1. The ground state of  has j^Π = 3/2^-1 The first excited state at 67.4 keV has j^Π = 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 j^Π = 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

where Y is the spherical harmonic, and R is the radial part of the wave function. They are normalized. i.e.

(a) Since s m_j = m₁ + m₂, what must m₁ and m_s be if m_j = j? 

(b) Show that the quadrupole moment, when m = j, is given by

To do this start with the quadrupole moment given by .

Write the quadrupole moment operator in terms of Y20 and use the integral,

(c) Apply this result to the ground state of , which has j^Π = 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 A^1/2 . 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 j^Π = 5/2⁺. Compare your result to the measured quadrupole moment of 5.8 ± 0.4 fm².