Level density within a micro-macroscopic approach
A.G. Magner, A.I. Sanzhur, S.N. Fedotkin, A.I. Levon, S. Shlomo
aa r X i v : . [ nu c l - t h ] J un Level density within a micro-macroscopic approach
A.G. Magner a, ∗ , A.I. Sanzhur a , S.N. Fedotkin a , A.I. Levon a , S. Shlomo b a Institute for Nuclear Research, 03680 Kyiv, Ukraine b Cyclotron Institute, Texas A & M University, College Station, Texas 77843, USA
Abstract
Statistical level density ρ ( E , A , M ) is derived for nucleonic system with a given energy E , particle number A andangular momentum projection M to a symmetry axis in the mean-field approximation beyond the standard saddle-point method (SPM). This level density reaches the two limits; the well-known SPM grand-canonical ensemble limitfor a large entropy S related to large excitation energies, and the finite micro-canonical limit for a small combinatoricalentropy S at low excitation energies. The inverse level density parameter K as function of the particle number A inthe semiclassical periodic orbit theory, taking into account shell e ff ects, is calculated and compared with experimentaldata. Keywords: level density, shell e ff ects, Thomas-Fermi approach, periodic orbit theory, neutron resonances.
1. Introduction
Many properties of heavy nuclei can be to large extent described in terms of the statistical level density [1–9]. Usually, the level density ρ ( E , A , M ), where E , A , and M are the energy, nucleon number and orbital angularmomentum projection, is calculated by the inverse Laplace transformation of the partition function Z ( β, α A , α M ) of thecorresponding Lagrange multipliers. Within the grand canonical ensemble (GCE), the standard saddle-point method(SPM) was used for the integration over all variables including β , which is related to the total energy E . This methodassumes a large excitation energy U , so that the temperature T can be defined through a well-determined saddlepoint in the integration variable β for a finite Fermi system of large particle numbers. However, many experimentaldata are related also to the low-lying part of the excitation energy U , where such a saddle point does not exist. Forpresentation of experimental data on nuclear spectra, the cumulative level-density distribution N ( U ) – cumulativenumber of quantum levels below the excitation energy U – is conveniently often used. We point out the uniformnuclear spectra presented through the unfolding procedure for a statistical analysis [10–12] of the experimental dataon the collective excitation energies of rare-earth and actinide nuclei excited in two-neutron transfer (p,t) reactions[13, 14]. For calculations of the cumulative level density N ( U ), one has to integrate the level density over a largeinterval of the excitation energy U from small values where there is no thermodynamical equilibrium to large valueswhere the standard GCE can be successfully applied in terms of the temperature T in a finite Fermi system. Therefore,to simplify the level density, ρ ( E , A , M ), calculations we are going to integrate over the Lagrange multiplier β in theinverse Laplace transformation of the partition function Z ( β, α A , α M ) more exactly beyond the SPM.A micro-macroscopic approximation (MMA) which unifies micro- and macroscopic ensembles, for the statisticallevel density ρ was suggested in Ref. [15]. A simple expression of ρ in terms of the modified Bessel function ofthe entropy variable was obtained for a small heat excitation energy U as compared to the rotational excitations E rot within the mean field approach. The yrast line was defined as a minimal excitation energy U (minimum of thestatistical level density ρ ) for a given angular momentum within the cranking model [1, 16]. The shell-correctionmethod (SCM) [17], extended to the description of nuclear rotational bands [18], was applied [15] for studying theshell e ff ects in the moment of inertia (MI) at the yrast line. For a deeper understanding of the correspondence between ∗ Corresponding author
Email address: [email protected] (A.G. Magner) he classical and the quantum approach, especially their applications to high-spin physics, it is worthwhile to analyzethe shell components of the MI within the semiclassical periodic-orbit theory (POT) [19–23].In the present study we extend the MMA problem [15] for the description of shell e ff ects in terms of the leveldensity itself for larger excitation energies U . The level density parameter a is one of the key quantities underthe intensive experimental and theoretical discussions [3, 5–7, 9]. Smooth properties of the inverse level density K = A / a , as function of the nucleon number A have been studied within the framework of the self-consistent ExtendedThomas-Fermi (ETF) approach [5, 7]. However, shell e ff ects in the statistical level density is still an attractive subject,especially important near nuclear magic numbers.
2. Micro-macroscopic approach
For the statistical description of level density of a nucleus in terms of the conservation variables – E , A , and M ,one can begin with the micro-canonical expression for the level density in terms of the inverse Laplace transformationof Z ( β, α A , α M ) over the corresponding Lagrange multipliers β, α A and α M [1, 3, 15], ρ ( E , A , M ) = (2 π i ) − R d β d α A d α M Z ( β, α A , α M ) exp h β E − A α A − M α M i ≈ ρ MMA ( S ) ∝ S − ν I ν ( S ) , (1)where I ν is the modified Bessel function of order ν ( ν = is convenient to introduce, instead of thepartition function Z , the potential, Ω = − ln Z /β , for any value of the integration variables. Then, one can recognisein (1) the entropy S in terms of the potential Ω , S = β ( E − Ω ) − α A A − α M M .As usual [1], the level density (1) is calculated within the GCE by using the SPM for integrations over all variables β, α A , and α M . These variables correspond to the additive integrals of motion E , A , and M through the saddle point(SP) conditions which ensure their conservation. The SPM assumes, at least, a large excitation energy U with respectto a distance between levels. In this case, the temperature T is determined by a good SP, 1 /β SP , over the integrationvariable β for such a finite system as a heavy nucleus. Taking Ω at all SPs values, if exist, Ω GCE = Ω (cid:16) β SP , λ SP , ω SP (cid:17) with λ = α A /β and ω = α M /β ~ at the SP, can be defined, as usual; E = Ω GCE + ( β ∂ Ω /∂β ) SP . In this case, β SP = / T with T being the system temperature, λ SP is the chemical potential, and ω SP the rotation frequency.We calculated the integrals in Eq. (1) over the restricted set of Lagrange multipliers α A and α M related to A and M , respectively, by the SPM at the SPs. The potential Ω ( β, λ SP , ω SP ) contains the two contributions – the heat intrinsicexcitation, U ( β ), related to the entropy production, and the rotational energy, E rot ( ω ), excitation. Below we omitsubscript SP and consider all quantities at the SP, λ = λ SP and ω = ω SP . Assuming a small heat-excitation energy, U ∝ /β ( ∝ T in the asymptotic excitation-energy limit), with respect to rotational ones, E rot ( ∝ ω in the adiabaticapproximation) but large as compared to a mean distance between level energies for validness of the statistical andsemiclassical arguments, one obtains the analytical MMA level density ρ MMA in Eq.(1). This expression for ρ MMA is validfor excitation energies U ∼ > / ˜ g ( λ ) , where ˜ g ( λ ) is approximately a smooth (TF) part of the single-particle (s.p.) leveldensity g ( ε ), accounting for the spin or spin-isospin degeneracy, at ε = λ , g TF ( λ ) = A / λ . It should be also smallerthan a distance between major shells, D sh ≈ λ/ A / , in the adiabatic approximation for rotational excitations. At thesame time, under the condition U ≈ a /β ≪ E rot = Θ ω /
2, where Θ is the moment of inertia (MI), we neglected the β dependence of the Jacobian (two-dimensional determinant) J ( ∂ Ω /∂λ, ∂ Ω /∂ω ; λ, ω ). This appears because of theSPM intergrations over two variables, α A and α M , in Eq. (1) to arrive at ρ MMA for ν = U ∼ < ~ ω ∼ < . U ≪ λ , for the same (adiabatic) rotational energies E rot , that leads to Eq. (1) at ν = . ∼ < U ∼ <
10 MeV).The upper limit is in agreement with disappearance of shell e ff ects, ( U / a ) / = ( UK / A ) / ≈ − λ =
50 MeV, A = K = A / a = S , according to Eq. (1) for ρ MMA , one has ρ ≈ ρ MMA ( S ) ∝ exp( S ) S ν √ π S " + − ν S + − ν + ν S + O S ! . (2)2his approximation at zero order in expansion over 1 / S is identical to that obtained directly from (1) by the SPM. Atsmall entropy, S ≪
1, one obtains also from Eq. (1) the finite combinatoric power expansion: ρ ≈ ρ MMA ( S ) ∝ − ν Γ ( ν + " + S ν + + O (cid:16) S (cid:17) , (3)where Γ ( x ) is the Gamma function.By applying approximately the Fermi-gas (mean-field) expression for the relationship between the entropy S and excitation energy U , S = √ a U , one obtains the excitation energy dependence in Eq. (1-3). In the adiabaticmean-field approximation for a , one finds a = π g ( λ ) , g ( λ ) = ˜ g ( λ ) + δ g ( λ ) , (4)where ˜ g ( ε ) is the (E)TF level density, and δ g ( ε ) the oscillating (shell) component, both taken approximately at theFermi energy, ε = λ ≈ ε F . Its ETF approximation, g ETF , is in good agreement with the Strutinsky averaged quantumlevel density ˜ g [22]. For calculations of δ g ( ε ), one can use the shell correction method [17] and, analytically, thesemiclassical POT [21–23]. We need also an averaging over spectra near the Fermi surface within the interval Γ larger than 1 / ˜ g ( λ ), but smaller than the distance between major shells, D sh , to keep shell e ff ects. The value of Γ can be taken from a required energy resolution of the problem, e.g., Γ ∼ U , one finds U = E − E − Θ ω / E (cid:27) ˜ E + δ E is the intrinsic energy, ˜ E is the Strutinsky smooth energy, ˜ E ≈ E ETF , E ETF isthe ETF energy, and δ E the energy shell correction [19, 21, 22]. For Θ one has a similar decomposition: Θ (cid:27) ˜ Θ + δ Θ ,˜ Θ ≈ Θ ETF .Thus, for a large entropy S ≫
1, one finds the well-known SPM result for the GCE, ρ ∝ e S , but with the pre-exponential factor of the inverse power expansion (2) over S . However, for the level density ρ MMA ( S ) (1) at smallentropy, S ≪
1, one obtains the combinatoric power expansion (3). These two limits are connected continuouslyanalytically through the Bessel function I ν ( S ) for S ∼ U .Thus, there is no divergence of the level density ρ ( U ) in the limit U →
0, in contrast to the SPM case. This resultis identical to the leading asymptotics (2) of Eq. (1) at aU ≫ S ≫
1) when we neglect all corrections in powersof 1 / S . The constant in the full SPM is proportional to the Jacobian (three-dimensional determinant) taken at the SPincluding β = β SP = / T , where T is the temperature, 1 / T = ( ∂ S /∂ U ) SP = √ a / U , i.e., U = aT . There are othermethods to overcome divergence of the full SPM for low excitation-energy limit U → ρ MMA ( S ) (1) (solid) with ν =
3, up to a proportionality constant, onthe entropy variable S and its di ff erent asymptotics for a small [ S ≪
1, frequent dashed curve, Eq. (3)] and large[ S ≫
1, Eq. (2)] entropy S . For large S ≫ / S to see a convergence of “0” (dotted) and“2” (heavy dashed lines) asymptotics to the exact result (1) with increasing inverse powers of S of zero and secondorder, respectively. Notice that there is a smooth analytical short-range transition at S ≈ −
4, from a small to largeasymptotical behavior of the density if we take into account the second correction (see the curve “2”) term of the orderof 1 / S in square brackets of (2) at ν =
3. This transition takes place at a little larger S than that for ν =
2. Thefirst correction at ν = S ∼ >
1. For both ν values, almost a parallelconstant shift of the simplest, ρ ∝ exp( S ) / S ν + / , SPM asymptotic approximation at large S (frequent dots “0” inFig. 1), S ∼ >
4, with respect to the solid curve of the exact MMA result (1) clarifies a phenomenological back-shiftedFermi gas (BSFG) model for the level density [4, 6].From Eq. (1) one can calculate the level density ρ ( E , A , I ) with a given energy E and the total angular momentum I in terms of the Bessel functions, ρ ( E , A , I ) = − ∂ρ ( E , A , M ) /∂ M at M = I + / S ≪ S ≫ S of the Bessel functions, one finds a finite combinatoric and GCE limit expressions.The main term of these expressions for S ≫ I andlarge excitation energy U = E − E , I ( I + ~ / Θ U ≪ ν = f ( I ), and excitation-energy, U , factors, ρ ∝ f exp (cid:16) √ aU (cid:17) a / U / , f = I + Θ exp − I ( I + ~ √ a Θ √ U ! . (5)3 s ρ S>>1 S<<102
Figure 1: Level density ρ , Eq. (1) at ν = S for di ff erent approximations: S ≫
1, where “0”, and “2” are expansions (2) over 1 / S up to zero, and second order terms, respectively, and S ≪
1, given by Eq. (3).
The power dependence of the pre-exponent level density ρ ( E , A , I ) on the excitation energy E − E di ff ers from thatof ρ ( E , A , M ), while one has the same exponential dependence ρ ∝ exp(2 √ a ( E − E ) for a large excitation energy E − E and small angular momentum I .The decomposition of Ω in terms of a smooth part, ˜ Ω ≈ Ω ETF , and shell correction δ Ω , Ω (cid:27) Ω ETF + δ Ω , wasused successfully within the POT SCM in [15]. We applied this approximation for the calculation of the pre-exponentcoe ffi cient proportional to the Jacobian factor J − / . In these derivations we used also expansion over a small rotationfrequency ω up to quadratic terms. The frequency ω can be eliminated with the help of the SP conservation relations,which can be written in the same quadratic approximation in ω as M = Θ ω , where Θ is the MI, decomposed interms the smooth Θ ETF and oscillating δ Θ components. These components for small excitation energies and majorshell-structure averaging, g − TF ≪ Γ ≪ D sh , of δ g are much smaller than the average rigid-body value Θ TF , δ Θ / Θ TF ≈ δ g / g TF . In these estimations we used the POT evaluations of the distance between major shells, D sh ≈ λ/ A / [19, 21], determined by a mean period of the most short and degenerate POs [19, 21–23].
3. Results and discussions
Fig. 2 shows the inverse level-density parameter K = A / a , with a is of Eq. (4), as function of the particle number A in the semiclassical POT approximation. The result of these calculations are largely in a qualitative agreement with thenew experimental data [6], which are much beyond those on neutron resonances [2] and include many other reactionswith nuclear excitation energies being significantly smaller than the neutron separation energy. Complete low-energylevel schemes of 310 nuclei from F to Cf were used in the analysis. The sets of levels in the limited energy rangebelow the neutron binding energy with reliable completeness were selected for each nucleus, and neutron resonancedensities were included in the analysis. We added the smooth self-consistent ETF values of a for the KDE0v1 [24]and the SkM ∗ [25] Skyrme force from Ref. [7] to their shell corrections [ δ g ( λ )] through the total s.p. level density g ( λ ) of Eq. (4). Its oscillating component δ g ( λ ) was approximated by the analytical POT trace formula [20] for theinfinitely deep spherical square-well potential. This formula almost identically reproduces the SCM quantum resultsfor the s.p. level density for the same potential [22]. The major shells in Fig. 2 are clearly seen as maxima of K ( A ) ,which correspond to minima of the density parameter a , or the oscillating level density component δ g ( λ ), see Eq. (4).4 A K [ M e V ] Figure 2: The inverse level-density parameter K = A / a (solids “1” for SKM ∗ and “2” for KDE0v1 forces) is shown as function of the particlenumber A . The smooth part in the ETF approach is taken from Ref. [7] for these two versions of the Skyrme forces SKM ∗ (“3” dashed) andKDE0v1 (“4” dashed). The solid oscillating curves are obtained by using the semiclassical POT approximation [19, 20] for the level density shellcorrections at Gauss width averaging parameter γ = .
3, and dashed curves “3” and “4” for a smooth part, both including the e ff ective mass.Experimental values are shown by solid points, taken from Ref. [6]. The relationship between the chemical potential λ , through k F R , where ~ k F = (2 m λ ) / ≈ (2 m ε F ) / is the Fermimomentum, m is the nucleon mass, and particle numbers A for this potential [19, 20, 22] is given by A = − ∂ Ω ∂λ ! SP ≈ Z λ d ε g ( ε ) . (6)In this transformation, k F R to A , one can conveniently use the quantum SCM level density [17]. The Gaussianaveraging width of the oscillating level density δ g in Fig. 2 is the same γ = . Γ ≈ − λ =
50 MeV, r = .
14 fm, A = − δ g ( ε ) in Eq. (4). Mean value of oscillating K ( A ) in Fig. 2 is about 8 MeV, ~ / mr ≈
15 MeV,as predicted in Ref. [5]. This is in accordance with the ETF (SkM ∗ or KDE0v1) value, accounting for the e ff ectivemass m ∗ . As shown in Ref. [7], the e ff ect of the e ff ective mass m ∗ on the inverse level density K is strong, decreasingof K by factor about 2, that leads approximately to mean values of the experimental data [6]. However, we should notexpect that for the infinitely deep spherical square-well potential the positions of the maxima (minima of a , i.e., of thes.p. level density, δ g ( ε ) at ε = λ , related to magic nuclei in that potential) can be correctly reproduced in such shell-correction calculations, first of all, because of neglecting the spin-orbit interaction. As shown in Refs. [21, 23] forsemiclassical explanation of the positions of the magic nucleus Pu, we should shift the curves K ( A ) along the A axis[through k F R , Eq. (6)]. Therefore, we shifted the semiclassical curves in Fig. 2 in about ∆ A =
20 along the particlenumber A axis. This shift is of the order of a half of the distance D sh between major shells near the Fermi surface inthe particle number variable A . According to the POT estimations for the period D sh and TF level density g TF [19],one finds for the period of the major shell structure, A sh , in the particle number variable, A sh ≈ D sh g TF ≈ A / / A = − A sh ≈ −
50, which is of the order of the realistic period of the nuclear major shellstructure. The position of a maximum over the particle number variable, as related to the value of k F R is the onlyone parameter to adjust the semiclassical solid curves with the experimental data. This is similar to the discussionsin Ref. [23] where we obtained semiclassically the magic number for Pu by using a similar shift. Therefore, three5inima of the major shell closures in the semiclassical calculations at A ≈ −
150 in Fig. 2 correspond to theexperimentally obtained maxima. In spite of a very simple explicitly given analytical formula [19, 20, 22] for thes.p. level density shell corrections in the spherical cavity, one obtains largely good agreement of the semiclassicalapproximation, which is almost identical to the quantum SCM result for the same cavity, to experimental data in thisrange of nuclear particle numbers A . Magnitudes of periods and amplitudes for the oscillations of K ( A ) are basicallyin good agreement.However, there is a discrepancy between experimental and theoretical results for K ( A ) near the double magicnuclei Ca and
Pb. One of possible reasons is that di ff erent approximations for the statistical level density ρ areused in fitting procedure to obtain the experimental prediction of K with respect to those of this MMA approach(1). New information on a , the energy parameter E (intrinsic energy E ) and the spin-cuto ff parameter σ ( σ =Θ √ U / a / ~ in our notations) were obtained applying the SPM BSFG formula for the level density to the experimentalspectra in Ref. [6]. The specific reason of the discrepancy might be that the density parameter a (or K ) was obtainedby three parameter ( a , E and σ ) fitting of these experimental data, including excitation energies below neutronresonances, to the level density, ρ ∝ exp( S ), in the BSFG approximation. This approximation is not valid for smallexcitation energies U , in contrast to the MMA approach with the correct zero excitation energy limit. Assuming asmall angular momentum I and large excitation energy, U ≫ / g TF ( λ ), one finds the result of Eq. (5) for ρ ( E , A , I )which is similar to the BSFG approximation if we assign another notation E in Ref. [6] for the intrinsic energy E here. Notice nevertheless that another source of the discussed discrepancy can be a di ff erent pre-exponent dependenceof ρ ( E , A , I ) on the excitation energy U (at zero angular momentum I ) and on the level density parameter a , cf. Eqs.(1) with (2) and (4) of Ref. [6] and Eq. (5) here. Our prediction for the reason of discrepancy is based on Fig. 11 ofRef. [6] where relatively large values of E , i.e., small excitation energy U are shown near the double magic nuclei Ca and
Pb. Large deformations and pairing correlations of the rare earth and actinide nuclei should be also takeninto account to improve the comparison with experimental data.
4. Conclusions
We derived the statistical level density ρ ( S ) as function of the entropy S within the micro-macroscopic approxi-mation using the mixed micro- and grand-canonical ensembles. This function can be applied for small and, relatively,large entropies S or excitation energies U of a nucleus. For a large entropy (excitation energy), one obtains the stan-dard exponential asymptotics, however, with the significant inverse 1 / S power corrections. For small S one finds theusual finite combinatoric expansion in powers of S . The transition from small to large S is su ffi ciently rapid in asmall region of S ≈ −
4. Major shell oscillations of the inverse level density parameter K are compared with newexperimental data, in particular, for neutron resonances. Thus, the divergences at small excitation energies U of theSPM approach were removed within the MMA. We found qualitatively good agreement between semiclassical POTand quantum-mechanical results with experimental data on the inverse level density parameter K , after overall shift ofall K ( A ) curves by only one parameter because of the spin-orbit interaction, between particle numbers A ≈ − Caand
Pb.As perspectives, the neutron-proton asymmetries, large angular momenta and deformations of the rare earth andactinide nuclei should be also taken into account to improve the comparison of the theoretical evaluations with exper-imental data on the level density parameter beyond the neutron resonances. Several applications of our approach tothe statistical analysis of the experimental data on collective states in rare-earth and actinide nuclei will be presentedin forthcoming work.
Acknowledgement
The authors gratefully acknowledge D. Bucurescu, R.K. Bhaduri, M. Brack, A.N. Gorbachenko, and V.A. Plujkofor creative discussions. This work was supported in part by the budget program ”Support for the development ofpriority areas of scientific researches”, the project of the Academy of Sciences of Ukraine, Code 6541230. S. Shlomois supported in part by the US Department of Energy under Grant no. DE-FG03-93ER-40773.6 eferences [1] Bohr Aa and Mottelson B R
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