--- title: "Density of states and occupancy statistics" subject: "Electrical Engineering Material" chapter: 1 key: "semester_3/electrical_engineering_material:density-of-states-statistical-tool" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/density-of-states-statistical-tool site: BCT Engineering --- # Density of states and occupancy statistics Electrical Engineering Material · Chapter 1: Theory of Metals - Asked in: 2072 Kartik, 1+4 marks - Video: [Density of States Derivation Part 1](https://www.youtube.com/watch?v=z7YGS67GETo) — Jordan Edmunds Chetty - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/density-of-states-statistical-tool - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-1 ## Question What is density of states? Describe any statistical tool used in quantum mechanics to predict number of energy states being occupied by an electron. ## Short answer Density of states g(E) gives the number of states per unit energy per volume; for free electrons g(E) ∝ √E. Fermi–Dirac statistics gives occupation probability f(E)=1/[1+exp((E−E_F)/kT)]; at T=0 it is a step function, and it smooths for T>0. ## Answer #### Density of states The **density of states** $g(E)$ is the number of available electron states per unit energy per unit volume at energy $E$. So $g(E)\,dE$ is the number of states (per m³) between $E$ and $E + dE$. For free electrons in a metal $$ g(E) = \frac{8\sqrt{2}\,\pi\,m^{3/2}}{h^3}E^{1/2} \propto \sqrt{E} $$ #### Fermi–Dirac statistics $g(E)$ only says how many states exist. To know how many are **occupied**, quantum mechanics uses **Fermi–Dirac (FD) statistics**, which applies to electrons because they are identical, indistinguishable particles that obey the Pauli exclusion principle (at most one electron per state). The probability that a state of energy $E$ is occupied at temperature $T$ is $$ f(E) = \frac{1}{1 + \exp\left(\dfrac{E - E_F}{kT}\right)} $$ where $E_F$ is the Fermi energy and $k$ is Boltzmann's constant. **Properties:** - At $T = 0$ K: $f(E) = 1$ for $E < E_F$ and $f(E) = 0$ for $E > E_F$ (a sharp step). - At $T > 0$: $f(E_F) = \frac{1}{2}$ at every temperature; the step becomes smooth over a range of a few $kT$ around $E_F$. - For $E - E_F \gg kT$, $f(E) \approx e^{-(E - E_F)/kT}$ (Boltzmann approximation, used for semiconductors). ```text f(E) 1 +----------. T = 0 K: step | \ 0.5|- - - - - - x T > 0: smooth | \___ 0 +----------|-------------> E E_F ``` #### Number of occupied states The number of electrons per unit volume per unit energy is the product: $$ n_E(E) = g(E)\,f(E) $$ and the total electron concentration is $n = \int_0^\infty g(E)f(E)\,dE$. At 0 K this gives $E_F = \frac{h^2}{8m}\left(\frac{3n}{\pi}\right)^{2/3}$. Example: probability of occupation of a state $3kT$ above $E_F$ is $1/(1 + e^3) = 0.047$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does g(E) dE represent? A. Number of electrons actually present between E and E+dE B. Number of available states per unit volume between E and E+dE C. Probability that a state is occupied D. Total energy of electrons Answer: B. g(E) counts available states, not occupied ones. (Key point: Meaning of g(E)) 2. Why are electrons described by Fermi-Dirac statistics? A. They are distinguishable and many can share a state B. They are identical, indistinguishable and obey the Pauli exclusion principle C. They have integer spin D. They have no charge Answer: B. Identical particles with Pauli exclusion allow at most one electron per state. (Key point: Why Fermi-Dirac statistics) 3. For T > 0, what is f(E) at E = E_F? A. 0 B. 1 C. 1/2 D. 1/e Answer: C. At E = E_F the exponent is 0, so f = 1/(1+1) = 1/2 at every temperature. (Key point: Value at EF) 4. How is the number of electrons per unit volume per unit energy obtained? A. g(E) + f(E) B. g(E) f(E) C. g(E)/f(E) D. f(E)/g(E) Answer: B. Available states times probability of occupancy: n(E) = g(E) f(E). 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