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Index/Atomic physics/ATM-18

ATM-18

Natural line width

Γ = ħ / τ, Δν = A / 2π. Lorentzian core of a line.

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SpectraNatural width

Governing equation

Γ=ℏ/τ,Δν=A/2π\Gamma=\hbar/\tau,\quad\Delta\nu=A/2\pi

τ∈[0.01,5000]\tau \in \left[0.01, 5000\right]

where

\tau
Lifetime (ns)
\Gamma
Energy width (neV)
\Delta\nu
Frequency width (MHz)

The figure, part by part

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ATM-18 · spectrum
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①

Governing law — red

Γ=ℏ/τ,Δν=A/2π\Gamma=\hbar/\tau,\quad\Delta\nu=A/2\pi

τ∈[0.01,5000]\tau \in \left[0.01, 5000\right]

τ=0.01\tau = 0.01Γ=15915.5\Gamma = 15915.5

②

Domain of the variable — blue

τmin⁡=0.01,τmax⁡=5000\tau_{\min} = 0.01,\quad \tau_{\max} = 5000

τ∈[0.01,5000]\tau \in \left[0.01, 5000\right]

τ=0.01\tau = 0.01τ=0.01\tau = 0.01

③

Response curve — green

Γ=ℏ/τ,Δν=A/2π\Gamma=\hbar/\tau,\quad\Delta\nu=A/2\pi

τ∈[0.01,5000]\tau \in \left[0.01, 5000\right]

τ=0.01\tau = 0.01Γ=15915.5\Gamma = 15915.5

④

Rate of change — amber

dΓdτ\frac{\mathrm{d}\Gamma}{\mathrm{d}\tau}

τ∈[0.01,5000]\tau \in \left[0.01, 5000\right]

τ=31.46\tau = 31.46Γ′=−253\Gamma^{\prime} = -253

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-18 — Natural line width) is the form associated with Natural width. Working symbols: τ\tau →\rightarrow Γ\Gamma, Δν\Delta\nu. The Fourier transform of exponential decay. Usually dwarfed by Doppler and pressure.

Purpose

Purpose: compute Γ\Gamma, Δν\Delta\nu from τ\tau in Atomic physics via Γ=ℏ/τ,Δν=A/2π\Gamma=\hbar/\tau,\quad\Delta\nu=A/2\pi Γ = ħ / τ, Δν = A / 2π. Lorentzian core of a line. 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 τ=10.000 ns\tau = 10.000\,\mathrm{ns}, the governing relation Γ=ℏ/τ,Δν=A/2π\Gamma=\hbar/\tau,\quad\Delta\nu=A/2\pi yields Γ=65.8212 neV\Gamma = 65.8212\,\mathrm{neV}, Δν=15.9155 MHz\Delta\nu = 15.9155\,\mathrm{MHz}. A Lorentzian spike, a lifetime. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • Energy width \Gamma65.8212 neV
  • Frequency width \Delta\nu15.9155 MHz
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ATM-18 · spectrum
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Narration of this film

A Lorentzian spike, a lifetime.

The Fourier transform of exponential decay. Usually dwarfed by Doppler and pressure.

Reading speed

Watch on YouTube