Chapter 8 relates a line's spontaneous rate to its absorption oscillator strength by

PHYSICS · Multi-part · Fermi University Physics · Atomic, Molecular & Optical Physics · Radiative transitions, external fields and collisions · Radiative transitions, external fields and collisions

Deriving the Spectrum, Ch Atoms in electric fields

Problem

A_{21}=\dfrac{2\omega^2e^2}{m_e\epsilon_0c^3}\dfrac{g_1}{4\pi g_2}f_{12}

, where

g_1

and

g_2

are the degeneracies

2J+1

of the lower and upper levels. With the classical radiative rate

\gamma_{\rm cl}=e^2\omega^2/(6\pi\epsilon_0m_ec^3)

this reads

1/\tau=3(g_1/g_2)f_{12}\gamma_{\rm cl}

for an upper level with a single route down. Sodium's two D lines both sit at angular frequency

\omega=3.20\times10^{15}\,\mathrm{rad\,s^{-1}}

. The

3p_{3/2}

level (upper level of D2) has a measured lifetime

\tau=16.2\,\mathrm{ns}

, and the

3p_{1/2}

level (upper level of D1) has

\tau=16.3\,\mathrm{ns}

; both decay only to

3s_{1/2}

. ((a)) Find the oscillator strengths of the D2 and D1 lines. ((b)) Treating the two D lines as sodium's only significant oscillator strength, estimate its static polarisability

\alpha_{\rm pol}/4\pi\epsilon_0

in units of

a_0^3

, using

\alpha_{\rm pol}=(e^2/m_e)\sum f/\omega^2

. ((c)) A sodium Rydberg level whose effective quantum number, from its quantum defect (Chapter 5), is

n^*=20

behaves hydrogenically. Estimate the field, in

\mathrm{V\,cm^{-1}}

, that would classically ionise it, using

\mathcal E_{\rm ion}=3.21\times10^8/n^{*4}\,\mathrm{V\,cm^{-1}}

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