Textbook Errors, 98 Denis Quane East Texas State University Commerce, Texas 75428
Crystal Lattice Energy and the Madelung Constant
Equations for calculating the crystal lattice energy, U , of ionic solids generally contain, as their principal component, an energy term based on a model in which the ions are considered as point charges, occupying fixed positions in a regular geometric array, with coulombic forces acting between the ions (the Madelung Energy, U M ) . Differences between various equations for the lattice energy arise from the way they take into account those interactions which result from the fact that ions are not motionless point charges: overlap repulsive forces, induced dipole forces, vibrational energy, etc. (1-5). The oldest and simplest of these equations, and the one seen most often in textbooks, is the Born-Lande equation (4). Since the points to be raised here concern the Madelung energy term, which is the same in most lattice energy equations, the following discussion of the Born-Lande equation applies equally well to other equations of the same type. This equation may appear in textbooks in one of three forms
us-- NM%"i"' r
-
4)
(1)
I n these three equations e is the charge on the electron, and r the shortest distance between oppositely charged ions; -ez/r would be the Coulombic energy of a M+X- ion pair. An ion in a crystal lattice is not only affected by a single oppositely charged ion, but is in a potential field determined by all other ions in the lattice. All these attractions and repulsions can be expressed by an infinite series characteristic of the particular geometric array. Procedures for finding the sum of such a series were first described by Madelung in 1918 (5). The result is a constant, M, called the Madelung constant, characteristic of the crystal lattice. I t is itself unitless, not depending on the units used for electric charge and interionic distance. This "conSuggestions of material suitable for this column and guest cob. ~tmnssoitable for publication directly should be sent with as many details as possible, and particularly with reference to modem textbooks, to W. H. Eberhardb, School of Chemistry, Georgia Institute of Technology, Atlanta, Georgia 30332. Since the purpose of this column is t,o prevent the spread and continuation of errors and not the evaluation of individual texts, the sources of errors discussed will not be cited. I n order to be presented, an error must occur in a t least two independent recent standard books.
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stant," however, is not always constant for a particular crystal structure. A[ multiplied by -e2/r gives the potential energy of a single univalent ion located in a particular crystal lattice. Multiplication by N , Avagadro's Number, gives the energy of a mole of ion pairs in a crystal lattice. The expression 1 - (l/n), which contains the Born repulsion exponent, n, corrects for the repulsion energy. With crystal lattices other than those involving only univalent ions, a difference in treatment by various authors appears. Some consider that the Madelung constant should be concerned only with the geometry of the crystal structure, ionic charge being considered separately. This results in eqn. (1) where (z,e) and (z,e) are the charges on the cation and anion, respectively. The geometrical arrangement of the ions in the crystal lattice is not, however, completely independent of ionic charge. A crystal having the rutile structure must involve a compound having MXz stoichiometry with a 2: 1 ratio between the charges on the positive and negative ions. This basic ratio is often included in the Madelung constant. The result is eqn. (2) where z is defined as the highest common factor of the charges on the two ions. z = 1 for NaC1, CaF2,A1203,etc., 2 for ZnS, TiO*,etc., 3 for AlN, etc; zdiffers fromunity only when the charges on the ions are some multiple of the basic ratio determined by the stoichiometry. For AB crystal lattices eqns. (1) and (2) are identical, for all other lattices the two equations are not identical. Since eqn. (2) includes part of the integral charge in the Madelung constant, it is but a short step to including i t completely, giving eqn. (3). This equation is identical to eqn. (1) only for compounds with AB stoichiometry and univalent ions; it is identical with eqn. (2) only for lattices for which z, as defined above, equals unity. The series of papers by Born and Lande in 1918-19, in which the lattice energy equation was first developed, gave thc equation in a form not identical with those given above, but which resembled eqn. (3) in that the integral ionic charges were not separated from the Madelung constant. This is the case not only for the univalent ion, rock salt lattice, discussed in the earliest papers (4, 6), but also for the fluorite (7) and zinc blende (8)lattices, which have non-univalent ions. Born and Lande's original equations gave the "Madelungsche Gitterpotential" for the unit cell, rather than for a mole of the ionic compound. This potential was calculated to be 13.94e2/6 for the alkali halides (41, 38.7e2/6 for the fluorite structure (7),and 61.2e2/6 for
the zinc blende structure (a), where 6 is the length of a side of the unit cell. The numbers 13.94, 38.7, and 61.2, which take the place of M in these expressions, include not only the integral charges on the ions (for CaFz and ZnS) but also the number of ion pairs per unit cell and the relationship between r and 8. Except for crystal lattices having AB stoichiometry and univalent ions, the value of the Madelung constant depends, not only on the geometry of the crystal lattice, but also on whether all or part of the ionic charge is included in the constant, that is, on which equation is to be used. Values of the Madelung constant of common crystal structures for each of the three equations are given in the table. Values of the Madelung Constant Defined in Terms of the Three Equations for Various Common Crystal Structures
Structure
M defined M defined for for Uu = UM = NMz,zies/ NMznen/r r
rock salt (AB)
1.74756
1.74756"
cesium chloride (AB) zinc blende (AB)
1.76267
1.76267'
1.63805
1.63805"
wurtzite (AB)
1.64132
1.64132°
fluorite (A&)
2.51939
5,03878
rutile (ABd
2.3850
4.7701
@-quartz (AB2)
2.201
4.402
cuprite (A2B) corundum (A.Bs)
2.22124 4.44248" 4.040 24.242
M defined for
U M = NMeVr 1.74756 (MfX-)
". -"---
fi 4Qn74 \IM*+X*-) .. --
1.76267 (M +X-) ' 7.05068 (Ma+X2-) 1.63805 (M+X-) 6.55220 (M'+Xa-) 1.64132 (M+X-) 6.56528 (M2+Xa-) 5.03878 (Mz"2X-) 20.15512 (M4+2X2-) 4.7701 (M112X-) 19.0803 (M'YXP-)b 4.402 (M2%X-) 17.609 (M4t2X2-)L 4.44248 (2MtXP-) 24.242 (2M"3X2-)b
* Reference (1). a Reference (it), values for the rutile, @-quartz,and corundum
structures were calculated specifically for TiOz, SiOs and AI@a, respect,ively; a. mare accurate value for other compounds hsvmg one of these structures would take into account the axial ratio (c/a) of the particular compound (P). Values quoted for these three structures in most textbooks are older values from reference (9).
All other values in the table are calculated fiom these literature values.
Most current textbooks discuss lattice energy in terms of eqn. (1) (or an equivalent form of the BornMayer (9) equation, in which the correction for repulsion takes a different form), apparently since with such an equation, it is easier to explain the role of ionic charge in the calculation. Tabulations of values for the Madelung oonstant in the review literature are almost invariably in terms of eqn. (2) (1-3). Most articles reporting calculations of the lattice energy use equations having this form (lo), although occasionally eqn. (1) will be found (11) and a t least one tabulation of calculated values of the constant is in terms of eqn. (3) (18). I n view of this difference between the treatment of the lattice energy equation in textbooks and in the literature, it is not surprising that a considerable degree of confusion exists in textbooks concerning the definition and value of the Madelung constant. Situations where such confusion is present are as follows The values quoted for the Madelung wnstant for various struetures may not he consistent with the equstion used. Typically s n equation similar to eqn. (1) is developed, but values of the Madelung constant for non-AB structures sppropriate to eqn.
(2) are quoted from the literature. It might be noted that this error is d m found in one standard review article on the subject. A less commou mistake, but one found in several texts, is development of an equation similar to eqn. (31, with values of the Madelung constant quoted which do not take into account the fact that with this equation, the Madelung oonstant may take different values for crystals having the same structure, but with different ionic charges. Thus 1.638 and 4.816 are validvalues of the Madelung constant (in terms of eqn. (3)) for some compounds having the zinc blende and rutile slruclures, respectively, but not for ZIIS sndTiOn. A number of texts which present equations having a form similar to eqn. (2) either do not define z a t all, or define i t as the valency of the ions, thus limiting application of the equation to AB Iatt,ices. . This is particularly confusing if values of the Madelung wnstant we then quoted for non-AB lattices. I n one text, z is defined as "the absolute value of the charge on the negat,ive ions of the lattice" which (provided that the term "absolute value of the charge" is taken as meaning the integral charge) is equivalent to the usual definition for many structures, but not all, e.g., cuprite (Cu1~0) and corundum (AbOs) have z = 1. Some texts after developing the Madelung equation in a form similar t o eqn. ( I ) give no values for the Madelung constant other than for AB structures, such as the rock salt, cesium chloride, or wurtzite structures, for which the vslues appropriat,e t o eqns. (1) and (2) m e identical. This evasion of the issue, while avoiding error, has an indirect tendency to mislead. A student going from such a text to a literature tabulation, seeing that the vslues in his text for AB structures are the same a s those tabulated, is not likely to he aware that the values tabulated for non-AB structures cannot he used with the equation given in his text.
Of twenty-five textbooks in inorganic and physical chemistry surveyed, only five present the lattice energy equation with all terms clearly and correctly defined, together with sufficient consistent values of the Madelung constant to make application of the equation clear. Two of theseuse an equation of form (1) (13, Id), three of form (2) (16-1 7). One of the former (IS) commendably adds a warning that other texts tabulate values of the Madelung constant inconsistent with its own definition. As we have seen with the discussion of Born and Lande's original work, the lattice energy equation may be set up in terms of lattice parameters other than the shortest interionic distance. Rewriting eqn. (2) in more general form, we obtain
where 1 is any lattice parameter, and M , is a value of the Madelung constant consistent with this parameter. Thus, values of the Madelung constant can he found tabulated in terms of the cube root of the molecular volume (8,IX) or of the length of a side of the unit cell (I,@. Surprisingly, this has caused less confusion than might be expected. Almost all texts set up the equation in terms of 7 , and use values of the Madelung constant calculated for that parameter. Only one text does not follow this procedure, setting up the lattice energy equation in terms of the length of a side of the unit cell, a. Values of the Madelung constant for the CsCl and NaCl structures are then stated to be 1.7626 and 1.7476, respectively; these are correct values of M,; Ma for these structures would be 2.03536 and 3.49513, respectively (I). It would he advisable for authors of textbooks to: Jirst, develop the lattice energy equation in a form equivalent to eqn. (2), since tabulated values of the Madelung constant will he found to be most often conVolume 47, Number 5, May 1970
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sistent with such an equation; second, define all terms carefully, particularly z, the meaning of which is not likely to be clear unless sufficient examples are presented; third, give warning that values of the Madelnng constant found in the literature may not always be consistent, and care should be taken in using them. Literature Cited (1) Tow. M.P., Solid Slalc Phys., 16.1 (1964). (2) WIDDINGTON, T.C.. Aduon. Inore. Chem. Radiochem., I, 158 (1959). Chem. Rou.. 11.93 (1932). (3) SXERMAN.J., (4) Born, M..A N D LANDE, A , , S i l d e i . Deul. A t o d . Wiss.,Berlin, 45, 1048 (1918). ( 5 ) M ~ o m . u n o , E . , P h y s i kZ., . 19,524 (19181. M..A N D LAND=, A,, VwhondL Deul. PhyrikGes., 20,210 (1918). (6) BORN, ( 7 ) LANDE. A,, V w h m d . Deut. Physik. Ccs., 20,217 (1918). E., Verhandl. Deul. Physii. Ces., 21, 733 (8) B O R N , M.,A N D BORMANN. (1919).
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~ , R., A N D PORTEB.G. B., "Intmdu~tion t o P h m i d Inor(13) H A R Y E K. ganic Chemistry," Addison-Wealey Publishing Co., Reading, Mass.. 1963, p. 29. 35. (14) G m s w o m . E., "Chemiosl Bonding and Structure," Raytheon EducationCo.. 1958, p. 115-8. (15) Douab*s. B. E., A N D MCDINIEL. D. H.. "Conoepts hndModels of Inorganic Chemistry,"Blhisdell Publishing Co., Waltham, Mass., 1965, P. 113-5. L.. "The Nature of the Chemical Bond:' 3rd ed.. Cornell (16) PAULINO, U. P., Ithsos, N. Y., 1960,~.507-9. (17) PHILLIPB, C. S. G . . A N D WILLIAMB,R. J. P.."Inorganic Chemistry." Oxford U. P., New York, 1965, Vol. I , p. 147-9. E. B.. J . P h y ~ .Chem., 69, 3611 (18) Bn~orelm,T. E . . nlro BRACKETT. (1965).
