Madelung constant

The Madelung constant allows for the calculation of the electric potential V_i of all ions of the lattice felt by the ion at position r_i

${\displaystyle V_{i}={\frac {e}{4\pi \epsilon _{0}}}\sum _{j\neq i}{\frac {z_{j}}{r_{ij}}}\,\!}$

where r_ij = |r_i − r_j| is the distance between the ith and the jth ion. In addition,

z_j = number of charges of the jth ion

e = 1.6022×10⁻¹⁹ C

4πϵ₀ = 1.112×10⁻¹⁰ C²/(J⋅m).

If the distances r_ij are normalized to the nearest neighbor distance r₀ the potential may be written

${\displaystyle V_{i}={\frac {e}{4\pi \epsilon _{0}r_{0}}}\sum _{j}{\frac {z_{j}r_{0}}{r_{ij}}}={\frac {e}{4\pi \epsilon _{0}r_{0}}}M_{i}}$

with ${\displaystyle M_{i}}$ being the (dimensionless) Madelung constant of the ith ion

${\displaystyle M_{i}=\sum _{j}{\frac {z_{j}}{r_{ij}/r_{0}}}.}$

Another convention is to base the reference length on the cubic root of the unit cell volume, which for cubic systems is equal to the lattice constant. Thus, the Madelung constant then reads

${\displaystyle {\overline {M}}_{i}=\sum _{j}{\frac {z_{j}}{r_{ij}/w}}=M_{i}{\frac {r_{0}}{w}}.}$

The electrostatic energy of the ion at site ${\displaystyle r_{i}}$ then is the product of its charge with the potential acting at its site

${\displaystyle E_{el,i}=z_{i}eV_{i}={\frac {e^{2}}{4\pi \epsilon _{0}r_{0}}}z_{i}M_{i}.}$

There occur as many Madelung constants ${\displaystyle M_{i}}$ in a crystal structure as ions occupy different lattice sites. For example, for the ionic crystal NaCl, there arise two Madelung constants – one for Na and another for Cl. Since both ions, however, occupy lattice sites of the same symmetry they both are of the same magnitude and differ only by sign. The electrical charge of the Na⁺ and Cl⁻ ion are assumed to be onefold positive and negative, respectively, ${\displaystyle z_{Na}=1}$ and ${\displaystyle z_{Cl}=-1}$. The nearest neighbour distance amounts to half the lattice constant of the cubic unit cell ${\displaystyle r_{0}=a/2}$ and the Madelung constants become

${\displaystyle M_{\text{Na}}=-M_{\text{Cl}}={\sum _{j,k,\ell =-\infty }^{\infty }}^{\prime }{{(-1)^{j+k+\ell }} \over {(j^{2}+k^{2}+\ell ^{2})^{1/2}}}.}$

This graph demonstrates the non-convergence of the expanding spheres method for calculating the Madelung constant for NaCl as compared to the expanding cubes method, which is convergent.

The prime indicates that the term ${\displaystyle j=k=\ell =0}$ is to be left out. Since this sum is conditionally convergent it is not suitable as definition of Madelung's constant unless the order of summation is also specified. There are two "obvious" methods of summing this series, by expanding cubes or expanding spheres. The latter, though devoid of a meaningful physical interpretation (there are no spherical crystals) is rather popular because of its simplicity. Thus, the following expansion is often found in the literature:[2]

${\displaystyle M=-6+12/{\sqrt {2}}-8/{\sqrt {3}}+6/2-24/{\sqrt {5}}+\dotsb =-1.74756\dots .}$

However, this is wrong as this series diverges as was shown by Emersleben in 1951.[3][4] The summation over expanding cubes converges to the correct value. An unambiguous mathematical definition is given by Borwein, Borwein and Taylor by means of analytic continuation of an absolutely convergent series.

There are many practical methods for calculating Madelung's constant using either direct summation (for example, the Evjen method[5]) or integral transforms, which are used in the Ewald method.[6]

Examples of Madelung constants

Ion in crystalline compound	${\displaystyle M}$ (based on ${\displaystyle r_{0}}$)	${\displaystyle {\overline {M}}}$ (based on ${\displaystyle w}$)
Cl− and Cs+ in CsCl	±1.762675	±2.035362
Cl− and Na+ in rocksalt NaCl	±1.747565	±3.495129
S2− and Zn2+ in sphalerite ZnS	±3.276110	±7.56585
F− in fluorite CaF2	1.762675	4.070723
Ca2+ in fluorite CaF2	-3.276110	−7.56585

The continuous reduction of ${\displaystyle M}$ with decreasing coordination number ${\displaystyle Z}$ for the three cubic AB compounds (when accounting for the doubled charges in ZnS) explains the observed propensity of alkali halides to crystallize in the structure with highest ${\displaystyle Z}$ compatible with their ionic radii. Note also how the fluorite structure being intermediate between the caesium chloride and sphalerite structures is reflected in the Madelung constants.

A fast converging formula for the Madelung constant of NaCl is

${\displaystyle 12\,\pi \sum _{m,n\geq 1,\,\mathrm {odd} }\operatorname {sech} ^{2}\left({\frac {\pi }{2}}(m^{2}+n^{2})^{1/2}\right)}$[7]

It is assumed for the calculation of Madelung constants that an ion's charge density may be approximated by a point charge. This is allowed, if the electron distribution of the ion is spherically symmetric. In particular cases, however, when the ions reside on lattice site of certain crystallographic point groups, the inclusion of higher order moments, i.e. multipole moments of the charge density might be required. It is shown by electrostatics that the interaction between two point charges only accounts for the first term of a general Taylor series describing the interaction between two charge distributions of arbitrary shape. Accordingly, the Madelung constant only represents the monopole-monopole term.

The electrostatic interaction model of ions in solids has thus been extended to a point multipole concept that also includes higher multipole moments like dipoles, quadrupoles etc.[8][9][10] These concepts require the determination of higher order Madelung constants or so-called electrostatic lattice constants. The proper calculation of electrostatic lattice constants has to consider the crystallographic point groups of ionic lattice sites; for instance, dipole moments may only arise on polar lattice sites, i. e. exhibiting a C₁, C_1h, C_n or C_nv site symmetry (n = 2, 3, 4 or 6).[11] These second order Madelung constants turned out to have significant effects on the lattice energy and other physical properties of heteropolar crystals.[12]

The Madelung constant is also a useful quantity in describing the lattice energy of organic salts. Izgorodina and coworkers have described a generalised method (called the EUGEN method) of calculating the Madelung constant for any crystal structure.[13]

• Glasser, Leslie (2012). "Solid-state energetics and electrostatics: Madelung constants and Madelung energies". Inorg. Chem. 51 (4): 2420–2424. doi:10.1021/ic2023852. PMID 22242970.
• Sakamoto, Y. (1958). "Madelung constants of simple crystals expressed in terms of Born's basic potentials of 15 figures". J. Chem. Phys. 28 (1): 164–165. Bibcode:1958JChPh..28..164S. doi:10.1063/1.1744060.
• Sakamoto, Y. (1958). "Errata 2: Madelung constants of simple crystals expressed in terms of Born's basic potentials of 15 figures". J. Chem. Phys. 28 (6): 1253. Bibcode:1958JChPh..28.1253S. doi:10.1063/1.1744387.
• Zucker, I. J. (1975). "Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures". J. Phys. A: Math. Gen. 8 (11): 1734–1745. Bibcode:1975JPhA....8.1734Z. doi:10.1088/0305-4470/8/11/008.
• Zucker, I. J. (1976). "Functional equations for poly-dimensional zeta functions and the evaluation of Madelung constants". J. Phys. A: Math. Gen. 9 (4): 499–505. Bibcode:1976JPhA....9..499Z. doi:10.1088/0305-4470/9/4/006.
• Weisstein, Eric W. "Madelung Constants". MathWorld.
• OEIS sequence A085469 (Decimal expansion of Madelung constant (negated) for face-centered cubic lattice)