Nghiên cứu thực nghiệm cấu trúc phổ năng lượng kích thích của các hạt nhân 172Yb và 153Sm trên kênh nơtron của lò phản ứng hạt nhân Đà Lạt

Introduction Level structure of atomic nuclei, including nuclear level scheme (NLS), nuclear level density (NLD), and radiative strength function (RSF), are all together important quantities, which carry fundamental information on the structure and properties of excited nuclei. For NLS, its completeness plays important roles for the study of not only nuclear reaction and statistical model calculations but also adjustment of the nuclear level density parameters. Ideally, the completeness of NLS can be obtained by studying the spectroscopic data from non-selective reactions. However, in practice, it is almost impossible due to experimental conditions and the complex of gamma spectrum [1]. Therefore, the only way to obtain a complete NLS is to study through various experiments and combine their results together. Most of the NLS data were compiled in the ENSDF library [2] containing about 187,067 datasets for 3,312 atomic nuclei collected from various experiments including beta decay, electron capture, neutron inelastic scattering, compound nuclear reactions induced by ions such as ( 3 He, He’ ), (, ’), (p, d), (d, t), etc. However, information on the excited states and their corresponding primary and/or secondary transitions of many nuclei in the intermediate energy region, where the thermal neutron capture (n th 3 , ) reaction was mostly employed to extract the data, is still sparse and incomplete. Regarding the NLD and RSF, although they are important quantities for the study in many fields such as low-energy nuclear reactions, astrophysical nucleonsynthesis, nuclear energy production, transmutation of nuclear waste, nuclear reactor design, there is still lacking a lot of experimental data in the literature, in both

1.1 Compound nuclear reaction 11

1.1.1 Bohr-independence hypothesis 11

1.1.2 Reciprocity theorem 13

1.2 Nuclear level scheme 13

1.3 Nuclear level density 16

1.3.1 Fermi-gas model 18

1.3.1.1 Systematics of the Fermi-gas parameters 21

1.3.1.2 Parity ratio 24

1.3.2 Constant temperature model 25

1.3.3 Gilbert-Cameron model 26

1.3.4 Generalized superfluid model 27

1.3.5 Microscopic-based models 29

1.4 Radiative strength function 33

1.5 Conclusion of chapter 1 37

2 Experiment and data analysis 39

2.1 Experimental facility and experimental method 39

2.1.1 Dalat Nuclear Research Reactor and the neutron beam-port No.3 39

2.1.2 The γ− γ coincidence method 41

2.1.3 γ− γ coincidence spectrometer 44

2.1.3.1 Electronic setup and operation principle 44

2.1.3.2 Main properties 46

2.1.4 Experimental setup and target information 49

2.1.5 Sources of “systematic” errors in γ− γ coincidence method 51 2.2 Data Analysis 56

2.2.1 Pre-analysis 57

2.2.2 Two-step cascade spectra 61

2.2.3 Determination of gamma cascade intensity 65

2.2.4 Construction of nuclear level scheme 66

2.2.5 Determination of gamma cascade intensity distributions 67

2.2.6 Extraction of nuclear level density and radiative strength function 69

2.2.6.1 Basic ideas and underlying assumption 69

2.2.6.2 Determination of the functional form of the γ-rays transmission coefficient 72

2.2.6.3 Determination of nuclear level density 76

2.2.6.4 Determination of radiative strength function 78

2.3 Conclusion of chapter 2 79

3 Results and discussion 81 3.1 Nuclear level scheme of172Yb and153Sm 81

3.1.1 172Yb 81

3.1.2 153Sm 92

3.2 Gamma cascade intensity distributions of172Yb 97

3.3 Nuclear level density and radiative strength function of172Yb 105

3.3.1 Comparison with other experimental data 108

3.3.2 Comparison with theoretical models 111

3.3.2.1 Nuclear level density 111

3.3.2.2 Radiative strength function 111

3.4 Conclusion of chapter 3 114

2.17 Summation spectrum for 152Sm(n,2γ) reaction E1 + E2 is sum ofenergies measured from two detectors Energies (in keV) of thefinal levels in the cascades are pointed near the peaks of the fullabsorption energy 602.18 Explanation of the input variables used in the procedure of obtain-ing the TSC spectra given in Fig 2.19 622.19 Detail procedure for obtaining the TSC spectra 642.20 a experimental TSC spectrum; b simulated TSC spectrum, c un-resolved TSC spectrum with noise line, d unresolved TSC spec-trum without noise line corresponding to the decays from the com-pound state to the ground state of172Yb 682.21 Procedure of extracting the NLD and RSF 712.22 Illustration of the shifting procedure for172Yb nucleus with Em =3.625 MeV, Em0 = 3.875 MeV and Emax

f = 1.198 MeV The perposed energy range is between the two vertical arrows Thecurve (1) simulates the standard dataset (circle), while the curve(2) models the to-be-shifted dataset (triangle) The k factor is theratio between the area under the curve (1) and that under thecurve (2) The two curves have the form of an exponential func-tion C0exp(C1E), whose parameters (C0, C1) are obtained via thefitting to the corresponding datasets 732.23 The final dataset describes the functional form of γ-rays transmis-sion coefficient of 172Yb nucleus in the energy region from 0.5 to7.5 MeV The line is the average values over an 250 keV energyinterval 752.24 Comparison of the γ-rays transmission coefficients of 172Yb nu-cleus obtained by different starting excitation-energy bins His-togram with black color is the average of the γ-rays transmissioncoefficients obtained by all the starting excitation-energy bins from2.125 MeV to 5.375 MeV The corresponding uncertainties are given

su-by upper and lower lines 763.1 TSC spectrum corresponding to the ground state of172Yb 883.2 TSC spectrum corresponding to the final level with the energy Ef =78.8 keV of172Yb 89

3.3 Experimental level scheme of 172Yb obtained within the gammacascades from compound state to six distinct low-lying discretelevels with spin from 0¯h to 2¯h Explanation of the figure is given intext 913.4 TSC spectrum corresponding to the final levels with energies Ef =

0 and 7.8 keV of153Sm 953.5 TSC spectrum corresponding to the final level with the energy Ef

= 35.8 keV of153Sm 953.6 NLS of 153Sm obtained within this present work Explanation ofthe figure is the same as in Fig 3.3 963.7 The gamma cascade intensity distributions of 172Yb obtainedwithin the present work 983.8 The extracted NLD and RSF of 172Yb obtained within the presentwork 1063.9 Comparison between the experimental gamma cascade intensitydistributions and the calculated one obtained from the extractedNLD and RSF 1073.10 Comparison between the NLD obtained within the present workand the other experimental data Explanation for this figure isgiven in text 1093.11 Comparison between the RSF obtained within the present workand the other experimental data Explanation of this figure is given

in text 1103.12 Comparison between the NLD obtained within the present workand a few common theoretical models 1123.13 Comparison between RSF obtained within this work and few the-oretical models See explanation of the figure in text 113

List of Tables

2.1 Selected parameters of the electric modules 462.2 Parameters of the relative efficiency functions 493.1 Primary and secondary gamma-ray energies and absolute inten-sities obtained from the 171Yb(nth, γ) reaction The experimentalvalues are compared with the ENSDF data 813.2 Primary and secondary gamma-ray energies and absolute inten-sities obtained from the 152Sm(nth, γ) reaction The experimentalvalues are compared with the ENSDF data 933.3 The gamma cascade intensity distribution of172Yb and the contri-bution of the resolved cascades 99

Level structure of atomic nuclei, including nuclear level scheme (NLS), nuclearlevel density (NLD), and radiative strength function (RSF), are all together im-portant quantities, which carry fundamental information on the structure andproperties of excited nuclei For NLS, its completeness plays important roles forthe study of not only nuclear reaction and statistical model calculations but alsoadjustment of the nuclear level density parameters Ideally, the completeness ofNLS can be obtained by studying the spectroscopic data from non-selective re-actions However, in practice, it is almost impossible due to experimental condi-tions and the complex of gamma spectrum [1] Therefore, the only way to obtain

a complete NLS is to study through various experiments and combine their sults together Most of the NLS data were compiled in the ENSDF library [2]containing about 187,067 datasets for 3,312 atomic nuclei collected from variousexperiments including beta decay, electron capture, neutron inelastic scattering,compound nuclear reactions induced by ions such as (3He,3He’γ), (α, α’), (p, d),(d, t), etc However, information on the excited states and their correspondingprimary and/or secondary transitions of many nuclei in the intermediate energyregion, where the thermal neutron capture (nth,γ) reaction was mostly employed

re-to extract the data, is still sparse and incomplete

Regarding the NLD and RSF, although they are important quantities for the study

in many fields such as low-energy nuclear reactions, astrophysical thesis, nuclear energy production, transmutation of nuclear waste, nuclear reac-tor design, there is still lacking a lot of experimental data in the literature, in both

nucleonsyn-low- and high-energy regions.

It has been well-known that the γ − γ coincidence method [3] can be used tostudy the NLS, NLD and RSF Within this method, the cascade events, whichare obtained from the decay of the initial compound state to the different finalstates, are separated into different Two-Step-Cascade (TSC) spectra Particularly,only correlated gamma transitions are detected, therefore the number of γ-rayscontributed to the TSC spectra is less than that presented in a normal promptgamma spectrum, leading to the significant reduction of the overlapping γ-rays

as well as improving the detecting ability of this method In addition, ent from the normal gamma spectra, the TSC spectra obtained using the γ − γcoincidence method, after applying the background subtraction algorithm, havealmost no Compton background Therefore, the detection limit of the coincidencemethod is much improved in comparison with the normal gamma spectra anal-ysis method Beside that, the state from which a secondary gamma transition isdecayed can be determined in the coincidence method if one of the two γ-rays inthe cascade is a known primary transition Based on these above advantages, the

differ-γ − γ coincidence method is appropriate for the determination of excited stateswith low spin in the energy region from 0.5 MeV to Bn− 0.5 MeV (Here, Bnis theneutron binding energy) Furthermore, the γ− γ coincidence method can also beused to determine the gamma cascade intensity distributions, which are related

to the NLD and RSF [see e.g Eq (2.1)] Therefore, it is possible to determine theexperimental NLD and RSF based on this method

172Yb and153Sm are two deformed and rare earth nuclei Their NLS are absolutelynecessary for either confirming or enhancing the predictive powers of the nuclear

models for heavy nuclei The adopted nuclear levels and gamma transitions of

172Yb and 153Sm are given in Ref [4] and Ref [5], respectively A summary ofimportant investigations on the NLS of these two nuclei is presented as follows

For 172Yb

NLS of172Yb has been thoroughly studied in different methods such as beta decay

of 172Tm [6], electron capture decay of 172Lu [7], neutron inelastic scattering forlow-lying states in 172,174Yb [8], (n, n’γ) reactions using fast neutrons from reac-tors [9],170Er(α,2n)172Yb reaction for high-spin states [10],171Yb(n,γ) reaction forlow-spin states [11, 12] Additional methods, which are also able to provide thelevel scheme of172Yb based on the compound nuclear reactions induced by lightions, include 172Yb(3He,3He’γ) [13], 172Yb(α,α’) [14], 173Yb(p,d) [15], 171Yb(d,p)[16], 173(d,t) [17], 173Yb(3He,γ), (3He,αγ) [17, 18], 170Yb(t,p) [19], and elastic andinelastic proton scatterings [20, 21] Furthermore, lifetimes of a number of levelshave been also determined via the 172Yb(γ, γ’) reactions using the nuclear reso-nance fluorescence [22] and Coulomb excitation [23, 24] methods Through theseexperiments, the low-lying discrete level scheme of 172Yb in the low-energy re-gion (E < 2.4 MeV) has been well understood [4] In this low-energy region,energies of the levels were determined with the accuracy of ten to hundred eV,whereas spins and parities were also identified for a majority of levels However,information on the excited states and their corresponding primary transitions inthe intermediate energy region (2.4 MeV < E < 5 MeV), where the thermal neu-tron capture reactions (nth,γ) were mostly employed to extract the data, is sparseand incomplete

In particular, based on the neutron capture reaction with both thermal and 2

keV neutrons, Greenwood et al [11] have detected, by using the Ge(Li) tor, in total 127 primary gamma transitions including their intensities from theprompt gamma spectrum of172Yb At the same time, the prompt gamma yield of

detec-172Yb has also been determined from the abundance of171Yb in natural ytterbiumand their relative thermal-neutron capture cross sections In addition to that, 136gamma-ray transitions, whose energies are less than 2.5 MeV, have been also re-ported in this paper Using the same thermal neutron capture reaction with theuse of the pair formation spectrometer, Gellety et al [12] have measured the pri-mary transitions of172Yb, whose energies and relative intensities were found ingood agreement with those reported by Greenwood Although the results ob-tained from Gellety have improved the level scheme of172Yb, which had previ-ously been constructed by Greenwood using the Riz combination principle, thesignificant differences between the absolute intensities of gamma transitions ob-tained within those works have not yet been explained It is obvious that thenumber of detectable gamma rays in the normal gamma-ray spectrum dependsupon the energy resolution of the detectors as well as the number of excited statesexisting in an interval of excitation energy Thus, there has been a certain limita-tion on the results of Refs [11, 12] due to the restricted energy resolution of theGe(Li) detectors used in those works as well as the large number of excited states

of172Yb in the energy region from 3 MeV to 5 MeV, where the discrete region ofthe level density interferes with the continuous one

In fact, the gamma-gamma coincidence technique was used to measure the levelscheme of172Yb in Ref [12], however, it was set to cover the energy range from 0

to 2 MeV only This method was later used to measure the TSC intensities of172Ybfrom171Yb(nth, γγ) reaction in Ref [25] The results obtained were then comparedwith the statistical-model calculations which base entirely on the experimentallevel density and gamma strength function extracted from the primary gamma

spectra of173Yb(3He, αγ)172Yb reaction On the other hand, in Ref [25] the troscopic data were also presented and compared with the results of Refs [11, 12]but these data were not shown in detail because of their low statistics, in whichonly 4000 cascade events corresponding to the decays from the compound state

spec-to the ground state are collected

For 153Sm

NLS of 153Sm has been investigated in a thorough manner in various methodssuch as beta decay of153Pm produced from252Cf fission [26], isomeric transitiondecay of 153Sm [5], 152Sm(n,γ) reaction for low-spin state [27, 28], transfer reac-tions such as152Sm(d,p) [29],154Sm(p,d) [30],152Sm(α,3He) [31],154Sm(d,t) [28,29]and recent 151Sm(t,p) [32] Through these experiments, the low-lying discretelevel scheme of153Sm in the low-energy region (E < 2.2 MeV) has been well un-derstood [5] In this low-energy region, information about 203 excited states,including excitation energies, spins, and parity, were determined However, inthe high-energy region (2.2 MeV < E < 4 MeV), although the number of excitedstates is expected to be large, most of the states were reported without providinginformation on their spins and parities Moreover, the uncertainty of the excita-tion energies was from 10 to 17 keV, which is rather high in comparison with theuncertainty of gamma energies obtained by using the HPGe gamma spectrome-ters

Particularly, based on the neutron capture reaction with thermal neutron, Smither

et al [27] have detected, by using the bent-crystal spectrometer for the low-energyregion and the Ge(Li) detector for the high-energy region, in total 251 gammatransitions in the energy region from 28 keV to 1041 keV and 23 additional transi-tions between 4.5 and 5.9 MeV Using the same thermal neutron capture reaction,

Bennette et al [28] have measured the prompt gamma spectrum of153Sm by means

of both Ge(Li) and Si(Li) detectors The results obtained were in good agreementwith those reported in Ref [27], and also a number of new levels was reported

As can be seen in Ref [5], the number of excited states of 153Sm is rather large,therefore it is obvious that there has been a certain limitation on the results ofRefs [27, 28] due to the restricted energy resolution of the detectors used withinthose works

On the another hand, Blasi et al [30] have detected, based on the154Sm(p,d) tion, 170 gamma transitions in the energy region below 2.2 MeV The result ob-tained by Blasi et al confirms not only the existence of many excited states given

reac-in Refs [27, 28] but also provides a number of new excited states, especially reac-inthe energy region from 1 MeV to 2.1 MeV Regarding the high-energy region (E

> 2.2 MeV), most excited states were determined through the 152Sm(d,p) tion [28, 29] However, the uncertainty of excited state energies proposed in theseinvestigations is larger than 10 keV

reac-Experimental nuclear level density and radiative strength function

From the experimental side, the NLD has been studied within several methodssuch as counting of discrete levels for the very low-energy region [33,34], neutronresonance spacing at the neutron binding energy [35], and evaporation spectrafor the high-energy region (above the particle threshold) [36] For the RSF, itsinformation can be extracted from the photoabsorption cross section [37] as well

as the radiative neutron capture reactions [38] and reactions with charged ticles [39] Recently, the Oslo group at Oslo University (Norway) has proposed

par-an advpar-ance technique, called Oslo’s method, which allows them to simultpar-ane-ously extract both NLD and RSF from the measured gamma-decay spectra ob-tained via the transfer and/or inelastic scattering reactions [40, 41] Since afterthat the study of NLD and RSF has become particularly attractive for the world-wide nuclear physics community However, due to the limitation of the ion-beamsources, the Oslo’s method has been performed only for about 60 nuclei, whosedata are accessible through Ref [42] In fact, the NLD and RSF can be extractednot only from the ion-induced compound reactions as by the Oslo group, butalso from the gamma spectra obtained from the (nth,γ) reaction The latter, whichwas popularly employed by the Dubna’s group [43–45], is performed based onthe gamma cascade intensity distributions obtained from the two-step cascade(TSC) measurement, which also depends on both NLD and RSF, similar as theOslo’s method (see e.g, Eq (2) of Ref [45]) Originally, the Dubna’s group pro-posed a Monte-Carlo method to direct extract the NLD and RSF from the gammacascade intensity distributions [43], however this method yields unambiguousresults To solve this problem, A.M Sukhovoj [44] proposed a new model to si-multaneously describe the NLD and RSF By fitting the model to the experimentalgamma cascade intensity distributions, its parameters are determined However,the NLD and RSF extracted by the Dubna’s group are very much different fromthose reported by the Oslo, for example in case of 96Mo [46] It can be seen thatthe main difference between the Dubna’s and Oslo’s methods is that within theOslo’s method, both NLD and RSF are varied freely to obtain the best fit to thefirst generation of the experimental gamma spectra [40, 41], whereas within theDubna method, the NLD and RSF are fixed by using given functional forms Inaddition, within the Oslo’s method, after the fittings to the experimental spectra,the obtained NLD and RSF should be normalized to the known data, namely theNLD data at low excitation energy taken by counting the number of discrete lev-els, the NLD data at the neutron binding energy taken from the average neutron

simultane-resonance level spacing, and the average radiative neutron capture width for theRSF, whereas the Dubna’s method does not apply any normalization.

co-incidence method in Vietnam

In Vietnam, the first γ−γ coincidence spectrometer has been successfully set up atDalat Nuclear Research Institute (DNRI) since 2004 Thereafter, there has been anumber of researches being carried out using this spectrometer together with thethermal neutron source from Dalat Nuclear Research Reactor (DNRR) However,these researches were primarily focused on optimizing the electronic parameters

of the spectrometer as well as improving the neutron facility at DNRI In terms ofnuclear structure study, these researches had provided some preliminary experi-mental results for the energy levels of several nuclei, such as49Ti,52V,59Ni,153Sm,

172Yb [47–49]

Particularly, the gamma cascades of 172Yb and 153Sm have been investigated inRef [49], in which the gamma cascades from171Yb(n,γ) and152Sm(n,γ) reactionswere measured within 400 and 600 hours, respectively Despite the long exper-imental times, the obtained statistics are rather low (see Figs 3.1a and 3.1b inRef [49]) It is probably due to the low quality of the used 171Yb and 152Smsamples The impurities existed in these samples can be easily seen via the veryhigh Compton background under the summation peaks in Figs 3.1a and 3.1b ofRef [49] Moreover, the HPGe detectors, which were used in Ref [49], only haverelative efficiencies of 15% and 20% Because of the above-mentioned shortcom-ings, there was not enough information to construct the NLS of172Yb and153Sm,

and therefore only the raw data on the gamma cascade energies and relative tensities were reported in Ref [49] Furthermore, the NLD and RSF were also notexamined within this work Consequently, in order to determine the NLS, NLDand RSF of172Yb and 153Sm, further investigations and/or experiments must beperformed.

in-Goal of the dissertation

The goals of the present dissertation are:

• To provide the updated information on the NLS of 172Yb and 153Sm, based

on the spectroscopic data obtained by using the γ − γ coincidence trometer These updated information is determined based on the compar-ison between the experimental data and those extracted from the ENSDFlibrary [2]

spec-• To solve the discrepancy between the Oslo and Dubna’s methods by bining the Dubna’s technique (using the experimental gamma cascade in-tensity distributions) with the Oslo one (normalization to the known data),and thus, to provide a new method to extract the NLD and RSF from thegamma cascade intensity distributions The preliminary test will be per-formed using the experimental gamma cascade intensity distributions of

Structure of the dissertation

The present dissertation is organized as follows Chapter 1 introduces theoriesrelated to this dissertation including the compound nuclear reaction, NLS, NLD,

and RSF models Chapter 2 presents the experimental facility, γ− γ coincidencemethod, as well as the data analysis The new method proposed to extract theNLD and RSF from the gamma cascade intensity distributions is also included

in this chapter Chapter 3 provides the results obtained for the NLS of172Yb and

153Sm and the extracted NLD and RSF of 172Yb In this chapter, a comparisonbetween the obtained NLS and those extracted from the ENSDF library is alsogiven Conclusions are given in the end of each chapters, whereas the last sec-tion, Summary and Outlook, summarizes all the results and proposes plan forthe forthcoming studies

The present dissertation has 130 pages including 38 figures and 5 tables

Chapter 1

Theory

The compound nuclear reaction is defined as a nuclear reaction in which tion of the incident particle with the target causes the production of a compoundnucleus [50] The compound nuclear reactions play an important role in the basicand applied nuclear physics They provide a prime example of chaotic behav-ior of a quantum-mechanical many-body system [51, 52] and their cross sectionsare required for nuclear astrophysics and nuclear application The compoundnuclear reaction is based on the assumption of Niels Borh, Borh-independencehypothesis [53]

According to the Bohr-independence hypothesis [53], after an incident particlecollides with a target, a compound nucleus is formed and then decays by emittingthe particles or γ-rays The compound nucleus has a relatively long lifetime incomparison with interaction time of the direct reaction (∼ 10−21 s) [53] Duringthat time, the “memory” of the entrance channel is “lost” and only the conservedquantities such as energy, total angular momentum, J, and parity, Π, play a keyrole in the compound nucleus

In order to describe a compound nuclear reaction, it is useful to divide the cess into two phases: (a) the formation of the compound system C, and (b) thedisintegration of the compound system into the products of the reaction as:

pro-x + X −→ C(a) −→ y + Y ,(b) (1.1)

where x and X are respectively incident particle and target, C is an excited pound nucleus decaying into the particle y and product nucleus Y The twophases (a) and (b) can be treated as independent processes in the sense that themode of disintegration of the compound system depends only on its energy, an-gular momentum, and parity, but not on the specific way in which it has beenproduced The cross section of the compound reaction based on the Bohr hy-pothesis can be expressed as:

com-σ(x+X → y+Y ) = σ(x+X → y+Y )P (C → y+Y ) = σ(x+X → C)ΓC→y+Y

It can be clearly seen from Eq (1.3) that the terms related to the entrance and exitchannels are separated

1.1.2 Reciprocity theorem

The partial radiative decay width, Γ, can be expressed in term of the cross section

of the entrance channel as follows

where ρ(EC∗) is the state density at excitation energy E∗

C of the compound state,and k2 is the square of the wave number, defined as k2 = 2µε/¯h2 with µ and εbeing the reduced mass and the center-of-mass kinetic energy, respectively Con-sidering a compound nuclear reaction and its inverse reaction corresponding tothe cross section σa,band σb,a, respectively, one has:

The complete knowledge of the nuclear level scheme (NLS) is required for lations of nuclear reactions and statistical models because it is needed to specify

calcu-all the possible outgoing reaction channels and to calculate the partial cross tion The knowledge of discrete levels is also necessary for adjusting the leveldensities, which are used to replace for the unknown discrete level scheme athigher excitation energy Consequently, the completeness of NLS is particularlyimportant When a complete level scheme of a given nucleus is defined up to

sec-a certsec-ain excitsec-ation energy, sec-all the discrete levels sec-are observed sec-and chsec-arsec-acterized

by the unique energy, spin and parity values Furthermore, the information ofgamma transitions such as energy, intensity, transition-type, and initial and finalstates are also required

It is obvious that studies based on the comprehensive spectroscopy of selective reactions can provide a complete NLS The statistical reactions, such

non-as (n, n0γ) and averaged resonance capture, are especially suitable for the study

of NLS due to their non-selective excitation mechanism The information of NLSfrom those reactions is extracted by means of the γ-ray spectroscopy [54] How-ever, for practical reasons, many nuclei are not able to be studied by such means.Therefore, generally, the complete NLS is constructed based on the informationprovided by various methods such as beta decay, electron capture decay, neu-tron induced reaction, ion induced reaction, etc Each method provides a certainamount of information on NLS and a combination of these information allows

us to construct the complete NLS For this reason, the ENSDF library was lished Up to September, 2016, the ENSDF library contains 187,067 datasets cor-responding to NLS of about 3,312 nuclei collected from various experiments [2].This library is continuously updated based on the reports of new levels or newtransitions and the recommendations to correct or reject the existed values Anillustration of NLS extracted from the ENSDF library is given in Fig 1.1

estab-It is noted that the γ − γ coincidence method (see Sec 2.1.2) is an efficientway to study the NLS, particularly in the energy region from 0.5 MeV up to

stable 0+

1332.508 0.9 ps2+

14.70 2u

0.12

1942

2158.61 2+

≥14.2 2u

0.000

670

2505.748 0.3 ps4+

7.512 99.88

317.88

Level Scheme Intensities: relative I γ

1173.328 E2(+M3) 99.85

347.14 0.0075

2158.57

0.0012 826.10 D+Q 0.0076

1332.492

E2 99.9826

FIGURE 1.1: Nuclear level scheme of 6028Ni32 from6027Co33 β−-decay

with T1/2=1925.8 days extracted from ENSDF library [2]

Bn− 0.5 MeV It is obvious that this method is not able to construct a completeNLS because it only measures two-step cascades The other types of transitionsuch as multi-step cascades and direct transitions are absent from the γ − γ co-incidence method Moreover, due to the fact that type of the gamma transitionswithin the γ − γ coincidence experiment can be E1, M1, E2, or a mixture of thesetypes [3], the parity of the determined nuclear states is not able to be determinedwithin the experiment1

However, given the advantages of achieving the very low Compton backgroundand identifying the correlated gamma transitions, the γ− γ coincidence methodcan determine energies and intensities of a lot of gamma cascades, of which the

1 In fact, in case that transition type can be taken from other experiments, the γ − γ coincidence method is able to deduce the parity of the measured nuclear states using the selection rules for gamma transition.

other method is lack Additionally, based on the transition rules, this method canprovide a range of possible spin for its obtained nuclear states [55].

For the purpose of constructing a complete NLS, the data obtained within the

γ− γ coincidence method must be combined together with ones obtained withinthe other methods/experiments, such as beta decay, electron capture, and neu-tron scattering experiments for the low-energy region, angular momentum mea-surements using multiple detectors for the determination of spin, parity, andmulti-step cascades, prompt gamma neutron activation analysis for measuringhigh energy transitions as well as direct transitions, Coulomb excitation andMossbauer methods for the determination of the lifetime of nuclear states [2].Thus, the γ− γ coincidence method does not itself construct a complete NLS butconfirms the NLS obtained from the other methods and provides updated in-formation Consequently, within the present dissertation, we compare the NLSobtained by the γ − γ coincidence method with one extracted from all the othermethods/experiments, which are presented in the ENSDF library, in order to pro-pose updated information

Besides, according to Ref [54], the experimental data of the discrete levels may

be complemented and/or cross-checked with the theoretical level schemes andlevel density model predictions It exists a vital interconnection between nuclearreaction model calculations and discrete level scheme that should be explored by

an iterative way

The models, which are used to describe the nuclear level density (NLD), can

be categorized into the phenomena-based and microscopic-based ones The

phenomena-based models provide functions with few free parameters on the sis of theoretical ideas to describe the NLD Those parameters are determined

ba-by fitting the function to the experimental data, whereas the microscopic-basedmodels take into account the nucleon-nucleon interaction in form of single-particle level scheme and deformation parameters to calculate the thermody-namic quantities and deduce the NLD It is noted that in some microscopic-basedmodels, explicit treatments of pairing, vibration and rotation states are also in-cluded [1]

The most simple and common phenomena-based model is the Fermi-gas one [56],which assumes that the equally spaced single particle states are filled with non-interacting fermions Another phenomena-based model is the constant temper-ature one [57], which bases on a assumption that the phase transition in nucleioccurs without changing the temperature when a nucleus gains energy, thus, thenuclear temperature is independent of excitation energy [58] Due to the fact thatneither the Fermi-gas model nor the constant temperature succeeds in describ-ing the NLD in the whole energy range, a new model, called Gilbert-Cameronmodel [35], was proposed The Gilbert-Cameron model uses the constant tem-perature model to describe the NLD in the low-energy region (below neutronbinding energy) and the Fermi-gas model for the high-energy region (above neu-tron binding energy) Generally, it is well-known that these models can predictthe overall behavior of the NLD over a wide energy range using a simple ap-proach

It is noted that our current understanding on the structure of low-lying nuclearlevels based on some important concepts including shell effects, pairing corre-lations, and collective phenomena The generalized superfluid model, which isdeveloped from a microscopic-based model proposed by A V Ignatyuk [59], is

an additional phenomena-based approach that takes into account all the above

theoreti-of cumulative levels taken from the NLS and the average level spacing data atthe neutron binding energy Recently, N Quang Hung et al [61] propose a newmicroscopic approach, which bases on the exact solutions of the pairing problemembedded into the canonical ensemble in order to construct the partition functionand to calculate the NLD after combining with the independent particle model.One of the merits of this approach is that it does not employ any parameter fit-ting at different excitation energy Furthermore, the NLD calculated within thisapproach can be directly compared to the experimental data without any normal-ization such as in case of the Hartree-Fock-BCS-based model mentioned above.

The Fermi-gas model [56] is proposed for the first time by Bethe in 1936 and it isone of the most used NLD models In this model, a nucleon inside the nuclear vol-ume is regarded as an element of a non-interacting fermion gas and each fermioncan occupy only one single-particle state due to the Pauli-principle The basicassumption of the Fermi-gas model is that each excited single state is equallyspaced and collective effects are not involved

The Fermi-gas NLD, ρF(E, J), for a given spin J at excitation-energy E is givenas

where ρF(E) is the Fermi-gas NLD including all spin (total NLD), which is fined as

de-ρF(E) =

√π12

The term σ in Eq (1.9) is the spin cutoff parameter, which characterizes the width

of the distribution of the z-component of the total angular momentum The tity σ2 can be calculated with the formula derived by Ericson [63] as

quan-σ2 = ghm2it = 0.0888atA2/3 , (1.10)

where, t is the nuclear temperature;hm2i is the mean square of the spin projection

on the z axis near the Fermi level, which is expressed in terms of the mass number

A ashm2i = 0.146A2/3[64] The level density parameter a in Eqs (1.8) and (1.10)

ρSF(E) =

√π12

the fact that the pairing interaction creates a shifted ground state with additionalbinding energy Therefore the real excitation energy, E, in the Fermi-gas model

is replaced by an effective excitation energy U = E − ∆ in the shifted Fermi-gasmodel Because a virtual ground state at a finite excitation energy depends onwhether the nucleus is odd-odd, odd-even, or even-even, the energy shift ∆ isgiven by the relations

δZ+ δN for even-even nucleus,

δZ(δN) for even-Z(N) nucleus,

0 for odd-odd nucleus,

Because the level density parameter a, the spin cutoff parameter σ, and the parityratio Π are known as the ambiguous ones, the determination of these parameters

is very important for the valid prediction of the NLD

1.3.1.1 Systematics of the Fermi-gas parameters

A linear relationship between the level density parameter a and the mass number

A can be found based on the model of independent particles This model assumesthat the level density increases as the number of nucleons participating in theexcitation increases, namely

where α is a parameter adjusted to the experimental data This formula describesthe experimental trend well when the residual interactions, shell effects, and de-formations are negligible Some systematics of the level density parameter a aredescribed as follows:

• The author of Ref [67] found a relationship between a and A as

σ2 = 0.145aA2/3p(E − ∆)/a , (1.16)

where ∆ is calculated by using the odd-even mass differences as in Ref [35].

• S I Al-Quraishi et al [68] proposed two different formulas for the level sity parameter a The first one assumes that the level density reaches amaximum for a given A when N = Z = A/2 This formula is given as

σ2 = 0.0145A5/3p(E − ∆)/a (1.19)

The second formula is obtained by calculating the square of the spin tions on the Z axis averaged over the single particle states It reads

projec-σ2 = 0.01416A2/3p(E − ∆)a (1.20)

Here, the way to calculate ∆ as well as their values are also presented in Ref.[68] The differences between the spin cutoff parameter predicted by Eq.(1.19) and that obtained at lower energies from known levels is from∼12%

up to∼30% depending on the mass number range Although, this formuladoes not predict the actual spin cutoff parameter for the known discretelevels accurately, it still provides a reliable approximation as a classical limitwith no consideration of shell and collective effects

• Unlike the two models above, T Von Egidy and D Bucurescu assumed thatthe spin cutoff parameter depends on both energy and mass number [71].The new empirical formula is obtained by fitting the experimentally known

7202 discrete levels of 227 nucleus in the mass region between Fluorine andCalifornium The spin cutoff parameter is then determined by the followingformula

With this empirical formula of σ2 (see Eq (1.21)), the corresponding leveldensity parameter a, is also obtained by using the same set of known dis-crete levels Thus, a is expressed as [71]

a = (0.199 + 0.0096S0)A0.869 , (1.24)

where S0 = S + 0.5P a0

and S is the shell correction in the mass formuladefined as S = Mexp− MLDwith Mexpand MLDbeing the experimental massand the mass calculated from a macroscopic liquid-drop formula, respec-tively

1.3.1.2 Parity ratio

The ratio of densities of positive and negative parity states at certain excitationenergy is an important quantity of the NLD This ratio is evaluated through theparity formula defined as

Π(E) = ρ+(E)

where ρ− and ρ+ are the densities of positive and negative parity levels cally, the Fermi-gas model predicts that the parity ratio reaches 0.5 smoothly asthe excitation energy increases However, a disagreement between the Fermi-gasmodel prediction and the experimental results was reported [73] According toRef [73], this discrepancy is due to a mathematical limitation of the formalism inthe Fermi-gas model

Typi-S.I Al-Quraishi et al [68] proposed an another formula of the parity ratio by fittingthe experimental parities of discrete levels in the mass range from 20≤ A ≤ 110

This formula is given as

where c = 3 MeV-1 is a constant and a0, a1, a2 are the three fitting parameters forthe shift δpwhose values are assigned in the following ways

a0 = 1.34, a1 = 75.22, a2 = 0.89 for even-even nuclei,

a0 =−0.88, a1 = 75.22, a2 = 0.89 for even-odd nuclei,

a0 =−0.42, a1 = 75.22, a2 = 0.89 for odd-even nuclei,

a0 =−0.90, a1 = 75.22, a2 = 0.89 for odd-odd nuclei

(1.27)

Equation (1.26) together with the fitting parameters given in Eq (1.27) show aconsistent trend for the parity ratio, which approaches 0.5 fastest for odd-oddnuclei, somewhat slower for even-odd and odd-even nuclei, and slower still foreven-even nuclei

The observed energy dependence of the cumulative number of nuclear levels can

be described rather well by using the following function [35, 57]:

N (E) = exp [(E− E0)/T ] , (1.28)

where N (E) is the cumulative numbers of nuclear levels in the energy range up

to E Whereas E0 and T are adjustable parameters which are determined viathe fitting to the experimental data The NLD is simply calculated as the first

derivative of N (E) with respect to E:

ρCT M(E) = dN

dE =

1

T exp [(E− E0)/T ] , (1.29)

where, the parameter T is defined as the nuclear temperature Since the value

of this parameter is assumed to be unchanged over the energy range considered,this model is called the constant temperature

It is noted that both the constant temperature and Fermi-gas models cannot produce the experimental NLD in a wide excitation energy region Therefore, A.Gilbert and A G W Cameron proposed a composite model, in which the con-stant temperature model, ρCT M, describes the NLD at the low excitation energy,whereas the Fermi-gas model, ρF, represents the NLD at the high excitation en-ergy [35]

re-The fundamental idea of the Gilbert-Cameron model is to have a smooth nection between the two different NLD functions in the low and high excitationenergies as well as their first derivatives at certain excitation energy EM by vary-ing the level density parameters T, E0, and EM These constraints are expressed

con-by the following set of equations:

EM = T1

(1.30)

Since there are three unknown parameters T, E0, and EM, we need one more dition, that is, the theoretical calculation within the constant temperature modelshould fit the experimental discrete levels in the low-lying energy region Using

con-the above three conditions, we are able to determine all con-the adjustable ters [35] From the set of Eq (1.30), the nuclear temperature is deduced:

parame-1

T =pa/EM − 3

2EM

where a is level density parameter in the Fermi-gas model

It can be clearly recognized that both the Fermi-gas and the constant temperaturemodels provide simple and convenient formulas for parameterizing the experi-mental NLD However, these models do not give any explanation for the shifts ofexcitation energies and shell changes of the level density parameters Therefore,

it is necessary to reveal a more rigorous model that takes into the shell geneities of single-particle level spectra and the superfluid and collective effectsproduced by the residual interaction of nucleons A detailed discussion of suchmodel was presented in Ref [59] Nevertheless, rigorous microscopic methods tocalculate level densities are extremely complicated, and thus limit their applica-tion to the experimental data analyses For this reason, a consistent phenomeno-logical description of NLD, called the generalized superfluid model, that takesinto account the basic ideas of microscopic approaches to the structure of highlyexcited nuclear levels has been developed by many authors

inhomo-Considering the shell and collective effects coming from the excited nuclear levelstructure, the NLD can be defined as

where ρqp is the NLD caused by the quasi-particle excitation only, and Kvibr and

Krot are the corresponding enhancement coefficients caused by the vibrationaland rotational excitations, respectively

The rotational enhancement of NLD in the adiabatic approximation depends onthe nuclear shape symmetry, and can be written as [74]

1 for spherical nuclei,

I⊥t for deformed nuclei

(1.33)

where I⊥ is the moment of inertia relative to the perpendicular axis and t is clear temperature This formula is obtained if mirror and axial symmetry of adeformed nuclei is assumed The most stable nuclei of the rare earth elements(150 ≤ A ≤ 190) and the actinides (A ≥ 230) have this shape The rotationalenhancement of the NLD becomes larger for nuclei with non-axial shapes [74].The vibrational enhancement coefficient is determined within the microscopicapproach using the relationship

corre-In general, the success of the generalized superfluid model is attributed to theinclusion of the well-known major components of nuclear theory: paring cor-relations, shell effects and collective excitations Some complexity in the modelseems to be justified by the mutual consistency of the parameters obtained fromthe various experimental data, and by the close relationship between the theoret-ical concepts used to describe the structure of low-lying nuclear levels and thestatistical properties of highly excited nuclei.

A more rigorous description of the level densities and other statistical teristics of excited nuclei can be obtained from the model calculations using therealistic schemes of the single-particle levels Such calculations are discussed indetail in Ref [59] The thermodynamic functions of an excited nucleus are writtenas

i

gi

hp(i− λτ)2+ ∆2

where Nris the number of protons or neutrons in a nucleus, λris the ing chemical potential, and Gr is the pairing force constant Equations (1.36) de-termines the proton and neutron correlation functions for the ground state of anucleus at t = 0

correspond-For given schemes of the single-particle levels, Eqs (1.35) and (1.36) permit thethermodynamic functions and the NLD to be calculated without any additionalparameters Codes for the above microscopic calculations of the NLD are in-cluded in the RIPL-2 [1] Collective effects are also included in these codes onthe basis of the same approximations as for the phenomenological generalizedsuperfluid model The single-particle level schemes of Moller et al [75] are rec-ommended for such calculations because they were also used to determine therecommended nuclear binding energies, shell corrections and deformations inRIPL-2

An alternative description of NLD has been proposed by Goriely [76] based onthe Extended-Thomas-Fermi plus Strutinksy-Integral model for the ground stateproperties (single-particle level schemes and pairing strengths) Although thisapproach represents the first global microscopic formula that could reasonablyreproduce the experimental neutron resonance spacings, some large deviationshave been found (for example, in the neutron binding energy region) These de-ficiencies have been removed by using the Hartree-Fock-BCS-based model [60],

which predicts all the experimental resonance spacings with an accuracy parable with the equivalent data obtained by the phenomenological back-shiftedFermi-gas formula This microscopic model is based on the Hartree-Fork-BCSpredictions of the ground-state structure properties [60] and includes a consis-tent treatment of the shell effects, pairing correlations, deformation effects andcollective excitations All details of the level density calculations were presented

com-in Ref [60], com-in which the Hartree-Fock-BCS scom-ingle-particle schemes is used andthe basic nuclear structure properties listed as following, must be known

• the single-particle energies εk

q up to an energy cut-off Λ,

• the pairing strength G,

• the deformation parameters β2, β4 (and consequently δ ' β2/1.056), and

• the moment of inertia I⊥

The Hartree-Fock-BCS-based model has been normalized to experimental data(278 neutron resonance spacings and 1210 low-lying levels) to account for theavailable experimental information, and can consequently be used for practicalapplications with a certain degree of confidence

Recently, a new microscopic approach to the NLD, which takes into account boththe effect of exact thermal pairing and independent particle motion, has been pro-posed by N Quang Hung et al [61] In this approach, thermal pairing is treatedbased on the exact eigenvaluesES, obtained by diagonalizing the pairing Hamil-tonian H [77] at zero temperature and different numbers of unpaired particles (se-niorities) S, embedded into the canonical ensemble (CE) (see Eq (7) of Ref [78]).Due to the limitation on the size of the matrix to be diagonalized, these exact so-lutions are limited to the truncated levels (levels around the Fermi surface) only.Therefore, in order to find the total partition function of the whole system, the

exact CE partition is then combined with those obtained within the particle model (IPM) [79] for the remain single-particle levels The sum of allthese partition functions is the total partition function Knowing the total parti-tion function, one can calculate all thermodynamic quantities such as free energy

independent-F, total energy E, entropy S, heat capacity C, and thermal pairing gap ∆ can beeasily calculated [80, 81] The density of state ω(E∗) at excitation energy E∗

is tained by using the inverse Laplace transformation of the partition function [63]as

perpen-ρ(E∗) = kvibrkrotω(E∗)/(σ||

deforma-1.4 Radiative strength function

Photon emission is one of the most important channels in the nuclear excitation processes and accompanies most nuclear reactions Both gamma de-cay and photo-absorption can be described by the radiative strength function(RSF) The latter provides the vital information related to nuclear structure such

de-as proton and neutron distributions de-as well de-as nuclear shape There are two types

of RSF: “downward” (←−

f ) and “upward” (−→

f ) corresponding to the gamma cay and photo-absorption processes, respectively In the present dissertation, weonly considers the “downward” RSF (←−

de-f ), which determines the average tive width of the gamma decay For a given multipolarity XL, where X can beeither electric (E) or magnetic (M ) resonance and L is the multipolarity of gammatransition, the “downward” RSF can be expressed as [1]

wherehσXL(E)i is the photo-absorption cross section and gL= 2L + 1 According

to the time reversal theorem, the “downward” and “upward” RSFs are identical.The Brink-Axel hypothesis plays an important role on the understanding of the