Set 6

\[ \newcommand\v[1]{\boldsymbol{#1}} \newcommand\vdot[2]{\v{#1}\cdot\v{#2}} \]

\[ \newcommand{\dd}[2][]{\mathrm{d}^{#1}#2} \]

\[ \newcommand\f[2]{\dfrac{#1}{#2}} \newcommand\da[2]{\dfrac{d #1}{d #2}} \newcommand\daa[2]{\dfrac{d^2 #1}{d #2^2}} \newcommand\pa[2]{\dfrac{\partial #1}{\partial #2}} \newcommand\pal[3]{\left.\dfrac{\partial #1}{\partial #2}\right|_{#3}} \newcommand\paa[2]{\dfrac{\partial^2 #1}{\partial #2^2}} \newcommand\pab[3]{\dfrac{\partial^2 #1}{\partial #2 \partial #3}} \def\half{\frac{1}{2}} \]

\[ \newcommand\ket[1]{\left|#1\right\rangle} % |x> \newcommand\exp[1]{\left\langle#1\right\rangle} % <x> \newcommand\lr[1]{\left(#1\right)} % (x) \newcommand\lrs[1]{\left[#1\right]} % [x] \newcommand\lra[1]{\left|#1\right|} % |x| \newcommand\lrc[1]{\left\{#1\right\}} % {x} \newcommand\lrl[3]{\left.#1\right|_{#2}^{#3}} % x_limits \]

\[ \def\ie{\text{i.e.}\quad} \]

\[ \newcommand{\me}[3][\rightarrow]{{#2}#1{#3}} % Typical reaction \newcommand{\nr}[3]{{#1}\,({#2})\,{#3}} % Nuclear reactions \newcommand{\ele}[4]{_{#2}^{#3}{\textbf{#1}}_{#4}} \newcommand{\el}[2]{^{#2}\text{#1}} \newcommand{\til}[2][\mathbf{\sim}]{\underset{#1}{{#2}}} \newcommand{\b}[1]{\overline{#1}} \]

\[ \newcommand{\coulomb}[1]{\dfrac{#1}{4\pi\epsilon_0}} \def\degree{^*} \]

\[ \def\amu{\text{ amu}} \def\J{\text{ J}} \def\K{\text{ K}} \def\s{\text{ s}} \def\second{\text{ s}} \def\Psecond{\text{ s}^{-1}} \def\m{\text{ m}} \def\mm{\text{ mm}} \def\km{\text{ km}} \def\fm{\text{ fm}} \def\mum{\text{ $\mu$m}} \def\mus{\text{ $\mu$s}} \def\mPERs{\text{ ms$^{-1}$}} \def\eV{\text{ eV}} \def\keV{\text{ keV}} \def\GeV{\text{ GeV}} \def\MeV{\text{ MeV}} \def\MeVc{\text{ MeV/c$^2$}} \def\KeVc{\text{ KeV/c$^2$}} \newcommand\ttp[1]{\times 10^{#1}} % To The Power \]

\[ \def\en{\nu_e} \def\Aen{\b{\nu_e}} \def\mun{\nu_\mu} \def\Amun{\b{\nu}_\mu} \]

\[ \def\uraniumA{\ele{U}{92}{233}{}} \def\uraniumB{\ele{U}{92}{235}{}} \def\uraniumC{\ele{U}{92}{238}{}} \def\carbonB{\ele{C}{6}{14}{}} \def\nitrogenB{\ele{N}{7}{14}{}} \def\oxygenA{\ele{O}{8}{14}{}} \def\oxygenB{\ele{O}{8}{16}{}} \]

\[ % \def\com{\text{\scriptsize CM}} \def\com{\mathrm{\scriptsize CM}} \]

\[ \def\l{l} \]

\[ \def\g{\gamma} \]

TipInstructions

• 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 6.1(Compton Edge)

1. Sketch the energy spectrum from a gamma detector receiving a single photon with energy \(E_\gamma\).
   • Please highlight the following features:
     • Photopeak
     • Compton Edge
     • Compton Continuum
     • Single Escape Peak
     • Double Escape Peak
   • Indicate the values for each of these peaks in terms of \(E_\gamma\).
2. Show that the gap between the photopeak and the Compton edge approaches \(\approx256\) keV in the limit of high-energy photons.

Problem 6.2(The Fermi Gas Model)

For the Fermi Gas Model of the Nucleus

1. Starting with equation Equation 13.18, derive equation Equation 13.19

2. For a fixed \(A\), show that the most stable configuration for the Fermi model occurs when \(N=Z\).

3. Show that the difference in energy \(\Delta E\) between a configuration with \(N\neq Z\) and \(N=Z\) can be written as:

\[ \Delta E = \f{3}{10m}\f{\hbar^2}{R_0^2}\lr{\f{9\pi}{2^{7}}}^{2/3} A\lrs{\lr{1+\f{\delta}{A}}^{5/3}+\lr{1-\f{\delta}{A}}^{5/3}-2} \]

Where \(\delta = N-Z\)

4. With the help of the binomial expansion(see below) to show that this reduces to: \[ \Delta E \approx \f{1}{3m}\f{\hbar^2}{R_0^2}\lr{\f{9\pi}{2^{6}}}^{2/3} \f{\delta^2}{A} \]

NoteBinomial Theorem for Fractional Exponents

For \(|x| < 1\) and when \(\alpha\) is a rational (e.g. \(3/4\) not \(\sqrt{2}\)) number it follows that \[ \def\a{\alpha} (1+x)^\a = 1+ \a x + \f{\a(\a-1)}{2!}x^2+ \f{\a(\a-1)(\a-2)}{3!}x^3+\ldots = \sum_{n=0}^\infty \binom{\a}{n}x^n \]

Where: \[ \binom{\a}{n} =\f{\a(\a-1)(\a-1)\cdots\lr{\a-(n-1)}}{n!} \]

Problem 6.3(Some Electrostatic Results)

This question is related to several results from electrostatics that we used in some of the discussions.

1. Show that the energy necessary to assemble a uniformly charged sphere with radius \(R\) and total charge \(Q\) is given by: \[ E = \f{3}{5}\coulomb{1}\f{Q^2}{R} \]

2. Show that the potential inside a uniformly charged sphere with radius \(R\) and total charge \(Q\) is given by: \[ U(r) = -\coulomb{1}\f{Q}{R}\lrs{\f{3}{2}-\half\lr{\f{r}{R}}^2} \]

Problem 6.4(RMS of a Charged Sphere)

Show that the mean square charge radius, \(\exp{r^2}\), for a uniformly charged sphere of radius \(R\) comes out to be \(\f{3}{5}R^2\).

Problem 6.5(Nuclear Charge Density)

The spherically symmetric charge density of a nucleus can be modelled by

\[ \rho(r)=\frac{\rho_0}{1+e^{(r-R)/a}}, \]

where \(\rho_0\) is the interior charge-density scale and \(a>0\). Assume \(R\gg a\), so that \(\rho(0)\approx\rho_0\).

1. Show that \(\rho(R)=\rho_0/2\). Hence explain the physical meaning of \(R\). Does the charge distribution end at \(r=R\)?

2. Let \(r_{90}\) and \(r_{10}\) denote the radii at which the charge density is \(0.9\rho_0\) and \(0.1\rho_0\), respectively.

   1. Show that the surface thickness is

   \[ t=r_{10}-r_{90}=4a\ln 3. \]

   2. For a measured surface thickness of \(2.4\,\mathrm{fm}\), calculate \(a\).

   3. Show that \(R=(r_{90}+r_{10})/2\). Sketch \(\rho(r)/\rho_0\), labelling \(R\), \(r_{90}\), and \(r_{10}\).

3. Consider a second charge-density profile with the same values of \(\rho_0\) and \(R\), but a larger value of \(a\). Assume that \(R\gg a\) still holds.

   1. Add this profile to your existing sketch. Explain how increasing \(a\) changes the nuclear surface.

   2. Find the slope \(\mathrm d\rho/\mathrm dr\) at \(r=R\). How does your result support your explanation?

4. Extension: the sharp-surface limit

   Describe the limiting charge distribution as \(a\rightarrow0\). Show that, in this limit, a nucleus with total charge \(Ze\) satisfies

   \[ Ze=\frac{4\pi}{3}\rho_0R^3. \]

   If \(\rho_0\) is approximately the same for different nuclei, what scaling between \(R\) and \(Z\) follows?