About ionic crystal and Madelung constant

In summary, the conversation discusses the derivation of the Madelung constant for ionic crystals as described in Kittel's Introduction to Solid State Physics. The constant takes into account both the repulsive potential and the interaction energy between nearest neighbors. The sum of the interaction energy involves all interactions involving the ion i, with the exception of i=j. The Madelung constant also includes the contribution of both nearest and other neighbors, and the coulomb potential is not split into separate sums.
  • #1
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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 [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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  • #2
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.
 

Related to About ionic crystal and Madelung constant

1. What is an ionic crystal?

An ionic crystal is a type of crystal structure that is composed of positively and negatively charged ions held together by strong electrostatic forces. Common examples of ionic crystals include salt (NaCl) and quartz (SiO2).

2. How is the Madelung constant calculated?

The Madelung constant is calculated by summing the electrostatic potential energies of all the ions in an ionic crystal lattice. This includes the contributions from neighboring ions in all directions, taking into account their charges and distances from each other.

3. What is the significance of the Madelung constant?

The Madelung constant is important in understanding the stability and lattice energy of ionic crystals. It also helps to predict the properties of these crystals, such as their melting points and solubility.

4. How does the Madelung constant affect the properties of an ionic crystal?

The Madelung constant directly affects the lattice energy of an ionic crystal, which in turn affects its physical properties. A larger Madelung constant results in a stronger lattice and higher melting point, while a smaller constant leads to a weaker lattice and lower melting point.

5. Can the Madelung constant be used to compare different ionic crystals?

Yes, the Madelung constant can be used to compare the stability and properties of different ionic crystals. Generally, crystals with higher Madelung constants will have stronger lattices and higher melting points, making them more stable and less soluble.

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