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Index/Atomic physics/ATM-10

ATM-10

Reduced-mass Rydberg

μ = m M / (m+M), R_μ = R_∞ μ / m_e. Finite-mass shift of the spectrum.

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StructureReduced mass

Governing equation

μ=mM/(m+M),Rμ=R∞μ/me\mu=mM/(m+M),\quad R_\mu=R_\infty\mu/m_e

M∈[5×10−4,240]M \in \left[5\times 10^{-4}, 240\right]

where

M
Nuclear mass (u)
\mu/m_e
Reduced mass (—)
R_\mu
Rydberg (1/m)

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ATM-10 · spectrum
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①

Governing law — red

μ=mM/(m+M),Rμ=R∞μ/me\mu=mM/(m+M),\quad R_\mu=R_\infty\mu/m_e

M∈[5×10−4,240]M \in \left[5\times 10^{-4}, 240\right]

M=5×10−4M = 5\times 10^{-4}μ/me=5.23×106\mu/m_e = 5.23\times 10^{6}

②

Domain of the variable — blue

Mmin⁡=5×10−4,Mmax⁡=240M_{\min} = 5\times 10^{-4},\quad M_{\max} = 240

M∈[5×10−4,240]M \in \left[5\times 10^{-4}, 240\right]

M=5×10−4M = 5\times 10^{-4}M=5×10−4M = 5\times 10^{-4}

③

Response curve — green

μ=mM/(m+M),Rμ=R∞μ/me\mu=mM/(m+M),\quad R_\mu=R_\infty\mu/m_e

M∈[5×10−4,240]M \in \left[5\times 10^{-4}, 240\right]

M=5×10−4M = 5\times 10^{-4}μ/me=5.23×106\mu/m_e = 5.23\times 10^{6}

④

Rate of change — amber

dμ/medM\frac{\mathrm{d}\mu/m_e}{\mathrm{d}M}

M∈[5×10−4,240]M \in \left[5\times 10^{-4}, 240\right]

M=1.51M = 1.51μ/me′=1.9×106\mu/m_e^{\prime} = 1.9\times 10^{6}

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-10 — Reduced-mass Rydberg) is the form associated with Reduced mass. Working symbols: MM →\rightarrow μ/me\mu/m_e, RμR_\mu. Hydrogen versus deuterium: the Balmer lines shift by about 0.17 nm. Positronium has μ = m/2.

Purpose

Purpose: compute μ/me\mu/m_e, RμR_\mu from MM in Atomic physics via μ=mM/(m+M),Rμ=R∞μ/me\mu=mM/(m+M),\quad R_\mu=R_\infty\mu/m_e μ = m M / (m+M), R_μ = R_∞ μ / m_e. Finite-mass shift of the spectrum. 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 M=1.008 uM = 1.008\,\mathrm{u}, the governing relation μ=mM/(m+M),Rμ=R∞μ/me\mu=mM/(m+M),\quad R_\mu=R_\infty\mu/m_e yields μ/me=0.999456 —\mu/m_e = 0.999456\,\mathrm{—}, Rμ=1.1e+7 1/mR_\mu = 1.1e+7\,\mathrm{1/m}. Two masses, a reduced μ, a slightly smaller R. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Outputs

  • Reduced mass \mu/m_e0.999456 —
  • Rydberg R_\mu10967761.4 1/m
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ATM-10 · spectrum
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Narration of this film

Two masses, a reduced μ, a slightly smaller R.

Hydrogen versus deuterium: the Balmer lines shift by about 0.17 nm. Positronium has μ = m/2.

Reading speed

Watch on YouTube