---
title: "Energy band formation from N lithium atoms"
subject: "Electrical Engineering Material"
chapter: 1
key: "semester_3/electrical_engineering_material:energy-bands-lithium-atoms"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/energy-bands-lithium-atoms
site: BCT Engineering
---

# Energy band formation from N lithium atoms

Electrical Engineering Material · Chapter 1: Theory of Metals

- Asked in: 2079 Bhadra, 6 marks; 2073 Chaitra, 6 marks; 2072 Kartik, 6 marks; 2070 Chaitra, 6 marks; 2069 Asar, 6 marks; 2068 Chaitra, 6 marks (asked 6 times)
- Video: [Formation of Energy Bands in solids](https://www.youtube.com/watch?v=ORPNkHt2rG8) — Engineering Physics by Sanjiv
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/energy-bands-lithium-atoms
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-1

## Question

Explain how energy bands are formed in solids taking the example of N number of Lithium atoms for the explanation.

## Short answer

When N lithium atoms form a solid, their 2s orbitals overlap and split into N closely spaced levels, creating a half-filled 2s band (only N valence electrons occupy 2N states). The full, narrow 1s band lies below; overlapping 2p band provides empty states, making lithium a good conductor.

## Answer

An **energy band** is a group of very closely spaced allowed energy levels that forms when a large number of atoms come together to make a solid. It forms because the outer orbitals of neighbouring atoms overlap and, by the Pauli exclusion principle, each atomic level must split into many slightly different levels.

#### Lithium atom
Lithium (Z = 3) has the configuration $1s^2\,2s^1$. The 1s shell is full; the 2s orbital has one electron and room for one more.

#### Two Li atoms
When two Li atoms come close, their 2s orbitals overlap. The two atomic 2s orbitals combine (LCAO) into two molecular orbitals:
- a **bonding** orbital, with lower energy than the atomic 2s level, and
- an **antibonding** orbital, with higher energy.

So the single 2s level splits into **2 levels**. The 1s orbitals lie close to the nuclei and hardly overlap, so they split very little.

#### N Li atoms
For $N$ atoms (about $10^{23}$ per cm³ in a solid) the 2s level splits into **N** separate levels. They are so close together (spacing of order $10^{-22}$ eV) that they form an almost continuous **2s band**.

```text
 Energy
   ^         separated atoms     solid (N atoms)
   |  2p .........................######  2p band
   |                       ######### (overlap)
   |  2s ..................#########  2s band
   |                       #########  (N levels)
   |
   |  1s ..........................-   1s band
   |                                  (narrow, full)
   +-----------------------|-----------------> r
                          r0     interatomic distance
```

- Each level holds 2 electrons (spin up/down), so the 2s band has $2N$ states.
- N Li atoms supply only $N$ valence electrons, so the **2s band is only half filled**.
- The 1s band ($2N$ states, $2N$ electrons) is full and very narrow because the 1s orbitals overlap very little.
- At the equilibrium spacing $r_0$, the 2s and 2p bands **overlap**, giving one wide band with many empty states just above the filled ones.

#### Result
Because the highest band is partly filled, electrons can easily move into empty states just above them when a small field is applied. This is why lithium (and other alkali metals) are **good conductors**. The band width depends on how much the orbitals overlap: outer orbitals give wide bands, inner orbitals give narrow ones.

## Check yourself

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

1. Why does the single 2s level of Li split into many closely spaced levels when N atoms form a solid?
   A. Outer orbitals of neighbouring atoms overlap and, by the Pauli principle, identical levels cannot coincide
   B. The nuclei of the atoms fuse together
   C. The 1s electrons are pushed into the 2s level
   D. The temperature of the solid rises
   Answer: A. Overlap of outer orbitals plus the Pauli exclusion principle forces each atomic level to split into slightly different levels. (Key point: Why levels split)

2. For N lithium atoms, into how many levels does the 2s level split, and how many electron states does the 2s band hold?
   A. N levels, N states
   B. 2N levels, 2N states
   C. N levels, 2N states
   D. 2 levels, 2N states
   Answer: C. The 2s level splits into N levels and each holds 2 electrons (spin up/down), giving 2N states. (Key point: Number of levels)

3. Why is the 2s band of lithium only half filled?
   A. Half of the levels are forbidden
   B. The N atoms supply only N valence electrons for 2N available states
   C. The 1s band takes half the electrons
   D. Spin allows only one electron per level
   Answer: B. Each Li atom gives one 2s electron, so N electrons fill only half of the 2N states. (Key point: Half-filled band)

4. What makes lithium a good conductor according to the band picture?
   A. Its 1s band is full
   B. Its highest band is partly filled, so electrons find empty states just above
   C. It has a wide forbidden gap
   D. Its 1s band is very narrow
   Answer: B. A partly filled band lets electrons move into nearby empty states under a small field. (Key point: Conduction in Li)

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