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

$n \in \left[1, 12\right]$

where

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

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

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

$n \in \left[1, 12\right]$

$n = 1$ $a_0 = 0.529$

Along the curve

②

Domain of the variable — blue

n_{\min} = 1,\quad n_{\max} = 12

$n \in \left[1, 12\right]$

$n = 1$ $n = 1$

Along the curve

③

Response curve — green

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

$n \in \left[1, 12\right]$

$n = 1$ $a_0 = 0.529$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}a_0}{\mathrm{d}n}

$n \in \left[1, 12\right]$

$n = 1.069$ $a_0^{\prime} = 1.132$

Along the curve

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