Question 1: A certain nucleus has the following levels below 3 MeV:
E (keV) IΠ t1/2
0 0+ stable 992
2+ 1.9 ps 1799
2+ 2.0 ps 1910
0+ 1.0 ns 2307
4+ 0.8 ps 2609
0+ 0.2 ps 2737
4+ 1.5 ps 2794
2+ 0.05 ps 2980
3+ 0.30 ps 2999
3- 0.15 ps
(a) What is the most likely (Z, N ) for this nucleus: (odd, odd), (odd, even), (even, odd), or (even, even)? Give reasons.
(b) Based on this information, are these levels most likely to be
i. single-particle shell model states;
ii. vibrational states;
iii. rotational states; or
iv. none of the above?
Explain your answer. If there is more than one reason for your answer, please state all of them.
(c) There are two different states with I = 3, but similar energy. Briefly explain the likely structure of these two states.
(d) What are all of the possible multipoles (order L and type: E or M) for γ-ray transitions from the 992 keV level to the ground state? Give reasons.
(e) For the dominant multipole, estimate the half-life based on the figure in Lecture 7 slide 7 (this is Lilley Figure 3.2). Is the observed half-life shorter than, similar to, or longer than this estimate?
(f) The integral to derive the = 1 MR2 result for a solid cylinder or disk was shown in lectures. Show the integral that derives the = 1 ML2 result for a thin rod, with an axis perpendicular to the rod, and passing through its centre (see the figure from lectures).
Question 2: The short-lived (t1/2 = 27.4 s) nuclide 220Fr, with Iπ = 1+ in its ground state, undergoes alpha decay to many levels of 216At. The first six levels are shown in the table, together with the fraction of the alpha decays from 220Fr that populate each level:
E (keV) IΠ α fraction (%) 0.0 1- 65.8
44.6 2- 12.2
57.1 4- 1.6
105.9 0- 9.0
122.0 5- not seen
153.4 2- 2.5
(a) Calculate the Q value and the alpha particle energy for each of these decays.
(b) What are the possible values of the angular momentum Aα for each of these decays?
(c) The fraction of alpha decays to the 57.1 keV level is lower than that to the neighbouring levels, and alpha decays to the 122.0 keV level are not seen. Explain these facts using the results of the previous two questions.
E (keV) IΠ
0.0 0+
178.5 2+
410.1 4+
413.0 1
(d) A small fraction (0.35%) of 220Fr decays are β- decays to levels of 220Ra. Calculate the Q value for decays to the ground state.
(e) The energies and Iπ values of the four lowest-energy levels in 220Ra are shown in the table. State the type of the beta decay (i.e. allowed or Nth-forbidden) from 220Fr to each level, and give brief reasons.
(f) The second and third excitation are very closely-spaced in en- ergy. Decay is seen to one of these levels but not the other. State which level you expect this to be, and why.
Question 3: The medical isomer 87mSr is the first excited state of 87Sr, with an excitation energy of 0.3885 MeV. It has Iπ = 1 - and a half-life of 2.8 hours, decaying 99.7% of the time to the Iπ = 9 + ground state.
(a) Do you expect internal conversion to be
i. negligible compared to gamma decay
ii. less important than gamma decays
iii. comparable to gamma decay
iv. more important than gamma decay, or
v. completely dominant
for this transition? Give brief reasons.
(b) 87mSr is produced in electron capture and β+ decays of 87Y, with a half-life of 80 hours. Assuming that no 87mSr is initially present, calculate the activity of 87mSr, as a fraction of the activity of 87Y, after 1 hour, after 5 hours, and after 10 hours.
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