Department of Molecular Physics, Urals State Technical University, Yekaterinburg, Russia
Characteristics of physical processes, such as gas - solid surface interactions, are determined by intermolecular forces. Thus, solubility and migration coefficients of gases in solids depend upon the energies of coupling between the gas atoms and their neighborhood. Since ab initio calculations of the interaction energies are complicated, there is a reason for development of techniques which infer characteristics of interaction of gas atoms and their environment in solids from experimental measurements.
It is known, that many properties of ionic crystals, including formation energies of point defects in the lattice, can be reproduced with models that allow only pair interactions between the ions that constitute these crystals. Present work studies the plausibility of description of interaction between neutrals (in particular, helium atoms) dissolved in ionic crystals and the closest ions in terms of pair potential functions U(R) and direct determination of these potentials from experimental values of the solution and migration energies E s
and E
M
,
A crystal lattice provides a variety of structural positions for a gas atom. Equilibrium concentrations of the atoms in the interstices ("i") and the vacant sites (V) of the lattice are dependent on the solution energies, E st
and E sv respectively, which are those required to move the atom from gas phase into a position of the specified type. The height of a potential barrier ("n"), separating two equivalent positions, is given by the migration energy E
M
,
The energies £^ and E
M
can be acquired from experiment in a relatively simple way and, consequently, with a high accuracy. For example, given that number and characteristics of the structural positions are independent of temperature T, the equilibrium concentration C(T) of the atoms, penetrating into a crystal from gas phase, in these positions is an exponential function kT
The same relation links the migration energy E
M
and the migration coefficient of the gas atoms in a crystal D:
The values E, which are equal to the energies of the neutral atoms in the equilibrium positions E s
or on the potential barriers E s
+ £?, can be used to express pair potentials of interaction between these atoms and the ions that form their first coordination sphere. Within the simplest model of an ionic crystal as a periodic system of rigid ions
^ - } ) , ( 3 )
CP585, Rarefied Gas Dynamics: 22 nd
International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
354

where R,AR are the coordinates of the atom "A" and all the ions "i" and the displacements that result from deformation of the crystal in the vicinity of the atom. E
Def
' is the crystal energy change due to this deformation.
Depending on symmetry of the crystal, the closest neighbors of the dissolved atom are n identical ions at the same separations R+AR. The value U(R+AR) of the pair potential function of interaction between the atom and each of these ions is given by expression
1
/
E— y u A
Being applied to the experimental data that corresponds to the same neutral-atom pair and a series of ionic crystals and atom positions, this analysis would give the potential U as a function of the separation R+AR within the appropriate range of distances.
The considered method of determination of interaction potentials is especially accurate when applied to the systems in which ion displacements during the relaxation caused by dissolved neutral atom are negligible. If energies of interaction of this atom with all the ions except the closest ions are also relatively small, then t / ( / 0 ~ - > (5) n which means, that potential is directly linked to the experimental data. For example, this approximation is valid for interstitial dissolution of helium atoms in crystals with the fluorite structure (section 3). However, it is always not sufficient at the potential barriers.
To be valid in general case, the method requires an independent calculation of correction terms, which include the ionic displacements, the deformation energies and also the energies of interaction between the atom and all the distant ions. To calculate the displacements {A/?^} and the energies E
De f • ({AR i
}) we have used the method of molecular statics (see, for example, (1)) with periodic boundary conditions. This means, that coordinates of the ions, corresponding to relaxation, were chosen such as to minimize unit potential energy of the infinite crystal containing aperiodic system of the extrinsic atoms. We surrounded the neutral atom with 300-500 ions, because increasing this number did not significantly affect the calculations.
The model of rigid ions, which has been used to obtain Equations (3-5), is not always sufficient for calculation of deformation energies, for it does not account for polarization of the ions. Thus we have used the shell model (2) for the rare-gas-configuration ions. In this model the ions are considered as being constituted of an outer spherical shell and a core consisting of the nucleus and the remaining electrons. In an electric field the shell retains its spherical charge distribution but moves as a whole with respect to the core. There is also a harmonic restoring force of spring constant &, which acts between the core and shell. The shell charges Qf and spring constants ki are parameters of the shell model, which determine polarizabilities and dipole moments of the ions and can be obtained from dielectric properties of the crystals. Since non-coulomb short-range potentials, such as U&j(R), in the shell model act between the shells, Equations (3-5) remain unchanged, but the coordinates in them correspond to the shells, and the energy of polarization is a part of the deformation energy E
Def
• (^f , R? ])• This simple shell model does not include multiparticle interactions, and we neglect them in this study.
We have applied the approach stated above to acquire the potentials of interaction between helium atoms and ions F, Cl" and Br~ from the data on helium ion mobility and solubility in CaF
2
, SrF
2
, BaF
2
, KC1, RbCl H KBr crystals (2-6). The fluorides CaF
2
' SrF
2
'and BaF
2
are cubic crystals with the same fluorite structure that is shown on
Figure 1. In this structure cations Me
2+
occupy half of the interstices of the simple cubic anion sublattice. The wide unoccupied interstices of this anion sublattice are suitable positions for helium atoms. We use the energies E^for helium atoms in these crystals and the interstitial migration activation energies E
Mt
in CaF
2
and SrF
2
(3) to obtain the potential U[He-F']. Crystals KC1, RbCl, and KBr have NaCl-structure (Figure 2). An interstitial position for a helium atom is in the center of each structural cubic cell. Migrating atom jumps through the face center of the cell.
The energies I? t
and E
Mt
for helium atoms in crystals KC1 and RbCl (4, 5) have been used to get the potential U[He-
Cl']. Recently, we have obtained an experimental value of the helium solution energy in cationic vacancies of KBr
355

crystal E sv
. We utilize this value together with known energies E st
and E
Mt
(6) for helium atoms in KBr to infer the potential U[He-Bf].
FIGURE 1. Helium atom in a fluorite crystal. Here 1 - Me
2+
ion, 2 - F" ion, 3 - helium atom, 4 - position on the potential barrier
FIGURE 2. Helium atom in a crystal of NaCl-structure. Here 1 - positive ion, 2 - negative ion, 3 - helium atom, 4 - position on the potential barrier
Interactions between the ions, which form the crystals, were modeled with empirical potentials and the shell model parameters obtained in the studies (7) and (8). In order to check and prove quality of the models of the crystals, we have calculated formation energies of point defects in these crystals. The relative errors of the calculated
Frenkel and anti-Frenkel defect formation energies with respect to best experimental values do not exceed 5 %, so we consider the models of the crystals to be reasonable for determination of the interaction potentials.
A significant contribution to the energies of the migrating atom at the potential barriers may come from its coupling with the closest positive ions. The data, which is studied in this work, does not allow determination of these contributions from the experiment. We have used exponential potentials
U(R) = A'Qxp{-B-R} (6) with parameters yielded from mobility of the cations in helium gas phase (9) to describe helium short-range repulsive interaction with the alkaline ions. The ranges of separations, where these potentials are reliable according to (9), are given in Table 1, along with the parameters. Pair potentials U[He-Me
2+
] we have calculated using the LCAO technique and approximated with the "two exponents" potential functions
- 2 • e - - " (7) in the range of separations (1. 8-^2.4)- 10~
10 m. Values of the parameters, which we have used, are listed in Table 1.
TABLE 1. The parameters for short-range repulsive potentials of interaction between helium atoms and cations ____
Ion A, eV B, l/(10'
10 m) ____ e, eV ____ |3, l/(10'
10 m) R m
, 10'
10 m range of separations, IP'
1
K
+
651
Rb
+
136
3.84
2.86
—
—
—
_
—
_
1.9-2.1
2.1-2.5
Ca
2+
—
Sr
2+
—
— 0.0828
— 0.0791
2.126
1.991
2.374
2.572
1.8-2.4
1.8-2.4
356

We have considered the relatively long-range dispersion attraction between the helium atom and all the ions, with exception of the closest neighbors, because of the quenching of this interaction with overlap of the electron shells.
We use the formula of Slater and Kirkwood (10)
, (8)
2
^Hel
N
He+^il
N i where a
0
= 0.5292- 10~
10 m, a
He
= 0.205 -lO'^m
3
, a/ are polarizabilities of the helium atom and the ions in the crystal, which we have calculated from the shell model parameters, N
He
= 2, N t
are effective numbers of electrons on the outer shells of the ions (see, for example, (11)), corresponding to the rare gases with the same shell structure. The parameters are listed in Table 2.
Ion
K
+
Rb
+
Ca
2+
Sr
2+
Ba
2+
F
Br
TABLE 2. The parameters of dispersion interaction between the ions and helium atom
a I<r
30
1.06
1.56
1.01
1.54
2.43
0.8
3.2
4.5
m
3
N
6.106
7.305
6.106
7.305
7.901
4.455
6.106
7.305
C
Disp
'[He-I]
4.62
6.42
4.48
6.37
8.95
3.5
9.9
13
Since the solution and migration energies have been determined under high temperatures, we have used high temperature lattice parameters calculated with empirical coefficients of thermal expansion (at 600 K for KC1, RbCl and KBr crystals and 1100 K for the fluorides).
The ion displacements that enter Equation 4 are in turn affected by the potential U(R), values of which are given by this relation. So, the displacements and the potentials were adjusted by iterations i
F A
& — 7 v
A
, | MV -i- zi /v, r
— i\
A
11 — ^ n ^1 r^; 11 (9)
The values U(R+AR), received on each step k, were approximated by smooth potential functions, Equations (6) or
(7), which were than used to improve the displacements. The final parameters of the potential functions and the values U(R+AR) are given in Tables 3 and 4. The relative deviations of the values from the smooth curves do not exceed 5 %. The parameters of the potential functions obtained may not be valid outside the ranges of the separations
R+AR corresponding to the crystals.
TABLE 3. The parameters for short-range repulsive potentials of interaction between helium atoms and anions_____
Ion A, eV B, l/(10'
10 m)____e, eV____p, l/(10'
10 m) R m
, 10'
10 m range of separations, 10'
10 m
F 65
8
2.64 —
7.91 —
— —
— —
1.8-2.8
2.9-3.05
Br — — 0.29
0.36 5.1
2.9-3.75
CaF
2
"n"
SrF
2
"n"
CaF
2
"i"
SrF
2
"i"
BaF
2
"i"
KC1 "n"
RbCl "n"
KC1 "i"
RbCl "i"
KBr "n"
KBr "i"
TABLE 4. The experimental data and the calculation results for the potentials U[He-F], U[He-O'] and U[He-Br']
R, m'
10
___________E/n, eV_________R+AR, I(r
10 m_______U(R+AR), eV
1.39
1.47
2.41
2.55
2.75
2.25
2.36
2.76
2.89
2.36
2.89
0.88010.005
0.63010.005
0.10610.005
0.06410.005
0.03510.005
0.3410.02
0.29510.01
0.07010.005
0.057510.003
0.31510.01
0.072510.003
1.80
1.88
2.45
2.58
2.785
2.89
2.93
2.97
3.03
2.90
3.17
0.64
0.38
0.11
0.070
0.040
0.088
0.069
0.040
0.030
0.16
0.023
357

The results confirm the assumption that interaction of a neutral atom and the closest neighbor ions in ionic crystals could be described in terms of pair potential functions and that these potentials can be inferred from the data on diffusion and solubility of the gases in ionic crystals. The main difficulties in the processing of the measurements are associated with use of the migration energies, because the atom-ion separations at the potential barriers and corresponding values of the potentials must be calculated with respect to relaxation of the lattice. Table 5 shows that absolute values of the displacements achieve 0.6 angstrom, and the relative displacements are 30% of the initial separations (before relaxation) The relaxation contributes up to 40 % of the atom's energy at the barrier. Interaction between the atom and its second neighbors (in this work these are positive ions) at the potential barriers also may be strong. So, inaccuracy of the presented method is determined by the quality of description of transition of the atom through the barrier. It is possible to use additional data to obtain the potentials of interaction of the atom with positive and negative ions simultaneously.
Contribution of the dispersion forces to the energies of helium atoms in ionic crystals are relatively small, but of the same order as values of the inferred potentials at the large distances, corresponding to the interstitial positions.
Since Formula (8) of Slater and Kirkwood is an approximation, this makes us unable to predict from the experimental data, which has been used, the separations and depths of "wells" for potentials U[He-F~] and
U[He-Cl~] with considerable accuracy.
The potentials of interaction between helium atoms and negative ions (at the least, U[He-F~] and U[He-Ct]) are universal, being valid for different crystals. Figure 3 shows the potentials acquired in this work in comparison with these resulting from measurements of mobility of the ions in the gas (9). The comparison yields a very good agreement for U[He-F] and does not show any fundamental disagreement for U[He-Br~] in the area of relatively high positive interaction energies covered in the work (9). The discrepancy observed in the latter case is, probably, due to the not quite correct description of the lattice relaxation at the potential barrier, which can be improved. It is difficult to compare the present gaseous and in-crystal potentials U[He-Ct], for these are known in separated ranges of distances.
However, the coincidences observed on Figure 3 prove, that the neutral-ion short- and intermediate-range potentials determined by the presented method can be used in a wide range of systems. This method has enabled us to extend (with respect to (9)) the ranges of energies and distances, at which the empirical potentials U[He-F~],
U[He-Cl~] and U[He-Bf] are known. Especially important extensions are those into the area of greater distances, as it has been difficult to establish an unambiguous relation between the gaseous transport coefficients and the potentials in that area. Importance of experimental methods is justified by the fact, that the gaseous and in-crystal potentials U[He-F~] at larger separations are much closer to each other than to the calculated potential.
The solution energy E sv
that we have measured for helium atoms in cationic vacancies of KBr crystal is extremely negative (-0.9 eV) and corresponds to the deep minimum found for the He-Br" potential. If this value is correct, it means, that helium atoms form chemical bonds in ionic crystals. This result requires additional experimental and theoretical investigations.
358

U , e V 0.8
0.6
0.4
0.2
0.0
1.5
2.0
2.5
3.0
3.5
r, lO'
1
'
4.0
m
FIGURE 3. Helium atom - halide anions interaction potentkls. Here l-s-3 are the values for the potentkls U[He-F~], U[He-Cl~] and U[He-Bf] respectively, obtained from Equation (4), 4^-6 are the corresponding potentkls from the study (9), and 7 are the approximations from Table 4. 8 is U[He-F~] calcukted with the electron-gas method (12). All the potentkls are shown within the ranges of their experimental determination.
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359