Question
Determine the expected shell-model quadrupole moment of .
Answer :
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The nuclear quadrupole moment in the shell model for a single unpaired nucleon is given by: [ Q = \frac{3K^2 - I(I+1)}{(I+1)(2I+3)} , Q_0 ] where (I) is the nuclear spin, (K) is the projection of (I) on the symmetry axis (for a single particle, (K = I)), and (Q_0) is the intrinsic quadrupole moment of the nucleon orbit. For a single proton outside a closed shell, the intrinsic quadrupole moment (Q_0) is given by: [ Q_0 = \frac{2j-1}{2j+2} , e , \langle r^2 \rangle ] where (j) is the total angular momentum of the unpaired nucleon, and (\langle r^2 \rangle) is the mean-square radius of the orbital. For (^ {209}\text{Bi}), the unpaired proton occupies the (h_{9/2}) orbital, so (j = 9/2), (I = 9/2), (K = 9/2). Step 1: Calculate the factor (\frac{2j-1}{2j+2}): [ 2j-1 = 2 \cdot \frac{9}{2} - 1 = 9 - 1 = 8 ] [ 2j+2 =
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