---
title: "Effective mass of electron in gold"
subject: "Electrical Engineering Material"
chapter: 1
key: "semester_3/electrical_engineering_material:gold-effective-mass-justify"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/gold-effective-mass-justify
site: BCT Engineering
---

# Effective mass of electron in gold

Electrical Engineering Material · Chapter 1: Theory of Metals

- Asked in: 2078 Bhadra, 4 marks
- Video: [Effective mass of Electron (Solid state Physics) PHYSICS (in हिन्दी)](https://www.youtube.com/watch?v=__nOVPc7FIU) — Gautam Varde
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/gold-effective-mass-justify
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-1

## Question

"The effective mass of electron in Gold is 1.1 times the mass of electron". Justify.

## Short answer

The effective mass m* = ħ²/(d²E/dk²). In gold, the conduuction band curvature near the Fermi level is slightly smaller than for free electrons, giving m* ≈ 1.1 m_e. This means an electron accelerates about 0.91 times as fast as a free electron, as if it were 10% heavier.

## Answer

The statement means that a conduction electron in gold responds to an applied force as if its mass were $m^* = 1.1\,m_e$ instead of the free-electron mass $m_e$. This is correct and is explained by the **effective mass** concept.

#### Reason
A conduction electron in a metal is not truly free. Besides the external force $F_{ext}$ (e.g. $-eE$), it feels the periodic force of the positive ion cores and other electrons, $F_{int}$:

$$
F_{ext} + F_{int} = m_e a
$$

Since $F_{int}$ cannot be written simply, its whole effect is put into a new mass:

$$
F_{ext} = m^* a, \qquad m^* = \frac{\hbar^2}{d^2E/dk^2}
$$

- For a perfectly free electron $E = \hbar^2k^2/2m_e$ and $m^* = m_e$.
- In gold, the E–k curve of the conduction band near the Fermi level is **slightly less curved** than the free-electron parabola, because of the interaction with the lattice. Smaller $d^2E/dk^2$ gives larger $m^*$: here $d^2E/dk^2 = \hbar^2/(1.1\,m_e)$, so $m^* = 1.1\,m_e$.

```text
 E
 ^        free electron   gold band
 |        (m* = me)      (m* = 1.1 me,
 |          \   /         flatter)
 |           \ /   .  .
 |            V  .      .
 +---------------------------> k
```

#### Physical meaning
Under the same electric field, an electron in gold accelerates only $1/1.1 \approx 0.91$ times as fast as a free electron, as if it were 10 % heavier; the lattice "holds it back" slightly. The value is found from experiments such as electronic specific heat or cyclotron resonance. Since 1.1 is close to 1, the free electron model works quite well for gold (and other noble metals).

## Check yourself

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

1. 'm* = 1.1 m_e for gold' means:
   A. Gold electrons actually have 10% more mass
   B. An electron responds to the applied force as if its mass were 1.1 m_e
   C. The electron mass changes with temperature
   D. Gold has 1.1 times more electrons
   Answer: B. m* absorbs the effect of lattice forces into an apparent mass. (Key point: Meaning of m*)

2. Why does m* differ from m_e in gold?
   A. Because of the interaction with the periodic lattice potential
   B. Because gold is a heavy atom
   C. Because electrons are heated
   D. Because of the external field
   Answer: A. The internal lattice force is included by defining m* from the E-k curvature. (Key point: Why m* differs)

3. Compared with the free-electron parabola, the E-k curve of gold near the Fermi level is:
   A. More curved, giving smaller m*
   B. Straight, giving infinite m*
   C. Identical
   D. Slightly less curved, giving larger m*
   Answer: D. Smaller d^2E/dk^2 means larger m*. (Key point: Curvature)

## Related in Electrical Engineering Material

- [Effective mass of a free electron](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Afree-electron-effective-mass) — Electrical Engineering Material, chapter 1, asked 3×
- [Effective mass can be positive or negative](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aeffective-mass-positive-negative) — Electrical Engineering Material, chapter 1, asked 4×
- [Infinite effective mass and its E-k explanation](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Ainfinite-effective-mass-ek) — Electrical Engineering Material, chapter 1, asked 4×
- [Negative and infinite effective mass](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Anegative-infinite-effective-mass) — Electrical Engineering Material, chapter 1
- [Negative effective mass at top of valence band](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Anegative-mass-valence-band-top) — Electrical Engineering Material, chapter 1
- [Concept of effective mass in a crystal](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aeffective-mass-concept-crystal) — Electrical Engineering Material, chapter 1, asked 4×

## Related in other subjects

- [Mass of particle from de Broglie ratio](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adebroglie-ratio-mass-of-particle) — Engineering Physics, chapter 10