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Coupling of the nuclear electric-quadrupole moment to the electronic electric-field gradient. Its first-order splitting requires appropriate nonzero nuclear and electronic tensor moments; the usual I and J must each be at least one. It can produce deviations from a pure magnetic-dipole Landé interval rule.

The interaction of a nuclear magnetic moment with the electronic magnetic field at the nucleus. Within an isolated electronic J manifold, rotational symmetry gives the displayed effective Hamiltonian for angular momenta in hbar units. Its adjacent F intervals obey the Landé interval rule.

The electric quadrupole moment of an atomic nucleus, conventionally specified in its stretched nuclear spin state. It measures a nonspherical nuclear charge distribution. Its interaction with the electronic electric-field gradient contributes to hyperfine structure and generally changes the magnetic-dipole Landé interval rule. A permanent rank-two moment vanishes for nuclear spin less than one.

The closed $1s^{2}2s^2$ shells have $L=S=0$, leaving the equivalent $2p^2$ electrons. Addition of angular momentum and the Pauli exclusion principle permit the atomic terms ${}^{3}P,{}^{1}D,{}^{1}S$. Their term symbols use ${}^{2S+1}{L}_J$. Hund's rules put the triplet state below the singlet states; for a less-than-half-filled $p$ shell the smallest $J$ in ${}^{3}P$ is lowest. ThusAll have even parity. These assignments agree with independent tabulated carbon energy levels.^[ref] The rounded source values should not be forced to satisfy an exact Landé interval rule; pure first-order LS coupling need not account for every measured fine structure correction.

Let the dimensionful angular momentum operators obey $\mathbf{F}=\mathbf{I}+\mathbf{J}$ and take the interaction Hamiltonian as $H^{\prime }=A\mathbf{I}\cdot \mathbf{J}$. Sincethe energy eigenvalues at fixed $I,J$ arewhere $F=|I-J|,\dots ,I+J$. Subtraction gives the Landé interval ruleMany atomic conventions use dimensionless angular momentum operators and call the energy-valued constant itself $A$; then the right-hand side is $AF$. Each energy level has $2F+1$ magnetic sublevels with the same energy in zero field.

For fine structure, the effective interaction is $a_{LS}\mathbf{L}\cdot \mathbf{S}$ with dimensionless angular momentum operators. It originates in the spin-orbit interaction, including the magnetic interaction of electron spin with orbital motion and its relativistic correction. Within an isolated LS coupling atomic term, the labels are the electron configuration, $L,S,J,M_J$, and the interval between adjacent $J$ energy levels is $a_{LS}J$. This is smaller than the electrostatic separation of atomic terms.

For magnetic-dipole hyperfine structure, the nuclear magnetic dipole moment interacts with the magnetic field generated by the electrons. Contributions include the Fermi contact interaction of penetrating $s$ electrons, orbital magnetic fields, and the dipolar field of electron spin. Projected onto one electronic $J$ energy level, their rotationally invariant magnetic-dipole part has the form $a_{IJ}\mathbf{I}\cdot \mathbf{J}$. The labels are $I,J,F,M_F$, together with the approximate electronic labels such as $L,S$. This splitting is generally smaller than fine structure, reflecting the much smaller nuclear magnetic dipole moment. An electric nuclear quadrupole moment can add a different interaction and violate the simple Landé interval rule.

At the larger electrostatic energy scale, an effective two-spin exchange interaction can also be written $a_s{\mathbf{s}}_1\cdot {\mathbf{s}}_2$ within fixed orbitals. Its origin is the Coulomb interaction combined with the Pauli exclusion principle, rather than a large direct magnetic interaction. It separates singlet state and triplet state atomic terms; before spin-orbit coupling, useful labels include the electron configuration, $L,M_L,S,M_S$. General electrostatic atomic term separations need not obey a single Landé interval rule as $L$ varies.

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.

First make the units explicit. Use angular momentum operators in units of $\hslash$, so their squared eigenvalues are $J(J+1)$ rather than ${\hslash }^{2}J(J+1)$; the coefficient $A$ then has units of energy. The operator identitygives, at fixed $J_1,J_2$,Subtracting the expression for the adjacent allowed value $J-1$ leavesThis is the Landé interval rule. Its content is that consecutive gaps grow in proportion to the higher total-angular-momentum quantum number, with the same coefficient within an isolated coupled multiplet. A negative $A$ reverses the ordering of the levels, while positive observed transition frequencies alone measure gap magnitudes.

If the source symbols instead denote physical angular momenta with units of action, the Hamiltonian written literally as $A{\mathbf{J}}_1\cdot {\mathbf{J}}_2$ gives the gap $A{\hslash }^{2}J$. Recovering the printed form then requires absorbing ${\hslash }^2$ into the coefficient or writing $H=(A/{\hslash }^2){\mathbf{J}}_1\cdot {\mathbf{J}}_2$. The convention is essential to dimensional consistency rather than an extra dynamical assumption.