UNIVERSITY OF SOUTH AFRICA (UNISA)
College of Science, Engineering and Technology
⋄
ASSIGNMENT 2
Semester 1 — Due June 2026
⋄
Module Code: PHY3707
Module Name: Solid State Physics
Assignment No.: Assignment 2
Due Date: June 2026
Semester: Semester 1, 2026
Unique Number: 197028
Submitted in partial fulfilment of the requirements for PHY3707 Solid State Physics
at the University of South Africa.
,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.
1.1 Setting Up the Problem
Consider an infinite linear chain of alternating positive and negative ions, each separated by
a distance a. Take one positive ion at the origin as the reference ion. The ions on either side
alternate in sign, so the arrangement looks like:
· · · (+) (−) (+) (−) (+) (−) (+) · · ·
|{z}
reference
The potential energy of the reference ion due to all others is:
e2 M
U =− ·
4πε0 a
where M is the Madelung constant.
1.2 Writing the Coulomb Sum
The reference negative ion at the origin interacts with:
• Two positive ions at distance a (nearest neighbours, one on each side)
• Two negative ions at distance 2a
• Two positive ions at distance 3a
• And so on.
The total Madelung sum is:
1 1 1 1
M =2 − + − + ···
1 2 3 4
The factor of 2 accounts for the fact that both sides of the chain contribute equally.
Page 1 of 12
,UNISA | PHY3707 Solid State Physics – Assignment 2
1.3 Identifying the Series
Recall the standard Taylor series expansion of the natural logarithm:
x2 x3 x4
ln(1 + x) = x − + − + ··· for |x| ≤ 1
2 3 4
Setting x = 1:
1 1 1
ln(1 + 1) = ln 2 = 1 − + − + ···
2 3 4
This is precisely the Leibniz-Gregory series, which converges to ln 2.
1.4 Completing the Proof
Substituting the series result back into the Madelung sum:
1 1 1
M = 2 1 − + − + ···
2 3 4
M = 2 ln 2 ≈ 1.3863
Therefore, the Madelung constant for a one-dimensional alternating ion chain with spacing a
between successive ions is indeed 2 ln 2.
Key Distinction
The Madelung constant encodes the geometric arrangement of all ions relative to a ref-
erence ion. In one dimension, M = 2 ln 2 ≈ 1.386. For real three-dimensional crystals
such as NaCl (face-centred cubic), M ≈ 1.748, which reflects the much more complex
geometry of three-dimensional packing (Kittel, 2005:20).
Page 2 of 12
,UNISA | PHY3707 Solid State Physics – Assignment 2
Question 2: Cohesive Energy per Ion-Pair of LiCl and KCl
Question: Calculate the cohesive energy per ion-pair of LiCl and KCl crystals using the fol-
lowing data, assuming that the repulsive potential energy is negligibly small.
Table 1: Given Data for LiCl and KCl
Parameter LiCl KCl
Madelung constant M 1.748 1.748
Li+ /K+ – Cl− spacing r0 2.57 Å 3.14 Å
Ionisation energy Li/K 5.4 eV 4.34 eV
Critical Consideration
This question asks for cohesive energy per ion-pair, not per mole. The repulsive term
is ignored, so the lattice energy reduces purely to the attractive Madelung contribution
minus the energy cost of forming the ions.
2.1 Physical Basis
The cohesive energy per ion-pair is defined as the energy released when two free ions (one pos-
itive, one negative) come together from infinity to form one ion-pair in the crystal. When the
repulsive term is neglected, it is given by:
M e2
Ecoh = − Eion + EEA
4πε0 r0
where:
• M is the Madelung constant
• r0 is the nearest-neighbour ion spacing
• Eion is the ionisation energy of the metal (energy cost to create the cation)
• EEA is the electron affinity of Cl (energy gained by the anion)
The electron affinity of Cl is EEA = 3.61 eV (Kittel, 2005).
2.2 Useful Constant
e2
The fundamental electrostatic constant in practical units:
4πε0
Page 3 of 12
, UNISA | PHY3707 Solid State Physics – Assignment 2
e2
= 14.4 eV Å
4πε0
This means that if r0 is in angstroms, the Madelung energy is directly obtained in electron-
volts.
2.3 Calculation for LiCl
Step 1: Madelung attractive energy
M · e2 1.748 × 14.4 eV Å
EMad = =
4πε0 r0 2.57 Å
25.171 eV Å
EMad = = 9.794 eV
2.57 Å
Step 2: Net energy balance
Cohesive energy per ion-pair = Madelung energy − Ionisation energy + Electron affinity
Ecoh (LiCl) = 9.794 − 5.4 + 3.61
Ecoh (LiCl) = 8.004 eV per ion-pair
2.4 Calculation for KCl
Step 1: Madelung attractive energy
1.748 × 14.4 eV Å 25.171 eV Å
EMad = = = 8.016 eV
3.14 Å 3.14 Å
Step 2: Net energy balance
Ecoh (KCl) = 8.016 − 4.34 + 3.61
Page 4 of 12
College of Science, Engineering and Technology
⋄
ASSIGNMENT 2
Semester 1 — Due June 2026
⋄
Module Code: PHY3707
Module Name: Solid State Physics
Assignment No.: Assignment 2
Due Date: June 2026
Semester: Semester 1, 2026
Unique Number: 197028
Submitted in partial fulfilment of the requirements for PHY3707 Solid State Physics
at the University of South Africa.
,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.
1.1 Setting Up the Problem
Consider an infinite linear chain of alternating positive and negative ions, each separated by
a distance a. Take one positive ion at the origin as the reference ion. The ions on either side
alternate in sign, so the arrangement looks like:
· · · (+) (−) (+) (−) (+) (−) (+) · · ·
|{z}
reference
The potential energy of the reference ion due to all others is:
e2 M
U =− ·
4πε0 a
where M is the Madelung constant.
1.2 Writing the Coulomb Sum
The reference negative ion at the origin interacts with:
• Two positive ions at distance a (nearest neighbours, one on each side)
• Two negative ions at distance 2a
• Two positive ions at distance 3a
• And so on.
The total Madelung sum is:
1 1 1 1
M =2 − + − + ···
1 2 3 4
The factor of 2 accounts for the fact that both sides of the chain contribute equally.
Page 1 of 12
,UNISA | PHY3707 Solid State Physics – Assignment 2
1.3 Identifying the Series
Recall the standard Taylor series expansion of the natural logarithm:
x2 x3 x4
ln(1 + x) = x − + − + ··· for |x| ≤ 1
2 3 4
Setting x = 1:
1 1 1
ln(1 + 1) = ln 2 = 1 − + − + ···
2 3 4
This is precisely the Leibniz-Gregory series, which converges to ln 2.
1.4 Completing the Proof
Substituting the series result back into the Madelung sum:
1 1 1
M = 2 1 − + − + ···
2 3 4
M = 2 ln 2 ≈ 1.3863
Therefore, the Madelung constant for a one-dimensional alternating ion chain with spacing a
between successive ions is indeed 2 ln 2.
Key Distinction
The Madelung constant encodes the geometric arrangement of all ions relative to a ref-
erence ion. In one dimension, M = 2 ln 2 ≈ 1.386. For real three-dimensional crystals
such as NaCl (face-centred cubic), M ≈ 1.748, which reflects the much more complex
geometry of three-dimensional packing (Kittel, 2005:20).
Page 2 of 12
,UNISA | PHY3707 Solid State Physics – Assignment 2
Question 2: Cohesive Energy per Ion-Pair of LiCl and KCl
Question: Calculate the cohesive energy per ion-pair of LiCl and KCl crystals using the fol-
lowing data, assuming that the repulsive potential energy is negligibly small.
Table 1: Given Data for LiCl and KCl
Parameter LiCl KCl
Madelung constant M 1.748 1.748
Li+ /K+ – Cl− spacing r0 2.57 Å 3.14 Å
Ionisation energy Li/K 5.4 eV 4.34 eV
Critical Consideration
This question asks for cohesive energy per ion-pair, not per mole. The repulsive term
is ignored, so the lattice energy reduces purely to the attractive Madelung contribution
minus the energy cost of forming the ions.
2.1 Physical Basis
The cohesive energy per ion-pair is defined as the energy released when two free ions (one pos-
itive, one negative) come together from infinity to form one ion-pair in the crystal. When the
repulsive term is neglected, it is given by:
M e2
Ecoh = − Eion + EEA
4πε0 r0
where:
• M is the Madelung constant
• r0 is the nearest-neighbour ion spacing
• Eion is the ionisation energy of the metal (energy cost to create the cation)
• EEA is the electron affinity of Cl (energy gained by the anion)
The electron affinity of Cl is EEA = 3.61 eV (Kittel, 2005).
2.2 Useful Constant
e2
The fundamental electrostatic constant in practical units:
4πε0
Page 3 of 12
, UNISA | PHY3707 Solid State Physics – Assignment 2
e2
= 14.4 eV Å
4πε0
This means that if r0 is in angstroms, the Madelung energy is directly obtained in electron-
volts.
2.3 Calculation for LiCl
Step 1: Madelung attractive energy
M · e2 1.748 × 14.4 eV Å
EMad = =
4πε0 r0 2.57 Å
25.171 eV Å
EMad = = 9.794 eV
2.57 Å
Step 2: Net energy balance
Cohesive energy per ion-pair = Madelung energy − Ionisation energy + Electron affinity
Ecoh (LiCl) = 9.794 − 5.4 + 3.61
Ecoh (LiCl) = 8.004 eV per ion-pair
2.4 Calculation for KCl
Step 1: Madelung attractive energy
1.748 × 14.4 eV Å 25.171 eV Å
EMad = = = 8.016 eV
3.14 Å 3.14 Å
Step 2: Net energy balance
Ecoh (KCl) = 8.016 − 4.34 + 3.61
Page 4 of 12
