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

vii

Contents
Declaration of Authorship

iii

Acknowledgements

v

List of Figures

xi

List of Tables

xv

List of Abbreviations

xvii

Introduction
1

1

Theory

11

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


viii
2

Experiment and data analysis

39

2.1

39

Experimental facility and experimental method . . . . . . . . . . . .
2.1.1

2.2

2.3
3

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

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

Conclusion of chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . .

79

Results and discussion
3.1

Nuclear level scheme of 172 Yb and 153 Sm

81
. . . . . . . . . . . . . . .

81

3.1.1

172

Yb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

3.1.2

153

Sm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

3.2

Gamma cascade intensity distributions of 172 Yb . . . . . . . . . . . .

97

3.3

Nuclear level density and radiative strength function of 172 Yb . . . 105


ix

3.4

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

Conclusion of chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . 114

Summary and outlook

115

List of publications

117

References

118


xi

List of Figures
1.1

2.1
2.2
2.3
2.4
2.5
2.6

2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16

β − -decay with
from 60
Nuclear level scheme of 60
27 Co33
28 Ni32
T1/2 =1925.8 days extracted from ENSDF library. . . . . . . . . . . . .
The horizontal cross-section view of the DNRR. . . . . . . . . . . .
The detail structure of the neutron beam-port No. 3. . . . . . . . .
Two-step gamma cascades corresponding to the decays from a
compound state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The γ − γ coincidence electronics. . . . . . . . . . . . . . . . . . . . .
The TAC amplitude spectrum measured with 60 Co. . . . . . . . . .
The channel-energy relationship of the two detectors. Data of the
detector B has an offset of 1000 on y-axis. . . . . . . . . . . . . . . .
The energy resolution of the two detectors in the energy range from
0.5 MeV to 8 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The relative efficiencies of the two detectors. . . . . . . . . . . . . .
Experimental setup for measuring the γ − γ coincidences. . . . . .
Experimental system at the neutron beam-port No. 3 of the DNRR.

An illustration for a three-step cascade. . . . . . . . . . . . . . . . .
Illustration of the cross talk effect. BS: Compton backscattered photon; Ann: annihilation photon. . . . . . . . . . . . . . . . . . . . . .
Data analysis procedure. . . . . . . . . . . . . . . . . . . . . . . . . .
The discrepancy between two datasets. Dataset A is collected from
detector A and dataset B is collected from detector B. . . . . . . . .
Dataset B is corrected according to dataset A. . . . . . . . . . . . . .
Summation spectrum for 171 Yb(n,2γ) reaction. E1 +E2 is sum of energies measured from two detectors. Energies (in keV) of the final
levels in the cascades are pointed near the peaks of the full absorption energy. The notations SE and DE correspond to the single- and
double-escape peaks, respectively. . . . . . . . . . . . . . . . . . . .

15
40
40
42
45
47
48
48
49
50
51
52
54
57
58
59

60

xii
2.17 Summation spectrum for 152 Sm(n,2γ) reaction. E1 + E2 is sum of
energies measured from two detectors. Energies (in keV) of the
final levels in the cascades are pointed near the peaks of the full
absorption energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.18 Explanation of the input variables used in the procedure of obtaining the TSC spectra given in Fig. 2.19. . . . . . . . . . . . . . . . . .
2.19 Detail procedure for obtaining the TSC spectra. . . . . . . . . . . . .
2.20 a. experimental TSC spectrum; b. simulated TSC spectrum, c. unresolved TSC spectrum with noise line, d. unresolved TSC spectrum without noise line corresponding to the decays from the compound state to the ground state of 172 Yb. . . . . . . . . . . . . . . . .
2.21 Procedure of extracting the NLD and RSF. . . . . . . . . . . . . . . .
2.22 Illustration of the shifting procedure for 172 Yb nucleus with Em =
3.625 MeV, Em = 3.875 MeV and Efmax = 1.198 MeV. The superposed energy range is between the two vertical arrows. The
curve (1) simulates the standard dataset (circle), while the curve
(2) models the to-be-shifted dataset (triangle). The k factor is the
ratio between the area under the curve (1) and that under the
curve (2). The two curves have the form of an exponential function C0 exp(C1 E), whose parameters (C0 , C1 ) are obtained via the
fitting to the corresponding datasets. . . . . . . . . . . . . . . . . .
2.23 The final dataset describes the functional form of γ-rays transmission coefficient of 172 Yb nucleus in the energy region from 0.5 to
7.5 MeV. The line is the average values over an 250 keV energy
interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.24 Comparison of the γ-rays transmission coefficients of 172 Yb nucleus obtained by different starting excitation-energy bins. Histogram with black color is the average of the γ-rays transmission
coefficients obtained by all the starting excitation-energy bins from
2.125 MeV to 5.375 MeV. The corresponding uncertainties are given
by upper and lower lines. . . . . . . . . . . . . . . . . . . . . . . . .
3.1
3.2

TSC spectrum corresponding to the ground state of 172 Yb. . . . . .
TSC spectrum corresponding to the final level with the energy Ef =
78.8 keV of 172 Yb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60
62
64

68
71

73

75

76
88
89


xiii
3.3

3.4
3.5
3.6
3.7
3.8
3.9

3.10

3.11

3.12
3.13

Experimental level scheme of 172 Yb obtained within the gamma
cascades from compound state to six distinct low-lying discrete
levels with spin from 0¯
h to 2¯h. Explanation of the figure is given in
text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
TSC spectrum corresponding to the final levels with energies Ef =
0 and 7.8 keV of 153 Sm. . . . . . . . . . . . . . . . . . . . . . . . . . .
TSC spectrum corresponding to the final level with the energy Ef
= 35.8 keV of 153 Sm. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
NLS of 153 Sm obtained within this present work. Explanation of
the figure is the same as in Fig. 3.3. . . . . . . . . . . . . . . . . . . .
The gamma cascade intensity distributions of 172 Yb obtained
within the present work. . . . . . . . . . . . . . . . . . . . . . . . .
The extracted NLD and RSF of 172 Yb obtained within the present
work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison between the experimental gamma cascade intensity
distributions and the calculated one obtained from the extracted
NLD and RSF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison between the NLD obtained within the present work
and the other experimental data. Explanation for this figure is
given in text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison between the RSF obtained within the present work
and the other experimental data. Explanation of this figure is given
in text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison between the NLD obtained within the present work
and a few common theoretical models. . . . . . . . . . . . . . . . . .
Comparison between RSF obtained within this work and few theoretical models. See explanation of the figure in text. . . . . . . . .

91
95
95
96
98
106

107

109

110
112
113


xv

List of Tables
2.1
2.2

Selected parameters of the electric modules. . . . . . . . . . . . . . .
Parameters of the relative efficiency functions. . . . . . . . . . . . .

3.1

Primary and secondary gamma-ray energies and absolute intensities obtained from the 171 Yb(nth , γ) reaction. The experimental
values are compared with the ENSDF data. . . . . . . . . . . . . . .

Primary and secondary gamma-ray energies and absolute intensities obtained from the 152 Sm(nth , γ) reaction. The experimental
values are compared with the ENSDF data. . . . . . . . . . . . . . .
The gamma cascade intensity distribution of 172 Yb and the contribution of the resolved cascades. . . . . . . . . . . . . . . . . . . . . .

3.2

3.3

46
49

81

93
99


1

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, 3 He’γ), (α, α’), (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 (nth ,γ) 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


2
low- and high-energy regions.
It has been well-known that the γ − γ coincidence method [3] can be used to
study the NLS, NLD and RSF. Within this method, the cascade events, which
are obtained from the decay of the initial compound state to the different final
states, are separated into different Two-Step-Cascade (TSC) spectra. Particularly,
only correlated gamma transitions are detected, therefore the number of γ-rays
contributed to the TSC spectra is less than that presented in a normal prompt
gamma spectrum, leading to the significant reduction of the overlapping γ-rays
as well as improving the detecting ability of this method. In addition, different from the normal gamma spectra, the TSC spectra obtained using the γ − γ
coincidence method, after applying the background subtraction algorithm, have
almost no Compton background. Therefore, the detection limit of the coincidence
method is much improved in comparison with the normal gamma spectra analysis method. Beside that, the state from which a secondary gamma transition is
decayed can be determined in the coincidence method if one of the two γ-rays in
the cascade is a known primary transition. Based on these above advantages, the
γ − γ coincidence method is appropriate for the determination of excited states
with low spin in the energy region from 0.5 MeV to Bn − 0.5 MeV (Here, Bn is the
neutron binding energy). Furthermore, the γ − γ coincidence method can also be
used 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 the
experimental NLD and RSF based on this method.

Nuclear level scheme of 172Yb and 153Sm
172

Yb and 153 Sm are two deformed and rare earth nuclei. Their NLS are absolutely

necessary for either confirming or enhancing the predictive powers of the nuclear


3
models for heavy nuclei. The adopted nuclear levels and gamma transitions of
172

Yb and

153

Sm are given in Ref. [4] and Ref. [5], respectively. A summary of

important investigations on the NLS of these two nuclei is presented as follows.

For 172 Yb
NLS of 172 Yb has been thoroughly studied in different methods such as beta decay
of

172

Tm [6], electron capture decay of

low-lying states in

172,174

172

Lu [7], neutron inelastic scattering for

Yb [8], (n, n’γ) reactions using fast neutrons from reac-

tors [9], 170 Er(α,2n)172 Yb reaction for high-spin states [10], 171 Yb(n,γ) reaction for
low-spin states [11, 12]. Additional methods, which are also able to provide the
level scheme of 172 Yb based on the compound nuclear reactions induced by light
ions, include
[16],

173

172

Yb(3 He,3 He’γ) [13],

(d,t) [17],

173

172

Yb(α,α’) [14],

Yb(3 He,γ), (3 He,αγ) [17, 18],

170

173

Yb(p,d) [15],

171

Yb(d,p)

Yb(t,p) [19], and elastic and

inelastic proton scatterings [20, 21]. Furthermore, lifetimes of a number of levels
have been also determined via the

172

Yb(γ, γ’) reactions using the nuclear reso-

nance fluorescence [22] and Coulomb excitation [23, 24] methods. Through these
experiments, the low-lying discrete level scheme of

172

Yb 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 in

the intermediate energy region (2.4 MeV < E < 5 MeV), where the thermal neutron capture reactions (nth ,γ) were mostly employed to extract the data, is sparse
and incomplete.
In particular, based on the neutron capture reaction with both thermal and 2


4
keV neutrons, Greenwood et al [11] have detected, by using the Ge(Li) detector, in total 127 primary gamma transitions including their intensities from the
prompt gamma spectrum of 172 Yb. At the same time, the prompt gamma yield of
172

Yb has also been determined from the abundance of 171 Yb in natural ytterbium

and their relative thermal-neutron capture cross sections. In addition to that, 136
gamma-ray transitions, whose energies are less than 2.5 MeV, have been also reported in this paper. Using the same thermal neutron capture reaction with the
use of the pair formation spectrometer, Gellety et al [12] have measured the primary transitions of

172

Yb, whose energies and relative intensities were found in

good agreement with those reported by Greenwood. Although the results obtained from Gellety have improved the level scheme of

172

Yb, which had previ-

ously been constructed by Greenwood using the Riz combination principle, the
significant differences between the absolute intensities of gamma transitions obtained within those works have not yet been explained. It is obvious that the
number of detectable gamma rays in the normal gamma-ray spectrum depends
upon the energy resolution of the detectors as well as the number of excited states

existing in an interval of excitation energy. Thus, there has been a certain limitation on the results of Refs. [11, 12] due to the restricted energy resolution of the
Ge(Li) detectors used in those works as well as the large number of excited states
of 172 Yb in the energy region from 3 MeV to 5 MeV, where the discrete region of
the level density interferes with the continuous one.
In fact, the gamma-gamma coincidence technique was used to measure the level
scheme of 172 Yb 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 of 172 Yb
from 171 Yb(nth , γγ) reaction in Ref. [25]. The results obtained were then compared
with the statistical-model calculations which base entirely on the experimental
level density and gamma strength function extracted from the primary gamma


5
spectra of 173 Yb(3 He, αγ)172 Yb reaction. On the other hand, in Ref. [25] the spectroscopic 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 which
only 4000 cascade events corresponding to the decays from the compound state
to the ground state are collected.

For 153 Sm
NLS of

153

Sm has been investigated in a thorough manner in various methods

such as beta decay of 153 Pm produced from 252 Cf fission [26], isomeric transition
decay of

153

Sm [5],

152

Sm(n,γ) reaction for low-spin state [27, 28], transfer reac-

tions such as 152 Sm(d,p) [29], 154 Sm(p,d) [30], 152 Sm(α,3 He) [31], 154 Sm(d,t) [28,29]
and recent

151

Sm(t,p) [32]. Through these experiments, the low-lying discrete

level scheme of 153 Sm in the low-energy region (E < 2.2 MeV) has been well understood [5]. In this low-energy region, information about 203 excited states,
including excitation energies, spins, and parity, were determined. However, in
the high-energy region (2.2 MeV < E < 4 MeV), although the number of excited
states is expected to be large, most of the states were reported without providing
information on their spins and parities. Moreover, the uncertainty of the excitation energies was from 10 to 17 keV, which is rather high in comparison with the
uncertainty of gamma energies obtained by using the HPGe gamma spectrometers.
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-energy
region and the Ge(Li) detector for the high-energy region, in total 251 gamma
transitions in the energy region from 28 keV to 1041 keV and 23 additional transitions between 4.5 and 5.9 MeV. Using the same thermal neutron capture reaction,


6
Bennette et al [28] have measured the prompt gamma spectrum of 153 Sm by means
of both Ge(Li) and Si(Li) detectors. The results obtained were in good agreement
with 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

153

Sm is rather large,

therefore it is obvious that there has been a certain limitation on the results of
Refs. [27, 28] due to the restricted energy resolution of the detectors used within
those works.
On the another hand, Blasi et al [30] have detected, based on the 154 Sm(p,d) reaction, 170 gamma transitions in the energy region below 2.2 MeV. The result obtained by Blasi et al confirms not only the existence of many excited states given
in Refs. [27, 28] but also provides a number of new excited states, especially in
the 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

152

Sm(d,p) reac-

tion [28, 29]. However, the uncertainty of excited state energies proposed in these
investigations is larger than 10 keV.

Experimental nuclear level density and radiative
strength function
From the experimental side, the NLD has been studied within several methods
such as counting of discrete levels for the very low-energy region [33,34], neutron
resonance spacing at the neutron binding energy [35], and evaporation spectra
for the high-energy region (above the particle threshold) [36]. For the RSF, its
information can be extracted from the photoabsorption cross section [37] as well
as the radiative neutron capture reactions [38] and reactions with charged particles [39]. Recently, the Oslo group at Oslo University (Norway) has proposed

7
an advance technique, called Oslo’s method, which allows them to simultaneously extract both NLD and RSF from the measured gamma-decay spectra obtained via the transfer and/or inelastic scattering reactions [40, 41]. Since after
that the study of NLD and RSF has become particularly attractive for the worldwide nuclear physics community. However, due to the limitation of the ion-beam
sources, the Oslo’s method has been performed only for about 60 nuclei, whose
data are accessible through Ref. [42]. In fact, the NLD and RSF can be extracted
not only from the ion-induced compound reactions as by the Oslo group, but
also from the gamma spectra obtained from the (nth ,γ) reaction. The latter, which
was popularly employed by the Dubna’s group [43–45], is performed based on
the gamma cascade intensity distributions obtained from the two-step cascade
(TSC) measurement, which also depends on both NLD and RSF, similar as the
Oslo’s method (see e.g, Eq. (2) of Ref. [45]). Originally, the Dubna’s group proposed a Monte-Carlo method to direct extract the NLD and RSF from the gamma
cascade intensity distributions [43], however this method yields unambiguous
results. To solve this problem, A.M. Sukhovoj [44] proposed a new model to simultaneously describe the NLD and RSF. By fitting the model to the experimental
gamma cascade intensity distributions, its parameters are determined. However,
the NLD and RSF extracted by the Dubna’s group are very much different from
those reported by the Oslo, for example in case of 96 Mo [46]. It can be seen that
the main difference between the Dubna’s and Oslo’s methods is that within the
Oslo’s method, both NLD and RSF are varied freely to obtain the best fit to the
first generation of the experimental gamma spectra [40, 41], whereas within the
Dubna method, the NLD and RSF are fixed by using given functional forms. In
addition, 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 the
NLD data at low excitation energy taken by counting the number of discrete levels, the NLD data at the neutron binding energy taken from the average neutron


8
resonance level spacing, and the average radiative neutron capture width for the
RSF, whereas the Dubna’s method does not apply any normalization.

Experimental study of gamma cascades using γ − γ coincidence method in Vietnam

In Vietnam, the first γ−γ coincidence spectrometer has been successfully set up at
Dalat Nuclear Research Institute (DNRI) since 2004. Thereafter, there has been a
number of researches being carried out using this spectrometer together with the
thermal 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 of
nuclear structure study, these researches had provided some preliminary experimental results for the energy levels of several nuclei, such as 49 Ti, 52 V, 59 Ni, 153 Sm,
172

Yb [47–49].

Particularly, the gamma cascades of

172

Yb and

153

Sm have been investigated in

Ref. [49], in which the gamma cascades from 171 Yb(n,γ) and 152 Sm(n,γ) reactions
were measured within 400 and 600 hours, respectively. Despite the long experimental times, the obtained statistics are rather low (see Figs. 3.1a and 3.1b in
Ref. [49]). It is probably due to the low quality of the used

171

Yb and

152

Sm

samples. The impurities existed in these samples can be easily seen via the very
high Compton background under the summation peaks in Figs. 3.1a and 3.1b of
Ref. [49]. Moreover, the HPGe detectors, which were used in Ref. [49], only have
relative efficiencies of 15% and 20%. Because of the above-mentioned shortcomings, there was not enough information to construct the NLS of 172 Yb and 153 Sm,


9
and therefore only the raw data on the gamma cascade energies and relative intensities were reported in Ref. [49]. Furthermore, the NLD and RSF were also not
examined within this work. Consequently, in order to determine the NLS, NLD
and RSF of 172 Yb and 153 Sm, further investigations and/or experiments must be
performed.

Goal of the dissertation
The goals of the present dissertation are:
• To provide the updated information on the NLS of 172 Yb and 153 Sm, based
on the spectroscopic data obtained by using the γ − γ coincidence spectrometer. These updated information is determined based on the comparison between the experimental data and those extracted from the ENSDF
library [2].
• To solve the discrepancy between the Oslo and Dubna’s methods by combining the Dubna’s technique (using the experimental gamma cascade intensity distributions) with the Oslo one (normalization to the known data),
and thus, to provide a new method to extract the NLD and RSF from the
gamma cascade intensity distributions. The preliminary test will be performed using the experimental gamma cascade intensity distributions of
172

Yb.

Structure of the dissertation
The present dissertation is organized as follows. Chapter 1 introduces theories
related to this dissertation including the compound nuclear reaction, NLS, NLD,

10
and RSF models. Chapter 2 presents the experimental facility, γ − γ coincidence
method, as well as the data analysis. The new method proposed to extract the
NLD and RSF from the gamma cascade intensity distributions is also included
in this chapter. Chapter 3 provides the results obtained for the NLS of 172 Yb and
153

Sm and the extracted NLD and RSF of

172

Yb. In this chapter, a comparison

between the obtained NLS and those extracted from the ENSDF library is also
given. Conclusions are given in the end of each chapters, whereas the last section, Summary and Outlook, summarizes all the results and proposes plan for
the forthcoming studies.
The present dissertation has 130 pages including 38 figures and 5 tables.


11

Chapter 1

Theory
1.1

Compound nuclear reaction

The compound nuclear reaction is defined as a nuclear reaction in which interaction of the incident particle with the target causes the production of a compound
nucleus [50]. The compound nuclear reactions play an important role in the basic
and applied nuclear physics. They provide a prime example of chaotic behavior of a quantum-mechanical many-body system [51, 52] and their cross sections
are required for nuclear astrophysics and nuclear application. The compound
nuclear reaction is based on the assumption of Niels Borh, Borh-independence
hypothesis [53].

1.1.1

Bohr-independence hypothesis

According to the Bohr-independence hypothesis [53], after an incident particle
collides with a target, a compound nucleus is formed and then decays by emitting
the particles or γ-rays. The compound nucleus has a relatively long lifetime in
comparison with interaction time of the direct reaction (∼ 10−21 s) [53]. During
that time, the “memory” of the entrance channel is “lost” and only the conserved
quantities such as energy, total angular momentum, J, and parity, Π, play a key
role in the compound nucleus.


12

Chapter 1. Theory

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

(b)

x + X −→ C −→ y + Y ,

(1.1)

where x and X are respectively incident particle and target, C is an excited compound nucleus decaying into the particle y and product nucleus Y . The two
phases (a) and (b) can be treated as independent processes in the sense that the
mode of disintegration of the compound system depends only on its energy, angular momentum, and parity, but not on the specific way in which it has been
produced. The cross section of the compound reaction based on the Bohr hypothesis can be expressed as:

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

ΓC→y+Y
, (1.2)
Γ

where σ(x + X → y + Y ) is the cross section for the formation of the compound
nucleus and P (C → y + Y ) represents the decay probability to the exit channel
y + Y . The probability P can be defined in term of the ratio between partial
(ΓC→y+Y ) and total (Γ) radiative widths. Let’s σa,b stand for the cross section of the
(a)

(b)

reaction x + X −→ C −→ y + Y in which the subscripts a and b correspond to
the entrance and exit channels, respectively. Equation (1.2) is then simplified as
follows
σa,b = σa Pb .

(1.3)

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


1.2. Nuclear level scheme

1.1.2

13

Reciprocity theorem

The partial radiative decay width, Γ, can be expressed in term of the cross section
of the entrance channel as follows
σ k2
Γ(k) = 2
,
2π ρ(EC∗ )

(1.4)

where ρ(EC∗ ) is the state density at excitation energy EC∗ of the compound state,
and k 2 is the square of the wave number, defined as k 2 = 2µε/¯h2 with µ and ε
being the reduced mass and the center-of-mass kinetic energy, respectively. Considering a compound nuclear reaction and its inverse reaction corresponding to
the cross section σa,b and σb,a , respectively, one has:
(a)

(b)

(a)

(b)

x + X −→ C −→ y + Y ,

(1.5)

x + X ←− C ←− y + Y .
These two reactions form the same compound nucleus, C, at the same excitation
energy, EC∗ . By combining Eq. (1.4) with Eq. (1.2), we can deduce the reciprocity
theorem:
σb,a
σa,b
=
.
kb2
ka2

(1.6)

The reciprocity theorem therefore provides the relationship between the cross section of a compound nuclear reaction and its inverse.

1.2

Nuclear level scheme

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


14

Chapter 1. Theory

all the possible outgoing reaction channels and to calculate the partial cross section. The knowledge of discrete levels is also necessary for adjusting the level
densities, which are used to replace for the unknown discrete level scheme at
higher excitation energy. Consequently, the completeness of NLS is particularly
important. When a complete level scheme of a given nucleus is defined up to
a certain excitation energy, all the discrete levels are observed and characterized
by the unique energy, spin and parity values. Furthermore, the information of
gamma transitions such as energy, intensity, transition-type, and initial and final
states are also required.
It is obvious that studies based on the comprehensive spectroscopy of nonselective reactions can provide a complete NLS. The statistical reactions, such
as (n, n γ) and averaged resonance capture, are especially suitable for the study
of NLS due to their non-selective excitation mechanism. The information of NLS
from those reactions is extracted by means of the γ-ray spectroscopy [54]. However, for practical reasons, many nuclei are not able to be studied by such means.
Therefore, generally, the complete NLS is constructed based on the information
provided by various methods such as beta decay, electron capture decay, neutron induced reaction, ion induced reaction, etc. Each method provides a certain
amount of information on NLS and a combination of these information allows
us to construct the complete NLS. For this reason, the ENSDF library was established. Up to September, 2016, the ENSDF library contains 187,067 datasets corresponding 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 new
transitions and the recommendations to correct or reject the existed values. An
illustration of NLS extracted from the ENSDF library is given in Fig. 1.1.
It is noted that the γ − γ coincidence method (see Sec. 2.1.2) is an efficient
way to study the NLS, particularly in the energy region from 0.5 MeV up to


1.2. Nuclear level scheme

15
Level Scheme

5+

0.0

60
27 Co33

Intensities: relative Iγ

1925.28 d

Eβ −

Iβ −

317.88 99.88

Log ft
7.512

670

0.000

≥14.22u

1942

0.12

14.702u

4+
2+

2+

25

%β − =100

05
.6
11 92
73 E4
.3
0.
0
34 28
E2 000
7.
1
(
4
+ 020
21
0.
M
58

0
07 3)
82 .57
99
5
6. 0
.8
10 .0
5
13
0
D 12
32
+
.4
Q
92
0.
00
E2
76
99
.9
82
6

Q− =2823.95

0+

2505.748

0.3 ps

2158.61

1332.508

0.0

0.9 ps

stable

60
28 Ni32

60
−
F IGURE 1.1: Nuclear level scheme of 60
28 Ni32 from 27 Co33 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 complete
NLS because it only measures two-step cascades. The other types of transition
such as multi-step cascades and direct transitions are absent from the γ − γ coincidence method. Moreover, due to the fact that type of the gamma transitions
within the γ − γ coincidence experiment can be E1, M1, E2, or a mixture of these
types [3], the parity of the determined nuclear states is not able to be determined
within the experiment1 .
However, given the advantages of achieving the very low Compton background

and identifying the correlated gamma transitions, the γ − γ coincidence method
can 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.


16

Chapter 1. Theory

other method is lack. Additionally, based on the transition rules, this method can
provide 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 within
the other methods/experiments, such as beta decay, electron capture, and neutron scattering experiments for the low-energy region, angular momentum measurements using multiple detectors for the determination of spin, parity, and
multi-step cascades, prompt gamma neutron activation analysis for measuring
high energy transitions as well as direct transitions, Coulomb excitation and
Mossbauer methods for the determination of the lifetime of nuclear states [2].
Thus, the γ − γ coincidence method does not itself construct a complete NLS but
confirms the NLS obtained from the other methods and provides updated information. Consequently, within the present dissertation, we compare the NLS
obtained by the γ − γ coincidence method with one extracted from all the other
methods/experiments, which are presented in the ENSDF library, in order to propose 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 and
level density model predictions. It exists a vital interconnection between nuclear
reaction model calculations and discrete level scheme that should be explored by
an iterative way.

1.3

Nuclear level density

The models, which are used to describe the nuclear level density (NLD), can
be categorized into the phenomena-based and microscopic-based ones. The


1.3. Nuclear level density

17

phenomena-based models provide functions with few free parameters on the basis of theoretical ideas to describe the NLD. Those parameters are determined
by fitting the function to the experimental data, whereas the microscopic-based
models take into account the nucleon-nucleon interaction in form of singleparticle level scheme and deformation parameters to calculate the thermodynamic quantities and deduce the NLD. It is noted that in some microscopic-based
models, explicit treatments of pairing, vibration and rotation states are also included [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 noninteracting fermions. Another phenomena-based model is the constant temperature one [57], which bases on a assumption that the phase transition in nuclei
occurs without changing the temperature when a nucleus gains energy, thus, the
nuclear temperature is independent of excitation energy [58]. Due to the fact that
neither the Fermi-gas model nor the constant temperature succeeds in describing the NLD in the whole energy range, a new model, called Gilbert-Cameron
model [35], was proposed. The Gilbert-Cameron model uses the constant temperature model to describe the NLD in the low-energy region (below neutron
binding energy) and the Fermi-gas model for the high-energy region (above neutron binding energy). Generally, it is well-known that these models can predict
the overall behavior of the NLD over a wide energy range using a simple approach.
It is noted that our current understanding on the structure of low-lying nuclear
levels based on some important concepts including shell effects, pairing correlations, and collective phenomena. The generalized superfluid model, which is
developed from a microscopic-based model proposed by A. V. Ignatyuk [59], is
an additional phenomena-based approach that takes into account all the above

18

Chapter 1. Theory

concepts.
Concerning the microscopic-based models, beside the Ignatyuk model [59] mentioned above, the Hartree-Fock-BCS-based model [60] is also widely used with
a certain degree of confidence as its prediction is made by fitting the theoretical NLD with the experimental data for two excitation-energy regions. The latter consist of the data in the low-energy region taken by counting the number
of cumulative levels taken from the NLS and the average level spacing data at
the neutron binding energy. Recently, N. Quang Hung et al [61] propose a new
microscopic approach, which bases on the exact solutions of the pairing problem
embedded into the canonical ensemble in order to construct the partition function
and 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 fitting at different excitation energy. Furthermore, the NLD calculated within this
approach can be directly compared to the experimental data without any normalization such as in case of the Hartree-Fock-BCS-based model mentioned above.

1.3.1

Fermi-gas model

The Fermi-gas model [56] is proposed for the first time by Bethe in 1936 and it is
one of the most used NLD models. In this model, a nucleon inside the nuclear volume is regarded as an element of a non-interacting fermion gas and each fermion
can occupy only one single-particle state due to the Pauli-principle. The basic
assumption of the Fermi-gas model is that each excited single state is equally
spaced and collective effects are not involved.
The Fermi-gas NLD, ρF (E, J), for a given spin J at excitation-energy E is given
as
ρF (E, J) = ρF (E)g(E, J) ,

(1.7)