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A localized magnetic moment can polarize a metal's conduction electrons, whose response changes the energy of a second moment. This indirect exchange interaction can be long ranged and oscillate in sign with separation because of the Fermi surface. In the three-dimensional isotropic free-electron asymptotic limit its range dependence is proportional to $\cos (2k_{F}r)/r^3$, with coupling-dependent prefactor and sign convention. The original calculation for nuclear moments used their contact hyperfine interaction with conduction electrons; it was motivated by nuclear magnetic resonance line widths in metallic silver.^[ref]

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 areTheir ratios are approximately $4:5:6$, sinceThe 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 $10.60-7(0.706)\simeq 5.66$ and then $5.66-8(0.706)\simeq 0.01\mathrm{GHz}$. They identify peaks $i$ and $l$ unambiguously. The full assignment for ${}^{151}\mathrm{Eu}$ isWith $J=11/2$, these endpoints requireEquivalently, 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.

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.

The remaining six positions give the other isotope's sequenceThe adjacent differences divided by $4,5,6,7,8$ are $0.3125,0.3140,0.3133,0.3143,0.3138\mathrm{GHz}$, respectively. Thus $a_{\nu ,153}\simeq -0.314\mathrm{GHz}$ and the same endpoint argument gives $\boxed{I_{153}=5/2}$.

For the same electronic energy level, the magnetic-dipole hyperfine structure constant is proportional to ${\mu }_I/I$, to leading order in the small isotope-dependent change of the electronic wavefunction. Since the two nuclear spins are equal,Using the full spans averages over the approximate Landé interval rule: ${\sum }_{F=4}^{8}F=30$, soThe constants have the same sign, so the nuclear magnetic dipole moments have the same relative sign. Their absolute sign cannot be obtained from their ratio without the electronic field's sign. Finite nuclear-size corrections to the hyperfine interaction, often called the hyperfine anomaly, and electric-quadrupole corrections are neglected at this accuracy.

For a nucleus of spin $I>0$ and electronic $J=1/2$, the magnetic-dipole hyperfine interaction is $A\mathbf{I}\cdot \mathbf{J}$, using dimensionless angular-momentum operators. The electric-quadrupole hyperfine contribution vanishes for $J=1/2$. In zero field, $F=I\pm 1/2$ labels the levels:Their signed separation is $\Delta =A(I+1/2)$. For $A>0$, the larger-$F$ multiplet lies higher; negative $A$ reverses this order. If $I=0$ there is no such hyperfine splitting.

Let the field define $z$ and writeHere $g_I$ is the signed nuclear Zeeman coefficient expressed in Bohr-magneton units, including its nuclear-to-Bohr-magneton factor; it is not an unconverted nuclear $g$ factor. Total projection $m=m_I+m_J$ is conserved, while $F$ need not be. In the basis $|m-1/2,+1/2⟩$, $|m+1/2,-1/2⟩$, the two-state block isFor $A>0$, define $x=(g_J-g_I){\mu }_{B}B/\Delta$. Diagonalization gives the Breit-Rabi formula,For either sign of $A$, the block itself remains valid; its ordered eigenvalues are $-A/4+g_I{\mu }_{B}Bm\pm \frac{1}{2}\sqrt{{\Delta }^2+2Am(g_J-g_I){\mu }_{B}B+(g_J-g_I{)}^2{\mu }_B^2{B}^2}$.

For a stretched hyperfine Zeeman state at projection $m=\pm (I+1/2)$ there is only one state each. Their energies are exactlyAssigning two branches to each stretched projection would double-count states.

In weak fields the shift is $g_F{\mu }_{B}mB$, with $g_F=g_I\pm (g_J-g_I)/(2I+1)$ for $F=I\pm 1/2>0$; an $F=0$ state has no first-order Zeeman shift. At strong hyperfine-decoupling fields the eigenstates approach product states withThis is the hyperfine regime of the Paschen-Back effect, still assuming the field is weak enough not to invalidate the electronic $J=1/2$ manifold. States of the same $m$ mix and avoid crossing when their off-diagonal coupling is nonzero; different-$m$ levels can cross.

Ground-state electronic angular momentum $J=1/2$ couples to nuclear spin $I=3/2$, giving $F=1,2$. For contact hyperfine interaction $A\mathbf{I}\cdot \mathbf{J}/{\hslash }^2$, their zero-field energies are $-5A/4$ and $3A/4$. The splitting is $2A$. Electronic weak-field contributions give $g_{F=1}\simeq -1/2$ and $g_{F=2}\simeq +1/2$; the small nuclear contribution refines these numbers.