- #1
KFC
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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?
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?
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