Coulomb Excitation Experiments

The amount of (quadrupole, hexadecapole) deformation can be inferred from measured values of matrix elements of the quadrupole- and hexadecapole- (E2 and E4) operator. These matrix elements can be determined in Coulomb excitation (Coulex) experiments. The projectile energy in Coulex experiments is always chosen such that the projectiles cannot enter the region of nuclear forces. Thus nuclei can be excited by the electric-field pulse that is generated by the positive charged projectiles which are scattered by the electric field of the nuclei under investigation. Since the excitation is caused solely by electromagnetic forces, the only nuclear properties that enter into the theory of Coulomb excitation are matrix elements of the electromagnetic multipole moments. In the 1950's Coulex experiments using alpha- and proton-projectiles established the existence of rotational band structures in the rare earth and actinide region of the table of nuclides. These early experiments used gamma-ray spectroscopy with NaI-scintillation detectors to determine the excitation cross sections and extract the transition quadrupole matrix elements. Their large value confirmed the rotational collectivity of these transitions.

In 1966 pioneering work was started at NPL to develop a new and much more accurate method to measure excitation probabilities as a function of the scattering angle and the charge of the projectile. It was based on high energy-resolution heavy-ion spectroscopy using the world's first large solid-angle high-resolution Enge split-pole spectrometer in combination with position sensitive Si detectors. This allowed for a very precise determination of the ratio of the number of in-elastically scattered projectiles to the number of elastically scattered projectiles as a function of the scattering angle and type of projectile. Heavy ion beams are especially well suited for Coulex experiments since the excitation probabilities increase rapidly with projectile charge, resulting in high order excitation effects. Figure E.1 shows the spectrum resulting from the scattering of 48 MeV ¹⁶O ions from a ¹⁶²Dy target. It illustrates that the excitation probability for exciting the first 2⁺ state at 167 kev is larger than the probability for elastic scattering and the two step excitatation probability for the 4⁺ state at 488 keV is quite pronounced, demonstrating the importance of second- and higher order effects. Of particular interest are interference effects between first-order and second-order excitations that provide unique tools to measure the quadrupole moments of short lived excited states and hexadecapole (E4) transition matrix elements.

Quadrupole Moments of Excited States

Two methods were used to determine the quadrupole moment Q from Coulex experiments. In the first, the excitation probabilities of the excited state under investigation were measured at a forward angle and a backward angle using a heavy ion beam such as ¹⁶O. In the second, the excitation probabilities were determined at a fixed backward angle using two different projectiles, a light one like ⁴He and a heavy one such as ¹⁶O or heavier. The effect of the quadrupole moment on the excitation probability is largest for heavy ion projectiles at large backward angles and smallest for light ions and small forward angles. Its value is extracted from the experiments by comparing measured ratios of excitation probabilities with the corresponding ratios obtained from coupled channel calculations.

¹¹⁴Cd

The first experiment was carried out on Cd which is near the Z=50 closed shell (ref. 1, 2). Its low energy level scheme, shown in Fig. E.2 is characteristic for that of near-closed shell nuclei. At the time, these level schemes were interpreted in terms of collective harmonic quadrupole vibrations about a spherically symmetric shape. A consequence of the harmonic assumption is, that the quadrupole moment Q_2⁺ of the first excited 2⁺ state must be zero. It was therefore a big surprise when first Coulex experiments (ref. 3) suggested an unexpected large value -0.85 ≤ Q_2⁺ ≤ -0.35 eb, that is in clear conflict with the model of a harmonic quadrupole vibration about a spherical equilibrium shape. It should be noted that in these experiments the excitation probabilities were determined via gamma-ray spectroscopy using NaI scintillation detectors. Precise extraction of excitation probabilities from this type of experiments is however quite difficult because of significant background in the γ-ray spectra, poor energy resolutions and calibration problems. It is these concerns and the totally unexpected result, that motivated the NPL group to carry out an experiment using the new method of high energy resolution heavy ion spectroscopy.

In the experiment 42 MeV ¹⁶O ions where scattered from a ¹¹⁴Cd target. Energy spectra of the scattered ions were accumulated with the Enge split-pole spectrometer at 7 angles between 45° and 142.8°. Fig.E.3 shows spectra for 142.8° and 48°. The analysis of the data via the coupled channel calculations leads to four possible results for Q_2⁺, the largest of which is Q = -0.45 ± 0.09eb and the smallest Q = -0.71 ± 0.09eb. It should be noted that there are a total of nine matrix elements that were used in the coupled channel calculations, and the relative signs (phases) between some of them were unknown, leading to the above ambiguity. These results were, within the uncertainties in good agreement with the earlier experiments, confirming the conflict with a model based on harmonic quadrupole vibration about a spherical equilibrium shape.

Shape Evolution in Transitional Nuclei

In view of the discovery of large quadrupole moments of the first 2⁺ excited state in ¹¹⁴Cd it seemed of particular interest to investigate these moments for nuclei in the transition region between the deformed rare-earth region and the region near the double-closed shell nucleus ²⁰⁸Pb. An additional motivation was a new theory by Kumar and Baranger (ref. 4) based on an exact diagonalization of Bohr's collective Hamiltonian whose parameters were obtained from microscopic calculations based on the pairing-plus-quadrupole model . This theory predicted large quadrupole moments for the lowest two 2⁺ states (2, 2) in this region and a change in the sign of these moments between the isotopes ¹⁹²Os and ¹⁹²Pt. This sign change corresponds to a change in the nuclear shape from prolate to oblate. Figure E.4 shows the excitation energy of the first and the second 2⁺ states as a function of the atomic and mass number in the mass region 184 ≤ A ≤ 198, illustrating the evolution that occurs in the level scheme as the nuclear shape changes from prolate to oblate.

In 1968 the NPL group started a systematic study of the electromagnetic properties of low lying states of even-even nuclei in this mass region. First experiments were carried out on ^194,196,198Pt and ^188,190,192Os (ref. 5, 6, 7) using high-resolution heavy-ion spectroscopy to determine the quadrupole moments of the first 2⁺ states. These early experiments were later complemented using particle-γ coincidence spectroscopy. The combination of the two experimental techniques provided more detailed information about the electromagnetic properties of the low energy states in these nuclei, including for the first time quadrupole moments of the second 2⁺-state 2 (ref. 8). Figures E.5 to E.8 from (ref. 9) show the experimental values of the quadrupole moments for the 2 and 2 states, the experimental branching ratios B(E2;2 → 2)/B(E2;2 → 0⁺) of transition probabilities, and the ratios Q(2)/M_E2(0⁺, 2) where Q(2) and M_E2(0⁺, 2) are the quadrupole moments of the first 2⁺ state and the E2-transition matrix element between the ground state and the second excited 2⁺ state. Also shown in the figures are the predictions of several models as indicated in the figure caption. The pairing-plus-quadrupole model PPQ (ref. 11) and the boson expansion theory BET (ref. 10) seem to provide the best overall fit to the data. Both models also favour an interpretation in which the 2 states are γ-vibrations. This is in contrast to the assumptions made in the asymmetric rotor model (ARM) (ref. 12) that assumes a rigid triaxial shape. The IBA2 model (ref. 13) has difficulties in fitting the quadrupole moments in Pt nuclei.

Determination of Hexadecapole Deformations

Relaxing the simplifying but artifical constraint β₄ = 0 can have major effects on the equilibrium shape of deformed nuclei. Calculations have shown that even small β₄ values can result in a shape change from oblate to prolate. Hexadecapole deformations play also a crucial role in a number of phenomena such as the behavior of rotational nuclei at very large angular momentum, the physics of fission, and the properties of super-heavy nuclei. It was therefore of considerable interest to find a method to determine the hexadecapole transition-matrix elements < 0⁺ || M(E4) || 4⁺ > between the ground state and the first 4⁺ state from which the β₄ parameters can be extracted. Coulomb-excitation

experiments provided again a model-independent method for determining the E4-transition matrix elements. However, in contrast to experiments measuring quadrupole moments it is necessary to choose conditions under which excitation probabilities are considerably smaller. First experiments at NPL (ref. 14) were carried out in 1972 on ^152,154Sm using α-beams between 10.5 and 12 MeV. The scattered α-particles were analyzed in the Enge split-pole spectrometer at a scattering angle of 143°. In subsequent experiments (ref. 15) back scattered α-particles were detected in a cooled Si(Li)-detector at a scattering angle of 170° resulting in higher sensitivity to E4 effects. The parameters extracted from the data were the ratios of the differential cross sections dσ(4⁺)/dσ(0⁺) and dσ(2⁺)/dσ(0⁺). The matrix elements were determined by comparing the experimental data with the results of coupled-channel calculations. The results of a number of experiments on nuclei in the rare earth region (ref. 15) are shown in figure E.9 and compared with a theoretical calculation based on a deformed Woods-Saxon potential (solid lines) (ref. 16) and a modified harmonic oscillator potential (ref. 17). Both models reproduce the fall off in β₄ with increasing atomic number A. It appears that the deformed Woods-Saxon potential is overall somewhat closer to the experimental results. A separate experiment (ref. 18) on ²³²Th gave β₄ = 0.130 ± 0.020, the largest value among the experiments.