PHY3707 Assignment 2 solutions 2026
, Question 1
Show that the Madelung constant for a one-dimensional array of ions of alternating sign
with distance 𝑎between successive ions is 2ln 2.
Understand the arrangement
Consider a one-dimensional chain of alternating ions:
positive, negative, positive, negative,……………
The distance between neighbouring ions is 𝑎.
Take a positive ion as the reference ion.
The electrostatic potential energy due to all other ions is:
𝛼𝑒 2
𝑈=−
4𝜋𝜀0 𝑎
where:
• 𝛼= Madelung constant
• 𝑒= electronic charge
We must determine 𝛼.
Write contributions from neighbouring ions
The neighbouring ions occur at distances:
𝑎, 2𝑎, 3𝑎, 4𝑎, …
Because the signs alternate:
• ion at 𝑎: opposite sign → attractive
• ion at 2𝑎: same sign → repulsive
• ion at 3𝑎: opposite sign → attractive
etc.
Thus the Madelung constant becomes:
1 1 1 1
𝛼 = 2 (1 − + − + −⋯ )
2 3 4 5
The factor 2appears because ions exist on both sides of the reference ion.
, Question 1
Show that the Madelung constant for a one-dimensional array of ions of alternating sign
with distance 𝑎between successive ions is 2ln 2.
Understand the arrangement
Consider a one-dimensional chain of alternating ions:
positive, negative, positive, negative,……………
The distance between neighbouring ions is 𝑎.
Take a positive ion as the reference ion.
The electrostatic potential energy due to all other ions is:
𝛼𝑒 2
𝑈=−
4𝜋𝜀0 𝑎
where:
• 𝛼= Madelung constant
• 𝑒= electronic charge
We must determine 𝛼.
Write contributions from neighbouring ions
The neighbouring ions occur at distances:
𝑎, 2𝑎, 3𝑎, 4𝑎, …
Because the signs alternate:
• ion at 𝑎: opposite sign → attractive
• ion at 2𝑎: same sign → repulsive
• ion at 3𝑎: opposite sign → attractive
etc.
Thus the Madelung constant becomes:
1 1 1 1
𝛼 = 2 (1 − + − + −⋯ )
2 3 4 5
The factor 2appears because ions exist on both sides of the reference ion.
