The capability of the new 3-stage Van De Graaff accelerator to produce heavy ion beams at energies up to 70 MeV opened the way for a large number of novel experiments. Examples are the systematic studies by Saladin and his collaborators of the shapes, shape vibrations,and collective rotations of nuclei via Coulomb excitation, and later the investigation of the physics of nuclei at very high spin by means of fusion-evaporation reactions.

Studies of Nuclear Shapes, Shape Vibrations, and Collective Rotations

Shapes, shape vibrations and collective rotations are fundamental concepts in nuclear structure physics. A series of experiments starting in 1966 at NPL made seminal contributions to the study of shapes and shape related phenomena. The development of new experimental techniques made it possible for the first time to:

• Determine the ellipsoidal (quadrupole) deformation of short-lived (10^-16 sec) excited states
• Distinguish between prolate and oblate deformations
• Distinguish between static and dynamic triaxial shapes
• Determine the magnitude and sign of higher order (hexadecapole) deformations of short lived excited states.
• Study the relation between charge- and mass-deformation

Parametrization of Nuclear Shapes

Axially symmetric nuclear shapes are typically expressed in terms of spherical harmonics, Y_{λ μ}(θ,φ). The expansion is dominated by the terms of multipolarity λ=2 (quadrupole) and λ=4 (hexadecapole). For axially- and reflection-symmetric shapes the first two terms of such an expansion is given by

R(θ) = R₀ 1 + β₂ (3cos²θ - 1) + β₄ (35cos⁴θ - 30cos²θ + 3)  (S.1)

where β₂ and β₄ are the deformation parameters defining the shape and R₀ = 1.2 x A^1/3 x 10^-13cm. Figures 1 to 4 illustrate the dependance of the shape on the parameters β₂ and β₄ assuming A = 166.

Click on the image to see all figures.

Figures 1 to 4.
Notation: In the above figures β₄ = 0b, 0.15r, -0.15g indicates that the black line corresponds to β₄ = 0, the red line to &beta₄ = 0.15, and the green line to β₄ = -0.15 .

Triaxial reflection symmetric shapes can be introduced by modifying the β₂ term in equation S.1. Omitting for simplicity the β₄-term, these shapes can be expressed by

R(θ,φ,β₂,γ) = R₀ 1 + β₂ cos γ(3cos²θ - 1) + sin γ sin²θ cos2φ (S.2)

Click on the image to see both figures.

Figures 5 and 6.
Three cross sections of such a shape assuming β₂ = 0.35 and γ = 30° are shown below. The red line in the left figure corresponds to the y-z cross section and the black line to the x-z cross section.

Here R₀ and β₂ have the same meaning as in equation S.1, θ and φ are the standard spherical coordinates. The parameter γ is a measure for the deviation of the shape from axial symmetry; γ = 0 and 60° correspond to axially symmetric prolate and oblate shapes. It should be noted that in this representation β₂ is always ≥ 0. Three cross sections of such a shape assuming β₂ = 0.35 and γ = 30° are shown in figures 5 and 6. The red line in the left figure corresponds to the y-z cross section and the black line to the x-z cross section.