Joint Determination of Several Interaction Potentials for
Gas-Ion Pairs from Measurements of the Gas Diffusion and
Solubility in Ionic Crystals
Nekrassov K. A. and Kupryazhkin A. Ya.
Department of Molecular Physics, Urals State Technical University, Yekaterinburg, Russia
Abstract. The work proposes a procedure for the joint recovery of several neutral atom – ion interaction potentials from
the data on diffusion and solubility of the gases in ionic crystals. The procedure is featured with flexible determination of
the form of the potentials achieved by the direct use of the equilibrium forces acting on the closest neighbors of the solute
atom from the crystal in the relaxed configuration. The pair interaction potentials U[He-Cl-] and U[He-K+] are recovered.
These potentials at small distances coincide with the potentials measured from the ion transport coefficients in the gas
phase and extend the latter potentials into the region of larger separations and small interaction energies.
1. INTRODUCTION
Microscopic description of rare gas – crystal surface interactions requires knowledge of interaction potentials for
gas atom – ion pairs. Since capabilities of existing methods of recovery of the potentials are limited, there is a reason
to develop new independent techniques.
In earlier work [1] we have proposed a method of recovery of “static” ion – neutral atom potentials based upon
the solution and migration energies of gases in ionic crystals. The treatment used in [1] allowed determination of
only one unknown potential provided that all the other interactions for the solute atom and ions, which formed
considered crystals, were negligible or known. However, within an ionic crystal gas atom interacts with ions of at
least two types (positive and negative ions) and the both interaction potentials may be known not quite well. In
contrast to [1], this work presents an improved version of the method, which allows simultaneous determination of
the pair potentials for the solute gas atom and ions of several types.
2. CALCULATION OF THE GAS SOLUTION AND MIGRATION ENERGIES IN
IONIC CRYSTALS
Similarly to [1], we use the ionic model of the crystals and the approximation of pair interactions between the
electron shells of the solute atom and the ions. This approximation means that we neglect the influence of the
environment on the internal structure of the particles. Numerous works demonstrate, that the fully ionic approach
with pair interactions allows quantitative calculation of the cohesive energies and the intrinsic defect formation
energies of ionic crystals (see [2-4] as an example). Since the rare gas atoms have the same closed shell structure as
the ions forming ionic crystals, it is possible to use the pair potential approximation to model the solution and
migration of these gases in ionic crystals. Consequently, it is also possible to recover the potentials from the
diffusion and solubility measurements.
At low concentrations the atoms dissolved in crystals do not substantially interact with each other and form the
ideal solution. The solution energy corresponding to a particular type of positions of the atoms in the lattice ES can
be calculated as
r
r
r
r
(1)
E S = ∑ U Ai Ri + ∆Ri − R A + E Def . ∆Ri + 3 ⋅ h ⋅ ω 0 2 .
(
)
({ })
i
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
r
r
Here, Ri are the coordinates of the ions in the vicinity of an empty position, R A are the coordinates of the atom
r
placed into the position, ∆Ri are the displacements of the ions resulting from the relaxation of the lattice around the
atom. UAi(R) are the interaction potentials for all the atom – ion pairs. The deformation energy of the lattice
r
EDef({ ∆Ri }) is calculated as the total change in the energy of interaction of the ions with each other after the
relaxation. Unlike [1], in this work we consider the zero-point oscillations energy of the atom, which is estimated by
the term 3 ⋅ h ⋅ ω 0 2 . The cyclical frequency of the oscillations is calculated as
ω0 =
(∂
2
)
E A ∂x 2 m A ,
(2)
where ∂ 2 E A ∂x 2 is the second-order partial derivative of the potential energy of the atom by one of the coordinates,
mA is mass of the atom.
It can be shown, that the migration energy of a solute atom is determined by the minimal change in potential
energy of the crystal when the atom moves onto the potential barrier separating two equilibrium positions. The
migration energy is calculated as
(
)
({ })
r
r
r
r
E M = ∑ U Ai Ri + ∆Ri − R A + E Def . ∆Ri − E S ,
(3)
i
where ES is the solution energy, and the other terms give the change in the energy when the atom is introduced into
the crystal and placed on the potential barrier. The notation has the same meaning as in (1). The displacements take
account of the relaxation of the crystal to minimize the energy.
Since the effective solutionand migration energies measured in experiments are the temperature independent
components of the full energies, the calculations (1-3) should correspond to zero absolute temperature.
3. THE ORDER OF RECOVERY OF THE INTERACTION POTENTIALS
Let us consider the experimental solution and migration energies of a particular gas, corresponding to a series of
ionic crystals and/or different positions of the atoms. To reproduce all the measured energies, a set of the pair
interaction potentials should satisfy the corresponding system of equations (1, 3), which has general form
E k = n A,k ⋅ U A,k (R A, k + ∆R A,k ) + nC ,k ⋅ U C ,k (RC ,k + ∆RC ,k ) + ∆E k .
(4)
Here the values of the index k correspond to the particular gas – crystal systems characterized with the known
solution and migration energies. E k are the measured quantities which are equal to ES (from (1)) or ES+EM (from
(3)). UA,k(R) and UC,k(R) are the atom – anion and atom – cation interaction potentials in the system k, RA,k, RC,k,
∆RA,k, ∆RC,k are the atom – closest anions, atom – closest cations separations “before” the relaxation and the changes
in these separations “after” the relaxation, nA,k, nC,k are numbers of the closest anions and cations. The last term ∆Ek
includes all the other terms from (1) or (3). These terms cannot be independently determined from the data
considered in this work. We calculate them using the shell model [5] and the lattice statics approach.
In [1], we recovered single potentials of interaction between the solute helium atom and its closest neighbors
using the system of equations analogous to (4) and considering all the other interactions to be negligible or known.
Since the displacements of the ions and the deformation energies are determined by the used version of the potential
recovered, we applied an iterative procedure. During this procedure, the displacements and the deformation energies
were calculated with the current approximation of the potential and then used to refine it. The calculations were
based on the computer simulation of the crystal at zero temperature in the shell model approximation. The potentials
were recovered as simple analytical functions, such as the Buckingham potential. The parameters of these functions
were adjusted to reproduce the solution and migration energies within the model.
The simultaneous utilization of the gas solution and migration energies allows recovery of the interaction
potentials in a wide range of separations of the order of 1 Å. In such a range it is not always possible to approximate
the interaction with the same analytical potential of one of the conventional forms. So, in this work we abandon the
use of the iterative procedure and the analytical forms. To control the form of the potentials on the adjustment stage
we directly use the condition that the relaxed configuration of a crystal with the solute atom at zero temperature
corresponds to the potential energy minimum:
∂E k ∂∆R A,k = n A,k ⋅ ∂U A,k (R ) ∂R + ∂∆E k ∂∆R A,k = 0, at R = R A,k + ∆R A,k
∂E k ∂∆RC ,k = nC ,k ⋅ ∂U C ,k (R ) ∂R + ∂∆E k ∂∆RC ,k = 0, at R = RC ,k + ∆RC , k
(5)
This means, that for each of the simultaneously recovered potentials Ui(R) in all the considered crystals and
positions in the relaxed configurations
dU i (R ) dR = (− ∂∆E k ∂∆R ) ni ,k = Fi*,k (R + ∆R ) ,
(6)
where Fik* is the force acting on the ion of type i in the direction of the atom at the relaxed configuration of the
position or crystal k from the other ions of the crystal. The relations (6) form a system of equations, which contains
one equation for each of the recovered potentials in each of the considered crystals and atom positions. The
potentials should be adjusted such as to match the systems (4) and (6) simultaneously, which provides high
unambiguity of the recovered set.
The explicit formulation of the conditions (6) has allowed us to use the equilibrium separations (Ri,k+∆Ri,k) as the
parameters varied in the procedure of determination of the best set of the potentials. Assigning a set of displacements {∆Ri,k} for all the closest to the atom i-type ions in all the positions and the crystals, one can calculate the
corresponding forces {Fik*} acting on the displaced ions from the rest ions of the crystals. As in [1], we used the
computer simulation of the crystals in the simple shell model approximation to conduct the necessary calculations.
Since in the different positions the values of Fik* correspond to the different equilibrium atom – ion separations, it is
possible to carry out the numeric integration and find the interaction potential
U i (R ) =
R
R
dU i
dR
U
Fi* (R ) ⋅ dR + U i 0 ,
+
=
i0
∫
∫
dR
R min
R min
(7)
where Rmin is the minimum equilibrium atom – ion separation in the current set {Ri,k+∆Ri,k}, Ui0 is the arbitrary
constant of the integration. To make the integration (7), we approximated the calculated sets {Fik*} by various
analytical functions and by the cubic splines.
The potentials (7) can be used to calculate the solution and migration energies according to the system (4).
Quality of the examined set of the displacements is determined by the difference of the calculated and the experimental energies (4) when the best arbitrary constants Ui0 are used. As the objective function to be minimized we
used the sum of squares of the differences between the calculated and the measured solution and migration energies:
(
F0 = ∑ E iCalc − E iExp
)
2
=
k
R A , k + ∆R A , k
RC , k + ∆RC , k


= ∑  n A,k ⋅ ∫ FA* (R ) ⋅ dR + n A, k ⋅ U 0 A + nC ,k ⋅ ∫ FC* (R ) ⋅ dR + nC ,k ⋅ U 0C + ∆E k − E kExp 


k 
R min[ A ]
R min[ C ]

(8)
Here the indexes “A” and “C” denote the interactions of the atom and the closest anions and cations, respectively, as
in (4). If there is m potentials to be recovered then differentiation of the objective function by all the constants U0i
and setting the derivatives equal to zero leads to the system of m equations
∑n
k
i,k
R A , k + ∆R A , k
RC , k + ∆RC , k


*

⋅ n A,k ⋅ ∫ FA (R ) ⋅ dR + n A,k ⋅ U 0 A + nC ,k ⋅ ∫ FC* (R ) ⋅ dR + nC ,k ⋅ U 0C + ∆E k − E kExp  = 0, i = 1K m


R min[ A ]
R min[ C ]


(9)
To increase the controllability of the recovery procedure we did not solve the system (9) and fixed the constants
U0i using additional variable parameters σI, namely the separations of “zeros” of the potentials (Ui(σi) = 0). Then
U 0i = −
σi
∫ F (R ) ⋅ dR .
*
(10)
i
R min[i ]
The described order of recovery of the potentials is illustrated on Fig. 1. Provided that all the other potentials are
fixed, the given potential must be tangent to the series of yk = f(xk) curves, defined for all the crystals and the atom
positions in the following way. If the unknown potential is for the atom–anion pair then for any xk ≡ Ri,k+∆Ri,k
y k (x k ) =
1
n A, k
(
)
⋅ E kExp − nC , k ⋅ U C ,k (RC ,k + ∆RC ,k ) − ∆E k (x k ) .
(11)
These curves for the recovered He-Cl- potential are shown on Fig. 1 as the dotted lines.
4. SIMULTANEOUS RECOVERY OF THE POTENTIALS U[HE-CL-] AND U[HE-K+]
Implementation of the joint determination of the potentials and the modification of the recovery procedure have
enabled us to improve determination of the potential U[He-Cl-] with respect to [1], recovering it simultaneously
U[He-K+] with the U[He-K+] potential.
0.200
U, eV
KCl “n”
-1
-2
RbCl “n“
-3
0.100
KCl “i“
0.000
RbCl “i“
-0.050
2.400
2.700
R+∆R, Å
3.000
FIGURE 1. Recovery of the interaction potential U[He-Cl-]. In order to reproduce all the interstitial helium
solution and migration energies in KCl and RBCl crystals, the potential should pass through the centers of the
circles. The slope of the diameters shows the required derivatives of the potential at these points. The potential was
adjusted 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 (11) 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-Cl-] must be tangent to all the dotted lines. The deviation of the RbCl “i” point from the potential
is probably due to inaccuracy of the used He-Rb+ potential, which had not been recovered in this work.
0.400
U, eV
-1
-2
-3
0.200
0.200
KCl “n“
RbCl “n“
KCl “i“
0.000
-0.050
2.200
RbCl “i“
3.000
2.600
R+∆R, Å
FIGURE 2. The recovered U[He-Cl-] in comparison with the potential recovered from the mobility of the ions in helium gas
phase. Here 1 are points, at which the potential should pass to reproduce the experimental solution and migration energies, 2 is
the smooth approximation of the potential, 3 is the potential determined from the ions mobility in helium gas phase [10], “n”
indicates a saddle point position, “i” indicates an interstitial position.
0.300
-1
U, eV
-2
-3
0.150
KCl “n“
KBr “n“
KCl “i“
KI “n“
0.000
-0.050
2.000
KBr “i“
KI “i“
2.550
R+∆R, Å
3.100
FIGURE 3. The recovered U[He-K+] in comparison with the potential recovered from the mobility of the ions in helium gas
phase. Here 1 are points, at which the potential should pass to reproduce the experimental solution and migration energies, 2 is
the smooth approximation of the potential, 3 is the potential determined from the ions mobility in helium gas phase [10], “n”
indicates a saddle point position, “i” indicates an interstitial position.
To acquire the potentials we used the data on helium interstitial solubility and diffusion in KCl, RbCl, KBr and
KI crystals [6-9]. These crystals have the same NaCl structure. The closest neighbors of the atom placed into the
interstitial position are 4 anions and 4 cations at the separations a⋅√3/4, where a are the periods of the crystals. The
saddle points of the interstitial migration for all the crystals were taken to be at the center of the face of the cubic cell
formed by the 4 anions and 4 cations closest to the interstitial atom.
Interionic interactions in the crystals were modeled with empirical potentials and the shell model parameters
obtained in [4]. Short-range potentials for He-Rb+, He-Br- and He-I- pairs could not be determined from the data
used in this work, so these interactions were taken from measurements of mobility of these ions in helium gas phase
[10]. It was supposed that the ions of the 3rd and the further coordination spheres interact with the atom by the
dispersion forces, which were calculated using the formula of Slater and Kirkwood [11]. The simultaneously
recovered potentials U[He-Cl-] and U[He-K+] are shown on Fig. 2, 3.
5. CONCLUSION
The work presents a method of simultaneous recovery of a set of neutral atom – closed shell ions interaction
potentials from the data on diffusion and solubility of the gas in ionic crystals. The simultaneous recovery is
important because in the saddle points the migrating atoms intensively interact with the anions and the cations at the
same time. This is also true for the interstitial positions in crystals of the NaCl structure. The flexibility and
controllability of the recovery procedure proposed are provided by the direct use of the equilibrium condition (6) in
determination of the forms of the potentials. The recovered potentials U[He-K+] and U[He-Cl-] reproduce almost all
the considered helium solution and migration energies and coincide with the potentials recovered from the gaseous
transport coefficients. The deviation of the calculated interstitial solution energy in RbCl from the measured value is
probably caused by the use of the U[He-Rb+] potential which has been recovered from the mobility of the ions in the
gas for the range of the separations (2.1 ÷ 2.5) Å [10]. In this range the potential U[He-Rb+] has the exponential
form U = 136⋅exp(-2.86⋅R). To describe He-Rb+ interaction in the interstitial position we had to extrapolate this
exponent into the region of greater separations and this most probably led to overestimation of the interaction
energies. This conclusion follows from the comparison of the potentials U[He-Rb+] and U[He-K+]. The latter
potential, which has been recovered in this work, at the separations R ≥ 2.5 Å is negative and cannot be
approximated with an exponent. The method proposed in this work does, in principle, allow refinement of the
potentials such as U[He-Rb+] at the medium separations using additional experimental data, because it is not
constrained by the ambiguous relation between the gaseous transport coefficients and the potentials when the
interaction energies have the low absolute values.
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