Set 8

Instructions

Here are some additional practice questions and activities for you to work on independently.
Some of these questions are from previous exams and tests.
I will not provide written solutions, as I want to avoid them being circulated to future batches, which would render the practice less effective. However, I am more than happy to discuss and assist you as you work through them.
Please feel free to email me at (chammika[at]nus.edu.sg) to organise a meeting.

Problems

Problem 8.1(Coulomb and Symmetry Energy Parameters)

Determine the Coulomb and symmetry energy parameters of the Weizacker semi-empirical formula from the binding energies of three members of the $A = 27$ isobar $(A, Z) = (27, 12), (27, 13)$ and $(27, 14)$.

Problem 8.2($n$ and $p$ Separation Energies)

Show that the neutron and proton separation energies are given by the following expressions.

\[ \begin{align} S_n &= \lrs{M(\ele{X}{Z}{A-1}{N-1}) -M(\ele{X}{Z}{A}{N}) +m_n }c^2\\ S_p &= \lrs{M(\ele{X}{Z-1}{A-1}{N}) -M(\ele{X}{Z}{A}{N}) +M(^1\text{H}) }c^2 \end{align} \]

$M$ represents the corresponding atomic masses.

Problem 8.3(Weizacker Semi-Empirical Mass Formula)

The Weizacker semi-empirical mass formula is useful to obtain global distribution of binding energies as a function of $A$, $Z$, and $N$. Use this formula to show that the $Q$-value, i.e., kinetic energy released, in fission is positive only for heavy nuclei.

Problem 8.4(Mass Defect and Binding Energy)

Find the mass defect and the binding energy of a nucleus of $^{16}\text{O}$ in MeV.

Problem 8.5(Nuclear Mass Excess and Decay)

Use a table of mass excesses to investigate the following nuclear reactions. State whether the tabulated values refer to atomic or nuclear masses, and account consistently for any electron masses.

Formation of an alpha particle

Calculate the energy released when two free protons and two free neutrons combine to form an $\alpha$-particle. Explain how this energy is related to the binding energy of ${}^{4}\mathrm{He}$.
Alpha decay of uranium-238

Consider the decay of a stationary ${}^{238}\mathrm{U}$ nucleus to the ground state of its daughter nucleus.
1. Write down the decay equation and calculate its $Q$-value. Hence show that the decay is energetically allowed.
2. Using conservation of energy and momentum, calculate the kinetic energy of the emitted $\alpha$-particle. Explain why it receives less than the full decay energy.
Hint: You may treat the recoil motion nonrelativistically.
Neutron-induced fission of uranium-235

A stationary ${}^{235}\mathrm{U}$ nucleus absorbs a thermal neutron and undergoes fission. Neglect the incident neutron’s kinetic energy.
1. Choose a plausible fission channel producing two medium-mass nuclei and any emitted neutrons. Write a balanced reaction equation, conserving both charge and nucleon number.
2. Use the mass excesses to estimate the total energy released in your chosen channel. State your assumptions.
3. Must all of this energy appear as kinetic energy of the two fission fragments? Explain briefly.

Problem 8.6($Z_{min}$ for fixed $A$)

\[ \def\aa{a_\text{volume}\,} \def\ab{a_\text{surface}\,} \def\ac{a_\text{coulomb}\,} \def\ad{a_\text{symmetry}\,} \def\ae{a_\text{pairing}\,} \def\af{\delta_\text{pairing}} \begin{align} M(Z,A) &= Z(m_p+m_e)+ (A-Z)m_n - B(Z,A)\\ B(Z,A) &= \aa A - \ab A^{2/3} - \ac Z(Z-1)A^{-1/3} - \ad \f{(A-2Z)^2}{A} +\af \end{align} \]

Starting from the SEMF shown above, show that the minimum proton number $Z_{min}$ corresponding to the the mass chain of $A$ is given by: \[ Z_{\text{min}} = \f{m(n) - m(^1\text{H}) + \ac A^{-1/3} + 4\ad}{2\ac A^{-1/3} + 8\ad A^{-1}} \]
Argue that the previous result maybe approximated to:

\[ Z_{\text{min}} \approx \lr{\f{A}{2}}\f{1}{1+\frac{1}{4} A^{2/3}\ac/\ad} \]

Problem 8.7(A Deformed Nucleus)

We may modify the SEMF to describe a deformed nucleus as follows:

\[ \def\aa{a_1\,} \def\ab{a_2\,} \def\ac{a_3\,} \def\ad{a_4\,} \def\ae{a_5\,} \def\af{\delta} B(Z,A) = \aa A - \ab A^{2/3}\lr{1+\f{2}{5}k^2} - \ac \f{Z(Z-1)}{A^{1/3}}\lr{1-\f{1}{5}k^2} - \ad \f{(A-2Z)^2}{A} +\af \]

\[ \af = \begin{cases} +\ae A^{-3/4} & \text{Both $Z$ and $N$ even}\\ 0 & \text{One of $Z$ and $N$ odd}\\ -\ae A^{-3/4} & \text{Both $Z$ and $N$ odd}\\ \end{cases} \]

$k$ is the deformation parameter where $k=0$ corresponds to a spherical neuclus.

Discuss the physical basis for the various terms in the above equation.
The neutron separation energy is given by \[ S_n = B\lr{\ele{X}{Z}{A}{N}}-B\lr{\ele{X}{Z}{A-1}{N-1}} \]

Consider a heavy even $Z$, even $N$ nucleus with a non-zero deformation parameter $k$. If this nucleus were spherical, it is known that the neutron separation energy is approximately $k a_5 A^{-3/4}$ greater than it is for a nearby even $Z$, odd $N$ nucleus.

Find the value of $k$.