For large times the vacuum is reached, i. Again the vacuum is reached for large times t? Just like in the case of a number state we see that the resulting density matrix evolves from a pure state to an statistical mixture. We can use the series representation of the Wigner function 76 , to calculate the Wigner function and look at the dynamics of the decay of the cat. This decoherence has been observed experimentally by Brune et al.
In Section 6 we will ask ourselves about the possibility to reconstruct quasiprobability distribution functions despite such bad effects. Fokker—Planck and Langevin equations Before applying superoperator methods in the solution of the above equation, let us show how it may be casted into a Fokker—Planck equation. The Fokker—Planck equation can be also obtained from the Kramers—Moyal [60,61] expansion, a Langevin equation that does not stop after second order derivatives see for instance .
Solution to the master equation The formal solution of Eq. A non-zero entropy then describes additional uncertainties above the inherent quantum uncertainties that already exist. Let us for simplicity study the atomic 26 H. Both function are shown to have the same qualitative form. Here we follow Phoenix , to perform the calculation of the entropy. It may be seen that immediately after the interaction with the environment, the cat state losses it purity going to a statistical mixture of coherent states see Fig.
It treats the possibility of obtaining complete information of a quantum state by means of quasiprobability distribution functions. There are several methods to achieve such reconstruction either in ideal or lossy cavities. We go on to study a more direct reconstruction in an ideal cavity, through the interaction with atoms, and then combine both situations, i. This procedure will enable us to obtain information about all the elements of the initial density matrix from the diagonal elements of the time-evolved displaced density matrix only. As diagonal elements decay much slower than off-diagonal ones, information about the initial state stored this way becomes robust enough to withstand the decoherence process.
We will now show how this robustness can be used to obtain the Wigner function of the initial state after it has started to decay. Experimental setup to produce superposition of atomic states. Atoms leave the oven in excited states, pass through a Ramsey zone to produce the proper superposition. This is the main result of our paper; the reconstruction is made possible even under the normally destructive action of dissipation. We would like to stress that the identity in Eq. We can use the set up of Fig. Reconstruction in an ideal cavity We now show two methods to realize the quantum state reconstruction in ideal cavities.
Direct measurement of the Wigner function Let us look again at Eq. Fresnel approach Another possibility to reconstruct quasiprobability distributions, is the so-called Fresnel representation of the Wigner function see  , in which it is used the fact that, if we integrate Fresnel transform Eq.
We show in Fig. This is studied next. Note that in the above equation we are dividing by a superoperator, however this is not a problem because the numerator we have a function of the same operator which, if developed in a Taylor series, has powers of this operator and cancels it. However, one can determine completely the state by noting that an s-parametrized quasiprobability may be reconstructed exactly.
Review articleFull text access. Dynamical correlations. Umberto Balucani, M. Howard Lee, Valerio Tognetti. Pages Download PDF. Article preview. Read the latest articles of Physics Reports at ykoketomel.ml, Elsevier's leading platform of peer-reviewed scholarly literature. Volume , Issue 6.
One may see that the reconstructed quasiprobability distribution is not as negative as the Wigner function, Fig. Of course, for greater values of the decay parameter the effect would be stronger. However, even in the case of dissipation one would measure a negative quasiprobability distribution, but should be stressed that not the Wigner function. Usually these equations are solved by transforming them to Fokker—Planck equations  which are partial differential equations for quasiprobability distribution functions typically the Glauber—Sudarshan Pfunction and the Husimi Q-function.
Another usual approach is to solve system-environment problems is through the use of Langevin equations, this is stochastic differential equations that are equivalent to the Fokker—Planck equation . We can have a different approach to the solution by again using superoperators. This process is known as parametric down conversion.
It is generally accepted that nonclassical effects emerge as a consequence of quantum interference, therefore the decay of quantum coherences results gives rise to the deterioration of nonclassical effects. By expanding Eq. Schneider and Milburn  have solved Eq. We now solve Eq.
The deterioration of revivals may be seen in Figs. Conclusions Through this report we have solved master equations for different quantum optical systems by using superoperator techniques. We have shown in all those cases that it is possible to solve them by using superoperator methods rather than translating them into partial differential Fokker—Planck or Stochastic differential Langevin equations. Acknowledgments I would like to thank useful discussions with A.
Klimov, R. References  A. Caldeira, A. Leggett, Phys. A 31 Zurek, Phys. Today 36 Unruh, W. D 40 ; in: J. Wheeler, W. Zurek Eds. Barnett, P. Knight, Phys. A 33 Jaynes, F. Cummings, Proc. IEEE 51 Eberly, N. Narozhny, J. Dutra, H. Moya-Cessa, P. A Lutterbach, L.
Davidovich, Phys. Lougovsky, E. Solano, Z. Zhang, H. Walther, H. Mack, P. Schleich, Phys. Dutra, Eur. Milburn, Phys. A 48 Wigner, Phys. Hillery, R. Scully, E. Allen, J. Klimov, L. A 61 Susskind, J. Glogower, Physics 1 49; See also R. Lynch, Phys. Glauber, Phys. Dodonov, I. Malkin, V. Yurke, D. Stoler, Phys. Stayanarayana, P. Rice, R. Vyas, H. Carmichael, J. B 6 Cirac, L. A 40 Milburn, J. Moya-Cessa, A. Vidiella-Barranco, J. Fleischauer, W.
A 47 Yuen, Phys. Loudon, P. Knight, J. Optics 34 Abramowitz, I. Gradshteyn, I. Gea-Banacloche, Phys. A 44 Knight, S. Phoenix, Phys. A 45 Rempe, F. Schmidt-Kaler, H. Walther, Phys. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J.
Raimond, S. Haroche, Phys. Vidiella-Barranco, V. Moya-Cessa, J. Filipowicz, J. Javanainen, P. Meystre, Phys. A 34 Rempe, H. Walther, N. Klein, Phys. Wineland, C. Monroe, W. Itano, D. Leibfried, B. King, D.
Meekhof, J. Mancini, D. Vitali, H. Moya-Cessa, Phys. B 71 Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. Raimond, M. Brune, S. Haroche, Rev. Kim, P. Knight, V. Buzek, Phys. A 41 Meystre, M. Lovsky, S. Babichev, Phys. A 66 Leibfried, D. Meekhof, B. King, C. Wineland, Phys. Walls, G. Sudarshan, Phys. Husimi, Proc. Stone, Pac. Alicki, K. Brune, E. Dreyer, X.
Maitre, A. Maali, C. Wunderlich, J. Scully, M. Gardiner, in: Handbook of Stochastic Methods, 2nd ed. Kramers, Physica 7 Moyal, J. Araki, E. Lieb, Commun. Phoenix, P. Knight, Ann. A 68 ; R. Moya-Cessa, I. A N. Milburn, C. Holmes, Phys. Gardiner, M. Collet, Phys. Kim, V. Zubairy, Opt. Vidiella-Barranco, L. Moya-Cessa, Int. Golden, Phys. A 46 Moya-Cessa, V. Schneider, G. Rev A 57 States close to the decay thresholds are of particular interest, as clustering becomes dominant. Recent studies of loosely bound light neutron-rich nuclei have focused attention on structures based on clusters and additional valence neutrons, which give rise to covalent molecular binding effects.
These nuclear molecules appear only at the extremes of deformation, in the deformed shell model they are referred to as super- and hyper-deformed. Further nuclear molecules consisting of unequal cores and also with three centres can be considered. These arise in the isotopes of neon and carbon, respectively. Examples of recent experiments demonstrating the molecular structure of the rotational bands in beryllium isotopes are presented. Further experimental evidence for bands as parity doublets in nuclei with valence neutrons in molecular orbits is also analysed.
Work on chain states nuclear polymers in the carbon isotopes is discussed. Clustering in neutron-rich nuclei. Clusters and valence nucleons in light nuclei. Clustering and deformed shells. Valence nucleons and nuclear molecules. Cluster collisions and molecular potentials. Nucleus—nucleus potentials, relation to threshold diagrams. The nucleon exchange potential, elastic transfer. Nuclear molecular orbitals and the two-centre shell model. Hybridisation and Coriolis couplings. Chain states. Resonant structure in 24 Mg. Theoretical approaches, recent developments.
Overview of microscopic cluster models. Bloch—Brink alpha-cluster model. Generator coordinate method. Antisymmetrized molecular dynamics. Clusters of different size. Covalent binding for asymmetric systems. Experimental results for symmetric two-centre systems. The structure of beryllium isotopes—complete spectroscopy. Electromagnetic decay properties in beryllium isotopes. Models for three-centre systems.
Chain states in nuclei: nuclear polymers. Cluster states of triangular shapes. Results for three-centre systems. Oblate and prolate states in carbon isotopes. Complete spectroscopy in 13 C. Cluster states in 14 C. Complete spectroscopy for 14 C. The proposed oblate rotational bands. The proposed prolate rotational bands.
Future perspectives. Clustering at the drip line. Molecular structures in heavier nuclei. Introduction 1. In nuclear physics, clusterisation enhances, in certain circumstances, the binding energy of the system. The binding energy per nucleon W. Cluster structures are predicted to appear close to the associated decay thresholds. These energies needed for the decomposition of the normal nucleus into the structures are indicated in MeV, adapted from . The Ikeda diagram [,] is shown in Fig. The clear prediction, which is borne out experimentally, is that cluster structures are mainly found close to cluster decay thresholds.
Under the assumption of spherical symmetry this gives rise to the nuclear shell structure.
Importantly, clustering gives rise to states in light nuclei which are not reproduced by the shell model. The nuclear shell model does, however, play an important role in the emergence of nuclear clusters, and also in the description of special deformed nuclear shapes, which are stabilised by the quantal effects of the many-body system, namely the deformed shell gaps as opposed to the spherical shell gaps.
This connection is illustrated by the behaviour of the energy levels in the deformed harmonic oscillator , shown in Fig. The numbers in the circles correspond to the number of nucleons, which can be placed into the crossing points of orbits. At zero deformation there is the familiar sequence of magic numbers which would be associated with spherical shell closures, and the associated degeneracies.
At a deformation of the potential, where the ratios of the axes are , these same magic degeneracies reappear, but are repeated twice. This establishes an explicit link between deformed shell closures and clustering. Energy levels of the deformed axially symmetric harmonic oscillator as a function of the quadrupole deformation oblate and prolate, i.
Degeneracies appear due to crossings of orbits at certain ratios of the length of the long axis the symmetry axis to the short perpendicular axis. This concept, fundamental for the understanding of the appearance of clustering within the nucleus, has been discussed before in detail [3,54,,,] see also Section 2. In particular the work of Rae was seminal in crystallising the discussion.
To illustrate this point, we show in Table 1 the compilation made by Rae , following an examination of the properties of the deformed harmonic oscillator as shown in Fig. For example, for super-deformed structures the magic numbers have a decomposition into two magic numbers, of two spherical clusters, e. These structures give rise to not only rotational bands, but also to exotic vibrational modes, e.
In the present review, it will become evident that additional valence neutrons do not destroy these structures, instead interesting nuclear structures described by molecular concepts will emerge. They are then related to octupole shapes [3,54,]. The octupole deformations give rise to the observation of rotational bands with parity inversion doublets [46,54].
Clustering in neutron-rich nuclei The interest in nuclear clustering has been pushed strongly due to the study of neutron-rich and of exotic weaklybound nuclei. Valence neutrons can exist in molecular orbitals, their role becomes analogous to that of electrons in covalent bonds in atomic molecules. In the nuclear case, these covalent neutrons stabilize the unstable multi-cluster states. The form of the covalent orbits for p-states is illustrated in Fig.
These concepts can also be used to describe the exchange of valence neutrons between cluster cores on the nuclear scale. A new threshold diagram is required, in order to describe the structure of non-alpha conjugate nuclei, i. The states close to the thresholds for the decomposition 48 W. Here the z-direction is aligned with the separation axis of the two centres indicated by the black dots.
This extended Ikeda-diagram appears in Fig. The relevant threshold energies for the decomposition into the constituents [,] are given. This review concentrates on recent work on the structure of excited states in light nuclei related to molecular structures consisting of clusters and valence nucleons. It is organised as follows. In Section 2 we give a historical introduction.
In Section 3 the important question of the nucleus—nucleus potentials is addressed, the potentials have special properties for strongly bound clusters and the presence of valence neutrons gives important effects. This is extended to three centre systems in carbon isotopes in Section 7 and in Section 8. Section 9 presents perspectives for future research in weakly bound exotic nuclei. Clusters and valence nucleons in light nuclei In this Section some basic principles of molecular concepts in nuclear physics are shown. Extended threshold diagram as in Fig. Some molecular structures with clusters and covalent valence neutrons are shown.
The schematic shapes are given with the threshold energies for the decomposition into the constituents. Following the work of Bethe et al. Such a description produced good agreement with the binding energies of the so-called saturated nuclei.
These binding energy systematics are shown in Fig. Their ideas were based upon the structure of the 5 He nucleus in which the last neutron was in a p-orbit. The binding energy of 5 He with respect to the neutron and 50 W. The plot shows the change in binding energy versus the number of bonds in an alpha-particle model.
For example, 8 Be has 1 bond, 12 C—3, 16 O—6, 20 Ne—9, etc. The two-centre wave functions for the system, 9 Be, are obtained by linear combinations of the single centre states see also the more detailed discussion in Section 3. These systematics agreed remarkably well with the available experimental data. Since the s the ideas underpinning the appearance of clustering have evolved considerably.
Clustering and deformed shells We resume the discussion of the relationship between the shell model and the cluster model. This is done on the basis of the harmonic oscillator HO , which has long been the basis of many calculations of the properties of nuclei. Most relevant for systems described here is the deformed harmonic oscillator important in the deformed shell model, or the Nilsson model .
The large degree of degeneracy present in the spherical oscillator is lost as the potential is deformed. This decrease in degeneracy reduces the number of correlated particles and thus the stability of the system. However, at a deformation represented by the axis ratio , and similarly for , etc. Thus, at the deformed shell closures quasi-stable structures appear. At the degeneracies of the spherical oscillator are repeated twice i. This fact was already mentioned in the introduction, where it was shown that the degeneracy pattern may be interpreted in terms of clusterisation Table 1.
This observation was formulated more mathematically by Nazarewicz and Dobaczewski . They demonstrated, using group theory, that for deformations with axis ratios n:1, there was a decomposition into n groups. Thus, it is possible to view the deformed harmonic oscillator at a deformation with the ratio , as consisting of two degenerate harmonic oscillator potentials. Similarly for deformations corresponding to axis ratios of , three identical potentials are superimposed.
By the addition of a microscopically derived shell correction term to a liquid drop energy, a measure of the potential energy of the nucleus can be determined as a function of deformation. This scheme is called the Nilsson—Strutinsky NS method [47,,]. This study found a series of deformed secondary minima for nuclei such as 24 Mg.
The same approach was used by Ragnarsson, Nilsson and Sheline [,] to predict a whole series of new magic numbers which coincide with a variety of cluster sub-structures. The underlying link between cluster structure and the deformed HO is a strong indication for the existence of clustering in the deformed minima. The separation of the cluster-like components within the deformed HO at a deformation of can be performed within the framework of the double-centre oscillator or the two-centre shell model .
This description allows the evolution of the quantum numbers from separate clusters to a fused system to be traced. The two linear combinations produce symmetric and asymmetric states. This conserves the number of nodes along the axis of separation in the original W. Minima are found at particular deformations. Energy levels in a double centre oscillator. For large separations the associated nz quantum numbers in the two potentials are degenerate. This may readily be extended to an arbitrary number of clusters or potentials fused along a single axis, i.
This extension of the deformed harmonic oscillator permits the decay properties of a predetermined cluster state to be described. Similarly, for the formation of chain states in nuclear reactions, where particular orbits must be populated, the Harvey rules have to be considered. These rules are equivalent to the fractional parentage rules in multi-nucleon transfer reactions. It should be noted that the correspondence found by Harvey are exactly those contained within the SU3 description explored by Nazarewicz and Dobaczewski .
The algebraic features of the SU3 are also at the basis of the RGM- models for molecular resonances and the work of Hess et al. Valence nucleons and nuclear molecules The deformed harmonic oscillator DHO framework also provides a useful starting point for the description of nuclear molecules with valence neutrons. A valence neutron can then occupy the 1,0 or 0,2 orbits. The wave-functions of these orbits have a strong overlap with those shown earlier in Figs.
In other words these two DHO orbits contain molecular properties. To illustrate these points Fig. The orbits with maximum number of quanta on the Z-axis gain energy if the system becomes more elongated. Thus, the ideas developed in terms of the two-centre shell model, or the double-centred oscillator, are useful in developing molecular wave functions. Energy diagram of orbitals of nucleons in an axially deformed harmonic oscillator, adapted from .
The open dots mark the positions of valence nucleons, they lie on the perpendicular orbitals x or y. For the two-centre system the energy E is obtained using a variational method see also Ref. This is analogous to the analysis performed by Hafstad and Teller . This latter condition is often used for more complex systems with many centres and restricts the neutron exchange to be with nearest neighbours only.
This process can be repeated for more complex nuclei. Cluster collisions and molecular potentials 3. Nucleus—nucleus potentials, relation to threshold diagrams There is a strong relationship between clustering, molecular resonances and the properties of the nucleus—nucleus potentials. In the Hauser—Feshbach HF picture, with a binary channel consisting of two clusters, the formation or decay of resonances is governed by the real and imaginary parts of the optical potential.
The imaginary part is responsible for the width or the life time of the cluster resonances. In part this was realized in the early stages of heavy-ion scattering by Feshbach , see also Ref. In the early work mainly shallow potentials have been adopted in order to obtain resonant behaviour. In addition, the observation of resonant molecular structure at the thresholds can be related to a small imaginary part. The latter is mainly observed for systems consisting of strongly bound clusters.
The aim is to describe both bound and resonant states in the same potential. In the original Ikeda diagram the 12 C nucleus was included. Thus, 12 C may not be treated as an inert cluster. As stated, for the imaginary potential surface transparency was required, which is well accounted for by the model of number of open channels NOC by Haas and Abe .
The strongly-bound clusters giving the smallest NOC. This model was further developed and applied to the decay of various binary decays [31,32]. The centrifugal potential for larger angular momenta gives a repulsive effect at smaller distances, and effective potentials with a pocket result.
This approach is contrasted by the more recent work based on the double folding model. When applied to the scattering of 4 He and of 16 O nuclei, it gave very deep real potentials [—,]. This phenomenon is generally observed for combinations of strongly bound clusters [—,], for which all reaction Q-values are very negative, giving a weak imaginary potential. We repeat here the basic facts of rainbow scattering [48,,]. The rainbow structure appears if the real nuclear potential is strong enough, i.
Such trajectories are shown in Fig. In the region of the rainbow angle several partial waves contribute to the same angular region and a particular Fig. The maximum negative angle corresponds to the rainbow angle R , see also Fig. These patterns are referred to as Airy-structures [48,]. More recent analysis of the elastic scattering of 4 He and 16 O projectiles on nuclei was based on the more advanced double folding model. In this approach very deep potentials are obtained, which are able to describe all details of the differential cross sections at many energies [—,] and over a large angular range covering many orders of magnitude.
This self-consistent double-folding model uses an effective density dependent nucleon—nucleon interaction adjusted to the properties of nuclear matter [—]. This gives a potential which is very deep in accordance with the model independent analysis of the experimental results. These studies allowed the determination of the nucleus—nucleus potential over a large range of the inter-nuclear distances. Some of the data are shown in Fig.
These correspondingly higher order maxima and minima are referred to as 2nd and 3rd-order Airy-structures, respectively. In contrast to this, the strong regular maxima and minima in these data were formerly interpreted as resonant molecular structures [91,]. The self-consistent double-folding model appears to be valid down to rather low energies, where the Pauli principle would inhibit the penetration of the two clusters.
A repulsion in the potential energies has been proposed to result . This validity of the approach can be understood because the effective relative momentum of nucleons in the overlap region remains quite large for the deep potentials. This effect of self consistency strongly reduces the effect of the Pauli principle, as described by Subotin et al.
The deep potentials have many bound states which are not physically relevant, and are not allowed by the Pauli principle. This problem was solved by D. The resulting potentials can be considered to be of molecular type. The result is a potential with a shallow attractive part and a strong repulsion at small distances. These are shown in Fig. For the earlier systematic work on resonances e. For the lowest energies there is still the concept of super-deformed molecular states in 32 S, consisting of two 16 Onuclei which has been described by Ohkubo and Yamashita  with the deep potential model, and also by Kimura and Horiuchi  with the AMD method.
Actually, the local potentials between clusters, which are repulsive at small distances and otherwise weakly attractive, are needed for nuclei with additional neutrons in order to build covalently bound nuclear cluster structures which are discussed below. The nucleon exchange potential, elastic transfer Scattering of two nuclei consisting of two clusters and valence nucleons have played an important role in the development of the molecular orbital model of nucleons.
In the scattering systems with two nuclei differing only by 58 W. Due to the coherent superposition of the two amplitudes a pronounced interference structure is observed in the intermediate angular range in the angular distributions. Because the projectile and target nuclei are interchanged in this process, a backward rise in the elastic channel is observed.
Different partial waves are shown. Figure adapted from Ref. In the optical potential the effect of nucleon exchange can be described by the antisymmetrisation of the whole system, which gives rise to a parity dependent nucleus—nucleus potential [26,27,—]. For the description of such scattering systems the two-centre harmonic oscillator model and the GCM have been pursued by Baye and coworkers [4,26,27]. On the other hand, the approach based on the molecular orbital method for the valence particles has been developed by Imanishi and von Oertzen [,—].
This was extended later  to the method of rotating molecular orbitals RMO. The valence particle transfer between nuclei has also been treated in the two-centre shell model by Park et al. The parity dependence of the nucleus—nucleus potential can also be obtained in the coupled reaction channels approach in which various channels corresponding to the exchange of nucleons are included. In this approach, again via the introduction of local and phase equivalent potentials, the dependence on parity and on angular momentum appears.
In using such potentials for the cluster—cluster interaction the analysis of the elastic channel can directly predict the occurrence of parity-split bands in nuclei with a particular cluster structure as discussed by Baye . Nuclear molecular orbitals and the two-centre shell model We come back to a more detailed discussion of the wave functions of two-centre systems. The two-centre state with one valence particle, mentioned in Section 2. The text books on physical chemistry contain the details for these approaches, where the cores the atomic nuclei and the valence particles electrons are treated without antisymmetrisation in their region of overlap.
The method in atomic physics is called the linear combination of atomic orbitals LCAO , which has also been used in nuclear physics as the linear combination of nuclear orbitals LCNO for valence neutrons [,,]. The microscopic approach for the nuclear structure, using the LCNO, was extended and applied by Itagaki et al. The success in the description of the rotational bands, and the spin values where they terminate, may thus vary from model to model. We use coordinates as shown in Fig. The coordinates for the two-centre orbitals of neutrons as described in the text.
If we use as the coordinates Fig. This is borne out in the plot of the molecular orbitals in the correlation diagram shown in Fig. The amplitudes C1 and C2 will give a measure of the sharing of the neutron between the two asymptotic wave functions, e. For identical cores these amplitudes are identical. In the more general case, with two different centres, with different binding energies and quantum numbers, or for the same centres with different single particle states, the molecular wave function has different properties.
The two amplitudes Ci will become very different and an additional parity projection is needed. In this case the valence particle may concentrate at one centre and we could obtain ionic binding as in atomic physics, however, because the neutron carries no charge, this situation has no molecular binding properties in nuclear physics. The overlap of the single particle states determines the non-orthogonality, K,p R , and it depends on the distance R. The correlation diagram of molecular orbitals for a two-centre system .
The quantum numbers K, parity and the related gerade and ungerade g and u property for the various two-centre orbits are shown. Actually, the integral with the wave functions of Eq. The diagonalisation of all interactions in Eq. This method has been pursued in ion—atom collisions and for nuclear reactions in the approach of Park et al. These couplings produce two kinds of effects: the rotational Coriolis coupling, giving rise to 62 W. These correlation diagrams Fig. For the axial symmetry the quantum numbers are conserved for all distances and merge close to the united nucleus limit at small distances into the quantum numbers of the Nilsson model.
It comes thus as no surprise that the behaviour of the lowest orbits in correlation diagram in Fig. This is described by the two-centre shell model TCSM approach, which was implemented 30 years ago . The diagram shown in Fig. Placing nucleons into these orbitals one has to include the residual pairing interaction, which will change the two-centre system, once several valence nucleons are introduced.
The rotational band structure in 9 Be and the structure of other beryllium isotopes see Section 6. Hybridisation and Coriolis couplings Two important features of molecular physics appear in the elastic scattering of two nuclei with valence particles, including the elastic transfer of the valence particle.
Features, which are also important in the rotational bands of nuclei and for the covalently bound systems discussed below: i the hybridisation of valence particle orbitals [37,,,] and, ii the Coriolis coupling between states of different K-quantum numbers in the rotating frame [46,]. This gives rise to the distorted orbitals, the hybridized bonds which are responsible for binding in carbon polymers. In nuclear physics again the vicinity of the s-orbit and the p-orbit is observed in light nuclei, for example in carbon isotopes [,]. Most conspicuous is the case of 13 C.
The hybridisation effect is illustrated in Fig. At large distances With decreasing distance and with larger overlap the distortion sets in, creating a strong increase of the density of the valence neutron on the axis between the two centres. Because of the additional binding energy the latter shows a lower barrier if compared to the case without valence neutron.
The resulting distortion of the density of the valence particle along the line connecting the two centres is known to give rise to enhanced binding in atomic systems and can play also the same role in nuclear W. Left part: schematic illustration adapted from Ref. Right part: the potential between two 12 C-nuclei with and without—the higher of the two potentials the presence of the valence neutron.
There a strong enhancement of the density and of the transfer probability is observed . Two asymptotic channels, with the possible channel coordinates r1 or r2 , can be used . This result can also be understood in the usual coupled reaction channels CRC approach as arising from the combined coherent action of the transfer and inelastic interactions see Ref. The second effect, the Coriolis coupling, is a well known problem for rotating nuclei, but is particularly strong for valence particles in a two-centre system.
It arises from the coupling of the individual angular momentum of the valence particle with the collective motion of the core. It introduces in rotational bands a mixing of states with different K-quantum numbers. This leads to a staggering in the energy sequence of states in the bands see Fig. In order to understand this we must examine the interactions. The single-particle potentials at the two centres give rise to the bound states. But at the same time these interactions, Vn,C Ri , are responsible for the transfer from one core to the other and for the single particle excitation within one mass partition.
If the coordinates, shown in Fig. This arises as for the different mass splits two different asymptotic coordinates have to be chosen, which show the non-locality. In the RMO calculation a channel wave function of 64 W. The different CRC calculations show the increasing effect with the inclusion of more excited single particle states of 37 Cl. The rotational Coriolis coupling gives rise to K-mixing, and the radial couplings induce the transitions between individual channels, which are eigenstates of the two-centre system.
In the RMO-approach of Imanishi and von Oertzen the diagonalisation for the two-centre wave-functions is done including the Coriolis coupling, and the concept of RMO with mixed orbitals is obtained. The radial couplings remain, they become particularly strong at avoided crossings of two levels with the same quantum numbers, but belonging to different asymptotic states. The transitions connected to these avoided crossings are known as Landau—Zener transitions [,,]. Complete quantum mechanical models showed that such sharp resonances are not possible due non-local effects.
Thus subsequent experimental work  by the same group corrected this result and showed that the experimental resonances were due to a target contaminant. Subsequent W. The general expectation of smooth structures as function of energy is explained in Refs. In the following example this effect is illustrated directly with the single-particle states of a valence proton in 37 Cl. In this case the sd - and the fp -shells are involved, which will be the basis for mixed-parity orbitals. The proton transfer and the inelastic transfer and the inelastic scattering populate the single particle states in 37 Cl.
With the difference of one proton the role of the cores is exchanged in the transfer process. The intrinsic states in this case are two-centre states as mixed linear combinations of the valence nucleon orbitals in the single particle states of 37 Cl. The hybridisation of the ground state and the excited states in 37 Cl via transfer and inelastic excitations leads to an enhanced proton transfer probability in the ground state, see Fig.
The enhanced proton transfer cross section is only reproduced if all channels in particular the fp-shell are included in the calculation. For the complete 6-channel calculation the differential cross section is strongly enhanced. This can be understood with the increased density of the valence proton between the two centres induced due to mixing of orbitals differing in parity. The detailed study of this system [37,] showed that the data can only be reproduced if the mixing of all six individual orbits of the proton in excited states of 37 Cl is included.
The same distinct non-linear effect of the mixing interactions is also observed in the CRC-calculations . There it is also due to the combined action of inelastic transitions and the transfer interactions. Here, we record a few of the developments made in these areas.
Renewed interest 66 W. Intensive experimental and theoretical investigation ensued [58—60,,,,,,,,]. Actually a state in which the rotational members are energetically degenerate, owing to the large deformation, cannot be characterised uniquely by a single angular momentum but by shape alone.
Rather more mundanely the resonance was found to belong to a sequence of resonances extending to much higher energies see below. As an example, the 7. It would appear that in all likelihood that linear-chain structures do not exist and this is largely due to the instability against the bending mode, in which the chain collapses into a more compact object. Calculations by Itagaki et al. It is expected, as in the case of the beryllium isotope, 9 Be, that the addition of valence neutrons can stabilize the chain-like structures in the carbon and oxygen isotopes .
Experimental evidence for such structures is given in Section 8. Calculations using the molecular orbital approach appear to demonstrate that for particular mixed orbitals of neutrons with hybridisation in 14 C and 16 C a very distinct stabilisation of the bending mode  can be expected. We note that as in molecular science with atoms, that the covalently bound chains can also be stabilised by the centrifugal forces in their rotational bands. The most recent efforts have sought to expand the systematics of the resonances both in terms of decay channels and the centre-of-mass energy.
The corresponding excitation energy range of 44—64 MeV in 24 Mg is given, from . The inelastic channels leading to the excitation of particle-bound states had previously been measured by Cormier et al. The stability of these structures has been tested within the various models. This may be understood in terms of the coupling of the broad quasimolecular band—doorway states—to excited states of the scattering system. In the weak coupling picture the doorway states are associated with a pocket in the potential of the scattering system, and thus the two 12 C nuclei couple only weakly to 24 Mg.
Such ideas form the basis of the band-crossing model . In this description the resonances associated with the pocket in the scattering potential couple most strongly with the aligned inelastic molecular band. Such an approach provides a good description of the experimental data . In the preceding discussion, the broad resonances were associated with a secondary minimum in the 24 Mg deformed potential, and the alignment is associated with the intrinsic structure, and the decay channels populated would be those with a strong structural link, i.
The fragmentation is then generated by the coupling to states at smaller deformations just as predicted for the 32 S superdeformed case using the AMD framework . The latter values are extrapolated solid line to the cross-over with the cluster band larger circles. For example, the spectrum of decays of 24 Mg excited states sampled using this method [,] is shown in Fig.
References to peer-reviewed publications, theses and reports describing in detail and applying the methods implemented in the pyunicorn package. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D. In Physics Reports , vol. Costa, F. Rodrigues, G. Travieso, P.
Villas Boas. In Advances in Physics , vol. Watts and S. In Nature vol. In Physical Review E vol. In Physical Review Letters , vol. Arenas, A. Cabrales, A. In Social Networks , vol 27 no. Soffer and A. In Physical Review E , vol. Holme, S. Park, B. Kim, C. Tsonis, K. Swanson, G. Ueoka, T. Suzuki, T. Ikeguchi, Y. Donges, H. Schultz, N. Marwan, Y. Zou, J. Heitzig, J. Donges, Y. Zou, N. Marwan, J. Diploma thesis, Free University, Berlin Wiedermann, J. Donges, J. In Europhysics Letters , vol. Zemp, M.
Kurths, A. Rammig, J. Bretherton, C. Smith, J. In Journal of Climate , vol.
Diploma thesis, Humboldt University, Berlin Review of climate network analysis in Chinese! In Complex Systems and Complexity Science , vol. Tominski, J. Donges, T. Radebach, R. Donner, J. Runge, J. Tsonis and P. Swanson, P. In Bull. Gozolchiani, K. Yamasaki, O. Gazit, S. Tsonis and K. In Physical Review Letters vol , doi In Journal of Climate vol. Yamasaki, A. Gozolchiani, S. Diploma thesis, University of Potsdam Advisor: Prof.