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Rotational Damping and Compound Formation in Warm Rotating Nuclei

The rotational damping width \Gamma_{rot} and the compound damping width \Gamma_{comp} are two fundamental quantities that characterize rapidly rotating compound nuclei having finite thermal excitation energy. A two-component structure in the strength function of consecutive E2 transitions reflects the two widths, and it causes characteristic features in the double and triple gamma-ray spectra. We discuss a new method to extract experimentally values of \Gamma_{rot} and \Gamma_{comp}. The first preliminary result of this method is presented.


INTRODUCTION
Quasicontinuum gamma-rays emitted from compound nuclei formed by heavy-ion fusion reactions carry information on rapidly rotating nuclei which have thermal excitation energy U of an order of 1-8 MeV. The warm nuclei are characterized by a high density of energy levels, which is a consequence of a large number of manyparticle many-hole (np-nh) states associated with thermal excitations. In addition, the residual two-body interaction acting among particles and holes leads to formation of compound states. Energy levels of compound states are very different from the levels of shell model character observed in the cold region (U<1MeV); they are complicated mixtures of many np-nh configurations. The compound states at U>8 MeV are known as a typical example of quantum chaos [1,2,3] since their statistical properties of level and strength fluctuations exhibit the generic laws [4] governed by random matrix theories. We thus expect in the region of U=1-8 MeV an interesting order-to-chaos transition, i.e., the transition from simple shell model states to the complicated compound states.
The collective rotation of deformed nuclei exhibits a transition from rotational band structure to rotational damping with increasing thermal excitation energy U [5]. Analysis of quasicontinuum gamma-rays has revealed that the energy levels forming the rotational band structures characterized by strong E2 transitions within band members are limited only in the cold region U<1 MeV, whereas the E2 decay from levels in the warm region U>1 MeV are spread over many decay branches, implying the damping of collective rotation [6,7]. The onset of rotational damping is a consequence of the formation of compound states and it is closely related to the order-to-chaos transition expected in the same energy region. We thus expect that the quasi-continuum E2 gamma-rays can be used as a probe to study the formation of compound states in rotating nuclei. We report here our recent theoretical and experimental efforts to this end.

ROTATIONAL AND COMPOUND DAMPING WIDTHS
The rotational damping width Γrot and the compound damping width Γcomp are two fundamental quantities. Γrot measures the spreading of the E2 strength associated with gamma-decays from an energy level exhibiting rotational damping [5]. It is defined as the FWHM of the B(E2) strength distribution measured by the gamma-ray energy ( Fig.1 left). This quantity is related to a time scale of decoherence in the damped collective rotation. The compound damping width Γcomp, on the other hand, is a measure of the compound state formation. Taking a typical np-nh state at spin I, we consider how the strength of this state is distributed among the compound energy levels at the same spin I. The energy interval of this strength distribution is Γcomp (Fig.1 right). In other words, Γcomp is the spreading width of np-nh states, which is caused by configuration mixing due to the residual interaction. Its reciprocal is related to a relaxation time for np-nh states to form compound states. Γcomp is expected to increase with thermal excitation energy U.
The two widths Γrot and Γcomp play central roles in the dynamics of warm rotating nuclei [5]. Rotational band structure is realized if Γcomp, Γrot < D, where D is the mean level spacing (the notion of damping becomes meaningless in this case). The opposite condition Γcomp, Γrot > D characterizes the onset of configuration mixing and the onset of rotational damping. If Γcomp > Γrot > D, the strong mixing brings about the motional narrowing effect [5], which gives counter-intuitive decrease of Γrot as a function of U. In an extreme case of Γcomp >> D > Γrot, there may emerge the ergodic rotational bands, where the strongly chaotic compound energy levels form rotational bands instead of exhibiting rotational damping [8]. It is therefore important to evaluate Γrot and Γcomp to understand the warm rotation.  Theoretical predictions of Γrot and Γcomp can be provided by the microscopic cranked shell model calculations [9,10]. In this model the intrinsic excitation is described on the basis of particle-hole excitations in a uniformly rotating deformed mean-field whereas the rotational angular momentum is treated approximately. Each np-nh configuration is ascribed to an unperturbed rotational band. As the thermal excitation energy U increases, configuration mixing among np-nh unperturbed bands becomes strong. Using the shell model diagonalization technique, we obtain energy levels and their wave functions for U< a few MeV with a reasonable choice of cut-off energy of basis truncation. Near the yrast line, the energy levels after diagonalization still retain a rather pure cranked shell model character and form rotational bands. The rotational damping sets in around U> 1 MeV, leaving only 20-30 rotational bands in the case of a typical deformed nuclei 168 Yb [10], in agreement with the experimental observation [6,7].
Γrot is governed by rotational alignment of the single-particle orbits and by the compound damping width Γcomp. If the mixing is not very strong (Γcomp < Γrot), the rotational damping width Γrot is dominated by the single-particle alignments [5]. The alignments exhibit a strong shell effect which varies depending on the rotational frequency, the deformation, and the position of the Fermi level (i.e. the neutron and proton numbers) on top of a smooth dependence on these quantities [11]. The cranked shell model calculation predicts Γrot =100-300 keV (increasing with spin) in a typical rare-earth nucleus 168 Yb, but it may vary within the very wide interval Γrot =30-500 keV, depending on nuclear species [12]. An extremely small value Γrot ~30keV is calculated for the superdeformed 192 Hg, where the shell effect is very strong [11].
Experimentally the value of Γrot has been difficult to measure. A previous analysis [13] using the technique of the rotational plane mapping of the 3D spectra of triple coincident gamma-rays has suggested an order of 50-80 keV in 168 Yb, but at the same time it was pointed out that the spectra are not consistently described by the simple assumption of a single Gaussian or Lorentzian profile of E2 strength functions, that is adopted in the analysis. On the other hand, a recent analysis [14] using a different technique reports Γrot =200-300 keV in 168 Yb, in good agreement with the theory.
Concerning Γcomp, the model predicts that it is rather independent on nuclear species, deformations and spins. A typical value is Γcomp ~ 50 keV in the energy region U~1-2 MeV [12,15]. Experimentally, however, there has been no direct measurement so far, not only in rotating nuclei but also in connection with other phenomena. This difficulty arises from a fact that there is no direct way to excite shell model np-nh states. Nevertheless, we have recently shown [15] that it should be possible to measure both Γrot and Γcomp by a properly designed analysis. Indeed in the present work we extract for the first time experimental value, though preliminary, of the compound damping width Γcomp

NARROW COMPONENT IN 2D AND 3D SPECTRA
To probe Γrot and Γcomp, we utilize a theoretical observation [12,15] that the E2 strength function reflects both Γrot and Γcomp through a doorway mechanism of rotational damping (although this is only valid when Γrot > Γcomp).
In an illustration in Fig.2(a), we suppose that an initial level at spin I is a mixture of three unperturbed np-nh bands. The damped E2 transition from this level follows the unperturbed transition energies of these unperturbed bands. The unperturbed np-nh bands, however, are not energy eigenstates, and mix with other np-nh states at spin I-2, leading to smearing by the spreading width Γcomp. As a consequence, the E2 strength function exhibits fine structures which are characterized by Γcomp whereas the overall profile is determined by Γrot. Considering further two-dimensional (2D) distribution of E2 strength for two consecutive E2 decays γ1 and γ2 passing I → I-2 → I-4, the fine structure brings a characteristic correlation along the diagonal line in the Eγ1×Eγ2 plane ( Fig.2(b)). It is a remnant of the rotational correlation in the unperturbed np-nh bands, but the correlation is smeared by Γcomp. Cutting the 2D strength distribution in the direction perpendicular to the diagonal, the strength plotted in the Eγ1-Eγ2 axis exhibits a two-component structure centered around the ridge position Eγ1-Eγ2=4/I≈70 keV(I being the moment of inertia) with a narrow distribution with width Γnarrow, and a wide distribution with width Γwide (Fig.2(c)). The narrow component arises from the doorway mechanism. One can relate the width Γnarrow and the intensity Inarrow of the narrow component to the compound damping width by Γnarrow ~ 2Γcomp, and Inarrow ~ 2D/Γcomp. The width Γwide of the wide component can be related to the rotational damping width Γwide~r ot 2Γ assuming a Gaussian shape for the overall E2 distribution (Γwide ~2 Γrot if we assume a Lorentzian shape). See also [13].
Actual spectra contain not only the consecutive gamma-rays but also non-consecutive ones which form distributions around the second, third etc. ridge positions Eγ1-Eγ2=70×n keV (n=2,3, …). Symmetric combinations exchanging γ1 and γ2 should be included also. Fig.3(a) shows a simulated spectrum calculated with use of the cranked shell model and a simple simulation of cascades neglecting the E1 contribution. Here a sizable structure observed around the first ridge Eγ1-Eγ2=±70keV originates from the narrow component.
The narrow component creates more prominent features in the 3D spectra of coincident three gamma-rays, as shown in Fig.3(d). The figure is a 2D representation of the spectra projected on the 'transversal plane', defined by Eγ1+Eγ2+Eγ3=const, which is perpendicular to the diagonal line in the Eγ1×Eγ2×Eγ3 cube (Fig.3(c)). A volcano-like structure characterized by a crater rim and six peaks around, and ridges of volcano extending from the peaks is seen. The rim-ridge structure arises from the narrow component.
The displaced rotational plane cut, which we have developed recently [16,17], is a method to map the volcanolike structure of the 3D spectra. The displaced rotational plane, defined by Eγ1+Eγ2-2Eγ3=δ, is perpendicular to the transversal plane (Fig.3(f)). The plane without a displacement (i.e. δ=0) intersects the volcano at the center. By displacing the plane with variation of δ, the volcano-like structure can be scanned. See [16,17] for results of this method.

EXTRACTING TWO DAMPING WIDTHS
Motivated by these studies, we recently have developed another new analysis aiming at extracting the rotational and compound damping widths from experimental quasi-continuum gamma-ray spectra. For this purpose we introduce a parameterized spectrum function which incorporates the two-component structure. The strength function for the consecutive two E2 gamma rays is given as ) is a strength function for single step decay from states at spin I (and I-2), which is a simple Gaussian function whose FWHM is set to the rotational damping width Γrot. The second term represents the uncorrelated part in the consecutive transitions and corresponds to the wide component. The first term on the other hand represents the narrow component which is correlated along the diagonal line and has a narrow width to the Eγ1-Eγ2 direction (cf. Fig.2(b)). We parameterize ) , ( For multiple steps of E2 gamma decays, we consider a multidimensional strength function ) , , , given by a conditional product of the two-step functions (1). Spectra for double and triple coincident gamma-rays are obtained by integrating unobserved gamma-rays. These procedures can be performed analytically, leading to final functional forms of 2D and 3D spectra which consist of many Gaussian functions. The spectra thus obtained contain four parameters; two damping widths Γrot and Γnarrow (related to Γcomp), the intensity Inarrow of the narrow component, and the number of decay steps considered. We confirm that the parameterized spectra describe very well the essential features of the cranked shell model spectra, as seen by comparing Fig.3(a) and (b), and (d) and (e). By using a multidimensional fitting of the parameterized spectrum to the experimental E2 spectrum, we are able to extract the best fit values of Γrot, Γnarrow, and Inarrow. As a first attempt, we applied this method to the perpendicular cut of the 2D spectrum in 168 Yb [18] at different gamma-ray energies Eγ=(Eγ1+Eγ2)/2. The known discrete peaks are subtracted from the data. The E1 contribution to quasi-continuum spectra is evaluated with use of a spectrum function ) / ( exp  This preliminary result is compared with the cranked shell model calculation in Fig. 5, where the recent report by the Berkeley group [14] is also included. In the model calculation, the rotational damping width Γrot and the width of narrow component Γnarrow are evaluated from the Eγ1-Eγ2 correlation (cf. Fig.2(c)) calculated for the lowest 300 energy levels for each spin and parity, covering approximately the region up to U~2.3MeV. (Γrot is obtained with use of Γwide and the relation Γrot=Γwide/ 2 ). In this figure, the gamma-ray energy is converted to the rotational spin by using the experimental average gamma-ray energy [14].
The extracted values of the rotational damping width Γrot agree fairly well with the model. In a previous analysis [13], we estimated Γrot =50-80 keV as a lower limit in the same region of gamma-ray energy. The present values are much larger than the previous ones. It should be noted that in the previous analysis we have used a different function which neglects the presence of the narrow component.
The experimentally extracted value of Γnarrow, which we obtain for the first time, is within the range of 20-100 keV, in qualitative agreement with the model prediction (Fig.5 right). Possible contribution from unresolved discrete rotational bands could influence results. An analysis including 3D spectra is in progress. The aim is to subject the two-component parameterization to more stringent tests and to obtain more firm results.  [14] is also shown with dotted line for Γrot.

CONCLUSIONS
Compound nuclei with high rotational angular momentum and with finite thermal excitation energy are studied both theoretically and experimentally by analyzing the quasicontinuum E2 gamma-rays spectra. There exits two fundamental quantities; the rotational damping width Γrot which measures the energy spreading in rotational E2 transitions, and the compound damping width Γcomp which is the spreading width of many-particle many-hole states carrying thermal excitation. The strength function of consecutive E2 transitions exhibits a two-component structure made of wide and narrow distributions. The width of the narrow component is directly related to Γcomp whereas the wide component reflects Γrot. The presence of narrow component causes characteristic features in the 2D and 3D spectra of double and triple coincident gamma-rays. We introduced a simple function that parameterizes these features. By fitting the new function to the spectra, it is possible to extract Γrot and Γcomp experimentally. We made the first preliminary application of this technique, and derived experimental values of Γcomp for the first time.