School of Physics & Astronomy

Nuclear Physics

PHYS11041 (SCQF Level 11)

Thursday 9th May, 2019 14:30 - 16:30

(May Diet)

Please read full instructions before commencing writing.

Examination Paper Information

Answer TWO questions

Special Instructions

• Only authorised Electronic Calculators may be used during this examination.

• A sheet of physical constants is supplied for use in this examination.

• Attach supplied anonymous bar codes to each script book used.

Special Items

• School supplied Constant Sheets

• School supplied barcodes

Chairman of Examiners: Prof A S Trew

External Examiner: Prof A De Santo

Anonymity of the candidate will be maintained during the marking of this

examination.

Printed: Tuesday 23rd April, 2019 PHYS11041

Nuclear Physics (PHYS11041)

1. The ground state of 22Na (Jpi = 3+ and lifetime τ = 2.6 y) decays to the ground state

of 22Ne by positron emission, i.e. β+ decay, (with decay probability 0.1%), as well as to

the first excited state (Jpi = 2+) of 22Ne by positron emission (with probability 90.4%) or

electron capture (with probability 9.5%). Once populated, the first excited state of 22Ne

further decays to the ground state of 22Ne by emitting a γ ray of 1274 keV. Mass excesses

for 22Na and 22Ne are −5.182 MeV and −8.024 MeV, respectively. From the information

provided, answer the following:

(a) Briefly describe the processes of nuclear positron emission, electron capture and

gamma decay. A couple of sentences in each case should be enough. [3]

(b) Sketch a comprehensive level diagram of the decay of 22Na with appropriate energies,

Jpi labelling, and branching ratios. [2]

(c) Calculate the end-point energies of the positrons emitted in the two β+ decays. [3]

(d) Use angular momentum and parity conservation selection rules to classify the β+

decays to the ground state and first excited state of 22Ne, respectively. Fully justify

any assumption you make and comment on the correctness of your classification

based on the given decay probabilities. [4]

(e) Use angular momentum and parity conservation laws to determine the multipolarity

and nature of the emitted γ rays. [3]

(f) Estimate the half-life of the first excited state in 22Ne, knowing that the appropriate

Weisskopf decay rate estimate (in units of s−1) is λ = 7.3×107A4/3E5γ , with A being

the mass number of the emitting nucleus and Eγ the γ-ray energy in MeV. [2]

(g) Sketch the expected γ-ray spectrum that you would obtain with a medium-sized

germanium detector. Label your sketch appropriately to indicate the origin of each

prominent feature of the spectrum. [8]

Printed: Tuesday 23rd April, 2019 Page 1 Continued overleaf. . .

Nuclear Physics (PHYS11041)

2. In nuclear physics, frequent use is made of the time-reversal approach to determine the

cross section of a reaction of interest from the cross section measured for its inverse

process. For a generic reaction 1+2 →3+4 the cross sections for the direct (σ12→34) and

inverse (σ34→12) reactions are related, according to the reciprocity theorem:

σ12→34

σ34→12

=

A3A4

A1A2

E34

E12

(2J3 + 1)(2J4 + 1)

(2J1 + 1)(2J2 + 1)

where Eij are the centre-of-mass energies corresponding to the same total energy in the

compound system, and Ai and Ji are the mass number and total spin of the nuclei involved

in the reaction, respectively.

Consider the 14O+α ⇀↽ p+ 17F direct and inverse processes proceeding through an excited

state at Ex = 6.150 MeV and J

pi = 1− in the compound nucleus 18Ne. Mass excesses,

expressed in MeV, for the nuclei involved are: 7.289 (1H), 2.425 (4He), 8.007 (14O), 1.952

(17F), and 5.315 (18Ne). Use the information provided to answer the following:

(a) Draw an appropriate level diagram, indicating all relevant energies and Jpi values of

the nuclei involved. The ground state of 17F has Jpi=5/2+. [5]

(b) Use the reciprocity theorem to calculate the cross section for the direct reaction

14O(α,p)17F assuming a measured cross section of 15 µb for the inverse reaction

17F(p,α)14O. [2]

(c) Use angular momentum and parity conservation rules to determine the relative or-

bital angular momentum required in both the direct and inverse channels to populate

the Jpi = 1− state in the compound nucleus, 18Ne. [4]

(d) Explain why only the mj = 0 substate of the J

pi = 1− resonance is populated

through the direct reaction 14O + α → p + 17F. Sketch the angular distribution

that you would expect for the γ rays emitted in the de-excitation process to the

18Ne ground state. Fully justify your answer. [8]

(e) Explain why you would expect Γγ Γα Γp for the partial widths of the

6.150 MeV state, thus rendering the observation of 6.150 MeV γ rays highly unlikely.

Fully justify your answer. [6]

Printed: Tuesday 23rd April, 2019 Page 2 Continued overleaf. . .

Nuclear Physics (PHYS11041)

3. Rutherford Backscattering Spectrometry (RBS) is a well-established analytical technique

used to determine the structure and composition of surfaces and thin films by measuring

the energies of ions scattered at large angles.

The energy of an ion, of mass m and initial energy E0, elastically scattered at 180

o by an

initially stationary nucleus of mass M > m is given by the non-relativistic equation:

E(180o) = E0

(

M −m

M +m

)2

,

while the scattering probability is described by the Rutherford formula (modified for a

finite mass M of the scatterer), as:

dσR

dΩ

= 1.296

(

zZ

E0

)2 [ 1

sin4(θ/2)

−

(

m

M

)2]

mb sr−1

where z and Z are the atomic numbers of the incident ion and target nucleus, respectively,

θ is the scattering angle in the laboratory system and the incident ion’s energy E0 is given

in units of MeV. From the information provided, answer the following:

(a) A 5-MeV α-particle source is used for RBS analysis on samples containing equal

amounts of C, Si, and Ni (mass numbers A = 12, 28 and 58, respectively). Calculate

the energies of backscattered α particles in each case. [4]

(b) What detector energy resolution (in keV) for α particles would be needed to resolve

∆A = 1 for each of the above mass regions? [4]

(c) With reference to the Rutherford backscattering cross section, draw a qualitative

sketch of the expected spectrum (counts vs Eα) for α-particles scattered by each

element in the sample. Label and briefly explain all features in your sketch. [4]

(d) Draw another qualitative sketch to show how you would expect the spectrum to

change for an increased sample thickness. [Note that the initial energy E0 diminishes

as α particles penetrate deeper into the sample.] [5]

(e) A thin target is bombarded with a 10-nA beam of 10-MeV α particles in a backscat-

tering experiment to determine the level of lead contamination. After 5 minutes, 30

counts corresponding to scattering from lead in the target are recorded in a detector

placed near 180o, which subtends a solid angle of 5×10−2 sr at the target. Calculate

the number of lead (Z = 82, A = 208) nuclei (in atoms/cm2) present in the target. [8]

Printed: Tuesday 23rd April, 2019 Page 3 End of Paper

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