View source for Madelung constant

{{Short description|Constant in crystallography}}
[[File:NaCl-ionlattice-madelung.png|right|thumb|The Madelung constant being calculated for the NaCl ion labeled 0 in the expanding spheres method. Each number designates the order in which it is summed. Note that in this case, the sum is divergent, but there are methods for summing it which give a converging series.]]
The '''Madelung constant''' is used in determining the [[electrostatic potential]] of a single [[ion]] in a [[crystal]] by approximating the ions by [[point charge]]s. It is named after [[Erwin Madelung]], a German physicist.<ref>{{cite journal | author = Madelung E | year = 1918 | title = Das elektrische Feld in Systemen von regelmäßig angeordneten Punktladungen | journal = Phys. Z. | volume = XIX | pages = 524–533 }}</ref>

Because the [[anions]] and [[cations]] in an [[ionic compound|ionic solid]] attract 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]].

== Formal expression ==
The Madelung constant allows for the calculation of the [[electric potential]] {{mvar|V{{sub|i}}}} of the ion at position {{mvar|r{{sub|i}}}} due to all other ions of the lattice

:<math>V_i = \frac{e}{4 \pi \varepsilon_0 } \sum_{j \neq i} \frac{z_j}{r_{ij}}\,\!</math>

where <math>r_{ij} = |r_i-r_j|</math> is the distance between the {{mvar|i}}th and the {{mvar|j}}th ion. In addition,
*{{mvar|1=z{{sub|j}} =}} number of charges of the {{mvar|j}}th ion
*{{math|1=''e'' =}} the [[elementary charge]], 1.6022{{e|&minus;19}} [[coulomb|C]]
*{{math|1=4''πε''{{sub|0}} =}} {{val|1.112|e=-10|u=C<sup>2</sup>/(J⋅m)}}; {{math|''ε''{{sub|0}}}} is the [[permittivity of free space]].

If the distances {{mvar|r{{sub|ij}}}} are normalized to the nearest neighbor distance {{math|''r''{{sub|0}}}}, the potential may be written

:<math>V_i = \frac{e}{4 \pi \varepsilon_0 r_0 } \sum_{j} \frac{z_j r_0}{r_{ij}} = \frac{e}{4 \pi \varepsilon_0 r_0 } M_i</math>

with {{mvar|M{{sub|i}}}} being the ([[dimensionless]]) Madelung constant of the {{mvar|i}}th ion

:<math>M_i = \sum_{j} \frac{z_j}{r_{ij}/r_0}.</math>

Another convention is to base the reference length on the cubic root {{mvar|w}} of the unit cell volume, which for cubic systems is equal to the [[lattice constant]]. Thus, the Madelung constant then reads

:<math>\overline{M}_i = \sum_{j} \frac{z_j}{r_{ij}/w}=M_i \frac{w} {r_0}.</math>

The electrostatic energy of the ion at site {{mvar|r{{sub|i}}}} then is the product of its charge with the potential acting at its site
:<math>E_{el,i} = z_ieV_i = \frac{e^2}{4 \pi \varepsilon_0 r_0 } z_i M_i.</math>

There occur as many Madelung constants {{mvar|M{{sub|i}}}} in a [[crystal structure]] as ions occupy different lattice sites. For example, for the ionic crystal [[Sodium chloride|NaCl]], there arise two Madelung constants &ndash; 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 {{chem2|Na+}} and {{chem2|Cl-}} ion are assumed to be onefold positive and negative, respectively, {{math|1=''z''{{sub|Na}} = 1}} and {{math|1=''z''{{sub|Cl}} = –1}}. The nearest neighbour distance amounts to half the lattice constant of the cubic [[unit cell]] <math>r_0=\tfrac a 2</math> and the Madelung constants become

:<math>M_\text{Na}=-M_\text{Cl} = {\sum_{j,k,\ell=-\infty}^\infty}\!\!\!\!^\prime \ \ {{(-1)^{j+k+\ell}} \over \sqrt{j^2 + k^2 + \ell^2}}. </math>

[[File:NaClMadelungConstant.png|thumb|alt=Madelung Constant for NaCl|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 <math>j=k=\ell=0</math> 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. Although the latter is often found in the literature,<ref>Charles Kittel: ''[[Introduction to Solid State Physics]]'', Wiley 1995, {{ISBN|0-471-11181-3}}</ref>

:<math>M = -6 + \frac{12}{\sqrt 2} - \frac{8}{\sqrt 3} + \frac{6}{2} - \frac{24}{\sqrt 5} + \dots</math>

it fails to [[Convergent series|converge]], as was shown by Emersleben in 1951.<ref>{{cite journal | last1 = Emersleben | first1 = O. | year = 1951 | title = Das Selbstpotential einer endlichen Reihe neutraler äquidistanter Punktepaare | journal = [[Mathematische Nachrichten]] | volume = 4 | issue = 3–4 | page = 468 | doi = 10.1002/mana.3210040140 }}</ref> The summation over expanding cubes converges to the correct value, although very slowly. An alternative summation procedure, presented by [[David Borwein|Borwein]], [[Jonathan Borwein|Borwein]] and Taylor, uses [[analytic continuation]] of an absolutely convergent series.<ref>{{cite journal |last1=Borwein |first1=D. |last2=Borwein |first2=J. M. |last3=Taylor |first3=K. F. |year=1985 |title=Convergence of Lattice Sums and Madelung's Constant |journal=J. Math. Phys. |volume=26 |issue=11 |pages=2999–3009 |bibcode=1985JMP....26.2999B |doi=10.1063/1.526675 |hdl-access=free |hdl=1959.13/1043576}}</ref>

There are many practical methods for calculating Madelung's constant using either direct summation (for example, the Evjen method<ref>{{cite journal | last1 = Evjen | first1 = H. M. | year = 1932 | title = On the Stability of Certain Heteropolar Crystals | journal = Phys. Rev. | volume = 39 | issue = 4| pages = 675–687 | doi=10.1103/physrev.39.675| bibcode = 1932PhRv...39..675E | url = https://authors.library.caltech.edu/4091/1/EVJpr32a.pdf }}</ref>) or [[integral transform]]s, which are used in the [[Ewald summation|Ewald method]].<ref>{{cite journal | last1 = Ewald | first1 = P. P. | year = 1921 | title = Die Berechnung optischer und elektrostatischer Gitterpotentiale | url = https://zenodo.org/record/1424363| journal = Annalen der Physik | volume = 64 | issue = 3| pages = 253–287 | doi = 10.1002/andp.19213690304 | bibcode = 1921AnP...369..253E }}</ref> A fast converging formula for the Madelung constant of NaCl is

:<math>12 \, \pi \sum_{m, n \geq 1, \, \mathrm{odd}} \operatorname{sech}^2\left(\frac{\pi}{2}(m^2+n^2)^{1/2}\right)</math><ref>{{cite journal|last1=Bailey|first1=David|last2=Borwein|first2=Jonathan|last3=Kapoor|first3=Vishaal|last4=Weisstein|first4=Eric|title=Ten Problems in Experimental Mathematics|journal=The American Mathematical Monthly|date=March 9, 2006|volume=113|issue=6|pages=481|url=http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/tenproblems.pdf|doi=10.2307/27641975|jstor=27641975}}</ref>

{| class="wikitable" border="1"
|+ '''Examples of Madelung constants'''
! Ion in crystalline compound !! <math>M</math> (based on {{math|''r''{{sub|0}}}}) !! <math>\overline{M}</math> (based on {{mvar|w}})
|-
! Cl<sup>−</sup> and Cs<sup>+</sup> in [[Cubic crystal system#Caesium chloride structure|CsCl]]
| ±1.762675 || ±2.035362
|-
|-
! Cl<sup>−</sup> and Na<sup>+</sup> in [[rocksalt| rocksalt NaCl]]
| ±1.747565 || ±3.495129
|-
! S<sup>2−</sup> and Zn<sup>2+</sup> in [[sphalerite| sphalerite ZnS]]
| ±3.276110 || ±7.56585
|-
!F<sup>−</sup> in [[fluorite structure|fluorite CaF<sub>2</sub>]]
| 1.762675 || 4.070723
|-
! Ca<sup>2+</sup> in [[fluorite structure|fluorite CaF<sub>2</sub>]]
| -3.276110 || −7.56585
|}

The continuous reduction of {{mvar|M}} with decreasing [[coordination number]] {{mvar|Z}} for the three cubic AB compounds (when accounting for the doubled charges in ZnS) explains the observed [[Cation-anion radius ratio|propensity]] of [[alkali halide]]s to crystallize in the structure with highest {{mvar|Z}} compatible with their [[ionic radius|ionic radii]]. Note also how the fluorite structure being intermediate between the caesium chloride and sphalerite structures is reflected in the Madelung constants.

== Generalization ==
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 (mathematics)|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.<ref name= Kana1955>{{cite journal |author1=J. Kanamori |author2=T. Moriya |author3=K. Motizuki |author4=T. Nagamiya  |name-list-style=amp | title = Methods of Calculating the Crystalline Electric Field | journal = J. Phys. Soc. Jpn. | volume = 10 |issue=2 | pages = 93&ndash;102 | year = 1955 | doi = 10.1143/JPSJ.10.93|bibcode = 1955JPSJ...10...93K }}</ref><ref name= Nijb1957>{{cite journal | doi = 10.1016/S0031-8914(57)92124-9 |author1=B. R. A. Nijboer  |author2=F. W. de Wette  |name-list-style=amp | title = On the calculation of lattice sums | journal = Physica | volume = 23 |issue=1–5  | pages = 309&ndash;321 | year = 1957 |bibcode = 1957Phy....23..309N |hdl=1874/15643 |s2cid=122383484 |hdl-access = free }}</ref><ref name= Bert1978>{{cite journal | author = E. F. Bertaut | title = The equivalent charge concept and its application to the electrostatic energy of charges and multipoles | journal = J. Phys. (Paris) | volume = 39 | issue = 2 | pages = 1331&ndash;48 | year = 1978 | doi = 10.1016/0022-3697(78)90206-8|bibcode = 1978JPCS...39...97B }}</ref> 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 ''C''<sub>1</sub>, ''C''<sub>1''h''</sub>, ''C''<sub>''n''</sub> or ''C''<sub>''nv''</sub>  site symmetry (''n'' = 2, 3, 4 or 6).<ref name= ZPB1995a>{{cite journal | author = M. Birkholz | title = Crystal-field induced dipoles in heteropolar crystals – I. concept | journal = Z. Phys. B | volume = 96 | issue = 3 | pages = 325&ndash;332 | year = 1995 | doi = 10.1007/BF01313054 |bibcode = 1995ZPhyB..96..325B | url=https://www.researchgate.net/publication/227050494| citeseerx = 10.1.1.424.5632 | s2cid = 122527743 }}</ref> These second order Madelung constants turned out to have significant effects on the [[lattice energy]] and other physical properties of heteropolar crystals.<ref name= ZPB1995b>{{cite journal|author = M. Birkholz|title = Crystal-field induced dipoles in heteropolar crystals&nbsp;– II. physical significance|journal = Z. Phys. B|volume = 96|pages = 333–340|year = 1995|doi = 10.1007/BF01313055|bibcode = 1995ZPhyB..96..333B| url=https://www.researchgate.net/publication/226272268 |issue = 3|s2cid = 122393358}}</ref>

== Application to organic salts ==
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.<ref name= Izgorodina2009>{{cite journal | author = E. Izgorodina| title = The Madelung Constant of Organic Salts | journal =Crystal Growth & Design | volume = 9 | issue = 11 | pages = 4834&ndash;4839 | year = 2009 | doi = 10.1021/cg900656z|display-authors=etal}}</ref>

== References ==
<references />

==External links==
* {{ cite journal| first1=Leslie | last1=Glasser
|title=Solid-state energetics and electrostatics: Madelung constants and Madelung energies| doi=10.1021/ic2023852 | journal=Inorg. Chem. | year=2012
|volume=51 | issue=4
| pages=2420–2424| pmid=22242970
}}
* {{cite journal|first1=Y. |last1=Sakamoto | title=Madelung constants of simple crystals expressed in terms of Born's basic potentials of 15 figures
|journal=J. Chem. Phys. | volume=28|issue=1 |year=1958 | pages=164–165|doi=10.1063/1.1744060 
|bibcode = 1958JChPh..28..164S }}
* {{cite journal|first1=Y. |last1=Sakamoto | title=Errata 2: Madelung constants of simple crystals expressed in terms of Born's basic potentials of 15 figures
|journal=J. Chem. Phys. | volume=28|issue=6 |year=1958 | pages=1253|doi= 10.1063/1.1744387 
|bibcode = 1958JChPh..28.1253S |doi-access=free }}
*{{cite journal| first1=I. J. | last1=Zucker | title=Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures
|journal= J. Phys. A: Math. Gen. |volume=8 |number=11 |pages=1734–1745
|year=1975 | doi=10.1088/0305-4470/8/11/008
|bibcode = 1975JPhA....8.1734Z }}
*{{cite journal |first1=I. J. | last1=Zucker | title=Functional equations for poly-dimensional zeta functions and the evaluation of Madelung constants
|journal= J. Phys. A: Math. Gen. |volume=9 |number=4 |pages=499–505
|year=1976 |doi=10.1088/0305-4470/9/4/006
|bibcode = 1976JPhA....9..499Z }}
* {{MathWorld|urlname=MadelungConstants|title=Madelung Constants}}
* {{OEIS el|1=A085469|2=Decimal expansion of Madelung constant (negated) for NaCl structure}}

[[Category:Crystallography]]
[[Category:Physical constants]]
[[Category:Physical chemistry]]
[[Category:Solid-state chemistry]]
[[Category:Theoretical chemistry]]