statis- tical mixing through correlations

-

static or time dependent. But the problem remains complicated enough an6 the numerical problems grow considerably with the nodel of the neighourhood of the guest molecule. We have started with a model where only the nearest neighbours could relax. This model turned out to be to rigid: the global lattice con- stant changed too nuch, the frequencies changed partly in a wrong di- rection. Therefore, we introduced another model with re1axations.u~ to the fourth neighbour shell. All static and dynamic displacements were restricted not to violate the cubic synnetry.

Since hydrogen forms quantum crystals, the short range correla- tions have to be taken into account. wethave done this with Jastrow- factors of an appropriate kind.

The whole calculation turned out to be a voluminous one to be solved self consistently with several variational parameters: the lat- tice constant a, the static displacenents from the lattice points of molecules in four shells in the ne'ighbourhood of a guest molecule A R i j

the width parameters y . of the displacement-displacement correlation

11

function, and the parameters in the Jastrowfactor. The iteratTon pro- cedure turned out to be very difficile and special tricks had to be used to reach convergence. Calculations were performed mostly on the iJe-(p-H2) system, a few on p-B2-(Ne) and on p-H2-(H atoms). The table contains the static structural changes, the figures show the spectra of the crystals, sone Jastrowfactors and the conparison of the mean lattice constant from which follows, that the nodel with relaxation of four shells is necessary to describe the local displacement field of the point defect formed by a guest molecule.

1 1 51

rdil+lillC 5 5 . 1 9 2 7 0.0111 0.0166

I

6

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