Nuclear quadrupole vibration ($E_n-E_0=n\hslash \omega$)

A collective oscillation of the quadrupole shape coordinates of an atomic nucleus. Near spherical equilibrium, the ideal harmonic description has five components, corresponding to the five magnetic substates of angular momentum two. Each quantum carries $I^{\pi }=2^{+}$; it is an excitation of the many-nucleon system rather than an extra constituent. Its frequency is $\omega$, and its energy spacing $\hslash \omega$ follows the quantum harmonic oscillator.

One quantum produces a $2^{+}$ level at $\hslash \omega$. Two identical quadrupole quanta can couple to $I=0,2,4$ in a symmetric state, giving a degenerate $0^{+},2^{+},4^{+}$ multiplet at $2\hslash \omega$ in the ideal model. Odd total $I$ would be antisymmetric under their exchange. Thus the first $4^{+}$ to first $2^{+}$ excitation ratio is $2$, rather than the nuclear rotational band value $10/3$. Real nuclear quadrupole vibrations can be anharmonic and their multiplets split: the ratio is a diagnostic of a limiting model, not a universal classification rule.

There is a useful state-counting check. Two identical quanta distributed among five substates span $5(5+1)/2=15$ symmetric states. The proposed multiplet contains $(2\cdot 0+1)+(2\cdot 2+1)+(2\cdot 4+1)=1+5+9=15$. This verifies the angular-state bookkeeping independently of the chosen energy scale.

Three identical quadrupole quanta occupy the third symmetric power of the five-dimensional spin-two space. Its dimension is $\left(\genfrac{}{}{0ex}{}{7}{3}\right)=35$, and its irreducible representations have $J=0,2,3,4,6$, each once. All have positive parity. This counts allowed angular states in the ideal harmonic nuclear quadrupole vibration; anharmonic interactions can split their energies.