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Experimental Study of the reaction9640Zr + 124
50Sn at 530 MeV using theGASP array
Wilmar Rodrıguez Herrera
Universidad Nacional de Colombia
Facultad de Ciencias
Departamento de Fısica
Bogota, Colombia
2014
Experimental Study of the reaction9640Zr + 124
50Sn at 530 MeV using theGASP array
Wilmar Rodrıguez Herrera
Master’s thesis submitted in partial fulfillment of the requirements for the degree of:
Magister en Fısica
Supervisor:
Ph.D., Diego Alejandro Torres Galindo
Research area:
Nuclear structure
Research group:
Grupo de Fısica Nuclear de la Universidad Nacional
Universidad Nacional de Colombia
Facultad de Ciencias
Departamento de Fısica
Bogota, Colombia
2014
Aknowledgments
The contribution of the accelerator and target-fabrication staff at the INFN Legnaro Na-
tional Laboratory is gratefully acknowledged. I would also like to thank the scientific and
technical staff of Gasp and Prisma/Clara.
I would like to thank all the staff of the nuclear physics group for their support along the
performance of this thesis. I specially thank to professor Fernando Cristancho the director
of the group whose teachings have been applied during the performance of this thesis.
I specially thank to Cesar Lizarazo for the discussions of different topics of the thesis
that allows me to clarify some issues.
I have studied undergraduate physics as well as masters studies in physics department.
The professors and administrative staff are also acknowledged for their teaching and support
given.
I thank to “Direccion academica” from “Universidad Nacional de Colombia” for the
scholarship (Asistente Docente) that gives me the economical support which allow me to
carry out my master studies.
Finally the supervision of professor Diego Torres is gratefully acknowledged.
ix
Abstract
In this thesis an experimental study of the binary nuclear reaction 9640Zr + 124
50Sn at 530
MeV using the Gasp and Prisma-Clara arrays at Legnaro National Laboratory (LNL),
Legnaro, Italy is presented. The experiments populate a wealth of projectile-like and target-
like binary fragments, in a large neutron-rich region below the magic number Z = 50 and at
the right side of the magic number N = 50, using multinucleon-transfer reactions. The data
analysis is carried out by γ-ray spectroscopy.
The experimental yields of the reaction in each one of the experiments, is presented.
Results on the study of the yrast and near-yrast excited states of 9541Nb are presented, along
with a comparison of the predictions by shell model calculations.
Keywords: Gamma-ray Spectroscopy, Shell Model, Neutron-Rich Nuclei, Deep
Inelastic Reactions, Nuclear Structure.
Resumen
En este trabajo se muestra una caracterizacion experimental de la reaccion nuclear 9640Zr+
12450Sn
a 530 MeV usando los arreglos experimentales Gasp y Prisma-Clara ubicados en el labo-
ratorio nacional de Legnaro (LNL), Legnaro, Italia. En estos experimentos se poblaron una
gran cantidad de fragmentos binarios de tipo proyectil y de tipo blanco en una gran area
de nucleos ricos en neutrones con numero de protones menores al numero magico Z = 50 y
numero de neutrones mayor al numero magico N = 50, usando reacciones de transferencia
multiple de nucleones. El analisis de los datos es realizado mediante espectroscopıa de rayos γ.
La produccion experimental de los nucleos en cada uno de los experimentos es presen-
tada. Resultados en el estudio de estados yrast y yrast-cercanos para 9541Nb son presentados
junto con una comparacion con predicciones hechas por calculos de modelo de capas.
Palabras clave: Espectroscopıa de rayos Gamma, Modelo de Capas, Nucleos Ricos
en Neutrones, Reacciones Deep Inelastic, Estructura Nuclear
x
Preliminary results of the present work were presented in the conferences:
XXXVI Brazilian Meeting on Nuclear Physics, Study of the Evolution of Shell
Structure of Z<50 Neutron-rich Nuclei near the N=82 Closed Shell Using the 96Zr +124Sn Reaction at 576 MeV with the Gasp Array. (1 - 5 September 2013, Maresıas,
Sao Paulo, Brazil)
http://sbfisica.org.br/∼rtfnb/xxxvi-en/
X Latin American Symposium on Nuclear Physics and Applications, Expe-
rimental study of the 9541Nb level scheme using the 96
40Zr + 12450Sn reaction with Gasp
and Prisma-Clara arrays. (1 - 6 December 2013, Montevideo, Uruguay)
http://www.fing.edu.uy/if/lasnpa/
The following articles have been produced:
Study of the Evolution of Shell Structure of Z<50 Neutron-rich Nuclei near the N=82
Closed Shell Using the 9640Zr +
12450Sn Reaction at 576 MeV with theGasp Array. Annual
report contribution at Legnaro National Laboratory.
http://www.lnl.infn.it/∼annrep/index.htm
Experimental study of neutron-rich nuclei near the N = 82 closed shell using the 9640Zr
+ 12450Sn reaction with Gasp and Prisma-Clara arrays. Sent to publish at: AIP Conf.
Proc.
http://www.sbfisica.org.br/∼rtfnb/xxxvi-en/index.php?option=com
content&view=article&id=72&Itemid=198
Experimental study of the 9541Nb level scheme using the 96
40Zr + 12450Sn reaction with
Gasp and Prisma-Clara arrays Sent to publish at Proceedings of Science.
http://pos.sissa.it/
Note: Copy of the articles are presented at the appendices of this document.
Contents
Acknowledgments VII
Abstract IX
1. Introduction 2
2. Preliminary concepts on nuclear structure 4
2.1. Chart of nuclides and the region under study . . . . . . . . . . . . . . . . . . 4
2.2. Production of neutron-rich nuclei using grazing reactions . . . . . . . . . . . 6
2.3. The nuclear shell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1. The mean field potential . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2. Ground state predictions . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.3. Predictions for excited states . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.4. Shell model calculations . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4. Spins and parities of excited states . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.1. Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.2. Multipolar radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3. The 95Nb nucleus 21
4. Experimental methods 24
4.1. Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1.1. The Prisma-Clara experiment . . . . . . . . . . . . . . . . . . . . 24
4.1.2. The Gasp experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2. Gamma-ray detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2.1. Energy resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2.2. Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5. Data analysis 34
5.1. Construction of a level scheme from Gasp experiment . . . . . . . . . . . . . 34
5.1.1. γγ coincidence matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.1.2. γγγ coincidence matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.1.3. Angular correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Contents 1
5.2. Products of the reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2.1. The Prisma-Clara experiment . . . . . . . . . . . . . . . . . . . . 42
5.2.2. The Gasp experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6. Results 53
6.1. Products of the reaction from the Gasp and the Prisma-Clara experiments 53
6.2. Level scheme of 95Nb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.3. Shell model calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7. Conclusions and perspectives 66
A. Appendix: Contribution to the Legnaro National Laboratory. 67
B. Appendix: Contribution to the proceedings of the XXXVI RTFNB 69
C. Appendix: Contribution to the X LASNPA proceedings 73
Bibliography 80
1. Introduction
The interaction between two nucleons (protons or neutrons) mediated by the strong nuclear
force, has not a complete theoretical explanation yet. The nuclear force depends not only on
the relative separation of the two nucleons, but also on their intrinsic degrees of freedom. The
dependence with the relative separation does not have a simple mathematical expression,
moreover different attempts trying to give an analytic expression for the strong nuclear force
includes around 9 terms with more than 10 parameters which have to be fitted experimen-
tally, see for example Ref. [1]. Because of this complexity of the nuclear force, different nuclei
have different properties, and the characterization of a nuclear region implies an enormous
task. As an example of that, in the experiments described in this thesis more than 100 nuclei
were created.
The number of particles in the nuclear system is not low enough to try to solve the
system by use of ab-initio calculations, and it is also not large enough, for most of the
nuclei, so that models do not provide a complete explanation of nuclear properties. Many
models have been proposed since the discover of the nuclear force, for example the Fermi
gas, the liquid drop model and the nuclear shell model. The shell model is one of the most
successful, in terms of the number of correct predictions made for nuclei near the so called
magic numbers. The nuclear shell model was proposed in 1949 by Eugene Paul Wigner,
Maria Goeppert-Mayer and J. Hans D. Jensen, who shared the Nobel Prize in Physics for
their contributions in 1963 [2]. Currently, the nuclear shell model continues being tested
experimentally in order to improve the model or to identify its limits. To succeed in this
goal different nuclei, in several mass regions, must be characterized because predictions of
the shell model are different for different nuclei. For instance the region approaching N ≥ 50
and Z ≈ 40 is a very interesting region for both, nuclear structure and nuclear astrophysics,
due to the possibility to study shell closures and sub-closures in the neutron-rich region, and
for the opportunity to increase our knowledge on nuclei in the path of the rapid neutron
capture r-process nucleosynthesis, respectively.
The neutron-drip line, where neutrons can no longer bind to the rest of the nucleus,
is not well define by the existent nuclear model, and it is the challenging frontier that
experimentalist are looking forward to reach. Recent experimental progress has been made
in the theoretical side to describe the structure of neutron-rich nuclei [3, 4, 5], and large
γ-ray arrays [6] coupled to fragment mass separators [7, 8] have provided with outstanding
structural information of neutron-rich nuclei [9].
During the last decade experimental studies of neutron-rich nuclei have been conducted
3
using deep inelastic reactions using dedicated experimental setups, such as the Prisma-
Clara array at Legnaro National Laboratory, Italy. Due to the large acceptance of the
Prisma magnetic spectrometer, and its use in conjunction with the high-resolution gamma-
ray detector array Clara in thin target experiments, a clear identification of the sub-
products of the reaction is possible. More detailed spectroscopy information can be obtained
if partner thick target experiments are performed using highly efficient γ − ray arrays,
such as Gasp. The latter may allow the obtention of pivotal information for a complete
characterization of the nuclear states in neutron-rich nuclei. The results obtained in this
work will contribute with structural information of the 95Nb nucleus, and it is a first step
toward a systematic study of isotopic chains of neutron-rich nuclei in the region.
A description of the region of interest in this thesis along with an explanation of the
shell model will be presented in Chapter 2. A brief description of the production of neutron-
rich nuclei using grazing reactions will be also presented. Chapter 3 is a summary of the
main properties of 95Nb, which was the object under study in this thesis, as well as the
latest studies carried out about 95Nb level scheme. In Chapter 4 the experimental methods
used to perform the Gasp and Prisma-Clara experiments are exposed. The data analysis
performed over the data from both experiments is explained in Chapter 4. Finally in Chapter
5 the results obtained from characterization of the reaction from Gasp and Prisma-Clara
experiments, along with the level scheme of 95Nb proposed in this work, are presented.
2. Preliminary concepts on nuclear
structure
The atoms are the components of ordinary matter. They are formed by electrons and a
nucleus with neutrons and protons inside. The electrons are bound to the atom by the
Coulomb force generated between the electrons and the protons in the atomic nucleus. The
atoms have an order size of ∼ 10−10 m ≡ 1 angstrom (A). However the nucleus in the atom
has a size experimentally proved to be 1.2A1/3 fm, with A the mass number. Thus the nuclear
dimensions are ∼ 10−15 m ≡ 1 fm. It means that the nucleus in the atom has a size five
orders of magnitude lower than the size of the complete atom. Despite the difference of sizes
between the complete atom and its nucleus, most of the mass in the atom is contained in
the atomic nucleus. The mass of an electron is ∼ 0.5 MeV/c2 and the mass of protons and
neutrons approximately the same is ∼ 1000 MeV/c2. For example in the case of the hydrogen
atom (1 proton and 1 electron) the atomic nucleus has approximately 2000 times the mass
of the electron. All these facts means that the nucleus has a very high density of ∼ 1017
Kg/m3.
Due to the Coulomb force the number of protons determines the number of electrons of
an atom, and the electrons are responsible for the chemical properties of the atoms. For this
reason, depending on the number of protons, the nucleus and the atom have a chemical name.
Several nuclei with the same number of protons and different atomic masses can generate a
bound system. These types of nuclei are called isotopes. Some isotopes are stables but most
of them are unstable and decay by different ways. In the next subsection is exposed the chart
of nuclides which is a tool to visualize all the nuclei, as well as the region of interest in this
work.
2.1. Chart of nuclides and the region under study
There are less than 300 known stable nuclei, and more than 3000 radioactive isotopes have
been produced in the laboratory, so far. The way to visualize all those nuclei is to sort them
in the so called “chart of nuclides”, shown in Figure 2-1, The Y-axis indicates the number
of protons and the X-axis indicates the number of neutrons.
Figure 2-1 also shows the neutron and proton drip lines, which indicate the limits in the
number of protons or neutrons for which certain nucleus could generate bound states. While
the proton drip-line has been experimentally explored during the last decades, with the use of
2.1 Chart of nuclides and the region under study 5
Figure 2-1.: Chart of nuclides. The magic numbers for protons and neutrons and different decay
modes are shown as well as the proton and neutron drip lines. The region of interest
in this work is also highlighted. Modified from the original at [10]
fusion-evaporation reactions, the neutron drip-line is more difficult to access experimentally.
The region of interest in this work is highlighted in Figure 2-1. Figure 2-2 shows with more
detail the relevant area for this work in the chart of nuclides.
In Figure 2-2 can be seen bars enclosing the magic numbers Z = 50 and N = 50. The
target and the projectile are the stable isotopes of Z = 40 and Z = 50 with the highest
number of neutrons. It can also be seen that the 95Nb nucleus, that will be the subject of
study in this work, is near to the N = 50 magic closed shell. In this work the region of
interest corresponds to neutron-rich nuclei with A ∼ 100. These nuclei lie on the pathway
of the rapid neutron capture process (r-process) [11], so there is also a nuclear astrophysical
interest in the structure of such nuclei. The r-process is a nucleosynthesis event that occurs
in core-collapse supernovae and is responsible for the creation of approximately half of the
neutron-rich atomic nuclei heavier than iron. Neutron-rich nuclei decays by β− decay (n →
p + e− + νe ), it is, a neutron is exchanged by a proton. The r-process entails a succession of
rapid neutron captures (hence the name r-process) by heavy seed nuclei and these neutrons
get the nucleus faster than the β− decay occurs. Heavy elements (those with atomic numbers
6 2 Preliminary concepts on nuclear structure
Figure 2-2.: Chart of nuclides in region of interest. The target, 12450 Sn and the beam 96
40Zr of the
reaction are located as well as the 95Nb and the 125In.
Z > 30) are mainly synthesized by r-process and their isotopic abundances (Z > 56) are
regarded as the main r-process [12]. In this thesis an experimental study of neutron-rich
nuclei in the A ∼ 100 region is performed. The nucleus 95Nb is expected to be populated
trough the reaction 2-3 and this nucleus will be study in this thesis.
From the experiments analyzed in this work, it is expected that most of the nuclei below
of stable nuclei shown in Figure 2-2 had been populated. This region contains isotopes with
more neutrons than the stable nuclei. These nuclei are called neutron-rich. The production of
nuclei is carry out colliding some nuclei against each other and in this way, different reactions
can occur and produce different nuclei. Neutron-rich nuclei are usually populated by mean
of grazing reactions, a type of mechanism explained in the following subsection.
2.2. Production of neutron-rich nuclei using grazing
reactions
Neutron-rich nuclei are difficult to produce. Currently one of the most efficient methods to
populate neutron rich nuclei is using grazing reactions which could be deep inelastic and
multinucleon-transfer reactions. Both type of mechanism are binary, which means that the
projectile and target exchange few nucleons and the products of the reactions maintain
some resemblance to the initial products. After the occurring reaction, a couple of nuclei are
produced, one similar to the projectile (projectile-like) and another one similar to the target
(target-like). This situation is shown in Figure 2-3 for the reaction 9640Zr +
12450 Sn at Elab =
530 MeV.
2.2 Production of neutron-rich nuclei using grazing reactions 7
Figure 2-3.: Scheme of the process in a grazing reaction. The grazing angle at 530 MeV is θ = 38.
If the excitation energy of the ejectiles is larger than 20 MeV the reaction is called
deep inelastic, due to the large amount of kinetic energy in the beam that is converted
to excitation energy, otherwise the binary reaction is called multinucleon-transfer reaction.
When the energy increases, the excitation energy does the same, but it has a limit imposed by
the binding energy of the nucleus in the beam. The couple of products is generated in ∼ 10−22
seconds, which is too short time to be discriminated by the electronics. Experimentally nuclei
already formed can be observed. It is due to the electronics time of response is ∼ 10−8 s and
the typical lifetime of the excited nuclear states is ∼ 10−12 s.
Grazing reactions are expected to generate more neutron-rich nuclei than other types
of reactions (Inelastic or fusion-evaporation reactions). Angle with the largest cross section
for the grazing reactions is called “grazing angle”. This angle is produced when the distance
of maximum closest equals the sum of the radii of both nuclei implied in the reaction. The
distance of closest approach is deduced in [13, 14] and is given by
d =
(
ZpZt
4πǫ0Ek
)(
1 + csc
(
θ
2
))
, (2-1)
where Zp and Zt are the number of protons in the projectile and the target respectively. Ek
is the kinetic energy of the beam.
Experimentally it is found that the nuclear radius of a nucleus with A nucleons has a
value given by r = 1.2 ·A1/3. So the sum of the radii of the two nuclei implied in the reaction
is given by,
d = 1.2(
A1/3p + A
1/3t
)
. (2-2)
In Equation (2-2) Ap and At are the number of nucleons in the projectile and the target
respectively. In this work the reaction used was,
9640Zr +
12450 Sn at Elab = 530 MeV. (2-3)
8 2 Preliminary concepts on nuclear structure
The grazing angle for this case calculated from Equations (2-1), (2-2), (2-3) is 38, as is
noted in Figure 2-3.
From a theoretical point of view only the transfer of a single nucleon can be explained,
this due to the complexity of the nuclear force. When the number of transferred nucleons
increases, the calculations get extremely complex and, for this reason, the theoretical studies
of this phenomenon have not given a complete explanation. This is the case of the code
“GRAZING” by G. Pollarolo [15]. In this work the numerical code is used to simulate the
total cross section for the most important yields of the reaction (2-3). The results will be
shown in Chapter 6 along with a comparison of the experimental data. The nuclei generated
in the reaction have excitation energies which produce a de-excitation process. In cases when
this energy exceeds the bounding energy of a neutron, the nucleus will emit neutrons, this
process is known as neutron emision. In the cases when the excitation energy is lower than
the bound energy of a neutron, then the nucleus will decay emitting γ-rays and this γ-rays
gives the information about the excited states of the nucleus. When a nucleus is close to
the magic numbers in the chart of nuclides it is expected that its first excited states can be
described by shell model that will be explained in the next subsection.
2.3. The nuclear shell model
The nucleus is a system of A particles which interacts under the potential generated by the
strong nuclear force. The hamiltonian for such system can be written as
HExact =A∑
i=1
Ti +1
2
A∑
i=1
A∑
j=1j 6=i
Vij(|~ri − ~rj|). (2-4)
In Equation (2-4), Ti, is the kinetic energy of each nucleon and A is the number of nucleons.
The second part in Equation (2-4) which corresponds to the potential, contains A(A− 1)/2
terms, each one corresponds to the nucleon-nucleon potential. This potential is schematically
shown in Figure 2-4. At large distances the potential in Figure 2-4 is explained by the
Yukawa potential which can be obtained solving the Klein-Gordon Equation for the exchange
of a pion and taking the potential proportional to its wave function. At short distances the
potential is repulsive.
The A(A− 1)/2 terms of the second part of Equation (2-4) have the functional behavior
shown in Figure 2-4. To date in the laboratory has been generated nuclei with number of
nucleons, A, larger than 200. This made the calculations of Equation (2-4) a very complex
problem even for a computer. Thus a model had to be developed in order to simplify the
hamiltonian in Equation (2-4). The shell model was developed in 1949 by several independent
works by Eugene Paul Wigner, Maria Goeppert-Mayer and J. Hans D. Jensen [16, 17]. The
model consists in calculate the following approximation for the nuclear potential
2.3 The nuclear shell model 9
0.0 0.5 1.0 1.5 2.0 2.5r (fm)
−50
0
50
V N−N
(MeV
)
VN−N(r) Schematic
Figure 2-4.: Scheme of the shape of nucleon-nucleon potential.
1
2
A∑
i=1
A∑
j=1
Vij(|~ri − ~rj|) ≈A∑
i=1
V (ri). (2-5)
Equation (2-5) replaces the interaction that acts over each nucleon due to the presence of
the other ones as an interaction that depends just on the position operator, r, of each nucleon.
It is assumed that the potential has a spherical symmetry. The hamiltonian proposed in this
model, HSM , in this first approximation of a spherical nucleus is
HSM =A∑
i=1
Ti +A∑
i=1
V (ri). (2-6)
From Equation (2-6) the following aspects have to be noted:
This expression propose that nucleons inside the nucleus can be modeled as non-
interacting particles and particles just interacts with a mean field potential, V (r). This
potential is the same for all the nucleons and depends just on the position operator,
ri, of each nucleon.
The dependence of the potential results kind of counter-intuitive due to the absence
of a center in the nucleus. This model had been proposed before to study the energy
levels of the electrons in the atoms with several electrons. However in the atomic case
was expected that the mean field potential had such a dependence because most of the
interaction that acts over the electrons is central. It is due to the coulomb interaction
made by the protons in the nucleus that defines the center of the atom. However this
approximation also works in nuclear case.
10 2 Preliminary concepts on nuclear structure
This model is coherent with the experimental data to predict excited states and g-
factors among others. However the predictions are not always correct due to the fact
that Equation (2-6) is an approximation to the real hamiltonian of Equation (2-4).
The difference between the exact hamiltonian and the model proposed in Equation (2-6)
is called the residual interaction, Hresidual.
Hresidual =1
2
A∑
i=1
A∑
j=1
Vij(|~ri − ~rj|)−A∑
i=1
V (ri). (2-7)
If the model is suitable to describe the nucleus it is expected that
〈Hresidual〉 ≪ 〈HSM〉. (2-8)
2.3.1. The mean field potential
The dependence of the potential, V (r), in Equation (2-6) must be coherent with experimental
observations. The nuclear potential has short range and it drops quickly a few fermis away
from the nucleus. This potential cannot have strong variations inside the core and in fact
should be approximately constant. Taking this into account three different types of potential
have been proposed being consistent with these statements.
Square well =⇒ V (r) =
−V0, if r ≤ R0
0, if r > R0
(2-9)
Harmonic oscillator =⇒ V (r) = −V0
[
1−
(
r
Roa
)2]
(2-10)
Woods Saxon =⇒ V (r) =−V0
1 + exp[
r−R0
a
] . (2-11)
The values of R0 and Roa in Equations 2-9, 2-10 and 2-11 as well as the functional shape of
these potentials, are shown in Figure 2-5.
The harmonic oscillator potential allows an analytical solution of the energy levels, these
are given by
ǫnℓ = hω0
[
2(n− 1) + ℓ+3
2
]
= hω0
[
N +3
2
]
. (2-12)
Spin-orbit interaction is also present in nuclei and it is very important to understand the
so called ”magic numbers”. The shell model without spin-orbit interaction does not predict
all the magic numbers. The inclusion of the spin-orbit interaction in the shell model was
proposed by Maria Goepert Mayer [18, 19] and can be included in the model
Hℓs = V ′0
1
r
dV (r)
dr~L · ~S = V0~L · ~S, (2-13)
2.3 The nuclear shell model 11
Figure 2-5.: Representation of harmonic oscillator, square well and Woods-Saxon potentials.
where L is the angular momentum of the nucleus and S is the spin of a nucleon. There is no
analytic expression for V0 in Equation (2-13). However it can be measured experimentally
and its sign can be also determined. It is found that
V0 ≤ 0. (2-14)
Thus the hamiltonian of the shell model including spin-orbit interaction is
H ′SM =
A∑
i=1
Ti +A∑
i=1
[
V (ri)− |V0| ~L · ~S]
. (2-15)
The potential, V (r), of the Equation (2-15) can be written as
V (r) =
V + |V0|12(ℓ+ 1), if j = ℓ− 1
2
V − |V0|12ℓ, if j = ℓ+ 1
2.
(2-16)
This term in the potential produces a splitting of each energy level with angular momentum
ℓ 6= 0. One schematic example of the splitting generated by the spin-orbit interaction is
presented in Figure 2-6. This splitting allows the shell model to predict the magic numbers
that are the numbers for which some energy levels called “Shells”, of the model are full
following the Pauli exclusion principle. The shells that generate the magic numbers are the
ones with high gap energy between the next one.
12 2 Preliminary concepts on nuclear structure
|n, ℓ〉
|n, J = ℓ− 1/2〉
|n, J = ℓ+ 1/2〉
∆ǫℓs
Figure 2-6.: Splitting of an energy level with quantic numbers n and ℓ generated by the spin-orbit
interaction.
The energy levels of the harmonic oscillator potential given by equation (2-12) can be
written including the spin-orbit interaction as
ǫnℓ = hω0
[
N +3
2
]
−ℓ
(ℓ+ 1)
j = ℓ+ 12
j = ℓ− 12.
(2-17)
The harmonic oscillator potential has an analytical solution, however the Woods-Saxon po-
tential generates a better description of the nucleus. A modification over the harmonic osci-
llator potential can be done in order to try to generate a potential similar to Woods-Saxon
with an analytical solution. The modified harmonic oscillator potential is given by
HMO =1
2hω0ρ
2 − κhω0
[
2ℓ · s+ µ(
ℓ2 − 〈ℓ2〉N)]
with ρ =
(
Mω0
h
)1/2
r and κµ = µ′.
(2-18)
The energy levels generated by the potential of Equation (2-18) are given by
ǫN,ℓ,j = hω0
[
N +3
2− κ
]
ℓ
−(ℓ+ 1)
− µ′
(
ℓ(ℓ+ 1)−N(N + 3)
2
)
j = ℓ+ 12
j = ℓ− 12,
(2-19)
where κ and µ′ are parameters which must be fitted experimentally and they are different
for different mass regions [20]. κ gives a measure of the strength of the spin-orbit interaction.
µ′ is the parameter which gives information about the skin of the nucleus, hω0 ≈ 41 · A1/3,
with A the number of nucleons. These parameters also determine the energy level scheme
and the first excited states of some nuclei which can be considered to have a single-particle
behavior.
Figure 2-7 shows the distribution of the energy levels for the harmonic oscillator potential
with and without spin-orbit interaction and also the energy levels generated by the modified
harmonic oscillator potential. The energy labels in Figure 2-7 refers to the quantum numbers
ℓ and J , the orbital and the total angular momenta respectively. The equivalence in angular
momentum for the letters in the labels of Figure 2-7 are, s ≡ 0, p ≡ 1, d ≡ 2, f ≡ 3, g ≡ 4
and h ≡ 5. For example the level 1g9/2 refers to a level with orbital angular momentum
2.3 The nuclear shell model 13
20
28
50
82
N = 2
N = 3
N = 4
κ = 0.08µ’ = 0.0
κ = 0.075µ’ = 0.0263
µ’ = 0.024κ = 0.06
-µ′hω0
(
ℓ2 − N(N+3)2
)
Harmonic −2κhω0ℓ · soscillator
1h11/22d3/23s1/21g7/22d5/2
2p3/2
1f7/2
1d3/2
2s1/2
3s
2d
1g
2p
1f
2s+ 1d
1g9/2
2p1/21f5/2
1d5/2
Figure 2-7.: Energy levels produced by harmonic oscillator potential. At the left the levels gene-
rated by a pure harmonic oscillator potential. At the middle the modification of the
potential is introduced. At the right the spin-orbit interaction is added.
ℓ = 4 ≡ g and total angular momentum J = 9/2. Each energy level of Figure 2-7 is called “a
shell”. In each shell can be placed 2(J+1) nucleons according with Pauli exclusion principle.
Neutron-rich nuclei
One of the research frontiers in nuclear structure is the experimental study of the neutron-
rich nuclei, which are isotopes with larger number of neutrons than the stable nuclei. These
nuclei have shown a strong variation of the κ and µ′ parameters when they are compared
with the stable nuclei. For example 4020Ca, which is a stable nucleus, has an energy gap of 7
MeV between the shells 1d3/2 and 1f7/2 of the Figure 2-7, and on the other hand, 288O has
and energy gap of 2.5 MeV. The 28O nucleus has 10 neutrons more than the stable isotopes
of 188O, so it is a neutron rich nucleus. Neutron-rich nuclei allow us to explore the behavior
of matter with excess of neutrons, like neutron stars. Most of the nuclei generated in the
experiments studied in this work are neutron-rich nuclei.
14 2 Preliminary concepts on nuclear structure
The magic numbers
If a nucleus has an even number of protons and neutrons its total angular momentum J is
coupled to 0, because this coupling generates a lower energy state than states with other
configurations. This lower energy is called “pairing energy” and it is bound energy generated
when two nucleons with equal angular momenta J and opposite angular Jz-component are
coupled into the same shell. When the number of protons or neutrons fills completely some
shell, it is said that we have a “closed shell” in protons or neutrons. Nuclei with closed shells
have bound energy larger than its neighbors due to the pairing energy.
The numbers that are shown in blue in Figure 2-7 corresponds to the number of nucleons
needed to fill the levels below these numbers. 20, 28, 50 and 82 are located between a couple
of levels which have energy separation larger than other near levels. This energy separation
means that it is more difficult to promote one nucleon in that shell to another one. These
types of numbers are called “magic numbers”. Nuclei with number of protons or neutrons
equal to a magic number have bound energy larger than its neighbors. For these reasons the
number of stable isotopes is larger for nuclei with a magic number of protons. Magic nulcei
are very well described by the shell model.
There are shells in Figure 2-7 with large energy separation between them. This is the
case of the 2p1/2 shell which has 40 nucleons for the close shell. For this reason 40 is known
as a semi-magic number.
2.3.2. Ground state predictions
Figure 2-7 can be used to make predictions about the spin and parities of the ground state.
It has been proved that these predictions are in agreement with the experimental data for
stable nuclei and its neighbors. As it was stated a nucleus with even number of protons and
neutrons has a total angular momentum J = 0 for its ground state. If a nucleus has an even
number of neutrons and an odd number of protons then the total angular momentum is
given by the shell in which is located the unpaired proton. All other protons are coupled by
pairs to a total angular momentum of 0. The nucleus of interest in this work is 95Nb, with
54 neutrons and 41 protons. As the low energy state is generated when nucleons are coupled
by pairs of angular momentum with Jz-component opposite, then the angular momentum
J is given by the unpaired proton that can be located making the filling of the shells in
Figure 2-7. In this case it is located in the shell 1g9/2. Thus the ground state of 95Nb is
expected to have a total angular momentum J = 9/2. The parity is given by
π = (−1)ℓ. (2-20)
In this case ℓ = 4 ≡ g, so the parity of the ground state of 95Nb will be positive. It is written
using the typical notation in nuclear physics as
Jπ = 9/2+. (2-21)
2.3 The nuclear shell model 15
2.3.3. Predictions for excited states
Some nucleons can be promoted to the higher shells in order to generate excited states. For
these processes, however, there are some nucleons in closed shells with high bound pairing
energy that are difficult to promote to other shells. For example the first excited state for9541Nb
54 nucleus could be generated by the promotion of the proton into the shell 1g9/2 to
the higher shell 2d5/2 (see Figure 2-7) however the gap energy between these two shells is
larger than, for example, the gap between the shells 2p1/2 (with two protons) and 1g9/2. This
nucleus has 4 neutrons in the 2d5/2 shell and the energy gap between this shell and the next
one, 1g7/2, is very low. Depending on the pairing energy of the two protons in the shell 2p1/2and the pairing energy of the neutron in the 2d5/2 shell, different possible configurations are
possible for the first excited state of 9541Nb54 nucleus. Different configurations implie that the
angular momentum of the all unpaired nucleons has to be combined in order to construct
the angular momentum of the excited state.
2.3.4. Shell model calculations
Excited states of nuclei near magic and semi-magic numbers in the chart of nuclides are well
described by shell model calculations made on the basis that excited states can be produced
by promotion of nucleons between different shells in the model. These excited states are
formed by “single-particle excitations”.
Shell model calculations can be made to predict the energy of some excited states. These
calculations are based on the fact that a nucleus with a closed shell has higher bound energy
than neighbor nuclei. Some nuclei can be considered as a sum of an inert core and some
valence nucleons which could be promoted to some valence orbitals to generate excited
states. These concepts can be defined and illustrated with an example of the particular case
of 9541Nb54 nucleus.
Inert core; the nucleus composed by nucleons filling completely lower shells. For 9541Nb54,
the inert core can be 8838Sr50.
Valence nucleons; nucleons in higher shells than the ones of the inert core 8838Sr50.
9541Nb54
has 4 valence protons and 3 valence neutrons.
Valence space; the energy levels available for valence nucleons. They are energy levels
above the ones filled by the inert core. Neutron valence space for the 4 valence neutrons
of 9541Nb54 is composed by the shells 2d5/2, 1g7/2, 3s1/2, 2d3/2 and 1h11/2. Proton valence
space for the 3 valence protons are 2p1/2 and 1g9/2.
External orbitals; the remaining orbitals that are always empty.
Figure 2-8 shows the concepts defined above for the case of 9541Nb54 considered as a
sum of the 8838Sr50 inert core plus 4 valence neutrons and 3 valence protons. A particular
16 2 Preliminary concepts on nuclear structure
20
28
50
82
20
28
50
82
2p1/21f5/22p3/2
1f7/2
1d3/2
2s1/2
1d5/2
1g9/2
1h11/22d3/23s1/21g7/22d5/2
2p1/21f5/22p3/2
1f7/2
1d3/2
2s1/2
1d5/2
1g9/2
1h11/22d3/23s1/21g7/22d5/2
ProtonsNeutrons
Valence space
Valence protons
Inert core
Valence space
Valence neutrons
Inert core
External space
Figure 2-8.: Inert core, valence neutrons and protons, and valence spaces for the case of 9541Nb54
nucleus.
selection of the inert core and valence space must be made based on the shell model energy
levels from Figure 2-7. A suitable selection of an inert core will be a nucleus with a magic
number of protons and neutrons and the valence orbitals will be the higher shells. Once
the inert core, valence orbitals and valence nucleons has been selected, an effective nucleon-
nucleon interaction must be introduced. The success of the calculations suggest that the
simple free nucleon-nucleon interaction can be regularized in the valence space. Thus there
are different effective interactions for different valence spaces. Effective interactions between
pair of nucleons are generated from the empirical values [21] which are then compared with
experimental data in order to obtain better effective interactions which can describe the
nuclei in some particular region. Some of the purposes of the experimental study of the
excited states of the nuclei are to improve the determination of an effective interaction. The
exact solution of the real interaction can be approximated by the solution of the effective
2.4 Spins and parities of excited states 17
interaction in the valence space such that
Hψ = Eψ → Heffψeff = Eψeff , (2-22)
where Heff and ψeff are the effective halmitonian and wavefunctions in the valence space.
The single particle energy levels in Figure 2-7 must be also found experimentally and they
are needed to make the calculations.
In this work an experimental study of the 95Nb excited states will be presented. These
data will contribute to the determination of an effective interaction in the valence space
described in Figure 2-8.
2.4. Spins and parities of excited states
When the nucleus decays from an excited state it emits γ-rays which have some multipora-
larity. Depending on the multipolarity of the emitted γ-ray, spins and parities of the excited
states can be determined.
2.4.1. Selection rules
In a transition between an initial state with spin and parity Jπi
i and a final state with spin
and parity Jπf
f , a γ-ray can be emitted with a total angular momentum jγ and parity πγ.
This process is illustrated in Figure 2-9.
J iπi
Jfπf
jγπγ
Ei
Ef
Eγ = Ei − Ef
Figure 2-9.: Quantum numbers in a γ transition. Ei and Ef are the enegies of the initial and the
final state. Jπi
i and Jπf
f are the spin and parity of the initial and the final state. jγ ,
πγ and Eγ are the angular momentum, parity and energy of the emitted γ-ray.
The quantum numbers of the final state are calculated by the composition of the quantum
numbers Jπf
f and jγ , πγ . The angular momentum conservation is
Ji = Jf + jγ . (2-23)
Equation (2-23) implies an angular momentum composition which produces a selection rules
on the quantum numbers jγ and Ji,
|Ji − Jf | ≤ jγ ≤ Ji + Jf (2-24)
|jγ − Jf | ≤ Ji ≤ jγ + Jf . (2-25)
18 2 Preliminary concepts on nuclear structure
The electromagnetic decay preserves parity thus,
πi = πfπγ(Xjγ). (2-26)
In Equation (2-26), jγ indicates the angular momentum of the radiation and X indicates
the character of the radiation, X = E for an electric transition and X = B for a magnetic
transition. Notation used in Equation (2-26) is widely used in nuclear physics, for example
an E2 transition represents an electric quadrupole transition and a M1 transition represents
a magnetic dipole transition, etc. The parity of the electromagnetic radiation is given by
(−1)j for an electric multipole, (2-27)
(−1)j+1 for a magnetic multipole. (2-28)
Depending on the angular momentum of the γ-ray emitted and taking into account the
section rule (2-26), the character of the radiation X can be determined. To illustrate how
works the selection rules [(2-25), (2-26), (2-28)], let us consider the transition in Figure 2-10.
9/2+
J iπi
E2
Figure 2-10.: Transition with an emission of a E2 γ-ray to an state of spin and parity 9/2+.
The situation illustrated in Figure 2-10 is an example of a typical experimental result
where the spin and parity of the ground state is known and the multipolarity character of
the γ-ray emitted is measured. The objective will be to assign the spin and parity of the
excited state. To do that the selection rules [(2-25), (2-26), (2-28)] must be considered. If Jiis the spin of the initial state in Figure 2-10 then the selection rule (2-25) gives
|2− 9/2| ≤ Ji ≤ 2 + 9/2 (2-29)
5/2 ≤ Ji ≤ 13/2. (2-30)
The selection rule (2-27) gives the parity of the initial state in Figure 2-10. The γ-ray
is of E2 type, so its parity is (−1)2 = +1, thus the parity of the initial state must be
πi = (+1)(+1) = +1. (2-31)
According to Equation (2-30) there are several possibilities for the spin and parity of the
initial state from Figure 2-10,
Jπi
i = 5/2+, 7/2+, 9/2+, 11/2+, 13/2+ (2-32)
2.4 Spins and parities of excited states 19
The comparison with shell model calculations may help to determine which value given
by (2-32) is the correct value.
As it was stated the multipolarity character of the radiation can be measured, this will
be exposed in the next subsection.
2.4.2. Multipolar radiation
The γ radiation emitted by a nucleus can have either a electric or a magnetic nature. Electric
and magnetic transitions are due to the redistribution of the multipole magnetic and electric
moments of the nucleus, respectively. The γ-ray angular distribution depends on the multi-
polarity order of the emitted radiation. This angular distribution dependence for a multipole
of the order ℓ,m is given by
Zℓ,m =1
2
[
1−m(m+ 1)
ℓ(ℓ+ 1)
]
|Yℓ,m+1|2 +
1
2
[
1−m(m− 1)
ℓ(ℓ+ 1)
]
|Yℓ,m−1|2 +
m2
ℓ(ℓ+ 1)|Yℓ,m|
2, (2-33)
where Yℓ,m are the spherical harmonics.
For example the angular distribution of the intensity of the radiated energy by a dipole,
and a quadrupole are given by Equations [(2-34), (2-35)]. The angular distribution generated
by these Equations are represented in Figures [2-11, 2-12].
Z1,0(θ) =1
2|Y1,−1|
2 +1
2|Y1,1|
2 = |Y1,1|2 =
3
8πsin2(θ) (2-34)
(a) 2D (b) 3D
Figure 2-11.: Angular distribution of the emitted γ radiation of the order ℓ = 1 y m = 0. The red
arrow indicates the multipole orientation.
Z2,0(θ) =1
2|Y2,1|
2 +1
2|Y2,−1|
2 = |Y2,1|2 =
15
8πcos2(θ) sin2(θ) (2-35)
As can be seen from Figures [2-11, 2-12] the angular distribution of the energy radiated
is different for different multipoles. These differences in the angular distributions allow the
20 2 Preliminary concepts on nuclear structure
(a) 2D (b) 3D
Figure 2-12.: Angular distribution of the emitted γ radiation of the order ℓ = 2 y m = 0. The red
arrow indicates the multipole orientation.
experimental determination of the multipolarity of the emitted radiation. In Chapter 5 the
experimental technique utilized to determine the multipolarity of radiation will be explained
and finally in Chapter 5 the results obtained for the γ-rays emitted from 95Nb nucleus will
be shown.
The following subsection describes the current state of the excited states of 95Nb measu-
red by γ-ray spectroscopy. These excited states are represented in nuclear physics as a level
scheme.
3. The 95Nb nucleus
9541Nb nucleus has a radioactive half-life of T1/2 = 35.991(6) days [10] and decays from the
ground state via β− to the stable 95Mo. The number of protons of 9541Nb is 41 protons, just
one proton to the semi-magic number 40 and the number of neutrons is 54, 4 neutrons to
the 50 closed shell. Due to its proximity to 8838Sr nucleus, which is emplyed as a standard
closed-core shell [5], a single-particle behavior is expected.
Previous experimental studies of 95Nb nucleus have been performed using β decay [22],
which did not populate high excited states, and also by fusion-evaporation reactions which
populates high-spin states [23]. The most recent experimental results of 95Nb reported more
than 10 different excited states with proposed spin and parity for levels close to the ground
state [23]. For the latter work data from three experiments were analyzed. The first two
utilized the fusion evaporation reactions
12Ca +82 Se at Elab = 38 MeV (3-1)16O+82 Se at Elab = 48 MeV. (3-2)
The γ-rays produced in these reactions were detected by an array of just three Ge
detectors. The low statistics generated in these experiments had to be complemented by a
third experiment that made use of 16O and 12C contaminants from the target of the reaction
82Se +192 Os at Elab = 470 MeV, (3-3)
the γ-rays were detected using the detector array Gasp [6] (for specific details of the Gasp
array see Chapter 4). Based on the Gasp experiment the level scheme of Figure 3-1 was
proposed. In Figure 3-1 the spins and parities proposed by Bucurescu et al [23] are also
shown.
As it was mentioned in section 2.3.2, the predicted spin and parity of the 95Nb ground
state are
Jπ = 9/2+. (3-4)
These spin and parity were measured experimentally by Rahman and Chowdhury [24], they
found that predictions by shell-model calculations to their ground state are also correct.
In the report made by Bucurescu et al., [23] two problems were reported in the cons-
truction of this level scheme. Firstly, the intensities of the γ-rays at each side of the energy
23
level of 5643 keV were the same between the uncertainty range, like happened with the γ
rays coming in and going out from the energy level of 4071 keV. Secondly, the experiment
using the gasp array made use of the contaminants in the target and no the target itself.
These contaminants could not be uniformly distributed which could cause difficulties in the
assignment of the intensities of the γ-rays. These problems do not give confidence in the
arrangement of the levels proposed in Figure 3-1, as stated in the report.
The reasons presented above encourage the performance of a new experimental study of
the 95Nb nuclei, and motivates the present work. To allow that, two experiments were carried
out at Legnaro National Laboratory, Legnaro, Italy. These experiments are described in the
following Chapter.
4. Experimental methods
4.1. Experiments
In order to study properties from nuclear states, the nucleus has to be created. To do this an
accelerator must collide the nuclei in the beam with the nuclei in the target. The beam at
Legnaro was initially accelerated by the Tandem and finally by the linear accelerator ALPI.
As a result of the reaction, excited nuclei are generated and they decay emitting γ-rays,
which will be the subject of our study. Those γ-rays will provide information about the
properties of the nuclei. An array of Ge-detectors will collect information of energy and time
of γ-rays emitted by the nuclei produced in the reaction. In this thesis two arrays in two
different experiments: Prisma-Clara [7] and Gasp [6], were used.
In Prisma-Clara experiment was utilized a thin target in order to allow the projectile-
like fragments to reach the spectrometer Prisma. On the other hand a thick target was
utilized for the Gasp experiment. It made the projectile-like fragments stop inside the Gasp
multidetector array. A complete description of the experiments will be done in the next
subsections. A summary of the experimental details of both experiments is shown in Table 4-
1.
Table 4-1.: Target thickness and beam energy of the Prisma-Clara and Gasp experiments.9640Zr +
12450Sn
Prisma-Clara Gasp
Target (12450Sn) thickness (mg/cm2) 0.3 8
Thickness of the backing target (mg/cm2) 0.04 of 12C 40 of 208Pb
Beam energy (MeV) 530 570 a
Number of working detectors 25/25 38/40
aThe beam energy at the middle of the 124
50Sn target was 530 MeV.
4.1.1. The Prisma-Clara experiment
For the Prisma-Clara experiment [8, 25] the binary fragments produced in the reaction are
separated in the target. The target-like products remains in its initial position, meanwhile
4.1 Experiments 25
the projectile-like fragments continue moving through Prisma which have several stages as
shown in Figure 4-1.
Target Start detector
Magnetic
Magneticdipole
Focal
Projectile−likedetectors
quadrupole
detectorplane
Target−like
96Zr530 MeV
124Sn∆E-E
Beam
Clara detectorarray
Figure 4-1.: Prisma-Clara set-up correlating the coincidence signals at the focal plane of Prisma
with the γ-ray transitions detected by CLARA.
Figure 4-2.: Prisma-Clara array at the Legnaro National Laboratory.
The magnetic quadrupole is used to focus the beam. The start detector and the focal
plane detector gives the time of flight information which together with the length of the
26 4 Experimental methods
trajectory enable us to calculate the velocity v of the beam. After the nucleus cross the
magnetic dipole the beam is separated in different trajectories with a radius given by
ρ =mv
Z. (4-1)
The incident velocity v is the same for all the nuclei on the beam, so they are separated by
their charge-mass relation. When the nucleus pass trough the detectors labeled as ∆E − E
in Figure 4-1 [26, 27], they loose energy depending on the width of the detector so that
dE
dx∝mZ2
E, (4-2)
where m and Z are the mass and the number of protons of the nucleus. From Equation (4-2)
can be seen that the nuclei are separated by their charge, which make possible a complete
identification of a nucleus.
Prisma and Clara were linked at a laboratory grazing angle of 38. However this link
has an angular acceptance of ∆θ ∼ 12 and ∆φ ∼ 22. Being φ the azimuthal angle with
respect to the beam direction and θ the polar angle. Thus, Prisma is detecting just the
nuclei produced between these angles. Besides Clara detected just the γ-rays which were
in coincidence with the γ-rays emitted by the nuclei produced at these angles. This way,
just the radiation produced by the nuclei produced at angles near to the grazing angle were
detected. This is an important fact that will be discussed later.
The Prisma-Clara experiment has the advantage of select products of the reaction at
an specific angle, besides, due to Prisma magnetic spectrometer, this experimental set-up
can select the radiation produced by an specific nucleus. However due to Prisma covering
solid angle of 80 msr, this experimental set-up has the setback of the low yield production. To
solve this problem a complementary experiment was conducted and it is called here “Gasp
experiment”.
4.1.2. The Gasp experiment
Gasp [6] is an array of 40 High-Resolution Ge-detectors, each one equipped with BGO Com-
pton suppressor detectors which suppress most of the Compton events using a coincidence
technique as shown in Figure 4-7. Figure 4-7 shows a Ge-detector surrounded by BGO
Compton suppressor detector. If Compton event occurs in the Ge-detector it could be also
detected by the high efficiency BGO detector, and this event can be suppressed. On the
other hand, if an event getting the detector produces photoelectric effect, depositing all the
energy of the γ-ray in the crystal, then the event does not produce a BGO detector signal,
and it will be a valid event as shown in Figure 4-7. Gasp is a spherical array covering a
solid angle close to 4π that has a total of 40 Ge-detectors distributed in 11 rings with the
central ring hosting 8 detectors. A transversal cut of the central ring is shown in Figure 4-3.
4.1 Experiments 27
Target
BGO Comptonsupressor detectors
Ge Detectors
20 cm
Beam96Zr at 574 MeV
124Sn
Figure 4-3.: Gasp central ring Set-up.
Figure 4-4.: Gasp Set-up real image.
Figure 4-3 shows also the distance between target and the position of the detectors.
γ-rays from Gasp and Prisma-Clara experiments were detected using Ge-detectors su-
rrounding by BGO detectors. The characteristics of such detectors will be explained in the
28 4 Experimental methods
next subsections.
4.2. Gamma-ray detectors
A detector is a device that is constructed with the objective of convert all the radiation
that impact over it, into an electronic signal. However this is not always possible. Different
detectors have been developed for different purposes. In this work just the γ-ray detectors
are of interest. These detectors could be divided in three different types:
Plastic: This type of detectors emits light when the radiation inside over it, but they
cannot distinguish between the energy of the radiation. These detectors spend a very
low time forming the signal, for this reason they are called fast detectors.
Scintillators: When the radiation hit these detectors it excites the atoms and the mo-
lecules in the crystal making possible the light will be emitted in the de-excitation
process. This light is transmitted to the photomultiplier which convert it into a weak
electric current that is amplified by an electronic system. This type of detectors has a
relatively low time detection of ∼400ns (rise time of the signal after the preamplifier).
On the other hand the energy resolution of these detectors is relatively low compa-
red with semiconductor detectors. The most known scintillator detectors are the NaI
(sodium iodide) and the BGO (bismuth germanate).
Semiconductor detectors: these types of detectors need a BIAS voltage which polarizes
a junction n-p in the crystal, generating a depletion zone in which a γ-ray can generate
a cascade of electrons proportional to the energy of the γ-ray. This type of detectors has
a very high energy resolution compared with scintillator detectors. On the other hand
these detectors have a very low time of response ∼5µs. The most common detectors of
this type are Ge-detectors.
When a γ-ray reach a detector three different type of processes can occur, they are,
Compton effect, photoelectric effect and pair production. Compton effect could occur in the
electrons of the crystal. In this case the γ-ray losses energy and is also defected, this way,
it could escape from the detector without loss all its energy, thus, the detector will register
a count for a value of energy which is lower than the one of the initial γ-ray. Photoelectric
effect could also occur. In this case the γ-ray losses all its energy inside the detector and it
will generates a count in a value of energy which corresponds with the γ-ray energy. The
cross section, σ, of each one of these processes depends on which it is called the attenuation
coefficient µ in the following way
σ =ω
NA
(
µ
ρ
)
. (4-3)
4.2 Gamma-ray detectors 29
Where ω is atomic weight, NA is the Avogrado’s number, µ is the attenuation coefficient
and ρ is the density of the material. As it can be seen from Equation (4-3) the cross section
depends on the factor(
µρ
)
which has units of(
cm2
g
)
. Ge-detectors are widely used in nuclear
structure experiments and for this reason is important to know how important is each process
when γ-radiation interacts with germanium. Figure 4-5 shows the(
µρ
)
factor of cross section
of the different processes when γ-radiation interacts with germanium.
10-4
10-3
10-2
10-1
1
10
102
103
104
10-3 10-2 10-1 1 10
µ/ρ
(cm
2 /g)
Energy (MeV)
ComptonPhotoelectric
Pair productionTotal
Figure 4-5.:(
µρ
)
factor (proportional to the cross section) of different processes in γ-germanium
interaction.
When a γ-ray of energy Eγ interacts by Compton effect with an electron, the energy E ′γ
of the γ-ray after the interaction is given by
E ′γ =
Eγ
1 + ǫ(1− cos(θ))with ǫ =
Eγ
mec2. (4-4)
From Equation (4-4), me represent the electron mass and c is the velocity of light. The
energy, Er, registered by the detector will be the difference between the initial and final
energy of the γ-ray.
Er = Eγ − E ′γ = Eγ
ǫ(1− cos(θ))
1 + ǫ(1− cos(θ)). (4-5)
The energy, Er, reach its maximum value when θ = 180, this value is given by Equa-
tion (4-6).
Er−max = Eγ2ǫ
1 + 2ǫ. (4-6)
30 4 Experimental methods
Figure 4-6.: Spectrum of a 60Co source took with a Ge-detector the Compton edge energies for the
two energies of the peaks (1173 and 1332) are labeled.
Because of Compton effect is present in the detection process, a typical γ spectrum of a
Ge-detector is like what it is shown in Figure 4-6 for a 60Co source which emits two γ-rays
at energies of 1173 and 1332 keV.
In Figure 4-6 the peak corresponds to photoelectric effect and for this reason it is called
photopeak. The counts in the photopeak are located at the energy of the γ-ray that hits
the detector. In this case the γ-ray leaves all its energy inside the detector. The counts in
the region labeled as “Compton region” correspond to the energy that the γ-ray losses when
the Compton effect takes place, it is, Er, from Equation (4-5). In this case the γ-ray does
not leave all its energy in the detector, and a count is added in an undesired region of the
spectrum. The edge of the “Compton region” is given by the Equation (4-6). For γ-rays at
energies of 1173 and 1332 keV, as the ones in Figure 4-6, the values of Er are 963 and 1118
keV respectively. These values are located in the spectrum of Figures 4-6 and 4-8.
The Compton region can be suppressed using a technique in which a γ-ray, that is de-
flected by Compton effect, can be detected by another detector surrounding the Ge-detector,
in the way that is shown in Figure 4-7.
An incident γ-ray that is deflected by Compton effect (red line in Figure 4-7) can be
detected by a BGO detector. This detector is connected in coincidence with the Ge-detector,
that way, the events detected by the Ge-detector in coincidence with an event detected in
the BGO detector will be suppressed from the final spectrum. The difference between a
spectrum took by a Ge-detector when is used a Compton suppressor is shown in Figure 4-8.
From Figure 4-8 can be seen that the Compton region has less counts when a suppressor
is used. However in this last case the effect is still present. These counts could be due to
a multiple scattering in the Ge-detector or it could be due to the γ-ray deflected, was not
4.2 Gamma-ray detectors 31
Valid event
Ge Detectors
supressor detectorsBGO Compton
ComptonPhotoelectric
PhotoelectricSupressedevent
Figure 4-7.: Compton suppressor.
Figure 4-8.: Spectrum of a 60Co source with a Ge-detector. At left the complete spectrum of the
comparison with and without the use of a Compton suppressor detector. At right the
Compton region is shown in more detail.
detected by the BGO suppressor. BGO detectors have a high efficiency and Ge-detectors have
high energy resolution. These are the reason because the Compton suppressor detector from
Figure 4-7 use a BGO as a suppressor and a Ge as detector to register the final spectrum.
The following subsections will explain the concepts of energy resolution and efficiency of a
detector.
4.2.1. Energy resolution
The resolution in energy varies for different types of detectors. In nuclear structure it is
needed to have high energy resolution because of the high interference that is present in
the spectra produced in a experiment. Depending on the reaction, many excited states of
the same nucleus are populated. This fact combined with the large amount of nuclei that
are produced in the experiments, produce a lot γ-rays which have energies that are between
a few keV’s and go up to around 3000 keV (due to bound energy of the nucleons). In the
experiments analyzed in this work around 100 nucleus were created and more than 2000
32 4 Experimental methods
different energy γ-rays were emitted. Ge-detectors has up to date the better resolution in
energy for γ-ray detection. Figure 4-9 shows a comparison between the spectra generated
by NaI-detector and Ge-detector.
1100 1150 1200 1250 1300 1350 1400Energy (keV)
0
50
100
150
200
250Co
unts 1332 (keV)
1173 (keV)
NaIGe
Figure 4-9.: Comparison between spectra of a 60Co source with a Ge and NaI detectors.
From Figure 4-9 you can see the difference in energy resolution between the Ge and NaI
detectors. Ge-detector has clearly higher resolution than the NaI. To quantify the resolution
in energy, the FWHM (Full Width at Half Maximum) could be used. Figure 4-10 shows
a comparison between the FWHM of Ge, NaI and BGO detectors. For NaI two different
dimensions in the crystal of the detector are shown. 2×2 and 3×3, where the first number
indicates the length of the cylinder and the second number indicates the diameter of this
cylinder. Both values are indicates in inches.
From Figure 4-10 can be seen that the Ge-detector has the lower FWHM, so, it has the
highest energy resolution. However this type of detectors has a low efficiency compared with
scintillator detectors.
4.2.2. Efficiency
For γ-ray detectors could be defined two different types of efficiency, the first one is called
“geometric efficiency (ǫgeo)”. It is due to the fact that not all the γ-rays emitted by a source
reaches the detector. Usually the detector just cover a few degrees of solid angle, as is shown
in Figure 4-11. When a γ-ray hits the detector it has a probability to deposit its energy inside
the detector and, eventually, this γ-ray could be not detected. For this reason the second
type of efficiency is defined as “intrinsic efficiency (ǫint)(Eγ)”. This last type of efficiency
depends on the γ-ray energy. ǫgeo and ǫint are defined below.
4.2 Gamma-ray detectors 33
0
20
40
60
80
100
120
140
160
0 200 400 600 800 1000 1200 1400
FW
HM
(ke
V)
Energy (keV)
GermaniumNaI(3X3)NaI(2X2)
BGO
Figure 4-10.: Comparison between FWHM of the Ge, NaI and BGO detectors.
Figure 4-11.: Solid angle ∆Ω covered by a detector.
ǫgeo =number of γ-rays that reach the detector
number of γ-rays emitted by the source. (4-7)
ǫint(Eγ) =number of γ-rays registered by the detector for an specefic energy
number of γ-rays that reach the detector for an specefic energy. (4-8)
And the total efficiency, ǫtot, is given by
ǫtot = ǫgeo · ǫint =number of γ-rays registered by the detector
number of γ-rays emitted by the source. (4-9)
The effects of the intrinsic efficiency have to be corrected when the spectra produced in
the experiments will be analyzed.
In order to obtain physical results from Gasp experiment the raw data has to be read
and sorted. This process will be explained in the following section as well as the analysis
needed to construct a level scheme and to obtain the products of the reaction in Gasp and
Prisma-Clara experiments.
5. Data analysis
5.1. Construction of a level scheme from Gasp
experiment
Once the end of the experiment is reached it is necessary to perform an offline data analysis.
The raw data must be read at first. It implies the implementation of a numerical code which
enables to watch the raw data. The experiment is performed in the so-called “runs”. A “run”
is a data set taken during a space of time during which the experimental setup is not mo-
dified. Different runs are created in order to check if the experiment is stable and working,
and to generate files which do not have a large size. However, the data of each “run” have a
size ∼ 1 GB and the computer is not able to process all this data at the same time. Then
the reading process has to be performed by data blocks with size of ∼ 32 kilobytes, in order
to process only one block at time. Each experimental set-up in nuclear physics has its own
data format. The beginning of the header of each block in Gasp experiment looks as follow
E B E V E N T D
Record ID = 7531
Run number = e2
header length = 0
Tape number = 0
Record length = 0x7fc2
Record length = 32706.
The first line says “E B E V E N T D” which means “EuroBall Event Data”. The se-
cond line gives an identifier of the record, the third line give us the information about which
block is being reading at that moment, and the last two lines gives the information about
the length of the block which is ∼ 32000, this number could vary slightly After this header
comes the data from the experiment itself. These data look as follows.
f1ff 1a00 0000 0000 0616 7b05 5802 0816
e904 5802 2616 0000 8502 f1ff 2000 0000
0000 0316 0000 7e02 0a16 ed01 e201 1a16
8501 1102 2416 c705 e901 f1ff 1a00 0000
5.1 Construction of a level scheme from Gasp experiment 35
0000 0816 f204 ec01 1016 0205 5702 2516
f502 6f01 f1ff 2000 0000 0000 0016 0000
Each “word” here has a length of 2 bytes expressed as a hexadecimal number. The first
word f1ff is a separator between events. An event is recorded if at least 2 detectors regis-
tered a count, this is call the trigger signal. It is an important fact because these events will
allow the construction of the γγ and γγγ coincidence matrices on which physical properties
of the nuclei will be studied. The number after the separator gives the length of each event.
The next two words are always 0. When a detector is shot it registers the information of
the identifier of the detector, the energy of the γ-ray detected and the time when it was
detected. So the first event has the following words.
f1ff = Event separator
1a00 = (Length of event in bytes)
0000
0000
(06)16 = (Identifier of the first germanium detector)16
7b05 = Energy registered by the first germanium detector
5802 = Time registered by the first germanium detector
(08)16 = (Identifier of the second germanium detector)16
e904 = Energy registered by the second germanium detector
5802 = Time registered by the second germanium detector
(26)16 = (Identifier of the third the germanium detector)16
0000 = Energy registered by the third germanium detector
8502 = Time registered by the third germanium detector
The number 16 is used to check if the word is an identifier of a detector. The number
of detectors shot in each event can be calculated starting from the length of the event. The
number after the separator indicates 2 times the length of the event in bytes. To calculate
the number of germanium detectors that were shot in that particular event this number must
be divided by 2 (bytes of each word), then subtracting the bytes of the next two words, it is,
subtracting the number 4 and finally dividing by 3 because each event has 3 different words
(Identifier, energy and time). For example for the first event the length is 1a00 = 26, so
the number of detectors shot will be,
Number of detectors shot =26/2− 4
3= 3. (5-1)
This data format is very complicated to work with and, in addition to this, the energy
registered is not calibrated as well as the time information. For these reasons the first step
is to generate another files of data that will suppress information that is no needed such as
36 5 Data analysis
the headers of the blocks. This reduction of data is called presort. These new data set will
be calibrated in energy and time. These data has the following look
ffff 0002 0011 07ea
0299 0024 0870 0235 ffff 0004 0009 06ea
0279 0014 0ebe 0240 0018 0fce 0268 0026
0639 028c ffff 0003 0003 02e6 027b 000a
0893 0236 0020 089d 020b ffff 0003 0003
02e6 01a0 0010 0893 025b 0018 089d 0212
For example the first event is
ffff = Event separator
0002 = Length of event in bytes
0011 = Identifier of the first germanium detector
07ea = Energy registered by the first germanium detector
0299 = Time registered by the first germanium detector
0024 = Identifier of the second germanium detector
0870 = Energy registered by the second germanium detector
0235 = Time registered by the second germanium detector.
With these new data the spectra of the individual energy and time signals for each one
of the 40 Ge detectors, has to be generated. The identification word by word of the data in
an event will allow to understand the construction of the γγ and γγγ coincidence matrices,
that will be explained in the following subsection.
5.1.1. γγ coincidence matrix
Data from each event (defined in the previous section) can be sorted by pair of coincidences.
When a nucleus is generated in the reaction it has an excitation energy and it will decay
emitting γ-rays. When these γ-rays are detected, the energy and time information is obtained.
When at least two γ-rays are detected into the time window, it is called, a simple coincidence.
If more gamma rays are detected in an event, then there is a multiple coincidence. With the
coincidences sorted by pairs, a 2-dimensional matrix can be generated adding a count in
coordinates (Eγ1 ,Eγ2) and also a count in the coordinates (Eγ2 ,Eγ1).
For example if the 3 different events shown in Figure 5-1 were detected from the γ
radiation emitted by the same nucleus. Then the γγ coincidence matrix is constructed as it
is shown in Figure 5-1.
This matrix is symmetric by definition and has a great importance in nuclear structure
studies. The time that requires a nucleus to decay from its excited states to its ground state
is the order of few nanoseconds. The time response of the electronics is slightly faster than it.
5.1 Construction of a level scheme from Gasp experiment 37
Figure 5-1.: Scheme of γγ coincidence matrix constructed from the 3 events shown.
So two events in coincidence have a high probability of belong to the same nucleus. From this
matrix the events which are in coincidence with any energy can be selected, and a spectrum
can be generated. A particular selection of an energy in the γγ coincidence matrix is called a
“gate”. In this thesis the matrices were analyzed using the UPAK [28], GASPware [29] and
Radware [30] package analysis codes.
As an example, if a γ-ray energy of 1750 KeV is selected and if the spectrum of all
coincident γ-rays is generated, then the spectra shown in Figure 5-2b is obtained.
The 1750 KeV transition corresponds to the most intense line emitted by 96Zr which is
the nucleus in the beam of the experiment. The γ-ray energies shown in the spectrum of
Figure 5-2b are emitted by the 96Zr nucleus. On the other hand the time window is small
enough to see only transitions of one nucleus at the time when the coincidence technique is
used. Figure 5-3 shows the level scheme of 96Zr taken from [31], the energies are observed in
the spectrum of Figure 5-2b. The γγ coincidence technique is a powerful tool to study one
specific nucleus generated in a reaction due to the possibility to distinguish the γ radiation
of an specific nucleus. There is another technique which can give more detailed information
about radiation emitted by a nucleus, this is the generation of the γγγ coincidence matrix.
5.1.2. γγγ coincidence matrix
If at least three different detectors are fired in the time window then a count on the 3-
dimensional matrix with coordinates (Eγ1,Eγ2,Eγ3) is added to generate the γγγ coincidence
matrix. When a triple coincidences matrix is built, a cleaner spectra of a nucleus can be
38 5 Data analysis
200 400 600 800 1000 1200 1400 1600Energy (keV)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Coun
ts × 106
1750 (96Zr)
Total spectrum
(a) Total spectrum of radiation.
(b) γ-rays in coincidence with a γ-ray of 1750 keV,
from the γγ coincidence matrix.
(c) Double gated spectra at energies of 1750 and
915 KeV, from the γγγ coincidence matrix.
Figure 5-2.: Gamma-ray spectra showing transitions belonging to the 96Zr nucleus.
5.1 Construction of a level scheme from Gasp experiment 39
1750
1107
915
617
518
831
508
215
361
1185 1222
364
1115751
456 336
906
1095
146
0 0
2 1751
4 2857
6 3772
8 4389
(10 ) 4907
(11 ) 5738
(12 ) 6246
(13 ) 6461
(14 ) 6822
31897
4
3082 53119
63483
74234
10469011
(10 ) 5484
96Zr
Figure 5-3.: Level scheme of 96Zr proposed in Ref. [31]. Most of the transitions here are observed
in Figures 5-2b and 5-2c
generated, showing just a band of the level scheme. A band in a level scheme is form by a
set of states correlated temporally via γ-decay, in which all the states have the same parity.
If from this matrix, the events which are in coincidence with two selected energies in a
band of a nucleus are selected, then all the other energies in this band should be visualized,
as long as the counts are inside of the defined time window. Figure 5-2c shows a double gated
spectrum at energies 1750 and 915 from the γγγ coincidence matrix. The other energies of
this band in the level scheme in Figure 5-3 can be seen in Figure 5-2c.
The stable 96Zr nucleus was the beam of the reaction which makes experimentally con-
venient the study of this nucleus as a first test of the data. For this reason the results from
40 5 Data analysis
Figures [5-2b- 5-2c] show that the results of the experiment are working as it is expected.
In this work the products of the reaction has to be founded in order to determine which
nuclei are suitable to study with the data obtained from the experiments. Then the objec-
tive is to choose a nucleus and use the γγ and γγγ coincidence matrices to increase the
information of the current level scheme in that nucleus that in this case will be 95Nb.
5.1.3. Angular correlations
In Section 2.4.2 was shown how different multipolar radiation produces different angular
distributions of the energy emitted. Here the experimental technique to measure the multi-
polarity of the radiation will be explained. If the nuclear spin is aligned along one particular
axis, the γ-ray angular distribution associated to a specific state, with a given spin, could
provide information about the multipolarity of the γ-ray radiation. However, in grazing reac-
tions this is not expected to happen, but the γγ coincidence technique can be utilized to
solve this problem. Consider three successive γ-rays as shown in Figure 5-4. If these γ-rays
are observed in coincidence, then they were probably emitted by the same nucleus.
I3
I0
I1
I2
Randomly populated(unoriented state)
Oriented state
Intermediateoriented state
γ1
γ2
γ0
Figure 5-4.: Three successive γ-rays emitted from the same nucleus and the definition of an oriented
state.
The presence of γ0 in Figure 5-4 ensures that the orientation of the lower substates I1and I2 is the same. This important fact together with the coincidence technique allows the
determination of the multipolarity of the emitted radiation. As it was stated in Chapter 2
the angular distribution of the radiation emitted by any multipole has azimuthal symmetry,
so the angular dependence can be expressed as function of just the angle θ.
5.1 Construction of a level scheme from Gasp experiment 41
For a cascade of three successive γ-rays I0γ0−→ I1
γ1−→ I2
γ2−→ I3 as it is shown on Figure 5-
4, three γγ coincidence matrices can be generated with a common gate on the γ0 energy.
The presence of γ0 in each event to be included in the analysis generates an alignment of
the lower sub-states and it is also useful to resolve the interference between closely spaced
transitions.
For the analysis of the radiation emitted by the 95Nb nucleus three two-dimensional
coincidence matrices were generated with a common gate on γ0, when the γ-rays γ1 and
γ2 were detected in a pair of detectors with separation angles θ. For the first matrix the
separation angle correspond to θ = (90 ± 10). The second matrix contains the sum of the
events detetected at separation angles of θ = (120± 10) and θ = (60± 10). For the third
matrix the events with θ = (120 ± 10) and θ = (60 ± 10), were registered. Then making
a gate on energy of γ2 the number of counts of γ1 were calculated. This precedure gives
the numbers N(90), N(120) and N(150). The angular distribution of γ1 is described by the
function
W (t1, t2, θ) =λmax∑
λ
qλAλ(t1, t2)Pλ(cos(θ)), (5-2)
where t1, t2 denotes the properties of the transitions γ1 and γ2 and the spins of the levels
that they connect. As transitions with multipolarity higher than λ = 4 are very unlikely
to happen in the states populated trough grazing reactions, here λmax = 4. The function
W (t1, t2, θ) used for the analysis in this thesis was
W (t1, t2, θ) = q0A0(t1, t2) + q2A2(t1, t2)Pλ(cos(θ)) + q4A4(t1, t2)Pλ(cos(θ)). (5-3)
The coefficients Aλ were calculated in Ref. [32] for different types of transitions and have
the values shown in Table 5-1.
Table 5-1.: Predicted angular correlations coefficients for cascades Q-Q and D-Q. Q, denotes a
quadrupole transition and D, denotes a dipolar transition. Taken from Ref. [32]
Cascade A2/A0 A4/A0
4Q−→ 2
Q−→ 0 0.102 0.009
3D−→ 2
Q−→ 0 -0.071 0
The attenuation coefficients qλ in Equation (5-3) were calculated in Ref. [33] using a well
known E2-E2 cascade and fitting the numbers N(90), N(120) and N(150) to the theoretical
function (5-3) for a E2-E2 cascade. This procedure gives the numbers q0 = 1.0, q3 = 0.909,
q0 = 0.602 for the Gasp array and they take into account the finite size of the detectors
and the effects of choosing ± 10 as the range for the separation angle between pair of
detectors. A cascade of three well known E2 γ-rays emitted by the 96Zr nucleus (the beam
42 5 Data analysis
of the experiment) at energies of 617, 915 and 1107 keV were used to obtain normalization
factors for the values N(θ). The coefficients A2/A0 were calculated for different γ-rays the
level scheme of 95Nb. The results will be shown in Chapter
5.2. Products of the reaction
As it was stated in the Chapter 2, after the reaction (2-3) occurs a couple of nuclei are
produced. Different couples of nuclei are produced and some of them are produced in lar-
ger amounts than others. The production of each nucleus depends on the nuclear reaction.
Despite the reaction was the same in Prisma-Clara and Gasp experiments, the experi-
mental setups are very different and it is expected to found differences in the products of
the reaction. The information about what nuclei were produced in each experiment gives a
powerful tool to combine both experiments in an optimal way and start to study the nuclear
structure of the produced nuclei. For these reasons it is important to do an experimental
characterization of the reaction over each one of the experiments. The next subsection shows
the case of the Prisma-Clara experiment.
5.2.1. The Prisma-Clara experiment
In this experiment the yields of the reaction are provided directly by the Prisma magnetic
spectrometer when a value of charge is selected in the ∆E−E detectors shown in Figure 4-1.
The products of the reaction with the highest production in this experiment are presented
in Figure 5-5.
From Figure 5-5 it can be seen that the number of counts is lower for the products
which are losing or capturing one proton compared with the isotopes of the beam. The
number of isotopes produced decreases with the number of protons losed or acquired. The
number of counts in Figures 5-5 is proportional to the production of each nucleus. This
production represents how many data it are available to the study of each nucleus. Thus,
the characterization gives information of which nuclei are more suitable to study from the
data of this experiment.
In order to compare the production of the nuclei in the Prisma-Clara experiment with
Gasp experiment, a fitted over each Gaussian in Figure 5-5 must be done. These fits are
shown in red in Figure 5-6 for the niobium (Nb) isotopes.
The integral of each one of the Gaussians in Figure 5-6 gives a number which is propor-
tional to the production of each nucleus. This production can be compared with the products
of the reaction in Gasp experiment that will be shown in the following subsection.
5.2 Products of the reaction 43
Figure 5-5.: Products of the reaction with the highest production in Prisma-Clara experiment.
Figures in the upper part correspond to the isotopes produced when the projectile,9640Zr, losses one or two protons in the reaction. Figures in the lower part represent the
production of the isotopes which are obtained when the projectile captures one or two
protons.
44 5 Data analysis
92 93 94 95 96 97 98 99 100 101Mass
0
2
4
6
8
10
12
14
16
18
Counts × 103
Z = 41 (Nb)
Figure 5-6.: Mass distribution in Prisma-Clara experiment for Niobium (Nb) isotopes. The red
lines are Gaussian fits to each one of the peaks in the spectrum.
5.2.2. The Gasp experiment
The determination of the yields of the reaction in the Gasp experiment starts from the total
spectrum of the experiment which contains the events registered in all experiment. From this
spectrum the background has to be removed. This background is present in the experiment
for several reasons such as electronic noise and non Compton suppressed signals. Figure 5-7
shows this background subtraction.
From Figure 5-7 you can identify the energies 1133 and 1750 keV which are the γ-rays
that goes to the ground state for 124Sn and 96Zr, the target and the beam of the reaction
respectively. From Figure 5-7 you can also see that the spectrum after background subs-
traction is a flat spectrum and the protuberance at low energies has disappeared. From this
last spectrum the efficiency correction has to be made, and for that purpose the efficiency
calibration of the experiment is required. Sources of 152Eu and 133Ba were used in the ca-
libration. Figures 5-8 and 5-9 show the spectra and the energies used for the efficiency
calibration.
Some energies from the 133Ba and 152Eu sources were selected for the calibration taking
into account energies distributed along all the spectrum. Several energies in the region of
150-300 keV were also selected because in this region the efficiency function has a strong
variation. This will be shown later. To make the efficiency calibration it is needed to know
the branching of decay (bγ), it is, the probability that a γ-ray transition had to be emitted
compared with other possible transitions. This probability is calculated dividing the number
of γ-rays with an specific energy that were emitted by the nucleus between the total events
of decay. These branching ratios are shown in Table 5-2 for the energies of the γ-rays utilized
5.2 Products of the reaction 45
Figure 5-7.: Subtraction of background from the total spectrum of the Gasp experiment. At left
the total spectrum along with the background are shown. At rgiht the spectrum after
the background subtraction is shown. The γ-rays emitted when the target, 124Sn, and
the beam, 96Zr, go to the ground state are located.
0 100 200 300 400 500Energy (keV)
0
1
2
3
4
5
6
7
8
9
Coun
ts × 106
80
160223
302276
386
356
133Ba source
Figure 5-8.: Spectrum of a 133Ba source, took using Gasp experiment. The lines used for the
efficiency calibration are identified.
46 5 Data analysis
0 200 400 600 800 1000 1200 1400Energy (keV)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Coun
ts ×
106
121
244
344
411
778
867
964
1112
1408
1212
152Eu source
Figure 5-9.: Spectrum of a 152Eu source, took using Gasp experiment. The lines used for the
efficiency calibration are identified
in 133Ba and 152Eu sources.
With the branching ratios of the Table 5-2 the total efficiency (ǫtot) of the Equation (4-9)
can be calculated from each source separately using the expression (5-4),
ǫtot(Eγ1) =number of γ1 detected
A∆tΩbγ1. (5-4)
In Equation (5-4) the number of γ1 detected cab be calculated as the Gaussian integral of a
peak in Figures 5-8 and 5-9. bγ1 are the numbers in Table 5-2 for each energy. A represent
the source activity,
A =number of decays
unit of time, (5-5)
∆ t represent the time of measure and Ω represents the solid angle covered by the complete
detector array. The quantity A∆tΩ is the same for all the energies emitted from the same
source. So it will be called K,
K = A∆tΩ (5-6)
Thus the number,
ǫrel(Eγi) =number of γi detected
bγi, (5-7)
gives a measure about the relative efficiency of different γ-rays emitted at different energies
by the same source. However the value of K in Equation (5-6) is different for each source,
because the activity, A, of the sources of 133Ba and 152Eu, were not the same.
5.2 Products of the reaction 47
Table 5-2.: Decay branching for each energy utilized in the efficiency calibration. At the top, ener-
gies used from the 133Ba source are shown. At the bottom, energies from the 152Eu
source are shown.
Source Eγ (keV) bγ (%)133Ba 80 34.1
160 0.7
223 0.5
276 7.1
302 18.3
356 61.9
383 8.9152Eu 121 28.4
244 7.5
344 26.6
411 2.2
778 12.9
867 4.2
964 14.6
1112 13.5
1212 1.4
1408 20.8
In this work just the relative efficiency is of interest. The values for the 133Ba and 152Eu
sources can be normalized choosing the relative efficiency at an specific energy, in both cases,
and calculating the ratio,
ǫrel(Eγi)(133Ba)
ǫrel(Eγi)(152Eu)
. (5-8)
If each value, i, of relative efficiency obtained from the 133Ba source (ǫrel(Eγi)(133Ba)) is multi-
plied by the number (5-8), the resulting values will be relative efficiency of this source norma-
lized to the relative efficiency of the 152Eu source. These vales will be called ǫrel(Eγi)(133Ba)Eu,
it is,
ǫrel(Eγi)(133Ba)Eu =
ǫrel(Eγ1)(133Ba)
ǫrel(Eγ1)(152Eu)
(ǫrel(Eγi)(133Ba)). (5-9)
A function can fitted to the values ǫrel(Eγi)(133Ba)Eu and ǫrel(Eγ1)(
152Eu) in order to
obtain a number for the relative efficiency to any energy. The result of this fit is shown in
Figure 5-10. The functional shape of the curve in Figure 5-10 is given in Ref. [34],
Eff(Eγ) = exp
[
(
(
a+ bE1 + cE21
)−g+(
d+ eE2 + fE22
)−g)− 1
g
]
(5-10)
48 5 Data analysis
0 500 1000 1500 2000Energy (keV)
0.0
0.2
0.4
0.6
0.8
1.0
Efficiency
152Eu and 133BaEfficiency (GASP)
Figure 5-10.: Relative efficiency for the Gasp experiment. The line is the efficiency fit to fun-
ction (5-10)
with E1 = log
(
Eγ
100
)
and E2 = log
(
Eγ
1000
)
.
In Equation (5-10) a, b, c, d, e, f, g are parameters which were fitted. This function is just
the one that has shown fitted correctly the efficiency of germanium detectors. The relation
between the mathematical form of the Equation and the characteristics of the crystal is an
study area in solid state physics. In this work just to obtain the relative efficiency for any
value of energy is important.
From Figure 5-10 can be seen that the γ-rays with low energies are more efficiently
detected than the γ-rays at high energies. However this is not true for energies lower than
∼ 200 keV where is located the maximum of efficiency in Figure 5-10. If the number of
counts of a spectrum is divided by the function in Figure 5-10, then the spectrum will be
corrected by efficiency and the effect to detect some energies more efficiently than other
ones, is corrected. If the efficiency correction is performed over the total spectrum from the
Figure 5-7, it is obtained the spectrum shown in Figure 5-11.
The effect of apply an efficiency correction can be visualized better if a zoom at high
energies is performed. This zoom is shown in Figure 5-12. In Figure 5-12 can be seen that
the number of counts increase after the efficiency correction is made. This effect is due to
the value of the efficiency function for these energies is lower than 1 (see Figure 5-7) and
the spectrum corrected by efficiency is divided by the function in Figure 5-7.
The determination of the products of the reaction in this experiment will be made over
the spectrum in Figure 5-12. The idea is identify the energy of the γ-ray that is emitted
when some nucleus goes to the ground state. For example 92Sr nucleus emits a γ-ray of
5.2 Products of the reaction 49
500 1000 1500 2000 2500 3000 3500Energy (keV)
0
1
2
3
4
5
6
Counts × 106
96Zr (Beam)
124Sn (Target)
With efficiency correction
Figure 5-11.: Total spectrum in Gasp experiment obtained applying the function in Figure 5-10
to the spectrum at the right in Figure 5-7. The γ-rays emitted when the target,124Sn, and the beam, 96Zr, go to the ground state are located.
1620 1640 1660 1680 1700 1720 1740 1760Energy (keV)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Counts ×
106
96Zr (Beam)
1676 KeV 95Zr
1640 KeV 97Nb
Without efficiency correctionWith efficiency correction
Figure 5-12.: Comparison between the total spectrum in Gasp experiment with and without effi-
ciency correction. Just the high energies are shown.
50 5 Data analysis
815 keV, as can be seen from the partial level scheme of Figure 5-13. In the case of 92Sr
Figure 5-13.: Partial level scheme of 92Sr according [35]. The transition which goes to the ground
state has an energy of 815 keV.
the peak which correspond to 815 keV has to be found in the spectrum of Figure 5-11. A
Gaussian has to be fitted to this peak, but the spectrum in Figure 5-11 contains all the γ-rays
emitted by all the nuclei produced in the experiment. So this is a spectrum which has a lot of
interference in most of the energies. For this reason a superposition of several Gaussians have
to be generated in order to fit some region of the spectrum. A region surrounding 815 keV
is initially fitted to the three Gaussians shown in Figure 5-14. One of the three Gaussians
804 806 808 810 812 814 816 818Energy (keV)
0
2
4
6
8
10
12
14
Counts × 105
Too wide!!
813 94Zr 815 92Sr
Figure 5-14.: Region of the spectrum surrounding the 815 keV. Three Gaussians are fitted to this
region of the spectrum.
in Figure 5-14 seems to be wider than the other ones. A very good fit to this spectrum
has been found (see the red line over the spectrum in Figure 5-14), however this fit could
be produced by Gaussians which not corresponds just to one γ-ray energy. To try to solve
5.2 Products of the reaction 51
this problem the FWHM calibration can be made. This gives the correct width of each peak
depending on its energy. The FWHM for the Gasp detector array is shown in Figure 5-15.
0 500 1000 1500 2000Energy (keV)
1.0
1.5
2.0
2.5
3.0
FWHM (KeV)
152Eu and 133BaFWHM (GASP)
Figure 5-15.: FWHM of the Gasp detector array.
Applying the calibration of Figure 5-15 to the fitted code, such that, Gaussians with
the correct fit for each energy are utilized, a new fit can be obtained. This fit is shown in
Figure 5-16. The fit in Figure 5-16 contains four Gaussians, it is, one more than the previous
fit in Figure 5-14. The first peak is actually produced by two γ-rays of different energies
(807 and 809 keV). The integral of a peak which corresponds to γ-ray emitted when the
nucleus goes to its ground state is proportional to the production of that nucleus, because
all the nuclei are produced at excites states and any excited state in any nucleus always
decays to its ground state. The production of the projectile-like products of the reaction in
Gasp experiment can be obtained making one fit, similar to the one in Figure 5-16, for
each projectile-like nucleus produced in the reaction. This production can be compared with
the production in Prisma-Clara experiment. This comparison is analyzed in the following
Chapter.
52 5 Data analysis
804 806 808 810 812 814 816 818Energy (keV)
0
2
4
6
8
10
12
14
Counts × 105
807 116Cd
809 89Rb
813 94Zr 815 92Sr
Figure 5-16.: Region of the spectrum surrounding the 815 keV. Four Gaussians, taking into account
the calibration in Figure 5-15, are fitted to this region of the spectrum..
6. Results
6.1. Products of the reaction from the Gasp and the
Prisma-Clara experiments
The production of the projectile-like products of the reaction obtained in both experiments
can be visualized making a plot for each different set of isotopes (different values of the Z
number). The Zr (Z=40) isotopes produced in both experiments as well as a comparison
with the production of the expected yields in a pure deep inelastic reaction at a grazing
angle (which was calculated using the code GRAZING [15]) are shown in Figure 6-1.
85 90 95 100 105 110A
0.2
0.4
0.6
0.8
1.0
Prod
uctio
n (A.U.)
Z = 40 (Zr)Prisma/CLARAGASPGRAZING code1
Figure 6-1.: Comparison between the production of isotopes of Zr in both experiments. A compa-
rison with a numeric calculation is also shown.
In order to make a direct comparison between both experiments, and also with the
GRAZING calculations, a normalization has been conducted over the mass distributions in
Figure 6-1. All the production values have been divided by the production of the nucleus
with the larger production in each case. For this reason the y-axis in Figure 6-1 is shown
in arbitrary units (A. U.). Figure 6-1 shows that there is a shift between the centers of the
mass distribution for the Prisma-Clara and the Gasp experiments. The mass distribution
54 6 Results
from the Gasp experiment shifts to the left, to the less neutron-rich side, with respect to the
Prisma-Clara distribution. At the same time, the distributions shown by Prisma-Clara
present a good agreement with the grazing code predictions. In the initial proposal of this
thesis it was expected to study the Rb isotopes 90,92,94Rb. this was a proposed based on the
predictions of the GRAZING [15] code calculations, which give the result shown with red
dashed line in Figure 6-2.
80 85 90 95 100 105A
0.2
0.4
0.6
0.8
1.0
Prod
uctio
n (A.U.)
Z = 37 (Rb)(Zr -3p)
Prisma/CLARAGASPGRAZING code1
Figure 6-2.: Comparison between the production of isotopes of Rb in both experiments. A compa-
rison with a numeric calculation is also shown.
All the mass distributions from Figure 6-2 are shifted. The distribution from the Gasp
experiment is shifted to the left of all of them, this is, to the less neutron rich side. In the
middle is located the mass distribution from the Prisma-Clara experiment; and finally, to
the right, it is located the prediction made by GRAZING calculations. Based on GRAZING
calculations the Rb isotopes 90,92,94Rb must have a high production. However the results
of Figure 6-2 shows very small values for the production of these isotopes in the Gasp
experiment. Besides looking at the γγ and γγγ coincidence matrices, the transitions which
belongs to these nuclei does not have good statistics.
Both experiments were conducted using the same reaction, 9640Zr +
12450 Sn at Elab = 530
MeV. However Gasp is a thick target experiment and Prisma-Clara is a thin target
experiment. GRAZING calculations are made for products of the reactions that are produced
in pure deep inelastic reaction at a grazing angle. The nuclei produced at a grazing angle
are the ones with the largest number of neutrons. Because the angular acceptance of the
Prisma-Clara experiment (∆Θ ∼ 12 and ∆φ ∼ 22) just the nuclei, and its respective
γ-rays, that were produced at angles surrounding the grazing angle (See section 3.2.1.) were
detected. Gasp experiment detects the γ rays produced by the nuclei that were produced
6.1 Products of the reaction from the Gasp and the Prisma-Clara experiments 55
at any angle. The differences between the results of Prisma-Clara arrays and the Gasp
set-up is a clear example of the power of the Prisma-Clara for the selection of neutron-rich
channels of the reaction, due to the possibility of selecting specific grazing angle with a very
good A/∆A discrimination.
Figure 6-2 also shows a shift between the GRAZING predictions and the results from
Prisma-Clara experiment. This effect is due to the angular acceptance that makes that
some nuclei, with less neutron that the expected at a grazing angle, were produced. This
behavior is also observed for the other projectile-like products of the reaction as it is shown
in Figure 6-3.
Figure 6-3.: Yields of the reaction for some of the projectile-like products. The production was
normalized and it is shown in arbitrary units (A.U.). Figures in the upper part corres-
pond to the isotopes produced when the projectile, 9640Zr, captures one or two protons
in the reaction. Figures in the lower part represent the production of the isotopes
which are obtained when the projectile loses one or two protons.
Figure 6-3 shows the projectile-like products of the reaction with the highest production
56 6 Results
in both experiments. For the Gasp experiment some isotopes are difficult to distinguish, due
to its low production and the interference in the spectrum due to transitions with higher
intensities from different isotopes. As example, see the isotopes of Nb (Z=41) with a mass
number lower than 94 or higher than 97. Other example is the 99Mo nucleus.
The shift to the left of the distributions of the Gasp, the Prisma-Clara experiments
and the GRAZING code calculations respectively are also present for all these products.
However, the distributions shown by Prisma-Clara present very low shift when compared
with the grazing code predictions, especially the isotopes that capture protons from the
beam, Nb (Z = 41) and Mo (Z = 42).
The experimental characterization of the reaction gives information about what nuclei
and in which amount were produced in both experiments. This information is a powerful
tool to establish a criterion of which nuclei are suitable to study in each experiment, and
also which nuclei have the better combination in data of both experiments. For example at
the top right of the Figure 6-3 are located the isotopes of Nb (Z=41). From this Figure it
can be seen that the isotope with the highest production in the Gasp experiment is 95Nb.
A study of this nucleus will be shown in the next section.
6.2. Level scheme of 95Nb
Figure 6-4 shows the double-gated spectrum at 870 keV and 873 keV which belong to the95Nb level scheme. From the spectrum in this figure the background has been subtracted and
Figure 6-4.: Double-gated spectra from the Gasp experiment. The double gate was performed at
γ-ray energies of 870 keV and 873 keV. γ-rays energies at 795 and 1233 keV are new
lines added to 95Nb level scheme, in this work.
6.2 Level scheme of 95Nb 57
it is also corrected by efficiency. From this spectrum it can be seen most of the lines reported
in Ref. [23], as well as two more peaks at energies of 795 keV and 1233 keV. These two lines
have two possible explanations. Either they could be emitted by the partner nucleus of 95Nb,
as well as its neighbors, or they could also be new transitions found for this nucleus. The
intensity of these lines is lower than most of the other lines in the spectrum. It is possible
that it cannot be seen in previous experiments because of this low intensity. To try to assign
these lines to the 95Nb nucleus the Prisma-Clara experiment can be used. When a mass
number, A, and a charge, Z, are selected in the Prisma-Clara experiment a spectrum of
γ-rays detected can be constructed. Figure 6-5 shows this spectrum for 95Nb nucleus. Two
regions of the same spectrum have been selected in order to visualize the lines of interest.
Figure 6-5.: Spectrum from Prisma-Clara experiment gated at Z=41 and M=45. At left energies
lower than 1000 keV are shown. At rigth energies higher than 1000 keV are shown.
The spectrum of Figure 6-5 shows most of the lines in the level scheme of Figure 6-6.
It confirms that these lines belongs to the 95Nb nucleus. The two lines observed at energies
of 795 and 1233 keV cannot be seen clearly in the Prisma-Clara spectrum from Figure 6-
5. The low intensity of these two lines combined with the setback of the Prisma-Clara
experiment of low statistics, generates that they cannot be seen in the spectrum of Figure 3-
1. However γ rays emitted from the partner nucleus and neighbor nuclei were checked to
ensure that the γ rays belong to 95Nb. In this work it is propose to place the γ rays in the
positions shown in Figure 6-6.
The width of the lines in the level schemes in Figure 6-6 represent the intensity of
each γ-ray energy. This intensity is calculated by adding all the counts of each γ-ray from
the Gasp experiment. γ-rays at the top of the level schemes have lower intensity than γ-
rays at the bottom of the level scheme. This difference in intensities is due to the nuclei
are generated at different excitation energies, and nuclei at high excitation energies have a
smaller probability to be generated than the low excitation energies. Figure 6-6 shows the
58 6 Results
Figure 6-6.: Right: Level scheme as proposed in the present work. Left: Level scheme proposed in
Ref. [23].
6.2 Level scheme of 95Nb 59
level scheme proposed in this thesis. Two new transitions with energies of 1233 keV and 795
keV were added. Transitions with energies at 676.2 and 872.6 keV proposed in Ref. [23] were
not observed in this work.
From the data of the Gasp experiment spins and parities can be proposed making
angular correlations from the detected γ-rays [32]. Using the technique exposed in section
4.1.3 the angular distribution of the γ-rays in the level scheme in Figure 6-6 can be measured
to determine spins and parities of the levels. Figure 6-7 shows the angular distribution of
the radiation for two successive γ-rays at different energies in the level scheme of Figure 6-6
as well as the theoretical prediction for a two successive γ-rays from the type E2.
From Fig. 6-7 it can be seen that the shape of the angular distribution of the 1068-
873 keV, 873-870 keV, and 825-825 keV γ-rays is the same, within uncertainties, to the
function which describes two successive transitions of the type E2-E2. However for the Figure
describing the cascade 870-679, the theoretical prediction for a E2-E2 cascade present the
greatest difference with the experimental curve. From 95Nb nucleus two successive γ-rays are
emitted at energies of 870 and 873 keV. These are two values very close which can produce
interference in the determination of the number of counts for each angle. Nevertheless the
curve has a value of A2/A0 positive, so it can not be a cascade from the type E2-E1. In
Table 6-1 the energy levels, energy γ-rays and γ-ray intensities are shown.
Ef Ei Eγ Iγ
0.0 825.3 825(1) 100
825.3(12) 1650.6(12) 825(1) 100(3)
1650.6(13) 2328.6(13) 678(1) 62(2)
2328.6(13) 3199.0(13) 870(1) 31(2)
3199.0(13) 4072.3(13) 873(1) 21(2)
4072.3(14) 5140.7(14) 1068(1) 10(2)
5140.7(14) 5644.0(14) 503(1) 9(4)
5644.0(14) 6487.0(14) 843(1) 15(4)
6487.0(15) 7492.0(15) 1005(1) 11(4)
7492.0(15) 8694.9(15) 1203(1) 10(4)
4072.3(16) 5305.4(16) 1233(2) 1.0(7)
5644.0(16) 6439.4(16) 795(1) -
Table 6-1.: Energies of excited states of 95Nb together with transition energies (Eγ),and relative
intensities (Iγ) of γ-rays placed in the level scheme in Fig. 6-6.
In Table 6-2 The ratios A2/A0 are shown for the lowest γ-rays placed in the level scheme
in Fig. 6-6.
The coefficients A2/A0 were calculated for different γ-rays in the lowest states of the
level scheme of 95Nb. Highest γ-rays of this level scheme cannot be analyzed because of the
low statistics. A special discussion of the ratios A2/A0 in Table 6-1 is carried out below, for
6.3 Shell model calculations 61
Ef Ei Eγ1 Eγ2 A2/A0a γ-ray multipolarity Jπ
f Jπi
γ1 γ2
0.0 825.3 825(1) 825(1) 0.076(25) E2 E2 9/2+ 13/2+
825.3(12) 1650.6(12) 678(1) 825(1) – (E2) E2 13/2+ 17/2+
1650.6(13) 2328.6(13) 870(1) 678(1) 0.049(30) E2 (E2) 17/2+ (21/2+)
2328.6(13) 3199.0(13) 873(1) 870(1) 0.092(50) E2 E2 (21/2+) (25/2+)
3199.0(13) 4072.3(13) 1068(1) 873(1) 0.152(70) E2 E2 (25/2+) (29/2+)
Table 6-2.: Energies of excited states of 95Nb together with transition energies (Eγ1 and Eγ2),
A2/A0 ratios, γ-ray multipolarity, spins and parities of the levels of the lowest γ-rays
placed in the level scheme in Fig. 6-6.
aTheoretical value of A2/A0 for transitions of the type E2-E2 conecting levels with spins 2 −→ 0 is 0.102
each γ-ray, to explain the spins and parities proposed in Table 6-1.
825 keV : The behavior of the angular correlations for the γ-ray doublet can be seen in
Fig 6-7. This figure together with the value of A2/A0 reported at the top in Table 6-1 allow
us to asign spins and parities of the first two excited states as 13/2+, 17/2+. These values
are also in agreement with the shell model calculations shown in Table 6-4.
678 keV : The presence of a γ-ray doublet of 825 keV in the lower states of the level
scheme does not allow us to determine angular correlation for this specific γ-ray. For this
reason the predictions of the spins and parities for excited states above excitation energy of
2328.6 keV, must be confirmed. However the consistency with the values of the two neighbor
γ-rays could indicate an E2 transition.
870 keV : For this γ-ray the angular correlation with the 678 keV γ-ray transition gives
a value of 0.049(30), a bit far to the theoretical value for two successive E2-E2 transitions
(0.102). However angular correlations between the γ-rays at energies of 870 and 873 keV
gives a value of 0.092(50), which leads us to propose an E2-E2 cascade for γ-rays at energies
of 870 and 873 keV. In the same way spins and parities of (21/2+), (25/2+) for the levels at
energies of 2328.6, 3199.0 are proposed.
873 and 1069 keV : The values for these two γ-rays are in agreement with transitions
from an E2-E2 cascade. Spins and parities proposed are shown in Table 6-1.
6.3. Shell model calculations
The 9541Nb nucleus is located 4 neutrons to the right of the neutron magic number 50 and one
proton up of the proton semi-magic number 40. For these reasons a single particle behavior
is expected. To make shell model calculations for the 9541Nb nucleus it was assumed 88
38Sr50as an inert core. Thus the 95
41Nb54 nucleus is considered to have 3 valence protons and 4
valence neutrons. The valence space used is shown in Figure 6-8. The valence orbitals used
62 6 Results
20
28
50
82
20
28
50
82
2p1/21f5/22p3/2
1f7/2
1d3/2
2s1/2
1d5/2
1g9/2
1h11/22d3/23s1/21g7/22d5/2
2p1/21f5/22p3/2
1f7/2
1d3/2
2s1/2
1d5/2
1g9/2
1h11/22d3/23s1/21g7/22d5/2
ProtonsNeutrons
Valence space
Valence protons
Inert core
Valence space
Valence neutrons
Inert core
External space
Figure 6-8.: Inert core, valence neutrons and protons, and valence spaces for the case of 9541Nb54
nucleus.
for protons and neutrons as well as the single particle energies relative to 8838Sr50 (taken from
Refs. [36, 37]) are shown in Table 6-3.
Valence neutron Single particle Valence proton Single particle
orbitals energies (keV) orbitals energies (keV)
2d5/2 -6359 2p1/2 -6160
1g7/2 -3684 1g9/2 -7069
3s1/2 -5327
2d3/2 -4351
1h11/2 -4280
Table 6-3.: Single particle energies relative to the 8838Sr50 inert core (taken from Refs. [36, 37]) for
the valence space used to study the 9541Nb nucleus.
The Oslo code was utilized to make the shell-model calculations [38]. The effective in-
6.3 Shell model calculations 63
teraction was constructed based in the CD-Bonn nucleon-nucleon interaction described in
Ref. [39].
The levels with the lowest energy for given angular momentum J form the so called
yrast line (yrast is a swedish word which means the dizziest). Most of the nuclear reactions
populate just the yrast and near yrast sates. Grazing reactions in particular are one of them.
The yrast and near yrast energy levels calculated from shell model calculations are shown
in Figure 6-9.
Figure 6-9.: Excited states predicted by shell model calculations using the code described in
Ref. [38].
From Figure 6-9 different transitions are possible but other ones are forbidden by se-
lection rules of angular momentum composition. Some transitions have higher probability
than other ones depending on the angular momentum difference between levels and the mul-
tipolarity character of the possible emissions. Some of the predicted levels and the reduced
probability transitions, B(E2), found are shown in Table 6-4.
The energies of the first two excited states, reported in Table 6-4, are in very good agree-
ment with the experimental values in Table 6-1. These values are even in better agreement
than the previous ones calculated in Ref. [23]. However higher excited states are not well
predicted by the calculations made in this work, in Table 6-4 the three first excited states
are shown. Effective interaction utilized as well as the presence of other effects like pair brea-
king in some shell of the 8838Sr50 nucleus may generate the disagreement of the calculations
64 6 Results
Theoretical Experimental
Ei Ef Eγ B(E2 ↓) (W.u.) Ei Ef Eγ Jπi Jπ
f
0.0 850.6 850.6 23.5 0.0 825.3 825.3 9/2+ 13/2+
850.6 1733.8 883.2 26.6 1650.6 825.3 825.3 13/2+ 17/2+
2694.0 1733.8 960.2 2.3 2328.6 1650.6 678.2 21/2+ 17/2+
Table 6-4.: Level energies (keV) together with transition energies (Eγ (keV)) for the γ-rays with
the highest B(E2), predicted by shell model calculations.
with the experimental data. Further efforts in this direction should be made to clarify this
disagreement.
In Table 6-5 the average occupation number for valence protons and neutrons in the
valence orbitals are shown. From this table the evolution of these numbers can be extracted.
Almost all the numbers have a general behavior with the excitation energy for the first excited
states in Table 6-5. The numbers for the protons in the orbital 2p1/2 decreases because
protons goes to the higher energy orbital 1g9/2 in order to generate the first excited states.
On the other hand the occupation numbers for neutrons in the 1h11/2 2d5/2 orbitals decreases
meanwhile the neutrons in the 1g7/2, 2d3/2 and 3s1/2, increases, when higher excited states
are generated. These general behaviors are found for almost all the excited states reported in
Table 6-5, being the state at energy of 2694.0 the only one that presents a different behavior
in the neutron orbitals 1g7/2, 2d5/2 and 2d3/2. In Figure 6-10 it can be seen the comparison
of the energy levels between the experimental and the theoretical values. From this figure it
is possible to see the good agreement of the calculations with the experimental data for the
first two excited states. This agreement justify the selection of the 8838Sr50 as an inert core for
the 9541Nb54 nucleus. On the other hand the higher energy level reported in Table 6-5, has the
larger disagreement with the experimental values. There are several possible explanations
for this difference. The pairing energy included in the code can be different of the real value.
The 8838Sr50 inert core could be broken, as well as the valence space for neutrons and protons
can change at excitation energies higher than 1733.8. Future analysis are require to confirm
or rule out these hypothesis.
Protons Neutrons
Level energy (keV) J 1g9/2 2p1/2 1h11/2 1g7/2 2d5/2 2d3/2 3s1/2
0.0 9/2 2.785 0.215 0.223 0.792 2.227 0.434 0.324
850.6 13/2 2.884 0.116 0.166 0.845 2.131 0.485 0.374
1733.8 17/2 2.947 0.053 0.140 1.003 2.059 0.486 0.312
2694.0 21/2 2.944 0.056 0.145 0.921 2.253 0.420 0.261
Table 6-5.: Occupation number for valence protons and neutrons in the valence orbitals for pre-
dicted excited states using the Oslo code [38].
6.3 Shell model calculations 65
Figure 6-10.: Comparison of excited states obtained from the Gasp experiment and the theore-
tical values obtained by shell model calculations. At the left the predicted excited
states obtained by the calculations carried out in this work is shown. The effective
interaction used is described in Ref. [39]. At the middle are the experimental excited
states obtained in this work. At the right are the predicted axcited states using the
effective interaction described in Ref [40]. The values in green represent the difference
between the predicted energy level by the calculations carried out in this work and
the experimental value. The numbers in red represent the difference between the pre-
dicted energy level by the calculations carried out in Ref. [23] and the experimental
value
7. Conclusions and perspectives
In this work an experimental study of the reaction 96Zr + 124Sn using two different expe-
rimental arrays, Prisma-Clara and Gasp respectively, was presented. The capabilities of
the Prisma-Clara array were utilized, in a thin target experiment, to populate neutron
rich nuclei using deep-inelastic transfer reactions. Neutron-rich nuclei under study cover a
wide area in the south-west intersection of Z = 50 and N = 82 region. An additional thick
target experiment was performed using the Gasp array to complement the spectroscopic
information obtained with the Prisma-Clara setup.
The 95Nb nuclei, with N = 54 and Z = 41, provides a good example of the combination
of Prisma-Clara and Gasp. The nucleus is close to the sub-shell closure Z = 40 and the
closed shell N = 50, these provide a good opportunity to explore the magicity of the 88Sr
(Z = 38, N = 50). In this work a level scheme is proposed for 95Nb and the results are
interpreted with the help of shell-model calculations using a 88Sr closed core. In this work
the 95Nb was extended, two new transitions were added to the previous level scheme and
spin and parities were confirmed for the first two excited states.
The characteristics of the experiments provides excellent opportunities to perform furt-
her investigations, it is worth to mention that
the data analysis of the experiments should continue in order to perform an experi-
mental characterization of the target-like products of the reaction, and to explore other
effects, such as the neutron emission.
The first excited states of 95Nb are well predicted by shell model calculations using a88Sr as a core. Other Niobium isotopes also have a high production and are located
near magic and semi-magic numbers N = 50 and Z = 40. A further analysis can be
done over these isotopes in order to study the evolution of the shell model in the region
with Z = 41 and N ≥ 52.
Around 100 nuclei in the region could be studied from the data of the Gasp and the
Prisma-Clara experiments. The characterization of the reaction of both experiments,
along with the γγ and the γγγ coincidence matrices that were generated in this work,
can be utilized in further analysis. It will allow to perform an experimental study of
the evolution of shell structure in the region.
B. Appendix: Contribution to the
proceedings of the XXXVI RTFNB(XXXVI Reuniao de Trabalho sobre Fısica Nuclear no Brasil)
C. Appendix: Contribution to the X
LASNPA proceedings(Latin American Symposium on nuclear physics and applications)
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