Odd nuclei

A nucleus with an odd number of nucleons: the unpaired nucleon occupies a Nilsson orbital on the deformed core and couples to it with \(\mathbf I=\mathbf R+\mathbf j\). Rotational bands on the band head \(K\), decoupling term for \(K=1/2\) and Coriolis rotational alignment. Nucleus tables from the Python backend (via gw2py).

All previous nuclear pages deal with even-even cores; here the odd nucleon is added: half-integer angular momentum, bands on K and the signature staggering.

A nucleus with an odd number of nucleons: the unpaired nucleon couples to the rotating deformed core.

Particle + rotor

The even-even core (deformed, prolate) provides a potential in which the odd nucleon occupies a Nilsson orbital, characterized by the projection \(\Omega\) of its angular momentum on the symmetry axis and by the asymptotic quantum numbers \(\Omega[N\,n_z\,\Lambda]\). The total angular momentum is \(\mathbf I=\mathbf R+\mathbf j\): collective rotation of the core \(\mathbf R\) plus the particle's momentum \(\mathbf j\).

Strong coupling: the band on K

In the strong-coupling limit the particle is locked to the deformed shape: \(K=\Omega\) is a good band constant and the band head has \(I=K\). The rotational band is

\[ E(I)=\frac{\hbar^2}{2\mathcal J}\big[I(I+1)-K(K+1)\big],\qquad I=K,K+1,K+2,\dots \]

²⁴¹Am has ground state \(5/2^-[523]\) (a proton from the \(h_{9/2}\) subshell): the band is \(5/2^-,7/2^-,9/2^-,\dots\) on the ²⁴⁰Pu core, the same droplet that in the fission page forms the double-humped barrier.

K=1/2 bands: decoupling

For \(K=1/2\) the Coriolis interaction survives even at low spin and adds a decoupling term:

\[ E(I)=\frac{\hbar^2}{2\mathcal J}\Big[I(I+1)+a\,(-1)^{I+1/2}\big(I+\tfrac12\big)\Big], \]

with the decoupling parameter \(a\). This splits the band into two signature partners (\(\alpha=\pm1/2\)): the states \(I=\tfrac12,\tfrac52,\tfrac92,\dots\) shift relative to \(\tfrac32,\tfrac72,\tfrac{11}2,\dots\) → the characteristic odd-even staggering.

Odd nucleus

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Method

Band \(E(I)=A[I(I+1)-K(K+1)]\) on the Nilsson band head \(K=\Omega\); for K=1/2 the decoupling term \(a(-1)^{I+1/2}(I+\tfrac12)\). Moment of inertia \(A=\hbar^2/2\mathcal J\) and \(a\) adopted/characteristic, from the backend table. The visualization shows the coupling \(\mathbf I=\mathbf R+\mathbf j\) and the transition towards rotational alignment.

Angular-momentum coupling

I j (particle)

rotational alignment (Coriolis)

speed drag to rotate · wheel to zoom

Rotational band

Band head Kᵖ

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Nilsson orbital

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A = ℏ²/2𝒥

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Core

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Decoupling a

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

Two special functions

The Nilsson orbitals are the eigenstates of the 3D anisotropic harmonic oscillator (with spin-orbit and \(l^2\)) — the same family as the harmonic page, deformed: the asymptotic numbers \([N\,n_z\,\Lambda]\) come from the Hermite functions along the axes. The rotational part is instead a Wigner D function \(D^I_{MK}(\phi,\theta,\psi)\) (the rotation matrices, which reduce to Jacobi polynomials): it is the wave function of the symmetric rotor. The odd-A wave function is the (symmetrized) intrinsic⊗rotational product.

A new branch of the model

All previous nuclear pages (collective, fission, GDR, pear) deal with even-even cores. Here the odd nucleon is added: the core stays the same (²⁴⁰Pu under ²⁴¹Am), but the half-integer angular momentum and the particle-rotation coupling open up the phenomenology of signature, decoupling and rotational alignment.

Strong coupling vs rotational alignment

At low frequency strong coupling holds (\(\mathbf j\) along the symmetry axis, good \(K\)). As rotation increases, the Coriolis force detaches \(\mathbf j\) from the symmetry axis and aligns it with the rotation axis (in the visualization: the “alignment” slider). It is strongest for high-\(j\), low-\(K\) intruder orbitals, where the signature staggering becomes large.

What is missing (honesty)

This is the phenomenological particle-rotor model with good \(K\); it neglects Coriolis mixing among different \(\Omega\) orbitals and backbending. The values here (\(\mathcal J\), \(a\)) are adopted/characteristic. The microscopic description is cranked HFB (one-quasiparticle bands with blocking), runnable by the backend.

References

S. G. Nilsson, «Binding states of individual nucleons in strongly deformed nuclei», Mat. Fys. Medd. Dan. Vid. Selsk. 29, 16 (1955).
A. Bohr, B. R. Mottelson, Nuclear Structure, Vol. II (Benjamin, 1975) — particle-rotor model, decoupling.
G. A. Leander, Y. S. Chen, «Reflection-asymmetric rotor model of odd-A nuclei», Phys. Rev. C 37, 2744 (1988). doi.
A. V. Afanasjev et al., «Pairing and rotational properties of actinides», Phys. Rev. C 88, 014320 (2013) — one-quasiproton bands in ²⁴¹Am. doi.

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

Keywords: odd nuclei, particle-rotor, Nilsson orbitals, rotational band, decoupling, signature, Coriolis, americium