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1. Assume an infinite one-dimensional (1D) NaCl lattice as shown in Figure 1. a + + Na CI Na CI Na CI™ Na CI™ Na+ Figure 1: Infinite 1D NaCl lattice with nearest neighbor separation a. (a) Choose the basis and the unit cell for this lattice. (b) Derive an analytical expression for the Madelung constant. (c) Assume this crystal to be of a finite size varying from 1 to 50 unit cells and plot the Madelung constant as a function of the crystal size; plot the corresponding interaction energy evolution (use the nearest-neighbor separation for three-dimensional NaCl, i.e., a = 2.82 Å). (d) Is there a condition related to the repulsive/attractive force balance determining the potential realization of such crystals? (Comment: no metaphysical or philosophical analysis is required; disregard any consideration of the substrate used). Assume that the Pauli repulsion is similar to that for three-dimensional NaCl, i.e., X 1 keV and p = 0.32 Å, and plot the equilibrium separation ao between Na+ and Cl- ion as a function of the crystal size. =