= Solution
{c}
Write $a_\nu=a/h$, so the <Landé interval rule> for <hyperfine structure> intervals in ordinary <frequency> is $\nu_F-\nu_{F-1}=a_\nu F$. For the first four selected positions, the successive positive differences are
$$
21.19-18.37=2.82,\quad18.37-14.84=3.53,\quad14.84-10.60=4.24\quad\mathrm{GHz}.
$$
Their ratios are approximately $4:5:6$, since
$$
\frac{2.82}{4}=0.7050,\qquad\frac{3.53}{5}=0.7060,\qquad\frac{4.24}{6}=0.7067\quad\mathrm{GHz}.
$$
The decreasing <frequencies> therefore correspond to increasing $F$, with $a_\nu\simeq-0.706\,\mathrm{GHz}$, if increasing printed position means increasing physical <frequency>. Continuing the intervals predicts additional positions near $10.60-7(0.706)\simeq5.66$ and then $5.66-8(0.706)\simeq0.01\,\mathrm{GHz}$. They identify peaks $i$ and $l$ unambiguously. The full assignment for $^{151}\mathrm{Eu}$ is
$$
\boxed{\begin{array}{c|rrrrrr}
\text{peak}&a&b&c&e&i&l\\\hline
F&3&4&5&6&7&8\\
\nu\ (\mathrm{GHz})&21.19&18.37&14.84&10.60&5.63&0.00
\end{array}}
$$
With $J=11/2$, these endpoints require
$$
I+J=8,\qquad |J-I|=3,\qquad\boxed{I_{151}=5/2.}
$$
Equivalently, a complete set of six <hyperfine levels> gives $2I+1=6$ when $I<J$. This inference assumes that every upper <hyperfine level> contributes an observed position and the negligible lower <hyperfine structure> lets coincident transitions be grouped by their upper <energy>. A line count without these completeness assumptions would not alone fix <nuclear spin>.
\b[The supplied positions obey the <Landé interval rule> only approximately.] For example, $\nu_e-\nu_i=4.97\,\mathrm{GHz}$, whereas $7(\nu_a-\nu_b)/4=4.935\,\mathrm{GHz}$. Even allowing each printed position a rounding error of $0.005\,\mathrm{GHz}$ does not fully reconcile these two relations. A least-squares dipole-only fit has a largest position residual of about $0.017\,\mathrm{GHz}$. Small additional <hyperfine interactions>, measurement uncertainty beyond the displayed precision, or an error in the tabulated data could account for this; it is inappropriate to claim an exact identity. The grouping and <nuclear spin> are nevertheless robust. Reversing the convention for the printed <frequency> axis reverses the inferred signs of both constants, without changing the following ratio.
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