Evidence for the reduction of nuclear level density away from the β -stability line
Pratap Roy, K. Banerjee, T. K. Rana, S. Kundu, S. Manna, A. Sen, D. Mondal, J. Sadhukhan, M. T. Senthil Kannan, T. K. Ghosh, S. Mukhopadhyay, Deepak Pandit, G. Mukherjee, S. Pal, D. Paul, K. Atreya, C. Bhattacharya
aa r X i v : . [ nu c l - e x ] D ec Evidence for the reduction of nuclear level density away from the β -stabilityline Pratap Roy , , ∗ K. Banerjee , , T. K. Rana , S. Kundu , , S. Manna , , A. Sen , D. Mondal ,J. Sadhukhan , , M. T. Senthil Kannan , T. K. Ghosh , , S. Mukhopadhyay , , DeepakPandit , G. Mukherjee , , , S. Pal , D. Paul , , K. Atreya , , and C. Bhattacharya , Variable Energy Cyclotron Centre, 1/AF, Bidhan Nagar, Kolkata - 700064, India Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai - 400094, India
7, Ambiganagar I street, Thudiyalur, Coimbatore -641034, Tamil Nadu, India.The isospin dependence of nuclear level density has been investigated by analyzing the spectra ofevaporated neutrons from excited
Sn and
Te nuclei. These nuclei are populated via p + Inand He +
Sn reactions in the excitation energy range of 18 - 26 MeV. Because of low excitationenergy, the neutron spectra are predominantly contributed by the first-chance decay leading to the β -stable Sn and neutron-deficient
Te as residues for the two cases. Theoretical analysis ofthe experimental spectra have been performed within the Hauser-Feshbach formalism by employingdifferent models of the level density parameter. It is observed that the data could only be explainedby the level density parameter that decreases monotonically when the proton number deviates fromthe β -stable value. This is also confirmed by performing a microscopic shell-model calculationwith the Wood-Saxon mean field. The results have strong implication on the estimation of thelevel density of unstable nuclei, and calculation of astrophysical reaction rates relevant to r - and rp -processes. PACS numbers: 25.70.Jj, 25.70.Gh, 24.10.Pa
One of the primary objectives of current nuclearphysics research is to reproduce the observed abun-dances of different elements in the universe, and tounderstand the underlying physical processes behindthe synthesis of these elements at different astro-physical sites. Although it is reasonably well under-stood how the elements up to Fe (Z=26) are pro-duced within the stars through fusion reactions, theunderstanding of the synthesis of heavier elements(Z >
26) has been evasive. The mysteries regard-ing the astrophysical sites, as well as the nucleardata needed to describe the heavy-element nucle-osynthesis, persist even after intense investigationsover the past decades [1–4]. Many of the astrophysi-cal reactions related to the nucleosynthesis of heavyelements involve either neutron-rich ( r -process) orproton-rich ( rp -process) nuclei, making it difficultto carry out direct experimental measurements fordetermining the reaction rates. Since most of therelevant cross-sections are not available experimen-tally, the usual approach is to calculate them withinthe statistical Hauser-Feshbach (HF) framework [5].One of the most critical inputs in the HF calcu-lations of nuclear reaction cross-sections is nuclearlevel density (NLD). Experimental information onNLD is available mostly for isotopes near stability,and for most of the unstable nuclei, only theoreticalestimates are available, which tend to be highly un-certain. Therefore, experimental data to understand ∗ Email: [email protected] the variation of level density from stable nuclei to thenuclei away from the stability line is of high impor-tance.The simplest and most widely used description oflevel density is given in terms of the non-interactingFermi gas model (FGM) [6], ρ ( E ) = 112 √ σ exp 2 p a ( E − ∆) a / ( E − ∆) / , (1)where E is the excitation energy, ∆ is the pairingenergy shift [7], σ is the spin cut-off factor, and a isthe level density parameter (LDP) which is directlyrelated to the density of single-particle states nearthe Fermi surface. Several important phenomeno-logical refinements to Eq. 1 have been introducedin due course to include the angular momentum [8],shell [9], and collective effects [10–12]. On the otherhand, another crucial factor, the iso-spin effects inNLD is somewhat neglected largely because theseeffects are expected to be small for nuclei at the val-ley of stability. However, the iso-spin effects in NLDcan be profound for unstable nuclei which, in turn,can significantly influence the calculated astrophys-ical reaction rates.The Fermi gas model gives a smooth dependenceof the level density parameter on the mass number( A ) which can be expressed as a = αA (2)where the proportionality constant α is taken eitherfrom global systematics [13–16] or adjusted to matchthe experimental data [17–23]. The iso-spin, as wellas the single-particle binding energy can give a de-pendence of the level density parameter a on neutron( N ), and proton ( Z ) numbers rather than merely onthe mass number A . Two alternative forms of Eq. 2that include the N , Z dependence have been sug-gested by Al-Quraishi et al. [24, 25] a = αA exp [ β ( N − Z ) ] (3) a = αA exp [ γ ( Z − Z ) ] (4)where α , β and γ are empirical constants. Thepresence of the ( N − Z ) factor in Eq. 3 causes thelevel density to be maximum for N = Z = A/ A and to decrease as the neutron-protonasymmetry (iso-spin) increases. On the other hand,the ( Z − Z ) factor in Eq. 4, where Z is the atomicnumber of the β -stable isotope for the mass number A , reduces the level density as the nucleus movesaway from the β -stability line. The expression for Z is obtained from the fit of the semiempiricalmass formula. The arguments for the reduction ofthe level density parameter away from the stabilityline can be found in Refs. [26–31]. For the lightermass nuclei (A ≤
40) where Z ≃ Z ≃ N ≃ A/ a give similar results, and thedifference is expected to become prominent onlybeyond A = 40.In Refs. [24, 25] the validity of the different expres-sions for the level density parameter was tested fornuclei in the mass range of 20 ≤ A ≤ a as given in Eqs. 2 - 4. The analysis suggestedthat the ( Z − Z ) form provides somewhat betterreproduction of the experimental data compared tothe other formulations. However, the distinctionwas not overwhelming mainly because the completelevel schemes those were used to test the modelswere limited to low energies (3 - 4 MeV for A ≥ | Z − Z | . Z − Z form have also been found in some of thelow energy particle evaporation studies [32–34].However, no such evidence was observed by Charity et al. [35] in the high energy fusion evaporationmeasurement around A ≈ i.e. | Z − Z | &
2, is highly demandingfor carrying out further tests of the different N , Z dependent parametrization of a .The analysis of the evaporation spectra of lightparticles can be used as an excellent tool forcarrying out such tests in a wide excitation energyand angular momentum range. The light-ioninduced reactions are particularly advantageous inpopulating low excitation energies and restrictingthe number of effective decay channels compared tothe heavy-ion (HI) route [17, 18, 20].In this Rapid Communication, we report theexperimental study on the N , Z dependence of NLDinvestigated using the neutron evaporation spectrafrom Sn, and
Te compound nuclei (CN)populated through the p + In, and He +
Snreactions, respectively. The CN were populated atlow excitation energies so that the neutron spectraare dominated by the first-chance neutron emissionleading to the residual
Sn,
Te nuclei in thetwo cases. The analysis of the neutron spectrawill allow us to simultaneously investigate the leveldensities of the two A = 115 isobars, the β -stable Sn ( Z ≈ Z ) and the neutron-deficient Te( | Z − Z | >
2) which will provide crucial informationin understanding the variation of NLD as a functionof N and Z .The experiment was performed using 9 and 12MeV proton and 28 MeV He beams from the K130cyclotron at VECC, Kolkata. Self-supporting foilsof
In (thickness ≈ ), and isotopically-enriched (99.6%) Sn [37] (thickness ≈ ) were used as targets. The compoundnucleus Sn ( p + In) was populated at twoexcitation energies E ∗ CN =18.2 and 21.2 MeV,whereas Te ( He +
Sn) was populated at E ∗ CN =26 MeV. The neutrons emitted duringthe compound nuclear decay process were detectedusing six cylindrical liquid scintillator based neutrondetectors (of 5-inch length and 5-inch diameter)placed at the laboratory angles of 55 ◦ , 85 ◦ , 105 ◦ ,120 ◦ , 140 ◦ , 155 ◦ at a distance of 1.5 m fromthe target. The neutron kinetic energies weremeasured using the time-of-flight (TOF) technique.The start trigger for the TOF measurement wasgenerated using the prompt γ -rays detected by a50-element BaF detector array [38], placed nearthe target position. The prompt γ - γ -peak in theTOF spectrum was taken as the time reference. Theefficiencies of the neutron detectors were measuredin the in-beam condition using a ≈ µ Ci Cfsource [39]. Neutron- γ discrimination was achievedby both the TOF and pulse shape measurements(PSD) [40]. The scattered neutron contributions inthe measured neutron spectra were estimated andsubtracted using the “shadow bar” technique [18].Further details on the experimental setup and dataanalysis techniques are available in Refs. [41, 42]The background-corrected neutron spectra mea-sured at various laboratory angles were transformedinto the compound nucleus center-of-mass (c.m.)frame using the standard Jacobian transformation.The spectral shapes at the backward angles werefound to be almost overlapping indicating the dom-inance of the compound nuclear contribution in themeasured spectra. The spectra measured at themost backward angle (155 ◦ ) have been consideredfor the statistical model analysis and for testing dif-ferent models of the level density parameter. Thetheoretical calculation of the neutron energy spec-tra was performed with the TALYS (v 1.9) code [43]using the statistical HF framework. For the leveldensity, the composite Gilbert-Cameron (GC) for-mulation [44] was used. In the GC model, the leveldensity at low energies (from 0 to a matching energy E M ) is approximated by a constant-temperature(CT) formula ρ CT ( E ) = 1 T exp E − E T (5)and for energies higher than E M the level density isgiven by the Fermi gas expression (Eq. 1). The con-stant temperature ( T ), and the energy shift ( E )are chosen such that the two prescriptions match to-gether smoothly at the matching energy which variesinversely with the mass number ( A ), and for thepresent case E M ≈ a ( U ) = ˜ a [1 + ∆ SU { − exp( − γU ) } ] (6)where, U = E − ∆, and ˜ a is the asymptotic value ofthe level density parameter obtained in the absenceof any shell effect. Here ∆ S is the ground stateshell correction, and γ determines the rate at whichthe shell effect is depleted with the increase inexcitation energy [43]. The transmission coefficientswere calculated using the optical model (OM) wherethe OM parameters for neutron and proton weretaken from the local and global parameterizationsof Koning and Delaroche [45]. For the α -particles,simplifications of the folding approach of S. Watan-abe [46] is used in the TALYS calculations. Itwas found that the variation of the optical modelparameters have very little effect in determining thespectral shape which is mainly decided by the valueof the level density parameter.Fig. 1 shows the experimental neutron spectra forthe p + In reaction at the two incident proton (b) E c.m. (MeV) Expt aA (N-Z)(Z-Z ) (a) E c.m. (MeV) d s / d W d E ( m b / M e V s r) -3 -2 -1 Expt a A(N-Z)(Z-Z ) FIG. 1: Experimental neutron double differential spectra(filled circles) for the p + In reaction measured at155 ◦ for the incident proton energies of (a) 9 MeV (b)12 MeV. Lines are the predictions of TALYS using threedifferent parametrizations of the level density parameter(see text). The arrows show the positions above whichthe spectra are entirely determined by the first-chanceemission. energies. It can be seen that the experimental dataare nicely reproduced by the TALYS calculation(dashed blue line in Fig. 1) using the standardform of the level density parameter given by theexpression ˜ a = αA where the proportionalityconstant α has been taken from global systematics( α = 0 . a = α A + α A / with α =0.69 and α =0.28 for the GC model). Sincethe level of non-linearity is not very large and thenon-linear form does not make much difference interms of explaining the experimental data we haveused the linear form for simplicity. A linearizationof the non-linear expression has been done to obtainthe same resultant level density parameter for thegiven mass number.Different N , Z dependent expressions for thelevel density parameter as described in Eq. 3, andEq. 4 have also been tested. The values of theparameters β, γ , and Z were taken from Ref. [25].It should be mentioned here that unlike Ref. [25]we have used same α -values in the three differentparametrizations of a . Further, the α -value usedin the present work is slightly higher comparedto the values used in Ref. [25] because of thedifference in choosing the energy-shift ∆. For theGilbert-Cameron prescription the energy back-shiftis chosen as ∆ = χ (12 / √ A ). Where χ = 0, 1 and2 for odd-odd, even-odd and even-even nuclei,respectively. For the p + In reaction, theneutron spectra at both the excitation energiesare predominantly determined by the first-chance(1 n ) neutron emission leading to Sn as theevaporation residue (ER). The position of the E c.m. beyond which the spectra are completely decided by d s / d s d W ( m b / M e V s r) -2 -1 TALYS (GC)Expt.data E c.m. (MeV) -2 -1 TALYS (BSFG)Expt. data TALYS (GSM)Expt.data
Expt. dataTALYS (HFBCS)TALYS (HFB+C) (a) (c)(b) (d)
FIG. 2: Experimental neutron spectra (filled circles) forthe He +
Sn reaction at 155 ◦ along with the pre-dictions of TALYS with phenomenological (a) GC, (b)BSFG, and (c) GSM level densities using the standard αA form of the level density parameter. The shaded re-gions in plots (a) - (c) corresponds to ±
15% variationin the standard value of α for each model. (d) The ex-perimental spectrum compared with the TALYS calcu-lation using microscopic HFBCS (continuous line), andHFB+C (dashed line) level densities as inputs. n emission have been indicated by arrows in Fig. 1;for energies below this point there are small ( ≈ n channel. For the β -stable Sn, Z ≈ Z and therefore, the ( Z − Z ) formprovide similar results to that of αA form as can beseen from Fig. 1. On the other hand, the ( N − Z )expression of the level density parameter could notreproduce the experimental data (dashed-dot linein Fig. 1) in this case.In contrast to the p + In reaction, thestandard αA form of the LDP could not reproducethe experimental data in case of the He +
Snreaction, as shown in Fig 2. The situation couldnot be improved by using different forms of thephenomenological level density formulations suchas the Back-shifted Fermi Gas (BSFG) [7], and theGeneralized Super-fluid Model (GSM) [47, 48] asshown in Fig 2 (b) and (c). Efforts were made to fitthe data by tuning the proportionality constant α .However, the data could not be explained by anyreasonable variation of α irrespective of the choiceof the particular phenomenological NLD model(Fig. 2(a)-(c)). The shaded regions in Fig. 2(a)-(c)correspond to ±
15% variation in α around thedefault (systematic) value for each NLD modelprovided in TALYS. The situation could not beimproved even with higher variations in α . Themicroscopic level density inputs obtained underthe Hartree-Fock BCS (HFBCS) [49], and Hartree-Fock-Bogolyubov plus combinatorial (HFB+C) [50]methods also failed to reproduce the experimentaldata (Fig. 2(d)). E c.m. (MeV) d s / d W d E ( m b / M e V s r) -2 -1 Expt a A Z-Z N-Z
FIG. 3: The same as Fig. 1 but for the He +
Snreaction.
Interestingly, for the He +
Sn reaction, itis observed that the experimental data could benicely explained by using reduced level densityparameters given by the ( Z − Z ) form as shown bythe continuous red line in Fig. 3. For this reactionat the present excitation energy (26 MeV), the mostsignificant contributions to the neutron spectrumarise from the 1 n and pn neutron channels leadingto Te and
Sb as residual nuclei, respectively.Besides, there are some small contribution ( < n and αn channels below E c.m. =5MeV. For the most significant ERs ( i.e. Te and
Sb) in the He +
Sn reaction Z − Z & Z − Z ) term has a strong effect,and there is a significant reduction of the resultantlevel density parameters as predicted by Eq. 4. Theexperimental results as evident from Fig. 3 clearlyimply that the level density is strongly reducedfor the neutron-deficient Te and
Sb whichare away from the stability line. For instance, byincorporating the level density parameters given byEq. 4 into the Fermi gas level density expression(Eq. 1) the estimated NLD of the neutron-deficient
Te becomes ≈
10 times lower than that of the β -stable Sn around the neutron separationenergy.The present experimental observation is in con-trast to the high energy fusion evaporation studiesof Charity et al. [35] and Moro et al. [51] performedto investigate the isospin dependence in NLD. WhileCharity et al. [35] did not found any convincingevidence for the neutron-proton asymmetry depen-dence of NLD, Moro et al. [51] showed that the N - Z prescription provide a somewhat better explanationof their data although the results do not discard anisospin independent form of the level density param-eter. It should be mentioned here, that an extensionof the Al. Qurashi parameterizations to very highenergies and high spins could be questionable. Thesituation at higher energies become complicatedas the asymptotic level density parameter itselfmay show significant energy dependence [35, 52].Besides, in high energy heavy-ion fusion reactionsthe level density parameter gets averaged outover a large number of effective decay channelsand may not correspond to one or two specificnuclei of interest. Therefore, light-ion induced lowenergy reactions could possibly be the most suitableprobe to investigate the neutron-proton asymmetrydependence of level density.To investigate the observed variation of the leveldensity parameter we have calculated the single-particle energy levels of several A = 115 isobarsaround the β stable Z . For this purpose, a micro-scopic shell-model [53] with the Wood-Saxon meanfield defined using the Rost parameters [54, 55] isused. Subsequently, the occupation probabilitiesof these single particle levels are calculated atdifferent temperatures ( T ) by following the Fermidistribution function and, as proposed in Ref. [6],the corresponding excitation energies ( E ) areextracted by adding the single-particle energies ofoccupied levels [56]. The level density parameter,defined within the Fermi gas model, is obtainedby employing the Fermi gas formula E = aT .Finally, the corresponding asymptotic values of thelevel density parameter are extracted by using theIgnatyuk formula (Eq. 6) [9]. We have used 26harmonic oscillator shells to define the basis states,and the choice of this basis size reduces the un-certainty in the level density parameter below 5%.The results have been plotted in Fig. 4 (red dashedline) and compared with the phenomenological Z − Z form (green continuous line) with originalparametrization of Ref. [25]. The shaded region inthe plot indicates the theoretical uncertainty of 5%as mentioned above.Evidently, the ˜ a , calculated microscopically, showsthe same trend as described by the phenomeno-logical form given in Eq. 4. Particularly, both ofthem reasonably agree for the nuclei investigated inthe present work. However, Eq. 4 underestimatesthe shell-model prediction as the system departsconsiderably from Z = Z . It suggests, the simpleempirical form of the level density parameter (Eq. 4)may not be sufficient to calculate level densities fornuclei that lie far away from the valley of stability.Moreover, the shell-model ˜ a shows a sharper fallat Z < Z compared to the slope at Z > Z .Further experimental confirmations are requiredto understand this asymmetric variation along theisospin axis.The present analysis clearly established that thelevel density parameter depends on N and Z independently rather than a simple function of A ,and its value reduces as N/Z changes from thevalue around the valley of stability. A significantreduction of level density for proton or neutron-rich
A =115
Atomic number (Z)
44 46 48 50 52 54 56 ã ( M e V - ) Shell ModelEqn. 4 Z = 49.45 Sn Te FIG. 4: The prediction of the Z − Z form (Eq. 4) ofthe level density parameter (continuous line) is comparedwith a microscopic shell-model calculation (dashed line).The shaded region corresponds to the theoretical uncer-tainty in the shell model prediction. The nuclei investi-gated in the present work are indicated by the symbols. nuclei compared to the stable ones as suggestedwould have a profound effect on the nucleosynthe-sis calculations, which typically involve ( p, γ ) or( n, γ ) reaction channels under conditions such thatsuccessive proton or neutron captures can occur.A substantial reduction of level density will try toinhibit repeated captures which take the nucleustowards the drip line. Such a condition wouldchange the balance between β -decay and captureand would eventually push the paths for rp - and r -process nucleosynthesis closer to the valley ofstability.In summary, the neutron energy and angulardistribution have been measured in the p + In,and He +
Sn reactions in the compound nuclearexcitation energy range of ≈
18 - 26 MeV. Statisticalmodel analysis of the backward angle neutronspectra was carried out to investigate the N , Z dependence of nuclear level density parameter.It was observed that experimental data for thetwo reactions could be explained simultaneouslyby using a parametrization of the level densityparameter that reduced its value as the nuclei moveaway from the valley of stability. Another formof a which lowered the level density parameter asiso-spin is increased at fixed A could not explain thedata. The observed variation of the LDP aroundthe β -stable Z has been supported by a microscopicshell model calculation. Thus the present studyprovided a clear evidence for the reduction ofnuclear level density away from the β -stabilityline. Further experimental data for neutron- orproton-rich nuclei in different mass regions will beuseful for the systematic understanding of the N , Z dependence of NLD.The authors would like to acknowledge the VECCCyclotron operators for smooth running of the accel- erator during the experiment. The authors are alsothankful to Dr. Haridas Pai for providing the Sntarget. [1] M. Arnould and S. Goriely, Physics Reports (2003) 1.[2] M. Arnould, K. Takahashi, Rep. Prog. Phys. (1999) 395.[3] B. Meyer, Annu. Rev. Astron. Astrophys. (1994)153.[4] M. Busso, R. Gallino, G.J. Wasserburg, Annu. Rev.Astron. Astrophys. (1999) 239.[5] W. Hauser and H. Feshbach, Phys. Rev. (1952)366.[6] H. A. Bethe, Phys. Rev. (1936) 332; Rev. Mod.Phys. (1937) 69.[7] W. Dilg, W. Schantl, H. Vonach, and M. Uhl, Nucl.Phys. A 217 (1973) 269.[8] T. Ericson, Adv. Phys. (1960) 425.[9] A.V. Ignatyuk, G.N. Smirenkin and A.S. Tishin,Sov. J. Nucl. Phys. (1975) 255.[10] R. Capote, M. Herman, P. Oblozinsky et al. , ”RIPL-Reference Input Parameter Library for calculationof nuclear reactions and nuclear data evaluation”,Nucl. Data Sheets (2009) 3107.[11] A.S. Iljinov, M.V. Mebel, N. Bianchi, et al. , Nucl.Phys. A 543 (1992) 517.[12] A.R. Junghans, M. de Jong, H.-G. Clerc et al. , Nucl.Phys.
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