An arc source is contaminated with three species: neutral hydrogen

PHYSICS · Multi-part · Fermi University Physics · Atomic, Molecular & Optical Physics · One-electron atoms and the origins of atomic structure · One-electron atoms and the origins of atomic structure

Reading the Spectrum, Ch The nuclear atom and the Bohr-Sommerfeld model

Problem

An arc source is contaminated with three species: neutral hydrogen, singly-ionised helium (He

^+

) and doubly-ionised lithium (Li

^{2+}

). A photographic plate records four lines, whose vacuum wavelengths are

434.17\,\mathrm{nm}

450.0\,\mathrm{nm}

468.7\,\mathrm{nm}

and

656.47\,\mathrm{nm}

. The first, third and fourth are already known:

434.17\,\mathrm{nm}

is hydrogen's Balmer line H

\gamma

(

n=5\to2

468.7\,\mathrm{nm}

is the He

^+

line (

n=4\to3

), and

656.47\,\mathrm{nm}

is hydrogen's H

\alpha

(

n=3\to2

). ((a)) Show, using hydrogenic scaling with

Z=3

and treating the lithium nucleus as effectively infinitely heavy so that

R_{\rm Li}\approx R_\infty = 109737.3\,\mathrm{cm^{-1}}

, which transition of Li

^{2+}

accounts for the remaining line at

450.0\,\mathrm{nm}

. ((b)) State briefly why using

R_\infty

rather than

R_{\rm H}

introduces negligible error for lithium but would not for hydrogen itself.

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