The heavy nucleus as a collective droplet

Collective nucleus

The Bohr–Mottelson collective model: the heavy nucleus as a deformable quantum droplet that vibrates and rotates. Fourteen selectable nuclei spanning the vertices of the shape triangle — U(5), SU(3), O(6) — and the critical points X(5)/E(5), whose spectra come from Bessel zeros. Tables and systematics from the Python backend (via gw2py).

The deformable nuclear surface that vibrates and rotates: ten actinide rotors, a vibrator, a γ-soft nucleus and the two critical points, from ²³⁸U to ¹³⁴Ba.

The Bohr–Mottelson collective model: the heavy nucleus as a deformable quantum droplet.

The nuclear surface

The surface radius is parametrized with spherical harmonics; the dominant mode in heavy nuclei is the quadrupole (\(\lambda=2\)):

\[ R(\theta,\varphi)=R_0\Big[1+\sum_{\mu}\alpha_{2\mu}Y_{2\mu}(\theta,\varphi)\Big]. \]

In the intrinsic frame the five \(\alpha_{2\mu}\) reduce to two shape variables \((\beta,\gamma)\) plus three Euler angles (the rotation):

\[ R(\theta,\varphi)=R_0\Big[1+\sqrt{\tfrac{5}{16\pi}}\,\beta\big(\cos\gamma\,(3\cos^2\theta-1)+\sqrt3\,\sin\gamma\,\sin^2\theta\cos2\varphi\big)\Big]. \]

\(\beta\) measures the elongation, \(\gamma\) the triaxiality: \(\gamma=0^\circ\) prolate (cigar), \(\gamma=60^\circ\) oblate (pancake), triaxial in between.

The Bohr Hamiltonian

Collective kinetic energy (vibrations of \(\beta,\gamma\) + rotation) plus a potential \(V(\beta,\gamma)\). In the \(\beta\) coordinate the kinetic energy is that of a radial problem in 5 dimensions (measure \(\beta^4\,d\beta\)).

The vertices of the shape triangle

Spherical vibrator U(5) (¹¹⁰Cd): spherical equilibrium, small quadrupole oscillations (phonons). \(E\propto N\), \(R_{4/2}=2.0\); the \(\beta\) eigenfunctions are those of the 5D harmonic oscillator (Laguerre polynomials).
Axial rotor SU(3) (²³⁸U): a rigid deformed shape that rotates → band \(E_I\propto I(I+1)\), \(R_{4/2}=3.33\).
γ-soft O(6) (¹⁹⁶Pt): a deformed shape with \(\gamma\) completely free (wandering over all triaxial shapes). \(E\propto\tau(\tau+3)\), \(R_{4/2}=2.5\).
X(5) (¹⁵²Sm): the critical point of U(5)→SU(3) (Iachello). \(\beta\) in a square well, \(\gamma\approx0\).
E(5) (¹³⁴Ba): the critical point of U(5)→O(6). Potential independent of \(\gamma\).

The link with Bessel functions

At the critical points \(V(\beta)\) is a square well: with \(f=\beta^{-3/2}\phi\) the \(\beta\) equation becomes the Bessel equation, and the boundary condition \(\phi(\beta_W)=0\) quantizes the energy through the zeros of \(J_\nu\):

\[ E_{s,L}\propto x_{\nu,s}^2,\qquad \nu=\begin{cases}\tau+\tfrac32 & \text{E(5)}\\[2pt]\sqrt{\tfrac{L(L+1)}{3}+\tfrac94}& \text{X(5)}\end{cases} \]

Hence \(R_{4/2}=2.20\) for E(5) and \(2.91\) for X(5): pure numerical signatures, dictated solely by Bessel zeros.

Nucleus

Select the nucleus to visualize

Even-even rotors (Th, U, Pu, Cm, Cf) · ¹¹⁰Cd U(5) vibrator · ¹⁹⁶Pt γ-soft O(6) · ¹⁵²Sm critical point X(5) · ¹³⁴Ba critical point E(5).

Show the collective rotation

Animation

Speed Visual exaggeration of the deformation = ×1.6

The real deformation (β≈0.2–0.3) is amplified to make it visible.

Nuclear surface (β,γ) evolving

drag to rotate · wheel to zoom · blue = hollows, gold = bulges

State

Model: —
Instantaneous shape: —
Spectrum: —
E(2⁺): —
R₄/₂ = E(4⁺)/E(2⁺): —

Level scheme

Collective vs microscopic, and where the special functions live.

Why the collective model (and not TDHF)

The microscopic dynamics of a heavy nucleus is time-dependent Hartree–Fock (TDHF): a self-consistent mean field, a 3D density that evolves — the gold standard for fission and heavy-ion collisions, but unapproachable in a browser preview (3D grids, hundreds of nucleonic wave functions). The collective model is its geometric counterpart: it captures the same phenomenology (rotations, vibrations, shape transitions) with just two coordinates \((\beta,\gamma)\), at negligible numerical cost.

The Casten triangle

The full triangle: the three vertices (¹¹⁰Cd U(5) vibrator, ²³⁸U SU(3) rotor, ¹⁹⁶Pt γ-soft O(6)) and the two critical points on the sides (¹⁵²Sm X(5) on U(5)→SU(3), ¹³⁴Ba E(5) on U(5)→O(6)).

The special function of this page

Just as Pöschl–Teller carries the \(_2F_1\) and H₂⁺ the confluent Heun, here the role belongs to the zeros of the Bessel functions \(J_\nu\). The order \(\nu\) is not an integer: \(\nu=\tau+\tfrac32\) (E(5)) or \(\nu=\sqrt{L(L+1)/3+9/4}\) (X(5)). The radial equation in \(\beta\), with its 5-dimensional centrifugal term, is of the same type the FD radial engine of this collection already handles.

Evidence

Ratios \(R_{4/2}=E(4^+)/E(2^+)\): U(5) 2.0 (¹¹⁰Cd exp. 2.34) · O(6) 2.5 (¹⁹⁶Pt 2.46) · SU(3) 3.33 (²³⁸U 3.30) · X(5) 2.91 (¹⁵²Sm 3.01) · E(5) 2.20 (¹³⁴Ba 2.32). The critical points have spectra from the zeros of \(J_\nu\) (cross-checked with scipy.special).

References

A. Bohr, B. R. Mottelson, Nuclear Structure, Vol. II, Benjamin, 1975.
F. Iachello, «Dynamic Symmetries at the Critical Point» (E(5)), Phys. Rev. Lett. 85, 3580 (2000). doi.
F. Iachello, «Analytic Description of Critical Point Nuclei» (X(5)), Phys. Rev. Lett. 87, 052502 (2001). doi.
R. F. Casten, N. V. Zamfir, «Evidence for a Possible E(5) Symmetry in ¹³⁴Ba», Phys. Rev. Lett. 85, 3584 (2000). doi.
R. F. Casten, N. V. Zamfir, «Empirical Realization of a Critical Point Description» (¹⁵²Sm), Phys. Rev. Lett. 87, 052503 (2001). doi.
P. Ring, P. Schuck, The Nuclear Many-Body Problem, Springer, 1980.

WebNIR · CNR-IFAC | demo interface — data and systematics from the Python backend (gw2py).

Keywords: collective nucleus, Bohr-Mottelson, quadrupole deformation, beta gamma, rotor, vibrator, X(5), E(5), Bessel zeros, Casten triangle