1. The ground state of
has jΠ = 3/2-1 The first excited state at 67.4 keV has jΠ = 1/2-1 and the second excited state at 283.0 keV has List the possible γ -ray transitions between these levels and give their type. Estimate the half-life of the state jΠ = 1/2- using the Weisskopf approximation. How does this compare to the measured half-life of 23 ps?
2. Quadrupole moments in the shell model. We will calculate an estimate of the quadrupole moment for the special case of a single proton moving in an orbital around a closed shell spherical core. So the only contribution to the quadrupole moment is from this single proton. We will also assume that the proton moves in an orbital with j = l + 1/2 . The space wave function of the proton is
![1282_Quadrupole moments in the shell model2.png](https://www.mywordsolution.com/CMSImages/1282_Quadrupole%20moments%20in%20the%20shell%20model2.png)
where Y is the spherical harmonic, and R is the radial part of the wave function. They are normalized. i.e.
![1691_Quadrupole moments in the shell model3.png](https://www.mywordsolution.com/CMSImages/1691_Quadrupole%20moments%20in%20the%20shell%20model3.png)
(a) Since s mj = m1 + m2, what must m1 and ms be if mj = j?
(b) Show that the quadrupole moment, when m = j, is given by
![1934_Quadrupole moments in the shell model4.png](https://www.mywordsolution.com/CMSImages/1934_Quadrupole%20moments%20in%20the%20shell%20model4.png)
To do this start with the quadrupole moment given by
.
Write the quadrupole moment operator in terms of Y20 and use the integral,
![1747_Quadrupole moments in the shell model6.png](https://www.mywordsolution.com/CMSImages/1747_Quadrupole%20moments%20in%20the%20shell%20model6.png)
(c) Apply this result to the ground state of
, which has jΠ = 7/2-1. Write the configuration for this ground state and confirm that the condition of j = l + ½ holds. Estimate (r2) using r = 1.2 A1/2 . Compare your result to the measured quadrupole moment of -0.156 ± 0.003 b.
(d) Apply this result to the ground state of 917F, which has jΠ = 5/2+. Compare your result to the measured quadrupole moment of 5.8 ± 0.4 fm2.