---
title: "Occupation probability 3kT above Fermi level"
subject: "Electrical Engineering Material"
chapter: 1
key: "semester_3/electrical_engineering_material:fermi-probability-3kt-above"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/fermi-probability-3kt-above
site: BCT Engineering
---

# Occupation probability 3kT above Fermi level

Electrical Engineering Material · Chapter 1: Theory of Metals

- Asked in: 2080 Baisakh, 4 marks
- Video: [Sample Problems on Fermi Dirac Distribution Function](https://www.youtube.com/watch?v=AWDeh4Y9X14) — Deepu Joseph
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/fermi-probability-3kt-above
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-1

## Question

Evaluate the probability that an energy state 3KT above the Fermi level will be occupied by an electron.

## Short answer

Fermi-Dirac distribution f(E)=1/(1+exp((E−E_F)/kT)). With E−E_F=3kT, f=1/(1+e³)=1/21.09=0.0474, i.e., 4.74%.

## Answer

The probability that a state of energy $E$ is occupied by an electron is given by the **Fermi–Dirac distribution function**:

$$
f(E) = \frac{1}{1 + \exp\left(\dfrac{E - E_F}{kT}\right)}
$$

**Given:** the state is $3kT$ above the Fermi level, so $E - E_F = 3kT$.

$$
\begin{aligned}
f(E) &= \frac{1}{1 + e^{3kT/kT}} = \frac{1}{1 + e^{3}} \\
&= \frac{1}{1 + 20.09} = \frac{1}{21.09} = 0.0474
\end{aligned}
$$

**Answer:** $f(E) = 0.0474$, i.e. about **4.74 %**.

The result does not depend on temperature, because the energy is given in units of $kT$. A state only $3kT$ above $E_F$ (about 0.078 eV at 300 K) is already rarely occupied; the probability that it is empty is $1 - 0.0474 = 0.9526$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which function gives the occupation probability?
   A. Maxwell-Boltzmann with E - Ef = 0
   B. f(E) = 1/(1 + exp((E - E_F)/kT))
   C. f(E) = exp(-E/kT) only
   D. f(E) = 1/(1 - exp((E - E_F)/kT))
   Answer: B. The Fermi-Dirac function gives the probability a state at E is occupied. (Key point: Function used)

2. What is f(E) when E - E_F = 3kT?
   A. 0.0474
   B. 0.9526
   C. 0.5
   D. 0.0498
   Answer: A. f = 1/(1 + e^3) = 1/21.09 = 0.0474. (Key point: Calculation)

3. Why does the answer not depend on temperature?
   A. Fermi energy is zero
   B. kT cancels because the energy is given in units of kT
   C. Temperature is fixed at 300 K
   D. Electrons do not feel temperature
   Answer: B. (E - E_F)/kT = 3 regardless of T, so f is always 0.0474. (Key point: Temperature independence)

## Close variants set in other exams

- [Non-occupation probability 5kT above Fermi level](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Afermi-probability-5kt-not-occupied) — Electrical Engineering Material, chapter 1, asked 2×

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