Giant dipole resonance

All protons oscillating against all neutrons: the giant dipole resonance (GDR), the isovector E1 collective mode. Hydrodynamic two-fluid model (Goldhaber–Teller / Steinwedel–Jensen), splitting in deformed nuclei and photoabsorption cross section. Energies and systematics from the Python backend (via gw2py).

A ~10–25 MeV photon sets the proton fluid oscillating against the neutron fluid: the dominant peak of nuclear photoabsorption, which splits in deformed nuclei.

The giant dipole resonance: all protons oscillating against all neutrons (the isovector E1 collective mode).

Two fluids in antiphase

A ~10–25 MeV photon sets the proton fluid oscillating against the neutron fluid: an oscillating dipole moment is created, which absorbs the radiation. This is the giant dipole resonance (GDR), the dominant peak of the photoabsorption cross section.

Goldhaber–Teller vs Steinwedel–Jensen

Goldhaber–Teller (1948): two rigid spheres (protons, neutrons) translating against each other; restoring force from the symmetry energy → \(E\propto A^{-1/6}\) (harmonic oscillator).
Steinwedel–Jensen (1950): the two fluids interpenetrate and the isovector density oscillates inside the fixed volume → a standing dipole wave, \(E\propto A^{-1/3}\).

The experimental systematics sits halfway and is summarized by:

\[ E_{\rm GDR} = 31.2\,A^{-1/3} + 20.6\,A^{-1/6}\ \text{MeV}, \]

The two terms are exactly Steinwedel–Jensen (\(A^{-1/3}\)) and Goldhaber–Teller (\(A^{-1/6}\)). Width \(\Gamma\simeq 1.11\sqrt{E}\) MeV.

Deformation splitting

In a deformed nucleus the oscillation along the long axis (low frequency) differs from the perpendicular one (high): the GDR splits into two peaks. The separation is proportional to the deformation:

\[ E_b-E_a = 11.1\,|\beta_2|\ \text{MeV},\qquad E_a+2E_b=3E_{\rm GDR}, \]

with the perpendicular mode (\(E_b\)) carrying double weight (two degenerate axes). This is the direct link with the deformation of the collective nucleus and fission pages: the same prolate shape that there rotates and fissions, here splits the resonance.

TRK sum rule

\[ \int \sigma_{\rm abs}\,dE \simeq 60\,\frac{NZ}{A}\ \mathrm{MeV\,mb}\quad(+{\sim}20\%). \]

Nucleus

Select the nucleus

Spherical nuclei have a single peak; deformed ones split the resonance.

Method

The backend computes the energies from the systematics \(31.2A^{-1/3}+20.6A^{-1/6}\), the splitting \(11.1|\beta_2|\), the widths \(1.11\sqrt{E}\) and the TRK sum. In the visualization the two ellipsoids (protons in blue, neutrons in orange) oscillate in antiphase; for deformed nuclei you choose the axis. The cross section is one (or two) Lorentzians.

Two fluids in antiphase

protons neutrons

speed drag to rotate · wheel to zoom

Photoabsorption cross section

Deformation β₂

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GDR peak(s)

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Width Γ

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TRK sum

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Mode

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The hidden special function, the threads to the other pages, and the limits of the hydrodynamic model.

The special function: spherical Bessel j₁

In the Steinwedel–Jensen model the isovector density obeys the Helmholtz equation inside the nucleus; the dipole mode (\(\ell=1\)) is a spherical Bessel function \(j_1(kr)\), and the boundary condition (no current through the surface) pins the frequency to the first zero of the derivative: \(j_1'(kR)=0\Rightarrow kR=2.0816\), whence \(E_{\rm SJ}\propto 1/R\propto A^{-1/3}\). It is the same Bessel family as the E(5)/X(5) critical points; the Goldhaber–Teller model is instead the harmonic oscillator of the harmonic page.

Experimental signature of the shape

One peak = spherical nucleus; two peaks with area ratio ~2:1 = axially deformed nucleus. The photoabsorption cross section is therefore a direct gauge of nuclear deformation, complementary to the rotational spectroscopy of the collective page.

What is missing (honesty)

The two-fluid model is hydrodynamic (macroscopic): it reproduces energies and splitting but neither the width nor the fine structure. The microscopic description is the RPA (superposition of particle-hole excitations) or time-dependent TDHF, which give damping and fragmentation — out of reach for a browser preview, but runnable by the production backend. The width here comes from the systematics \(\Gamma=1.11\sqrt{E}\).

References

M. Goldhaber, E. Teller, «On Nuclear Dipole Vibrations», Phys. Rev. 74, 1046 (1948). doi.
H. Steinwedel, J. H. D. Jensen, Z. Naturforsch. A 5, 413 (1950).
G. C. Baldwin, G. S. Klaiber, Phys. Rev. 71, 3 (1947); 73, 1156 (1948).
B. L. Berman, S. C. Fultz, «Measurements of the giant dipole resonance…», Rev. Mod. Phys. 47, 713 (1975). doi.
R. B. Firestone, «The Origin of the Giant Dipole Resonance», arXiv:2009.03356 (2020) — splitting \(11.1|\beta_2|\).

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

Keywords: giant dipole resonance, GDR, two fluids, Goldhaber-Teller, Steinwedel-Jensen, photoabsorption, splitting, TRK sum rule