Consider the 2D ionic crystal below. There are two types of ions, with charge te (cations: denoted by the black dots) an

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Consider The 2d Ionic Crystal Below There Are Two Types Of Ions With Charge Te Cations Denoted By The Black Dots An 1 (161.93 KiB) Viewed 34 times

Consider the 2D ionic crystal below. There are two types of ions, with charge te (cations: denoted by the black dots) and -e (anions: denoted by the white dots), and they are held together by ionic bonds. The equilibrium separation between the nearest neighbors is ro 1 (a) The Coulomb interaction between the ions. From the crystal structure, write down the separations between the nearest neighbors, next nearest neighbors, next-next nearest neighbors, ... up to the next-(7 times) nearest neighbors. Show that the coulomb energy per ion u, is given by e2 4 8 uc - 4 + + 2 (1) 2 4περο + other four terms + ...). V2 V5 There are also short-range repulsion between the nuclei when the ions are very close to each other. The repulsive interaction can be represented by a term of the form A/r62, where A is some constant. Thus, the energy per ion is given by 4A u = c + (2) (the factor of four in the repulsive term comes from the four nearest neighbors of an ion). (b) Consider the terms inside the bracket in Eq. (1). List a table for the value of the sum if we retain only one term (the first term), two terms, three terms, ... up to 8 terms. (c) (Optional] The sums in the bracket in Eq. (1) is called the Madelung Sums. If you look at the numbers you get in (b), the sum does not seem to converge to some