ATM-17

Doppler line width

Δν_D = (ν₀ / c) √(2 k T / m). FWHM is 2√(ln 2) times this 1/e half-width.

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SpectraDoppler width

Governing equation

\Delta\nu_D=(\nu_0/c)\sqrt{2kT/m}

$T \in \left[10, 20000\right]$

where

\nu_0: Line frequency (THz)
T: Temperature (K)
M: Mass (u)
\Delta\nu_D: 1/e half-width (GHz)
\mathrm{FWHM}: FWHM (GHz)

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

\Delta\nu_D=(\nu_0/c)\sqrt{2kT/m}

$T \in \left[10, 20000\right]$

$T = 10$ $\Delta\nu_D = 1.031$

Along the curve

②

Domain of the variable — blue

T_{\min} = 10,\quad T_{\max} = 20000

$T \in \left[10, 20000\right]$

$T = 10$ $T = 10$

Along the curve

③

Response curve — green

\Delta\nu_D=(\nu_0/c)\sqrt{2kT/m}

$T \in \left[10, 20000\right]$

$T = 10$ $\Delta\nu_D = 1.031$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\Delta\nu_D}{\mathrm{d}T}

$T \in \left[10, 20000\right]$

$T = 135.7$ $\Delta\nu_D^{\prime} = 0.017$

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-17 — Doppler line width) is the form associated with Doppler width. Working symbols: $\nu_0$ , $T$ , $M$ $\rightarrow$ $\Delta\nu_D$ , $\mathrm{FWHM}$ . The dominant broadening in hot, low-density gas. Turbulence adds in quadrature.

Purpose

Purpose: compute

\Delta\nu_D

\mathrm{FWHM}

from

\nu_0

T

M

in Atomic physics via

\Delta\nu_D=(\nu_0/c)\sqrt{2kT/m}

Δν_D = (ν₀ / c) √(2 k T / m). FWHM is 2√(ln 2) times this 1/e half-width. 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

\nu_0 = 455.000\,\mathrm{THz}

T = 5000.000\,\mathrm{K}

M = 1.000\,\mathrm{u}

, the governing relation

\Delta\nu_D=(\nu_0/c)\sqrt{2kT/m}

yields

\Delta\nu_D = 13.839\,\mathrm{GHz}

\mathrm{FWHM} = 23.044\,\mathrm{GHz}

. A thermal cloud, a Gaussian line. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Line frequency \nu_0455 THzTemperature T5000 KMass M1 u

Outputs

1/e half-width \Delta\nu_D13.839 GHz
FWHM \mathrm{FWHM}23.044 GHz

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Narration of this film

A thermal cloud, a Gaussian line.

The dominant broadening in hot, low-density gas. Turbulence adds in quadrature.

Reading speed

Watch on YouTube