The Balmer- line of 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

Deriving the Spectrum, Ch Fine structure and the Lamb shift

Problem

The Balmer-

\beta

line of hydrogen,

n=4 \to n=2

, has vacuum wavelength

486.1\,\mathrm{nm}

. Take

cR_{\rm H}=3.28805\times10^6\,\mathrm{GHz}

and

\alpha=1/137.036

. ((a)) From the fine-structure formula, find the

2P_{3/2}

2P_{1/2}

interval and the full fine-structure spread of the

n=4

level, from

j=\tfrac12

j=\tfrac72

, both in GHz. Keeping only the larger, state the predicted frequency spread of the Balmer-

\beta

fine-structure pattern. ((b)) In a hydrogen discharge at

300\,\mathrm{K}

, the Doppler width (FWHM) of the Balmer-

\beta

line is

7.6\,\mathrm{GHz}

. A grating spectrometer has resolving power

R = \lambda/\Delta\lambda = 1.0\times10^5

at this wavelength. Find the smallest frequency interval

\Delta\nu

the spectrometer alone can resolve, and decide whether the splitting from part (a) would appear as separate lines in this discharge. ((c)) The potassium resonance doublet, the

4p\ {}^2P

4s\ {}^2S

transition, has components at

766.49\,\mathrm{nm}

and

769.90\,\mathrm{nm}

(vacuum wavelengths). These two wavelengths differ only through the

4p\ {}^2P_{3/2}

4p\ {}^2P_{1/2}

splitting, since the

4s

level is unsplit. Calculate this measured splitting in

\mathrm{cm^{-1}}

, calculate the hydrogenic (one-electron,

Z=1

) fine-structure splitting for

n=4

l=1

from the fine-structure formula, and give the ratio of the measured potassium splitting to the hydrogenic one.

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