SCIENCE · INVESTIGATE AND EXPLAIN

Build an ionic array ledger without double-counting pairs

Distinguish nearest-pair attraction from a surrounding charge array.

Learning outcome: Capture one through four neighbor shells for z=1; label alternating attractive and repulsive additions. Repeat z=2 and derive the scaling before reading the ledger.

Starting recap

Review the atomic electron ledger, signed energy changes and dimensional conversions. Distinguish a mathematical parameter from a unique material identification.

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Try

With four neighbors on each side and z=1, the central sum is2(−1+1/2−1/3+1/4)=−7/6 in units E0.

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Predict

Does adding another neighbor shell always make the central interaction more negative?

Your prediction and reason

Experiment

Capture one through four neighbor shells for z=1; label alternating attractive and repulsive additions. Repeat z=2 and derive the scaling before reading the ledger.

Distinguish one ion pair from a surrounding array

Worked teaching model · synthetic values · A fixed one-dimensional alternating ±ze array at spacing r0 has one central positive ion. Central interaction energy in units E0=k e²/r0 is 2 z² Σ(−1)^j/j. This is a central-ion ledger, not whole-array energy per ion pair, a 3D lattice or a stable equilibrium model; short-range repulsion is omitted.

Neighbors on each side: 4 count Charge magnitude: 1 e

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Axes rescale with settings. Compare numerical values, not screen slopes. Discrete stems show steps, contributors or partial sums; connected samples show the declared parameter relationship, not a measured time trajectory.

Numerical diagram coordinates

neighbors included on each side	central interaction sum (E0)
1	-2
2	-1
3	-1.666667
4	-1.166667

Calculated values under declared assumptions
Quantity	Value	Unit
Central interaction sum	-1.166667	E0
Nearest left/right contribution	-2	E0
One nearest unlike pair	-1	E0
Included neighbors	8	ions

Opposite and same-sign neighbors contribute with different signs.

Alternating partial sums bracket a limit; adding neighbors does not make every contribution attractive. Summing every central-ion ledger over a finite array would double-count each pair, requiring a factor one half. This omitted-repulsion chain cannot predict an equilibrium spacing or a real 3D lattice enthalpy.

Capture contrasting evidence and retain the interpretation limits. These supplied models do not identify an actual material or certify practical laboratory skill.

Opposite and same-sign neighbors contribute with different signs.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

A crystal is not an isolated molecule

An ionic solid is an extended collection of cations and anions. Assigning a neutral formula unit does not mean that the crystal consists of independent isolated ion pairs. Each ion interacts with many neighbors, including like charges as well as unlike charges. A nearest-pair model captures only one part of that accounting.

Construct the deliberately simpler chain

Place a positive central charge +ze at the origin and fixed alternating charges at distances r0, 2r0, 3r0 and so on on both sides. Define E0=k e²/r0. Each pair with a neighbor contributes z²(−1)^j/j in these units. Including both sides gives the central interaction sum 2z²Σ(−1)^j/j. The model fixes positions and omits short-range repulsion; it is not a stable one-dimensional crystal simulation.

Evaluate the additions

At z=1, the first shell contributes −2E0, the second adds +E0 and the third adds −2E0/3. Four shells give −7E0/6. The partial sums alternate around a limiting value rather than becoming monotonically more negative. A single nearest unlike pair contributes −E0, which is neither the complete central ledger nor a whole crystal’s molar energy.

Avoid double counting

If a finite collection were analyzed by listing every ion’s interactions with all other ions, every pair would appear twice. Dividing the sum by two corrects that duplication. Our central-only ledger is not that all-ion sum and must not simply be renamed energy per formula unit. A real three-dimensional structure has a different geometry, coordination and reference convention.

Transfer the scaling safely

With positions unchanged, changing z from1 to2 quadruples every contribution. Doubling all spacings while holding charges fixed halves the dimensional Coulomb contribution. Neither change tells us a new equilibrium spacing: the omitted short-range repulsion is essential for an energy minimum. Ionic bonding is a useful model of charge organization, but actual electron density is not a set of mathematical points and does not imply perfectly integer local atomic charges.

Evidence and safety boundary

Conceptual sources checked 30 October 2026: Ionic bonding, Lattice energetics. All prose, numerical packets and diagrams are original; no textbook exercises or images are reproduced. The activity is supplied mathematical or synthetic evidence, not a performed materials experiment. No unknown substances, heating, high voltage or chemical synthesis are requested. Retain prior predictions and a changed-case explanation; numerical agreement alone does not certify independent understanding or practical competence.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Identify the missing repulsion, geometry and pair-counting information needed before claiming a real lattice energy.

Your explanation and limitations

Continue

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Previous: Distinguish bond depth, equilibrium and restoring slope Next: Infer a signed lattice term from a formation cycle

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