= Solution
{c}
The ground <electron configuration> of <caesium> has a single $6s$ <electron>, with $J=1/2$. <Addition of angular momentum> giving $F=3,4$ fixes the <nuclear spin> to $I=7/2$. With dimensionless <angular momentum operators>, write the <Fermi contact interaction> as
$$
H_{\rm hf}=A\mathbf I\cdot\mathbf J,\qquad
E_F=\frac A2[F(F+1)-I(I+1)-J(J+1)].
$$
The two <hyperfine levels> therefore differ by $4A$. In the convention $\boldsymbol\mu_I=g_I\mu_B\mathbf I$ and $g_s\simeq2$,
$$
A=\frac{2\mu_0}{3}g_sg_I\mu_B^2|\psi_{6s}(0)|^2,\qquad
\boxed{\omega_0=\frac{4A}{\hbar}.}
$$
This exposes a missing ingredient in a prediction from the supplied data: $Z$ and $g_I$ alone do not specify the valence <electron> contact density. <Screening effect> and penetration of the closed core strongly affect it. If $g_I$ instead denotes the total nuclear moment in units of $\mu_B$, it must first be divided by $I$ in the displayed convention.
An explicitly crude unscreened <hydrogen-like ion> model uses $|\psi_{6s}(0)|^2=Z^3/(\pi6^3a_0^3)$ and gives $\nu_0\simeq5.76\times10^{11}\,\mathrm{Hz}$, or $\omega_0\simeq3.62\times10^{12}\,\mathrm{s^{-1}}$. It describes neither the shielding nor the radial function of neutral <caesium>. Using a pure unit-charge $6s$ <hydrogen> <orbital> instead gives only $3.46\,\mathrm{MHz}$, illustrating why a unique useful estimate cannot follow from an unspecified effective charge.
For physical neutral $^{133}\mathrm{Cs}$, the independently known zero-<magnetic field> <clock transition> is
$$
\boxed{\nu_0=9\,192\,631\,770\,\mathrm{Hz},\qquad
\omega_0=2\pi\nu_0\simeq5.776\times10^{10}\,\mathrm{s^{-1}}.}
$$
This measured <frequency>, used to define the <second>, is supplementary physical information, not a consequence of $Z=55$ and the rounded $g_I$ alone.<SI base unit: second (s) by Bureau International des Poids et Mesures>[Definition: the unperturbed ground-state hyperfine transition frequency of caesium-133 is exactly 9 192 631 770 Hz]{ref} The direct <electronic transition> is in the microwave range and is a <magnetic dipole transition>, rather than a direct optical <electric dipole transition>. An optical implementation needs a two-field <Raman transition>.
Using the corrected <weak-pulse Ramsey spectrum> of part (b), the peak at $\delta=0$ has height $(\Omega\tau_p)^2$. The <quantum interference> fringes have spacing $2\pi/T$ in <angular frequency>, with zeros at $(2k+1)\pi/T$, and the <sinc function> envelope has its first zeros at $\delta=\pm2\pi/\tau_p$. For $T\gg\tau_p$, the central peak has approximately <full width at half maximum> $\pi/T$ in <angular frequency>. The <clock transition> between the two $M_F=0$ states has no first-order <Zeeman energy>, although a quadratic one remains.
\Image[/past-exam-of-the-atomic-and-laser-physics-course-of-the-university-of-oxford/2021-two-pulse-fringes.png]
{title=Corrected weak-pulse Ramsey spectrum for a pulse-start separation ten times the pulse duration}
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