PHY3707 Assignment 5 2025 (657543) Due Monday, 18 August 2025

PHY3707
Assignment 5
Unique number: 657543
Due 18 August 2025

, Student Name: PHY3707 Assignment 5


PHY3707 Assignment 5
Due Date: 18 August 2025


Solution to Problem 6.9 [10 marks]
We are tasked with showing that the fraction of electrons within kB T of the Fermi level
is equal to:
3kB T
2εF
given that the density of states is D(ε) ∝ ε1/2 .


Step 1: Density of States and Total Number of Electrons
For a free electron gas:
D(ε) = C ε1/2

where C is a constant. The total number of electrons is:
Z εF
2 3/2
N∝ C ε1/2 dε = C · εF .
0 3


Step 2: Electrons Within kB T of εF
The number of electrons in the range (εF − kB T, εF + kB T ) is:
Z εF +kB T
∆N ∝ C ε1/2 dε.
εF −kB T


Since kB T ≪ εF , we can approximate:
 
2 3/2 2 3/2
∆N ≈ C (εF + kB T ) − (εF − kB T ) .
3 3


Step 3: Taylor Expansion
kB T
Let x = εF
, which is small. Using the binomial expansion:
 
3/2 3/2 3
(εF ± kB T ) ≈ εF 1± x .
2

Substituting:    
2 3/2 3 3
∆N ≈ C · εF 1+ x − 1− x .
3 2 2


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