Determination of Potentials of Interaction between Rare
Gases and Multiply Charged Ions
Kupryazhkin A. Ya.,1 Nekrassov K. A.,1 Ryzhkov M. V.,2 Delley B.3
1
Department of Molecular Physics, Urals State Technical University, Yekaterinburg, Russia
2
Institute of Solid State Chemistry UrO RAN, Yekaterinburg, Russia
3
Paul Scherrer Institute, Zurich, Switzerland
Abstract. In this work we analyze experimental measurements and present quantum-chemical calculations of the
energies of interaction between rare gas atoms and multiply charged ions. The processing of the data concerning helium
diffusion and solubility in ionic crystals of the fluorite structure reveals that the solute helium atoms form with the double
charged cations of these crystals bonds, which are much stronger then the van der Waals attraction. The energies of the
coupling achieve -0.3 eV, which denotes the chemical binding of the atoms and the cations. This conclusion is confirmed
using the developed procedure of joint recovery of the several atom–ion interaction potentials. The recovered set of the
pair potentials U[He-F-], U[He-Ca2+] and U[He-Sr2+] simultaneously reproduces the energies of helium interstitial
solution and migration and of the helium solution in anionic vacancies in CaF2 and SrF2 crystals. The results of ab initio
Dmol calculations of a number of the pair atom – ion potentials are presented. The calculations also denote the possibility
of the chemical binding between neutral atoms and multiply charged ions.
1. INTRODUCTION
The absence of experimental data concerning rare gas atoms interactions with multiply charged ions, which is
due to the complexity of experimental studies and theoretical estimations, requires the development of new experimental and theoretical investigation methods for the determination of parameters of such interactions. The task of
obtaining this information is important for modeling the interactions of gases with ionic crystals, thin-film surfaces,
and surfaces of metals covered with oxide films. The potentials of interaction between neutral atoms and multiply
charged ions are necessary also to describe the exit of the radioactive gases from oxide fuel of nuclear reactors.
Using the example of helium - double charged ion systems, this work proposes an experimental technique for
this determination of interaction potentials of rare gas atoms with multiply charged ions from data on the diffusion
and solubility of the gases in ionic crystals, which is similar to [1]. The experimental data is compared with the
calculation of the pair atom – ion potentials with the quantum-chemical Dmol method [2, 3].
2. DETERMINATION OF THE EFFECTIVE GAS SOLUTION AND MIGRATION
ENERGIES IN CRYSTALS
The diffusion and solubility of the noble gases in ionic crystals were studied with the method of gas thermal
desorption from previously saturated samples (as an example see [4]). This technique involves saturation of the ionic
crystal samples with the investigated gas under required temperature T and pressure P, followed with abrupt cooling
of the saturated samples. The solubility C and the diffusion coefficient D of the gas are then determined from
kinetics of gas desorption into vacuum by processing of the experimental data. The temperature dependencies of the
parameters Ñ and D as a rule have the Arrenius form
{
}
S
C(T ) = C0 ⋅ exp − E eff
kT ,
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
(1)
{
}
M
D(T ) = D0 ⋅ exp − E eff
kT .
(2)
The gas solubility in a crystal is always small compared with concentration of the ideal lattice ions. This allows
one assume that the solute atoms do not interact with each other and form the ideal solution. Within this
approximation, it is not difficult to obtain the expression for the concentration of the atoms in the crystal at temperature T and the gas partial pressure P:
C( P, T ) = ∑ Ci* (T ) ⋅
i
{
}
Li ⋅ P ⋅ exp − EiS kT
{
1 + L i ⋅ P ⋅ exp − E iS kT
}.
(3)
Here quantities EiS are the energies of solution of the gas in the corresponding pre-existing position “i”, such as
vacancy or interstitial site. Factors Li account of the modification of the vibration spectrum of the crystal and the
linear component of the dependence of the solution energies on temperature.
The “partial” gas solubility Ci (P, T) is proportional to concentrations of the corresponding positions Ci* , which
can exponentially depend on temperature:
{
}
Ci* (T ) = C *i0 ⋅ exp − E *i kT ,
(4)
where Ei* is the effective energy of formation of the position.
The expression (3) describes the typical behavior of the solubility at low and high partial saturation pressures of
the gas. If the pressure is relatively small (does not exceed several atmospheres) then Li ⋅ P ⋅ exp{− E iS kT }<< 1 , and
{
}
{(
) }
Ci ( P, T ) = Ci* (T ) ⋅ Li ⋅ P ⋅ exp − EiS kT = Ci*0 ⋅ P ⋅ exp − Ei* + EiS kT .
(5)
So, at relatively low saturation pressures and one main solution mechanism the experimental effective solution energy EeffS is sum of the position formation energy Ei* and the energy of atom solution in the pre-existing position E iS .
At the sufficiently high pressures, almost all positions of the given type are filled with the gas atoms. This
corresponds to condition Li ⋅ P ⋅ exp{− E iS kT }>> 1 . If this condition holds then
{
}
Ci ( P, T ) = Ci* (T ) = C0 ⋅ exp − E *i kT .
Under the high saturation pressures, the effective solution energy EeffS
position formation E *i .
(6)
becomes equal to the effective energy of
3. HELIUM SOLUTION IN THE ANIONIC VACANCIES OF CAF2 , SRF2 AND PBF2
CRYSTALS
The crystals of the fluorite structure usually undergo intrinsic lattice disordering of the Frenkel type. At the high
temperatures CaF 2 , SrF2 and PbF2 crystals are featured with intensive formation of the interstitial anion – anionic
vacancy pairs. The approximation, in which these point defects behave as components of the ideal solution, gives
expression for the concentration of the vacancies:
CV* (T ) = CV* 0 ⋅ exp {− (E Fr 2 ) kT} .
(7)
Here EFr is the temperature independent part of the Frenkel pair formation energy. The effective anionic vacancy
formation energy Ei* is equal to EFr 2 .
In addition to the “intrinsic” vacancies, which are formed in the process of self disordering, ionic crystals contain
“extrinsic” vacancies to compensate the surplus effective charge of the impurities. At relatively low temperatures,
concentration of the “extrinsic” vacancies is higher than of the “intrinsic” defects. This concentration does not
depend on temperature. However, the vacancies can combine into complexes with the impurity ions of the opposite
effective charge. Increase in temperature leads to partial dissociation of the complexes, which become donors of the
vacancies. The concentration of the “extrinsic” vacancies released from the neutral complexes CV (T) within the ideal
solution approximation is given by formula
CV (T ) =

KV (T ) 
4⋅D
⋅  1+
− 1 .
2
KV (T ) 

(8)
Here
KV = K V 0 exp {− E Ds kT}
(9)
is the equilibrium constant of the quasi-chemical reaction of the dissociation, EDs is the energy required to release
the vacancy from the complex, D is the concentration of the complexes.
The equation (10) has two limits. The low temperature limit corresponds to condition KV << 4⋅D, which leads to
CV (T ) ≈ D ⋅ K V 0 ⋅ exp {− (E Ds 2) kT} .
(10)
If the released vacancies become positions of the solute atoms then at this low temperature limit Ei * = EDs /2.
Relatively high temperatures are those, at which KV >> 4⋅D. At this limit
CV ≈ D .
(11)
Therefore, at high temperatures, the complexes dissociate completely, and the concentration of the “extrinsic”
vacancies does not depend on temperature. This means, that if these “extrinsic” vacancies remain dominant positions
of the solute atoms then the effective low pressure gas solution energy is equal to EiS .
The elementary complexes contain one “extrinsic” vacancy coupled with one effectively charged impurity ion.
Large complexes may consist of several vacancies and admixed ions. Release of the vacancies from different
complexes may correspond to different temperature intervals.
Using the equations (3-11) it is possible to determine the gas solution energies in various positions within the
crystal from the dependence of the effective solubility (1) on temperature. The measured values of the helium
solution energies in the anionic vacancies of calcium, strontium and lead fluorides are quoted in Table 1.
TABLE 1. The Measured Energies of the Helium Vacancy Dissolution in the Fluorite Structure Crystals
Crystal
The Effective SoluThe Atom Solution
T, K
Type Of
Source
tion Energy ESeff, eV
Energy ESAV, eV
The Vacancy
CaF2
-0.5 ± 0.1
-0.5 ± 0.1
1000 ÷ 1245
a, E a
[4, 5]
CaF2
0.8 ± 0.1
-0.6 ± 0.1 b
930 ÷ 1363
a, I
[4, 5]
CaF2 + O2-0.46 ± 0.07
-0.46 ± 0.1
760 ÷ 920
a, E
[4]
CaF2 + O2-0.42 ± 0.07
-0.42 ± 0.1
760 ÷ 920
a, E
[4]
CaF2 + O2- + Gd3+
-0.5 ± 0.1
-0.5 ± 0.1
1100 ÷ 1245
a, E
[6]
CaF2 + Li+
-0.55 ± 0.10
-0.55 ± 0.10
750 ÷ 910
a, E
[7]
SrF2
-0.33 ± 0.03
-0.33 ± 0.03
714 ÷ 910
a, E
[4, 8]
SrF2
0.66 ± 0.04
-0.41 ± 0.1 b
910 ÷ 1245
a, I
[4, 8]
2SrF2 + O
-0.33 ± 0.03
-0.33 ± 0.03
714 ÷ 910
a, E
[4]
SrF2 + O20.66 ± 0.04
-0.41 ± 0.1 b
910 ÷ 1245
a, I
[4]
PbF2
-0.3 ± 0.1
-0.8 ± 0.1 c
>783
a, I
[9]
PbF2 + 0.5% HoF3
-0.2 ± 0.1
-0.7 ± 0.1 c
>714
a, I
[9]
a
The abbreviations (a, E) and (a, I) refer to the “extrinsic” and “intrinsic” anionic vacancies, correspondingly
b
The values are calculated with account of the measured Frenkel pair formation energies
c
The values are calculated with account of the effective vacancy formation energies determined from the ionic conductivity in
the region of the superionic transition [10]
4. THE INTERACTION OF HELIUM ATOMS AND DOUBLE CHARGED IONS IN
THE CRYSTALS OF THE FLUORITE STRUCTURE
Knowledge of the structure and the lattice periods of the crystals allows one to link the experimental solution and
migration energies with the interaction potentials of the solute atom and ions forming the crystal. Because the rare
gas atoms are neutral, a distinctive feature of the systems considered is that the atom interacts with ions via shortrange forces. This feature can be used to determine interaction potentials for the solute atoms and crystal ions. In this
work we have used the helium solution energies ES in anionic vacancies of CaF2 , SrF2 and PbF2 crystals to
determine the interaction potentials between the atoms and double charged cations.
The crystals of calcium, strontium and lead fluorides have the same fluorite structure. This structure includes two
sublattices. The face-centered sublattice of metal cations (Me2+) has the period a, which is equal to the lattice
constant. This sublattice is built into the simple cubic sublattice of fluorine anions (F-), which has the period a/2.
The cations occupy half of the interstitial positions of the anion sublattice. The first neighbors of the gas atom placed
into an anionic vacancy are 4 metal cations. The second neighbors are 6 fluorine ions. Separations between the atom
and these ions in the unrelaxed lattice are R[He-Me2+] = a√3/4 and R[He-F-] = a/2.
Within the approximation of pair interactions the change in the potential energy of the crystal resulted from
introduction of the atom into the vacancy can be calculated using equation
({ })
r
E = 4 ⋅ U He− Me (RC 0 + ∆RC ) + 6 ⋅ U He− F ( RA0 + ∆RA ) + E Def ∆R +
2+
−
(
)
r
r
r
3
U He− i Ri + ∆Ri − RHe + ⋅ h ⋅ ω0 . (12)
2
the rest ions
∑
Here, RÑ0 , RA0 are separations between the atom and the closest cations and anions, which take into account
relaxation of the ideal lattice around the vacancy, but do not include the additional relaxation caused by the atom. To
allow comparison of the calculated values of E with the experimental solution energies the separations should
correspond to zero absolute temperature. This is so because the measured effective values EeffS are temperatureindependent components of the solution energies. Further, ∆RC, ∆RA are increments of the separations resulting from
the lattice relaxation, EDef. is the energy of the lattice deformation due to placement of the atom into a pre -existing
vacancy. The fourth term is the sum of the atom interaction energies with all ions of 3rd and further coordination
spheres (which in this work were taken to be energies of the van der Waals attraction), and 3 2 ⋅ h ⋅ ω0 is the
estimation of the zero-point vibration energy.
Within the accuracy of the model, the calculated gas solution energies (12) should be equal to the measured
energies E SAV from Table 1. Therefore, the values of the potentials U[He-Me2+] can be found from the experimental
helium solution energies:
U He− Me
2+


−
(
)
r
r
r

3
U He−i Ri + ∆Ri − RHe − ⋅ h ⋅ ω0  . (13)

2
the rest ions

( RC 0 + ∆RC ) = 1 ⋅  E AVS − 6 ⋅ U He− F (R A0 + ∆R A ) − E Def. − ∑
4
An important contribution to the energy of the helium atom in the anionic vacancy site comes from its interaction
with the closest fluorine ions. The relaxation of the lattice around an unoccupied anionic site is such that these
anions become closer to the center of the vacancy than the closest cations Me2+. Within the corresponding range of
distances, the potential U[He-F-] can be accurately recovered from the data on helium interstitial solubility in a
series of crystals of fluorite structure [1, 11]. For the current work, this potential has been refined using an improved
recovery procedure (see the other our paper presented for this conference) and the helium interstitial solution energy
in lead fluoride. The refined potential within the range of separations R ∈ (2.4; 2.8) Å can be expressed in the
exponential form
U [ He − F − ] = 105 ⋅ exp {− 2 .895 ⋅ R} + 0.0035 eV, R ∈ (2.4; 2.8) Å.
(14)
Expression (13) depends upon unknown displacements of the ions around the solute atom. Nevertheless, it can
be used to estimate the depth of the He-Me2+ potential minimum. According to (14), the term 6⋅UHe-F-(RA0 +∆RA ) is
positive. The deformation energy EDef is equal to the change in the interaction energy of the lattice ions between
each other due to the relaxation. Because the relaxation results in displacement of the ions from the positions
corresponding to the minimum of the lattice energy, the value of EDef is positive. The zero-point vibration energy
3 ⋅ h ⋅ ω0 2 also is a positive defined component. The energy of van der Waals interaction between the atom and the
crystal lattice is negative. We have estimated this energy using the formula of Sleter and Kirkwood [12], and found
that it is about -0.05 eV. This contribution is small compared with the interaction of the atom and its closest
neighbors. The conclusion is that values of the He-Me2+ interaction potentials at the minimum can not be higher than
E SAV 4 .
The separations R0C and R0A differ from a√3/4 and a/2 because the anionic vacancy has positive effective charge.
Therefore, the cations move off the center of the vacancy, and anions approach the center. We have found R0C and
R0A for CaF2, SrF2 and PbF2 crystals using a lattice statics technique and the shell model [13]. The details of the
simulation have been described in [1]. This simulation enabled us to calculate the values U[He-Me2+], which
correspond to the solution of helium atoms in anionic vacancies of these crystals, disregarding additional relaxation
of the crystals caused by the atom (Table 2). At this stage, we could not consider the relaxation caused by the solute
atom, because it required recovery of the form of the He-Me2+ potentials.
TABLE 2. Experimental Interaction Energies for Helium and Doble Charged Ions
Crystal
ESAV, eV
RC0[He-Me 2+],
6· U[He-F-], eV
a· √ 3/4,
CaF2
-0.49 ± 0.05
2.36
2.53
0.53
SrF2
-0.33 ± 0.02
2.50
2.69
0.31
PbF2
-0.75 ± 0.1
2.55
2.74
0.27
UHe-Me++ (RC0), eV
-0.27
-0.16
-0.26
The results shown in Table 2 demonstrate that helium atoms dissolved in crystals of the fluorite structure form
relatively strong bounds with the double charged cations. This binding cannot be accounted for the van der Waals
interaction because the observed interaction is more intensive at least by an order. In addition, the polarization of the
atom should not be significant, because the neighborhood of the anionic vacancy is symmetrical. The energies
discussed by the value are congruent with the chemical interaction.
5. CALCULATION AND RECOVERY OF THE ATOM - ION INTERACTION
POTENTIALS
So far, the information on the potentials of interaction between neutral atoms and multiply charged ions could be
obtained only from the ab initio calculations. We have conducted calculation of the pair He-Ca2+, He-Sr2+ and HePb2+ potentials using the quantum-chemical Dmol method [2, 3]. It turned out, that the calculated potentials have
relatively deep minima close to -0.1 eV at the separations near RC0 . Although the minima of the calculated potentials
are not as deep as necessary to recover the experimental measurements discussed above, they cannot be accounted
for the van der Waals attraction. Therefore, it is possible to say that Dmol calculations do qualitatively support the
assumption that helium forms chemical bonds with multiply charged ions.
In order to find out how general the latter conclusion is and discover possible trends we calculated the pair
potentials of interaction between rare gas atoms and a number of other multiply charged ions in the region of the
potential well. In order to use these potentials in the computer simulation of the gas dissolution we have
approximated them with the “two exponents” function
U ( R )= ε ⋅ {exp [− 2 ⋅ β ⋅ (R − Rm )] − 2 ⋅ exp [− β ⋅ (R − Rm )]} ,
(15)
This analytical form describes all the calculated potentials within the given ranges of the interpaticle separations
to a very good accuracy. Its parameters are the absolute value of the potential at the minimum ε, the separation at the
minimum Rm and the steepness β. The parameters of the “two exponents” approximations are listed in Table 3. It is
seen, that attractive interaction between the neutral atoms and all the considered multiply charged cations is much
stronger then van der Waals interaction, binding energies reach tenth of electron-volt. Increase in the charge of the
cation involves strengthening of the interaction. The potential of interaction between helium atom and Y3+ atom,
which is isoelectronic to Sr2+, has the minimum of -0.31 eV. This minimum is deep enough to reproduce the
measurements discussed above. The calculations also predict significant attraction between rare gas atoms and
multiply charged negative ions (O2-).
We have developed a method of simultaneous recovery of a series of the interaction potentials for the solute
atom and the ions of several types from the data on gas solubility and diffusion in crystals. The algorithm of the
recovery is based on the condition that the relaxation of the crystal lattice around the atom results in minimization of
the solution energy. Therefore, at the equilibrium configuration the first derivatives of the energy (12) should be
zero. It implies that
[
]
4 ⋅ ∂ U He − Me2 + ∂ R = − ∂ E Def ∂ (∆RC ) − 3 2 ⋅ h ⋅ ∂ω 0 ∂(∆RC )
R= RC 0 + ∆RC
≡ 4 ⋅ F * ( RC 0 + ∆RC ) ,
(16)
where F* is the force acting on each of the closest cations in the direction of the solute atom in the relaxed
configuration from the crystal lattice.
TABLE 3. Parameters of the Calculated Potentials of Interaction between Rare Gas Atoms and Ions
Pair
Rm, Å
ε , eV
β , Å-1
He-O20.060
3.21
1.14
He-H0.009
3.73
1.13
He-F0.009
2.94
1.81
He-Cl0.00854
3.5008
1.6010
He-I0.0077
3.972
1.441
He-He
0.0075
2.513
2.14
He-Ne
0.0127
2.569
2.18
He-Ar
0.0116
3.053
1.94
He-Kr
0.0112
3.264
1.81
He-Xe
0.0099
3.53
1.73
He-Re
0.0097
3.63
1.68
He-Li+
0.0474
1.988
2.34
He-Cs+
0.020
3.186
1.85
He-Be2+
0.870
1.472
2.07
He-Cd2+
0.349
2.072
2.01
He-Ca2+
0.082
2.376
2.13
He-Pb2+
0.111
2.555
1.91
He-Sr2+
0.082
2.58
1.96
He-Ba2+
0.063
2.877
1.80
He-Al3+
1.46
1.70
1.78
He-Y3+
0.310
2.268
1.87
He-La3+
0.215
2.510
1.84
He-Gd3+
0.36
2.283
1.82
Ne-Ne
0.0187
2.660
2.32
Ar-O21.14
2.52
1.18
Ar-Cl0.0908
3.432
1.461
Ar-Br0.0854
3.579
1.431
Ar-Ar
0.0259
3.463
1.83
Kr-O21.10
2.42
1.30
Kr-Kr
0.027
3.82
1.64
Xe-O21.82
2.262
1.48
Xe-Xe
0.0534
4.21
1.35
Re-Re
0.032
4.39
1.47
∆ R, Å
1.50 ÷ 3.15
1.75 ÷ 4.00
1.50 ÷ 3.15
2.00 ÷ 4.00
2.30 ÷ 5.00
1.50 ÷ 3.15
1.50 ÷ 4.00
2.00 ÷ 3.65
2.00 ÷ 4.00
2.30 ÷ 4.00
2.30 ÷ 4.00
1.50 ÷ 3.15
2.00 ÷ 4.00
1.00 ÷ 2.85
1.25 ÷ 2.85
1.75 ÷ 3.15
1.75 ÷ 3.15
1.75 ÷ 3.15
1.75 ÷ 3.40
1.00 ÷ 2.85
1.50 ÷ 3.15
1.75 ÷ 3.15
1.50 ÷ 2.60
1.75 ÷ 4.00
1.20 ÷ 4.00
2.50 ÷ 5.00
2.50 ÷ 5.00
2.50 ÷ 5.00
1.50 ÷ 4.50
2.50 ÷ 4.50
1.50 ÷ 4.00
2.00 ÷ 5.00
2.80 ÷ 5.00
The general form of the condition (16) is
∂U ∂ R R = R + ∆R = F *
0
R= R0 +∆R
(17)
It can be shown, that the migration energy of the atom EM is equal to the minimum energy the crystal can have
provided that the atom is fixed in the saddle point. This means that condition (16) also applies at the recovery of the
interaction potentials from the migration energies. The forces F* corresponding to any given configuration of the
atom’s neighborhood can be calculated using computer simulation of the crystal.
0.700
U, ýÂ
CaF 2 “n“
-1
-2
-3
SrF 2 “n“
0.350
BaF 2 “n“
CaF2 “i “
SrF 2 “i “
BaF 2 “i“
0.000
-0.050
1.600
2.250
2.900
R+ ∆∆R, Å
FIGURE 1. Recovery of the interaction potential U[He-F-] at the absence the of helium atom – closest cations interaction at the
interstitial migration saddle points. In order to reproduce all the interstitial helium solution and migration energies in CaF2, SrF2
and BaF2 crystals the potential should pass through the centers of the circles 1. The slope of the diameters shows the required
derivatives of the potential at these points. We adjusted the potential in the form of integrated cubic spline of the derivatives.
Curve 2 is the best potential of this form which has monotone decreasing modulus of the derivative. Dotted lines 3 are the y = f(x)
curves corresponding to the interstitial (“i”) and saddle point (“n”) positions of the solute atom. To reproduce the solution and
migration energies the potential U[He-F-] must be tangent to all the dotted lines.
0.500
0.300
-1
-1
-2
U, ý Â
-2
U, ý Â
-3
-3
-4
-4
0.250
CaF2 “i”
0.000
SrF2 “i”
0.000
-0.180
CaF2 “n”
0.167
SrF2 “s”
CaF2 “s”
SrF2 “n”
-0.360
1.800
3.100
2.450
R, Å
(a)
-0.350
1.800
2.425
R, Å
3.050
(b)
2+
2+
FIGURE 2. The potentials U[He-Ca ] and U[He-Sr ] recovered with the regard for the large negative values of the helium
solution energies in the anionic vacancies. Curves 1 are the recovered potentials. Circles 2 and 3 indicate the points were the
potentials should pass to reproduce the solution and migration energies. The slope of the diameters shows the required derivatives
of the potentials at these points. We adjusted the potential in the form of integrated cubic spline of the derivatives. Curve 2 is the
best potential of this form which has monotone decreasing modulus of the derivative. Symbols “n” indicate the positions at the
saddle points corresponding to the interstitial migration energies. Symbols “s” and “i” indicate the positions at the anionic
vacancies and the interstitial sites. Dotted curves 4 are the corresponding Dmol potentials U U[He-Ca2+] and U[He-Sr2+].
The sought potential, which is intended to reproduce a particular gas solution or migration energy, must be
tangent to the curve y = f(x), which is specified by the following conditions: if x = R0 +∆R then y is equal to the right
hand side of the type (13) equation and dy/dx = F*(x). Since the method is described in detail in [9], here we go
straight on to the recovery of the interaction potentials for helium atom in crystals of the fluorite structure, which is
illustrated by Fig. 2, 3.
Let us consider recovery of the He - F- potential from the data on helium solubility and migration in the calcium,
strontium and barium fluorides. As it is seen from Fig. 2, the assumption of zero He – Me2+ interaction at the saddle
points makes it impossible to recover a smooth potential which could simultaneously reproduce all the considered
solution and migration energies. The potential, which reproduces the interstitial solution energies within the limits of
the experimental error, systematically overestimates the migration energy. This means that He – Me2+ interaction
energies at the saddle points of the interstitial migration should have negative values in the order of -(0.1 ÷ 0.2) eV.
This conclusion agrees with the results of the measurements of the energies of helium solution in the anionic
vacancies of CaF 2 , SrF2 and BaF2 crystals.
Fig. 3 shows the interaction potentials U[He-Ca2+] and U[He-Sr2+] recovered using the anionic vacancy solution
energies in conjunction with the potential U[He-F-]. In connection with insufficiency of the experimental data, we
have used a side condition to define the form of these potentials. The potentials were adjusted to coincide with the
Dmol calculation at the small distances. The corresponding potential U[He-F-] is close to the exponent (14).
6. CONCLUSIONS
This work has conducted a generalization of the experimental and calculation results concerning the interaction
of helium atoms and double charged ions at the gas dissolution in ionic crystals of the fluorite structure. It has been
shown, that in CaF 2, SrF 2 and PbF2 crystals the solute helium atoms intensively interact with the closest Ca2+, Sr2+
and Pb2+ cations forming bonds with the energy up to -0.3 eV. The analysis allows suggesting that in other ionic
crystals helium atoms would also form with multiply charged cations bonds that have energies of tenth of electron
volt. We have recovered the interaction potentials U[He-Ca2+] and U[He-Sr2+] in ionic crystals of the fluorite
structure. Together with the refined He-F- potential these potentials accurately reproduce the interstitial solution and
migration energies of helium in these crystals. This set of potentials does also reproduce the extremely large
negative energies of the helium solution in the anionic vacancies of CaF2 and SrF2 crystals. The similar potentials
are expected to describe the He-Ba2+ and He-Pb2+ interactions.
The results of the processing of the experimental data and of the potential recovery agree qualitatively with the
quantum-chemical calculations carried out with the Dmol method. In order to make a more general conclusion, we
have conducted Dmol calculations of pair interaction potentials for helium and a number of other positive and
negative ions. The calculated interaction energies show that for triple-charged cations, the energies of interaction
with helium and other rare gases have even larger negative values. The large negative interaction energies are
predicted also for the neutral atom – oxygen ion systems.
Thus, the processing of the experimental data and the calculation of the pair potentials carried out in this work
denote possibility that helium atoms, as well as the other rare gas atoms, form chemical bonds with multiply charged
ions when interact with ionic crystals.
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