problem1) The ground state of ^{61}_{28}Ni 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 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
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^{*}Rr^{2} 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} (3z^{2}-r^{2}) Ψ_{j,j} dv. prepare the quadrupole moment operator in terms of Y_{20} and use the integral,
∫Y^{*}_{l,l}Y_{2,0}Y_{l,l}
(c) Apply this result to the ground state of ^{41}_{21}Sc , which has j^{π}=7/2‾. prepare 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 ^{17}_{9}F , which has j^{π}=5/2^{+} Compare your result to the measured quadrupole moment of 5.8 ±0.4 fm^{2} (note that this measurement does not determine the sign of the quadrupole moment, only the magnitude).
problem3) The deuteron wave function may be written |Ψ_{D}> a|^{3}S_{1}> b|^{3}D_{1}>where as states are all normalized i.e. <Ψ_{D}|Ψ_{D}> <^{3}S_{1}|^{3}S_{1}> =1 and |^{3}D_{1}|^{3}D_{1}>=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'+1}L^{'}_{j}|Û|^{2s+1}L_{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}.
problem4) A “simple” model of the deuteron. Consider a 3-dim square well of radius r_{0} and depth V_{0}. 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
d^{2}u/dr^{2} + k^{2}u=0 where k^{2} =2μ/?^{2}(E-V(r))
and μ=m_{p}m_{n}/m_{p}+m_{n} is the reduced mass
and Ψ(r^{→})=(u(r)/r) Y_{00 }(Θ,φ)_{ }(u(r)/r)(1/√4π)
a) Show that u(r) Asin(k_{1},r) with k_{1} =√2μ(V_{0}-E_{B})/?
u(r)=B_{e}^{-k2r} with k_{2} =√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 (V_{0}) 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= [2k_{2} /(1+r_{0}k_{2})]^{1/2} and B A_{e}^{k2 r0} [(V_{0}-E_{B})/v_{0}]^{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?