Article pubs.acs.org/jchemeduc
Radius Ratio Rule Rescue Anna Michmerhuizen, Karine Rose, Wentiirim Annankra, and Douglas A. Vander Griend* Department of Chemistry and Biochemistry, Calvin College, 1726 Knollcrest Circle SE, Grand Rapids, Michigan 49546, United States S Supporting Information *
ABSTRACT: Making optimal pedagogical and predictive use of the radius ratio rule to distinguish between solid state structures that feature tetrahedral, octahedral and cubic holes requires several updated insights. A comparative analysis of the Born−Landé equation for lattice energy is developed to show that the rock salt structure is a suitable choice for radius ratios between 0.326 and 1.00. The lower bound is the ratio at which the energy of the rock salt structure matches that of the zinc blende structure, all other things being equal. The upper bound extends all the way to unity because the relative energy of the rock salt and CsCl structures are within 1% of each other for radius ratios above 0.717, which is the ratio at which the energy of the rock salt structure matches that of the CsCl structure. Geometric analysis of 154 1:1 ionic compounds that can potentially adopt the zinc blende, rock salt, or CsCl structure demonstrates that while existing ionic radii can be used for predictions with substantial success, a more applicable set of predictive ionic radii based on cubic lattice parameters can be deduced. KEYWORDS: Second-Year Undergraduate, Upper-Division Undergraduate, Graduate Education/Research, Inorganic Chemistry, Misconceptions/Discrepant Events, Solid State Chemistry
■
INTRODUCTION The radius ratio rule is an elegant and attractive idea to relate structure and composition for solid state compounds, and it is commonly taught as a way to predict structure type in binary compounds. According to Jensen, Victor Goldschmidt was the first to apply the concept of ionic-radius ratio rules to solid state lattices.1 Ions are considered to be hard spheres that interact with each other in strictly Coulombic ways. Ignoring any covalency to the bonding, the lowest energy arrangement of the ions should simply be dictated by the degree to which oppositely charged ions interact, specifically number and distance. The ratio of the radii of the ions can be used to assign which type of solid state hole is the most appropriately sized to maximize anion−cation interactions (while simultaneously minimizing cation−cation and anion−anion interactions) throughout the solid. Small radius ratios indicate that a structure with tetrahedral holes will be optimal, whereas larger radius ratios point to structures utilizing octahedral and then cubic holes. This purely geometric categorization can then often be combined with the stoichiometry of the compound to predict a single most appropriate solid state crystal structure for the compound. Of course, there are exceptions. Using the ideal hole size as the cutoffs for appropriate radius ratio ranges, L. Nathan estimates that about one-third of the 380 samples he catalogued were predicted incorrectly.2 The exceptions are © XXXX American Chemical Society and Division of Chemical Education, Inc.
accompanied by explanations that often point to limitations in the strictly ionic paradigm upon which the model is based, but it should not be assumed that this is the only reason for the existence of exceptions. Besides this lack of accuracy, the radius ratio rule suffers from a more fundamental flaw: ionic radii are a poor model of ion size. This is for the following reasons: first because atoms/ions are not always spherical, second because they do not have definite boundaries, and third because they do not even have consistent size from one compound to the next. Consequentially, many different sets of radii have been postulated. There are crystal ionic radii, effective ionic radii, promolecule radii,3 numerous quantum mechanical radii,4 covalent radii, thermochemical radii,5 and van der Waals radii. Furthermore, many have shown that the space occupied by an atom or ion in a solid state compound is a very specific function of that compound. Coordination number, geometry, and especially the chemical identity of nearest neighbors all impact the apparent size.6 Given these weighty issues, it might seem that the radius ratio rule should, at best, be limited to an introductory academic exercise and at worse should be discarded entirely. Received: December 15, 2016 Revised: July 13, 2017
A
DOI: 10.1021/acs.jchemed.6b00970 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Article
Figure 1. Three distinct cubic structure types for 1:1 compunds: (a) the zinc blende structure which features tetrahedral holes, (b) the rock salt structure which features octahedal holes, and (c) the CsCl structure which features cubic holes.
Figure 2. Idealized relative lattice energy of the three 1:1 cubic structures for ionic compounds based on the Born−Landé equation. Ideal ratios mark the points at which the smaller ion is ideally sized to occupy the interstitial hole amidst the larger ions. The crossover ratios mark the points at which the relative energy of the structures switch order of stability.
radius ratios are accommodated best by the zinc blende structure; medium ones by the rock salt structure; and large ones by the CsCl structure. The crux of the matter is the precise value of the radius ratio at the boundary of each of these three regimes. Historically, the ideal octahedral hole size of 0.414 is considered the upper limit of suitability for the zinc blende structure, and the ideal cubic hole size of 0.732 is considered the lower limit for the CsCl structure.7 These boundaries are incorrect because they are not the energetic crossover points between these three structure types. To find the radius ratios that represent the points where the different structures have the same energy will require a close look at how, in a strictly electrostatic sense, lattice energy depends on the structure and radius ratio. Consider the Born− Landé equation, which shows that the lattice energy of a crystalline ionic compound is proportional to the ion charges, the Madelung constant, and a compressibility factor, and inversely proportional to the distance between the two ions.
Herein we present four key insights that have the potential to re-establish the radius ratio rule as a viable concept for relating structure to composition in simple ionic solids. Our work is based on the comprehensive study of ambient pressure cubic binary phases with a 1:1 stoichiometry. Only three distinct crystal structures are therefore considered: zinc blende (sphalerite), which features tetrahedral holes; rock salt, which features octahedral holes; and CsCl, which features cubic holes (Figure 1). (Note: BCC, where the two atom types are disordered, was not considered and does not occur for ionic compounds.) The Supporting Information contains comprehensive and sortable spreadsheet lists of the 154 compounds for which a lattice parameter could be found in the literature. The overwhelming majority of the compounds adopt the rock salt structure (119), while 27 adopt the zinc blende structure and 8 adopt the CsCl structure. While this list involves elements from all over the periodic table, it is worthwhile to note that all 20 alkali halide salts are on this list, all of which adopt the rock salt structure except for CsCl, CsBr, and CsI which adopt the CsCl structure.
■
E lattice ∝
RESULTS
Z +Z −M ⎛⎜ 1⎞ 1− ⎟ rlittle + rbig ⎝ n⎠
For a given compound, the charges and n value are constant, so the equation reduces to
Insight 1: True Boundaries
The first insight regards the particular radius ratios that define the ranges over which each structure is energetically preferred solely on the basis of geometric arguments. As expected, small
E lattice ∝
M r 1 + rlittle big
B
(1) DOI: 10.1021/acs.jchemed.6b00970 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Article
Here the proportionality has been modified to create a ratio of the smaller radii and the larger one. This is of course the radius ratio. The result is that lattice energy is simply proportional to the appropriate Madelung constant and inversely proportional to 1 + rlittle/rbig. Figure 2 shows the relative theoretical lattice energy (defined endothermically) for each of the three structures of interest as a function of radius ratio. The figure is constructed so that more stable arrangements of the ions, with their correspondingly larger positive lattice energy values, will be lower on the graph (notice inverted scale on the vertical axis so that lattice energy is highest at the bottom of the graph). The relative lattice energy of each structure type arcs downward from the top right of the figure indicating that the stability of each structure-type increases with decreasing radius ratio. This is because high radius ratios lead to structures with gaps between the larger ions, which are pushed apart by the smaller ones. As the smaller ion shrinks relative to the other one, the entire structure contracts and the interionic distances decrease, increasing the lattice stability. However, each curve reaches a point of discontinuity because in ionic crystal structures there comes a point at which, even though the radius ratio decreases, the structure can no longer contract, and therefore, the average interion distance remains constant. This occurs when the larger ions finally touch each other. The relative lattice energy of the structure therefore plateaus even as the radius ratio continues to decrease. The discontinuity point occurs at the radius ratio corresponding to the ideal hole size mentioned above: 0.732 for cubic, 0.414 for octahedral, and 0.212 for tetrahedral. The three curves of Figure 2 all follow a similar hyperbolic arc from the right until the ideal hole size for the corresponding structure is reached. Because decreasing the radius ratio further than these values does not allow the ions to get any closer on average, the relative stability plateaus. It is clear from Figure 2 that whereas the ideal hole sizes mark the radius ratios below which the relative lattice energy remains constant for each of the three structures, they do not mark the radius ratios at which one structure type becomes energetically favorable over another. The points where the three curves intersect occur at different radius ratios. For example, consider a hypothetical 1:1 ionic compound with a radius ratio of 0.3. This means the smaller ion is too big to fit in a tetrahedral hole formed by four of the larger ions without forcing them apart, but it would “rattle” around in an octahedral hole formed by six of them. So is zinc blende or rock salt the energetically preferred structure in this case? The answer lies in relation to the radius ratio at which the two structures have the same relative energy: a crossover point. The radius ratios that correspond to the three crossover values, rCOn’s, can be readily calculated by solving for the intersection points of the curves for two different structure types. In each case an arched portion of one curve intersects with a plateau from a different curve. The arched portions are described by eq 1 with the pertinent Madelung constant, M (where MZnS = 1.63805, MNaCl = 1.74756, and MCsCl = 1.76267). The plateaus are also defined by eq 1 except with rlittle/rbig replaced with the ideal hole ratio (roctahedral = 0.414 and rcubic = 0.732). The position along the x-axis of the left-most (eq 2), middle (eq 3), and right-most (eq 4) crossover point are calculated according to the following three equations, respectively: M ZnS/(1 + rCO1) = MNacl /(1 + roctahedral) = 1.2357
M ZnS/(1 + rCO2) = MCsCl /(1 + rcubic) = 1.017678
(3)
MNaCl /(1 + rCO3) = MCsCl /(1 + rcubic) = 1.017678
(4)
Solving each of these equations for rCOn leads to the radius ratio values corresponding to the three crossover points, 0.326, 0.610, and 0.717, which are all shown in Figure 2. Though Figure 2 consists of three discontinuous curves, the lowest curve at any given radius ratio corresponds to the most stable 1:1 cubic structure. This is the zinc blende curve for small radius ratios, CsCl for large radius ratios, and rock salt for intermediate radius ratios. Only two crossover points occur along the bottom trace of graph. These are the crossover points between zinc blende and rock salt (0.326), and between rock salt and CsCl (0.717). Therefore, on the basis of the strictly geometrical considerations of the Born−Landé equation for the lattice energy of ionic compounds, the rock salt structure is predicted to be the most stable 1:1 structure for radius ratios between 0.326 and 0.717. The zinc blende is predicted for radius ratios lower than 0.326, and the CsCl structure is predicted for radius ratios above 0.717. Insight 2: NaCl vs CsCl
The first insight points to these crossover values to properly catalogue predicted structures. Another insight also follows from the graph of Figure 2: above a radius ratio of 0.717, whereas the CsCl structure is technically the most stable arrangement, the rock salt structure is nearly as stable over that entire end of the range because the two respective Madelung constants differ by less than 1%. This is the second major insight: the expected range of radius ratio for the rock salt structure should really be from 0.326 all the way to 1. Nathan uses crystal radii to calculate radius ratios, and of the 52 rock salt compounds he catalogued, 22 were outside the range 0.326−0.717. Not surprisingly, all of them were above the range. Furthermore, all 12 of the compounds he catalogued with the cesium chloride structure had appropriate radius ratios. Clearly, while each structure has a distinct range of appropriate radius ratio, for the range 0.717−1.00, the rock salt structure should be expected as much as the CsCl structure. In contrast, the CsCl structure should only occur for ratios above 0.717, and the zinc blende structure is expected to occur if and only if the radius ratio is below 0.326. Using these correct boundaries, Nathan’s catalogue of 118 compounds (he includes the NiAs structure and various polyatomic ions) improves from 58% to 77%, and almost all of the remaining exceptions occur because of radius ratios too large for zinc blende (or wurtzite) structures. This has a lot to do with the choice of radii and now leads to the next major insight regarding the applicability of the radius ratio rule. Insight 3: Predictability of the Alkali Halide Crystal Structures
A third major insight concerning the applicability of the radius ratio rule involves the choice of ionic radius to use. The two most commonly used are crystal radii and effective radii. Crystal radii were developed by Shannon to best correspond to the physical size of an ion in a solid; effective radii are closely related in that all the cations are 14 pm smaller while the anions are 14 pm bigger.8 This 14 pm adjustment yields a value for O2− that corresponds with Pauling’s radii. Radius ratios calculated from crystal radii are larger than those calculated from effective radii, as long as the cation is smaller than the anion.
(2) C
DOI: 10.1021/acs.jchemed.6b00970 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Article
Insight 4: Predictive Radii
In order to evaluate different options for ionic radii, we will focus on a well-behaved and complete subset of 1:1 ionic compounds, namely, the alkali halides. All nine possible ions (5 cations and 4 anions) have a noble gas electron configuration rendering them spherical. Every combination exists at ambient temperature and pressure, and all crystallize in a cubic structure: 17 as rock salt and 3 as the CsCl structure. Given that the cubic lattice parameters for all 20 compounds are known precisely,9 a given set of ionic radii can be evaluated on how well they combine to produce the measured lattice parameter given the appropriate geometric relationships for each structure. For example, in the compound NaCl, the sodium cations and the chloride anions touch along the edge of the unit cell. Consequently, double the sum of the two radii should be equal to the lattice parameter. The analogous deduction can be made along the body diagonal of the CsCl unit cell. It is important to note that if the radius ratio of the ions is less than the ideal size of the corresponding hole type (octahedral for rock salt and cubic for CsCl), then the lattice parameter depends only on the size of the larger ion. For example, using effective radii, LiBr has a radius ratio of rLi+/rBr− = 76/196 = 0.39, which is less than the ideal octahedral hole size of 0.414. This means the corresponding lattice parameter is 2√2rBr− = 554 pm. Now we have a way to evaluate a given set of radii. If the nine ions are modeled with crystal or effective radii, the 20 lattice parameters are off by as much as 3.8% (R2 = 99.42%). Can we do better? Is there a set of radii, one for each ion, which can be used to more accurately model all 20 lattice parameters? The answer is “Yes”. We call them predictive ionic radii. Table 1 lists the three different sets of ionic radii for direct comparison. The predictive radii were ascertained by least-
The fact that there does exist a set of nine radii that not only can be used to correctly predict structure for the alkali halides, but also relate tightly back to the readily measurable parameter of the lattice, strongly indicates that the radius ratio rule can be relied on to predict structures accurately on the basis of a consistent set of ionic radii. This was the third key insight. The final insight will explore the applicability of a more comprehensive set of ionic radii to predict structure types. The 154 compounds studied here and listed in the Supporting Information are composed of 57 different cations (some of which have multiple oxidation states) and 15 different anions. An analogous optimization process, whereby the 72 ionic radii are optimized to fit these compounds, could be carried out on this complete set. This is of course similar to the process by which Shannon developed his ionic radii. (It should be noted that crystal ionic radii do not presently exist for many of the ions in the data set.) Calculating ionic radii in this way can be easily done with a spreadsheet, but the specific set of compounds chosen will have a noticeable impact on the resulting ionic radii. For example, just including the hydrides LiH and NaH in the aforementioned optimization for the alkali halides results in shifting the radius of Li+ from 73 ± 2 to 68 ± 3 pm while leaving the other eight radii almost unchanged. The overall degree of fit also worsens considerably. For the purposes of this study, we are interested in generating radii that are useful in predicting structures. We already have limited ourselves to 1:1 ionic compounds, and can therefore optimize ionic radii for each distinct charge pairing, i.e., +1/−1, +2/−2, +3/−3, and +4/−4. For consistency, we will handle each charge-set the same. We will optimize radii using all members of the set, and then reoptimize each set after ignoring all compounds for which the predicted lattice parameter is more than 5% off. This turns out to be a very small number as ultimately there are only five compounds for which the calculated lattice parameter is more than 5% off from the literature value: AgF (9.2%), AgI (9.2%), CdO (13.3%), SnTe (12.0%), and DyTe (8.5%). The cations involved in these six compounds are all heavy metal atoms. Figure 3 shows the results for all the ions. It is important to note that 42 of the 75 distinct cations occur only once in the set of 154 compounds and therefore result in an exact fit for the lattice parameter with no possible estimate of error. Also, many metallic elements occur in more than one oxidation state. In almost every case the cation with the higher oxidation state has a larger radius. The lone exception to this is europium. This is no doubt coupled to the fact that the predictive radii of the pnictide anions are smaller than those of their chalcogenide neighbor. Perhaps this is because the anions with more charge are more polarizable. Finally, an important observation about the set of compounds that are nominally +4/−4: Since every cation is different, there is not enough information to refine the radii of the anions. Therefore, the radii of the tetra-anions of carbon and germanium were estimated to be smaller than those of the trianions of nitrogen and arsenic by the same margin that the trianions are smaller than the dianions of oxygen and selenium, respectively. In the case of silicon(−4), since it occurs with ruthenium in the CsCl structure with a lattice parameter of 290.9 pm, the radius of the anion can be inferred directly at a maximum of 146 pm since the cation is smaller than the minimal allowable cubic hole. This value is close to falling in line with the radii of the common anions of sulfur (170 pm) and phosphorus (162 pm).
Table 1. Three Different Sets of Ionic Radii (pm) for Alkali Cations and Halide Anions Ion +
Li Na+ K+ Rb+ Cs+ F− Cl− Br− I−
Crystal8
Effective8
90 116 152 166 181 119 167 182 206
76 102 138 152 167 133 181 196 220
Predictive 73 105 137 152 175 129 179 194 216
± ± ± ± ± ± ± ± ±
2 1 1 1 1 1 1 1 1
squares analysis with the 20 lattice parameters after taking into account the various structure types and only including the radius of the smaller ion when it was greater than the ideal hole size. These predictive radii, presented for the first time here, accurately predict the cubic lattice parameter for alkali halides to within 1.3% in all 20 cases (R2 = 99.83%). Furthermore, the standard error on each value can be properly estimated by examining the curvature with respect to the variance.10 All three sets of radii acceptably predict the correct structure in all 20 cases, remembering of course that the rock salt structure is acceptable from a radius ratio of 0.326 all the way to 1.00. The lowest radius ratio is for LiI at 0.34 for predictive radii. No matter which set of radii are used, there are 3 rock salt structures with radius ratios between 0.72 and 0.78, and 5 rock salt structures with radius ratio above 0.80. D
DOI: 10.1021/acs.jchemed.6b00970 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Article
Figure 3. Complete set of predictive radii (pm). The number in parentheses is the number of compounds from the list of 154 studied involving that ion. For any ion that occurs in at least 4 different compounds, a standard deviation was also estimated, based on the curvature intrinsic to the leastsquares optimization.10 b indicates a value set by comparison with the two anions directly to the right (N3− and O2−, or As3− and Se2−).
Figure 4. Histogram of the 154 1:1 ionic compounds grouped by their true structure type and binned by the ratio of predictive radii. X-axis values mark the upper limit of the bin range.
Using these predictive radii, it is straightforward to map the relationship between radius ratio and structure type. Figure 4 shows the categorical results of this complete analysis of the 154 1:1 ionic compounds with a cubic structure. Of the eight compounds with the CsCl structure, five exhibit a proper radius ratio (>0.717), ThTe is only 0.574, and two others are borderline (0.701 and 0.710). Of the 119 compounds with the rock salt structure, only LiI (0.315) exhibits a radius ratio below 0.326, and 70 of them exhibit a radius ratio above 0.717. The key for any general application of the radius ratio rule is the ability to sort out the rock salt structure from the zinc blende structure. This is not only because there are very few examples of binary compounds with the CsCl crystal structure, nearly all of which exhibit appropriate radius ratios for both crystal and effective radii, but also because, as shown previously, the rock salt structure is just about as stable geometrically as the CsCl for radius ratios above 0.717. The crossover between rock
salt and zinc blende, however, is quite stark in both directions, meaning the difference between the more stable and less stable structure on either side of 0.326 is significant (see Figure 2). Only a very limited amount of structural crossover could be tolerated. Of the 27 compounds with the zinc blende structure, only 10 exhibit the proper radius ratio (≤0.326). The other 17 exhibit radius ratios between 0.35 and 0.58. Indeed, most of the clear violations of the radius ratio rule seem to occur when weightier cations (atomic number ≥29), which are appropriately sized for octahedral holes, instead end up in a zinc blende structure with another weightier anion (atomic number ≥15). The exceptions are CuF, AlP and AlAs. This preference for the zinc blende structure, not predicted by the radius ratio rule, is understandably driven by nongeometrical factors such as covalency. In total, the structures of 87% of the 154 compounds studied match the prediction of the radius ratio rule in accord with the E
DOI: 10.1021/acs.jchemed.6b00970 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Article
first two insights above, and all of the significant exceptions occur in a common manner.
(3) Feth, S.; Gibbs, G. V.; Boisen, M. B.; Myers, R. H. Promolecule Radii for Nitrides, Oxides, and Sulfides: A Comparison with Effective Ionic and Crystal Radii. J. Phys. Chem. 1993, 97 (44), 11445−11450. (4) Parsons, D. F.; Ninham, B. W. Ab Initio Molar Volumes and Gaussian Radii. J. Phys. Chem. A 2009, 113 (6), 1141−1150. (5) Jenkins, H. D. B.; Roobottom, H. K.; Passmore, J.; Glasser, L. Relationships among Ionic Lattice Energies, Molecular (Formula Unit) Volumes, and Thermochemical Radii. Inorg. Chem. 1999, 38 (16), 3609−3620. (6) Johnson, O. Ionic Radii for Spherical Potential Ions. I. Inorg. Chem. 1973, 12 (4), 780−785. (7) (a) Woolf, A. A. Coordination and Radius Ratio: A Graphical Representation. J. Chem. Educ. 1989, 66 (6), 509. (b) Toofan, J. A Simple Expression between Critical Radius Ratio and Coordination Number. J. Chem. Educ. 1994, 71 (2), 147. (8) Shannon, R. D. Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. 1976, 32, 751−767. (9) Sirdeshmukh, D. B.; Sirdeshmukh, L.; Subhadra, K. G. Alkali Halides; Hull, R., Osgood, R. M., Sakaki, H., Zunger, A., Series Eds.; Springer Series in Materials Science; Springer: Berlin, 2001; Vol. 49. (10) σ = 10RMSE(optimal radius)/√[SSE(optimal radius + 0.1) − SSE(optimal radius)]. SSE ≡ sum of squares of errors; RMSE ≡ rootmean-square error. (11) (a) Ong, W.-L.; O’Brien, E. S.; Dougherty, P. S. M.; Paley, D. W.; Fred Higgs, C., III; McGaughey, A. J. H.; Malen, J. A.; Roy, X. Orientational Order Controls Crystalline and Amorphous Thermal Transport in Superatomic Crystals. Nat. Mater. 2016, 16, 83−88. (b) O’Brien, M. N.; Jones, M. R.; Lee, B.; Mirkin, C. A. Anisotropic Nanoparticle Complementarity in DNA-Mediated Co-Crystallization. Nat. Mater. 2015, 14 (8), 833−839. (c) Chen, Z.; O’Brien, S. Structure Direction of II−VI Semiconductor Quantum Dot Binary Nanoparticle Superlattices by Tuning Radius Ratio. ACS Nano 2008, 2 (6), 1219− 1229.
■
CONCLUSION In conclusion, we have provided four key insights into the radius ratio rule: (1) The limiting values for the radius ratio for the rock salt structure are 0.326−0.717. Zinc blende lies below this and CsCl above. (2) The rock salt structure should actually be expected all the way up to values of 1.00 due to its energetic similarity (
