3.014 Materials Laboratory Fall 2006 LABORATORY 2: Module β1 Radius Ratios and Symmetry in Ionic Crystals Instructor: Francesco Stellacci Objectives Discover principles of X-ray diffraction from crystalline materials Collect X-ray powder diffraction patterns and analyze using Powder Diffraction File (PDF) Explore relationship between relative ion sizes and crystal structure symmetries Tasks Calculate structure factors of materials investigated Prepare samples for X-ray powder diffraction Obtain X-ray powder diffraction patterns for 4-5 perovskite-structure oxides Compare patterns obtained to calculations and PDF Apply peak fitting routines to determine lattice parameters Relate composition, lattice parameter, ionic radius, radius ratio, and crystal symmetry Materials CaTiO3, BaTiO3, SrTiO3, PbTiO3, CaZrO3, PbZrO3 Introduction Many inorganic materials, such as halides like NaCl and oxides like MgO, TiO2 or Al2O3, exhibit strong ionic character in their atomic bonding. As a result, atom packing in these systems is dictated by electrostatic forces—the structures chosen by nature are those that maximize interactions between ions of opposite charge while minimizing contact between like-charged ions and maintaining electrical neutrality. Structural consideration of ionic solids begins with the Goldschmidt1 ionic model, which assumes that ions are essentially charged, incompressible, non-polarizable spheres with a definable radius. As a consequence of electrostatic interactions, ionic crystals create ordered arrangements of coordination polyhedra, in which cations are in contact with a maximum number of surrounding anions, the number depending on the ratio of the cation radius to the anion radius, rC/rA (Table 1), and to a lesser extent cation charge. A large 1highly charged cation (such as Ba2+ or U4+) can accommodate a larger number of anions around it. U4+ cations in UO2 are 8-coordinated by O2– anions in the fluorite structure Fig. 1. (8:4) Fluorite structure of UO2, with rC/rA ratio = 0.724. The U4+ cations form a cubic face-centered arrangement, but alternatively can be thought of as filling every other cube interstice in the simple cubic arrangement of O2– anions, or as [UO8] coordination cubes linked by sharing edges. In the (4:8) anti-fluorite structure of Na2O, the roles of anion and cation are reversed, with rA/rC = 0.697, Na2+ cations are four-coordinated to O2– anions,. and [ONa8] cubes sharing edges. (Fig. 1), while Ba2+ cations in perovskite-structure BaTiO3 (Fig. 5, see below) are 12-coordinated by O2– anions). Conversely, smaller and less-highly charged cations cannot accommodate so many anions around them (Li2O and Na2O adopt the anti-fluorite structure (Fig. 1) in which the Li1+ and Na1+ cations are 4-coordinated by oxygen). Table 1. Preferred Cation Coordination in Ionic Crystals Cation Coordination No. Anion arrangement Minimum stable rC/rA 8 corners of cube 0.732 6 corners of octahedron 0.414 4 corners of tetrahedron 0.225 3 corners of triangle 0.155 2 co-linear 0 2Of course, the anion point of view may equally be adopted. In the Na2O example just mentioned, eight (small) Na1+ cations surround each (larger) O2– anion. In some cases (like BaO), the cation could accommodate a larger number of anions around it (e.g. 8 or 12) than the 6 it has, but the anion cannot accommodate around itself the geometrically consequential number of cations dictated by stoichiometry. Table 2. Coordination-Dependent Ionic Radii (Shannon & Prewitt3) Ion Radius r (pm) CN = 12 Radius r (pm) CN = 8 Radius r (pm) CN = 6 Radius r (pm) CN = 4 Li1+ 76 59 Na1+ 118 102 99 K1+ 185 138 Rb1+ 161 152 Cs1+ 177 167 F1-135 133 Cl1-184 181 Mg2+ 72 Ca2+ 134 112 100 Sr2+ 144 126 118 Pb2+ 149 129 119 Ba2+ 161 142 135 Ti4+ 61 Nb5+ 64 Zr4+ 72 O2-142 140 138 Ionic radii were first computed by the crystal chemist and Nobelist Linus Pauling2 (also of X-ray crystallography and Vitamin C fame), but revised radii that take into account polarization of the ion cores, and thus depend on coordination, were calculated 3more recently by Shannon and Prewitt 3 and are those now generally used (Table 2). Some of the stablest, and therefore most pervasive, ionic structures are those in which radius-ratio criteria are well satisfied for both anions and cations. Classic examples are those binary equiatomic compounds that crystallize in the rocksalt (halite) structure (Fig. 2b)—among them NaCl, KCl, LiF, Kbr, CaO, SrO, BaO, CdO, VO, Fe1–xO, CoO, NiO, etc.—which have cation-anion radius ratios rC/rA near 0.5 (NaCl 0.563, MgO 0.514) and comprise cation (or anion) coordination octahedral (e.g. [NaCl6] octahedral) which share edges. For more similar ion sizes, the CsCl structure is preferred (CsCl itself has rC/rA = 0.96) in which Cs1+ ions sit in the centers of cubes of Cl– ions ([CsCl8] cubes) that share faces (Fig. 2). Fig. 2. (8:8) structure of CsCl, in which each ion is 8-coordinated by ions of the opposite charge, may also be thought of as [CsCl8] coordination cubes that share all faces. Linus Pauling’s rules for crystalline compounds (Table 3) codify these notions and provide rationalization for structural tendencies observed in systems with ionic bonding. Despite being couched in terms of ion size, these rules turn out to be essentially driven more by the consideration of minimizing electrostatic energy (which can be accounted in a proper Madelung summation), than by the geometric necessities of ionic radii, however represented 4Table 3. Pauling’s Rules for Crystalline Ionic Compounds Rule 1. Coordination. A coordination polyhedron of anions is formed around every cation (and vice versa) and is stable only if the cation is in contact with each of its neighboring anions. The distance between anions and cations is thus the sum of the their ionic radii, and the coordination number of the cation will be maximized subject to the criterion of maintaining cation-anion contact. Rule 2. Electrostatic Valency. The total strength of valency “bonds” that reach an anion form all of its neighboring cations equals the charge of the anion. Rule 3. Polyhedral Linking. Cation coordination polyhedra tend to be linked through sharing of anions, at corners first, then edges, then faces—in this order because of the electrostatic repulsion between cations. Rule 4. Cation Evasion. The electrostatic repulsion between cations is greatest for cations of high charge and small