Madelung Constants and Born Exponents, Common Crystal Lattices, Lattice Types

For a review of the solid state, see Chapter 7 of Concepts of Chemistry

Madelung Constants and Born Exponents

Radius Ratio
(Cation/Anion)		 Lattice Type CN of Cation CN of Anion Madelung
Constant		 Reduced
Madelung
Constant
A. 1:1 Stoichiometry of Salt(MX)
0.225-0.414 Wurtzite(ZnS)
Zinc Blende (ZnS)		 4 4 1.63805 1.63805
0.414-0.732 Rock salt (NaCl) 6 6 1.74756 1.74756
0.732-1.000 CsCl 8 8 1.76167 1.76267
B. 1:2 Stoichiometry of Salt (MX2)
0.225-0.414 Beta-quartz (SiO2) 4 2 2.201 1.467
0.414-0.732 Rutile (TiO2) 6 3 2.408* 1.605
0.732-1.000 Fluorite (CaF2) 8 4 2.51939 1.6796
C. 2:3 Stoichiometry of Salt (M2X3)
0.414-0.732 Corundum (Al2O3) 6 4 4.1719* 1.6688
*Exact value dependent on details of structure
 

Born Exponents and Electron Configurations of Ions
Born Exponent Principal Quantum Number
of Outermose Electrons of Ion		 Examples
5 1 H-, Li+
7 3 F-, Na+
9 3 Cl-, K+, Zn2+, Ga3+
10 4 Br-, Rb+,Cd2+, In3+
12 5 I-, Cs+, Au+, Tl3+

Common Crystal Lattices

Lattice Name Lattice Structure
Rock Salt
(NaCl)
Cesium Chloride
(CsCl)
Zincblende
(ZnS)
Fluorite
CaF2

Lattice Types

There are several lattices that are so simple and/or occur so frequently that you need to be familiar w their characteristics and w examples of each.
  1. a. 1:1 stoichiometry
    1. NaCl lattice

      2. Unit cell (uc) is ccp (fcc) Cl-, Na+ in Oh holes.
      # formula units per uc = 4.
      C.N. = 6
      Madelung Constant = 1.74756

      Examples:
      All alkali halides except CsCl, CsBr, CsI
      AgCl, AgBr, AgI
      Most of M2+ oxides
      Alkaline earth monosulfides except BeS.
      Many transition metal sulfides
      CaC2--has Ca2+ at Na+ positions, C22- at Cl- positions along Ca-Ca axis.
      NH4Cl (> 184o), NH4Br (> 138o), NH4I (> -17.6o)
      TlF (deformed)

    2. CsCl lattice

      3. uc is simple cubic Cl- ions, Cs+ at body centers in cubic holes.
      # formula units/uc = 1
      C.N. = 8
      Madelung Constant = 1.76267

      Example:
      CsCl, CsBr, CsI, CsCN, CsSH
      TlCl, TlBr, TlI, TlCN
      Be(H2O)42+SO42-
      K+SbF6-
      NH4Cl (< 184o), NH4Br (< 138o), NH4I (< -17.6o)

    3. ZnS lattices. There are two.
      1. Zincblende

        2. uc is ccp S2- w Zn2+ in 1/2 of Td holes
        # formula units/uc = 4
        C.N. = 4
        Madelung Constant = 1.63806

        Example:
        BeS, ZnS, CdS, HgS, AgI, CuCl, CuBr, CuI, BN, BP

      2. Wurtzite

        3. uc is hcp S2- w Zn2+ in half of Td holes.
        # formula units/uc = 6
        C.N. = 4
        Madelung Constant = 1.64132

        Example:
        ZnS, CdS, HgS, NH4F

    4. NiAs lattice

      5. uc is hcp As w Ni in Oh holes
      # formula units/uc = 6
      C.N. = 6

      Example:
      Transition metal sulfides, arsenides, selenides.

  2. 2:1 stoichiometry
    1. Fluorite (CaF2)

      2. uc is ccp Ca2+ w F- in all Td holes. or
      sc F- w Ca2+ in alternate cubic holes.
      # formula units/uc = 4
      C.N.(Ca2+) = 8; C.N.(F-) = 4
      Madelung Constant = 2.51939

      Example:
      TiH2, CaF2, SrF2, BaF2, CdF2, HgF2, PbF2, SrCl2, EuF2

      Related is the antifluorite lattice (M2X), w anion/cation roles reversed.

      Example:
      Li+, Na+, K+ oxides, sulfides, selenides, tellurides. Mg2Si and other group II-group IV cpds.

    2. Rutile (TiO2)

      3. uc is approximately bcc cations (Ti4+) w anions arranged to give Oh about body center cation.
      # formula units/uc = 2
      C.N.(Ti4+) = 6; C.N.(O2-) = 3
      Madelung Constant = 2.408, but can vary somewhat according to details of structure.

      Example:
      Metal difluorides and dioxides, but not sulfides or other dihalides. Lattice seems to be appropriate for highly ionic cpds.
      Difluorides of Mg, Cr, Mn, Fe, Co, Ni, Cu, Zn, Pd; CaCl2; CaBr2;SrCl2

    3. CdCl2 lattice

      4. uc is ccp Cl- with Cd2+ in half of Oh holes such that alternate planes of Cd2+ ions are missing.
      This gives a sandwich structure of the following type:

      ------anions------
      ------cations----- repeats
      ------anions------ >BR>
      # formula units/uc = 2

      Requires considerable covalence in cation-anion interaction.

  3. 3:1 stoichiometry
    1. Cryolite (Na3[AlF6]) lattice.

      2. uc is ccp anions w cations in all of Td and Oh holes. # formula units/uc = 4
  4. Complex lattices
    1. Spinels, AB2O4. Named for MgAl2O4 (spinel).

      2. uc is ccp O2- ions with

      A cations in 1/8 of Td sites (equiv to Ca2+ positions in CaF2).
      B cations in 1/2 of Oh holes, with alternating rows (not layers) missing.

      # formula units/uc = 2
      # O2- = 1/8 (8) + 1/2 (10) + 1/4 (4) + 1 = 8
      # A = 2
      # B = 1/4 (8) + 1/2 (2) + 1 = 4

      C.N.'s O2- = 4 (÷Td) (3B and 1A)
      A = 4 (Td)
      B = 6 (Oh)

      Example:
      MgAl2O4 (A2+B23+O4) Sum of cation charges = 8
      Na2WO4 (A6+B2+O4)
      Zn2SnO4 (A4+B22+O4)
      NiAl2O4 (A2+B23+O4)
       

      Inverse Spinels-- A and half the B trade places, so we have A in Oh holes, half the B in Oh and half in Td holes.

      In these cases, A is transition metal which would be HS in Td hole, LS in Oh hole.
      Example:
      Fe3O4 (Fe2+Fe23+O4)
      NiFe2O4
      CoFe2O4
      NiGa2O4

    2. Perovskites, ABO3

      3. uc is ÷ ccp array of O2- and B, with A in 1/4 of Oh holes so as to form a cube.
      # formula units/uc = 1
      C.N.'s: B = 12
      A = 6
      O = 6 (4B and 2A)

      Example: BaTiO3