In the Classroom
An In-Depth Look at the Madelung Constant for Cubic Crystal Systems Robert P. Grosso Jr., Justin T. Fermann, and William J. Vining* Department of Chemistry, University of Massachusetts, Amherst, MA 01003-4510; *
[email protected] Solid-state and structural chemistry need to become a more integral part of the undergraduate chemistry curriculum (1). The dearth of time and effort spent teaching solid-state chemistry is particularly noticeable because many of the most exciting recent scientific discoveries and applications pertain to solids and solid-state devices, from semiconductors to superconductors (2), from high-tech photonic devices to improved petroleum catalysts. It is further disconcerting that the few core solid-state chemistry concepts that are being taught are sometimes taught in an unclear way. This is especially true of the Madelung constant and series. The Madelung constant is worth understanding clearly. In addition to its role in the fields of theoretical physics and chemistry, the Madelung constant enjoys prominence in the fields of applied chemistry and materials science. It is important in determining properties of high Tc copper oxides (3), metal–oxide interfaces (4), and differences between bulk and surface electronic potentials and isoelectronic impurities (5). The Madelung constant with the accompanying Madelung energy is one of several factors (others being radius ratio [6 ], synthetic conditions, free energy of formation, etc.) that can help predict structure. The Madelung constant is one of the terms contained in the lattice energy formula for ionic solids. It takes into account the electrostatic interactions between all the ions in the solid. It accomplishes this by choosing an ion as a reference and then looking at the charge of the surrounding ions, the number of ions, and the distances of these ions from the reference ion. The result of taking into account all of the electrostatics is an infinite series, the Madelung series, in which each term represents a group of ions located at a particular distance from the reference ion. The series starts with terms that take into account the ions closest to the reference ion and progresses to ions at greater distances. Therefore, the further one goes in the Madelung series, the farther the ions are from the reference ion. Sodium chloride is typically chosen as an example. It is a relatively simple and straightforward structure and lends itself well to illustrate the Madelung series. All bonds between ions are orthogonal, an ion is present at all unit distances along all three axial directions, and there are no vacancies or distortions. However, this does not preclude the Madelung series from being presented in an abstruse way. It is common for the first six terms of the series to be presented. One can infer from these first six terms that there is some regularity in the denominators and that these denominators refer to ever increasing distances from the reference ion, but one remains benighted in regard to the origin of the numerators, which seem to be random. Clearly, presenting the first six terms does not elucidate how the series is generated. Students are generally taught what the Madelung constant and series are. They are taught that the series is a summation of all electrostatic interactions as one progresses to greater 1198
distances from a reference ion. They are also taught that the result of this infinite summation is the Madelung constant and is unique for every ionic crystal structure prototype. (Although, depending on which Born–Lande equation is used, even the “constant” can vary from structure to structure [7].) What remains obscure is what the terms of the Madelung series actually are and how they are generated. The first six terms of the sodium chloride Madelung series provide little insight into the generation and values of the rest of the terms. It would be beneficial to find a way to illustrate not only what the Madelung series is, but also what all the terms of the series are and how each term is generated.1 The following is an in-depth look at the Madelung constants for cubic crystal systems. Generating the Madelung Series The first six terms of the sodium chloride Madelung series are presented by way of example.
A = + 6 – 12 + 8 – 6 + 24 – 24 1 2 3 4 5 6 The reason the first six terms are presented and no more seems to be that the seventh term is the point at which the first anomaly manifests itself. If one were to try to infer what the seventh term is, based on the first six, one would guess it is some positive number divided by radical 7. In reality there is no radical 7 term, or, more accurately, the seventh term is 0/(7)1/2. Once one understands what the terms mean and how the series is generated, the value of this term becomes clear. The denominator in each term refers to distance from a reference ion. The bond distance is taken as unity, so the first distance is one, or one unit away. All distances in the denominators then follow. Each numerator refers to the number of ions at the particular distance. Lastly, the sign of each term shows whether the ions are of like or unlike charges. The sodium chloride crystal lattice easily lends itself to being mapped in three dimensions. It is a grid with three traditional orthogonal axes, x, y, and z, with ions located at integral unit distances along any and all of the axes. One can choose any ion in the lattice, and an x,y,z value with respect to a reference ion at (0,0,0) can describe its location. If one thinks of the sodium chloride lattice in this way, a collection of integral x,y,z points in three dimensions, the generation of each Madelung term becomes clear. The first term is at a distance of one unit from the reference ion at (0,0,0). This means one unit along any of the three axes in either direction, which translates to six points (Fig. 1). There are six ions located a distance of one unit. This accounts for both the numerator and denominator of the first term. The term is positive because the ions are of opposite charge to the reference ion.
Journal of Chemical Education • Vol. 78 No. 9 September 2001 • JChemEd.chem.wisc.edu
In the Classroom
generated. The square of the distance is incremented by one, an x,y,z value is found that corresponds to that distance, a multiplicity is calculated from the x,y,z value, and the sign of the term is determined (odd-numbered terms are positive and even-numbered terms are negative).
Figure 1. Left: the first 6 nearest neighbors and right: the first 6 (light gray) plus the next 12 nearest neighbors in the NaCl structure (dark gray). The 12 nearest neighbors are located at a distance of radical 2 from the reference ion in the center.
The second term is at a distance of radical 2 units, or one unit along one axis and one unit along another: for example, at the point (1, 1, 0). The advantage of thinking of the ions in the sodium chloride lattice as x,y,z points now becomes apparent. Distances can be calculated as the sum of the squares of the x,y,z values, since the reference ion is at (0,0,0). Since there are 12 different ways to go one unit along one axis and one unit along another, there are 12 ions located a distance of radical 2 (Fig. 1). Thus the first six terms of the series are generated, by going a certain distance corresponding to an x,y,z value from the reference ion and counting the number of ions at that distance. The reason there is no radical 7 term in the Madelung series for sodium chloride is the same as the reason for no hkl line number 7 in an X-ray spectrum. There are no x,y,z values such that x 2 + y 2 + z 2 = 7, so there are no ions at a distance of radical 7, and no radical 7 term in the Madelung series. The seventh term of the series is therefore 0/(7)1/2. Every term of the Madelung series can be generated by incrementing the square of the distance by one unit, seeing if there are any x,y,z values that correspond to that distance, and then counting the number of ions at that distance. Realistically, counting ions is a cumbersome and unnecessary exercise. Once an x,y,z value is found for the distance, the number of ions at that distance can be calculated. It is equivalent to the number of ways the x,y,z values, positive or negative, can be arranged. For example, there are six ways to arrange (1,0,0) to give a distance of one, namely, (1,0,0), (0,1,0), (0,0,1), (1,0,0), (0,1,0), and (0,0,1), so there are six ions located one unit away. Another way of stating this is to say that the multiplicity of x,y,z = (1,0,0) is 6. A multiplicity can be calculated from any x,y,z value from two quantities: the number of different values for x, y, and z (Ndiff) and the number of nonzero x, y, and z values (Nnonzero). Once these two quantities are known, the multiplicity of the x,y,z value can be calculated using the following formula:
Multiplicity=
3! × 2 Nnonzero 4 – N diff !
Using x,y,z = (1,1,0) as an example, the number of different values for x, y, and z is 2: 1 and 0. The number of nonzero values is also 2, so the multiplicity is
3! × 22 = 3 × 2 × 1 × 22 = 3 × 4 = 12 2×1 4–2 ! and so the second term of the Madelung series is 12/(2) . Thus each and every term of the Madelung series can be 1/2
Generating the Madelung Series for Cesium Chloride and Zinc Blende The Madelung series for cesium chloride and zinc blende can likewise be generated, since they are both cubic crystal systems. In these series the cube chosen would have ions located at the eight corners (for CsCl) or the four opposite corners (for zinc blende). In each case a new unit distance of radical 3 is chosen instead of unity. This puts the corner ions at the (1,1,1) position, as in the case for NaCl. The two Madelung series for these crystals are generated in a slightly different way from that for NaCl, but to discuss the details of these differences is beyond the scope of this paper. Suffice it to say that the same multiplicities apply to the same x,y,z positions and the entire Madelung series for CsCl or zinc blende can be generated to as many terms as desired. Convergence of the Madelung Series Once all the terms of a Madelung series can be generated, the next logical step is to sum the series and calculate the Madelung constant. If one tries to calculate these constants by summing the infinite series, one is confronted with the next problem with the Madelung series: it doesn’t converge (8–10). A random example from the sodium chloride series is shown in Table 1. After 346,000 terms the series isn’t close to converging. The same holds for the other Madelung series. The rigorous mathematical reasoning behind the convergence or lack of convergence of the series can be complicated and it is beyond the scope of this paper (11, 12). Simply put, the Madelung series is conditionally convergent; it converges only if certain conditions are imposed upon it. The condition these series lack is charge neutrality. As shells of ions are Table 1. Nonconvergence of the NaCl Madelung Series Calcd Madelung Constant
Term No.
Term
346,000
1152/ (346,000)1/2
5.277831078
346,001
+8832/ (346,001)1/2
9.736997604
346,002
4608/ (346,002)1/2
1.903185248
346,003
+1920/ (346,003)
1/2
5.167269230
346,004
5112/ (346,004)1/2
3.523341179
.
.
.
.
.
Table 2. Sample Terms from NaCl Madelung Series Showing Net Charge on Ion Cluster Term No.
Term
Calcd Madelung Constant
Charge
346,000
1152/ (346,000)1/2
5.277831078
4132
346,001
+8832/ (346,001)
1/2
9.736997604
4700
346,002
4608/ (346,002)1/2
1.903185248
92
346,003
+1920/ (346,003)1/2
5.167269230
2012
346,004
5112/ (346,004)1/2
3.523341179
3100
.
.
.
.
.
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conceptually added about the central reference ion, the cumulative charge on the entire cluster of ions is rarely zero. The only instances in which the charge would be zero are those in which the number of positive ions is equal to the number of negative ions. Table 2 shows the same terms for the sodium chloride series as before with an extra column showing the net charge on the cluster of ions being considered.2 It is evident that the charge varies greatly from the ideal value of 0, and it is this variance from neutrality that causes the calculated Madelung constant to be so erroneous and erratic. In fact, when the charge on the ion cluster approaches neutrality the calculated Madelung constant approaches the literature value of 1.74756. For example, term number 342,648 has a net charge of 2 and the calculated Madelung constant is 1.748042464, close to the ideal value. The same holds true for the other Madelung series. The problem now becomes: how does one sum up the Madelung series so that a neutral collection of ions is always being considered? The way the series have been generated thus far was to count the number of ions residing at greater and greater distances. Since these distances are in the denominators of the series, it would appear that the series should converge, the denominators becoming larger and larger as the series progresses. However, as one proceeds to greater distances, the corresponding multiplicities increase just as rapidly. The ions are farther away, but there are more of them. Since these multiplicities are contained in the numerators, the values of the terms do not decrease as the series progresses, so the series doesn’t converge; it merely oscillates randomly, giving no indication of the value of the Madelung constant. The need is clear for an imposition of charge neutrality on the series. Charge neutrality can be accomplished by summarizing concentric cubes around the center ion rather than concentric spheres (13). In the unconditional Madelung series the square of the distance is incremented by one unit; then a sphere having this distance as its radius is inspected and any ions lying on it are counted. Each ion is counted as a whole ion. As it turns out, ions considered in this way are always of the same charge, so the only time a neutral ion cluster is being considered is when coincidence leads the number of ions in the spherical shell being added to the cluster to cancel out the charge of the cluster. This is rare and becomes rarer as the series progresses. Adding concentric cubes, on the other hand, gives one more control over the charge of the ion cluster and leads to a convergent series. The Conditionally Convergent Sodium Chloride Madelung Series Sodium chloride provides the most straightforward example of imposing charge neutrality via concentric cube addition to create a conditionally convergent series. A center ion is chosen, but instead of considering the first six nearest neighbors in a spherical shell, a cubic shell is considered that has the first six nearest neighbors at the center of six cubic faces. The distance of each face from the center ion is one unit, making the length of each face two units. The first cubic shell would contain not only the first six nearest neighbors at the center of each face, but also the next 12 nearest neighbors a distance of radical 2 units from the center 1200
Figure 2. The first concentric cube around the reference ion in the NaCl structure. Faces of the cube are semitransparent to show fractional occupancy of ions located on the surface of the cube.
Figure 3. The second concentric cube in the NaCl structure. The first concentric cube is completely contained within the second. Again, faces of the cube are semitransparent to show fractional occupancy of ions on the surface of the cube.
ion at the center of each edge. Also included are the eight next nearest neighbors a distance of radical 3 units from the center ion at the eight corners. The first cubic shell considers a total of 26 ions instead of 6 (Fig. 2). However, in order to keep the cube of ions neutral, only ions, or portions thereof, within the cube are counted. Thus each face ion is counted as 1⁄2 an ion. Each edge ion is counted as 1⁄4 and each corner is counted as 1⁄8. Considering only the first cubic shell, the Madelung series becomes
A = 1 + 6 + 1 12 + 1 + 8 = 1.456029926 2 4 8 1 3 2 This value is closer to the actual value of the sodium chloride Madelung constant, 1.74756, than the value obtained from the first three terms of the unconditional Madelung series, which is 2.133520779. The charge of the series above is easy to solve:
Charge = 1 +6 + 1 12 + 1 +8 = +1 2 4 8 The charge value makes sense because the center ion, which would be of opposite charge, isn’t being counted. If it were, the charge would be 0. The total sum of all charges within the first cube is 0.
Journal of Chemical Education • Vol. 78 No. 9 September 2001 • JChemEd.chem.wisc.edu
In the Classroom
A look at the x,y,z values reveals a lot about how this new series is generated. The first term (faces) corresponds to the (1,0,0) position and all other multiplicities. The second term (edges) corresponds to (1,1,0), and the third term corresponds to (1,1,1). All the terms of the first cubic shell can be generated by considering all x,y,z values in which at least one of the values (x, y, or z) must be 1 and none of the values greater than 1. Likewise, to generate the second cubic shell, one must consider all x,y,z values such that at least one of the values is 2 and none is greater than 2. Thus the x,y,z values for the second cubic shell are (2,0,0) and all multiplicities, as is true for (2,1,0), (2,1,1), (2,2,0), (2,2,1), and (2,2,2). The x,y,z values also show which ions are located on the faces, the edges, and the corners of the second cube (Fig. 3). If only one of the x,y,z values is 2, this corresponds to a face ion. Thus (2,0,0), (2,1,0), and (2,1,1) are all face ion positions. If two values of x,y,z are 2, this corresponds to an edge ion: (2,2,0) and (2,2,1) are edge ions. If all three x,y,z values are 2, this is a corner ion: (2,2,2) is the corner position. Knowing this, it is possible to take a look at the series generated from the first two concentric cubes:
6 + 24 – 24 A = + 6 – 12 + 8 + 1 + 2 5 6 4 1 2 3 1 12 + 24 + 1 8 = 1.751769133 8 4 9 8 12 The ions in the first cube are included in the series, but now, since they are completely contained within the second cube, they are no longer modified with fractional coefficients. Notice also that the denominators are no longer linear. The radical 10 and radical 11 terms are missing because the x,y,z values corresponding to these distances are (3,1,0) and (3,1,1), respectively. Since the highest value for x, y, and z is 3, this would correspond to the third cubic shell. Also, the multiplicity for the radical 9 term is 24. This is because the multiplicity of x,y,z = (2,2,1) is 24. The full multiplicity for radical 9 is 30 because the (3,0,0) term is also included. It has a multiplicity of 6, which, when added to 24, yields the total multiplicity of 30. The (3,0,0) portion of the radical 9 term isn’t included in the second shell for the same reason as stated previously. The result of the summation is also worth looking at. It is obvious that after only two cubic shells the value from the series is already close to the literature value (1.74756). After a similar nine terms of the unconditional Madelung series the result is an erroneous 5.826046944. If the series is summed over three concentric cubes the result becomes 1.747041565. After only 90 or so cubic shells the series has completely converged to a value of 1.747564595. Thus the series converges rapidly, simply by maintaining charge neutrality by summing up concentric cubes. The conditionally convergent Madelung series for sodium chloride can be generated in the following way. 1. The number of cubic shells to be summed, n, is chosen. 2. Each term is incremented not by distance, but by x,y,z value. This ensures that one cubic shell is added at a time. In this way x,y,z = (1,0,0) would supply the first term. Following this would be terms from (1,1,0), (1,1,1), (2,0,0), (2,1,0), (2,1,1), (2,2,0), etc.
3. Multiplicities, distances, and signs of the terms are generated the same way as in the unconditional Madelung series until the nth cube is reached. 4. Upon reaching the nth cube, the outermost cubic shell, only a fraction of each of the ions is considered. The ions are located on either a cubic face, edge, or corner. If (x,y,z) = (n,y,z) the ion is located on a face, so its term is multiplied by 1⁄2. If x,y,z = (n,n,z) the ion is located on an edge, so the term is multiplied by 1⁄4. If x,y,z = (n,n,n) the ion is located on a corner and the term is multiplied by 1⁄8. This ensures that charge neutrality is maintained upon summation of the series.
The Conditionally Convergent Cesium Chloride and Zinc Blende Madelung Series The Madelung series for CsCl and zinc blende can likewise become convergent if concentric cubes are used to summarize each series. Each series, however, converges in its own particular way. When adding concentric cubes for the CsCl structure, one discovers that each cube contains ions all of like charge and the sign of the charge alternates with each concentric cube. This creates a double layer that has a potential associated with it (13, 14), and the conditionally convergent CsCl series converges upon two limits that differ by an amount equal to this double layer. If one averages the two limits, one negates the effect of the double layer on the series, and it converges after 980 terms upon the correct value of 1.762674773. The conditionally convergent zinc blende series converges upon three limits. This occurs because there are three different charges associated with the cluster of ions being considered after each concentric cube summation. Again, if one takes the average of these limits, the series converges upon the correct value of 1.638055062. Conclusion A simple, clear, and visual method by which the Madelung constant can be taught to students has been presented. Theoretically, this method can be expanded to include other cubic systems such as fluorite and perovskites (15). This is a first step to help students become more aware of the importance and practicality of solid state chemistry. Once some of the ways by which core concepts in solidstate chemistry are taught are improved, other concepts can be introduced. If these other concepts in solid-state chemistry are taught in a clear and interesting way, linking concepts to exciting new applications in high-tech industries, a new generation of scientists with a solid background in solid-state and structural chemistry will be available for the betterment of education and industry. Notes 1. We have submitted software that teaches the Madelung constant and series in an in-depth, visual, and interactive way to JCE Software. 2. Reference ions are not included in the calculation of net charge. Only ions interacting with the reference ion are considered. This allows the summation of the charge to be analogous to the summation of the Madelung series.
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Journal of Chemical Education • Vol. 78 No. 9 September 2001 • JChemEd.chem.wisc.edu
