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EXAMINATION
Semester 1 - End of Semester, 2017
PHYS3033_Semester 1 Nuclear Physics
This paper is for ANU students.
Examination Duration:
180 minutes
Reading Time:
15 minutes
Exam Conditions:
Central Examination
Students must return the examination paper at the end of the examination
This examination paper is available to the ANU Library archives
Any materials (including books, lecture notes, assignments, ANU library books etc). No electronic
devices such as laptops, tablets, phones. Programmable calculators are permitted.
Materials Permitted In The Exam Venue:
(No electronic aids are permitted e.g. laptops, phones)
Any materials (including books, lecture notes, assignments, ANU library books etc). No electronic
devices such as laptops, tablets, phones. Programmable calculators are permitted.
Materials To Be Supplied To Students:
1 x 20 page plain
Instructions To Students:
There are 8 questions of unequal value totalling 92 marks.
Attempt all questions.
Semester 1 - End of Semester, 2017
PHYS3033_Semester 1 Nuclear Physics
Useful constants:
• e2 ⇠ 1.44 MeV fm
• Atomic mass unit, 1 u ⇠ 1.66⇥10
27
kg
• Atomic mass unit, 1 u = 931.502 MeV/c2
• ~c ⇠ 197.3 MeV fm
• 1 mbarn = 10
27
cm2 , or 1 mbarn = 0.1 fm2 , or 1 barn = 100 fm2
1. [6 marks in total] The internuclear potential includes three terms, as shown:
V (r) =
Z 1 Z2 e 2
L2
+
+ Vnuc (r)
r
2µr2
(a) [2 marks] What is the origin of the first term in this equation? Over what range
does this term act?
(b) [2 marks] Sketch the nuclear interaction Vnuc between two colliding nuclei. Is this
potential attractive or repulsive? Over what range does it act?
(c) [2 marks] What e↵ect does the middle term have on the fusion barrier for a collision
with an impact parameter of (i) b=0, (ii) b >> 0?
2. [10 marks in total] Consider a reaction between 12 C and 232 Th. The experiment is
performed in regular kinematics, so 12 C, the projectile, has an energy of 58 MeV, and the
232
Th target is at rest.
(a) [2 marks] What is the excitation energy of the compound nucleus for this reaction?
You may need the following information:
0 MeV/c2
(12 C) =
(232 T h) =
35.45 MeV/c2
(244 Cm) = 58.45 MeV/c2 ,
where
12
C has Z=6,
232
Th has Z=90, and
244
Cm has Z = 96.
(b) [2 marks] Calculate an approximate value for the fusion barrier for this reaction,
assuming you can treat the two colliding nuclei as di↵use spheres with radii in units of
[fm] given by rnuc =1.7 A1/3 .
(c) [2 marks] What is the classical fusion cross section in the centre-of-mass frame for
this reaction? Use VB = 54 MeV (which should be close to but not the same as your
answer in (b).) Give your answer in units of mb.
2
(d) [2 marks] Approximating the potential near the Coulomb barrier region by an inverted parabola with curvature ~! leads to the following analytical expression for the
fusion cross-section including quantum tunnelling:
✓
✓
~! 2
2⇡
rb ln 1 + exp
(Ec.m.
f us (Ec.m. ) =
2Ec.m.
~!
VB )
◆◆
.
Using ~! = 2.7 MeV and VB = 54 MeV, this yields a fusion cross section of 141.0 mb.
How does this cross-section compare with the one you computed in (c)? Explain why the
classical and quantum cross sections are similar or di↵erent for this reaction.
(e) [2 marks] 232 Th is a deformed nucleus, and your calculation in (b) assumes both
nuclei are spherical. How might the deformation of 232 Th influence the position of the
fusion barrier for low energy (Ec.m. ⇠ VB ) collisions?
3. [12 marks in total] You’re exploring an Egyptian ruin when you come across a small
yellow statue not far from the position of a tomb. You know a yellow pigment made of
As2 S3 was heavily used during a particular time period, and want to see if this type of
pigment was used on the statue you discovered.
You have an alpha source that emits 15 MeV alpha particles and a vacuum chamber
that can contain the beam, the sample, and an alpha particle detector with an angular
coverage of ✓lab = 1 . You can place the alpha particle detector at any angle relative
to the beam axis. Assume there’s a very thin flake of the statue pigment you can use to
conduct your analysis, so you can treat the sample as thin relative to the alpha particle
range.
For the case of Arsenic (As), there is only one stable isotope with Z = 33, A = 75. For
Sulfur (S), the most abundant isotope has Z = 16, A = 32. You can ignore any other
Sulfur isotopes for this problem.
(a) [2 marks] Calculate the distances of closest approach at which you would begin to
see deviations from Rutherford scattering for collisions between the alpha particles and
75
As or 32 S. Assume you can treat the two colliding nuclei as di↵use spheres with radii in
units of [fm] given by rnuc =1.7 A1/3 . (Your answers should be on the order of ⇠ 10 fm.)
(b) [4 marks] Over what lab angle range will you see deviations from the Rutherford
scattering cross section for alpha collisions with each isotope? (Your answers should be
✓lab < ⇡/2).
(c) [3 marks] Let’s say you would like to use the scattered alpha particle energy to
confirm the isotopes present in the sample, and you would also like to use the relative
intensity of the peaks corresponding to each isotope to confirm the chemical composition
of the sample. The scattered alpha particle energies are shown in the figure as a function
of ✓c.m. with As denoted by the dotted line and S by the solid line.
Given this figure and your answer in (b), at what angle would you place your detector to
achieve the maximum separation in the energy of the scattered alpha particles?
3
15
scat [MeV]
Elab
14
13
12
11
10
9
0
50
100
150
θcm [deg]
(d) [3 marks] In the centre-of-mass frame, what do you expect the relative intensity of
the As and S peaks to be?
4. [27 marks in total] The Nilsson diagram at the end of the exam shows the behaviour of
the single particle levels for Z < 50 and N < 50 as a function of the nuclear deformation.
Use this diagram where necessary to answer the following questions.
(a) [3 marks] At ✏2 = 0, the levels are labelled by letters and numbers, for example, p3/2 .
What microscopic quantities do the letters and numbers signify and how are they related
to a third quantity?
(b) [3 marks] Away from ✏2 = 0, the level labels are more complex, for example, 5/2[312].
In these labels, what does the first (half-integer) number signify? Draw a diagram to
illustrate your answer.
(c) [4 marks] What are the angular momenta and parities of the ground states of the
following four nuclei?
•
•
•
•
18
8 O
19
8 O
49
20 Ca
41
20 Ca
(d) [3 marks] Only about 30 nuclei of 99
50 Sn have ever been created in the laboratory and
therefore no excited states are yet known. The figure below shows what the predicted
spins, parities and excitation energies might be for the ground state and first two excited
states in 99 Sn. What particle configurations give rise to these three states?
(e) [2 marks] The 52 state in 99 Sn can undergo two possible -ray decays. What are the
multipolarities of these two transitions?
(f) [5 marks] Ignoring the e↵ects of internal conversion, but allowing for the fact there
are two possible decays, estimate the lifetime of the 52 state in 99 Sn. Recall the Weisskopf
estimates in units of [s 1 ] for transitions with energy E in [MeV]:
4
•
•
•
•
•
•
•
•
(E1) = 1.0 ⇥ 1014 A2/3 E 3
(E2) = 7.3 ⇥ 107 A4/3 E 5
(E3) = 34A2 E 7
(E4) = 1.1 ⇥ 10 5 A8/3 E 9
(M 1) = 3.1 ⇥ 1013 E 3
(M 2) = 2.2 ⇥ 107 A2/3 E 5
(M 3) = 10.4 ⇥ A4/3 E 7
(M 4) = 3.3 ⇥ 10 6 A2 E 9
(g) [1 mark] If you included the e↵ect of internal conversion in (f), the calculated halflife
would change. Give a one sentence explanation justifying whether it would increase or
decrease.
(h) [3 marks] The lowest lying states in 98
48 Cd exhibit the excitation pattern shown in
the figure. Explain the configuration and mechanism by which these states are formed.
(i) [1 mark] The 8+ state in
why this state is long-lived?
98
Cd is a long-lived isomer. What is the dominant reason
(j) [2 marks] Using the meanlife of the 8+ state, estimate the lifetime of the 2+ state in
98
Cd.
97
5. [6 marks in total] In this question we will consider the nuclei 97
48 Cd and 47 Ag, neighbours
to 98 Cd.
(a) [2 marks] What is the particle configuration for the ground state of
angular momentum and parity?
5
97
Cd and its
(b) [1 marks] What is the maximum angular momentum you can make from rearranging
the proton and neutrons in the 97 Cd ground-state configuration?
(c) [2 marks] What is the particle configuration for the ground state of
angular momentum and parity?
97
Ag and its
(d) [1 marks] What is the maximum angular momentum you can make from rearranging
the proton and neutrons in the 97 Ag ground-state configuration?
6. [18 marks in total] Again you can use the Nilsson diagram where necessary to answer
the following questions.
(a) [3 marks] The nucleus 83
39 Y has a very low-lying, single-particle state with spin and
3
parity 2 that is prolate deformed with a rotational band sequence built upon it. What
is the particle configuration for the states in the rotational band and what is a lower limit
on the magnitude of the deformation? Briefly explain the logic that led to your answer.
(b) [2 marks] Name two other low-lying, single-particle configurations that might also
give rise to rotational bands in 83 Y.
(c) [4 marks] The rotational moment of inertia parameter for prolate deformed 83 Y is
~2
= 45 keV and the bandhead energy of the 32 state lies at 62 keV. Hence draw the
2=
level scheme (including energy, spin and parity) of the bandhead and the first 3 excited
states in the rotational band.
(d) [2 marks] The level scheme for 84 Zr is given in the diagram below. In this relatively
light nucleus, you can ignore internal conversion unless you are explicitly asked to do
so. The experimental transition strength, B(E2), for E2 transitions in units of e2 fm4 is
given by B(E2) = 8.161 ⇥ 10 10 E 5 P , where E is in units of [MeV] and P , the partial
gamma-ray transition probability, is in units of [s 1 ]. What is the B(E2) for the 723 keV,
4+ ! 2+ transition?
(e) [2 marks] The Weisskopf estimate for the transition probability of an E2 transition is
(E2) = 7.3⇥107 A4/3 E 5 . Hence evaluate the strength of the 723 keV, 4+ ! 2+ transition
in Weisskopf units. You should get an answer of the order tens of W.u.
6
(f) [1 mark] What does the magnitude of the transition strength for the 723 keV, 4+ ! 2+
transition tell you about the nature of this transition?
(g) [2 marks] Using the rotational model expression for the transition strength, B(E2) =
5 2 2
e Q0 hI1 K20|I2 Ki2 , and given that h4020|20i = 0.5345, evaluate the magnitude of the
16⇡
electric quadrupole moment of 84 Zr, eQ0 , in units of eb.
(h) [2 marks] What sorts of motion are likely to be responsible for the excited states at
1119 and 1244 keV, respectively, in 84 Zr?
7. [7 marks in total] While making anti-static brushes, a worker accidentally breathes
in an aerosol that contains 210 Po. A picogram of 210 Po is deposited in their lungs and
cannot be removed. The halflife of 210 Po is 138.4 days and it emits an alpha-particle
with an energy of 5.4 MeV. The daughter nucleus, 206 Pb, is stable. Assume the lungs are
uniformly irradiated and weigh 0.8 kg.
(a) [3 marks] What is the absorbed dose (in units of mGy) to the lungs over the first
276.8 days after the 210 Po has been inhaled?
(b) [1 mark] What is the equivalent dose (in units of mSv) to the lungs over the same
period?
(c) [2 marks] What is the e↵ective whole body dose (in units of mSv) given that the
tissue weighting factor for the lungs is wT = 0.12?
(d) [1 mark] Is this exposure a dangerous dose? Give a very brief reason for your answer.
8. [6 marks in total] The Semi-Empirical Mass Formula (SEMF) describes the approximate
binding energy of the nucleus as a function of N and Z. The form used in the present
course is:
M (A, Z)c2 ⇡
a1 A + a2 A2/3 + a3
Z(Z 1)
(N Z)2
+
a
± a5 A
4
A1/3
A
1/2
,
(1)
with an empirical fit giving values of a1 = 15.56, a2 =17.23, a3 = 0.7, a4 = 23.6 and
a5 = 11.2, all in units of [MeV].
(a) [2 marks] There are only a handful of stable odd-odd nuclei in nature. Verify which
term in the SEMF is responsible for this phenomenon and give a very brief explanation.
(b) [4 marks] Very heavy nuclei can spontaneously deform and then fission. What two
terms in the SEMF will change when heavy nuclei become deformed? As the nucleus
becomes deformed, state whether these two terms will increase or decrease the binding
energy, respectively.
END OF EXAMINATION
Nilsson diagram follows on next page
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