A common nuclear potential is the Woods-Saxon potential. This takes the form of: U(r): Uo 1+...

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A common nuclear potential is the Woods-Saxon potential. This takes the form of: U(r): Uo 1+ exp[(r Ro)/a] Uls 1 d 1 + rdr (1+ exp[(r- Ro)/a]) R)/a]) 1.; S A slight simplification for 160 gives the following formula: V(r) (-51)/(1+exp(r-1.27*16^(1/3)/.67))+(51*.44)/1.27^2*(1/r)*-(e^(r-4.75)/(e^(r-4.75)+1)^2) Plotting this from r = 0.1-7 fm with the 3 bound proton nuclear levels gives the following curve: V(r) (MeV) 1 2 3 4 5 6 7 r (fm) -10 -12.4 MeV (1p1/2) -20 -19 MeV (1p3/2)4 -30 -40 -43 MeV (1s1/2) -50 a) What is the particle in a box nuclear radius for each of the three levels illustrated here? (Calculate the answer, do not just make a guess from the figure, though you can "check" your answer effectively by comparing your calculation to the lines drawn in the figure). b) Assuming h/2 = AxAp, calculate Ax from the momentum of the proton, for each of the three levels, calculated at R=(roA 1/3)/2, assuming ro = 1.27. (Hint: Find the kinetic energy of the protons at each level, and convert to momentum, then assume Ap = p.) Does this make sense, given the size of the 160 nucleus? c) What is the relativistic total energy of 1 proton in each level? (Either J or MeV acceptable) d) Assuming that the neutron energy levels are different by a constant Coulombic repulsion shift such that En = E-Ec, and E = 3*1.44 77, where R = roA 1/3, calculate Ax for the Z 5 R' neutrons in the lowest energy level (V(1S1/2)). e) How do neutron and proton radii compare for the same orbitals? Why are they different? A common nuclear potential is the Woods-Saxon potential. This takes the form of: U(r): Uo 1+ exp[(r Ro)/a] Uls 1 d 1 + rdr (1+ exp[(r- Ro)/a]) R)/a]) 1.; S A slight simplification for 160 gives the following formula: V(r) (-51)/(1+exp(r-1.27*16^(1/3)/.67))+(51*.44)/1.27^2*(1/r)*-(e^(r-4.75)/(e^(r-4.75)+1)^2) Plotting this from r = 0.1-7 fm with the 3 bound proton nuclear levels gives the following curve: V(r) (MeV) 1 2 3 4 5 6 7 r (fm) -10 -12.4 MeV (1p1/2) -20 -19 MeV (1p3/2)4 -30 -40 -43 MeV (1s1/2) -50 a) What is the particle in a box nuclear radius for each of the three levels illustrated here? (Calculate the answer, do not just make a guess from the figure, though you can "check" your answer effectively by comparing your calculation to the lines drawn in the figure). b) Assuming h/2 = AxAp, calculate Ax from the momentum of the proton, for each of the three levels, calculated at R=(roA 1/3)/2, assuming ro = 1.27. (Hint: Find the kinetic energy of the protons at each level, and convert to momentum, then assume Ap = p.) Does this make sense, given the size of the 160 nucleus? c) What is the relativistic total energy of 1 proton in each level? (Either J or MeV acceptable) d) Assuming that the neutron energy levels are different by a constant Coulombic repulsion shift such that En = E-Ec, and E = 3*1.44 77, where R = roA 1/3, calculate Ax for the Z 5 R' neutrons in the lowest energy level (V(1S1/2)). e) How do neutron and proton radii compare for the same orbitals? Why are they different?