Wilbur B. Bridgman Worcester Polytechn~clnstltute Worcester, Massachusetts
Calculation of Madelung Constants by Direct Summation
M a n y textbooks in general and physical chemistry (1-4) introduce the Madelung constant in their considemtion of the lattice energy of ionic crystals. It may be thought of as a proportionality constant that measures the increased binding energy in a lattice of ions as compared to the coulomhic potential between a pair of ions. Thus the constant is often presented as the result of summing the repulsions and attractions acting on a particular ion in a given crystal structure. For example, in a cesium chloride type lattice, a cation considered to he a t the origin is surrounded by a spherical shell of 8 anions. Defining R as the distance between the centers of these nearest cation-anion neighbors, the next shell consists of 6 cations a t 1.155 R. Then there are 12 more cations a t 1.633 R, followed by 24 anions a t 1.915 R. The contributions of these shells to the Madelung constant are shown in the expression
The reader is often left with the impression that the Madelung constant can he evaluated readily by continuing this series. Actually, the summing of the above terms, along with subsequent ones, results in values that fluctuate widely about the accepted value of the Madelung constant for cesium chloride. The early workers in this field were aware of the failure of this series to converge, and resorted to other procedures for evaluating the Madelung constant. An excellent summary of the methods that have been used has been made by J. Sherman (5). One of the most widely used procedures was developed by Ewald (6). I n his book, Iiittel (7) gives the following interpretation of Ewald's method. I n addition to the terms arising from point charges a t the lattice points, other terms are included corresponding to Gaussian distributions of charge a t each lattice point. These additional Gaussian terms are both added and subtracted so as to produce no net effect on the total value of the series. However, with these additional terms included, the summation can be divided into two series, each of which converges very rapidly. By such methods highly precise values have been reported. The values 1.747558 for sodium chloride type lattices and 1.762670 for cesium chloride type lattices calculated by Emersleben (8) are extensively used in the literature. For those interested in more detail about these calculations, references (5) and (7) are recommended. The modern computer makes it relatively easy to
' Ack~~owledgmentis made of the cooperation of the Worcester Area College Computation Center in rurrning these progl.amu on t,he IBM 360-40 at their facility. 592
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continue the summation of a series such as that started for cesium chloride in the first paragraph. The author thought it would be instructive and interesting to see if reliable values of Madelung constants could be obtained by such a direct summation over a large number of ions. Fortran programs were set up for summations hased on hoth sodium chloride and cesium chloride lattices.' While they did not prove to he as satisfactory as the procedures referred to above for precise evaluation of the Madelung constant, the results did illustrate the behavior of these series and served to bring out some characteristics of the ion lattices used. For purposes of computation the terms in the series may he organized either on the basis of spherical shells (as in the above example) or, alternatively, on the basis of cubic shells. Calculations were made with each type of series for hoth lattices. Representotive Results from Madelung Constant Calculation for Sodium Chloride" Radiusb Catioils
Anious
Net chareec
Md
M1Cor.V
"Based on summat,ion aver all ions within a sohere abmt a. given cation. "Itadins of sphere expressed as mdtiple of distance between neishhors. centers of nearest ~~.~ " Excess of cations over anions Sum of all terms contributiug to Madelnng ooustmt. " Sum corrected to correspond to an electricdly neutral w i t . ~~
~~~~~~
The table shows representative results calculated for a sodium chloride type lattice with the series hased on spherical shells about a central cation. The next to the last column gives the sum obtained as the total effect of all ions within a given radius. It can he seen that these values fluctuate widely and show no tendency to approach the accepted value of the Madelung constant. A major cause of this discrepancy is that the spherical unit of any given radius is not electrically neutral. The fourth column in the table shows these net charges. There is considerable correlation between the values
in the fourth and fifth columns. This indicates that the results should he improved by considering electrically neutral units. The last column gives values for units that have been made electrically neutral by the arbitrary device of adding (or subtracting as required) additional cations located on the outer surface of the spherical unit. This does bring the values for the larger units closer to the accepted value, but there are still deviations of the order of +0.1 from Emersleben's value. Calculations were continued on this basis to include spheres containkg more than 17,000 ions, with results that still showed the same degree of variability as the sample given in the table. One concludes that the effect of the ious added to the surface of the sphere to obtain neutrality gives only a crude approximation to the effect of the bulk of the crystal. When cubic shells are used as the basis of carrying out the summation for a sodium chloride lattice, one finds that successive layers produce net charges that alternate between an excess of one cation and an excess of one anion. The corresponding uncorrected sums for the larger units fluctuate in a range between 1.72 and 1.78. The most successful procedure for correcting to an electrically neutral unit, was to weight the contributions of the ious in the surface layer by factors of 1/2, 1/4, or 1/8, depending on the respective locations of the ions on a face, an edge, or a corner of the cubic unit. This procedure has been previously used by Evjen (9). It corresponds to the customary weighting of the surface ions in determining the number of formula units per unit cell. Corrected to neutrality in this manner, a cubic unit of 4913 ions gave a value of 1.747559 in close agreement to Emersleben's value. However, this was apparently fortuitous, as the calculated value continued to decrease as the effects of more layers were included. For a cube composed of 68,921 ions, the calculated value was 1.747163. When calculations were made for a cesium chloride lattice using spherical shells, the results were comparable to the sodium chloride case. The calculated values of the Madelung constant fluctuated about the accepted value in a similar manner and with deviation of about the same magnitude as shown for the sodium chloride case in the table. For cubic units, alternate layers in a ccsium chloride
lattice are either all catinits of all anio~~s.As a result, the net charge alternates betwen positive arid ncgativc values. The magnitude of this nct charge increases rapidly with the size of the unit. The uncorrected values of the summation varied correspondingly. Use of Evjen's procedure to obtain elect,rical neutrality, again improved the results. However, in this case different values were obtained, depending on the composition of the surface layer. With a cat,ion a t the origin, cubic units with cation surfaces gave values that increased from 3.11960 to 3.12171 as the total number of ions increased from 2,331 to 17,261. The cubes with anions on the surface gave values that decreased from 0.40651 to 0.40344 as the number of ions varied from 1729 to 14,859. Averaging values from units with cations on the surface vith value from units having anions on the surface gives more reasonable results. Thus the average of the values for the two largest units given above is 1.76258 as compared to Emersleben's value of 1.762670. I n conclusion, these calculations have demonstrated that summations over fairly large units that are adjusted to electrical neutrality, will produce values of the Madelung constant,^ close to the accepted values. However, the uncertainty in these results is greater than that implied in the accepted values obtained by more sophisticated methods. The exercise of setting up and
.constant .
and in illustrating geometric properties of cryst,al lat,tices. Literature Cited ( 1 ) MAHAN, B. IT., "Univerrity Chemistry,'' Addison Wesley
Publishing Co., Reading, Mass. 1965, pp. 402404. "Inorg&nicChemistl.y," John Wiley & Sons, Inc., New $mk, 1952, p. 183. (3) Mooiw, W. J., "Physical Chemixt,ry," Prent,ice-IIall, Ioe., Englewood Cliffs, N. J., (3rd Ed.), 1962, p. 693. L., "The Nat,we of the Chemical Bond," Cornell (4) PAELING, U. Press, Ithacs, N . Y., (3rd Ed.), 1960, pp. 507-508. (6) S H ~ M I N J.,, C h m ~R . m , 11, 93 (1932). ( 6 ) E ~ A I . P. D ,P., Ann. Physik 64, 253 (1921). (7) K I . ~ ~ C., L , "InRodlrction 1.0 Solid State Physics," John Wiley & Sons, Inc., New York, (2nd Ed.), 1956, pp. 74-77, 571-57.5. (2)
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Volume 46, Number 9, September 1969
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