Rubidium ( ) has first ionisation energy ; its principal-series line ( ) lies at (vacuum)

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

Deriving the Spectrum, Ch The central field and the periodic table

Problem

Rubidium (

Z = 37

) has first ionisation energy

4.18\,\mathrm{eV}

; its principal-series

D_2

line (

5s \to 5p_{3/2}

) lies at

780.0\,\mathrm{nm}

(vacuum), and the measured

5p

fine-structure interval is

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

. ((a)) Using the ionisation energy and the

D_2

wavelength, find the effective principal quantum number

n^*

of the

5p_{3/2}

level. ((b)) Using Chapter 2's hydrogenic fine-structure formula

\Delta E_{\rm fs}(\mathrm H, n, l) = \alpha^2hcR_\infty/[n^3l(l+1)]

with the integer

n = 5

l = 1

, find the ratio of rubidium's measured

5p

interval to hydrogen's

n=5

l=1

interval. ((c)) Using Lande's formula

\Delta E_{\rm fs} \approx \alpha^2Z_i^2Z_o^2hcR_\infty/[n^{*3}l(l+1)]

with

Z_o = 1

and the

n^*

from (a), find the inner effective charge

Z_i

that reproduces rubidium's measured

5p

interval.

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