Level densities of transitional Sm nuclei

PHYSICAL REVIEW C 71, 064320 (2005) Level densities of transitional Sm nuclei R. Capote Nuclear Data Section, International Atomic Energy Agency, Vienna, Austria A. Ventura Ente Nuove Tecnologie, Energia e Ambiente, and Istituto Nazionale di Fisica Nucleare, Bologna, Italy F. Cannata Dipartimento di Fisica dell’ Universit` a and Istituto Nazionale di Fisica Nucleare, Bologna, Italy J. M. Quesada Departamento de F´ ısica Atomica, Molecular y Nuclear, Universidad de Sevilla, Spain (Received 11 March 2005; published 28 June 2005) Experimentally determined level densities of the transitional isotopes 148,149,150,152Sm at excitation energies below and around the neutron binding energy are compared with microcanonical calculations based on a Monte Carlo approach to noncollective level densities, folded with a collective enhancement estimated in the frame of the interacting boson model (IBM). The IBM parameters are adjusted so as to reproduce the low-lying discrete levels of both parities, with the exception of the odd-mass nucleus, 149Sm, where complete decoupling of the unpaired neutron from the core is assumed. DOI: 10.1103/PhysRevC.71.064320 PACS number(s): 21.10.Ma, 21.10.Pc, 21.10.Re, 27.70.+q I. INTRODUCTION Interest in the study of nuclear excitations in the continuum below the neutron binding energy Bnhas been revived in recent years by a new method of analysis of γ-ray transitions, which permits simultaneous extraction of the level density at the excitation energy to which the nucleus under study decays and of the γstrength function at the difference of the initial and final excitation energies [1]. For rare-earth nuclei, to which the method was applied for the first time [2], both energies are in the continuum region, thus allowing an investigation of various thermodynamical quantities, such as temperature and entropy, versus excitation energy. In particular, for the lanthanides analyzed in Ref. [2], 162Dy, 166Er, and 172Yb, temperature does not increase monotonically with excitation energy, but shows some broad oscillations, interpreted as a consequence of the breaking of nucleon pairs and, at higher energies, of the possible quenching of pair correlations. Detailed analyses of the experimental results for these nuclei were published in a serie of papers: In particular, total level densities versus excitation energy were obtained for 161,162Dy [3], 170,171,172Yb [3,4], and 166,167Er [5]. In Refs. [3,5], heat capacity versus temperature, deduced from total level density, indicates a phaselike transition from the pair-correlate ground state to an uncorrelated one at a critical temperature Tc≃0.5MeV. It has been known for years that, in addition to pairing and shell effects, the level densities of deformed nuclei, like the lanthanides analyzed in Refs. [2–5], are strongly enhanced by collective effects, mainly of rotational nature, that are already sizable at excitation energies of the order of Bn.Onthe additional assumption of adiabatic decoupling of noncollective and collective degrees of freedom, strictly valid at excitation energies that are much higher than the average collective energy, total level densities are simply obtained by folding noncollective level densities, obtained by a microscopic model, with collective level densities, usually given by some simple quantum model, such as the axially symmetric rotor or the harmonic oscillator, for rotations and vibrations, respectively. At excitation energies of the order of Bn, rotational effects play the main role: The relevant formalism [6] was applied long ago to the analysis of neutron resonance spacings of nuclei in the lanthanide and actinide regions [7]. Damping of collectivity with increasing energy [8] will be neglected in the low-energy region of interest to the present work. Recently, the method of Ref. [1] has been applied to weakly deformed nuclei in the same mass region and total level densities and γstrength functions have been extracted for 148,149Sm [9]. It is not expected that the collective effects in the level densities of even-even transitional nuclei are adequately reproduced by means of a simple model, like the axially symmetric rotor used for strongly deformed nuclei or the oscillator adopted for spherical nuclei. However, it is known that the low-lying discrete levels and related electromagnetic transitions for this kind of nuclei are well reproduced by the interacting boson model (IBM) [10], an algebraic model of collective excitations with U(6) symmetry. Sm isotopes constitute a transitional chain from spherical to axially deformed nuclei, corresponding to the IBM limiting symmetries U(5) and SU(3), respectively. By resorting to the coherent state formulation [10] of the IBM, it is possible to interpret the U(5) →SU(3) shape transition as a quantum phase transition, whose order and critical point can be determined by investigating the behavior of the derivatives of the nuclear ground-state energy with respect to a suitable control parameter. In the Sm chain, the best candidates to be critical are 150Sm, on the basis of a recent study that combines 0556-2813/2005/71(6)/064320(9)/$23.00 064320-1 ©2005 The American Physical Society R. CAPOTE, A. VENTURA, F. CANNATA, AND J. M. QUESADA PHYSICAL REVIEW C 71, 064320 (2005) coherent state formulation with catastrophe theory [11], and 152Sm [12], where selected collective bands of positive parity and related electromagnetic transitions are nicely reproduced in the frame of the X(5) critical symmetry [13], corresponding to an analytical solution of the Bohr collective Hamiltonian. Since the discrete spectra of the transitional even-even isotopes 148,150,152Sm show so many interesting features, it appears rather natural to extend the analysis to the continuum. However, the collective level density generated by the IBM cannot be compared directly with experimental data, like those of Ref. [9], which refer to the total level density: The latter can be estimated, for an even-even nucleus, by folding the IBM level density with the density of incoherent quasiparticle excitations, generated by a realistic microscopic model. Since we are also interested in the possible fluctuations of the level density connected with the breaking of Cooper pairs with increasing energy, the calculations will be carried out in the microcanonical ensemble. The paper is organized as follows: Sec. II describes the collective level density, Sec. III the noncollective level density and Sec. IV the total level density. Finally, Sec. V is dedicated to the comparison with experimental data and Sec. VI to conclusions and perspectives. A comparison of our formalism with traditional treatments of collective enhancement of nuclear level densities is worked out in detail in the Appendixes. II. COLLECTIVE LEVEL DENSITY As is known, the IBM commonly used in the analysis of collective excitations of non-magic even-even nuclei is formulated in the laboratory frame, so that diagonalization of the IBM Hamiltonian yields the energies Ec, of collective levels with good angular momentum Jand parity π.Inthe present work, we use the simplest version of the model, IBM-1, which does not distinguish between neutron and proton degrees of freedom. The number of proton pairs, Nπ, and the number of neutron pairs, Nν, in the corresponding valence shells define the effective boson number NB=Nπ+Nν, which is a conserved quantity that is the corresponding operator  N, commutes with the Hamiltonian  H. Since the bosons of the model simulate collective fermion pairs, the Pauli principle is approximately taken into account by defining the boson number as the number of particle pairs, or of hole pairs, whichever is less, in the given valence shells. The density of collective levels of given angular momentum and parity is simply written as a sum of Dirac delta functions centered on the collective energies Ec(J,π): ρcoll(E,J,π)= c δ[E−Ec(Jc,π c)]δJ,Jcδπ,πc,(1) and the total density of collective levels of given parity is obtained by summing Eq. (1) over Jfrom 0 to the maximum angular momentum Jmax(π) allowed by the finite boson number NB: ρcoll(E,π)= Jmax(π)  J=0 c δ[E−Ec(J,π)].(2) Since a level with angular momentum Jcorresponds to 2J+1 degenerate sublevels, or states, the collective state density is simply ωcoll(E,π)= Jmax(π)  J=0 (2J+1)  c δ[E−Ec(J,π)].(3) In the present work, we also need the density of collective states with given projection Mon an arbitrary quantization axis: ωcoll(E,M,π)= Jmax(π)  J=0 (2J+1) × c δ[E−Ec(J,π)]fcoll(M,π),(4) where the distribution in Mof states of given parity π,fcoll(M,π), can be normalized to unity in the finite range from −Jmax(π)to+Jmax(π) and is directly estimated from the calculated set of collective levels of given parity π. Collective levels of positive parity can be obtained by diagonalization of the standard IBM-1 Hamiltonian  Hsd, containing free-boson terms and two-body interactions of s and dbosons, which are the boson images of collective nucleon pairs coupled to angular momentum and parity Lπ=0+and 2+, respectively. The 1 +5 boson degrees of freedom span a dynamical U(6) algebra. Consistent extension of the model to collective states of negative parity and of parity changing electromagnetic transitions requires introduction of pand f bosons, with Lπ=1−and 3−, respectively, corresponding to 3+7 additional degrees of freedom, so that the dynamical algebra is enlarged to U(16) [14–17]. However, pbosons play a minor role in reproducing the energies of the low-lying levels of negative parity and are neglected in the present work, where the levels of negative parity are obtained by coupling only one fboson to sand dbosons. Therefore, Jmax(π=+)=2N and Jmax(π=−)=2N+1. The sdf Hamiltonian adopted in the present work is the same as that of Ref. [18] and can be written in the form  Hsdf =εdnd+εfnf+α P†sdf · Psdf +β Qsdf · Qsdf +γ Ldf · Ldf +δ Qf· Qsd + Ld· Lf.(5) Here, ndand nfare the number operators of dand f bosons, respectively, and the multipole operators are defined as follows:  P†sdf =−s†·s†+d†·d†+f†·f†,(6)  Qsd =[s†× d+d†×s](2) −√7 2[d†× d](2),(7)  Qf=−3 10√42[f†× f](2),(8)  Qsdf = Qsd + Qf,(9)  Ld=√10[d†× d](2),(10)  Lf=2√7[f†× f](1),(11)  Ldf = Ld+ Lf.(12) 064320-2 LEVEL DENSITIES OF TRANSITIONAL Sm NUCLEI PHYSICAL REVIEW C 71, 064320 (2005) TABLE I. Boson number NBand coefficients (in keV) of the sdf Hamiltonian for Sm isotopes. Nucleus NBdfαβγ δ ε 148Sm 8 696.5 1303.0 −1.0 −10.0 4.0 −136.1 −3.0 150Sm 9 546.5 1303.0 −1.0 −10.0 −3.0 −136.1 −3.0 152Sm 10 442.3 1406.5 4.0 −17.0 −1.0 −46.3 −1.0 In the collective Hamiltonian (5), the monopole-monopole interaction,  P†sdf · Psdf ,the quadrupole-quadrupole interaction,  Qsdf · Qsdf , and the square of total angular momentum,  Ldf · Ldf ,modify the standard s−dmultipole interactions by including fbosons, whereas the last two cross-multipole terms on the right-hand side of the same equation improve the fit of octupole bands in deformed nuclei. Hamiltonian (5) is thus more general than the one first used in computing negative parity states in Sm isotopes [19]. The eigenvalues of  Hsdf , whose coefficients (in keV) are given in Table I, are obtained by numerical diagonalization using a modified version of the Octupole code [20]. The experimental levels of both parities are satisfactorily reproduced up to an excitation energy E≃1.5 MeV and angular momentum Jmax =10. To use IBM results in a collective enhancement of intrinsic level densities, collective states have been calculated up to J=Jmax.TheMdistributions of collective states turn out to be rather well approximated by Gaussians centered at M=0, with characteristic FWHM values of the order of Jmax. Figure 1 shows the collective level densities of 148,150,152Sm calculated by IBM for positive and negative parities separately according to Eq. (2). The level densities steadily increase if one goes from the spherical nucleus 148Sm to the deformed nucleus 152Sm. It is interesting to remark that the increase is faster for negative parity than for positive parity; therefore the use of fbosons in the IBM is an important requirement for the proper description of the collective enhancement in the Sm isotopic chain. As a consequence, the collective enhancement factor of total level density turns out to be much higher in the deformed case than in the spherical one. III. NONCOLLECTIVE LEVEL DENSITY Since the collective level density is computed in the laboratory frame, so must be the noncollective one; thus, both collective and noncollective angular momenta are good quantum numbers in the present approach. In particular, singleparticle levels have been generated in a spherical Woods-Saxon potential with parameters appropriate to the Sm region. It is important to remark that the noncollective level density has a weak dependence on deformation, whose main effect is a redistribution of levels, without significant changes in the average density. Here and in the following, we shall usually refer to the noncollective level density as the intrinsic level density, since this is the traditional name under which it is known in the literature (see, e.g., Ref. [6]). In the present work, intrinsic is equivalent to noncollective and does not imply that calculations are done in an intrinsic reference frame. Several methods of generating excited multiparticle configurations of an atomic nucleus from given sets of singleparticle states were proposed in the course of years, both in the microcanonical and the grand-canonical frameworks, and are available in the literature. In particular, an exact recursive method of interest to the present work was originally formulated by Williams [21]: It is based on the use of exact recursion relations to expand the grand partition function, but it does not rely on the saddle-point approximation used in the traditional grand-canonical approach. It has the advantage of requiring much shorter computation time than a combinatorial approach, which counts all possible configurations, but cannot deal with residual interactions, at least in its original form. In the combinatorial methods, such as the odometer approach of Hillman and Grover [22], level densities are 0246 0 50 100 150 200 250 300 350 400 246 246 148Sm LEVEL DENSITY [MeV-1] (a) 150Sm (b) Collective energy E C [MeV] 152Sm (c) FIG. 1. Collective level densities of 148,150,152Sm vs excitation energy. Solid histograms: positive parity; dashed histograms: negative parity. 064320-3 R. CAPOTE, A. VENTURA, F. CANNATA, AND J. M. QUESADA PHYSICAL REVIEW C 71, 064320 (2005) computed numerically by performing an exhaustive counting of excited configurations. More recently, combinatorial calculations were carried out by means of the Densidad code [23], based on a model proposed by Herman and Reffo [24–26], to estimate level densities with a fixed number of particles and holes (excitons), for use in pre-equilibrium models of nuclear reactions. In the frame of an independent particle picture of atomic nuclei, this method is exact but very time consuming and becomes intractable at high excitation energies or for large shell model spaces (i.e., for medium and heavy nuclei). A residual pairing interaction can be included in the usual Bardeen-Cooper-Schrieffer (BCS) approximation, at the cost of greatly increased computer time. It becomes thus natural to resort to a Monte Carlo (MC) technique, like the one proposed by Cerf [27], to avoid exhaustive counting of excited configurations and reduce computer time, without sizable loss of accuracy. In general, MC algorithms are known to be very efficient in solving combinatorial problems. In this work the MC procedure is followed to extract a random sample of excited states in the considered energy range. This very small fraction of configuration space is assumed to be representative of the whole ensemble. Since the level density increases exponentially, it is difficult to do a MC sampling uniformly over the entire energy range of interest. In what follows we make use of the Metropolis sampling scheme [28] to achieve uniform sampling, as explained in detail by Cerf [27]. A MC estimator for the intrinsic state density of the N-neutron or Z-proton system with given angular momentum projection Mand parity πis given by ωN,Z intr (E,M,π)≃s L  j=1 δ(E−ECj)δM,Mjδπ,πj W(Cj),(13) where Cjare random single-particle configurations along a long enough random walk of length L, and W(Cj)arethe weight functions associated with each configuration. The scale factor s, which is needed to normalize the state density, is derived from an exact recursive calculation by means of the Totstade code [29], where the recursive method of Williams [21] is implemented. The recursive method, however, cannot yield spin-parity distributions, nor can it take into account a residual pairing interaction: Therefore, we just need it to normalize the sampled MC state density. The calculated recursive state density ωrec intr(E)isalsousedasthe weight function W(Cj)=W(ECj) of the Metropolis sampling scheme, where ECjis the energy of a given jth configuration [27]. To derive the normalization factor needed in Eq. (13), the total number of states, Nrec, is calculated up to the maximum excitation energy Emax: Nrec =Emax 0 ωrec intr(E)dE. (14) Using the normalization condition NN,Z rec =Emax 0 dE  M π ωN,Z intr (E,M,π) (15) and Eqs. (13)–(15) we obtain the following expression of the scale factor s: s=NN,Z rec N j=1 1 W(Cj) .(16) Residual pairing interactions treated in the BCS approximation are easily introduced in the MC formalism following again the approximate method of Ref. [27]: For each sampled excited configuration Cj, the usual BCS equations for energy gap and particle number are solved by blocking the orbits occupied by unpaired nucleons, and the computed pairing energy PCjis used to correct the energy ECjof the sampled configuration, replaced by ECj−PCjin the computation of the MC state density, Eq. (13), which is changed into ωBCS intr (E,M,π)≃s N  j=1 δ(E−ECj+PCj)δM,Mjδπ,πj W(ECj),(17) because of the pairing energy correction PCj. The energy ECj used to calculate the weight function does not contain the pairing correction, therefore the state density without pairing can be computed at the same time, so the scaling factor sis common to both formulations, with pairing and without it. In this approximate treatment of pairing, the excitations involving promoted pairs are neglected, but this helps to minimize the overlap with the collective excitations computed in the IBM, since the interacting bosons of the model simulate collective fermion pairs and the eigenstates of the IBM Hamiltonian can be considered as excitations resulting from promoted collective pairs. The intrinsic state density ωintr(E,M,π) of the whole nucleus with given angular momentum projection Mand parity πis obtained by separate calculations of neutron and proton state densities, ωN intr(E,M,π) and ωZ intr(E,M,π), respectively, by means of Eq. (13), and their subsequent convolution is performed according to the formula for the coefficients of the Cauchy product of two infinite series ωintr(E,M,π)= E  Q=0 πN πZ MN MZ ωN intr(Q, MN,π N)ωZ intr ×(E−Q, MZ,π Z)δπ,πNπZδM,MN+MZ. (18) IV. TOTAL LEVEL DENSITY On the assumption of complete decoupling of intrinsic and collective degrees of freedom, at low energy E, where both collective and intrinsic levels are discrete, the total density of states of given spin projection Mand parity πcan be put in the form ω(E,M,π)= πi πc Nc(πc)  c=0 Ni(πi)  i=0 Mi Mc δ[E−Ec(Mc,π c) −i(Mi,π i)]δMc+Mi,M δπcπi,π ,(19) where Ec,M c,π care the collective and i,M i,π iare the intrinsic excitation energies, spin projection and parities. The 064320-4 LEVEL DENSITIES OF TRANSITIONAL Sm NUCLEI PHYSICAL REVIEW C 71, 064320 (2005) ground-state collective and intrinsic energies, E0and 0, respectively, are both set to zero. According to formula (19), in particular, at E< 1, the lowest two-quasiparticle excitation, the total level density of an even-even nucleus coincides with the collective one. At excitation energy Emuch greater than Ec, the average collective energy, where the intrinsic state density can be well approximated by a continuum, it is more convenient to start from the folding of the intrinsic and collective state densities at given projection of total angular momentum M=Mi+Mc: ω(E,M,π)= πi πc Nc(πc)  c=0∞ 0 dEi Mi+Mc=M ωintr(Ei,M i,πi) ×ωcoll(E−Ei,M c,π c)δπcπi,π ,(20) which immediately reduces to formula (19) if the intrinsic and collective state densities are discretized again as ωintr(Ei,M i,π i)= j,Mj,πj δ[Ei−j(Mj,π j)] ×δπi,πjδMi,Mj,(21) ωcoll(E−Ei,M c,π c)= l,Ml,πl δ[E−Ei−El(Ml,π l)] ×δπc,πlδMc,Ml.(22) The observable level density of given spin Jand parity π,ρ(E,J,π), is obtained from the calculated state density ω(E,M,π) of the whole nucleus, given by Eq. (19), using a formula valid in the case of spherical symmetry (see, e.g., Ref. [30]), ρ(E,J,π)=ω(E,M =J,π)−ω(E,M =J+1,π). (23) The total level density is finally obtained by summing this expression over Jand π, ρ(E)= Jπ ρ(E,J,π).(24) All level density calculations carried out in the present work make direct use of formulas (19)–(24). However, comparison of our formalism with the traditional treatment of collective enhancement of deformed nuclei [7], albeit not necessary to the present work, allows us to gain a qualitatively interesting physical insight. To this aim, a more compact approximation to formula (20) is worked out in Appendix A and this is compared with the corresponding formalism for axially symmetric rotors in Appendix B. V. NUMERICAL RESULTS The hybrid formalism described in the preceding sections, where an intrinsic state density from MC calculations done in the microcanonical ensemble is folded with a collective state density computed by means of the interacting boson model, on the assumption of complete decoupling of intrinsic and collective degrees of freedom, is applied to four transitional Sm isotopes. The resulting total level density ρ(E) is plotted in Fig. 2 as a function of Eup to E=Bn, the neutron binding energy, and compared with experimental data in the energy range from 1 MeV to Bn−1MeVfor148,149Sm [9]. In the case of 149Sm we have folded the intrinsic state density of the odd-mass nucleus with the collective state density of its even-even core, 148Sm, because we have assumed complete decoupling of the odd neutron from the core. Not surprisingly, this approximation is particularly poor at low excitation energies, where the leading contribution to the total level density is the collective one. At such energies, the collective level density could be computed in the interacting boson-fermion model (IBFM) [31], where the coupling of the odd fermion with the even-even core of an odd-mass nucleus is properly taken into account. This kind of calculation, however, lies beyond the scope of the present work, because it would require additional work in its own, namely, the extension of the standard IBFM, where the odd fermion is coupled with sand dbosons only, to the coupling with negative-parity bosons. Other important pieces of information about level densities at the neutron binding energy are the average spacings of s-wave and p-wave resonances detected in neutron capture by a target nucleus with one less neutron than the nucleus under consideration. A recent comprehensive compilation of neutron resonance spacings is included in the Reference Input Parameter Library, Version 2 (RIPL-2) [32], which contains recommended values for 296 s-wave resonance spacings and 82 p-wave resonance spacings. Table II compares the values of the model-independent s-wave resonance spacings D0com- puted in the present approach by counting relevant resonances for the four Sm compound nuclei under consideration with the recommended values of the RIPL-2 compilation. By definition D0=[ρ(Bn+E/2,1/2+)]−1,I t=0+, ρBn+E/2,Iπ t−1/2π+ρBn+E/2,Iπ t+1/2π−1,I t= 0+,(25) where Iπ tis the ground-state spin parity of the target nucleus and E is the energy interval for counting the resonances. In the case of 152Sm, there is also an experimental value of D0recently obtained by the n TOF collaboration [33] in neutron capture by 151Sm, which is slightly higher than the RIPL-2 recommendation and our theoretical result. The agreement between theory and experiment is satisfactory in the case of 148,152Sm and underestimates the experimental values by a factor of 2 in the case of 149,150Sm. However, it is to be stressed again that our theoretical values are obtained on the crude assumption of decoupling of intrinsic and collective degrees of freedom, which is poor, in particular, for the 064320-5 R. CAPOTE, A. VENTURA, F. CANNATA, AND J. M. QUESADA PHYSICAL REVIEW C 71, 064320 (2005) 012345678 100 102 104 106 108 123456789 012345678 100 102 104 106 108 123456789 148Sm LEVEL DENSITY [MeV-1] (a) (b) 149Sm (c) 150Sm Energy [MeV] (d) 152Sm FIG. 2. Total level densities of 148,149,150,152Sm vs excitation energy (dashed histograms) and corresponding intrinsic level densities (solid histograms). The experimental data (empty circles) of 148,149Sm are taken from Ref. [9]. odd-mass nucleus 149Sm. Therefore, we consider the present results as an encouraging approximation; in any case, they emphasize the crucial role played by collective enhancement in level densities around E≃Bn. As can be seen from Fig. 2, without folding with the collective contribution, the noncollective values of D0would overestimate the experimental ones by a factor of 10–50 in this mass region (see also Ref. [34]). The main advantage of our treatment of collective effects is that the model can be equally well applied to spherical (148Sm), transitional (150Sm), and deformed nuclei (152Sm), whereas the model of Refs. [6,7], briefly described in Appendix B, is valid only in the last case and, in fact, it has already been applied to heavier deformed lanthanides measured by the Oslo group, such as 161,162Dy, 166Er, and 171,172Yb [35,36]. VI. CONCLUSIONS AND PERSPECTIVES The numerical results discussed in Sec. V appear to us, in spite of some simplifying model assumptions, significant enough to encourage improvement of the model in two basic steps: (i) extension to higher excitation energies; (ii) introduction of a proper coupling of intrinsic and collective degrees of freedom for odd nuclei. TABLE II. s-wave resonance spacings for Sm isotopes. Compound Bn(MeV) Iπ tD0exp.(eV) D0calc.(eV) Nucleus 148Sm 8.141 7/2−(5.1±0.5)a5.4±0.3 149Sm 5.871 0+(100.0±20.0)a53.0±2.0 150Sm 7.985 7/2−(2.1±0.3)a0.94 ±0.03 152Sm 8.257 5/2−(1.04 ±0.15)a1.2±0.1 (1.48 ±0.04)b aFrom Ref. [32]. bFrom Ref. [33]. Step (i) is relatively simple, if one replaces the microcanonical description adopted in the present work with a grand-canonical formalism, at the cost of smoothing out the level density fluctuations related to shell and pairing effects, which are less and less observable with increasing excitation energy. Under this assumption, one could compute the intrinsic state density ωintr(E,M,π) in the grand-canonical ensemble (see, e.g., Ref. [30]) and then make use of Eq. (A11) of Appendix A for the total state density ω(E,M,π), and derive total level densities ρ(E,J,π) by means of Eq. (23). The problem still to be solved in this approach is the damping of collective effects with increasing temperature, that is, the temperature dependence of the collective partition function Zcoll(T,π c) defined by Eq. (A6) of Appendix A, which is expected to go through a maximum at some finite temperature and then decrease to unity in the T→∞limit. The problem was already faced in Refs. [18,37] under the simplifying assumption that only the effective boson number NBchanges with temperature and goes to zero when Tgoes to infinity. Even in that case, however, a realistic microscopic calculation of NB(T) is needed, since it was recently noticed [38] that the simple BCS calculations at finite temperature done in Refs. [18,37] do not properly deal with quadrupole and octupole bosons, inducing, as a consequence, the transition from a deformed ground state to a spherical equilibrium shape at much lower temperature than the one predicted by selfconsistent mean-field calculations. In addition, the temperature dependence of the IBM Hamiltonian parameters, in particular the d- and f-boson energies, might be non-negligible over a broad temperature range, so that a consistent microscopic model of both boson number and Hamiltonian appears to be mandatory. Once step (i) is taken and computed level densities at high energy compare reasonably well with available experimental data, one might try step (ii), that is, inclusion of the coupling of intrinsic and collective degrees of freedom in the formalism, of particular importance to the treatment of odd-mass and odd-odd nuclei at low excitation energies. 064320-6 LEVEL DENSITIES OF TRANSITIONAL Sm NUCLEI PHYSICAL REVIEW C 71, 064320 (2005) The most viable computational approach in this case is the simultaneous MC treatment of both mean-field and residual interactions. A finite temperature version of the MC treatment of the intrinsic level density described in Sec. III was proposed by Cerf in Ref. [39] and applied there to the pairing interaction in a single jshell, to check numerical results with the exact analytical results of the seniority model. Since the calculation of thermodynamic quantities is done there in the canonical ensemble, and promoted pairs are treated like bosons, with an approximate inclusion of the Pauli principle, the extension to the boson interactions of the IBM might be feasible, but how to treat boson-fermion couplings is an open matter for investigation. A completely different approach is the shell model MC (SMMC) method [40,41], where correlation effects are taken into account by including fluctuations of the mean field through the Hubbard-Stratonovich transformation, which permits decomposition of the many body operator exp(−βH), considered as an imaginary-time propagator, in a set of one-body propagators of noninteracting particles moving in fluctuating auxiliary fields. The SMMC method has been extensively applied by Alhassid and coworkers [42] to the calculation of level densities of nuclei in the mass and energy range that can be described in a model space limited to the fpg9/2major shell (i.e., nuclei in the 50 <A<70 region) up to excitation energies of the order of 30 MeV, by using a residual fermion interaction made of an isovector pairing interaction and a number of isoscalar multipole-multipole interactions, which are all attractive, thus yielding a good-sign Hamiltonian, i.e., avoiding fluctuating signs of integrands in MC integrals over auxiliary fields. Recently, the method has been extended to much higher excitation energies, of the order of 90 MeV for nuclei in the same mass region [43], including also the contribution of scattering states, in addition to the bound states, to the nuclear partition function, which becomes significant for weakly bound nuclei in the temperature range from 2 to 4 MeV. At higher temperature the approximation of fixed mean field used in the SMMC calculations loses its meaning. However, it is not trivial how the method could be extended to medium-heavy nuclei (70 <A<240), where the IBM and IBFM work at their best in reproducing discrete levels and related electromagnetic transitions. ACKNOWLEDGMENTS It is a pleasure to thank M. Bruno and M. D’Agostino for useful discussions. One of us (R. C.) acknowledges partial financial support from the Physics Department of the University of Bologna and the Bologna Section of the Italian National Institute for Nuclear Physics during his stay in Bologna in 2003. APPENDIX A: HIGH-ENERGY APPROXIMATION If EEc, the average collective energy, we start from the convolution (20), where we introduce expression (4) for ωcoll(Ec,M c,π c): ω(E,M,π)= πi πc∞ 0 dEi Mi+Mc=M ωintr(Ei,M i,πi) × I,c (2I+1) δ[E−Ei−Ec(I,πc)] ×fcoll(Mc,π c)δπcπi,π = πiπc Mi+Mc=M I,c (2I+1)ωintr [E−Ec(I,πc),M i,π i]fcoll(Mc,π c)δπiπc,π . (A1) We now assume that the intrinsic state density ωintr(Ei,M i,π i) also factorizes, ωintr(Ei,M i,π i)=ωintr(Ei,π i)fintr(Mi,π i),(A2) and expand ωintr(E−Ec,π i) in a Taylor series to first order in the collective energy Ec, since we have assumed that EEc. Therefore ωintr[E−Ec(I,πc),π i]≃ωintr(E,πi)−Ec(I,πc) ×∂ ∂Eωintr(E,πi) =ωintr(E,πi)1−Ec(I,πc) T(E) ≃ωintr (E,πi)exp −Ec(I,πc) T(E). (A3) Here, the inverse temperature corresponding to excitation energy Eis, by definition, 1 T=1 ωintr(E,πi) ∂ ∂E[ωintr(E,πi)].(A4) Therefore, Eq. (A1) is transformed into ω(E,M,π)= πiπc MiMc ωintr(E,πi)fintr(Mi,π i)fcoll(Mc,π c) ×Zcoll[T(E),π c]δπiπc,π δMi+Mc,M ,(A5) where the colllective enhancement factor is Zcoll(T(E),π c)= Imax(πc)  I=0 c (2I+1)exp −Ec(I,πc) T(E), (A6) which has the form of a canonical partition function. Formula (A6) lends itself to the introduction of collectivity damping with increasing temperature, provided the collective model is modified by a temperature dependence of the Hamiltonian parameters. In particular, the behavior of the partition function (A6) on the assumption that only the effective boson number decreases with increasing temperature was first studied at the limiting symmetries of IBM-1 for positive parity only [37], and, more recently, for both parities in the same transitional Sm chain [18] considered in the present work. 064320-7 R. CAPOTE, A. VENTURA, F. CANNATA, AND J. M. QUESADA PHYSICAL REVIEW C 71, 064320 (2005) We now make another assumption, that both the intrinsic and collective Mdistributions are well approximated by Gaussians, fg(Mg,π g)=1 √2πσg exp −M2 g 2σ2 g,(A7) where index gmeans either intrinsic (i) or collective (c). It is to be stressed that, in general, the width σgis a function of parity πg. On these conditions the folding of Mdistributions can be evaluated analytically: f(M,πi,π c)= Mi+Mc=M fi(Mi,π i)fc(Mc,π c) = MiMc fi(Mi,π i)fc(Mc,π c)δ(M−Mi−Mc) ≃+∞ −∞ dMcfi(M−Mc,π i)fc(Mc,π c) =1 √2πσi 1 √2πσc+∞ −∞ dMcexp −M2 c 2σ2 c ×exp −(M−Mc)2 2σ2 i =1 √2πσi 1 √2πσc exp −M2 2σ2 i+∞ −∞ dMc ×exp −bM2 c+aMc,(A8) where a=M/σ2 iand b=1/2(σ2 i+σ2 c)/σ2 iσ2 c. The integral on the right-hand side is elementary, simplifying to +∞ −∞ dMcexp −bM2 c+aMc=π bexp a2 4b(A9) and the final results for fis f(M,πi,π c)=1 √2πσ2 i+σ2 c exp −M2 2σ2 i+σ2 c. (A10) Therefore, ω(E,M,π)= πiπc f(M,πi,π c)Zcoll[T(E),π c] ×ωintr(E,πi)δπiπc,π .(A11) To evaluate the level density at given angular momentum and parity ρ(E,J,π), we make use of Eq. (23), where the difference on the right-hand side can be further approximated by a derivative, in the limit of large J: ρ(E,J,π)≃− ∂ ∂Mω(E,M,π)|M=J+1 2 = πiπc 2J+1 2√2πσ2 i+σ2 c3 2 exp −J+1 22 2σ2 i+σ2 c ×Zcoll[T(E),π c]ωintr(E,πi)δπiπc,π .(A12) The standard expression of the intrinsic level density of spherical nuclei (see, e.g., Ref. [30]) is recovered from this formula when πc=+,σ c=0, and Zcoll ≡1; σiis identified with the parallel spin cutoff factor σ. A widely used expression of ρ(E,J,π) for deformed nuclei (see, e.g., Ref. [44]) neglects the folding of intrinsic and collective Mdistributions (σc=0) and uses, instead of Zcoll, a collective enhancement factor KrotKvib, where, differently from our Zcoll, rotational and vibrational degrees of freedom are decoupled. These approximations lead to a significant overestimate of collective enhancement, at least at low energy, where collectivity damping with increasing excitation energy can be neglected. APPENDIX B: COMPARISON OF IBM AND ROTATIONAL ENHANCEMENT FACTORS In the traditional treatment of rotational enhancement [6,7] of level densities of axially symmetric nuclei, the intrinsic excitations are labeled with the projection Kof total angular momentum on the nuclear symmetry axis, and the corresponding state density ωintr(Eintr,K,π) is folded with a simple rotational level density to give the following level density for specific angular momentum J: ρ(E,J,π)=1 2 J  K=−J ωintr[E−Erot(K,J),K,π],(B1) where Erot(K,J)=¯h2 2⊥ [J(J+1) −K2],(B2) and ⊥is the moment of inertia about an axis perpendicular to the nuclear symmetry axis. In the expression of ρ(E,J,π), valid for E>E rot(K,J), the factor of 1/2 accounts for the fact that, when K= 0, the intrinsic states with projection ±K combine to form a single band with J=|K|,|K|+1, etc., whereas, for K=0, only one half of the bands contribute, for agivenJ. If there are many independent intrinsic degrees of freedom, ωintr(E,K,π) can be assumed to be normal in K: ωintr(Eintr,K,π)=1 √2πσ exp −K2 2σ2 ωintr(E,π),(B3) where σis the parallel spin cutoff factor. By repeating for ρtot(E,J,π) the steps of Taylor expansion to first order in Erot and introducing of temperature T, one obtains the well-known expression ρ(E,J,π)=1 2√2πσ ωintr(E,π) × J  K=−J exp −J(J+1) 2σ2 ⊥−K2 21 σ2 −1 σ2 ⊥, (B4) where σ2 ⊥=⊥T ¯h2(B5) is the perpendicular spin-cutoff factor. The total level density ρ(E,π) is obtained by summing Eq. (B4) over J: ρ(E,π)=∞  J=0 ρ(E,J,π). 064320-8 LEVEL DENSITIES OF TRANSITIONAL Sm NUCLEI PHYSICAL REVIEW C 71, 064320 (2005) Formula (B4) is considerably simplified when the term containing K2is negligible with respect to the term containing J(J+1) in the argument of the exponential. 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