|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| [[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 to summing it which give a converging series.]]
| | Another day I woke up and realised - Today I have also been single for some time and after much bullying from buddies I now find myself signed-up for web dating. They guaranteed me that there are plenty of entertaining, pleasant and regular people to fulfill, therefore the pitch is gone by here!<br>My pals [http://lukebryantickets.hamedanshahr.com luke bryan concert dates 2014] and household are magnificent and luke bryan concert tickets 2014 ([http://www.netpaw.org http://www.netpaw.org]) hanging out with them at pub gigabytes or meals is always a must. As I locate you can never get a significant conversation with all the [http://www.bbc.co.uk/search/?q=sound+I sound I] haven't ever been into night clubs. I also have 2 unquestionably cheeky and very adorable dogs that are constantly eager to meet fresh people.<br>I make an effort to maintain as physically fit as possible staying at the gym many times weekly. I love my athletics and make an effort to [http://lukebryantickets.sgs-suparco.org luke bryan live concert] play or view as numerous a potential. Being winter I will regularly at Hawthorn matches. Note: I have observed the carnage of fumbling matches at stocktake sales, If you [http://lukebryantickets.asiapak.net meet and greet with luke bryan] really considered purchasing an athletics I really don't mind.<br><br> |
| 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 | url = | journal = Phys. Zs. | volume = XIX | issue = | pages = 524–533 }}</ref>
| |
|
| |
|
| Because the [[anions]] and [[cations]] in an [[ionic compound|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]].
| | Also visit my site :: luke bryan on sale dates - [http://www.ladyhawkshockey.org like it], |
| | |
| The Madelung constant shall allow for the calculation of the [[electric potential]] ''V<sub>i</sub>'' of all ions of the lattice felt by the ion at position ''r<sub>i</sub>''
| |
| | |
| :<math>V_i = \frac{e}{4 \pi \epsilon_0 } \sum_{j \neq i} \frac{z_j}{r_{ij}}\,\!</math>
| |
| | |
| where ''r<sub>ij</sub>'' =|''r<sub>i</sub>'' - ''r<sub>j</sub>''| is the distance between the ''i''th and the ''j''th ion. In addition,
| |
| :''z<sub>j</sub>'' = number of charges of the ''j''th ion
| |
| :''e'' = 1.6022{{e|−19}} [[coulomb|C]]
| |
| :4 π ε<sub>0</sub> = 1.112{{e|−10}} C²/(J m).
| |
| | |
| If the distances ''r<sub>ij</sub>'' are normalized to the nearest neighbor distance ''r<sub>0</sub>'' the potential may be written
| |
| | |
| :<math>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</math>
| |
| | |
| with <math>M_i</math> being the (dimensionless) Madelung constant of the ''i''th ion
| |
| | |
| :<math>M_i = \sum_{j} \frac{z_j}{r_{ij}/r_0}.</math>
| |
| | |
| The electrostatic energy of the ion at site <math>r_i</math> 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 \epsilon_0 r_0 } z_i M_i.</math> | |
| | |
| There occur as many Madelung constants <math>M_i</math> in a [[crystal structure]] as ions occupy different lattice sites. For example, for the ionic crystal [[Sodium chloride|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<sup>+</sup> and Cl<sup>−</sup> ion are assumed to be onefold positive and negative, respectively, <math>z_{Na}=1</math> and <math>z_{Cl}=-1</math>. The nearest neighbour distance amounts to half the [[lattice parameter]] of the cubic [[unit cell]] <math>r_0=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 { (j^2 + k^2 + \ell^2)^{1/2}}}. </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. 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:<ref>Charles Kittel: ''Introduction to Solid State Physics.'', Wiley 1995, ISBN 0-471-11181-3</ref>
| |
| | |
| :<math>M = -6 +12/ \sqrt{2} -8/ \sqrt{3} +6/2 - 24/ \sqrt{5} + \dotsb = -1.74756\dots.</math>
| |
| | |
| However, this is wrong as this series diverges as was shown by Emersleben in 1951.<ref>O. Emersleben: ''[[Mathematische Nachrichten]]'' 4 (1951), 468</ref><ref>D. Borwein, J. M. Borwein, K. F. Taylor: "Convergence of Lattice Sums and Madelung's Constant", ''J. Math. Phys.'' 26 (1985), 2999–3009, {{DOI|10.1063/1.526675}}</ref> The summation over expanding cubes converges to the correct value. An unambiguous mathematical definition is given by [[David Borwein|Borwein]], [[Jonathan 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<ref>H. M. Evjen: "On the Stability of Certain Heteropolar Crystals", ''Phys. Rev.'' 39 (1932), 675–687, http://link.aps.org/abstract/PR/v39/p675</ref>) or [[integral transform]]s, which are used in the [[Ewald summation|Ewald method]].<ref>P. P. Ewald: "Die Berechnung optischer und elektrostatischer Gitterpotentiale", ''Ann. Phys.'' 64 (1921), 253–287, {{DOI|10.1002/andp.19213690304}}</ref>
| |
| | |
| {| border="1"
| |
| |+ '''Examples of Madelung Constants'''
| |
| ! Ion in crystalline compound !! <math>M</math> (based on <math>r_0</math>) !! <math>\overline{M}</math> (based on <math>w</math>)
| |
| |-
| |
| ! Cl<sup>-</sup> and Na<sup>+</sup> in [[rocksalt| rocksalt NaCl]]
| |
| | ±1.748 || ±3.495
| |
| |-
| |
| ! S<sup>2-</sup> and Zn<sup>2+</sup> in [[sphalerite| sphalerite ZnS]]
| |
| | ±1.638 || ±3.783
| |
| |-
| |
| ! S<sup>-</sup> in [[pyrite| pyrite FeS<sub>2</sub>]]
| |
| | … || 1.957
| |
| |-
| |
| ! Fe<sup>2+</sup> in [[pyrite| pyrite FeS<sub>2</sub>]]
| |
| | … || -7.458
| |
| |}
| |
| | |
| == 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 | author = J. Kanamori, T. Moriya, K. Motizuki, and T. Nagamiya | title = Methods of Calculating the Crystalline Electric Field | journal = J. Phys. Soc. Jap. | volume = 10 | pages = 93–102 | year = 1955 | doi = 10.1143/JPSJ.10.93}}</ref><ref name= Nijb1957>{{cite journal | doi = 10.1016/S0031-8914(57)92124-9 | author = B. R. A. Nijboer and F. W. de Wette | title = On the calculation of lattice sums | journal = Physica | volume = 23 | pages = 309–321 | year = 1957 |bibcode = 1957Phy....23..309N }}</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 | pages = 1331–48 | 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. In their case, instead of the nearest neighbor distance <math> r_{0} </math> another standard length like the cube root of the unit cell volume <math>w=\sqrt[3]{V}</math> is appropriately used for purposes of normalization. For instance, the Madelung constant then reads
| |
| | |
| :<math>\overline{M}_i = \sum_{j} \frac{z_j}{r_{ij}/w}.</math>
| |
| | |
| 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 | pages = 325–332 | year = 1995 | doi = 10.1007/BF01313054 |bibcode = 1995ZPhyB..96..325B | url=http://www.mariobirkholz.de/ZPB1995a.pdf}}</ref> These second order Madelung constants turned out of having 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 – II. physical significance | journal = Z. Phys. B | volume = 96 | pages = 333–340 | year = 1995 | doi = 10.1007/BF01313055 |bibcode = 1995ZPhyB..96..333B | url=http://www.mariobirkholz.de/ZPB1995b.pdf}}</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 et al | title = The Madelung Constant of Organic Salts | journal =Crystal Growth & Design | volume = 9 | pages = 4834–4839 | year = 2009 | doi = 10.1021/cg900656z}}</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 | pages=2420-2424}}
| |
| * {{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|year=1958 | pages=164|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|year=1958 | pages=1253|doi= 10.1063/1.1744387
| |
| |bibcode = 1958JChPh..28.1253S }}
| |
| *{{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
| |
| |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
| |
| |year=1976 |doi=10.1088/0305-4470/9/4/006
| |
| |bibcode = 1976JPhA....9..499Z }}
| |
| * {{MathWorld|urlname=MadelungConstants|title=Madelung Constants}}
| |
| * {{OEIS|A085469}}
| |
| | |
| [[Category:Crystallography]]
| |
| [[Category:Mathematical constants]]
| |
| [[Category:Physical chemistry]]
| |
| [[Category:Solid-state chemistry]]
| |
| [[Category:Theoretical chemistry]]
| |
Another day I woke up and realised - Today I have also been single for some time and after much bullying from buddies I now find myself signed-up for web dating. They guaranteed me that there are plenty of entertaining, pleasant and regular people to fulfill, therefore the pitch is gone by here!
My pals luke bryan concert dates 2014 and household are magnificent and luke bryan concert tickets 2014 (http://www.netpaw.org) hanging out with them at pub gigabytes or meals is always a must. As I locate you can never get a significant conversation with all the sound I haven't ever been into night clubs. I also have 2 unquestionably cheeky and very adorable dogs that are constantly eager to meet fresh people.
I make an effort to maintain as physically fit as possible staying at the gym many times weekly. I love my athletics and make an effort to luke bryan live concert play or view as numerous a potential. Being winter I will regularly at Hawthorn matches. Note: I have observed the carnage of fumbling matches at stocktake sales, If you meet and greet with luke bryan really considered purchasing an athletics I really don't mind.
Also visit my site :: luke bryan on sale dates - like it,