ATM-16

Einstein A coefficient

A_{21} = 1/τ. Spontaneous emission rate from a lifetime.

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SpectraEinstein A

Governing equation

A_{21}=1/\tau

$\tau \in \left[0.01, 1000\right]$

where

\tau: Lifetime (ns)
A_{21}: Einstein A (1/s)
\Gamma: Natural width ħ (MHz)

The figure, part by part

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①

Governing law — red

A_{21}=1/\tau

$\tau \in \left[0.01, 1000\right]$

$\tau = 0.01$ $A_{21} = 1\times 10^{11}$

Along the curve

②

Domain of the variable — blue

\tau_{\min} = 0.01,\quad \tau_{\max} = 1000

$\tau \in \left[0.01, 1000\right]$

$\tau = 0.01$ $\tau = 0.01$

Along the curve

③

Response curve — green

A_{21}=1/\tau

$\tau \in \left[0.01, 1000\right]$

$\tau = 0.01$ $A_{21} = 1\times 10^{11}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}A_{21}}{\mathrm{d}\tau}

$\tau \in \left[0.01, 1000\right]$

$\tau = 6.299$ $A_{21}^{\prime} = -7.94\times 10^{9}$

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-16 — Einstein A coefficient) is the form associated with Einstein A. Working symbols: $\tau$ $\rightarrow$ $A_{21}$ , $\Gamma$ . Related to f by A = (2π e² ν² / ε₀ m c³) f. B coefficients follow from detailed balance.

Purpose

Purpose: compute

A_{21}

\Gamma

from

\tau

in Atomic physics via

A_{21}=1/\tau

A_{21} = 1/τ. Spontaneous emission rate from a lifetime. 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

\tau = 10.000\,\mathrm{ns}

, the governing relation

A_{21}=1/\tau

yields

A_{21} = 1.000e+8\,\mathrm{1/s}

\Gamma = 15.9155\,\mathrm{MHz}

. An excited atom, a ticking decay. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Lifetime \tau10 ns

Outputs

Einstein A A_{21}100000000.000 1/s
Natural width ħ \Gamma15.9155 MHz

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Narration of this film

An excited atom, a ticking decay.

Related to f by A = (2π e² ν² / ε₀ m c³) f. B coefficients follow from detailed balance.

Reading speed

Watch on YouTube