Clusters in nuclei

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Martin Freer (2010), Scholarpedia, 5(6):9652. doi:10.4249/scholarpedia.9652 revision #137031 [link to/cite this article]
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Figure 1: The arrangement of four alpha particle clusters in the nucleus 16O.

A traditional description of the nucleus is one in which there is a roughly homogeneous distribution of protons and neutrons. However, even at the inception of nuclear science it was known that conglomerates of nucleons (nuclear clustering) were extremely important in determining the structure of light nuclei. In many cases a more appropriate picture of the nucleus is the one shown in Figure 1.

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. 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.



Contents

Some 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) numbers of protons and neutrons.
Figure 3: (top) 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... (bottom) Proposed arrangements of the alpha-particles.
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 initiated the idea that clusters of nucleons (two protons and two neutrons) might be preformed prior to emission. If one examines a rather basic property of light nuclei, namely the binding energy per nucleon (BE/A) (see Figure 2) then one observes that binding energies are higher for systems with even-numbers of protons, and for nuclei with even and equal numbers of protons and neutrons the binding energy per nucleon is maximal (e.g. 4He, 8Be, 12C.....). It should be observed that all of these nuclei can be considered to be composed of \(\alpha\)-particles. 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 (see Figure 3), 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-particles, 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 emerge 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) - Figure 1. 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 later). 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 anticipated by earlier (Hafstad and Teller, 1938), though were now linked to excited states.

Symmetries and the Harmonic Oscillator

Figure 5: The energy levels of the deformed harmonic oscillator
Figure 6: Densities calculated within the harmonic oscillator at deformations 2:1, 3:1 and 4:1 corresponding to the degeneracy patterns 2+2, 2+2+2 and 2+2+2+2. The alpha-cluster structures can be clearly observed.

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 Hafstad 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. In many light nuclear systems a description in terms of the spherically symmetric harmonic oscillator is sufficient, but in light systems deformation plays an important role in determining nuclear structure. Correspondingly, a simple characterisation of this motion may be found within the deformed harmonic oscillator ( Figure 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 Figure 5).

At zero deformation the solutions to the harmonic oscillator yields a series of degeneracies (in the nuclear case how many protons or neutrons can occupy orbitals) of 2, 6, 12, 20 .... As the potential is deformed then energy levels which are associated with oscillations along the deformation axis are reduced in energy (the oscillation frequency is reduced), whilst those perpendicular to the deformation axis increase in energy. At deformations of 2:1 and 3:1 (for example) there are a series of level crossings giving rise to a sequence of shell gaps. What is noticeable is that at 2:1 the spherical degeneracies are repeated twice (i.e. 2, 2, 6, 6, 12, 12...) and at 3:1 they are repeated 3 times. This symmetry which appears within the deformed harmonic oscillator indicates that at 2:1 there should be two-clusters and at 3:1 three clusters. These new magic numbers were interpreted in terms of cluster structures associated with each deformation by Rae ((Rae, 1989)) as illustrated in the Table below; Superdeformation corresponds to a potential with an axis ratio of 2:1 and Hyperdeformation to 3:1.

Deformed magic numbers Spherical Magic Numbers Spherical Constituents
N Superdeformation - dimers
4 2 + 2 \(\alpha\)+\(\alpha\)
10 8 + 2 16O+\(\alpha\)
16 8 + 8 16O+16O
28 8 + 20 16O+40Ca
N Hyperdeformation, chains
6 2 + 2 + 2 \(\alpha\)+\(\alpha\)+\(\alpha\)
12 2 + 8 + 2 \(\alpha\)+16O+\(\alpha\)
24 8 + 8 + 8 16O+16O+16O
36 8 + 20 + 8 16O+40Ca+16O
48 20 + 8 + 20 40Ca+16O+40Ca
60 20 + 20 + 20 40Ca+40Ca+40Ca
N Oblate nuclei, pumpkins
8 6+2 12C+\(\alpha\)
12 6 + 6 12C+12C
18 12 + 6 24Mg+12C
24 12 + 12 24Mg+24Mg

This behaviour is not only observed in the degeneracies, but importantly is also in the densities calculated within the harmonic oscillator model - (Freer, 2007). For example Figure 6 shows the densities which would correspond to the degeneracy patterns 2+2, 2+2+2 and 2+2+2+2 which would appear at 2:1, 3:1 and 4:1, respectively. In each case the alpha-clusters are apparent in the density pattern. Although this is a relatively simple approach, more complex calculations of nuclear structure reveal the same symmetries (see (Freer, 2007) and (Freer and Merchant, 1997)). An excellent demonstration of the universality of the appearance of clustering in models of light nuclei are the ab initio calculations of 8Be - Wiringa 2000. In this instance, the calculation applies a free nucleon-nucleon interaction together with three-body forces. In this approach the \(\alpha+\alpha\) cluster structure of 8Be emerges naturally from correlations implicit in the nucleon-nucleon force.

The harmonic oscillator is a very simple approach to calculating the structural properties of nuclei and is one which reveals the underlying symmetries. If the details of the nuclear structure and binding and excitation energies are to be reproduced then more sophisticated methods are required. These include:

  • Antisymmetrized Molecular Dynamics (AMD) (Kanada En'yo, 2001): ab initio type approach using effective nucleon-nucleon interaction.
  • Fermionic Molecular Dynamics (FMD): as AMD but employing a tensor interaction FMD.
  • Resonating Group Method (RGM) and Generator Coordinate Method (GCM): microscopic approach describing reactions and structure in a unified manner (e.g. Baye and Descouvemont 1998).
  • Bloch–Brink cluster model: A multi-centre alpha-cluster model.
  • Local potential cluster models: A binary cluster model with a cluster-core potential.

Significantly, all of these models predict very similar types of cluster structures.

Experimentally Observed Clusters in Alpha-Particle Nuclei

Figure 7: a) resonances observed in the 12C(16O,24Mg*) breakup reaction - from two separate measurements - one performed with a 16O beam energy of 115 MeV and the other 160 MeV, b) the energy-spin systematics of the breakup resonances. The cluster states track from an excitation energy of 20 MeV at low spin. The smaller symbols and the solid line indicates the trend of the low excitation states in 24Mg.
Figure 8: The 12C+12C cluster structure of 24Mg. The image shows two interlocking triangular arrangements of alpha-particles. Each triangle is associated with a 12C nucleus.

Experimental evidence for the cluster structure of light nuclei is well documented (see Freer, 2007 and references therein). The simplest case is that of the two \(\alpha\)-particle system 8Be. The dumbbell-like structure gives rise to a rotational band, from which the moment of inertia (\(\Theta\)) is found to be commensurate with an axial deformation of 2:1. The binding energy of the \(\alpha\)-particle is so large (~28 MeV) that systems such as 6Li and 7Li display 4He+d and 4He+t cluster structures, respectively. Perhaps the most famous cluster state is the Hoyle-state in 12C which lies at 7.65 MeV. The state was predicted by Hoyle (Hoyle 1953) to account for the abundance of carbon in the universe and it was subsequently measured (Cook 1957) at an energy which was extremely close to that predicted. Carbon is synthesised in stellar environments through the triple-alpha process, whereby first two alpha-particles fuse momentarily to form 8Be and then before the system decays a third \(\alpha\)-particle is captured. This predominantly proceeds through the 7.65 MeV, 0+, state which then radiatively decays to the 12C ground state via the 2+ state at 4.43 MeV. It is the 3\(\alpha\) cluster structure of this state which ensures that it resides so close to the 4He+8Be decay threshold and strongly influences the capture probability. This state is known to possess an extremely large radius (volume), which is sufficient for the \(\alpha\)-particles to retain their quasi-free characteristics. Given the bosonic nature of the spin zero 4He nucleus, the state has been interpreted in terms of a Bose-Einstein condensate (THSR). Theory predicts such states in heavier systems such as 16O, or even 40Ca - the challenge remains to identify them experimentally.

One of the great advances in accelerator based nuclear physics in the 1960s was the development of heavy-ion beams. One of the first systems to be characterised was 12C+12C. These measurements involved varying the energy of incident beam energy and observing the reaction cross section. Remarkably, rather than a smooth variation a series of resonances were observed. The width of these resonances were ~100 keV, indicating the formation of a 24Mg intermediate system with a lifetime significantly longer than the nuclear crossing time. These resonances were subsequently interpreted as 12C+12C cluster states (Erb 1985). Since the 1960s there have been numerous attempts to characterise the nature of the 12C+12C resonances, either directly as in the original measurements, or indirectly. Indirect approaches would include reactions such as 12C(16O,24Mg[12C+12C])4He. Here the 24Mg nucleus is formed in a nuclear reaction and then subsequently decays into two 12C nuclei - in essence the time reverse of 12C+12C scattering measurements. The advantage of this latter approach would be that a range of cluster states can be accessed without the need for a time consuming scan of the beam energies. The type of excitation energy spectra that can be accessed using this technique are shown in Figure 7. The peaks in the spectra correspond to resonances (states) in 24Mg, which can be seen to range from excitation energies of ~20 to 60 MeV. States decaying to the 12C+12C final state should have a large cluster content. Using an experimental technique in which it is possible to measure the emission angles of the decay products the spins, J, of the 24Mg excited states can be established. These are plotted in Figure 7b, where the horizontal axis is J(J+1). The 12C+12C breakup states fall on a linear locus projecting back to an excitation energy of ~ 20 MeV for zero spin. The gradient of the slope is found to be significantly less than for 24Mg states associated with the ground state. For a rotational structure the energies of the states are characterised by \( E=\hbar^2/2\Theta J(J+1)\ .\) Thus, from the experimental data it is possible to extract the corresponding moment of inertia. Figure 8 illustrates the possible arrangement of the two 12C nuclei when they interact to form the 24Mg excited states - the moment of inertia of this structure is consistent with that found experimentally. In fact, the symmetry, and oblate deformation, of the 12C nuclei in their ground states reflects their overlap with a triangular arrangement of 3 \(\alpha\)-particles. Hence, the 24Mg excited states are likely to possess a configuration of six alpha-particles. Similar large scale cluster structures have been identified in systems such as 28Si(12C+16O) and 32S(16O+16O) - consistent with both the Ikeda and Harmonic Oscillator predictions.

Beyond these examples, many of the systems predicted in the Ikeda diagram ( Figure 1) have been experimentally observed. However, the experimental challenge lies in finding definitive evidence for the \(N\alpha\) systems in nuclei heavier than 12C, e.g. in 16O and 20Ne, etc.... There are a few classic examples of cluster structure in such systems that are well studied. The ground-state of 20Ne is one such case. The properties are strongly influenced by an alpha+16O cluster structure. This is intrinsically mass asymmetric and hence cannot be described in terms of either positive and negative parity states alone. Instead, linear combinations of positive and negative parity states are required. The rotational states are then associated with two rotational bands, one with positive parity and the other with negative parity. The energy splitting between the two components has been interpreted in terms of the probability for the \(\alpha\)-particle to tunnel between the two sides of the 16O-core. This is analogous to the NH3, ammonia, molecule which possesses a similar asymmetric structure (see Horiuchi 1968). Italic text

Nuclear Molecules

Figure 9: The formation of molecular orbitals in Beryllium isotopes. The red spheres (left images) illustrate the location of the alpha-particles and the blue dumbbell-like regions the densities associated with the neutron-orbitals. The neutron-orbitals can either be arranged parallel or perpendicular to the separation axis. Linear combinations of the p-orbitals produce \(\pi\) and \(\sigma\) molecular orbitals (right two images).
Figure 10: Rotational bands in the nucleus 9Be(and 8Be). The two bands associated with the 3/2- and 1/2+ states have molecular character.
Figure 11: Modified Ikeda diagram showing molecular structures and the associated excitation energy.
Figure 12: Molecular orbitals for three centre (a-c) and four centre (d-g) systems. The molecular orbitals are formed linear combinations of p-states aligned perpendicular to the separation axis - and so have pi-type character. The lowest energy configurations are the ones with the least nodes along the horizontal axis. The black dots indicate the alpha-particle centres.

The above examples of cluster structures involve what are termed alpha-conjugate nuclei, i.e. ones which can be decomposed into alpha-particle subunits. An interesting question is what happens to such systems when valence particles (either protons or neutrons) are added. In this instance there exists the possibility that the valence particles - most typically neutrons - may be exchanged between the alpha-particle cores. This type of exchange process is completely analogous to the exchange of electrons in atomic molecules. As in atomic systems, where the covalent exchange of an electron binds two protons into the H\(^-_2\) molecule, in nuclear systems the exchange of a neutron between two alpha-particles binds the nucleus 9Be in the ground-state. In the nuclear case the nature of the covalent orbits is closer to that of O2 where the molecular orbitals are constructed from linear combinations of p-orbitals. The two protons and two neutrons of an \(\alpha\)-particle fill the shell-model 1s\(_{1/2}\) orbital, the next neutron lies in the p-shell. Correspondingly, 5He has a ground-state spin and parity of 3/2-. Figure 2 shows the types of molecular orbital which may be formed with two 4He cores and valence neutrons. The p-orbitals can be spatially aligned either perpendicular or parallel to the separation axis of the two alpha-particles. The former gives rise to \(\pi\)-type molecular orbitals and the latter to \(\sigma\)-type orbitals. The exchange of the neutron binds the two alpha-particles (9Be is bound in nature, whereas 8Be is not). The behaviour of molecular structures in nuclei has been explored extensively by W. von Oertzen and co workers (see von Oertzen 1996, von Oertzen 2001, von Oertzen 2004 and other references in Freer 2007).

This model predicts that the nucleus 9Be should have a ground state with spin and parity 3/2- and a low-lying 1/2+ excited state corresponding to the \(\pi\) and \(\sigma\)-type orbitals, respectively. Indeed the ground-state of 9Be has \(J^\pi\)=3/2- and the 1/2+ lies at 1.68 MeV. The deformed nature of the corresponding molecular structures implies that they should behave rotationally, which is again experimentally confirmed. From the rotational bands it is possible to extract moments of inertia for the two configurations, which are consistent with the molecular interpretation. The experimental situation is shown in Figure 10. The rotational band with \(\pi\) character -associated with the ground-state - is shown by the solid circles. The deformation of this band is given by \(\hbar^2/2\Theta\)=0.525, which is similar to that of the two alpha-particle system 8Be; \(\hbar^2/2\Theta\)=0.48. The rotational band linked with the \(\sigma\) molecular orbital has a staggering effect which arises from Corriolis de-coupling. Nevertheless, the extracted value of \(\hbar^2/2\Theta\) is 0.386. This indicates a much larger moment of inertia than for the \(\pi\) configuration, which may be understood from the nature of the \(\sigma\)-orbital which resides between the two alpha-particle cores, increasing their separation. Similar structures have been found in other two-centre systems such as 10Be and 11Be. There is also good evidence for molecular behaviour in systems in which one of the alpha-particle cores is replaced with a 16O cluster - as in the neon isotopes studied by von Oertzen (von Oertzen 2001). Indeed the mass asymmetric structure of 20Ne (16O+\(\alpha\)) plays a very important role in determining the behaviour of molecular orbitals in nuclei such as 21Ne and 22Ne. Here the mass asymmetry gives rise to parity doublets in the molecular spectrum. In the case of alpha-particle and 16O cores the molecular orbitals are hybridized orbitals built from the p-orbital around the alpha-particle and the d-orbital around 16O (see von Oertzen 2001). Figure 11 shows a modified Ikeda diagram illustrating the variety of molecular structures which may exist, many of which have already been observed. The mass asymmetry which appears in the case of structures involving molecules formed from different cores also plays a role in systems composed of three alpha-particles. In the nucleus 13C, for example, with the three alpha-particles arranged linearly, then the valence neutron would reside between the centre alpha-particle and one other - generating the mass asymmetry. As with 20Ne this gives rise to two rotational bands with identical moments of inertia but with opposite parity. The energy splitting between the two bands being representative of the probability for the neutron to tunnel through the central alpha-particle. The associated molecular orbitals are shown in Figure 12a-c. The possibility of extending molecular structures from dimers to trimers has been investigated in detail - see Itagaki 2001 and Milin 2002. Here the neutrons would be exchanged between the three centres (alpha-particles). Examples of the possible molecular orbitals for two, three and four centre systems formed from the linear combinations of p-orbitals are shown in Figure 12. Here the neutrons would be exchanged between the N-centres (see McEwan 2004) . It is possible that the three alpha-particles could align themselves in a linear fashion, or alternative collapse into a triangular arrangement - in either case the neutrons being delocalised across the three centres. At present there is limited experimental evidence for such structures in 13C and 14C, but more experimental data is required. Possibly the best case for the linear arrangement - from a theoretical perspective is 16C, Itagaki 2001, though in this instance experimental data is very sparse.

Summary and Outlook

Figure 13: Pictorial representation of the energetic advantage of clustering for nuclear matter at the neutron drip-line in order that the valence neutrons (red) can maximise their interaction with the core nucleons (blue).


The structure of light nuclei is in no small part driven by the influence of correlations. These often manifest themselves as clusters. In systems with equal and even numbers of protons and neutrons alpha-particle clustering is the favoured mode. In neutron-rich nuclei, close to decay thresholds, then molecular type structures emerge - where neutrons are exchanged between the alpha-particle cores. The challenge is to demonstrate that it is possible to form 3-centre molecules, or even more complex systems. The limits of nuclear stability (the neutron drip-line) may be probed only for the lightest of nuclei. Already it is known that at this extreme that nuclei form themselves into a compact core embedded into a diffuse cloud of neutrons - so-called neutron haloes. This is a form of clusterisation. It is possible that this may be a more general form of matter at the neutron drip-line; namely clusters, or cores, of normal nuclear matter embedded in a sea of valence (delocalised) neutrons exchanged between the cores; Figure 13. This is a conjecture which remains to be tested.

References

  • Peacock, J A (2001). A measurement of the cosmological mass density from clustering in the 2dF Galaxy Redshift Survey. Nature 410: 169. doi:10.1038/35065528.
  • Morinaga, H (1956). Interpretation of Some of the Excited States of 4n Self-Conjugate Nuclei Phys. Rev. 101: 254. doi:10.1103/physrev.101.254.
  • Ikeda, K; Tagikawa, N and Horiuchi, H (1968). The Ikeda Diagram. Prog. Theo. Phys. Suppl. extra number: 464.
  • Brink, D M (1966). Alpha Cluster Model. Proceedings of the International School of Physics Enrico Fermi, Varenna Course 36: 247.
  • Rae, W D M (1966). Deformed Magic Numbers Proceedings of the Fifth International Conference on Clustering Aspects in Nuclear and Subnuclear Systems, Kyoto, Japan, Phys. Society of Japan K. Ikeda (Ed.): 80.
  • Freer, M (2007). The clustered nucleus—cluster structures in stable and unstable nuclei Reports on Progress in Physics 70: 2149. doi:10.1088/0034-4885/70/12/r03.
  • Freer, M and Merchant, A C (1997). Developments in the study of nuclear clustering in light even - even nuclei J. Phys. G 23: 261. doi:10.1088/0954-3899/23/3/002.
  • Wiringa, R B; Pieper, S C and Carlson, J (2000). Quantum Monte Carlo calculations of A equals 8 nuclei. Phys. Rev. C 62: 014001. doi:10.1103/physrevc.62.014001.
  • Kanada En'yo, Y and Horiuchi, H (2001). Structure of Light Unstable Nuclei Studied with Antisymmetrized Molecular Dynamics Progress of Theoretical Physics Suppl. 142: 205. doi:10.1143/ptps.142.205.
  • Chernykh, M; Feldmeier, H; Neff, T; von Neumann-Cosel, P and Richter, A (2007). Structure of the Hoyle State in 12C Phys. Rev. Lett. 98: 032501. doi:10.1103/physrevlett.98.032501.
  • Hoyle, F; Dunbar, D N F and Wenzel, W A (1953). Prediction of the 7.65 MeV state Phys. Rev. 92: 1095.
  • Erb, K A and Bromley, D A (1985). Heavy-Ion Resonances Treatise on Heavy Ion Physics ed. D. A. Bromley (Plenum Press, New York, 1985): vol. III.
  • Horiuchi, H and Ikeda, K (1968). A Molecule-like Structure in Atomic Nuclei of 16O* and 10Ne Prog. Theor. Phys. A 40: 277. doi:10.1143/ptp.40.277.
  • von Oertzen, W (2001). Covalently bound molecular structures in the alpha+16O system Eur. Phys. J. A 11: 403. doi:10.1007/s100500170052.
  • McEwan, P and Freer, M (2004). Characterization of molecular structures in the deformed harmonic oscillator J. Phys. G 30: 447. doi:10.1088/0954-3899/30/4/005.

Internal references

  • Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
  • Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.


Further Reading

Introductory articles:

  • Fulton, B R (1999). Clustering in nuclei: nuclear chains, nuclear molecules and other exotic states of nuclear matter. Contemporary Physics 40: 299. doi:10.1080/001075199181378.
  • Appell, D (1999). Of dumbbells and doughnuts. New Scientist 2184: 35.

Other Review Articles:

  • Wildermuth K and McClure W 1966 Cluster Representations of Nuclei (Springer Tracts in Modern Physics 41) (Berlin: Springer)
  • Wildermuth K and Tang Y C 1977 A Unified Theory of the Nucleus (New York: Academic)
  • Arima A, Horiuchi H, Kubodera K and Takigawa N 1972 Clustering in Light Nuclei Adv. Nucl. Phys. 5 345
  • Tang Y C, LeMere M and Thompson D R 1978 Resonating-group method for nuclear many-body problems Phys. Rep. 47 167
  • Langanke K and Friedrich H 1987 Microscopic Description of Nucleus-Nucleus Collisions Adv. Nucl. Phys. 17 223
  • Langanke K 1994 The Third Generation of Nuclear Physics with the Microscopic Cluster Model Adv. Nucl. Phys. 21 85
  • Furutani H, Kanada H, Kaneko T, Nagata S, Nishioka H, Okabe S, Saito S, Sakuda T and Seya M 1980 Study of Non-Alpha-Nuclei Based on the Viewpoint of Cluster Correlations Prog. Theor. Phys. Suppl. 68 193
  • Freer M and Merchant A C 1997 Developments in the study of nuclear clustering in light even - even nuclei J. Phys. G 23 261
  • von Oertzen W, Freer M and Kanada En'yo Y 2006 Nuclear clusters and nuclear molecules Phys. Rep. 432 43
  • Greiner W, Park J Y and Scheid W 1995 Nuclear Molecules (Singapore: World Scientific)
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