Valence maps

Fig. 11.8. The valence map of a (110) section through Cap2 showing possible locations for ions. The 1.0 vu contour is shown with a dot and dash line. Contours less than 1.0 vu are shown with a broken line, those larger than 1.0 vu with a solid line. The contour interval is 0.2 vu. Contours above 1.8 vu have been omitted for clarity. Atomic positions are shown by the element s3Tubol (one F has been omitted to show the contours at this site). X, Y, and Y are proposed sites for an interstitial F ion as discussed in the text.

Valence maps may alternatively be presented in a way which gives a direct impression of the atom s probability density function, a function which indicates the probability of finding the atoms at a particular point in space. This is calculated by inverting the valence function using eqn (11.3) ... 

Fig. 11.9. The valence map of Mg2Si04 transformed to simulate the probability distribution function of at (a) room temperature, (b) high temperature. The...

Valence maps are not restricted to inorganic materials but can be used in any situation where acid-base bonding is involved. For example, Nayal and Di Cera (1994) have used valence maps to locate the sites of Ca + ions on the surfaces of Ca-binding proteins as described in Section 13.6.2. 

A program for calculating bond valence maps has been published by Gonzales-Platas et al. (1999). 

An ideal tool to study the mechanism of ionic conduction is the valence map described in Section 11.2.3 since this locates all the points in the crystal at... 

Topological Analysis with Bond-Valence Maps... 

As with VT, identifying conduction pathways with bond valence maps provides accurate predictions for ionic conductors (both crystalline and amorphous) with a percolation mechanism of conductivity, but is less successful when modeling proton conductors. 

Voronoi tessellation Bond valence mapping Ab initio static calculations and MD MEM... 

Fig. 10.3. A D-map, calculated from the valence map with N =i, shows the places in forsterite that can accommodate Mg. The third contour represents the positions that provide the best bonding for Mg. The positions of the Si and O atoms are not shown...

As early as 1978 Waltersson [59] had created the first valence map [78 section 6.2] to find the location of a lithium atom in Li2W04 where the presence of the heavy tungsten atom masked the weak signal from lithium. Brown [29] later showed that such maps could be used to trace out ionic diffusion paths, an idea that has been exploited by Adams [60] and Adams and Swenson [61-65] in a series of papers exploring diffusion, not only in crystals but also in simulated amorphous structures. A program to calculate valence maps, VALMAP, has been written by Gonzales-Platas et al. [66]. 

Fig. 8 A valence map of fluorine in a (110) plane in Cap2. Dotted lines are negative contours greater than 1.8 not shown. X and Y show possible sites for interstitial F (Fig. 11.8 from [11] by permission of the Oxford University Press)...

The valence vector sum can also be treated as a field since it can be calculated for an atom placed at any point in the structure, not just the known site of an atom. Mathematically it represents the slope of the valence map, U r), at the position r ... 

The valence map can be used for locating weakly scattering and weakly bonding atoms such as lithium in cases where it has not been detected by X-ray diffraction, but more commonly it is used for tracing possible diffusion paths in crystalline and amorphous solids as discussed in [25]. 

Fig. 10 Valence map plot of A-site cation position for CaTiOs in tilt system a b a (space group Pnma). AX and AZ are the differences in the fractional position from the high symmetry position located at (1/2, 1/4, 1/2). The valence of A-site cation is shown as the free positional parameters are varied while holding the octahedral tilt angle at 14.60°. The circle is the SPuDS predicted position and the triangle is the experimental position...

Adams S (2006) From bond valence maps to energy landscapes for mobile ions in ion-conducting solids. Solid State Ionics 177 1625-1630... 

Ion Transport Pathways from Bond Valence Maps in Crystalline Cation... 

In most cases stable solid electrolytes are nearly densely packed so that sites with the matching valence will form well-defined minima in these bond valence maps. This unfortunately led some authors to the misconception that ions would generally... 

Moreover, it has to be kept in mind that bond valence maps (or more precisely, bond valence sum mismatch maps) use bond valence units scale rather than an energy scale, and attempts to link energy or probability density to the bond valence mismatch were difficult to achieve in a general transferable way as the calculation of effective bond valence mismatches according to Eq. 1 requires scaling between bond valence terms and cation-cation (or anion-anion) repulsions or (for open structures) to discriminate between sites of identical bond valence sum based on the degree to which the equal valence rule is fulfilled. 

Three-dimensional bond valence maps of Ag" ion conductors were constructed by summing up bond valence contributions to all anions up to a distance of 8 A for any point of a three-dimensional grid. It is also advisable to choose the number of grid points across a unit cell as a multiple of 12 or 24 so that common special positions at fractional coordinates become explicit grid points. Initially, the bond valence sum mismatch term was complemented by hard cutoff criteria to account for the repulsion between mobile and immobile cations. Thus, grid points at a distance to other cations M smaller than the sum of radii of Ag" and M were treated as inaccessible. It should however be noticed that the choice of these minimum... 

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