About ionic crystal and Madelung constant

[KFC]

Hi there,
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 $$\lambda \exp(-r/\rho)$$ and the interaction energy can be written as
$$U_{ij} = \lambda\exp[-r_{ij}/\rho] \pm q^2/r_{ij}$$

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

1) For nearest neighbors, $$U_{ij} = \lambda\exp[-R/\rho] - q^2/R$$
where R is the nearest-neighbor separation

2) For others neighbors, $$U_{ij} = \pm q^2/(p_{ij}R)$$,
where $$p_{ij} = r_{ij}/R$$

So, the sum $$U_i = \sum_j' U_{ij}$$ 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 $$U_{ij}$$, we actually have to perform the sum over two region (nearest -neighbor) and others

so
$$\sum_j' u_{ij} = \left(z\lambda\exp(-R/\rho) - zq^2/R\right) + \sum_j^{''} \pm \frac{1}{p_ij}$$
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, $$U_i = z\lambda\exp(-R/\rho) - \alpha q^2/R$$
and $$\alpha = \sum_j' \frac{(\pm)}{p_{ij}}$$ which is just Madelung constant.

My questions are
1) why is the term $$zq^2/R$$ gone?
2) Should the Madelung constant consider both the nearest-neighbor and others neighbor's contribution? If so, why the book still use $$\sum_j'$$ instead of double prime?
3) how come there is $$\pm$$ in the Madelung constant?

 

[nasu]

The Madelung constant contains all the neighbors. The coulomb part of the potential energy is not split into nearest neighbors and the other neighbors.
What make you think otherwise?
You better consider splitting the sum into two sums: the one for the repulsive, exponential, potential and the one for the coulomb potential.
In the first sum then you keep the nearest neighbors only.