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THEORETICAL DETERMINATION OF THE
CATIONIC DIFFUSION PATH IN THE ALKALI
HALIDE CRYSTALS
P. Guerin, A. Laforgue
To cite this version:
JOURNAL DE PHYSIQUE Colloque C7, supplement au n° 12, Tome 37, Decembre 1976, page C7-379
THEORETICAL DETERMINATION OF THE CATIONIC DIFFUSION PATH
IN THE ALKALI HALIDE CRYSTALS
P. GUERIN and A. LAFORGUE
Laboratoire de Mecanique Ondulatoire Appliquee, U. E. R. Sciences Exactes et Naturelles, B. P. 347, 51062 Reims Cedex, France
Résumé. — La voie de diffusion cationique est étudiée dans les halogénures alcalins dans le cadre de deux approximations négligeant le phénomène de polarisation.
L'approximation du réseau rigide met en évidence pour tous les cristaux considérés, à l'exception des fluorures, deux voies de diffusion indirectes. Pour les fluorures on retrouve le trajet direct habi-tuellement envisagé.
L'approximation du réseau relaxé, dans laquelle une partie du cristal peut se déformer, a été utilisée pour NaCl. Nous retrouvons les deux voies de diffusion précédemment observées. De plus, l'énergie de migration obtenue est de l'ordre de grandeur des mesures expérimentales.
Abstract. — The cationic diffusion path for the alkali halides is investigated by means of two
approximations neglecting polarization phenomena.
The rigid lattice approximation shows for all the crystals, except fluorides, two indirect diffusion paths. For the fluorides we obtain the usually accepted direct diffusion path.
The relaxed lattice approximation, in which the ions contained in a part of the crystal are allowed to be displaced, is used in the case of NaCl. We obtain the two indirect diffusion paths previously observed. The calculated and the experimental values of the migration energy are within the same order of magnitude.
1. Introduction. — The aim of this work is the deter-mination of the process governing the ionic jump from an occupied site to a vacancy, in an alkali halide crystal such as NaCl.
The migration path has never been determined rigou-rously. The straight path has never been contested except in 1957 [1] and in 1973 [2]. But none of these works answer undoubtly.
We have investigated this migration process within the framework of the theory of rate process. We have calculated, from an adiabatic point of view, the crystal energy as a function of the migrating ion position. The crystal energy has been calculated according two approximations :
— The rigid lattice approximation. — This approxi-mation has been developped in a previous paper [3]. We have improved the calculation of the electrostatic potential by adding a 12th degree term to the polyno-mial expansion. The coefficient of this term is evaluated by the use of the boundary conditions.
— The relaxed lattice approximation. — The ions contained in an arbitrary region of the crystal, contain-ing the defect, are allowed to be displaced. For the calculation of the repulsive energy we make use of the Born-Mayer-Verwey interaction.
2. Definition of the planes intersecting the
equipo-tential surfaces. — The straight path is represented
by the Ou axis (Fig. 1). Two symmetry planes of the crystal contain the Ou axis :
FIG. 1. — Definition of planes and axis.
—• (Oy, Ou) contains two ofthenearest^neighbours of the migrating ion at its initial site (10 1 plane).
— (Oz, Ox) contains the other four nearest neigh-bours (10 1 plane).
The migrating process is also symmetrical with respect to the plane (Cv, Cy) (1 0 1 plane).
These three planes are mutually orthogonal and the equipotential lines in these planes offer a satisfying illustration of the equipotential surfaces.
The distance between the anion and the cation is taken as a unit of length. The equipotential lines have been drawn starting from a network of potential energy values with an 0.02 unit step. The equidistance in energy of these lines is 0.1 eV.
3. Results within the rigid lattice approximation for the cationic migration in NaCl. - The distribution of the equipotential lines in the (Ox, Oz) plane (Fig. 2) shows that the M and M' points, which represent the initial and final equilibrium position of the migrating cation, are not located at the cristallographic sites.The lenght MM' is equal to 8/10 of the distance between the corresponding cristallographic sites. The C point is a saddle point located in energy at 2.07 eV above the M and M' points.
-
The set of equipotential lines in the (Oy, Ou)plane (Fig. 3) displays two saddle points (J and J') symmetrical with respect to C. Between these two points stands up a secondary potential barrier (0.45 eV) which top is at C.
So there are two separate paths in an adiabatic process. For each of these paths there is a diffusion complex at the corresponding saddle point (J or J'). These two paths are symmetrical with respect to the Ou axis.
Within this approximation (rigid lattice), the hypo- thesis of an indirect migration path is verified.
FIG. 2.
-
Cationic migration in NaCl within the rigid lattice approximation. Equipotential lines in the plane (0 1 0). The equidistance in energy of the lines is 0.1 eV, a is the lattice parameter.FIG. 3.
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Cationic migration in NaCl within the rigid lattice approximation. Equipotential .lines in the plane .(I 33).THEORETICAL DETERMINATION OF THE CATIONIC DIFFUSION PATH C7-381
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On the figure 4, it appears that the J and J' points define the only two possibilities for the migrating cation and correspond to an indirect path.FIG. 4. - Cationic migration in NaCl within the rigid lattice approximation. Equipotential lines in the plane (1 0 1). The equidistance in energy of the lines is 0.1 eV. a is the lattice para-
meter.
The height of the potential barrier associated with the indirect path is the difference between the height of the main (MCM') and secondary (JCJ') barrier. So for NaCl we obtain a value of 1.61 eV for the migration energy via the indirect path.
These results show clearly the occurence of an indi-
contains 80 ions which, by taking the symmetry into account, are described by means of 67 parameters (when the migrating cation is at C, it is sufficient to consider 35 parameters only). Figure 5 shows the main potential barrier starting from the energy minima as zero. For the relaxed lattice approximation, the mini- mum has been obtained with the region 1 centered at the vacancy with a radius value of 3 units (123 ions, 12 parameters).
FIG. 5.
-
Cationic migration in NaCl. Comparison between the main barriers in the rigid and in the relaxed lattice approxima- tion (-- -
rigid lattice approximation, - relaxed latticeapproximation).
rect diffusion path by means of two separate paths.
Notice that when taking the ionic relaxation into This conclusion can directly be obtained by calculating
account, the initial site is closer to the crystallographic the potential energy variation along the Cy axis. It
site and the height of the main barrier is lowered to a be very to if the same value of 1.02 We have neglected the vibrational would be obtained from a more sophisticated model.
energy, 4. Results within the relaxed lattice approxima-
tion for the cationic migration in NaCI. - Within this approximation the crystal is divided in two regions :
-
Region 1. - In this region containing the vacancy and the migrating cation, the ions can be displaced by the perturbation due to the migrating cation.4.2 COMPARISON OF THE SECONDARY POTENTIAL BARRIER FOR THE RIGID AND THE RELAXED LATTICE.
-
We use the same region 1 and the 80 ions are defined by 66 parameters. The two secondary barriers have been drawn on figure 6. The indirect path is confirmed within the relaxed lattice approximation but the J and J' points are closer to the C point. The height of the secondary barrier is lowered from 0.4 eV to 0.06 eV.-
Region 2.-
It is the hole crystal except the 4.3 CONCLUSIONS.-
The use of the relaxed lattice region 1. In this region we apply the rigid lattice approximation confirms the indirect path for cationicapproximation. migration in NaC1, although the secondary barrier
The crystal energy is evaluated us. the ionic displa- height should be considerably lowered. Although cements in region 1. This energy is minimized with polarisation phenomena have not been taken into account, we have obtained a migration energy value of respect to these displacements.
The minimisation process is run with Box's the order of magnitude of the experimental method [5]. The minimisation process stops when the determination.
C7-382 P. GUBRIN AND A. LAFORGUE
I""
FIG. 6. - Cationic migration in NaCI. Comparison between the secondary barriers in the rigid and in the relaxed lattice approximation (-
-
-
rigid lattice approximation, - relaxedapproximation).
5. Comparison between the alkali halides in the rigid lattice approximation. - In the case of NaF
we obtain a direct migration path (along the Ou axis) (Fig. 7). For KF, RbF, we obtain similar pictures for the equipotential lines in the plane (1 0
f).
This direct path is observed only for the fluorides.Figure 8 shows the equipotential lines in the plane (1 0
7)
for RbI. These lines are very similar to those obtained for NaCl. The saddle points (J and J') are a little more separated than in NaCl. But especially the secondary barrier is higher (1.96 eV) than in NaCl (0.46 eV). Even after the relaxation process, we can hope that the two indirect paths will be well distin- guishable. By using a plotter we have drawn the corresponding pictures for the set of the alkali halides. The obtained results are summarized in table I. Within the rigid lattice approximation, the heights of the potential barrier are quantitatively not significant. We must only consider the evolution of this parameter. We observe the following laws :- The cationic migration energy increases with the cation radius for the direct as well as for the indirect path.
-
The width of the bottom of the main barrier (MCM') increases both with the anion and cation radius.- The half-height width of the main potential barrier decreases with the increase of the cation radius.
FIG. 7. - Cationic migration in NaF. Equipotential lines in the plane (1 0
1).
The equidistance in energy of the lines is 0.1 eV.THEORETICAL DETERMINATION OF THE CATIONIC DIFFUSION PATH C7-383
Comparison of the main and secondary potential barrier for the alkali halides in the rigid lattice approximation
Height of the main potential barrier (direct path MCM') in eV
Width of the bottom of the main barrier (MM') in
A
Height of the secondary potentiel barrier (along JCJ') in eV. Na K Rb F 0.524 3 0.453 8 0.477 7 C1 0.397 4 0.356 0 0.279 3 Br 0.422 7 0.373 1 0.292 3 I 0.549 3 0.399 7 0.311 5
Half-height width of the main barrier in
A
From this table we notice that the most significant so that the two indirect diffusion paths become less series is the iodide series. From RbI to NaI (Fig. 9) separated. KT (Fig. 10) has an intermediate behaviour
the main and secondary potential barriers are lowered, between the two other iodides.
FIG. 9. - Cationic migration in NaI. Equipotential lines in the plane (1 01). The equidistance in energy of the lines is 0.1 eV.
C7-384 P. GUERIN AND A. LAFORGUE
6. Conclusions. - The plotting of the potential Between these two cases, we would have an inter- surfaces governing the cationic migration seems to mediate case where there are two distinct but not
imply the following results : independant paths.
For the fluorides the path is direct (along the Ou The resulting phenomena are evidently of a quantum axis). On the contrary for a compound such as RbI we nature. This phenomena arising from the existence of a have an alternative between two independant diffusion double potential valley seems to imply a new type of paths, separated by a secondary potential barrier. quantum effect.
References
[I] LAUREN?., J. F., B~NARD, J., J. Phys. & Chem. Solids 3 (1957) 7.
[2] RAMDAS, S., SHUKLA, A. K., RAO, C. N. R., Phys. Rev. B 8 (1973) 2975.
[3] G U ~ ~ R I N , P., LAFORGUE, A., J. Phy~ique Colloq. 34 (1973)
C 9-185.
[4] DE WETTE, F. W., NIJBOER, B. R. A., Physica 24 (1958) 1105.
/5J RICHARDSON, J. A., KUESTER, J. L., Comm. of the A. C. M. 16 (1973) 487.
DISCUSSION F. BBNI~RE.
-
HOW do you justify the neglect ofthe polarization energy in your calculations of the migration barrier ?
A. LAFORGUE. - The variation of the polarisation energy along the ionic migration process seems to be negligible. Within a macroscopic model, the ion moves, in an homogeneous and continuous medium without change of the values of charge and size. Consequently, we suppose that the polarisation energy of each ion varies but that these individual variations cancel each other in a first approximation.
C. R. A. CATLOW.
-
I believe, that your conclusion that indirect diffusion paths occur for cation diffusion in some alkali halides, is incorrect. Our own calcula- tions (see Catlow et al., this conference), which were based on a more extensive minimisation procedure and which used more sohphisticated lattice potentials, suggested that the direct path is operative in all alkali halide crystals. It is difficult to identify definitively the source of the difference between our results and those you have just reported. But I would suggest, that the omission of ionic polarisation from your model is a serious deficiency ; and that this may, in part, be res- ponsible for unadequate features of your results.A. LAFORGUE.
-
HOW could you be sure that your results imply the direct path ? We think, that the plott- ing of the equipotential surfaces is necessary for the determination of the migration path. Evidently, ourcalculation must be improved and a quantum model should replace the classical one. This quantum model will, with out supplementary assumption, include the polarisation phenomenon.
M. 5. NORGETT.
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Your results show that lattice relaxation greatly reduces the energy difference bet- ween the symmetric and asymmetric saddle points for cation vacancy migration. Because you have not used a proper boundary relaxation, your lattice is still constrained.This probably explains why your activation energies are large compared with experiment. I think that if the relaxation was extended, you would find, as we do, that the symmetric direct saddle point has the lower energy.
