An In-Depth Look at the Madelung Constant for Cubic Crystal Systems

The way in which the Madelung constant is presented to undergraduate chemistry students provides little insight into its meaning. Students are usually shown the first six terms of the sodium chloride Madelung series and told that the series is infinite and takes into account all electrostatic interactions of the ions arranged in that structure. There is usually little discussion of how the series is generated and what each term represents. This paper presents an in-depth look at the Madelung constant for cubic crystal systems. We show how the numerators of the terms of the series refer to multiplicities of x,y,z positions in space and how these multiplicities can be calculated. We also show how to transfer this method of generating the Madelung series from orthogonally bonded sodium chloride to other cubic systems such as cesium chloride and zinc blende. Convergence of the series is discussed, establishing that the condition of charge neutrality is crucial in getting the Madelung series to converge.