Master's Thesis · Nuclear structure theory

Four Phonon Anharmonicity and Angular Momentum Effects in Even-Even Spherical Nuclei

Aous Ahmad Abdo · with V. Zelevinsky

Spherical even-even nuclei vibrate in quantized surface ripples — quadrupole phonons. In the harmonic approximation their spectrum is a perfectly even ladder. This thesis treats the anharmonic corrections up to four phonons and solves the resulting Hamiltonian analytically, showing exactly how the degenerate multiplets split and shift.

The idea

Real nuclear vibrations interact. Writing the quartic phonon interaction in terms of the intermediate angular momenta of phonon pairs, and exploiting a hidden rotational symmetry in five-dimensional phonon space, the model becomes exactly solvable through a canonical transformation of the boson-pair operators.

The result

A closed-form energy spectrum. The ground quasi-rotational band takes the compact form

E(\nu,0)=\tfrac{1}{4}\left(3\omega_\nu+\frac{1}{\omega_\nu}\right)\left(\nu+\tfrac{5}{2}\right),\qquad \omega_\nu^{3}-\omega_\nu=4\left(\nu+\tfrac{k}{2}+1\right)\lambda

In the harmonic limit λ → 0 the spectrum is exactly equidistant, $E = \nu + \tfrac{5}{2}$ ; as the anharmonicityλ grows, the multiplets split and the level structure reorganizes — the effect made interactive in Chapter I.

Tested against 42 nuclei

The model is not just formal. The thesis carries it through to real nuclear data, comparing the predicted level-energy ratios R_Jagainst measured values for 42 even-even nuclei across six isotope chains —Se, Kr, Mo, Ru, Pd, Cd — over the full range of angular momentum J. Twenty-two pages of the thesis are these comparisons; two are shown here.

Plots of theoretical versus experimental level-energy ratios against angular momentum for palladium-100 and palladium-102 — **¹⁰⁰Pd and ¹⁰²Pd.**Predicted (solid) and measured (dash-dot) level-energy ratios R_J versus angular momentum J. The two curves are very nearly indistinguishable across the whole range — the anharmonic model reproducing the real spectrum.

Plots of theoretical versus experimental level-energy ratios against angular momentum for ruthenium-98 and ruthenium-100 — **⁹⁸Ru and ¹⁰⁰Ru.**The same comparison for the ruthenium chain. Theory tracks experiment closely, with the interesting deviations appearing only at the highest spins — where the simple phonon picture is expected to strain.

Download the full thesis (PDF)← Back to the phonon diagram