A faint emission line is measured in the far-ultraviolet spectrum of a hot subdwarf…

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 Coulomb spectrum from wave mechanics: quantum numbers and degeneracy

Problem

A faint emission line is measured in the far-ultraviolet spectrum of a hot subdwarf star at a vacuum wavelength of

164.1\,\mathrm{nm}

. A colleague proposes that it is the

n=3 \to n=2

transition of a hydrogen-like ion, either He

^+

(

Z=2

) or Li

^{2+}

(

Z=3

). ((a)) Using hydrogenic scaling of the radial equation (charge

Z

rescales lengths by

1/Z

and energies by

Z^2

, so wavelengths scale by

1/Z^2

), predict the vacuum wavelength of the

n=3\to2

transition for He

^+

and for Li

^{2+}

, starting from hydrogen's Balmer-

\alpha

vacuum wavelength of

656.5\,\mathrm{nm}

(from

\tilde\nu = R_{\rm H}(1/4-1/9) = 15233.0\,\mathrm{cm^{-1}}

). You may ignore the reduced-mass differences between the three species, none larger than about

0.05\,\%

, far too small to change which ion is identified. ((b)) State which ion produces the observed line.

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