Electric potential energy in an infinite 2d crystal

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The discussion centers on calculating the electric potential energy in an infinite 2D crystal with alternating positive and negative charges. The formula presented involves a constant k as the distance r approaches infinity, with the potential energy expressed as We = k (±e0)(±e0)/(4 pi ε0 r). The challenge lies in summing an infinite number of terms, which theoretically converges to a final result. Participants reference the Madelung constant as a key component for this calculation, specifically seeking its value for a 2D arrangement. The conversation emphasizes the mathematical complexities involved in this theoretical framework.
Istvan
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So.. we've got a 2D crystal, corresponding an infinite square net with +-e0 charges in each one (see the atached photo). Now I'd like to calculate the constant k..when r goes to infinity... We = k (+-e0)*(+-e0)/(4 pi eps0 r)..on the positive charge in the center caused by the negative and positive charges around him in the net. It's teoretically impossible to sum infinite number of terms, but the constant k will converge to the final result. So I'd like some help how to calculate that... thanks.
 

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'madelung constant' on wikipedia is a starting point
 
thanks

and what is the madelung constant for a 2d arrangement
 
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