UNIVERSITY OF SOUTH AFRICA
College of Science, Engineering and Technology
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
PHY3707: Solid State Physics
Assignment 2 — Semester 1, 2026
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
PHY3707
Module Code:
Solid State Physics
Module Name:
Assignment 2
Assignment Number:
197028
Unique Number:
June 2026
Due Date:
50
Total Marks:
Srivastava, J.P. 2014. Elements of Solid
Prescribed Text:
State Physics, 4th ed.
Submitted in partial fulfilment of the requirements for PHY3707 — UNISA 2026
, UNISA | PHY3707 Solid State Physics – Assignment 2
Question 1: Madelung Constant for a One-Dimensional Ion Array
Question: Show that the Madelung constant for a one-dimensional array of ions of alternat-
ing sign with a distance a between successive ions is given by 2 ln 2. [12]
1.1 Setup of the One-Dimensional Model
Consider an infinite linear chain of ions with alternating charges +e and −e, each separated
from its neighbour by a distance a. The goal is to compute the total electrostatic (Madelung)
potential energy at one reference ion due to all others in the chain (Srivastava, 2014:38).
Place the reference ion at the origin. Ions at positions ±a, ±2a, ±3a, . . . carry alternating
signs. The nearest neighbours (at ±a) are oppositely charged, the next-nearest (at ±2a) are
like-charged, and so on.
1.2 Writing the Electrostatic Potential at the Reference Ion
The Coulomb potential at the origin due to all other ions is:
∞ ∞
e X (−1)n+1 2e X (−1)n+1
V = ×2= (1)
4πε0 na 4πε0 a n
n=1 n=1
The factor of 2 accounts for contributions from both sides (left and right). The sign factor
(−1)n+1 ensures the nearest ions are attractive (opposite sign) and the next-nearest repulsive
(same sign).
Expanding the series explicitly:
∞
X (−1)n+1 1 1 1 1
=1− + − + − ··· (2)
n 2 3 4 5
n=1
1.3 Identifying the Series as ln 2
Recall the well-known Maclaurin series for the natural logarithm:
x2 x3 x4
ln(1 + x) = x − + − + ··· for − 1 < x ≤ 1 (3)
2 3 4
Page 2 of 14
College of Science, Engineering and Technology
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
PHY3707: Solid State Physics
Assignment 2 — Semester 1, 2026
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
PHY3707
Module Code:
Solid State Physics
Module Name:
Assignment 2
Assignment Number:
197028
Unique Number:
June 2026
Due Date:
50
Total Marks:
Srivastava, J.P. 2014. Elements of Solid
Prescribed Text:
State Physics, 4th ed.
Submitted in partial fulfilment of the requirements for PHY3707 — UNISA 2026
, UNISA | PHY3707 Solid State Physics – Assignment 2
Question 1: Madelung Constant for a One-Dimensional Ion Array
Question: Show that the Madelung constant for a one-dimensional array of ions of alternat-
ing sign with a distance a between successive ions is given by 2 ln 2. [12]
1.1 Setup of the One-Dimensional Model
Consider an infinite linear chain of ions with alternating charges +e and −e, each separated
from its neighbour by a distance a. The goal is to compute the total electrostatic (Madelung)
potential energy at one reference ion due to all others in the chain (Srivastava, 2014:38).
Place the reference ion at the origin. Ions at positions ±a, ±2a, ±3a, . . . carry alternating
signs. The nearest neighbours (at ±a) are oppositely charged, the next-nearest (at ±2a) are
like-charged, and so on.
1.2 Writing the Electrostatic Potential at the Reference Ion
The Coulomb potential at the origin due to all other ions is:
∞ ∞
e X (−1)n+1 2e X (−1)n+1
V = ×2= (1)
4πε0 na 4πε0 a n
n=1 n=1
The factor of 2 accounts for contributions from both sides (left and right). The sign factor
(−1)n+1 ensures the nearest ions are attractive (opposite sign) and the next-nearest repulsive
(same sign).
Expanding the series explicitly:
∞
X (−1)n+1 1 1 1 1
=1− + − + − ··· (2)
n 2 3 4 5
n=1
1.3 Identifying the Series as ln 2
Recall the well-known Maclaurin series for the natural logarithm:
x2 x3 x4
ln(1 + x) = x − + − + ··· for − 1 < x ≤ 1 (3)
2 3 4
Page 2 of 14
