What is being summed. Every ion sits on a point of a simple cubic lattice, and neighbouring sites carry opposite charges. Taking one ion as the centre, a site a distance d away contributes −z²e²/(4πε₀d) if its charge is opposite and +z²e²/(4πε₀d) if it is the same. Factoring out the nearest-neighbour distance r₀ leaves a pure number, the Madelung constant: A = Σ′ (−1)^{n_x+n_y+n_z+1}/√(n_x²+n_y²+n_z²), the prime meaning the central site itself is left out. The lattice energy then follows as E = −A·N_Az²e²/(4πε₀r₀).

Exact values. 1D gives 2 ln 2 = 1.386294, which is the alternating harmonic series 2(1 − ½ + ⅓ − ¼ + …). The 2D square lattice gives 1.615543 and the 3D simple cubic (rock-salt) lattice 1.747564. The last two have no closed form in elementary functions; the figures quoted here were checked against an Evjen sum taken to L = 400 and L = 60 respectively, and agree with the standard literature values to every digit shown.

Why the order of summation matters. A shell of radius d holds a number of ions growing as d² in 3D, while each contributes only 1/d. The individual shell totals therefore grow without limit, and the series converges only through cancellation between positive and negative terms. A series like that is conditionally convergent: rearranging it changes the answer, so "the sum" is not defined until the order is fixed. The three orders offered here make that concrete.

• Expanding sphere. The obvious chemist's choice: nearest neighbours first, then next-nearest, and so on. A sphere cut through an alternating lattice carries a large and erratic surface charge. In 3D the enclosed charge runs −5, +7, −1, +5, −19, +5, +17, −13 … as each shell is added, and over the 42 shells available here the partial sums range from −8.9 to +10.7 without settling. This order cannot be used to calculate the constant.
• Expanding cube, whole ions. A cube of 2L+1 ions on a side has an odd number of sites and so always carries exactly one surplus ion's worth of charge, whose sign flips with every step. The partial sums therefore straddle the limit, overshooting and undershooting alternately while the gap narrows. In 2D the surplus is +1 at every step rather than alternating, so the sum creeps up from below instead.
• Neutral cube (Evjen's method). The cube is drawn so its faces pass through the centres of the outermost ions, splitting each one's charge with the neighbouring block: an ion on a face counts ½, on an edge ¼, at a corner ⅛. The region is then a cube of side 2L·r₀ containing exactly (2L)³ ions, half positive and half negative, with no net charge at all. In 3D this reaches 1.7476 by the second step and four correct decimal places by the fourth.

The energy readout. This is the Coulomb term alone, −A·N_Ae²/(4πε₀r₀), evaluated at NaCl's nearest-neighbour distance of 282 pm with singly charged ions, using the running value of A. At the 3D limit it gives −861 kJ mol⁻¹. The experimental lattice energy of NaCl is about −787 kJ mol⁻¹: the Coulomb attraction is opposed by short-range repulsion between the filled electron shells, which the Born–Landé expression accounts for with a factor (1 − 1/n). The same number is shown for the 1D and 2D lattices. There it is the Coulomb energy one ion would have at the same spacing if the lattice ran in only one or two directions.

Simplifications. Point charges throughout: no polarisation, no short-range repulsion, no zero-point or thermal contributions, and no distinction between the sizes of the two ions, which is why the drawn ions are equal in size. The 3D lattice is capped at L = 7 (3,375 ions) and the sphere at d = 7r₀ so the view stays responsive; the convergence behaviour is already unambiguous well before those limits.