A singly charged ion of a hypothetical element has two lines studied across four even…

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Atomic, Molecular & Optical Physics · One-electron atoms and the origins of atomic structure

Deriving the Spectrum, Ch Hyperfine structure and isotope shifts

Problem

A singly charged ion of a hypothetical element has two lines studied across four even isotopes, with atomic masses

M_{40} = 39.9626\,\mathrm{u}

M_{42} = 41.9586\,\mathrm{u}

M_{44} = 43.9555\,\mathrm{u}

and

M_{48} = 47.9525\,\mathrm{u}

. Line 1 has frequency

\nu_1 = 755.22\,\mathrm{THz}

and line 2 has

\nu_2 = 346.00\,\mathrm{THz}

. Each isotope shift is

\delta\nu = \nu(M') - \nu(M)

, with

M = M_{40}

and

M'

the heavier isotope of the pair: Data: pair

(40,42)

has

\delta\nu_1=450.8\,\mathrm{MHz}

and

\delta\nu_2=242.2\,\mathrm{MHz}

; pair

(40,44)

has

\delta\nu_1=894.6\,\mathrm{MHz}

and

\delta\nu_2=448.1\,\mathrm{MHz}

; pair

(40,48)

has

\delta\nu_1=1756.3\,\mathrm{MHz}

and

\delta\nu_2=772.4\,\mathrm{MHz}

. Take

m_e = 5.4858\times10^{-4}\,\mathrm{u}

, and use the atomic masses as the masses

M

in the normal-mass-shift formula. ((a)) For each pair and each line, subtract the normal mass shift and form the modified shift

\delta\nu^{\rm mod}

, in

\mathrm{GHz\,u}

. ((b)) Draw the King line through the

(40, 42)

and

(40, 48)

points, with line 1 on the horizontal axis. Give its slope

\mathcal{F}_2/\mathcal{F}_1

and its intercept in

\mathrm{GHz\,u}

, and check whether the

(40, 44)

point lies on it within the rounding of the data. ((c)) An atomic-structure calculation gives line 1 the constants

\mathcal{F}_1 = -280\,\mathrm{MHz\,fm^{-2}}

and

\mathcal{K}_1 = 15.0\,\mathrm{GHz\,u}

. Find the change in mean square charge radius,

\delta\langle r^2\rangle

, between isotopes 40 and 48.

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