Madelung constant
Encyclopedia
The Madelung constant is used in determining the electrostatic potential of a single ion
in a crystal
by approximating the ions by point charges. It is named after Erwin Madelung
, a German physicist.
Because the anions and cations in an ionic solid
are attracting each other by virtue of their opposing charges, separating the ions requires a certain amount of energy. This energy must be given to the system in order to break the anion-cation bonds. The energy required to break these bonds for one mole of an ionic solid under standard conditions is the lattice energy
.
The Madelung constant shall allow for the calculation of the electric potential
Vi of all ions of the lattice felt by the ion at position ri
where rij =|ri - rj| is the distance between the ith and the jth ion. In addition,
If the distances rij are normalized to the nearest neighbor distance r0 the potential may be written
with
being the (dimensionless) Madelung constant of the ith ion
The electrostatic energy of the ion at site
then is the product of its charge with the potential acting at its site
There occur as many Madelung constants
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,
and 
. The nearest neighbour distance amounts to half the lattice parameter of the cubic unit cell 
and the Madelung constants become
The prime indicates that the term
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:
However, this is wrong as this series diverges as was shown by Emersleben in 1951. 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) or integral transforms, which are used in the Ewald method
.
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, quadrupole
s etc. These concepts require the determination of higher order Madelung constants or so-called electrostatic lattice constants. In their case, instead of the nearest neighbor distance
another standard length like the cube root of the unit cell volume 
is appropriately used for purposes of normalization. For instance, the Madelung constant then reads
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, C1h, Cn or Cnv site symmetry (n = 2, 3, 4 or 6). These second order Madelung constants turned out of having significant effects on the lattice energy
and other physical properties of heteropolar crystals.
Ion
An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. The name was given by physicist Michael Faraday for the substances that allow a current to pass between electrodes in a...
in a crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
by approximating the ions by point charges. It is named after Erwin Madelung
Erwin Madelung
Erwin Madelung was a German physicist.He was born in 1881 in Bonn. His father was the surgeon Otto Wilhelm Madelung. He earned a doctorate in 1905 from the University of Göttingen, specializing in crystal structure, and eventually became a professor...
, a German physicist.
Because the anions and cations in an ionic solid
Ionic compound
In chemistry, an ionic compound is a chemical compound in which ions are held together in a lattice structure by ionic bonds. Usually, the positively charged portion consists of metal cations and the negatively charged portion is an anion or polyatomic ion. Ions in ionic compounds are held together...
are attracting each other by virtue of their opposing charges, separating the ions requires a certain amount of energy. This energy must be given to the system in order to break the anion-cation bonds. The energy required to break these bonds for one mole of an ionic solid under standard conditions is the lattice energy
Lattice energy
The lattice energy of an ionic solid is a measure of the strength of bonds in that ionic compound. It is usually defined as the enthalpy of formation of the ionic compound from gaseous ions and as such is invariably exothermic. Lattice energy may also be defined as the energy required to completely...
.
The Madelung constant shall allow for the calculation of the electric potential
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...
Vi of all ions of the lattice felt by the ion at position ri
where rij =|ri - rj| is the distance between the ith and the jth ion. In addition,
- zj = number of charges of the jth ion
- e = 1.6022 C
- 4 π ε0 = 1.112 C²/(J m).
If the distances rij are normalized to the nearest neighbor distance r0 the potential may be written
with
being the (dimensionless) Madelung constant of the ith ionThe electrostatic energy of the ion at site
then is the product of its charge with the potential acting at its siteThere occur as many Madelung constants
in a crystal structureCrystal structure
In mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry...
as ions occupy different lattice sites. For example, for the ionic crystal NaCl
Sodium chloride
Sodium chloride, also known as salt, common salt, table salt or halite, is an inorganic compound with the formula NaCl. Sodium chloride is the salt most responsible for the salinity of the ocean and of the extracellular fluid of many multicellular organisms...
, 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,
and . The nearest neighbour distance amounts to half the lattice parameter of the cubic unit cell and the Madelung constants becomeThe prime indicates that the term
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:However, this is wrong as this series diverges as was shown by Emersleben in 1951. The summation over expanding cubes converges to the correct value. An unambiguous mathematical definition is given by Borwein
David Borwein
David Borwein is a Canadian mathematician of Ashkenazi Jewish ancestry, known for his research in the summability theory of series and integrals. He has also done work in measure theory and probability theory, number theory, and approximate subgradients and coderivatives. He has recently...
, Borwein
Jonathan Borwein
Jonathan Michael Borwein is a Scottish mathematician who holds an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. Noted for his prolific and creative work throughout the international mathematical community, he is a close associate of David H...
and Taylor by means of analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
of an absolutely convergent series.
There are many practical methods for calculating Madelung's constant using either direct summation (for example, the Evjen method) or integral transforms, which are used in the Ewald method
Ewald summation
Ewald summation, named after Paul Peter Ewald, is a method for computing the interaction energies of periodic systems , particularly electrostatic energies. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an...
.
| Ion in crystalline compound |  (based on  ) |  (based on  ) |
|---|---|---|
| Cl- and Na+ in rocksalt NaCl | ±1.748 | ±3.495 |
| S2- and Zn2+ in sphalerite ZnS Sphalerite Sphalerite is a mineral that is the chief ore of zinc. It consists largely of zinc sulfide in crystalline form but almost always contains variable iron. When iron content is high it is an opaque black variety, marmatite. It is usually found in association with galena, pyrite, and other sulfides... |
±1.638 | ±3.783 |
| S- in pyrite FeS2 Pyrite The mineral pyrite, or iron pyrite, is an iron sulfide with the formula FeS2. This mineral's metallic luster and pale-to-normal, brass-yellow hue have earned it the nickname fool's gold because of its resemblance to gold... |
… | 1.957 |
| Fe2+ in pyrite FeS2 Pyrite The mineral pyrite, or iron pyrite, is an iron sulfide with the formula FeS2. This mineral's metallic luster and pale-to-normal, brass-yellow hue have earned it the nickname fool's gold because of its resemblance to gold... |
… | -7.458 |
Generalization
It is assumed for the calculation of Madelung constants that an ion’s charge densityCharge density
The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...
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
Multipole moments
In mathematics, especially as applied to physics, multipole moments are the coefficients of a series expansion of a potential due to continuous or discrete sources . A multipole moment usually involves powers of the distance to the origin, as well as some angular dependence...
of the charge density might be required. It is shown by electrostatics
Electrostatics
Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....
that the interaction between two point charges only accounts for the first term of a general Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
describing the interaction between two charge distributions of arbitrary shape. Accordingly, the Madelung constant only represents the monopole
Monopole
Monopole may refer to:*Magnetic monopole, or Dirac monopole, a hypothetical particle that may be loosely described as a magnet with only one pole, or related concepts in physics and mathematics:...
-monopole
Monopole
Monopole may refer to:*Magnetic monopole, or Dirac monopole, a hypothetical particle that may be loosely described as a magnet with only one pole, or related concepts in physics and mathematics:...
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, quadrupole
Quadrupole
A quadrupole or quadrapole is one of a sequence of configurations of—for example—electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity.-Mathematical...
s etc. These concepts require the determination of higher order Madelung constants or so-called electrostatic lattice constants. In their case, instead of the nearest neighbor distance
another standard length like the cube root of the unit cell volume is appropriately used for purposes of normalization. For instance, the Madelung constant then readsThe 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, C1h, Cn or Cnv site symmetry (n = 2, 3, 4 or 6). These second order Madelung constants turned out of having significant effects on the lattice energy
Lattice energy
The lattice energy of an ionic solid is a measure of the strength of bonds in that ionic compound. It is usually defined as the enthalpy of formation of the ionic compound from gaseous ions and as such is invariably exothermic. Lattice energy may also be defined as the energy required to completely...
and other physical properties of heteropolar crystals.