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Collective rotation and vibration
Many nucleons can move together. A deformed nucleus rotates, and its lowest states form a band with energies close to I(I+1). A spherical one vibrates about its shape, and its states come in near-equal steps of one phonon. The ratio R4/2 = E(4+1)/E(2+1) tells the two apart: 10/3 for a rigid rotor, 2 for a harmonic vibrator, 5/2 for a γ-soft (Wilets-Jean) rotor.
Model, exactly
Data: the ENSDF adopted levels of every even-even nucleus. E(2+1) is the lowest level whose Jπ field reads 2+ (firm) or reads 2+ once its parentheses are removed, as in (2+) or (2)+ (tentative, marked t and drawn narrow); likewise for 4+, 6+, 3− and the second 0+, 2+. A level with an unresolved Jπ (a list such as (1,2+), a range such as (0+:4+), a bound such as GE 1, a value from systematics in square brackets such as [2+], a spin without parity, or none) is never taken; when one that could be the state lies lower, the value is flagged. Levels whose energy is an unconnected offset (X, 0+X, …) are skipped, and so are levels ENSDF marks as questionable (? in column 80), expected but not observed (S), or with an energy from systematics (SY) or calculation (CA): they are never taken, a questionable level with the Jπ sought lying lower is flagged, and the level panel shows them dashed. The yrast sequence is the lowest positive-parity level of each even spin, up to the first even spin that has none, so it can switch from one band to another; the level energies, not the gamma energies, give Eγ (a dashed segment where ENSDF lists no gamma between the two).
Rigid rotor: E(I) = (ħ²/2𝓘) I(I+1), with ħ²/2𝓘 fitted to E(2+1): E(I) = E(2+) I(I+1)/6, 𝓘 = 3ħ²/E(2+), R4/2 = 20/6. Harmonic quadrupole vibrator: E = n ħω with ħω = E(2+1); n = 2 is the triplet 0+, 2+, 4+ at 2ħω, R4/2 = 2. γ-soft rotor (Wilets-Jean): E ∝ τ(τ+3), 2+1 at τ = 1, 4+1 at τ = 2, R4/2 = 10/4.
Deformation from B(E2; 0+ → 2+1)↑: β2 = (4π/3ZR0²)[B(E2)↑/e²]½, R0 = 1.2 A1/3 fm, the convention of Pritychenko et al. 2016, Eq. (2); the β2 column of the NNDC B(E2) web table is shown beside the computed value, and a 2016 (ADNDT 107) row whose β2 does not follow from its own B(E2) (outside the printed rounding and by more than 0.5 %) is flagged. B(E2) gives |β|, the size of the E2 matrix element: a static deformation for a rotor, the rms of the vibration for a vibrator; it fixes neither the sign of β nor axial symmetry. A B(E2) row whose 2+ energy differs from the ENSDF 2+1 by more than 2 keV and 0.5 % is flagged.
Moments of inertia of the sharp surface R(θ) = R0(1 + β Y20(θ)), uniform density, mass A u, about an axis perpendicular to the symmetry axis, to leading order in β: rigid body 𝓘rig = (2/5) A u R0² (1 + √(5/16π) β); irrotational flow 𝓘irr = (9/8π) A u R0² β² (the flow that keeps the surface moving with the rotating shape and has no vorticity). Both are checked against quadratures in tests/check_collective.py.
Yrast band, transition I → I−2 with Eγ = E(I) − E(I−2): ħω = Eγ/2, the slope dE/dI at the mid spin I−1 (exact for a rotor); kinematic 𝓘(1) = ħ²(2I−1)/Eγ = ħ²(I−½)/ħω; dynamic 𝓘(2) = ħ² dI/d(ħω) = 4ħ²/(Eγ(I+2) − Eγ(I)), placed at ħω = (Eγ(I) + Eγ(I+2))/4. For a rigid rotor both equal 3ħ²/E(2+) at every spin. A backbend is where ħω falls while I rises: 𝓘(1) plotted against ħω turns back, 𝓘(2) changes sign. It is counted only where both gammas are in ENSDF; elsewhere the yrast sequence may pass to another band with no linking gamma. Eγ/I against I: which falls as E(2+)/I for a vibrator and rises to 4ħ²/2𝓘 for a rotor.
Rotation scene: the frequency of yrast state I is the central difference (E(I+2) − E(I−2))/4 (the last state uses Eγ/2), ω = ħω/ħ, slowed down by the factor shown. A 0+ state has no orientation in space; the spheroid at I = 0 is the intrinsic shape, not a picture of the state.