Potassium ( ) has ionisation energy from its ground state

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

Potassium (

Z = 19

) has ionisation energy

4.341\,\mathrm{eV}

from its

4s

ground state, and its isotope

{}^{39}\mathrm{K}

has nuclear

g

-factor

g_I = 0.2610

. Take

hcR = 13.606\,\mathrm{eV}

, hydrogen's ground-state hyperfine constant

1420.4\,\mathrm{MHz}

and the proton's

g_p = 5.586

. ((a)) Find

n^*

for the

4s

level. ((b)) Using the Fermi-Segre formula

|\psi_{ns}(0)|^2=Z/(\pi a_0^3n^{*3})

for a neutral atom (neglecting the small

d\delta/dn

term), find

|\psi_{4s}(0)|^2

in units of hydrogen's

1/\pi a_0^3

. ((c)) Estimate the ground-state hyperfine constant

A_{\rm hfs}

and compare it with the measured

230.9\,\mathrm{MHz}

. Would a charge smaller than

Z

, in place of

Z

, improve the estimate?

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