Nuclear level density and thermal properties of 115 Sn from neutron evaporation
Pratap Roy, K. Banerjee, T. K. Rana, S. Kundu, Deepak Pandit, N. Quang Hung, T. K. Ghosh, S. Mukhopadhyay, D. Mondal, G. Mukherjee, S. Manna, A. Sen, S. Pal, R. Pandey, D. Paul, K. Atreya, C. Bhattacharya
aa r X i v : . [ nu c l - e x ] D ec EPJ manuscript No. (will be inserted by the editor)
Nuclear level density and thermal properties of
Sn fromneutron evaporation
Pratap Roy , K. Banerjee , T. K. Rana , S. Kundu , Deepak Pandit , N. Quang Hung , T. K. Ghosh ,S. Mukhopadhyay , D. Mondal , G. Mukherjee , S. Manna , A. Sen , S. Pal , R. Pandey , D. Paul , K.Atreya , and C. Bhattacharya Variable Energy Cyclotron Centre, 1/AF, Bidhan Nagar, Kolkata - 700 064, India Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai - 400 094, India, Institute of Fundamental and Applied Sciences, Duy Tan University, Ho Chi Minh City 700000, Vietnam, Faculty of Natural Sciences, Duy Tan University, Danang city 550000, Vietnamthe date of receipt and acceptance should be inserted later
Abstract.
The nuclear level density of
Sn has been measured in an excitation energy range of ∼ In( p, n ) Sn reaction. The experimentallevel densities were compared with the microscopic Hartree-Fock BCS (HFBCS), Hartree-Fock-Bogoliubovplus combinatorial (HFB+C), and an exact pairing plus independent particle model (EP+IPM) calcula-tions. It is observed that the EP+IPM provides the most accurate description of the experimental data.The thermal properties (entropy and temperature) of
Sn have been investigated from the measuredlevel densities. The experimental temperature profile as well as the calculated heat capacity show distinctsignatures of a transition from the strongly-paired nucleonic phase to the weakly paired one in this nucleus.
PACS.
XX.XX.XX No PACS code given
Nuclear level density (NLD), is one of the most criticalinputs of the statistical Hauser-Feshbach (HF) calcula-tion [1] of compound nuclear reactions. NLDs are observedto have a strong impact on the calculated neutron-, andproton-capture rates relevant for the r -, and p -processesof heavy-element nucleosynthesis [2,3]. Besides, a precisedetermination of NLD is critical for many practical appli-cations in different areas such as the fusion-fission cross-section calculations for reactor simulations, modeling ofthe spallation reactions for the development of spallationneutron source and accelerator-driven sub-critical systems(ADSS) [4], and production yield estimation of radioiso-topes for medical application [5].Experimental NLD also acts as a testing ground for differ-ent nuclear structure models as well as provide crucial in-formation on the thermodynamic properties of atomic nu-clei [6,7,8,9,10,11,12,13,14,15,16]. Recent investigationson the thermal properties of several nuclei have revealedinteresting features like the existence of peak-like struc-tures in the microcanonical temperature [6,7,8,9,10], S-shaped heat capacity curve [13,14,15,16], which are iden-tified as signatures of a phase transition in a finite systemfrom a strongly paired phase to a phase with weak pairing a Email for correspondence: [email protected], [email protected] correlations [17,18]. The pairing effect in finite nuclei iswell known through the odd-even effect in nuclear masses,for example. On the other hand, in a macroscopic con-ductor, pairing leads to a phase transition from a normalmetal to a superconductor below a certain critical temper-ature. In the BCS theory [19] of superconductivity the nor-mal to superconducting phase transition is characterizedby a finite discontinuity of the heat capacity at the crit-ical temperature. However, in a finite nucleus where thepair coherence length is much larger than the nuclear ra-dius, the sharp discontinuity in heat capacity is smootheddue to large fluctuations. The smoothing of the superfluidto normal phase transition in finite nuclei caused by thenon-vanishing of the neutron and/or proton pairing gapsat the critical temperature has been widely discussed in e.g. , Ref. [20] and references therein.The Sn isotopes provide an excellent ground to observethe thermal signatures of the pairing interaction. Becauseof the Z = 50 shell-closure, the proton pair breaking isstrongly hindered, and the signature for the pure neutronpair-breaking becomes much more prominent comparedto other mass region [9]. In the recent works by the Oslogroup, evidences of nucleonic Cooper pair breaking at fi-nite temperates were observed in Sn,
Sn, and
Snnuclei [9,10]. Surprisingly, such features were not promi-nent for
Sn, which could be due to the poorer statisticsin this case [10]. It would be interesting to extend similar
Pratap Roy et al.: Nuclear level density and thermal properties of
Sn from neutron evaporation studies for relatively neutron-deficient Sn isotopes such as
Sn.The nucleus
Sn is also significant from the astrophys-ical point of view. It is known to be formed mostly bythe astrophysical p -process, however, some small contri-bution from the s -process also could not be ruled out [3].Presently, the nucleosynthesis calculations largely under-predict the observed abundance of Sn [21]. Althoughthe origin of the large underproduction could not be at-tributed to uncertainties in nuclear physics inputs alone,they are seen to influence the nucleosynthesis predictionsin a non-negligible way [3]. The experimental informationon critical quantities such as NLD in a wide excitationenergy range could be useful to reduce the uncertainty inabundance calculations from the nuclear physics side.In this paper, we report the experimental level densityof
Sn in the excitation energy range of ∼ p + In reaction at two compound nuclear excitationenergies E ∗ = 18.2 and 21.2 MeV. At these low excitationenergies, the compound nucleus ( Sn) predominantly de-cays by the first-chance (1 n ) neutron emission leading tothe residual nucleus Sn. Different microscopic models oflevel density have been tested by comparing with the ex-perimental data. The thermodynamic properties of
Snhave also been investigated.The article has been arranged in the following manner.The experimental arrangement has been described in Sec. 2.The results have been presented and discussed in Sec. 3;the experimental level density and thermodynamic quan-tities have been presented in Sec. 3.1 and Sec. 3.2, re-spectively. The microscopic EP+IPM calculation has beenbriefly described in Sec. 3.3. Finally, the summary andconclusion have been presented in Sec. 4.
The experiment was performed using proton beams of E lab = 9 and 12 MeV from the K130 cyclotron at VECC,Kolkata. A self-supporting foil of In (thickness ≈ ) was used as the target. The neutrons emittedduring the compound nuclear decay process were detectedusing six liquid scintillator based neutron detectors (size:5-inch × ◦ ,85 ◦ , 105 ◦ , 120 ◦ , 140 ◦ , 155 ◦ at a distance of 1.5 m fromthe target. The neutron kinetic energies were measuredusing the time-of-flight (TOF) technique. The start trig-ger for the TOF measurement was generated by detect-ing the low energy γ -rays in a 50-element BaF detectorarray [22] placed near the target position. In convertingthe neutron TOF to neutron energy, the prompt γ peakin TOF spectrum was used as the time reference. Theneutron and γ separation was achieved by using both theTOF and pulse shape discrimination (PSD) methods. Theexcitation energy-dependent efficiency, which is a crucialparameter, was measured in the in-beam condition using astandard Cf neutron source [23]. The scattered neutroncontribution in the measured spectra was estimated usingthe shadow bar technique and found to be around 2.5% Q c.m. (deg.)
40 60 80 100 120 140 160 1800.00.20.40.6
Expt.TALYS (CN)TALYS (CN+PE)
Expt.TALYS (CN)TALYS (CN+PE) d s / d W ( m b / s r) Expt.TALYS (CN)TALYS (CN+PE)
Fig. 1. (Colour online) Neutron angular distributions for theselected energy bins (indicated in each panel) for the p + Inreaction at E lab = 12 MeV. The experimental data are shownby the filled squares. The dashed and the continuous lines arethe theoretical TALYS prediction for the compound nuclear(CN) and compound plus pre-equilibrium (CN+PE) compo-nents, respectively. in the total neutron energy range. It should be mentionedthat the back-angle neutron spectra for the present sys-tem have been recently utilized to investigate the iso-spindependence of the level density parameter for A = 115isobars [25]. Additional information on the experimentalsetup and analysis techniques can be found in Ref. [25]. The background-corrected neutron spectra measured atvarious laboratory angles were transformed into the com-pound nucleus (CN) center-of-mass (c.m.) frame using thestandard Jacobian transformation. The angular distribu-tions in the c.m. frame at the highest bombarding en-ergy ( E lab =12 MeV) for selected neutron energy inter-vals have been shown in Fig. 1. It can be seen that the ratap Roy et al.: Nuclear level density and thermal properties of Sn from neutron evaporation 3 cross-section does not vary significantly as a function ofthe emission angle indicating the dominance of the com-pound nuclear emission process in determining the neu-tron spectra at the present incident energies. To investi-gate the emission mechanisms further, the experimentalangular distributions were compared with the theoreticalpredictions obtained using the
TALYS (v 1.9) computercode [26]. The theoretical TALYS results are shown forboth the compound nuclear (dashed lines in Fig. 1) andcompound plus pre-equilibrium (PE) (continuous lines inFig. 1) components. The compound nuclear calculationare performed using the Hauser-Feshbach framework [1]whereas the PE component has been estimated using theExciton model [27,28] in TALYS. It can be seen fromFig. 1 that the contributions of pre-equilibrium processesin the experimental data are small, and it is negligibleparticularly at the backward angles where the data canbe rather described by the CN component alone. The ex-perimental data at the most backward angle (155 ◦ ) is con-sidered almost free of the non-equilibrium component, andused for the statistical model analysis to extract the leveldensity.The spectra measured at 155 ◦ for the two incident protonenergies of 9 and 12 MeV are shown in Fig. 2. The double-differential cross-sections are multiplied by 4 π to comparewith the calculated angle-integrated spectra. The experi-mental spectra have been compared with the theoretical TALYS calculations performed within the statistical HF framework. For the level density, the composite Gilbert-Cameron (GC) [29] formulation has been used as input ofthe TALYS calculations.In the GC model, the level density at low energies from0 up to some matching energy E M is approximated by aconstant-temperature (CT) formula ρ CT ( E ) = 1 T exp E − E T (1)and for energies higher than E M the level density is givenby the Fermi gas (FG) expression [30], ρ F G ( E ) = 112 √ σ exp (2 p a ( E − ∆ )) a / ( E − ∆ ) / (2)where a is the level density parameter, σ is the spin cut-off factor, and ∆ is the pairing energy shift. The constanttemperature ( T ), and the energy shift ( E ) in Eq. (1) arechosen in such a way the two prescriptions (CT and FG)match together smoothly at the matching energy, whichis ∼ a [31] a ( U ) = ˜ a [1 + ∆SU { − exp( − γU ) } ] (3)where U = E − ∆ , and ˜ a is the asymptotic value ofthe level density parameter obtained in the absence ofany shell effect. Here ∆S is the ground state shell correc-tion, and γ determines the rate at which the shell effect (b) E c.m. (MeV) -2 -1 Expt. Data
TALYS (incl.)
TALYS (1 n) TALYS (2 n) (a) d s / d E ( m b / M e V ) -3 -2 -1 Expt. Data
TALYS (incl.)
TALYS (1 n ) Fig. 2. (Color online) The experimental neutron energy spec-tra (symbols) along with the
TALYS calculations (lines) attwo incident proton energies (a) 9 and (b) 12 MeV. The ar-rows show the position above which the spectra are entirelydetermined by the first-chance emission. Contributions fromdifferent stages of the decay have also been shown. is depleted with the increase in excitation energy. Theasymptotic level density parameter ˜ a has been tuned toobtain the best match to the experimental data, whereasthe energy-shift ∆ was taken from the systematics [26].The optimum values of the level density parameter alongwith other model parameters for the CN ( Sn) and ma-jor evaporation residues
Sn (1 n ) and Sn (2 n ) havebeen tabulated in Table 1. The contributions of the dif-ferent decay channels (1 n and 2 n ) in the total (inclusive)neutron spectra have been shown in Fig. 2. It can be seenfrom the figure that at E ∗ = 18.2 MeV ( E lab = 9 MeV)the spectrum is entirely determined by the 1 n emissionchannel whereas at E ∗ = 21.2 MeV ( E lab = 12 MeV) asmall contribution from the 2 n channel bellow E c.m. ≈ Sn The “experimental” level densities ρ exp ( E ) were extractedfrom the neutron spectra using the following relation [32,33] ρ exp ( E ) = ρ model ( E ) ( dσ/dE ) exp ( dσ/dE ) model (4)Here ( dσ/dE ) exp is the experimental differential cross-section and ( dσ/dE ) model is the differential cross-section Pratap Roy et al.: Nuclear level density and thermal properties of
Sn from neutron evaporation
Table 1.
Different model parameters used in the
TALYS calculation.Nucleus S n ˜ a ∆P E T E M ∆S γ (MeV) (MeV − ) (MeV) (MeV) (MeV) (MeV) (MeV) (MeV − ) Sn 9.563 15.08 2.228 1.436 0.538 5.04 1.078 0.089
Sn 7.546 14.95 1.119 0.566 0.511 3.49 1.008 0.089
Sn 10.303 14.82 2.248 1.116 0.599 3.49 0.682 0.089 calculated by the HF calculation using ρ ( E ) model as itsinput level density. Eq. 4 predicts the correct energy de-pendence (slope) of level density, and the local variationsin NLD are provided by the bin wise normalization factorgiven by the ratio of the calculated to the experimentaldifferential cross-section. However, the level density cannot be obtained in absolute terms by Eq. 4 alone becausethe ratio dσ/dE ) exp ( dσ/dE ) model depends not only on the ratio of ρ ( E ) exp , and ρ ( E ) model but also on the competition withother decay channels. The calculated cross sections de-pend on the ratio of the decay probabilities to variouschannels and an overall arbitrary factor multiplying theNLD for all involved nuclei cancels out in this ratio. Thusone can extract only the energy dependence of the NLDand not their absolute values from the statistical modelanalysis alone. Therefore, the experimental data obtainedusing Eq. 4 needs to be renormalized to some known NLDvalue obtained using other independent techniques. Usu-ally the normalization is done with the density of discreteenergy levels for a given nucleus at low energies [34,35].The number of low lying discrete levels provide a less am-biguous and model independent way of normalizing theNLD. However, in the present work, we could not go downto the lowest energies with sufficient statistics. So, the ab-solute normalization was done at the neutron separationenergy ( S n ) using the level density ( ρ ) obtained from themeasured S -wave neutron resonance spacing ( D ). Thelevel density at S n is related to D through the followingrelation [10] ρ ( S n ) = 2 σ D × [( I t + 1)exp − ( I t + 1) σ + I t exp − I t σ ] − (5)Here, I t is the spin of the target nucleus, and the spincut-off parameter σ is evaluated at S n using the energy-dependent parametrization prescribed by Egidy and Bu-curescu [36]. The experimental S -wave resonance spacingfor Sn is 0.286 ( ± ρ ( S n ) =1.268 × MeV − .The experimental level densities for Sn extractedfrom the neutron spectra have been tabulated in Table 2.The errors in the measured NLD represents both the sta-tistical and systematic uncertainties. A maximum system-atic error of 40% arising out of the uncertainty in the de-termination of the normalization point has been added inquadrature with the statistical errors. The experimental
Excitation Energy (MeV) L e v e l D e n s it y ( M e V - ) Expt. data r ( (cid:1)(cid:0) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7) r (cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28) Sn Fig. 3. (Color online) The experimental level density (filledsquares) along with the theoretical predictions (lines) of dif-ferent microscopic NLD models (see text). The normalizationpoint has been indicated by the arrow. level densities are also plotted in Fig. 3 by filled squares.The density of discrete levels ( ρ disc. ) obtained from theexperimental energy levels [37] of Sn is also plotted inFig. 3 by the short-dashed line. In extracting the leveldensities using Eq. 4, only the part of the experimentalspectra which is determined completely by the first-chanceneutron emission (shown by the arrows in Fig. 2) was used.The excitation energy of the residual nucleus ( E ) was cal-culated using the relation E = E ∗ − S n − E n − E R , where E ∗ is the compound nuclear excitation energy, S n is theneutron separation energy, E n and E R are the kinetic en-ergies of the emitted neutron and the residual nucleus, re-spectively. The residual nucleus ( Sn) being significantlyheavier than the emitted neutron its kinetic energy is neg-ligible. For the overlapping data points obtained from theneutron spectra measured at the two bombarding ener-gies, the average values of the level densities have beentaken.The experimental level densities provide an excellent op-portunity to test different microscopic approaches of NLD.The Hartree-Fock-BCS (HFBCS) [38,39], and Hartree-Fock-Bogoliubov plus combinatorial (HFB+C) [40] mod-els are the most frequently used microscopic methods thatprovide a global description of NLD. The experimentaldata have been compared with the predictions of HF-BCS and HFB+C methods, as well as an exact pairing ratap Roy et al.: Nuclear level density and thermal properties of
Sn from neutron evaporation 5
Table 2.
Total nuclear level densities in
Sn.Excitation energy ( E ) Level density ( ρ ( E ))(MeV) × (MeV − )1.90 0.016 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± plus independent particle model (EP+IPM), introducedrecently [41,42]. It is observed that the experimental dataare in excellent agreement with the prediction of the mi-croscopic EP+IPM (shown by the (red) continuous linein Fig. 3) which takes into account the thermal effectsof exact pairing in finite size systems [41]. The HFB+Ccalculation of Goriely et al. , [40] also explains the exper-imental data reasonably well except for a small disagree-ment at lower energies as shown by the dash-dotted line inFig. 3. On the other hand, the total level densities calcu-lated based on the single-particle level schemes determinedwithin the HFBCS model [38,39] deviates from the exper-imental data at higher energies (medium-dashed line inFig. 3). The agreement of EP+IPM with the experimen-tal data emphasizes the crucial role of thermal pairingin the description of the NLD. It should be mentionedhere, that unlike the HFBCS and HFB+C methods, theEP+IPM-predicted NLDs do not require any additionalnormalization while comparing with experimental data. Sn The thermal properties of
Sn were investigated usingthe measured level densities. The microcanonical entropy( S ), and temperature ( T ) were extracted from the mea- Excitation Energy (MeV) T e m p e r a t u r e T ( E ) ( M e V )
95% confidence bandExpt. data (cid:29)(cid:30) Sn E n t r opy S ( E ) ( k B ) Expt. data (a)(b)
Fig. 4. (Color online) The experimental (a) entropy and(b) temperature as a function of excitation energy. The solid(green) and the dashed (red) lines are the Gaussian and linearfits to the data, respectively. The gray shaded regions representthe 95% confidence band. sured NLDs following the prescription of several earlierworks [8,7,10]. The entropy is defined by S = k B ln (cid:20) ρ ( E ) ρ (cid:21) (6)where k B is the Boltzmann constant which is usually setto unity to make the entropy dimensionless. The constant ρ is adjusted to fulfill the condition of the third lawof thermodynamics S → T →
0. The ground statesof even-even nuclei represent a completely ordered sys-tem with only one possible configuration, and are char-acterized by zero entropy and temperature. The value of ρ = 0.135 MeV − thus obtained for the nearest even-even nuclei Sn [9] is also used in the present case. Themicrocanonical temperature ( T ) is defined by the relation T = (cid:18) ∂S∂E (cid:19) − (7)The experimental entropy and temperature as a functionof the excitation energy have been plotted in Fig. 4. Thenumber of accessible microstates increases as a functionof the excitation energy resulting in the gross increase ofentropy as shown in Fig. 4(a). Any possible fine-structurepresent in the entropy curve will be amplified in the tem-perature plot due to the derivative relation with entropy.The temperature profile, thus obtained is displayed in Pratap Roy et al.: Nuclear level density and thermal properties of
Sn from neutron evaporation
Fig. 4(b), which shows fluctuating nature characterizing asmall isolated system like the atomic nucleus, away fromthermodynamic limits. On top of the inherent fluctua-tions, there are a couple of distinct peak-like structures inthe temperature profile between E ≈ E ∼ ∼ , Sn [9].
To investigate the microscopic origin of the observed fea-tures in the temperature plot, we have calculated the tem-perature dependent neutron pairing gap ( ∆P n ) within theEP+IPM which provided the best description of the ex-perimental level density (Fig. 3). The EP+IPM formalismfor the calculations of different thermodynamic quantitieshas been described in detail in Refs. [13,41,42]. The calcu-lated pairing gap, as shown in Fig. 5(a), decreases rapidlyin the region between E ∼ T c ( ∼ T = 0.4 MeV isdue to the blocking effect at finite temperature in odd nu-cleon systems [43]. When more Cooper pairs are broken,the pairing gap decreases slowly, which is reflected in thenear constant nature of the experimental temperature inthe region beyond E ≈ Sn has been in-vestigated within the EP+IPM, and it is found that thedifferent regions of the excitation energy can be identifiedas, (i) E ∼ E ∼ E ∼ E ∼ Sn. In general, for odd mass nuclei ( e.g.
Sn and
Sn [9]) there exist two peaks in the microcanonical tem-perature curve below E < ∼ D P n ( M e V ) Temperature (MeV) H ea t ca p ac it y (a)(b) T (MeV) d D P n / d T -0.8-0.40.00.4 E~ 6 MeVE~ 0.5 MeV Fig. 5. (Color online) (a) The neutron pairing gap, and (b)the canonical heat capacity ( C v ) of Sn as a function of tem-perature obtained within the EP+IPM calculation. The rateof change of the neutron pairing gap as a function of the tem-perature is shown in the inset of the upper panel. neutron pairs.Along with the neutron pairing gap the canonical heatcapacity has also been calculated within the EP+IPM,and shown in Fig. 5(b). The calculated heat capacity showsa S-shape nature with a local maximum around the criticaltemperature. As described in recent theoretical works [17,18], the S-shaped heat capacity curve could be treated asa fingerprint of a phase transition in a finite system froma phase with strong pairing correlations to a phase withweak pairing correlations. Pronounced S-shapes in heatcapacity are generally expected for the even-even nuclei,whereas for neighboring odd-odd and odd-even nuclei suchstructures could be less prominent [8,13,16]. In the presentcase of the even-odd
Sn, there is a distinct enhancementin the associated heat-capacity in around T ≈ Sn.
The neutron energy spectra in case of p + In reac-tion have been measured at low incident energies, andcompared with the statistical Hauser-Feshbach calculationperformed using the
TALYS code. The nuclear level den-sity of the residual
Sn nucleus has been extracted inthe excitation energy range of ∼ ratap Roy et al.: Nuclear level density and thermal properties of Sn from neutron evaporation 7 been compared with the predictions of microscopic HF-BCS, HFB+C, and EP+IPM models. It is observed thatthe experimental data are best described by the EP+IPMcalculation highlighting the importance of thermal pairingin the description of NLD. The experimental temperatureprofile shows peak-like structures around E ∼ Sn showed a S-shapewhich provide a clear indication of the quenching of thenuclear pairing correlation at finite temperature.
Acknowledgments
The authors would like to acknowledge the VECC Cy-clotron operators for smooth running of the acceleratorduring the experiment. We are thankful to J. K. Meena,A. K. Saha, J. K. Sahoo and R. M. Saha for their helpduring the experimental setup. The authors also thanksJhilam Sadhukhan for the stimulating discussions.NQH’s works are funded by The National Foundation forScience and Technology Development of Vietnam throughGrant Number 103.04-2019.371.
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