---
title: "Non-occupation probability 5kT above Fermi level"
subject: "Electrical Engineering Material"
chapter: 1
key: "semester_3/electrical_engineering_material:fermi-probability-5kt-not-occupied"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/fermi-probability-5kt-not-occupied
site: BCT Engineering
---

# Non-occupation probability 5kT above Fermi level

Electrical Engineering Material · Chapter 1: Theory of Metals

- Asked in: 2075 Asoj, 4 marks
- Video: [What is the probability that an energy level 3kT above Fermi level is occupied ?](https://www.youtube.com/watch?v=qysmvNxLwgo) — Learn with Amna-B
- 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-5kt-not-occupied
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-1

## Question

Find the probability that an energy state 5KT above the Fermi level will not occupied by an electron.

## Short answer

Fermi-Dirac occupation probability f = 1/(1 + e^(E-E_F)/kT). For E - E_F = 5kT, f = 1/(1 + e⁵) ≈ 0.0067. Probability not occupied = 1 - f = e⁵/(1 + e⁵) ≈ 0.9933 (99.33%).

## Answer

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

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

The probability that it is **not occupied** (empty) is $1 - f(E)$.

**Given:** $E - E_F = 5kT$.

$$
\begin{aligned}
f(E) &= \frac{1}{1 + e^{5}} = \frac{1}{1 + 148.41} = \frac{1}{149.41} = 0.00669 \\
1 - f(E) &= 1 - 0.00669 = 0.99331
\end{aligned}
$$

Equivalently, $1 - f(E) = \dfrac{e^5}{1 + e^5} = \dfrac{148.41}{149.41} = 0.9933$.

**Answer:** the probability that the state is not occupied is **0.9933 (99.33 %)**.

A state $5kT$ above $E_F$ (about 0.13 eV at 300 K) is almost always empty; only 0.67 % of such states hold an electron. The answer is the same at every temperature because the energy is expressed in units of $kT$.

## Check yourself

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

1. The probability that a state at E is occupied is:
   A. 1/(1 + exp((E - E_F)/kT))
   B. 1/(1 + exp((E_F - E)/kT))
   C. exp(-(E - E_F)/kT)
   D. 1 - 1/(1 + exp((E - E_F)/kT))
   Answer: A. This is the Fermi–Dirac function f(E). (Key point: Fermi function)

2. Probability that a state is NOT occupied is:
   A. f(E)
   B. 1 - f(E)
   C. 1/f(E)
   D. f(E)^2
   Answer: B. Empty probability is the complement of occupation. (Key point: Not occupied)

3. For E - E_F = 5kT, f(E) is about:
   A. 0.993
   B. 0.5
   C. 0.0067
   D. 0.067
   Answer: C. 1/(1 + e^5) = 1/149.4 = 0.0067. (Key point: Value)

4. So the probability that the state is empty is:
   A. 0.0067
   B. 0.9933
   C. 0.5
   D. 0.67
   Answer: B. 1 - 0.0067 = 0.9933, independent of temperature when expressed in kT. (Key point: Final answer)

## Close variants set in other exams

- [Occupation probability 3kT above Fermi level](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Afermi-probability-3kt-above) — Electrical Engineering Material, chapter 1, asked 2×

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