Pear-shaped nuclei

Some heavy nuclei are not reflection symmetric: they have a pear shape (octupole deformation \(Y_{30}\)). Spectroscopic signature: an alternating-parity band \(0^+,1^-,2^+,3^-,\dots\) and a parity doublet — which is the tunnelling doublet of a double well in the \(\beta_3\) coordinate. Nucleus tables from the Python backend (via gw2py).

The same double well as ammonia inversion and proton transfer, on the nuclear scale: the two minima are “pear up” and “pear down”, parity is the reflection β₃ → −β₃.

Some heavy nuclei are not reflection symmetric: they have a pear shape (octupole deformation).

Beyond the quadrupole: the octupole

The nuclear surface is expanded in spherical harmonics. Besides the quadrupole deformation (\(\lambda=2\), the “rugby ball”), some nuclei add an octupole component (\(\lambda=3\)):

\[ R(\theta)=R_0\big[1+\beta_2\,Y_{20}(\theta)+\beta_3\,Y_{30}(\theta)\big], \]

with \(Y_{30}\propto P_3(\cos\theta)=\tfrac12(5\cos^3\theta-3\cos\theta)\). Since \(P_3\) is odd (\(P_3(-x)=-P_3(x)\)), the \(\beta_3\) term breaks reflection symmetry: one pole widens, the other sharpens → a pear shape.

Spectroscopic signature: the alternating-parity band

A rotating pear generates a rotational band with alternating parity: \(0^+,1^-,2^+,3^-,4^+,5^-,\dots\) all on the same \(E\propto I(I+1)\) curve. States of opposite parity are connected by strong E1 transitions (a “lightning-rod” effect: charge accumulates at the tip → the centre of charge separates from the centre of mass → intrinsic dipole moment) and by strong E3 moments.

Static or dynamic?

Static pear (rigid): fixed \(\beta_3\), a stable pear shape. The negative-parity band comes down until it nearly merges with the positive one → a near-degenerate parity doublet. Confirmed in ²²⁴Ra and ²²⁶Ra (Gaffney, Nature 2013).
Octupole vibrator (dynamic): the pear oscillates around the symmetric shape (\(\beta_3\) oscillates about 0). The negative band stays shifted upwards. This is the case of the radon isotopes (²²⁰Rn) and ²²⁸Th.

Why they matter

A pear with a dipole moment amplifies a possible atomic electric dipole moment (EDM): pear-shaped nuclei (e.g. ²²⁵Ra, parity doublet at 55 keV) are prime candidates in searches for symmetry violations beyond the Standard Model.

Nucleus

Select the nucleus

Method

Shape \(R(\theta)=R_0[1+\beta_2 Y_{20}+\beta_3 Y_{30}]\); the parity doublet is the tunnelling doublet of the double well in \(\beta_3\). Parameters (\(\beta_3\), splitting) characteristic/adopted for the static or dynamic regime, from the backend table. In the visualization you can vary \(\beta_3\) by hand.

Rotating pear shape

β₃

speed intrinsic dipole

Double well in β₃

Alternating-parity band

β₂ (quadrupole)

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β₃ (octupole)

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Intrinsic dipole

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Parity splitting

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Regime

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The special functions, the threads to the other pages and the limits of the model.

A double thread: Legendre P₃ and the double well

The shape is the Legendre polynomial \(P_3\) (spherical harmonic \(Y_{30}\)) — the same family as the quadrupole of the collective nucleus page, one degree up. And the parity doublet is the tunnelling doublet of a double well in the \(\beta_3\) coordinate: the two minima are “pear up” and “pear down”, parity is the reflection \(\beta_3\to-\beta_3\). Deep barrier (rigid pear) → near-degenerate doublet; low barrier (soft octupole) → large splitting. It is literally instanton tunnelling physics, on the nuclear scale.

The E3 moment, the reliable indicator

The E1 dipole moment arises from the small centre-of-charge / centre-of-mass separation (\(\propto\beta_2\beta_3\)) but suffers cancellations (in ²²⁴Ra and ¹⁴⁶Ba it is nearly zero). The E3 moment, which depends on the reflection-asymmetric charge distribution over the whole volume, is the reliable collective indicator of the pear.

Octupole regions

Pears appear where nucleon numbers are \(\approx 34,56,88,134\) (opposite-parity levels close to Fermi): the Ra/Th region (\(Z\approx88,N\approx134\)) and the Ba/Nd region (\(Z\approx56,N\approx88\), e.g. ¹⁴⁴Ba, ¹⁴⁸Nd).

What is missing (honesty)

This is the geometric collective picture: it reproduces the shape, the alternating-parity band and the static/dynamic logic, but the adopted values (\(\beta_3\), splitting) are characteristic/parametrized. The microscopic description is parity-projected HFB or the generator-coordinate method (GCM) on the quadrupole-octupole collective Hamiltonian — runnable by the backend, out of reach for a browser preview.

References

L. P. Gaffney et al., «Studies of pear-shaped nuclei using accelerated radioactive beams», Nature 497, 199 (2013). doi.
P. A. Butler, W. Nazarewicz, «Intrinsic reflection asymmetry in atomic nuclei», Rev. Mod. Phys. 68, 349 (1996). doi.
P. A. Butler, «Octupole collectivity in nuclei», J. Phys. G 43, 073002 (2016). doi.
P. A. Butler et al., «Observation of vibrating pear-shapes in radon nuclei», Nat. Commun. 10, 2473 (2019). doi.
G. A. Leander, Y. S. Chen, «Reflection-asymmetric rotor model», Phys. Rev. C 37, 2744 (1988). doi.

WebNIR · CNR-IFAC | demo interface — nucleus tables from the Python backend (gw2py).

Keywords: pear-shaped nuclei, octupole deformation, Y30, alternating-parity band, parity doublet, radium, E3 moment, EDM