ATM-03

Bohr radius

a₀ = 4π ε₀ ħ² / (m e²) ≈ 0.529 Å, r_n = n² a₀ / Z.

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StructureBohr radius

Governing equation

r_n=n^2 a_0/Z,\quad a_0=4\pi\varepsilon_0\hbar^2/me^2

where

Z: Atomic number (—)
n: Principal n (—)
a_0: Bohr radius (Å)
r_n: Orbit radius (Å)

Lecture brief

Historical brief

Balmer, Rydberg, Bohr (1913), fine structure, Zeeman and Stern–Gerlach built the spectrum of one atom. The lab computes levels, selection and magnetic splitting. This sheet (ATM-03 — Bohr radius) is the form associated with Bohr radius. Working symbols: $Z$ , $n$ $\rightarrow$ $a_0$ , $r_n$ . The only length that SI constants allow for hydrogen. Expectation ⟨r⟩_{nℓ} is a bit larger.

Purpose

Purpose: compute

a_0

r_n

from

Z

n

in Atomic physics via

r_n=n^2 a_0/Z,\quad a_0=4\pi\varepsilon_0\hbar^2/me^2

a₀ = 4π ε₀ ħ² / (m e²) ≈ 0.529 Å, r_n = n² a₀ / Z. Use it when a real atomic physics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

Z = 1.000\,\mathrm{—}

n = 1.000\,\mathrm{—}

, the governing relation

r_n=n^2 a_0/Z,\quad a_0=4\pi\varepsilon_0\hbar^2/me^2

yields

a_0 = 0.5292\,\mathrm{Å}

r_n = 0.5292\,\mathrm{Å}

. A nucleus, a growing n-shell. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Atomic number Z1 —Principal n n1 —

Outputs

Bohr radius a_00.5292 Å
Orbit radius r_n0.5292 Å

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Narration of this film

A nucleus, a growing n-shell.

The only length that SI constants allow for hydrogen. Expectation ⟨r⟩_{nℓ} is a bit larger.

Reading speed

Watch on YouTube