Empirical electronic polarizabilities of ions for the prediction and interpretation of refractive indices: Oxides and oxysalts

An extensive set of refractive indices determined at λ = 589.3 nm ( n D ) from ~2600 measurements on 1200 minerals, 675 synthetic compounds, ~200 F-containing compounds, 65 Cl-containing compounds, 500 non-hydrogen-bonded hydroxyl-containing compounds, and ~175 moderately strong hydrogen-bonded hydroxyl-containing compounds and 35 minerals with very strong H-bonded hydroxides was used to obtain mean total polarizabilities. These data, using the Anderson-Eggleton relationship α T = ( n D 2 − 1 ) V m 4 π + ( 4 π 3 − c ) ( n D 2 − 1 ) where α T = the total polarizability of a mineral or compound, n D = the refractive index at λ = 589.3 nm, V m = molar volume in A 3 , and c = 2.26, in conjunction with the polarizability additivity rule and a least-squares procedure, were used to obtain 270 electronic polarizabilities for 76 cations in various coordinations, H 2 O, 5 H x O y species [(H 3 O) + , (H 5 O 2 ) + , (H 3 O 2 ) − , (H 4 O 4 ) 4− , (H 7 O 4 ) − ], NH 4 + , and 4 anions (F − , Cl − , OH − , O 2− ). Anion polarizabilities are a function of anion volume, V an , according to α − = α − 0 ⋅ 10 − N o / V an 1 . 20 where α − = anion polarizability, α − o = free-ion polarizability , and V an = anion molar volume. Cation polarizabilities depend on cation coordination according to a light-scattering (LS) model with the polarizability given by α ( C N ) = ( a 1 + a 2 C N e − a 3 C N ) − 1 where CN = number of nearest neighbor ions (cation-anion interactions), and a 1 , a 2 , and a 3 are refinable parameters. This expression allowed fitting polarizability values for Li + , Na + , K + , Rb + , Cs + , Mg 2+ , Ca 2+ , Sr 2+ , Ba 2+ , Mn 2+ , Fe 2+ , Y 3+ , (Lu 3+ -La 3+ ), Zr 4+ , and Th 4+ . Compounds with: (1) structures containing lone-pair and uranyl ions; (2) sterically strained (SS) structures [e.g., Na 4.4 Ca 3.8 Si 6 O 18 (combeite), Δ = 6% and Ca 3 Mg 2 Si 2 O 8 (merwinite), Δ = 4%]; (3) corner-shared octahedral (CSO) network and chain structures such as perovskites, tungsten bronzes, and titanite-related structures [e.g., MTiO 3 (M = Ca, Sr, Ba), Δ = 9–12% and KNbO 3 , Δ = 10%]; (4) edge-shared Fe 3+ and Mn 3+ structures (ESO) such as goethite (FeOOH, Δ = 6%); and (5) compounds exhibiting fast-ion conductivity, showed systematic deviations between observed and calculated polarizabilities and thus were excluded from the regression analysis. The refinement for ~2600 polarizability values using 76 cation polarizabilities with values for Li + → Cs + , Ag + , Be 2+ → Ba 2+ , Mn 2+/3+ , Fe 2+/3+ , Co 2+ , Cu +/2+ , Zn 2+ , B 3+ → In 3+ , Fe 3+ , Cr 3+ , Sc 3+ , Y 3+ , Lu 3+ → La 3+ , C 4+ → Sn 4+ , Ti 3+/4+ , Zr 4+ , Hf 4+ , Th 4+ , V 5+ , Mo 6+ , and W 6+ in varying CN’s, yields a standard deviation of the least-squares fit of 0.27 (corresponding to an R 2 value of 0.9997) and no discrepancies between observed and calculated polarizabilities, Δ > 3%. Using n D = 4 π α ( 2 . 26 − 4 π 3 ) α + V m + 1 the mean refractive index can be calculated from the chemical composition and the polarizabilities of ions determined here. The calculated mean values of n D > for 54 common minerals and 650 minerals and synthetic compounds differ by In a comparison of polarizability analysis with 68 Gladstone-Dale compatibility index (CI) (Mandarino 1979, 1981) values rated as fair or poor, we find agreement in 32 instances. However, the remaining 36 examples show polarizability Δ values