= Nuclear rotational band
{title2=$E_I-E_0=\hbar^2I(I+1)/(2\mathcal J)$}
A sequence of nuclear <energy levels> interpreted as different total <angular momenta> built on approximately the same intrinsic structure. For an axially symmetric, positive-<parity>, $K=0$ <ground state> band of an <even-even nucleus>, the allowed sequence is $I=0,2,4,\ldots$. The simple constant-inertia rotor has
$$
H_{\mathrm{rot}}=\frac{\widehat{\mathbf I}^{\,2}}{2\mathcal J},\qquad
\widehat{\mathbf I}^{\,2}|IM\rangle=\hbar^2I(I+1)|IM\rangle,\qquad
E_I-E_0=\frac{\hbar^2}{2\mathcal J}I(I+1).
$$
Here $\mathcal J$ is an effective <moment of inertia>, and $\hbar$ the reduced <Planck constant>. The even-$I$ restriction belongs to this particular intrinsic <symmetry> and band, rather than to all nuclear excitations. The <angular momentum operator> eigenvalue supplies $I(I+1)$, rather than the classical replacement $I^2$.
Set $\epsilon=E_2-E_0$. Since $2(2+1)=6$, the model predicts
$$
\frac{E_4-E_0}{\epsilon}=\frac{20}{6}=\frac{10}{3},\qquad
\frac{E_6-E_0}{\epsilon}=\frac{42}{6}=7.
$$
For a chosen model scale $\epsilon=90\,\mathrm{keV}$, the excitation <energies> are $0,90,300,630\,\mathrm{keV}$. The corresponding successive <gamma ray> <energies> are approximately $90,210,330\,\mathrm{keV}$, neglecting <nuclear recoil>. Level <energies> and emitted line <energies> are different lists. These numbers illustrate the formula and are not a measured <isotope> spectrum.
\Image[/nuclear-rotor-and-quadrupole-vibrator.png]
{title=An ideal rotor and an ideal quadrupole vibrator have different level ratios}
{description=Original chosen limiting models, both normalized to the same first $2^+$ excitation. The <nuclear rotational band> has $E(4)/E(2)=10/3$; the harmonic <nuclear quadrupole vibration> has a two-quantum $0^+,2^+,4^+$ multiplet at twice the one-quantum <energy>. Arrows show selected <electric quadrupole transitions>; their widths do not represent measured strengths. Higher vibrational multiplets are omitted.}
The fitted $\mathcal J=3\hbar^2/\epsilon$ is a response parameter, not automatically the <moment of inertia> of a classical rigid charge distribution. <Nuclear pairing> changes this response; <Ben Roy Mottelson> connected observed inertias with correlations beyond independent-particle motion.<Elementary Modes of Excitation in the Nucleus by Ben Roy Mottelson>[continuation after p. 237, pairing correlations and Fig. 1 potential-energy surfaces]{ref}
At larger <angular momentum>, changing intrinsic structure or <nuclear pairing> can invalidate a single constant $\mathcal J$. A useful diagnostic is to compare $(E_I-E_0)/[I(I+1)]$ across several levels: constancy tests the simple formula, while a drift identifies the need for additional physics. A successful ratio alone neither specifies every microscopic motion nor measures a literal solid-body rotation.
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