High Spin Physics

Starting in the early 1980's fusion-evaporation reactions became the standard tool to populate very high spin states in nuclei that were not previously accessible to investigation. In these reactions a target is bombarded with heavy ions at an energy slightly above the Coulomb barrier, making possible the fusion of target and projectile. The compound system is formed with a large amount of excitation energy (50 to 70 MeV) and high angular momentum (typically ∼ 70ħ). High angular momentum states are reached, because the projectile tends to collide and fuse with the nucleus on a grazing trajectory, thus deposing its large momentum at a large distance from the center of the nucleus. The decay sequence of such a system is illustrated in figure H.1. At first this highly excited system cools down rapidly by evaporating a few neutrons and sometimes an additional proton or other light fragment. Once cooled below the separation energy of a neutron (about 8 MeV), the system cools down further by the emission of a few high energy "statistical" gamma rays. On the average very little angular momentum is removed from the system during this first, rapid cooling, period. The so called yrast line at the bottom of the figure is defined as the sequence of states with the lowest excitation energy for a given spin.

They form a rotational band whose excitation energy is dominated by the collective rotational energy. Somewhat above the yrast band are additional, excited, rotational bands. The high energy statistical gamma-ray transitions feed into these bands which subsequently dissipate the rotational energy and angular momentum via the emission of intra-band and inter-band gamma-rays.

These decay patterns can be studied via discrete gamma-ray spectroscopy. Figure H.2 shows the decay schemes established by the first experiments done by the Pittsburgh group on ⁷⁶Kr and ⁷⁸Kr (ref. H.1). The reactions used were ⁶³Cu(¹⁶O,p2n) Kr₄₀ and ⁶⁴Ni(¹⁸O,4n)⁷⁸Kr at a bombarding energy of 69 MeV and the data were obtained with the Pittsburgh multi-detector array (ref. H.2). This array consists of six Compton-suppressed, high purity Germanium gamma detectors (HPGe) and a Sum-Energy and Multiplicity Spectrometer (SMS) made up of 14 independent Bismuth Germanate (BGO) scintillation crystals.

The geometry of the Pitt array and a picture of it are shown in figures H.3 and H.4. Each HPGe's is a high energy resolution detector (≈ 2keV for gammas ∼ 1MeV) with a sixfold reduced background from Compton-scattered γ's. Decay schemes such as those in figure H.2 are obtained by detecting, simultaneously (within ∼ 20 nanosec.) two or more γ's. Figure H.5 shows a gamma spectrum showing the sum of all events in which two gammas were registered simultaneously. One sees that in addition to gamma lines from ⁷⁶Kr there are gamma lines from the isotopes ⁷⁶Br and ⁷³Se which were also produced in the fusion evaporation reaction. Figure H.6 shows a spectrum which contains only γ's in coincidence with the known 2⁺→ 0⁺ and/or 4⁺ → 2⁺ transitions of the ground-state band in ⁷⁶Kr. This restriction, by suppressing gammas from other bands and isotopes, enhances gamma lines that come from higher transitions (up to 22⁺ → 20⁺) of the ground-state band of ⁷⁶Kr.

Correlations between gamma ray energies are essential to the method, and the number of possible correlations increases rapidly with the number of detectors, rising to 15 for six detectors. Therefore it is important to have a large number of detectors. The decay of the compound nucleus will eventually feed into a state of the ground state band or an excited band at an initial spin I_i. If this is for example an I = 22⁺ state in of the ground state band, then I/2 = 11 gamma rays will be emitted in the decay to the ground state. Thus in the investigation of high spin states it is useful to select decays which start at a high spin and as a result emitt a large number of gammas. The SMS was used as a filter to select decays with large number of gammas. This is achieved by requiring that a given number of these detectors detect a gamma ray (of any energy) simultaneously with the detection gamma rays in two of the HPGe detectors. It should be noted that the 14 BGO-detectors of the SMS have poor energy resolution, but very high detection efficiency.

The study of decay schemes from fusion-evaporation reactions has greatly enhanced our understanding of nuclear structure. It has led to the discovery of collective rotational bands in regions beyond the rare earth and actinide regions. Most of the nuclei produced in fusion evaporation reactions are proton-rich radioactive isotopes, which at the time were not accessible to investigation by standard methods. The extreme high spin states can reach the limits of nuclear stability with respect to spin. The effort of the Pittsburgh group concentrated on the mass region 65 < A < 85. The following represents a thumb-nail sketch of the theoretical concepts and the interpretation of the experimental data in terms of these concepts, using as an example the results on ^{76, 78}Kr.

The collective rotation of a nucleus at high angular momentum (of say, ≈ 20 ħ) may typically correspond to an angular frequency, ω > 10²¹sec^-1. Although the lifetimes of such states are extremely short (∼ 10^-12sec.), the system may undergo some hundred million revolutions before making a transition to another state (in its decay to the ground state it will typically undergo more revolutions than the planet Earth has since its formation). The Coriolis and centrifugal forces in such a system are enormous, sometimes large enough to compete with the nuclear force. These inertial forces therefore serve as sensitive probes of the microscopic structures of these states.

Nuclear angular momentum can arise in two ways. The individual single particle orbitals carry intrinsic angular momentum. If several nucleons occupy orbitals with their individual angular momenta roughly aligned, a non-collective highspin state results. On the other hand, when a large fraction of the nucleons combine their contributions to the total angular momentum coherently, the resulting state is called collective. These two types of high angular momentum states are not completely distinct; rather they represent two extreme limits of nuclear rotation. The Coriolis and centrifugal forces which act in the "body-fixed" intrinsic frame of a rapidly rotating nucleus couple the microscopic (non-collective) and macroscopic (collective) degrees of freedom, allowing the study of the interplay of these two modes of excitation. As discussed in section "Coulex," collective rotation is intimately connected to the intrinsic shape of the nucleus.Because of the influence of the Coriolis and centrifugal forces on the intrinsic structure, the nuclear shape will depend on angular momentum.

The interpretation of decay schemes like those in fig. H.2 is based on the so-called Cranking Model. In this model the deformed nuclear potential is assumed to rotate at a fixed angular frequency ω around the x-axis of the laboratory frame of reference. The intrinsic structure of the deformed nucleus is then investigated in the rotating, body fixed frame of reference, taking into account the effect of the centrifugal and Coriolis interaction on the intrinsic structure.

If calculations are performed using a deformed one-body potential (for example a deformed Woods-Saxon potential) one obtains values for the moment of inertia that are close to that of a rigid classical rotor of the same shape. This is in conflict with experimental results which, at low angular momentum, give values about half of the rigid value.The problem is that a single one body potential is not a complete description of the interaction between nucleonsin a nucleus. There remain residual interactions between them which give rise to correlations in their motion. The most important of these correlations are due to pairing interactions. Because the effective interaction between nucleons is attractive, their energy is minimized by maximizing the overlap of their wave functions. Pairs of orbitals which differ only in the sign of the angular momentum, i.e. are in time-reversed states, provide the maximum overlap consistent with Pauli's exclusion principle. These effects can be taken into account with so-called Cranked Hartree-Fock-Bogoliubov (CHFB) calculations, a discussion of which is beyond the scope of this presentation. Pairing correlations are important in lowering the moments of inertia, because the Cooper pairs effectively behave as a superfluid.

Figure H.2 shows three rotational bands in ⁷⁶Kr and two in ⁷⁸Kr. Each band corresponds to a different intrinsic structure (configuration). It is important to note that the eigen states of the intrinsic CHFB Hamiltonian have only two good quantum numbers: the parity π and a new quantum number α, called signature.

In nuclei with an odd number of nucleons, α can assume the values +1/2 or -1/2. The states of rotational bands based on an intrinsic structure with α = +1/2 have spin sequences 1/2, 5/2, 9/2 ... , while the sequences with α = -1/2 are 3/2, 7/2, 11/2 ... . In nuclei with even numbers of nucleons α can assume the values 0 or 1 and the corresponding spin sequences are 0, 2, 4, 6 ... and 1, 3, 5 ,7 ... . The band head may have any spin of a sequence.

The parametrization of the shape in terms of the parameters β₂ and γ is similar to that used in section B (Paramatrization of Nuclear Shapes) and is illustrated in Figure H.7. Figure H.8 represents a contour map of the energy in the intrinsic frame for the ground state band of ⁷⁶Kr for angular frequencies corresponding to ħω = 0.0, 0.5, 0.6, and 0.8 MeV. The full circles indicate the minima which determine the intrinsic shape at various values of ħω and illustrate the shape changes in ^76,78Kr as a function of ħω. These and all the following CHFB calculations were carried out at the Pittsburgh Super Computer Center in collaboration with the creators of the CHFB code, J.Dudek, W. Nazarewicz and S. Cwiok.

Decay schemes such as those of figure H.2 can be interpreted in terms of the cranking model by extracting the angular frequency of rotation ω(I) as a function of the total angular momentum I (in units of ħ); the projection I_x(I) of the total angular momentum onto the axis of rotation as a function of I; and the kinematic moment of inertia ℑ⁽¹⁾(I). Below are the expressions used to extract these quantities from the level scheme that will subsequently allow a discussion of the changes in the intrinsic structure as a function of angular momentum.

In the following, E(I) is the excitation energy of a state with angular momentum I, and K is an approximate quantum number representing the projection of the intrinsic angular momentum onto the symmetry axis of the nucleus. It is usually taken to be the angular momentum of the lowest state of a band (the so-called band head):

ω(I) =

I_x(I) =

ℑ⁽¹⁾ =

Figure H.9 shows the kinematic moments of inertia for the rotational bands of ⁷⁶Kr and ⁷⁸Kr extracted from the level schemes shown in Figure H.2, using the above expressions. The yrast bands of both nuclei have kinematic moments of inertia which are quite small at low spin, increasing gradually at first, and showing a rapid increase "upbend" around ħω ≅ 650 keV for ⁷⁶Kr and ħω ≅ 560 keV for ⁷⁸Kr. For the odd parity bands in both nuclei, ℑ⁽¹⁾(I) remains quite constant at a value similar to that of a rigid rotor.The low value of ℑ⁽¹⁾ in the yrast bands at low spin reflects the effects of the pairing correlations. The gradual increase results from a gradual loss of pairing correlations, while the sharp up-bends indicate an almost complete loss of pairing correlations as a result of the breaking of a pair of nucleons that align part of their angular momenta along the axis of rotation. Plots of I_x versus ħω are shown in figure H.10.

Sharp up-bends can be seen in the yrast bands for both nuclei at the same energies as in figure H.9, which is again in contrast to the nearly perfect linear dependence of I_x observed in the negative parity bands. Thus if one subtracts the I_x values of the odd parity bands from those of the yrast bands one obtains the amount of aligned angular momentum i in the yrast band. This is illustrated in figure H.10 which shows for the yrast band of ⁷⁶Kr an alignment i ≈ 6ħ at ħω ∼ 640 keV; the corresponding values for ⁷⁸Kr are i ≈ 3.6ħ. Comparison with theoretical calculations and systematics in neighboring nuclei leads to the conclusion that the alignment in ⁷⁸Kr is the result of the partial alignment of a g _9/2 (quasi)-proton pair while that in ⁷⁶Kr is due to a nearly simultaneous alignment of a g_9/2 quasi proton pair and a g_9/2 quasi-neutron pair. Figure H.11 shows an impressive comparison between calculated and experimental aligned angular momentum for the ground-state sequence in ⁷⁶Kr. The shape parameters were obtained from the total energy surface calculations shown in figure H.8.

The results presented above are a representative example of the work performed in the time period between 1986 and 1998. Further experiments were carried out on ⁷⁵Kr (ref. H.4), ⁷⁶Br (ref. H.5, H.6), ⁷³Se (ref. H.7), ²¹⁹Ac (ref. H.8), ⁷⁸Br (ref. H.9), and ⁸⁰Sr (ref. H.10).

In 1990 an international conference was sponsored by the University of Pittsburgh and Carnegie Mellon University in Pittsburgh on "High Spin Physics and Gamma-Soft Nuclei" (ref. H.11).

In the 1980's the Pitt group started to collaborate in experiments at National Laboratories (Brookhaven, Oak Ridge, Berkely, Argonne and the National Superconducting Cyclotron laboratory at Michigan State University). The last experiments using the Pittburgh accelerator took place in 1989. The Pitt array was subsequently used at the accelerator facilities of the University of Notre Dame and Florida State University. At FSU it was combined with a local array and is presently (2004) still in operation.

In 1992 the nuclear physics community in the United States formed a committee to develop a proposal for a new National facility , Gammasphere, an array with 110 Compton-suppressed HPGE detectors that constitutes a γ-ray spectrometer of unparalleled detection sensitivity due to its high resolution, multiplicity and efficiency. The Gammmasphere project was funded by DOE and constructed from 1993 to 1995 at the Lawrence Berkeley National Laboratory. The Pitt group participated in a number of experiments at Gammasphere.

The Coulomb excitation and High Spin programs at the University of Pittsburgh produced 25 PhDs who are now in leading positions at national laboratories, universities, industry, and government agencies in the United States, Taiwan, Egypt and France.