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Shapes in Nuclei
Nuclei can assume various shapes to take advantage of shell effects. The famous "magic numbers" correspond to closed shells for the spherical shape and result in a large additional stability. Between these closed shells nuclei break this symmetry by assuming non-spherical shapes. "Superdeformed" nuclei have an especially large deformation due to a shell closure for shapes having a long:short axis ratio of about 2:1. For axially symmetric nuclei the shape can be parameterized by a radius varying as: R(q) = R0[1+S bl Pl(cosq)], where the Pl are Legendre polynomials. The most common non-spherical shape is roughly ellipsoidal and can be generated by a non-zero b2 in the radius expansion. These shapes are symmetric about a plane through the center of the nucleus perpendicular to the symmetry axis (the two ends are identical), and such reflection-symmetric shapes with higher-order deformations are known.
The odd values of l give rise to shapes that are not reflection symmetric, so that the two ends of such nuclei are different. The lowest non-vanishing odd term (l = 3) generates "octupole" shapes which resemble pears, as shown in the figure.
The low-energy states in 222Ra, shown in the figure, were recently determined using GAMMASPHERE. The prominent sequence of negative-parity states is due to octupole collectivity. Measurements of strong E3 transitions (10-30 times single-particle strength) in several nuclei near 222Ra have provided direct evidence for the collective octupole character of such bands. Nuclei around 222Ra also have strong collective quadrupole correlations and the shape drawn on the opposite page has non-zero values for both b2 and b3.
A key question about such nuclei is whether the octupole shape is static in the intrinsic frame (as is the HCl molecule) and corresponds to the upper potential-energy curve on the opposite page, or whether the shape corresponds to vibrational motion in a potential energy well like the lower curve. These can be distinguished by finding whether the odd-spin states fit into an I(I+1) energy relationship with the even-spin states or whether the odd-spin states are instead displaced by a constant vibrational energy (<I>h</I>w) from the even-spin ones. The recent GAMMASPHERE work showed clearly that in 222Ra and some neighboring nuclei the octupole shape is vibrational at the lowest spins and becomes stable (or static) as the spin increases.