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For a hyperfine multiplet governed by the magnetic-dipole hyperfine interaction, adjacent frequency gaps obey ${\nu }_F=|A|F/h$. The five measured gaps for the first isotope are approximately in the ratios $1:2:3:4:5$. Therefore the six levels have $F=0,1,2,3,4,5$: the first measured gap corresponds to higher $F=1$. In particular $F_{\min }=|I-J|=0$ and $F_{\max }=I+J=5$, givingThis inference uses both the number of intervals and their successive ratios. Merely seeing six levels would specify only $\min (I,J)=5/2$ and would not by itself establish equality.

Solving a linear least-squares problem for a fit through the origin, taking the rounded frequencies as equally weighted, givesThus the scale of the hyperfine structure is about $4.80\times 10^{-26}$ J, or $3.00\times 10^{-7}$ eV. The displayed frequencies have unequal rounding precision, so additional fit digits are not physically warranted. The gap pattern supports a leading dipole interval rule, with small residuals that do not establish higher-multipole constants without uncertainties.

The total width between $F=0$ and $F=5$ is $15|A_{55}|$, about $1.086$ GHz in frequency units. The sign of $A_{55}$, the absolute nuclear g-factor and the electronic field coefficient cannot be separated from these gap magnitudes alone. Measuring the level ordering or using another known magnetic property would supply additional information. The figure in b shows the positive-$A$ ordering as an illustration.

For the second isotope, consecutive gap differences are approximately $56.4$ MHz, but the smallest gap is about $1.5$ times this value. The higher-$F$ values for the five gaps are therefore $3/2,5/2,7/2,9/2,11/2$. The six levels run from $F=1/2$ to $11/2$. The electronic state is unchanged, so $J=5/2$ as in d; the endpoints giveFor example, fitting ${\nu }_F$ to $F|A|/h$ with the same equal-weight convention gives $56.408$ MHz. The nearly constant gap differences are compatible with the same leading model of the magnetic-dipole hyperfine interaction and the printed rounding.

Within an unchanged electronic state, and neglecting isotope-dependent changes of the electronic wave function and hyperfine anomaly, $A\propto g_I$. Thusor the reciprocal magnitude is about $1.284$. This is a ratio of nuclear g-factors, not directly a ratio of nuclear magnetic moments: ${\mu }_I=g_{I}I{\mu }_N$, so the moment-magnitude ratio would additionally contain $I_{56}/I_{55}=3/(5/2)$ and is about $0.935$.

The positive transition frequencies do not fix either coupling sign. If a signed ratio is required, equal signs must be supplied as an additional assumption or determined by an ordering-sensitive measurement. No absolute nuclear g-factor is inferred without a calibrated electronic field coefficient. The second ladder in b illustrates the inferred half-integer $F$ sequence without turning that sign ambiguity into a falsely measured ordering.