Set 7
\[ \newcommand\v[1]{\boldsymbol{#1}} \newcommand\vdot[2]{\v{#1}\cdot\v{#2}} \]
\[ \newcommand{\dd}[2][]{\mathrm{d}^{#1}#2} \]
\[ \newcommand\f[2]{\dfrac{#1}{#2}} \newcommand\da[2]{\dfrac{d #1}{d #2}} \newcommand\daa[2]{\dfrac{d^2 #1}{d #2^2}} \newcommand\pa[2]{\dfrac{\partial #1}{\partial #2}} \newcommand\pal[3]{\left.\dfrac{\partial #1}{\partial #2}\right|_{#3}} \newcommand\paa[2]{\dfrac{\partial^2 #1}{\partial #2^2}} \newcommand\pab[3]{\dfrac{\partial^2 #1}{\partial #2 \partial #3}} \def\half{\frac{1}{2}} \]
\[ \newcommand\ket[1]{\left|#1\right\rangle} % |x> \newcommand\exp[1]{\left\langle#1\right\rangle} % <x> \newcommand\lr[1]{\left(#1\right)} % (x) \newcommand\lrs[1]{\left[#1\right]} % [x] \newcommand\lra[1]{\left|#1\right|} % |x| \newcommand\lrc[1]{\left\{#1\right\}} % {x} \newcommand\lrl[3]{\left.#1\right|_{#2}^{#3}} % x_limits \]
\[ \def\ie{\text{i.e.}\quad} \]
\[ \newcommand{\me}[3][\rightarrow]{{#2}#1{#3}} % Typical reaction \newcommand{\nr}[3]{{#1}\,({#2})\,{#3}} % Nuclear reactions \newcommand{\ele}[4]{_{#2}^{#3}{\textbf{#1}}_{#4}} \newcommand{\el}[2]{^{#2}\text{#1}} \newcommand{\til}[2][\mathbf{\sim}]{\underset{#1}{{#2}}} \newcommand{\b}[1]{\overline{#1}} \]
\[ \newcommand{\coulomb}[1]{\dfrac{#1}{4\pi\epsilon_0}} \def\degree{^*} \]
\[ \def\amu{\text{ amu}} \def\J{\text{ J}} \def\K{\text{ K}} \def\s{\text{ s}} \def\second{\text{ s}} \def\Psecond{\text{ s}^{-1}} \def\m{\text{ m}} \def\mm{\text{ mm}} \def\km{\text{ km}} \def\fm{\text{ fm}} \def\mum{\text{ $\mu$m}} \def\mus{\text{ $\mu$s}} \def\mPERs{\text{ ms$^{-1}$}} \def\eV{\text{ eV}} \def\keV{\text{ keV}} \def\GeV{\text{ GeV}} \def\MeV{\text{ MeV}} \def\MeVc{\text{ MeV/c$^2$}} \def\KeVc{\text{ KeV/c$^2$}} \newcommand\ttp[1]{\times 10^{#1}} % To The Power \]
\[ \def\en{\nu_e} \def\Aen{\b{\nu_e}} \def\mun{\nu_\mu} \def\Amun{\b{\nu}_\mu} \]
\[ \def\uraniumA{\ele{U}{92}{233}{}} \def\uraniumB{\ele{U}{92}{235}{}} \def\uraniumC{\ele{U}{92}{238}{}} \def\carbonB{\ele{C}{6}{14}{}} \def\nitrogenB{\ele{N}{7}{14}{}} \def\oxygenA{\ele{O}{8}{14}{}} \def\oxygenB{\ele{O}{8}{16}{}} \]
\[ % \def\com{\text{\scriptsize CM}} \def\com{\mathrm{\scriptsize CM}} \]
\[ \def\l{l} \]
\[ \def\g{\gamma} \]
Problems
Problem 7.1(Meaning of the Electric \(Q\))
The electric quadrupole moment of a charge distribution describes how the charge is distributed along a chosen axis compared with directions perpendicular to it.
For the \(z\)-axis, it is defined as
\[ Q_z=\int \rho(\mathbf r)\,r^2\left(3\cos^2\theta-1\right)\,\mathrm d^3r =\int \rho(\mathbf r)\left(3z^2-r^2\right)\,\mathrm d^3r, \]
where the origin is at the centre of the distribution and \(\theta\) is the polar angle measured from \(z\).
Consider a uniformly positively charged ellipsoid with symmetry axis \(z\).
Using \(3z^2-r^2=2z^2-x^2-y^2\), explain why:
An ellipsoid elongated along \(z\) has \(Q_z>0\).
An ellipsoid flattened along \(z\) and extended in the perpendicular plane has \(Q_z<0\).
A spherical distribution has \(Q_z=0\).
Problem 7.2(Electric \(Q\) from Different Directions)
The electric quadrupole moment of a charge distribution depends on the axis relative to which it is defined. For a chosen \(z\)-axis,
\[ Q_z=\int \rho(\mathbf r)\left(3z^2-r^2\right)\,\mathrm d^3r, \]
where \(\rho(\mathbf r)\) is the charge density and \(r\) is the distance from the origin. Similarly,
\[ Q_{z'}=\int \rho(\mathbf r)\left(3z'^2-r^2\right)\,\mathrm d^3r. \]
Consider a charge distribution that is axially symmetric about \(z'\), with the origin at the centre of the distribution.
Using rotational symmetry about \(z'\), explain why the following integrals vanish:
\[ \int \rho(\mathbf r)\,x'z'\,\mathrm d^3r=0, \qquad \int \rho(\mathbf r)\left(x'^2-y'^2\right)\,\mathrm d^3r=0. \]
A second axis, \(z\), passes through the same origin and makes an angle \(\beta\) with \(z'\). Show that the quadrupole moments defined relative to these axes satisfy
\[ Q_z=\frac{1}{2}\left(3\cos^2\beta-1\right)Q_{z'}. \]
Hint: Choose the \(x'\)-axis so that \(z\) lies in the \(x'z'\)-plane. Then
\[ z=z'\cos\beta+x'\sin\beta. \]
Problem 7.3(Electric \(Q\) for an Ensemble of Nuclei)
The electric quadrupole moment of a nuclear charge distribution, measured relative to a chosen laboratory axis \(z\), is
\[ Q_{\mathrm{lab}} =\int \rho(\mathbf r)\left(3z^2-r^2\right)\,\mathrm d^3r, \]
where the origin is at the nuclear centre.
Consider identical nuclei with axially symmetric, prolate charge distributions, resembling rugby balls. Each nucleus has the same intrinsic quadrupole moment \(Q_0>0\), defined relative to its own symmetry axis \(z'\).
If \(z'\) makes an angle \(\beta\) with the laboratory axis \(z\), then
\[ Q_{\mathrm{lab}}(\beta) =Q_0\,\frac{3\cos^2\beta-1}{2}. \]
Find \(Q_{\mathrm{lab}}\) when the nuclear symmetry axis is:
Parallel to the laboratory axis.
Perpendicular to the laboratory axis.
Suppose the nuclear symmetry axes are randomly oriented. Show that the ensemble-averaged quadrupole moment is
\[ \langle Q_{\mathrm{lab}}\rangle=0, \]
even though every nucleus has \(Q_0>0\).
Hint: Let \(\mathbf n\) be a unit vector along the nuclear symmetry axis, so that \(n_z=\cos\beta\). For an isotropic distribution, symmetry gives
\[ \langle n_x^2\rangle=\langle n_y^2\rangle=\langle n_z^2\rangle. \]
Use this together with \(n_x^2+n_y^2+n_z^2=1\).
Does a zero ensemble-averaged quadrupole moment imply that the individual nuclei are spherical? Explain.
Problem 7.4(Angular Momentum & Electric \(Q\))
An axially symmetric nucleus has an intrinsic quadrupole moment \(Q_0\), defined relative to its symmetry axis. Recall that its quadrupole moment relative to a laboratory axis \(z\) is
\[ Q_z=\frac{1}{2}\left(3\cos^2\beta-1\right)Q_0, \]
where \(\beta\) is the angle between the symmetry axis and \(z\).
Use the semiclassical angular-momentum vector model, assuming that the nuclear symmetry axis lies along the spin vector \(\mathbf I\).
For a nucleus with spin quantum number \(I\), state:
The magnitude of its spin angular momentum.
The allowed values of the spin projection quantum number \(m_I\) and the corresponding values of \(I_z\).
Express \(\cos\beta\) in terms of \(I\) and \(m_I\).
For a nucleus with \(I=1/2\), show that both allowed values of \(m_I\) give
\[ \cos^2\beta=\frac{1}{3}. \]
Hence show that this model gives \(Q_z=0\), even if \(Q_0\ne0\).
A student concludes: “A spin-\(1/2\) nucleus must therefore have a spherical intrinsic charge distribution.” Does this conclusion follow? Explain.