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I am reading Kittel's Introduction to Solid State Physics and I am quite wondering how the Madelung constant was derived for ionic crystal as in the example. In the book, he consider the repulsive potential of the form [tex]\lambda \exp(-r/\rho)[/tex] and the interaction energy can be written as

[tex]U_{ij} = \lambda\exp[-r_{ij}/\rho] \pm q^2/r_{ij}[/tex]

BY only considering the repulsive interaction among nearest neighbors, we have

1) For nearest neighbors, [tex]U_{ij} = \lambda\exp[-R/\rho] - q^2/R [/tex]

where R is the nearest-neighbor separation

2) For others neighbors, [tex]U_{ij} = \pm q^2/(p_{ij}R)[/tex],

where [tex]p_{ij} = r_{ij}/R[/tex]

So, the sum [tex]U_i = \sum_j' U_{ij}[/tex] includes all interactions involving the ion i.

where, the prime mean to sum all index except for i=j. Since we divide the group into nearest neighbors and others. Then to sum [tex]U_{ij}[/tex], we actually have to perform the sum over two region (nearest -neighbor) and others

so

[tex]\sum_j' u_{ij} = \left(z\lambda\exp(-R/\rho) - zq^2/R\right) + \sum_j^{''} \pm \frac{1}{p_ij}[/tex]

where z is the total number of the nearest neighbors, double prime means only consider the non-nearest-neighbor pairs

But in Kittel's book, [tex]U_i = z\lambda\exp(-R/\rho) - \alpha q^2/R[/tex]

and [tex]\alpha = \sum_j' \frac{(\pm)}{p_{ij}}[/tex] which is just Madelung constant.

My questions are

1) why is the term [tex] zq^2/R[/tex] gone?

2) Should the Madelung constant consider both the nearest-neighbor and others neighbor's contribution? If so, why the book still use [tex]\sum_j'[/tex] instead of double prime?

3) how come there is [tex]\pm[/tex] in the Madelung constant?

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# About ionic crystal and Madelung constant

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