Novel rapid technique for the simulation of thermal vibration figures and diffusion routes in ionic solids

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Novel rapid technique for the simulation of thermal

vibration figures and diffusion routes in ionic solids

M. Dempsey, R. Freer

To cite this version:

Novel rapid technique for the simulation of thermal vibration figures

and diffusion routes in ionic solids

M. J. Dempsey (*) and R. Freer (**)

(*) Department of Geology, the University of Manchester, Manchester M13 9PL, U.K.

(**I Grant Institute of Geology, the University of Edinburgh, West Mains Road, Edinburgh EH9 3JW, U.K.

Rbsumb. - Les mkthodes habituelles de calcul des paramktres de defaut et de diffusion dans des solides ioniques simples necessitent des installations d'ordinateur importantes. Une nouvelle f a ~ o n d'aborder le probleme, basee sur les relations entre longueur des liaisons et resistance des liaisons a l'inttrieur de polyhtdres de coordination metal-oxygene individuels est developpte et produit des donnkes qualitatives rapides sur l'agitation thermique des ions. En construisant un rkseau a 3 dimensions autour de la position d'equilibre de I'ion a etudier, il est pos- sible de calculer 1'Cnergie (N) necessaire cet ion pour se deplacer vers chaque point du rCseau tandis que les autres ions sont fixes. Des contours correspondant

a

N = Cte peuvent &re tracks pour n'importe quelle direc- tion. Pour des petits dkplacements, les contours correspondent de f a ~ o n satisfaisante aux ellipsoides de vibration thermique dCterminCes expkrimentalement, pour des distances plus importantes, les graphes montrent des chemins de diffusion possibles. Le modele est testk dans plusieurs cas avec des donnees pour des oxides et silicates sClec- tionnes.

Abstract. - Convential procedures for the calculation of defect and diffusion parameters in simple ionic solids require extensive computational facilities. A novel approach based on bond length-bond strength relations within individual metal-oxygen coordination polyhedra has been developed which provides rapid qualitative information about the thermal motion of ions. By constructing a three dimensional grid around the equilibrium position of the ion of interest it is possible to calculate the energy (N) required to displace the ion to each grid point whilst the others are fixed. Contours of constant N may be plotted for any arbitrary orientation. At small displacements the contours correlate well with experimentally-determined thermal vibration ellipsoids, and at larger distances the plots suggest possible diffusion routes. The model is tested in several situations with data for selected oxides and silicates.

I . Introduction.

-

The transport properties of simple solids, particularly oxides, halides a n d silica- tes a r e not only of theoretical interest, but also of industrial a n d geological importance. Procedures for the direct measurement of self- and impurity- diffusion rates, mainly employing radiotracers, are well established, and considerable advances in theoretical approaches have been made in recent years. Although we have not yet reached the stage where experimentally-determined and predicted dif- fusion parameters are in perfect agreement, there exist several powerful techniques for the theoretical analysis of diffusion. The HADES computer pro- gram, for example, has been successfully used t o calculate defect energies in a variety of oxides and halides [ I , 21. Similarly, Monte Carlo procedures have been employed t o examine various problems associated with diffusion, such as correlation fac- tors, the role of non-stoichiometry, and diffusion routes [3, 41. Unfortunately, such investigations require extensive computational facilities which limits their availability. In contrast, the present

work describes a simple model which enables limi- ted qualitative information about the thermal motion of ions in any crystal structure t o be obtain- ed rapidly with the minimum of input data a n d computation. The model has been specifically applied t o thermal vibration figures a n d the analysis o f diffusion routes in ionic solids.

2. Theory. - Computer modelling of crystal structures is a relatively new technique which allows structural data for real and hypothetical phases t o be generated quickly and easily. Meier and Villiger

[S] introduced a particularly successful program called DLS (Distance - Least Squares). Their method is based o n the fact that the number of crystallogra- phically independent interatomic distances in most structures exceeds the number of variables (atomic coordinates and unit cell parameters), so that the latter may be determined by prescribing a sufficient number of the former. In this type of procedure the function N, given by equation (I), is minimized by

C6-258 M. J . DEMPSEY AND R. FREER adjusting the unit cell parameters and atomic coor-

dinates

Here

D,

are the prescribed interatomic distances,

Bi

the calculated interatomic distances, and w i weight

assigned to the ith interatomic distance. The sum is over the rn inequivalent distances in the structure. Dempsey and Strens [6] generalized the weighting scheme by using the Brown and Shannon [7] bond length-bond strength relation. The weight w (in valence units) associated with a cation-oxygen dis- tance R was defined to be

where so is the strength (in v.u.) of a bond of length

Ro,

and n is constant for a particular bond type. It is interesting to note for this expression that n

-

NB - 1, where NB is the Born exponent repre- senting interatomic repulsion. Brown and Wu [8] simplified equation (2) by taking so = 1 and Ro = K , (the bond length for unit valence), hence

They also gave values of ( R , , n ' ) for over sixty elements in various oxidation states, and so (3) was used to provide weighting factors for the present study.

3. The Model and its Applications. - Once a crystal structure has been defined, the value of N (given by (I)), for a specified polyhedron, can be investi- gated as the central cation is displaced from its equilibrium position. This is achieved most conve- niently by constructing a three dimensional grid arond the cation of interest and then calculating the value of N (sum-of-squares) as the cation is displac- ed to each grid point, the remaining ions being kept fixed. The values of N do in fact represent a potential energy as it may be seen that each term in (1) is proportional to the harmonic term in the clas- sical potential energy function. Consequently, when contours of constant N at small displacements from the equilibrium position are plotted there is a high degree of correspondence with the observed easy directions of movement, namely the thermal vibra- tion ellipsoids. Figure 1, for example, shows sec- tions through the M2 site of the chain silicate diop- side (CaMgSi206). The highly anisotropic plots of constant N are in excellent agreement with the observed thermal vibration figures [9]. Good agree- ment is also obtained for the MI site of diopside

and a variety of sites in other structures. Although this simple procedure, involving only the calculation of N at a series of grid points, appears to be capa-

ble of generating thermal vibration figures, it is still limited by the problem of scaling. Essentially, it is impossible to predict accurately the value of N which will coincide with the surface of the thermal vibration figure.

Fig. 1. - Comparison of calculated (solid) and observed (dash- ed) vibration figures for the Ca ion in the M2 site of diopside [9] for sections in planes (Fig. a) and g-b_ (Fig. b). Approxi- mate coincidence of figures occurs when N = 10-3-10-4.

Ohashi and Finger [lo] employed a Born-type potential energy expression to determine the orienta- tion and magnitude of thermal vibration ellipsoids. The agreement with the observed data was satisfac- tory, but their method involved considerably more computation than the present procedure.

As the function described by (1) is able to provide equipotential surfaces which are close to thermal vibration figures at small displacements from the equilibrium position, it is of interest to examine how the function behaves at larger displacements. This will take the method to its limit as (3) is only

accurate for coordination numbers in the range 2 to 12 approximately. As an example, figure 2 shows a plot for Mg in MgO (001 plane). Predictably, the N

Fig. 2.

-

Sum-of-squares plot for Mg in MgO (001 plane). The cation site is denoted by ( + ) and the neighbouring oxygen by

values rise rapidly as the cation is displaced towards the oxygen, but there exist channels in the directions leading to the nearest neighbour cation sites. Taking a section from the equilibrium position t o an adja- cent cation site in the same plane yields a characte- ristic migration energy plot with a saddle point. This would be equivalent to the direct diffusion route for NaCI-type structures.

A detailed study of diffusion routes in alkali hali- des with the NaCl structure (2) indicated that the differences in barrier height for cation migration by the direct route, or the alternative indirect route, were in some cases exceptionally small. Attempts to compare the two routes in MgO with the present model would be inappropriate because of its gross simplicity. However, when profiles for a series of cations in MgO (direct route) are examined, the ordering of the data (Fig. 3) is broadly in agreement with measured activation energies [I I]. The smallest

Fig. 3. - Diagonal section through sum-of-squares plots (001 plane) for a series of ions in the MgO structure. The horizontal axis represents the direct route for diffusion in MgO. All ions are divalent except Ge (4 +).

cation, Be, has the snnallebc saddle point value, and conversely Ca the largest for the group shown. Sr and Ba yield even sharper profiles wjth saddle point values in excess of 100. Most of the remaining diva- lent ions are clustered between Be and Ca. The pro- file for the only quadrivalent ion, Ge, is perhaps surprisingly low. Any further comparison with expe- rimental data is currently precluded because cation migration energies in even the simplest of oxides are not known to any degree of accuracy.

A study of non-cubic structures is more reward- ing as the model may elucidate possible diffusion routes. The geologically important mineral olivine [(Fe, M ~ ) , S ~ O ~ ] has a moderately simple structure containing chains of edge-sharing (Fe, Mg) octahe- dra parallel to 2 (Fig. 4). Measurements of Fe-Mg interdiffusion in this orthorhombic mineral have shown that diffusion rates parallel t o c are conside- rably faster than in the other two principle direc-

tions [12]. Ohashi and Finger 1131 used a random walk technique t o investigate diffusion in olivine and suggested that the anisotropy resulted not from single jumps along the chains parallel to the c-axis (Fig. 4) but from an interlayer sequence of MI-M2- MI-M2. A sum-of-squares analysis of the two sites confirmed the difficulty of the direct route parallel

Fig. 4.

-

Polyhedral representation of the olivine structure. The stipled region is at a / 2 , and the remainder at a = 0. Chains of octahedra (MI, M2) parallel to

c

contain (Fe, Mg) and the tetra- hedra Si.

to

c,

and indicated that some form of interlayer jump sequence was necessary. However, a compo- site sum-of-squares plot of a possible M 1 -M2-M I..

.

sequence is difficult to construct because of the dif- ferences in orientation between sites and the fact that areas outside a particular polyhedron will not be accurately represented by a single plot. As a compromise figure 5 shows three plots in the

b-g

plane for M I , M2 and M1 respectively. The figure

Fig. 5 . - Partially contoured sum-of-squares plots in the b-c plane for sites I, I1 and I 1 1 denoted in figure 4. The possibility of the M I - M Z M 1

...

route is apparent.

shows clearly the MI-M2-MI sequence, but it is impossible t o decide from this alone whether the route continues along

a,

or whether it is confined to the

b_-c

plane. Plots in a plane containing 2 confirm the latter, but are not shown for reasons of clarity. In a similar manner, other structures could be examined at any arbitrary orientatiol,.

4. Summary. - A simple sum-of-squares relation

with an a\sociated bond 51rength weighting scheme

C6-260 M. J. DEMPSEY AND R. FREER

has been employed to obtain qualitative information figures, and at larger displacements the contours about the thermal motion of ions within individual suggest possible diffusion routes. As yet the model coordination polyhedra. Data is obtained in terms cannot generate quantitative data, but it has the of N, the sum-of-squares parameter. At small dis- advantage of employing a minimum of computa- placements from the equilibrium position, contours tion, and could be handled by a programmable cal- of constant N correlate well with thermal vibration culator of average capabilities.

References

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