Consider a model lead isotope with (three neutron holes below the shell closure) and

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Nuclear Physics · Radioactivity and decay modes

The Nucleus Resolved, Ch Gamma decay: multipoles, isomers, internal conversion and the Mossbauer line

Problem

Consider a model lead isotope with

N=123

(three neutron holes below the

N=126

shell closure) and

A=205

, with ground state

J^\pi=5/2^-

and a metastable excited state at

E_x=1013.6\,\mathrm{keV}

with

J^\pi=13/2^+

. Suppose the isomer decays entirely by one gamma transition to the ground state, with photon partial half-life

T_{1/2}=4.60\,\mathrm{s}

. (The half-life and decay route are chosen for this exercise; they are not the evaluated data for

{}^{205}\mathrm{Pb}

.) ((a)) Below

N=126

the relevant neutron orbitals are the negative-parity

2f_{5/2}

3p_{3/2}

3p_{1/2}

and the positive-parity high-

j

intruder

1i_{13/2}

, all close in energy. Using this level order, explain why a nucleus with a few neutron holes below

N=126

can have a low-lying

13/2^+

state whose only route to a low-spin negative-parity ground state is a strongly hindered transition. ((b)) Use the triangle rule and the parity rule to identify the multipolarity of the

13/2^+\to5/2^-

transition. ((c)) The Weisskopf estimate for an electric multipole scales as

\lambda_W(E\lambda)=\lambda_W^{\rm ref}\left(\frac{A}{125}\right)^{2\lambda/3}\left(\frac{E_\gamma}{1\,\mathrm{MeV}}\right)^{2\lambda+1},

with reference value

\lambda_W^{\rm ref}(E4)=4.2\,\mathrm{s^{-1}}

A=125

E_\gamma=1\,\mathrm{MeV}

; the corresponding magnetic estimate is smaller by the factor

0.3A^{-2/3}

. Using

A=205

and

E_\gamma=1.0136\,\mathrm{MeV}

, compute the Weisskopf estimate for the multipolarity found in part (b), express the measured rate in Weisskopf units, and say what this implies about the origin of the isomerism.

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