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problem1) The ground state of 6128Ni has  jπ=3/2‾ . The first excited state at 67.4 keV has jπ=5/2‾ and the second excited state at 283.0 keV hasjπ=1/2‾ . List the possible γ-ray transitions between these levels and give their type. Estimate the half-life of the jπ=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 find out 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

Ψj,mj= R(r) Yl,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 Yl,ml dΩ=1 & R*Rr2 dr=1
a) Since mj = ml ms , what must ml and ms be if mj = j?
b) Show that the quadrupole moment, when mj = 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) Ψj,j dv. prepare the quadrupole moment operator in terms of Y20 and use the integral,

∫Y*l,lY2,0Yl,l
(c) Apply this result to the ground state of  4121Sc , which has jπ=7/2‾. prepare the configuration for this ground state and confirm that the condition of j = l + ½ holds. Estimate 2>using 1.2A1/3 . 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 jπ=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).
problem3) The deuteron wave function may be written |ΨD> a|3S1> b|3D1>where as states are all normalized i.e. <ΨDD> <3S1|3S1> =1 and |3D1|3D1>=1
Find b2 such that |ΨD> reproduces the known magnetic moment of the deuteron,μ=.857μN. Use the result that for J = 1,
2s'+1L'j|Û|2s+1Lj {1/2(Σ+1/2)+1/4(Σ-1/2)[S(S+1)-L(L+1)]}δs's L'LμN where (μp +μn)/μN.
problem4) 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=mpmn/mp+mn 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-EB)/?

u(r)=Be-k2r with k2 =√2μEB 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 Aek2 r0 [(V0-EB)/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?

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