aa r X i v : . [ nu c l - t h ] A ug Nuclear Level Density: Shell Model vs Mean Field
Roman Sen’kov ∗ and Vladimir Zelevinsky † Department of Natural Sciences,LaGuardia Community College,City University of New York,31-10 Thomson Ave., Long Island City, NY 11101, USA Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory,Michigan State University, East Lansing,Michigan 48824-1321, USA
The knowledge of the nuclear level density is necessary for understanding various reactions in-cluding those in the stellar environment. Usually the combinatorics of Fermi-gas plus pairing isused for finding the level density. Recently a practical algorithm avoiding diagonalization of hugematrices was developed for calculating the density of many-body nuclear energy levels with certainquantum numbers for a full shell-model Hamiltonian. The underlying physics is that of quantumchaos and intrinsic thermalization in a closed system of interacting particles. We briefly explain thisalgorithm and, when possible, demonstrate the agreement of the results with those derived fromexact diagonalization. The resulting level density is much smoother than that coming from theconventional mean-field combinatorics. We study the role of various components of residual inter-actions in the process of thermalization, stressing the influence of incoherent collision-like processes.The shell-model results for the traditionally used parameters are also compared with standard phe-nomenological approaches.
PACS numbers: 21.10.Ma, 21.10.Hw, 21.60.Cs
I. INTRODUCTION
The knowledge of the level density is an important ele-ment in understanding the behavior of a quantum many-body system of interacting particles in various physicalprocesses. In nuclear physics, this knowledge is necessaryfor the description of numerous reactions, including thoseof astrophysical or technological interest. The cross sec-tions can be very sensitive to the level density that typi-cally grows exponentially as a function of the excitationenergy and the number of constituents. In turn, the the-oretically predicted level density is sensitive to the statis-tics of particles, their specific interactions and availableorbital space that, in realistic computation, usually hasto be truncated. Apart from that, the level density in afinite self-bound system, such as the atomic nucleus, canbe different for the available classes of eigenstates charac-terized by different quantum numbers of exact constantsof motion (in nuclei total spin J , parity Π if we neglectweak interaction, and isospin T if we neglect interactionsviolating charge-independence).Below we consider a problem of practical microscopiccalculation of the level density for a nucleus describedby a Hamiltonian of the shell-model type. In this frame-work it does not matter if the Hamiltonian is derivedfrom a more fundamental approach or fit phenomenolog- ∗ email: [email protected] † email: [email protected] ically with the use of experimental data. We assume thatthe Hamiltonian describes the low-lying energy spectra,transition rates and other observables known from theexperiment reasonably well. Then the task is to predictthe level density of the system at higher excitation en-ergy, in the region beyond direct measurements resolvingindividual quantum states. Practically, the relevant fac-tual milestones, apart from the low-lying spectroscopy,are the regions of isolated neutron resonances near neu-tron separation energy and the results of the Oslo methodand related experimental approaches [1–4]. Of course,any practical shell-model Hamiltonian loses its validityoutside of the truncated orbital space where this Hamil-tonian was expected to work. At some excitation en-ergy, the states of the system come from the particle or-bitals not included in the model. However, with availablecomputational means, the space of validity of the modelHamiltonian can be broad enough, in particular includingthe excited states involved, for example, in astrophysicalreactions at a typical stellar temperature. We can alsohope to use the microscopic results for the nuclei far fromstability, where the level density is usually predicted bypure phenomenology [5].Another physical limitation arises from the obviousfact that the states involved in the reactions belong tothe continuum, while the standard shell-model calcula-tions work in the discrete spectrum. Instead of discretestates here one has to deal with resonances seen in var-ious reactions. Then the whole definition of the leveldensity becomes questionable and, strictly speaking, onehas to move to the complex plane of resonances. How-ever, the traditional approach is still meaningful if thetypical widths of the involved states are small comparedto the spacings between the states with the same quan-tum numbers. In what follows we limit ourselves by thissituation.Neglecting the continuum effects, the trivial solutionfor the level density generated by a certain Hamiltonianwould be a full diagonalization of the Hamiltonian matrixin an appropriate orbital basis. However, this is practi-cally only possible in sharply truncated orbital spaces,which might be only sufficient for relatively light nuclei,like in the sd -shell [6]. In many realistic cases of cur-rent interest, the dimensions of corresponding matrices,even in subspaces with given quantum numbers, are pro-hibitively large. Moreover, such a diagonalization is any-way superfluous because we do not need full informationon every excited state in spectral regions of high leveldensity. The level density is, by construction, a statisti-cal notion.In this situation, we are looking for the statistical so-lution of the problem. The physical justification of suchan approach lies in the fact that, at small level spac-ings, the stationary nuclear states are extremely compli-cated superpositions of simple determinantal states withinteger occupation numbers of definite orbitals. Gradu-ally switching on inter-particle interactions and going inthis process through multiple avoided crossings of vari-ous configurations we come to chaotic states [6, 7] withobservable properties smoothly changing along the spec-trum. Therefore our problem reduces to finding a realis-tic way to describe this smooth evolution.This purpose can be reached using the methods of statistical nuclear spectroscopy [8–10]. Already in theframework of a single partition (a certain configurationof independent particles occupying the mean field levels),the level density after including the particle interactionrapidly goes to the Gaussian limit with the increasingparticle number [11]. This is some kind of manifestationof the central limit theorem. The many-level, and there-fore many-partition, generalization should give a reliableimage of the total level density for an accepted orbitalscheme. This has to be done for each class of globalquantum numbers. This direction of theoretical searchhas a long history. We would like especially mention theworks in the direction of statistical spectroscopy appliedto shell-model Hamiltonians, see for example [12, 13].After several preliminary publications, our successful al-gorithm was constructed [14] and opened for public use[15]. The results of implementing this algorithm for thelevel density in sectors with given values of global con-stants of motion are practically identical to those fromthe full diagonalization when the latter is possible. Forwell-tested shell-model Hamiltonians, the results are ingood agreement with available experimental data.For many years, starting with the classical work byBethe [16], the nuclear level density was estimated us-ing the combinatorics based on the ideas of a Fermi gas.The influential review of earlier approaches of this type was given by Ericson [17], the later derivations can befound in [18–21], see also [22]. The recent achievementsin this direction [23–25] include the pairing correlationsconsidered as a part of the self-consistent mean fieldin the framework of the BCS theory or Hartree-Fock-Bogoliubov variational ansatz. The shell-model Monte-Carlo methods [26–28], being very demanding computa-tionally, work relatively well at least with some parts ofthe full shell-model interaction but require the projectionto the correct values of spin and parity.The chaotization of the dynamics mentioned aboveleads to the possibility of describing the physics of excitedstates at high level density in terms of statistical thermo-dynamics including temperature, entropy etc. This wasunderstood in application to nuclear reactions from theearly times of nuclear physics [16, 29, 30]. The detailedanalysis of atomic [31, 32] and nuclear [6, 33] chaoticstates supported an old idea [34] of thermalization in aclosed system driven by the interactions between the con-stituents, with no heat bath: the average over a genericchaotic wave function in a chaotic region is equivalentto the average over a standard equilibrium thermal en-semble [35]. Currently this idea, sometimes called the“eigenfunction thermalization hypothesis”, is extensivelydiscussed in the many-body physics community [36]. Oneof the purposes of the current publication is in compar-ison of the exact shell-model nuclear level density withphenomenological ideas based on the Fermi-gas picture atcertain temperature. We look at these ideas and basedon them equations from the viewpoint of our numeri-cal results. Our attention will be mostly concentratedon the usually cited empirical parameters of the leveldensity and their energy and spin dependence. Anotherpoint of interest is in the role of various components ofthe shell-model interactions in the formation of the leveldensity. One important result is that the considerationof the mean field, even with addition of the BCS-typepairing, is not sufficient. Incoherent components of theinteraction in a finite many-body system, as a rule ne-glected in the mean-field combinatorics, play a significantrole smoothing the energy behavior of the level density.In what follows we briefly explain the method and givethe examples of practical calculations. The results willbe compared with what would follow from traditionalphenomenological models. II. MOMENTS METHOD
We consider a finite system of interacting fermions de-scribed by the standard Hamiltonian H = X ǫ a † a + 14 X V a † a † a a (1)that contains the mean-field part with effective single-particle energies ǫ and the antisymmetrized two-bodyinteraction. The generalized numerical subscripts com-bine all quantum numbers of single-particle orbitals. Inthis form the method can be applied to nuclei, atomicor molecular electrons, atoms in traps etc., any systemwhere the residual interaction is sufficiently strong toproduce complicated eigenstates. The three-body forcescan be included in the same way although the computa-tions become more cumbersome. Using a phenomenolog-ical shell-model Hamiltonian we assume that many-bodyforces, at least partly, are included in fitted matrix ele-ments. The quality of the Hamiltonian is checked by theexplicit applications to individual low-lying states andcomparison with available experimental information.In a finite self-bound system, such as the atomic nu-cleus, the total angular momentum (nuclear spin) is ex-actly conserved supplying good global quantum numbers J and M . Therefore it is convenient from the very be-ginning to use a spherically symmetric basis of single-particle orbitals | jm ) which define, along with the orbitalmomentum ℓ , main quantum number ν , and isospin τ ,the quantum numbers combined in Eq. (1) into a unifiednumerical subscript. If the orbital space is sufficientlybroad, this spherical shell model can describe intrinsicdeformation without violating rotational symmetry [37].In practice, the operators in the Hamiltonian can be com-bined into pairs, such as ( a a ) L Λ with the angular mo-mentum quantum numbers of the pair, L Λ, fixed throughthe Clebsch-Gordan coefficients. In the same way, for theisospin-invariant forces, the isospin of the pair can be alsofixed, and then the interaction V in a restricted orbitalspace is defined through a finite number of pairwise ma-trix elements. If more convenient for computations, onecan as well use the M -scheme without vector coupling.The practical algorithm of calculations in the M -scheme follows from individual configurations, p , ( par-titions ) which are possible distributions ( n , n , ... ) ofavailable particles, P n = N , over single-particle or-bitals (here the index does not include the projection m ). The many-body states | α i possible for each parti-tion form a subspace where α combines total quantumnumbers N, M, T and parity. It is convenient [14] to usethe proton-neutron formalism.Let D αp be the dimension of the class of states withglobal quantum numbers α built on the partition p . Asshown in statistical spectroscopy [9, 11] and confirmed inmany examples by the exact shell-model diagonalization,the density of states for a given partition is close to theGaussian. Of course, this is the main assumption basedon a rich experience with the features of quantum chaosin mesoscopic systems. The characteristics of the Gaus-sian are defined by the moments (traces) of the actualHamiltonian. The centroid is just the mean energy valuefor a given partition, E αp = h H i αp = 1 D αp Tr ( αp ) H. (2)The dispersion of the Gaussian, σ αp , is the second mo-ment, σ αp = h H i αp − E αp ≡ D αp Tr ( αp ) H − E αp . (3) It is important to stress that the calculation of thesetraces does not require the diagonalization of large-scalematrices. The first moment (2) is the diagonal matrixelement of the Hamiltonian averaged over the partition,while the second moment (3) is the sum of squared off-diagonal elements along one line of the Hamiltonian ma-trix, again averaged over the lines corresponding to thepartition. It is known that the dispersion for each ba-sis state very weakly fluctuates within a partition [6, 38]even prior to the next averaging. The second momentincludes all interactions coupling the partitions. If thetraces were calculated in the M -scheme, we obtain the density of states counting all M -degenerate states withinthe multiplets. To obtain the level density for given spin J , we have in a standard way to find the difference oftraces for M = J and M = J + 1.The total level density is given by summing the con-tributions of partitions using the constructed Gaussians G αp ( E ) with their centroids (2) and widths (3), ρ ( E ; α ) = X p D αp G αp ( E ) . (4)As was understood earlier [39], it is better to use finiterange Gaussians , G αp ( E ) = G ( E − E αp + E g . s . ; σ αp ) , (5)where G ( x ; σ ) = C (cid:26) e − x / σ , | x | ≤ ησ , | x | > ησ . (6)Here E g . s . is the ground state energy to be defined sep-arately, C is the normalizing factor, R dx G ( x, σ ) = 1,and the cutting finite-range parameter η has to be foundempirically [40]; its actual value η ≈ . N . In this case, the standardformulation of the shell model includes cross-shell tran-sitions with unphysical excitations of the center-of-mass.These spurious states are to be excluded from the leveldensity. In some versions of the shell model, these statesare artificially shifted to high energies. Here, the sub-traction of ghost states is accomplished by renormalizingthe contaminated level density ρ ( E, J ; N ) through therecurrence relations. For example, while the N = 0 case,which will be called the ρ ◦ ( E, J ; 0) approximation, is freeof admixtures, the pure level density ρ ◦ at the next step(no admixtures of the single center-of-mass excitation) isfound as ρ ◦ ( E, J ; 1) = ρ ( E, J ; 1) − J +1 X J ′ ≤| J − | ρ ( E, J ′ ; 0) . (7)
20 40 60 80Excitation energy (MeV)050100 N u c l ea r l e v e l d e n s it y ( M e V - ) J=0
20 40 60 80Excitation energy (MeV)0100200300 N u c l ea r l e v e l d e n s it y ( M e V - ) J=1
20 40 60 80Excitation energy (MeV)0200400 N u c l ea r l e v e l d e n s it y ( M e V - ) J=2
20 40 60 80Excitation energy (MeV)0200400 N u c l ea r l e v e l d e n s it y ( M e V - ) J=3
FIG. 1: Nuclear level densities for Si, Π = +1, various spins; shell model for the sd -shell and USDB two-body interaction(solid curves) vs. moments method (dashed curves); finite-range parameter η = 2 . Here the sum goes over the intermediate angular mo-menta J ′ from | J − | to J + 1 since the center-of-massoperator is equivalent to a vector. If higher admixtures N > H with various intermediatestates slightly more involved [40, 43]. III. EXAMPLES OF LEVEL DENSITYA. Comparison with the exact solution of the shellmodel
The first natural check of the approach is in compar-ison of the resulting level density with the picture aris-ing from the full shell-model diagonalization in the caseswhere such a diagonalization is technically plausible. The sd -shell model for a long time is known as the best ex-ample of exact diagonalization. The model is completelyfixed by the effective single-particle energies d / , s / ,and d / , and 63 phenomenologically fitted matrix el-ements of two-body interaction. The model works ex-tremely well for practically all observables of sd -nucleiand not only for the lowest states. For example, both,the experiment [44], and the sd -shell model [42], indicate the existence of ten stationary states with J Π = 0 + upto excitation energy of 15 MeV in Si, therefore provid-ing the same average level density at least at not veryhigh energy; the mean level spacing between those 0 + levels is 0.95 MeV in the experiment and 1.02 MeV inthe shell-model calculation.Fig. 1 illustrates the results of the moments methodfor calculating the level density in Si, a typical objectof the sd -shell model applications that has served longago as a testing ground for quantum chaos [6]. We seethat the level density for different classes of states, here0 + , + , + , and 3 + , is always a smooth curve of the Gaus-sian type. Of course, as the calculations have been donein the restricted orbital space, the real physical resultthat can be juxtaposed to the experimental data and usedfor the reaction calculations always corresponds only tothe left hand side of the full graph and to the excitationenergy below the centroid maximum.Various specific quantum numbers (total spin and par-ity) produce the level density of the same qualitative be-havior, with the integral corresponding to exact multi-plicities of states with a given set of quantum numbersin a fixed orbital space. All examples look the same. Theagreement with the exact shell-model diagonalization isalmost perfect, with slightly more visible fluctuations forthe class J Π = 0 + that has a smaller total dimension. In N u c l ea r l e v e l d e n s it y ( M e V - ) J Π =1 − N u c l ea r l e v e l d e n s it y ( M e V - ) J Π =2 − FIG. 2: (Color online.) Comparison of the nuclear level densities of Ne in 1¯ hω -approximation, Π = −
1. The densities arecalculated within the shell model approach (stair-case curves) and with the moments method (solid curves). The black curveson the left and right graphs (both are higher at low energies) correspond to the densities with spurious states included; theblue curves (that are lower at low energies) correspond to the densities without spurious states; the red (dotted) curves presentthe spurious nuclear level densities. N u c l ea r l e v e l d e n s it y ( M e V - ) J Π =0 − N u c l ea r l e v e l d e n s it y ( M e V - ) J Π =1 − FIG. 3: (Color online.) Comparison of nuclear level densities calculated with the shell model for Mg in 1¯ hω -approximation,Π = −
1, and the moments method. The coloring scheme is the same as in Fig. 2. The small peaks around 140 MeV show thespurious densities. The inserts present the low-energy part. all cases we see a small deviation near the centroid whichsupposedly can be eliminated by taking into account thefourth moment of the Hamiltonian (but it makes no senseto go for such complicated and time-consuming calcula-tions to improve the results in the region outside thephysically relevant area).The smoothness and Gaussian behavior of results in allcases confirm the possible thermodynamic interpretationin terms of entropy S ( E ) (mean logarithm of the leveldensity) and temperature, dS/dE = 1 /T ( E ). Formally,the centroid of the level density for the finite orbital spacecorresponds to infinite temperature and the right halfof the curve to negative temperatures. The full shell-model analysis of the wave functions [6] has found thatthe same effective temperature can be extracted by thesingle-particle thermometer using the occupation num- bers of available spherical orbitals for individual station-ary many-body states . The interaction of the quasipar-ticles in the self-consistent mean field acts as the heatbath, and the chaotic mixing of the eigenstates leads tothermalization even in such a small Fermi system. B. Elimination of spurious states
As mentioned above, the full shell-model diagonaliza-tion in the cases with the presence of transitions betweenthe orbitals of opposite parity (excitations across the os-cillator shells) brings in ghost states related to the center-of-mass motion rather than to intrinsic excitations. Weexplained above the recurrent techniques used for elimi-nating these spurious states and obtaining the pure level N u c l ea r l e v e l d e n s it y ( M e V - ) Π=+1 N u c l ea r l e v e l d e n s it y ( M e V - ) Π=−1
FIG. 4: Comparison of the experimental nuclear level densities (stair lines) and the densities calculated with the momentsmethod (straight lines) for Si, all J , positive (the left graph) and negative (the right graph) parity. N u c l ea r l e v e l d e n s it y ( M e V - ) Π=+1 N u c l ea r l e v e l d e n s it y ( M e V - ) Π=−1
FIG. 5: Comparison of the experimental nuclear level densities (stair lines) and the densities calculated with the momentsmethod (straight lines) for Al, all J , positive (the left graph) and negative (the right graph) parity. density. The recipe frequently used in the shell model isthe brute-force shift of the undesired states to high en-ergy by adding to the Hamiltonian under diagonalizationa Lawson term [45] that in the harmonic oscillator fieldof frequency ω looks as ( β > H ′ = β (cid:20) H c . m . −
32 ¯ hω (cid:21) A ¯ hω . (8)As was shown long ago [6], this recipe indeed generatesa new branch of eigenstates shifted to high energy (byabout ∼ β N A ) but having essentially the same com-plexity (measured by the information entropy) as theirpredecessors without spurious admixtures.The separation of unphysical spurious states accordingto the recurrence relations (7) works well. The standardshell-model shift of these states to higher energy pro-duces exactly the same group of states as our procedure.Fig. 2 shows the level density calculated for Ne in thebroad space of s + p + sd + pf orbitals in the 1¯ hω ap- proximation (only one-step excitation of negative paritystates that include non-physical admixtures). The full(“blind”) calculation compared to that treated with re-currence relations (7) contains the excess shown by thesmall (red dotted) Gaussian-type curves at relatively lowenergy. We see that this curve of difference exactly coin-cides with the result of the shell-model shift according toEq. (8). This means that the method of recurrence rela-tions correctly removes the unphysical excitations. Thesame conclusion can be made for Mg, Fig. 3, where theinsets zoom out the region of low energy.
C. Comparison to experimental data
As stated earlier, the density of low levels with J Π =0 + in the favorite nucleus Si of the sd shell model is ingood agreement with the data. Figs. 4 and 5 compare thelevel density calculated with the moments method (solid N u c l ea r l e v e l d e n s it y ( M e V - ) exact SM (pf-shell, gx1a)moments methodmodel of Goriely et al. Fe, J Π = 0 + N u c l ea r l e v e l d e n s it y ( M e V - ) exact SM (pf-shell, gx1a)moments methodmodel of Goriely et al. Fe, J Π = 1 + N u c l ea r l e v e l d e n s it y ( M e V - ) exact SM (pf-shell, gx1a)moments methodmodel of Goriely et al. Cr, J Π = 0 + N u c l ea r l e v e l d e n s it y ( M e V - ) exact SM (pf-shell, gx1a)moments methodmodel of Goriely et al. Cr, J Π = 1 + FIG. 6: Nuclear level densities for Fe and Cr, Π = +1, different spins: shell model calculation (solid curves) vs . momentsmethod (dashed curves) and Hartree-Fock + BCS (dotted corves). Finite-range parameter η = 2 . pf -shell, GXPF1Atwo-body interaction. lines) with the experimental results (stair-cases). Thesolid stair-cases present an “optimistic” attitude whenall experimental levels of uncertain parity were counted.Opposite to that, the dashed lines exclude the levelswhose parity is given by the experiment only tentatively.The left graph of Fig. 4 shows the summed level den-sity of all positive parity states in Si calculated for the s + p + sd + pf space up to 14 MeV excitation energy.The same quality of comparison can be seen for the pos-itive parity states in Al, the left graph of Fig. 5, theodd-odd nucleus with the relatively well measured energyspectrum.For the negative parity states, the right parts of Figs.4 and 5, the shell model predicts more levels than untilnow have been found experimentally. This can be bothdue to the imperfection of the shell-model Hamiltonianand because of incompleteness of the data.
IV. SHELL-MODEL PREDICTIONS ANDMEAN-FIELD COMBINATORICS
The widely used standard road to the nuclear leveldensity is going through the mean-field representation ofthe nuclear dynamics. This traditional approach is based on the classical idea [16, 17] of the Fermi-gas where theexcited levels result from combinations of many particle-hole excitations. Practically, the combinatorics is used ofsingle-particle excitations from the fully occupied Fermisurface identified with the ground state population of thelowest individual orbitals. At low excitation energy, arenormalization of the level density is related to the gapdue to the Cooper pairing. In complex nuclei, the low-energy levels observed inside the gap can be interpretedas collective excitations, vibrational or/and rotational.As the collective phenomena of these types correspondtypically to the slow self-consistent motion of many par-ticles, it is natural to expect that such coherent combina-tions of single-particle excitations partly compensate thedeficit of levels at low energy due to the pairing gaps andgive rise to the so-called collective enhancement of thelevel density [22, 46] in comparison to the single-particlecombinatorics of independent particles and holes. Mod-ern refined approaches of this class account in variousforms for the pairing phenomenon that changes the exci-tation spectrum, especially in even-even nuclei [23–25].In the spirit of the mean-field combinatorics, one has toexpect the corresponding suppression of level density athigher excitation energy (damping of collective enhance-ment); the level density is just redistributed. When the N u c l ea r l e v e l d e n s it y ( M e V - ) Moments, pf-shellMoments, pf+g9/2-shellmodel of Goriely et al. Ge, J Π = 0 +
01 0 0 02 0 0 03 0 0 04 0 0 05 0 0 06 0 0 07 0 0 08 0 0 0 N u c l ea r l e v e l d e n s it y ( M e V - ) Moments, pf-shellMoments, pf+g9/2-shell Ge, J Π =0 + FIG. 7: Level densities for Ge, spin J = 0 and Π = +1. The solid curve presents the calculation in the pf shell with theGXPF1A interaction, the dashed curve corresponds to the calculation in the larger model space with the level g / added,the dotted curve on the left graph presents the results obtained using the HFB single-particle energies and the combinatorialmethod. Si, J=all, k =k ={0.1-1.0} Si, J=0, k =k ={0.1-1.0} FIG. 8: (Color online.) Level densities for Si, sd model space. Different curves correspond to different scale factors: k = k = { . , . , . , . , . } when the pairing and non-pairing parts of the interaction scale similarly. The left graphcorresponds to the total density with all J included, while the right graph describes the evolution of the J = 0 density. general level density grows, the vibrational modes be-come strongly mixed with simpler excitations of the two-quasiparticle and more complicated structure, as knownvery well from the widths of the giant resonances. Withsmoothing shell gaps, it is harder to distinguish betweenrotational and intrinsic motion. Recent experiments innuclei, where low-lying collective excitations are wellknown, did not find phenomena of collective enhance-ment and its fade-out [47].In not too heavy nuclei, the quasiparticle combi-natorics (on the base of the BCS or Hartree-Fock-Bogoliubov pairing description) reveals step-wise effectsof subshell occupation and pair breaking. This leads, atrelatively low energy, to the irregular picture of the leveldensity that clearly reflects these steps. The shell-modelHamiltonians, as a rule, contain all interaction matrix el- ements allowed by the selection rules. One of the mainconclusions of the full shell-model calculation is that thepresence of all interactions is significantly smoothing thewhole picture so that it is hard to see the traces of in-dividual families which could be still recognized only bythe special observables and selection rules for the indi-vidual transitions. We can recall that our algorithm stillstarts with the partitions formed by independent parti-cles which then overlap and lose their boundaries.The shell-model Hamiltonian contains all pairing ma-trix elements (and not in the simplified form with con-stant matrix elements) as well as the interaction pro-cesses responsible for multipole-multipole forces and de-formation. Therefore all collective effects mentionedabove are fully taken into account if the orbital spaceis sufficiently broad. Fig. 6 shows the comparison of Si, J=all, k =0.1, k ={0.1-1.5}k =1.0 Si, J=0, k =0.1, k ={0.1-1.5}k =1.0 FIG. 9: (Color online.) Level densities for Si, sd model space. The non-pairing interaction is always off: k = 0 .
1, while thepairing interaction scales, k = { . , . , . , . , . , . } . The left graph corresponds to the total density with all J included,while the right graph describes the evolution of the J = 0 density. Si, J=0, k =1.0, k ={0.1-1.0} FIG. 10: (Color online.) Level densities for Si, J = 0, sd model space. The black curve presents k = 1 . , k =0 .
1, then the remaining parts of the interaction are in-creased together with k up to the red curve ( k = k =1 .
0) that shows the density for the realistic interaction. Fe, J=all, k =k ={0.1-1.0} FIG. 11: (Color online.) Level densities for Fe, all J , pf model space. This figure and its color scheme are similarto the left panel on Fig. 8. the level densities for the states 0 + and 1 + in the nu-clei Fe and Cr of the pf -shell. The thin dotted linesgive the level density found with the mean-field combi-natorics built on the Hartree-Fock mean field and BCSpairing description. All irregularities of the level den-sity found through the mean-field combinatorics are com-pletely smoothed in the full moments calculation. Thisis a typical result encountered in all examples. Againwe see that the method under discussion produces thelevel density practically identical to the full shell-modeldiagonalization when the latter is possible.Fig. 7 illustrates the influence of the enlargement ofthe orbital shell-model space. The level density of states0 + in the N = Z nucleus Ge becomes sensitive to theinclusion of the next shell ( g / ) only at the excitationenergy greater than 14 MeV (see the right graph of Fig.7) which means that the region of neutron resonances could be reliably evaluated with the more narrow orbitalspace. This case has important ramifications for the as-trophysical consideration of the element abundance sincethis nucleus is considered to be a waiting point in the r -process of nucleosynthesis. V. COHERENT AND INCOHERENTINTERACTIONS
As the whole shell-model Hamiltonian contributes tothe traces defining the level density, we can explore theeffects of individual components of the effective interac-tions including the “incoherent” parts of the full Hamil-tonian which do not significantly contribute to the for-mation of the mean field. These parts of the interactiondetermine the finite lifetime of the simple quasiparticle0
20 30 40 50 60 70 80024681012 a - p a r a m e t e r All othersZ=20 IsotopesZ=21 IsotopesZ=22 IsotopesN=40 Isotones
20 30 40 50 60 70 80 mass number, A -505 c on s t a n t
20 30 40 50 60 70 80024681012 a - p a r a m e t e r Moments MethodFermi-Gas: RohrFermi-Gas: Al-Quraishi
20 30 40 50 60 70 80 mass number, A -505 c on s t a n t FIG. 12: (Color online.) Interpolation of the single-particle level density parameter a , Eq. (15). Left panel: different colorspresent different isotopes or isotones. Right panel: moments-method calculation with interpolation (black circles), fit usingthe experimental data on neutron resonances (Ref. [51], orange diamonds), and fit using experimental low-lying levels ([52],yellow squares). (or collective) modes and their fragmentation in terms ofgenuine complicated eigenstates of exceedingly entanglednature. In particular, these collision-like interactions areresponsible for the formation of chaotic states with highinformation entropy and the process of thermalization[6]. For example, it was shown [48–50] that the exactlyconsidered pairing interaction contains some chaotic fea-tures but they are still not sufficient for establishing thecomplete chaotic picture comparable to the predictionsof the Gaussian Orthogonal Ensemble.Below we show the evolution of the level density as afunction of the interaction modes included in the full cal-culation of the moments. Although it is not difficult tovary all individual matrix elements, here we use a simpli-fied approach, presenting the whole two-body interactionHamiltonian in the sd -shell model as consisting of twoparts with variable intensity, H = h + k V (pairing) + k V (non − pairing) . (9)Here h contains the single-particle energies, V (pairing)all matrix elements with the pairs of nucleons in thechannel J π T = 0 +
1, while all other shell-model matrixelements are attributed to the last term. The numericalcoefficients k and k are varied giving rise to differentversions of the shell model; the realistic case emerges at k = k = 1. It is easy to understand that in the mo- ments method (and in the full diagonalization, see [6])new independent components of the Hamiltonian add inquadratures to the final width of the shell-model leveldensity.The global evolution of the full level density in Sias the coefficients k = k are varied from 0.1 to theirrealistic values is shown in the left graph of Fig. 8. Thelow residual interaction obviously keeps untouched theindependent-particle partition structure of the Hilbertspace that reminds the results of the mean-field combi-natorics. As the parameters k , k grow, the next curvesshow the development of the final picture. The configu-rational structure is gradually washed out by the resid-ual interaction leading to the final smooth level densitydiscussed earlier. Let us stress that the observed evo-lution is not a consequence of the superposition of allsubspaces with different values of J . The individual sub-space J Π = 0 + , the right graph on Fig. 8, demonstratespractically the same evolution.Fig. 9 describes the situation when the non-pairingcomponents of interaction are suppressed, k = 0 . J (the left graph of Fig. 9), we see that thesmooth Gaussian-like curve is achieved only at the non-realistically high pairing strength. Again the picture isnearly the same when only the states J = 0 are con-1
20 30 40 50 60 70 80
Mass Number, A a - p a r a m e t e r fitting range: 1-5 MeVfitting range: 5-25 MeV FIG. 13: (Color online.) Level density parameter (11) fittedin the simple Fermi-gas model for sd and pf shell nuclei. Theempty circles (red color) present the fitting range 1-5 MeV,the filled black circles correspond to the fitting range 5-25MeV.
20 30 40 50 60 70 80
Mass Number, A -8-6-4-202468 % ( ∆ a / a ) δ = - 1.0 MeV δ = 1.0 MeV FIG. 14: The relative changes in the level density parameter a due to the pairing energy shift δ , Eq. (17), shown for sd -and pf -shell nuclei. The empty circles present the fit withthe energy shift δ = − δ = 1 MeV. sidered, the right graph of Fig. 9. The realistic pairingstrength, k = 1 .
0, at the absence of non-pairing interac-tions is still not sufficient for the fully smooth level den-sity. At low excitation energy <
20 MeV the evolution ofthe level density for J = 0 clearly shows the disappear-ance of the typical large oscillations with the growth ofpairing. Here, indeed, the pairing interaction shifts thenoticeable part of levels to higher energies. If the pairingmatrix elements are fixed at the empirical value, Fig. 10,the large bumps from the original partitions do not ap-pear but the incoherent interactions very much broadenthe final result. The generic character of this scenario isconfirmed by Fig. 11 for Fe.
VI. THERMODYNAMIC DESCRIPTION ANDCOMPARISON WITH PHENOMENOLOGYA. Simple Fermi-gas
The shell-model Hamiltonian (1) starts from non-interacting particles or quasiparticles, elementary exci-tations in the mean field of certain symmetry that de-termines the appropriate quantum numbers of excitedstates. For nuclei, the adequate image is that of the per-fect two-component Fermi gas. The ground state of thesystem is the filled Fermi sphere, and the excited statesare described by the particle-hole picture. In the realisticmany-body physics, this is just an initial step that has tobe followed by switching on the interaction between par-ticles and holes. However, already by this mechanism,the level density increases exponentially which justifiesthe traditional phenomenological approaches.The particle-hole phenomenology uses the steepest de- scent method to calculate the level density as a functionof excitation energy E through the Laplace transform ofthe partition function which leads to the standard resultfor one type of particles, ρ ( E ) = 14 √ E e √ aE , (10)where the level density parameter is a = π ν F (11)and ν F is the density of single-particle states at the Fermisurface. The generalization to a proton-neutron systemleads to a modified expression, ρ ( E ) = √ π ¯ a
12 ( aE ) / e √ aE = 6 / ¯ ν
12 ( ν F E ) / e √ aE , (12)where the parameters a and ¯ a now include the totalsingle-particle density of states at the Fermi surface, ν F = ν F ( n ) + ν F ( p ), and the effective single-particledensity ¯ ν = ν F / (2 p ν F ( n ) ν F ( p )) , correspondingly, seeEq. (11). The singularities at E → E ≫ ν F . (13)2 Excitation Energy [MeV] T e m p e r a t u r e [ M e V ] Si: J=0 Si: J=1 Si: J=2 Si: all J Fe: J=0 Fe: J=1 Fe: J=2 Fe: all J Si Fe FIG. 15: (Color online.) Thermodynamic temperature as afunction of excitation energy, Eq. (18), calculated in the mo-ments method. The top four curves (black color) correspondto the densities of Si and the bottom four curves (blue color)correspond to the densities of Fe.
20 30 40 50 60 70 8005101520 T e m p e r a t u r e [ M e V ] All othersZ=20 IsotopesZ=21 IsotopesZ=22 IsotopesN=40 Isotones
20 30 40 50 60 70 80
Mass Number, A -20246 c on s t a n t FIG. 16: (Color online.) The constant temperature T andthe “constant” (see Eq. (19)) fitted for the energy range 5-15MeV. Different sets of nuclei from the pf shell with completeand almost complete shells are presented by different colors: Z = 20 (red), Z = 21 (green), Z = 22 (blue), and N = 40(brown). In thermodynamic language, the nuclear temperature t is introduced through the Fermi-gas formula for the ex-citation energy, E = at . (14)In general, for the low excitation energy region it is al-ways recommended to use directly the available experi-mental information.If one tries to compare the thermodynamic level den-sity of the Fermi-gas type with experimental data, it ishard to expect the numerical agreement of the level den-sity parameter (11) with that required by data even ifthe exponential growth of the total level density takesplace. As mentioned in the Introduction, when the leveldensity grows, the residual interactions lead to multipleavoided crossings and mixing of many-body levels withthe same exact quantum numbers. This process of chao-tization evolves the level network considered as a functionof the interaction strength close to the aperiodic crystalwith a small average spacing. The whole set of stationarystates is becoming locally close to the predictions of theGaussian Orthogonal Ensemble of random matrices. Thewave functions here are quite complicated superpositionsof very many particle-hole states. The energy behavior ofthe level density is now close to a Gaussian [6] with thetotal width that is given by adding in quadratures theinitial width due to the mean-field quasiparticles and thedispersion of the off-diagonal matrix elements of residualinteractions.Fig. 12 shows the parameters a (the top panels) and constant (the bottom panels) found from the shell-model calculation of the total level density fitted byln[ ρ ( E, M = 0)] = 2 √ aE −
54 ln E + const . (15)Both the parameter a and the constant show the changeclearly correlated with the microscopic filling of the nu-clear shells. The parameter a reveals the maximum inthe middle of the shell occupation as it should have beenexpected from the construction of the model. The em-pirical estimates for the same nuclei are available fromthe level density at the neutron resonances energy [51],which do not show considerable shell effects, and by ex-trapolation from low-lying levels [52] where one can seevery weak shell effects in the region of the mass num-ber around A ≈
50. The constant in Eq. (15) is smallbut also shows in some cases the shell-model dependencewith a minimum in counter-phase with the parameter a .Fig. 13 shows the dependence of the level density pa-rameter a on the energy range for which the fitting wasperformed. The empty circles (red color) present the lowenergy fit, around 1-5 MeV, while the filled circles (blackcolor) present the standard 5-25 MeV energy range fit.We can see that at low energies the fitted level density pa-rameter a is slightly larger, but still it is not large enoughto be compared with the empirical estimates [51, 52]. B. Back-shifted Fermi-gas model
In this paper we do not discuss in many details thenuclear pairing correlations and its importance for thenuclear level densities leaving this for the future consid-eration. One of the standard phenomenological way to3
Excitation Energy [MeV] N u c l ea r L e v e l D e n s it y [ M e V - ] Si, Exp[E/T+const], fitting: 5-15 MeV Si, Exp[E/T+const], fitting: 5-25 MeV Si, ρ M (E,0), moments method FIG. 17: Comparison between global level densities for Sicalculated with the moments method (the solid curve) andusing the constant temperature for 5-15 MeV (the dottedcurve) and for 5-25 MeV (the dashed curve) energy range;fits according to Eq. (19). The inset presents the enhancedlow-energy region.
20 30 40 50 60 70 80
Mass Number, A F itt e d C on s t a n t T e m p e r a t u r e [ M e V ] fitting range: 1-5 MeVfitting range: 5-15 MeVfitting range: 5-25 MeV FIG. 18: (Color online.) Constant temperature fitted for sd and pf shell nuclei. The diamonds (red color) present thefitting range 1-5 MeV, the circles (black color) correspond to5-15 MeV and the squares (blue color) correspond to 5-25MeV fitting ranges. account for pairing correlation is to use the back-shiftedFermi-gas formula (BSFG) [18, 53], ρ BSF G ( E ) = √ π ¯ a
12 [ a ( E − δ )] / e √ a ( E − δ ) , (16)where the excitation energy is shifted by the pairing en-ergy parameter δ . This introduces a new free parameterthat should be fitted alongside with the level density pa-rameter a . The pairing energy parameter δ is in generaldifferent for even-even, odd-odd, and even-odd nuclei dueto the formation of Cooper pairs, and does not necessarilycoincide with the corresponding pairing gap parameters∆ or 2∆.Fig. 14 shows the sensitivity of the level density pa-rameter a to inclusion of the energy shift δ . The quantityplotted along y -axis is the relative change in a if we shiftthe excitation energy by δ ,100% × a − a BSF G a , (17)where a BSF G is the level density parameter fitted usingEq.(16) with a fixed value of the pairing energy δ . Wecan see that the a parameter varies only in a ±
6% rangewhen the shift changes from δ = − δ = 1 MeV (filled circles). C. Constant temperature model
The model of the energy dependence of the level den-sity different from the Fermi-gas phenomenology (12) be-ing suggested long ago [18, 53] gradually becomes popularamong practitioners. It is assumed that the level density,at least up to 10 MeV excitation energy, and maybe even higher [54–56], can be described by the constant tempera-ture T . This temperature is the single parameter defined,in the simplest version, according to the thermodynamicsas T = (cid:20) d ln ρ ( E ) dE (cid:21) − . (18)The philosophy behind this approach is usually explained[57] in terms of the first-order phase transition that goesthrough the latent heat at fixed temperature. Althoughtypically this assumes the melting of the Cooper pairs butin fact one can also talk about other types of correlatedstructures which are undergoing something similar to theliquid-gas phase transition or even the first stage on theroad to multifragmentation. In a more detailed descrip-tion, the effective temperature parameter can be differentfor the classes of states with different quantum numbers,although such a generalization does not look well fromthe viewpoint of the thermal equilibrium between vari-ous degrees of freedom. Such an effective temperatureparameter (plus a corresponding constant) could be fit-ted in a reasonable energy range to represent the partial(with certain spin and parity) or total nuclear level den-sities as ln[ ρ ( E, J )] = ET J + constant , ln[ ρ ( E )] = ET + constant . (19)It is immediately clear that globally the constant tem-perature model cannot be compatible with our shell-model calculations. In the truncated orbital space theglobal level density will always look as a Gaussian with4 L og [ ρ ( Ε , Μ )] E=5 MeVE=10 MeVE=15 MeVE=20 MeVE=25 MeV Si L og [ ρ ( Ε , Μ )] E=5 MeVE=10 MeVE=15 MeVE=20 MeVE=25 MeV Fe M -4-20246 L og [ ρ ( Ε , Μ )] E=5 MeVE=10 MeVE=15 MeVE=20 MeVE=25 MeV Ca M -4-20246 L og [ ρ ( Ε , Μ )] E=5 MeVE=10 MeVE=15 MeVE=20 MeVE=25 MeV Cr FIG. 19: (Color online.) Logarithm of level density, ln[ ρ ( E, M )], versus M , Eq. (23), calculated for Si, Fe, Ca, and Cr. Different colors present different excitation energies: 5 MeV (black), 10 MeV (red), 15 MeV (green), 20 MeV (blue),and 25 MeV (brown). Dots correspond to the moments method calculations, the solid lines are linear interpolations, see the text. the effective temperature (18) T eff = σ E E c − E , (20)changing with energy from positive to negative valueson different sides of the centroid energy E c . Here σ E is the effective width of the Gaussian that reflects thesummed contribution of all components of interactions.However, at relatively low energy the level density cangrow approximately exponentially effectively resulting inan approximately constant temperature.Here we give a couple of examples showing the ex-ponential fit to the level density of different j -classesand global level density. Fig. 15 shows how the actualtemperature (see Eq. (18)) calculated by the momentsmethod for Si and Fe depends on the excitation en-ergy. The temperatures calculated for certain spins J andfor all spins (the total density) are not constants, theyincrease with the excitation energy as suggested in Eq.(20). The corresponding constant temperature fit of Eq.(19) performed for the sd and pf nuclei is presented in Fig. 16. The effective constant temperatures in the figuredepend on the range of the excitation energies where thefit was performed, − the greater the excitation energythe higher the effective constant temperature.We note that the majority of the fitted temperatures inFigs. 15 and 16 are concentrated in small regions near 2-5MeV, while there are some exceptional cases of unreason-ably high temperatures of 10-20 MeV that correspond tothe nuclei with complete or almost complete shells for oneor two sorts of nucleons when the Fermi gas approxima-tion is obviously invalid. Finally, Fig. 17 shows how goodthe constant temperature approximation is. The dottedcurve (fitting energy range is 5-15 MeV) and the dashedcurve (the range is 5-25 MeV) present the correspond-ing constant-temperature level densities for Si. We cansee that these densities work pretty well inside the en-ergy interval where they were fitted compared to the “ex-act” density calculated within the moments method. Asthe excitation energy increases the constant-temperaturedensities stop working going too high very fast. Fig.18 shows the fitted temperatures when different energy5
20 25 30 35 40 45 50 55 60 65 70 75 8002468 α - p a r a m e t e r All othersZ=20 isotopesZ=21 isotopesZ=22 isotopesN=40 isotones
20 25 30 35 40 45 50 55 60 65 70 75 80 mass number, A -0.0500.05 β - p a r a m e t e r
20 25 30 35 40 45 50 55 60 65 70 75 8002468 α - p a r a m e t e r Moments MethodStatistical calcRigid sphere
20 25 30 35 40 45 50 55 60 65 70 75 80 mass number, A -0.0500.05 β - p a r a m e t e r FIG. 20: (Color online.) Interpolation of the spin cut-off parameter σ , Eq. (24). Parameters α and β are shown for different sd and pf nuclei. Left panel: different colors present different isotopes or isotones. Right panel: moments-method calculationwith interpolation (black circles), statistical calculation (orange diamonds), rigid sphere approximation (yellow squares). ranges are used: red diamonds present 1-5 MeV fittingrange, black circles and blue squares present 5-15 MeVand 5-25 MeV energy ranges correspondingly. It is nat-ural that at higher excitation energies the effective tem-perature increases thus reducing the rate at which thedensity grows, see Eqs. (19) and (20). VII. SPIN CUT-OFF PARAMETER
The distribution of levels with certain values of globalnuclear constants of motion (angular momentum, parity,isospin) is of special interests in all applications. Ourmethod directly supplies the required information for ev-ery set of those exact quantum numbers. The standardphenomenological approach to the angular momentumdependence of the level density assumes the random an-gular momentum coupling as a diffusion process in thespace of projections M . The total projection results fromthe random walk, and the fraction of states with a givenprojection M is Gaussian, ρ ( E, M ) ρ ( E ) = 1 √ πσ e − M / σ . (21)Then, as it was mentioned earlier, the density of stateswith a given value of J is just a difference ρ J ( E ) = ρ ( E, M = J ) − ρ ( E, M = J + 1) . (22) Assuming this random angular momentum coupling weexpect the linear M -dependence of the logarithm of thelevel density,ln[ ρ ( E, M )] = ln[ ρ ( E, M = 0)] − M σ . (23)The top two graphs of Fig. 19 show the M -dependencesof ln[ ρ ( E, M )] at different energies for Si and Fe. Thelines correspond to the best linear interpolation of thislogarithm. In these examples we see a very good linearbehavior and the smooth dependence on excitation en-ergy. This can be interpreted as an evidence for randomcoupling of angular momenta of individual particles.The situation is different in two next examples, thebottom graphs in Fig. 19, Ca and Cr. Here we donot have a regular energy dependence and, therefore aclearly defined parameter σ . This is what could be ex-pected from physical arguments. The whole idea of theGaussian random walk in the angular momentum spacebreaks down here because of the isospin limitations. Thenucleus Ca in the shell-model description has only fouridentical f / neutrons which allow for the isospin T = 2only and for the interaction in the particle-particle chan-nel with the total isospin T = 1. Therefore many val-ues of the total spin are forbidden. The second nucleus, Cr , in the pf -shell model has only four valence pro-tons with the same limitations of the angular momentum6coupling. In both cases, it is hard to expect the require-ments of the random spin coupling to be satisfied.The value of the spin cut-off parameter σ can be ex-tracted from the curves as in Fig. 19 where we observea good linear behavior. Using the thermodynamic lan-guage, we expect this parameter to be proportional totemperature, or, for Fermi-gas, to the square root of en-ergy. The corresponding parameterization can be takenas σ = α √ E (1 + βE ) . (24)The coefficient α can be taken [17] from Fermi gas sta-tistical mechanics as ∝ ν F T h M i , where ν F is the single-particle level density at the Fermi surface, or assumingthe angular momentum corresponding in average to therigid-body rotation with the moment of inertia ∝ T A / .The results of the shell-model calculations are shown inFig. 20. We define the parameters α and β , Eq. (24),from the energy region 5-25 MeV. Comparing two groupsof nuclei, sd -shell and pf -shell, we indeed see the averagegrowth of the spin-cut off parameter for two represen-tative groups as proportional to A / . It is impossiblehere to make a selection between the statistical estimateof the spin cut-off parameter and the estimate from themoment of inertia as both of them, being too crude toreflect shell efects inside each group which are certainlypresent, do not agree with the A / estimate and requirea more detailed analysis. The constant β from Eq. (24)is small but, at least for the pf -nuclei, may also containsome shell effects. VIII. CONCLUSION
In this article we collected, explained and overviewedthe first results of the improved method for statisticalcalculation of the nuclear level density for a given shell-model Hamiltonian. The method is physically based onthe chaotization of the intrinsic dynamics by the inter-particle interactions. In practice, one needs to calculateonly the lowest moments of the Hamiltonian partitionedin terms of mean-field configurations. The first two mo-ments turn out to be sufficient for the full agreement ofthe found level density with the result of the exact di-agonalization as it was checked by the cases when suchfull diagonalization was technically possible. The seriousimprovements compared to the previous attempts in thesame direction include the use of the finite-range Gaus-sian distributions and of the recurrence relation for elim-inating the spurious states. We did not discuss the de-termination of the ground state energy that is necessaryfor the appropriate positioning of the level density. Thereare special methods for doing this, including the exponen-tial extrapolation also based on the chaotic properties ofremote highly excited states [58]. The shell-model leveldensity can be calculated in any specific class of globalconstants of motion (proton and neutron numbers, to-tal spin, parity and isospin) as a function of excitation energy. This is essentially what is needed for practicalapplications to nuclear reactions including those in as-trophysics.The main conclusion that can be drawn from this ex-perience is that the shell-model level density that re-sults from the statistical calculation is a smooth functionof excitation energy in all classes of quantum numberscontaining a considerable number of states allowed bythe truncation of the orbital space (of course, the statewith the maximum possible total spin is frequently justunique). Giant oscillations of the level density predictedby the calculations based on the mean-field combinatoricsare almost completely erased by the presence of incoher-ent collision-like interactions which usually remain out-side of the mean-field models or parametrizations withthe so-called collective enhancement. Taking into ac-count all components of residual interactions, coherent(such as pairing) and incoherent, is necessary for the ad-equate description.The comparison with phenomenological Fermi-gas ap-proaches, including the models with constant tempera-ture, shows that, being less theoretically justified thanthe full direct calculation, in many cases they are nev-ertheless quite reasonable for practical use. The calcu-lation of the spin dependence of the level density and ofthe relevant spin cut-off parameter is more sensitive toassumptions, and there are cases when it is not in goodagreement with exact results. The model of constanttemperature, in our opinion, can be applied at relativelylow excitation energy but, most probably, it reflects thegeneral process of chaotization the dynamics rather thanjust breaking of Cooper pairs. Certainly, the accumu-lation of experimental data and new applications of thestatistical method are necessary for better understandingthe underlying physics.The whole approach unavoidably suffers from the gen-eral problems of the shell model. It is possible to be-lieve that the results will not be sensitive to the specificversions of the shell-model Hamiltonian as soon as thischoice agrees well with the low-lying spectroscopy. How-ever, the space truncation provides a natural limitationfor the applications of all such methods. The space canbe expanded (and many-body residual interactions canbe included) paying the price of longer computationaltime that can be cut off by the parallelization. But any-way for any choice of finite space there is a natural limitof applicability. Luckily enough, it seems that this limita-tion is not essential for many astrophysical applications.More theoretical work is necessary for understanding theresonance density for the states deeply in the continuumwhich again might not be critical for a typical stellar tem-perature when the resonance states under considerationstill are quite narrow.The whole development of the method was done incollaboration with M. Horoi. The discussions with B.A.Brown are acknowledged. The work on level density wassupported by the NSF grants PHY-1068217 and PHY-1404442.7 [1] A. Schiller et al. , Nucl. Instr. Meth. Phys. Res. A ,498 (2000).[2] A. V. Voinov, S. M. Grimes, C. R. Brune, M. J. Hornish,T. N. Massey, and A. Salas, Phys. Rev. C , 044602(2007).[3] A.C. Larsen et al. , Phys. Rev. C , 034315 (2011).[4] T. von Egidy and D. Bucurescu, J. Phys. Conf. Series, , 012028 (2012).[5] T. Kawano, S. Chiba, and H. Koura, J. Nucl. Sci. Tech. , 1 (2006).[6] V. Zelevinsky, B. A. Brown, N. Frazier, and M. Horoi,Phys. Rep. , 85 (1996).[7] I.C. Percival, J. Phys. B , L229 (1973).[8] J.B. French and K.F. Ratcliff, Phys. Rev. C , 94 (1971).[9] S.S.M. Wong, Nuclear Statistical Spectroscopy (Oxford,University Press, 1986).[10] V.K.B. Kota and R.U. Haq, eds.,
Spectral Distributionsin Nuclei and Statistical Spectroscopy (World Scientific,Singapore, 2010).[11] T.A. Brody, J. Flores, J.B. French, P.A. Mello, A.Pandey, and S.S.M. Wong, Rev. Mod. Phys. , 385(1981).[12] S.M. Grimes, S.D. Bloom, R.F. Hausman, and B.J. Dal-ton, Phys. Rev. C , 2378 (1979).[13] C.W. Johnson, J.-U. Nabi, and E. Ormand,arXiv:0105041 (2001).[14] R.A. Sen’kov, M. Horoi, and V.G. Zelevinsky, Phys. Lett.B , 413 (2011).[15] R.A. Sen’kov, M. Horoi, and V.G. Zelevinsky, ComputerPhysics Communications , 215 (2013).[16] H.A. Bethe, Phys. Rev. , 332 (1936); Rev. Mod. Phys. , 69 (1937).[17] T. Ericson, Adv. Phys. , 425 (1960).[18] A. Gilbert and A. G. W. Cameron, Can. J. Phys. ,1446 (1965).[19] H. Baba, Nucl. Phys. A159 , 625 (1970).[20] J.A. Holmes, S.E. Woosley, W.A. Fowler, and B.A. Zim-merman, Atomic Data and Nuclear Data Tables, , 305(1976).[21] C.A. Engelbrecht and J.R. Engelbrecht, Ann. Phys. ,1 (1991).[22] A. Bohr and B. Mottelson, Nuclear Structure , vol. 1(1969), vol. 2 (1975), Benjamin, New York.[23] S. Goriely, S. Hilaire, and A.J. Koning, Phys. Rev. C ,064307 (2008).[24] H. Uhrenholt, S. ˚Aberg, P. Moeller, and T. Ichikawa,arXiv:0901.1987.[25] S. Hilaire, M. Girod, S. Goriely, and A.J. Koning. Phys.Rev. C , 064317 (2012).[26] Y. Alhassid, G.F. Bertsch, S. Liu, and H. Nakada, Phys.Rev. Lett. , 4313 (2000).[27] Y. Alhassid, S. Liu, and H. Nakada, Phys. Rev. Lett. ,162504 (2007).[28] Y. Alhassid, A. Mukherjee, H. Nakada and C. Ozen, J.Phys.: Conf. Series , 012012 (2012).[29] Ya.I. Frenkel, Sov. Phys. , 533 (1936).[30] L.D. Landau, JETP , 819 (1937).[31] V.V. Flambaum, A.A. Gribakina, G.F. Gribakin, andM.G. Kozlov, Phys. Rev. A , 267 (1994).[32] V.V. Flambaum amd F.M. Izrailev, Phys. Rev. E ,5144 (1997). [33] V. Zelevinsky, Annu. Rev. Nucl. Part. Phys. , 237(1996).[34] L.D. Landau and E.M. Lifshitz, Statistical Physics , Perg-amon Press, Oxford, 1958.[35] M. Srednicki, Phys. Rev. E , 888 (1994).[36] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalat-tore, Rev. Mod. Phys. , 863 (2011).[37] S.T. Belyaev and V.G. Zelevinsky, Sov. J. Nucl. Phys. , 416 (1970).[38] N. Pillet, V.G. Zelevinsky, M. Dupuis, J.-F. Berger, andJ.M. Daugas, Phys. Rev. C , 044315 (2012).[39] M. Horoi, J. Kaiser, and V. Zelevinsky, Phys. Rev. C ,054309 (2003).[40] R.A. Sen’kov and M. Horoi, Phys. Rev. C , 024304(2010).[41] N. Frazier, B.A. Brown and V. Zelevinsky, Phys. Rev. C (1996) 1665; reprinted in Ref. [10], p. 557.[42] M. Horoi and V. Zelevinsky, Phys. Rev. Lett. , 262503(2007).[43] C. Jacquemin, Z. Phys. A , 135 (1981).[44] J. Brenneisen et al., Z. Phys. A , 352, 430, 443,(1995).[45] D. H. Gloeckner and R. D. Lawson, Phys. Lett. B ,313 (1974).[46] A.V. Ignatyuk, K.K. Istekov, and G.N. Smirenkin, Sov.J. Nucl. Phys. , 450 (1979).[47] S. Komarov, R. J. Charity, C. J. Chiara, W. Reviol, D. G.Sarantites, L. G. Sobotka, A. L. Caraley, M. P. Carpen-ter, and D. Seweryniak, Phys. Rev. C , 064611 (2007).[48] V.G. Zelevinsky, D. Mulhall and A. Volya, Phys. At. Nuc. , 525 (2001).[49] A. Volya, V. Zelevinsky, and B.A. Brown, Phys. Rev. C , 054312 (2002).[50] J.R. Armstrong, S. ˚Aberg, S.M. Reimann, and V.G.Zelevinsky. Phys. Rev. E , 066204 (2012).[51] G. Rohr, Z. Phys. A , 299 (1984).[52] S.I. Al-Quraishi, S.M. Grimes, T.N. Massey, and D.A.Ressler, Phys. Rev. C , 065803 (2001).[53] P.J. Brancazio and A.G.W. Cameron, Can. J. Phys. ,1029 (1969).[54] A.V. Voinov, B.M. Oginni, S.M. Grimes, C.R. Brune, M.Guttormsen, A.C. Larsen, T.N. Massey, A. Schiller, andS. Siem, Phys. Rev. C , 031301 (2009).[55] M. Guttormsen, B. Jurado, J. N. Wilson, M. Aiche, L. A.Bernstein, Q. Ducasse, F. Giacoppo, A. Grgen, F. Gun-sing, T. W. Hagen, A. C. Larsen, M. Lebois, B. Leniau,T. Renstrm, S. J. Rose, S. Siem, T. Tornyi, G. M. Tveten,and M. Wiedeking, Phys. Rev. C , 024307 (2013).[56] M. Guttormsen, L. A. Bernstein, A. Grgen, B. Jurado, S.Siem, M. Aiche, Q. Ducasse, F. Giacoppo, F. Gunsing,T. W. Hagen, A. C. Larsen, M. Lebois, B. Leniau, T.Renstrm, S. J. Rose, T. G. Tornyi, G. M. Tveten, M.Wiedeking, and J. N. Wilson, Phys. Rev. C , 014302(2014).[57] L.G. Moretto, A.C. Larsen, F. Giacoppo, M. Guttorm-sen, S. Siem, and A.V. Voinov, arXiv: 1406.2642.[58] M. Horoi, A. Volya, and V. Zelevinsky, Phys. Rev. Lett. , 2064 (1999); M. Horoi, B.A. Brown and V. Zelevin-sky, Phys. Rev. C65