Clusters in nuclei

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Author: Prof. Martin Freer, School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham

Figure 1: The arrangement of four alpha particle clusters in the nucleus ¹⁶O.

Even at the inception of nuclear science it was known that conglomates of nucleons (nuclear clustering) were extremely important in determining the structure of light nuclei.

The propensity for objects to congregate on all physical scales is striking. Underpinning this must be some significant reduction in potential energy or gain in stability. On the largest scale known to man, the universe, the survey of the 2dF galaxy redshift (Peacock et al., 2001) shows matter congregates into filament-like structures. In this case, these are gravitationally assembled structures which grew from inhomogeneities post "Big Bang". The assemblage of stars into galaxies or the gravitational binding of planets within the solar system involve further reductions in scale, but yet more clustering. On the more human level, many biological systems have developed strategies which recognise that some organisation into collective type behaviour offers some evolutionary advantage. For example, fish shoal to distract predators and predators hunt in packs to maximise success rates in kills. Subatomic systems also recognise the importance of order and symmetry. Atoms form molecules in the liquid or gas phase and crystals in the solid. Quarks find themselves confined within hadrons with only particular numbers of constituents (2 or 3). It would, therefore, be highly surprising if such a phenomenon did not extend to the nuclear domain.

1 A short Introduction
- 1.1 Early Developments
- 1.2 Symmetries and the Harmonic Oscillator
2 References
3 External links
4 See also

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A short Introduction

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Early Developments

Figure 2: The binding energies per nucleon for the light elements. The lines show common elements. The maxima for a given isotope chain coincides with nuclei with even (and equal) number of protons and neutrons.

Figure 3: The binding energies plotted against the number of possible alpha-particle bonds, as suggested by Hafstad and Teller (Hafstad and Teller, 1938). For ⁸Be there is one bond, for ¹²C there are 3 and for ¹⁶O 6 bonds etc...

Figure 4: The Ikeda diagram (Ikeda et al., 1968). The cluster structures appearing at each decay threshold (in MeV) are indicated.

The discovery of alpha-decay of heavy-nuclei initiaited the idea that clusters of nucleons (two protons and two neutrons) might be preformed prior to emission. In fact some of the earliest models of nuclei extended such a principle. The work of Hafstad and Teller (Hafstad and Teller, 1938) was seminal in this regard. An examination of the binding energies of $A=4n (n = 2,3,4,...)$ , $N=Z$ , nuclei as a function of the possible number of alpha-alpha bonds revealed a linear relationship, indicating the important role of the alpha-cluster in the ground-states of these nuclei. This simple picture is in essence correct, particularly when the cluster decay threshold (i.e. Q-value for separating the nucleus into its cluster constituents) lies close to the ground state, though is an oversimplification. In most of the ground states the cluster structure does not survive as separated alpha-particle, but rather the cluster structure becomes more compact and the clusters overlap (and are hence influenced by the Pauli Exclusion Principle). However, the symmetries articulated in the earlier picture survive. In the 1950s Morinaga had postulated, in a rather extreme prediction for the time, that it should be possible for the alpha-particles to arrange themselves in a linear fashion (Morinaga, 1956). The idea that the cluster should not be manifest in the ground-state but emerges as the internal energy of the nucleus is increased was realised to be key in the 1960s. For a nucleus to develop a cluster structure it must be energetically allowed. Asymptotically, when the nucleus is separated into its cluster components an energy equivalent to the mass difference between the host and the clusters must be provided. Thus, close to the point at which the clusters are in contact within the host a similar energy (modulo the interaction energy between the clusters) is required. In other words, the cluster structure would expect to be manifest close to, and probably slightly below, the cluster decay threshold.

In order to be fully formed the proximity of cluster states to the decay threshold is crucial. This has become encapsulated in what is known as the Ikeda diagram (Ikeda et al., 1968). This would predict that cluster structures are most obvious at an excitation which coincides with a particular decay threshold. Hence, the alpha+alpha cluster structure is found in the ground state of ⁸Be (which decays within 10^-16 seconds to two alpha-particles. The three alpha-cluster structure would be expected close to the three alpha-decay threshold (i.e. 7.27 MeV). It is believed that the well-known Hoyle-state at 7.65 MeV has a well developed 3alpha structure (see section ). Brink (Brink., 1966) employed an alpha-cluster model to explore the stable and quasi-stable structures of alpha-particle like systems. He proposed a number of geometrical, or crystalline, like structures of alpha-particle like nuclei, which in many respects resembled those antiscipated by earlier (Hafstad and Teller, 1938), though were now linked to excited states.

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Symmetries and the Harmonic Oscillator

Figure 5: The energy levels of the deformed harmonic oscillator

The Ikeda diagram illustrates the importance of the nuclear excitation energy in appearance of clustering - clusters appear at decay thresholds. Another driving force is that of symmetries. These were at the heart of the predictions of Hafstadt and Teller. The single-particle behaviour of nuclei is well-described in the context of the nuclear shell model, where nucleons move in a mean field which characterises the average interaction of a nucleon with all of its other constituents. In this model nucleons possess a mean free path which is considerably larger than the nuclear scale. A simple characterisation of this motion may be found within the deformed harmonic oscillator (Fig. 5). Here the energy levels are given by

$E=\hbar\omega_\perp n_\perp+ \hbar\omega_zn_z+\frac{3}{2}\hbar\omega_0$

where the characteristic oscillator frequencies for oscillations perpendicular ( $\perp$ ) and parallel ( $z$ ) to the deformation axis are now required. These are constrained such that $\omega_0=(2\omega_\perp+\omega_z)$ , and the quadrupole deformation is given by $\epsilon=\epsilon_2=(\omega_\perp - \omega_z)/\omega_0.$ If $\epsilon$ is positive then this implies a nucleus with a prolate (rugby ball) deformation, whilst a negative value indicates a oblate (pumpkin) like shape. If $\epsilon$ is zero then the intrinsic shape is spherical (see Fig. 5).