Section: 12 | The Madelung Constant and Crystal Lattice Energy |

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THE MADELUNG CONSTANT AND CRYSTAL LATTICE ENERGY

Ionic crystalline solids are formed by ionic bonds of positive and negative ions. The primary contribution to the binding energy of ionic crystals is electrostatic and is named the Madelung energy. The total crystal lattice energy is the energy needed to separate a crystal into individual ions at infinite distance from each other and is the sum of all interactions between individual ions. The total crystal lattice energy U can be written as the following.

$U = \frac{N M z_{i} z_{j} e^{2}}{r} (1 - 1 / n)$

Here, N is Avogadro’s number; z_i and z_j are the integral charges on the ions (in units of e); n is the Born exponent; and e is the charge on the electron in electrostatic units (e = 4.803 × 10^–10 esu). r is the shortest distance between cation–anion pairs in centimeters, and M is called the Madelung constant. Then U is in ergs (1 erg = 10^–7 J).

Table 1 contains the value for the Madelung constant for several different crystal forms and ion types.

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TABLE 1. Madelung Constant for Selected Crystals

Substance	Synonym	Formula	Ion type	Crystal form^a	M
Aluminum oxide (α)	Corundum	Al₂O₃	2M³⁺, 3X^{– –}	Rhombohedral	4.1719
Cadmium chloride	Cadmium dichloride	CdCl₂	M⁺⁺, 2X^–	Hexagonal	2.244^b
Cadmium iodide	Cadmium diiodide	CdI₂	M⁺⁺, 2X^–	Hexagonal	2.355^b
Calcium chloride	Hydrophilite	CaCl₂	M⁺⁺, 2X^–	Cubic	2.365
Calcium fluoride	Fluorite	CaF₂	M⁺⁺, 2X^–	Cubic	2.51939
Cesium chloride		CsCl	M⁺, X^–	BCC	1.76267
Copper(I) oxide	Cuprite	Cu₂O	2M⁺, X^{– –}	Cubic	2.22124
Magnesium fluoride	Sellaite	MgF₂	M⁺⁺, 2X^–	Tetragonal	2.381^b
Silicon dioxide (β-quartz)	Smoky quartz	SiO₂	M⁴⁺, 2X^{– –}	Hexagonal	2.2197^b
Sodium chloride	Halite	NaCl	M⁺, X^–	FCC	1.74756
Titanium(IV) oxide (anatase)	Anatase	TiO₂	M⁴⁺, 2X^{– –}	Tetragonal	2.400^b
Titanium(IV) oxide (rutile)	Rutile	TiO₂	M⁴⁺, 2X^{– –}	Tetragonal	2.408^b
Zinc oxide	Zincite	ZnO	M⁺⁺, X^{– –}	Hexagonal	1.4985^b
Zinc sulfide (sphalerite)	Sphalerite	ZnS	M⁺⁺, X^{– –}	FCC	1.63806
Zinc sulfide (wurtzite)	Wurtzite	ZnS	M⁺⁺, X^{– –}	Hexagonal	1.64132^b

^{^a}FCC = face-centered cubic; BCC = body-centered cubic.
^{^b}For tetragonal and hexagonal crystals the value of M depends on the details of the lattice parameters.

The total crystal lattice energy is a balance between the electrostatic (Madelung) potential and a short-term repulsive potential, which is proportional to 1/n, where n is the Born exponent. Table 2 gives values of the Born exponent for a number of different ion types. For a crystal with a mixed-ion type, an average of the values of n in this table is to be used (6 for LiF, for example).

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TABLE 2. Born Exponent n for Different Ion Types

Ion type	n
He, Li⁺	5
Ne, Na⁺, F^–	7
Ar, K⁺, Cu⁺, Cl^–	9
Kr, Rb⁺, Ag⁺, Br^–	10
Xe, Cs⁺, Au⁺, I^–	12

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THE MADELUNG CONSTANT AND CRYSTAL LATTICE ENERGY

TABLE 1. Madelung Constant for Selected Crystals

TABLE 2. Born Exponent n for Different Ion Types

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