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


1.3.1

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


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 . . . . . . . . . . . . . . . . .


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


. . .

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


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

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



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



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. . . . . . . . . . . . . . .

(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.


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],


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

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

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


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,

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)


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)


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


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

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


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