First-principles derivation of the total width $\Gamma$ of the giant dipole resonance

The 1950 letter of Steinwedel, Jensen, and Jensen introduced the interpenetrating proton-neutron fluid model of the giant dipole resonance (GDR): with the nuclear symmetry energy as restoring force it predicts the resonance frequency without free parameters, but the total width $\Gamma$ — damping of the collective dipole vibration by dissipation into disordered motion followed by nucleon emission — 'must be taken from experiments', so the photoabsorption line shape is fitted, not predicted. Derive the total GDR width, and hence the line shape, from nuclear theory whose inputs are not adjusted to the measured width. The later literature has settled the mechanism — $\Gamma$ is the sum of collisionless Landau fragmentation, an escape width from direct decay into the continuum, and a spreading width from coupling of the collective doorway state to 2p2h and more complex compound configurations — and has produced semi-quantitative microscopic calculations; but every quantitative treatment in the literature anchors some ingredient to empirical width data: phonon-damping models (which describe damping by coupling the resonance to quantized collective vibrations, called phonons) fit the zero-temperature width, transport calculations invert the problem and use the measured width to constrain in-medium cross sections and effective masses, second-RPA underpredicts, and beyond-RPA or real-time density-functional approaches rest on phenomenological interactions. The open core — a parameter-unfitted quantitative prediction of the total width and line shape — remains nonempty and is the object of active work through 2025-2026.

Background

Nuclei absorb gamma rays strongly in a broad electric-dipole (E1) resonance — the giant dipole resonance (GDR) — centered between roughly 13 and 25 MeV depending on mass number, which carries most of the dipole oscillator strength built on the nuclear ground state. Goldhaber and Teller (1948) interpreted it as a collective vibration in which all protons move as a rigid sphere against a rigid neutron sphere. Steinwedel, Jensen, and Jensen (1950) — in the one-page letter, with the detailed companion paper by Steinwedel and Jensen in Zeitschrift fuer Naturforschung — proposed the more realistic interpenetrating-fluids picture: proton and neutron fluids of densities $\rho_p$ and $\rho_n$ permeate each other at constant total density $\rho_0=\rho_p+\rho_n$ inside a fixed sphere of radius $R=r_0A^{1/3}$. The restoring force is the symmetry energy, the approximately quadratic penalty $K(Z-N)^2/A$ (with $K\simeq20$ MeV) that the nuclear binding energy pays for proton-neutron imbalance; written as a local energy density proportional to $(\rho_n-\rho_p)^2$, it turns the relative motion into a peculiar hydrodynamics with velocity $u=[8KZN/(A^2M)]^{1/2}\approx c/5$ for relative proton-neutron density disturbances ($M$ the nucleon mass, $A=Z+N$, with $Z$ and $N$ the proton and neutron numbers). The lowest dipole mode is the standing wave of this isovector sound — a compression wave of the proton-neutron density difference — with $\hbar\omega_0=2.08\,\hbar u/R$, giving resonance energies of the observed size and $A^{-1/3}$-type scaling without adjustable parameters.

The ideal-fluid model, however, is lossless: it predicts the resonance frequency and the integrated absorption strength (one half of the Goldhaber-Teller value), but not the line shape. To confront experiment the authors included the calculable radiative damping and wrote the total photoabsorption cross section in terms of $\omega_0$, the total width $\Gamma$, and a phase angle $\varphi$ defined by $\tan\varphi=\Gamma\omega/(\omega^2-\omega_0^2)$, stating that the total width $\Gamma$ 'must be taken from experiments'. Physically $\Gamma$ is the damping of the ordered collective dipole motion by dissipation into disordered single-particle motion ('heating by friction') followed by nucleon emission. The open question is whether this width can be derived rather than fitted.

Later work identified the damping mechanisms. The standard decomposition (Bertsch, Bortignon, and Broglia, 1983) writes $\Gamma=\Delta\Gamma+\Gamma^{\uparrow}+\Gamma^{\downarrow}$: Landau damping $\Delta\Gamma$, the collisionless fragmentation of the one-particle-one-hole (1p1h) strength over nearby particle-hole configurations in the mean field (the analogue of plasmon damping in an electron gas); the escape width $\Gamma^{\uparrow}$, direct decay of 1p1h configurations into the particle continuum; and the spreading width $\Gamma^{\downarrow}$, the coupling of the collective 1p1h 'doorway' state to two-particle-two-hole (2p2h) and then ever more complex npnh compound-nucleus configurations — the quantum version of dissipation into disordered motion ending in nucleon emission. The Random Phase Approximation (RPA), the standard small-amplitude oscillation theory of the nuclear mean field, produces essentially only the Landau part; escape requires an exact continuum treatment, and the spreading part requires physics beyond the mean field. Empirically the GDR width is about 4-5 MeV in nuclei with $A\gtrsim100$, larger in lighter nuclei, and splits into two peaks in statically deformed nuclei, where the dipole mode along the long nuclear axis oscillates at a lower frequency than along the short axis.

Problem Statement

Derive the total width $\Gamma$ of the giant dipole resonance of an atomic nucleus from the theory of its internal dynamics — that is, predict, rather than fit, the damping rate of the collective proton-against-neutron dipole vibration, and thereby make the total photoabsorption cross-section line shape a prediction. This is precisely the quantity that Steinwedel, Jensen, and Jensen (1950) had to take from experiments, which limits their formalism (and every successor that keeps $\Gamma$ as input) to fitting, not predicting, the line shape.

For a stated nucleus or set of nuclei, without per-nucleus recalibration, the derivation is to: (1) compute the dipole strength function $S(E)$ — the distribution of electric-dipole transition strength over excitation energy $E$ — or the total photoabsorption cross section around the GDR and extract $\Gamma$ under a stated convention (for example the full width at half maximum of a Lorentzian fit, or the moment width $\sqrt{m_2/m_0-(m_1/m_0)^2}$ built from the strength moments $m_k=\int E^k S(E)\,dE$), with numerical uncertainties; (2) account for the physics of all damping mechanisms contributing to the total width — identified in the later literature (Bertsch, Bortignon, and Broglia, 1983) as escape into the continuum, collisionless Landau fragmentation, and spreading through coupling to 2p2h and more complex configurations — whether computed separately or emerging from a single framework (for example a real-time simulation), consistently and without double counting; (3) use input parameters (effective interactions, coupling constants, transport coefficients) whose values are not adjusted to the measured GDR widths or photoabsorption line shapes of the nuclei considered; parameters fitted to independent data such as ground-state masses or few-nucleon scattering are admissible.

References

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