Calculating an estimate of the quadrupole moment

Question

Question1) The ground state of ⁶¹₂₈Ni has  j^π=3/2‾ . The first excited state at 67.4 keV has j^π=5/2‾ and the second excited state at 283.0 keV hasj^π=1/2‾ . List the possible γ-ray transitions between these levels and give their type. Estimate the half-life of the j^π=1/2‾state using the Weisskopf approximation. How does this compare to the measured half-life of 23 ps?
Quewstion2) 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 + ½ . The space wave function of the proton is

Ψ_j,mj= R(r) Y_l,m(Θ,φ)

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

∫Y^*_l,ml Y_l,ml dΩ=1 & R^*Rr² dr=1
a) Since m_j = m_l m_s , what must m_l and m_s be if m_j = j?
b) Show that the quadrupole moment, when m_j = j, is given by

Q=2>(2j-1/2j+2)dΩ= (5/4π)^1/2 l(/2l+3)
To do this start with the quadrupole moment given by Q ∫Ψ^*_j,j (3z²-r²) Ψ_j,j dv. Write the quadrupole moment operator in terms of Y₂₀ and use the integral,

∫Y^*_l,lY_2,0Y_l,l
(c) Apply this result to the ground state of  ⁴¹₂₁Sc , which has j^π=7/2‾. Write the configuration for this ground state and confirm that the condition of j = l + ½ holds. Estimate 2>using 1.2A^1/3 . Compare your result to the measured quadrupole moment of-0.156 ±0.003 b.
d) Apply this result to the ground state of ¹⁷₉F , which has j^π=5/2⁺ Compare your result to the measured quadrupole moment of 5.8 ±0.4 fm² (note that this measurement does not determine the sign of the quadrupole moment, only the magnitude).
Question3) The deuteron wave function may be written |Ψ_D> a|³S₁> b|³D₁>where as states are all normalized i.e. <Ψ_D|Ψ_D> <³S₁|³S₁> =1 and |³D₁|³D₁>=1
Find b2 such that |Ψ_D> reproduces the known magnetic moment of the deuteron,μ=.857μ_N. Use the result that for J = 1,
^2s'+1L^'_j|Û|^2s+1L_j {1/2(Σ+1/2)+1/4(Σ-1/2)[S(S+1)-L(L+1)]}δ_{s's L'L}μ_N where (μ_{p +}μ_n)/μ_N.
Question4) A “simple” model of the deuteron. Consider a 3-dim square well of radius r₀ and depth V₀. i.e. V(r) = –V0 for r < r0, V(r) = 0 for r > r0. The two particles interact via this potential and have a binding energy EB, so the total energy is E = –EB. Converting to a centre of mass system we find that the radial equation for S-states (l = 0) is

d²u/dr² + k²u=0 where k² =2μ/?²(E-V(r))

and μ=m_pm_n/m_p+m_n is the reduced mass

and Ψ(r^→)=(u(r)/r) Y₀₀ (Θ,φ) (u(r)/r)(1/√4π)

a) Show that u(r) Asin(k₁,r) with k₁ =√2μ(V₀-E_B)/?

u(r)=B_e^-k2r with k₂ =√2μE_B for r > r0
where A and B are constants.

(b) Apply boundary conditions to u(r) and obtain a relation between k1, k2, and r0 that does not involve A and B.

(c) If r0 = 2.4 fm (the diameter of the deuteron), how deep is the potential well (V₀) to give the experimental binding energy of 2.225 MeV. (Note: you will have to solve an equation of the form tan(ak) =bk for k. You can do this
graphically, by using successive approximations, or by using a program
such as maple, whatever works for you.)

(d) If the wavefunction Ψ(r^→ ) is normalized i.e. ∫ Ψ(r^→ )² dv=1 it can be shown that
A= [2k₂ /(1+r₀k₂)]^1/2 and B A_e^{k₂ r0} [(V₀-E_B)/v₀]^1/2 .
Find the probability that the nucleons will be found outside the range of the potential i.e. r > r0. Does the answer surprise you?