Experimental Study of the reaction - 40Zr + 124 GASP array

 
Experimental Study of the reaction - 40Zr + 124 GASP array
Experimental Study of the reaction
96Zr + 124Sn at 530 MeV using the
40      50
           GASP array

       Wilmar Rodrı́guez Herrera

          Universidad Nacional de Colombia
                Facultad de Ciencias
               Departamento de Fı́sica
                  Bogotá, Colombia
                        2014
Experimental Study of the reaction - 40Zr + 124 GASP array
Experimental Study of the reaction - 40Zr + 124 GASP array
Experimental Study of the reaction
96Zr + 124Sn at 530 MeV using the
40      50
           GASP array

                   Wilmar Rodrı́guez Herrera

 Master’s thesis submitted in partial fulfillment of the requirements for the degree of:
                                Magister en Fı́sica

                                     Supervisor:
                        Ph.D., Diego Alejandro Torres Galindo

                                   Research area:
                                  Nuclear structure
                                   Research group:
                 Grupo de Fı́sica Nuclear de la Universidad Nacional

                          Universidad Nacional de Colombia
                                Facultad de Ciencias
                               Departamento de Fı́sica
                                  Bogotá, Colombia
                                        2014
Experimental Study of the reaction - 40Zr + 124 GASP array
Experimental Study of the reaction - 40Zr + 124 GASP array
Dedicated

            to my parents

            Whose unconditional support has allowed me to
            reach this point.
Experimental Study of the reaction - 40Zr + 124 GASP array
Experimental Study of the reaction - 40Zr + 124 GASP array
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 “Dirección académica” 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.
Experimental Study of the reaction - 40Zr + 124 GASP array
Experimental Study of the reaction - 40Zr + 124 GASP array
ix

    Abstract

  In this thesis an experimental study of the binary nuclear reaction 96        124
                                                                        40 Zr + 50 Sn 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 95
                                                                   41 Nb 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 caracterización experimental de la reacción nuclear 96       124
                                                                                        40 Zr+ 50 Sn
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 área
de núcleos ricos en neutrones con número de protones menores al número mágico Z = 50 y
número de neutrones mayor al número mágico N = 50, usando reacciones de transferencia
múltiple de nucleones. El análisis de los datos es realizado mediante espectroscopı́a de rayos γ.

    La producción experimental de los núcleos en cada uno de los experimentos es presen-
tada. Resultados en el estudio de estados yrast y yrast-cercanos para 95
                                                                      41 Nb son presentados
junto con una comparación con predicciones hechas por cálculos de modelo de capas.

   Palabras clave: Espectroscopı́a de rayos Gamma, Modelo de Capas, Núcleos Ricos
en Neutrones, Reacciones Deep Inelastic, Estructura Nuclear
Experimental Study of the reaction - 40Zr + 124 GASP array
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
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

         95
3. The        Nb 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 95 Nb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   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 95 Nb 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 95 Nb, which was the object under study in this thesis, as well as the
latest studies carried out about 95 Nb 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 95 Nb 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 (Å). 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 95 Nb 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− + ν̄e ), 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, 124
                                                                    50 Sn and the beam
                                                                                         96 Zr
                                                                                         40      of the
             reaction are located as well as the 95 Nb and the 125 In.

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 95 Nb 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 96       124
                                                                       40 Zr + 50 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
                         
         Zp Zt               θ
d=                 1 + csc        ,                                                      (2-1)
        4πǫ0 Ek              2
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,
                      
                   1/3
d = 1.2 A1/3
           p   + A t     .                                                                  (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,
96
40 Zr   +124
         50 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

           A              A   A
           X            1X X
HExact =         Ti +               Vij (|~
                                          ri − r~j |).                                         (2-4)
                        2   j=1j6=i
           i=1            i=1

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

                                         50
                                                      VN−N(r) Schematic
                            VN−N (MeV)
                                         0

                                     −50

                                         0.0   0.5   1.0        1.5   2.0    2.5
                                                       r (fm)
                          Figure 2-4.: Scheme of the shape of nucleon-nucleon potential.

   A     A                                A
1 XX                       X
       Vij (|~
             ri − r~j |) ≈   V (ri ).                                                        (2-5)
2  j=1
  i=1                                    i=1

     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

          A
          X               A
                          X
HSM =              Ti +         V (ri ).                                                     (2-6)
             i=1          i=1

    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 .
                    A    A                  A
                 1 XX                       X
Hresidual =             Vij (|~
                              ri − r~j |) −   V (ri ).                                         (2-7)
                 2  j=1
                   i=1                     i=1

If the model is suitable to describe the nucleus it is expected that
hHresidual i ≪ hHSM i.                                                                         (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.
                                                     (
                                                       −V0 , if r ≤ R0
Square well =⇒                               V (r) =                                    (2-9)
                                                       0,     if r > R0
                                                          "          2 #
                                                                   r
Harmonic oscillator =⇒                       V (r) = −V0 1 −                           (2-10)
                                                                  Roa
                                                                   −V0
Woods Saxon =⇒                                      V (r) =                 .                (2-11)
                                                              1 + exp r−R
                                                                         0
                                                                        a

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

                                              
                                3              3
ǫnℓ = h̄ω0       2(n − 1) + ℓ +     = h̄ω0 N +     .                                          (2-12)
                                2              2
     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
            1 dV (r) ~ ~        ~ · S,
                                    ~
Hℓs = V0′            L · S = V0 L                                                             (2-13)
            r dr
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
          A            A h                         i
                                             ~   ~
          X            X
 ′
HSM   =         Ti +         V (ri ) − |V0 | L · S .                                     (2-15)
          i=1          i=1

The potential, V (r), of the Equation (2-15) can be written as
        (
          V + |V0 | 21 (ℓ + 1), if j = ℓ − 12
V (r) =                                                                                  (2-16)
          V − |V0 | 12 ℓ,       if j = ℓ + 12 .

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, J = ℓ − 1/2i

                              |n, ℓi                      ∆ǫℓs

                                                                   |n, J = ℓ + 1/2i

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
                             (
                                j = ℓ + 12
                 
                3      −ℓ
ǫnℓ = h̄ω0 N +                                                                       (2-17)
                2    (ℓ + 1)    j = ℓ − 1.2

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
                                                                  1/2
         1     2                      2     2                M ω0
                                                                        r and κµ = µ′ .
                                              
HM O = h̄ω0 ρ − κh̄ω0 2ℓ · s + µ ℓ − hℓ iN with ρ =
         2                                                      h̄
                                                                                        (2-18)

The energy levels generated by the potential of Equation (2-18) are given by
                                                                  (
                                                                     j = ℓ + 12
                                      
                  3           ℓ                       N (N   + 3)
ǫN,ℓ,j = h̄ω0 N + − κ                 − µ′ ℓ(ℓ + 1) −                                      (2-19)
                  2        −(ℓ + 1)                        2         j = ℓ − 1,   2

     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, h̄ω0 ≈ 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

                                                                      82      1h11/2
                                               3s                             2d3/2
                                                                              3s1/2
                                               2d                             1g7/2
                        N=4
                                                                              2d5/2
                        κ = 0.06               1g
                        µ’ = 0.024                                    50
                                                                              1g9/2

                                                                              2p1/2
                                               2p                             1f5/2
                        N=3                                                   2p3/2
                        κ = 0.075              1f
                        µ’ = 0.0263                                   28
                                                                              1f7/2

                                                                     20
                                                                              1d3/2
                        N=2                 2s + 1d                           2s1/2
                        κ = 0.08                                              1d5/2
                        µ’ = 0.0

                                                              
                        Harmonic -µ′ h̄ω0 ℓ2 −      N (N +3)
                                                                   −2κh̄ω0 ℓ · s
                        oscillator                     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 40  20 Ca, 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, 288 O has
and energy gap of 2.5 MeV. The 28 O nucleus has 10 neutrons more than the stable isotopes
of 188 O, 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 95 Nb, 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 95 Nb 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 95 Nb will be positive. It is written
using the typical notation in nuclear physics as

J π = 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 for
95    54
41 Nb    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 2p1/2
and the pairing energy of the neutron in the 2d5/2 shell, different possible configurations are
possible for the first excited state of 95
                                        41 Nb54 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 95
   41 Nb54 nucleus.

      Inert core; the nucleus composed by nucleons filling completely lower shells. For 95
                                                                                        41 Nb54 ,
                             88
      the inert core can be 38 Sr50 .

      Valence nucleons; nucleons in higher shells than the ones of the inert core 88        95
                                                                                  38 Sr50 . 41 Nb54
      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 95
         41 Nb54 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 95 41 Nb54 considered as a
           88
sum of the 38 Sr50 inert core plus 4 valence neutrons and 3 valence protons. A particular
16                                              2 Preliminary concepts on nuclear structure

                 Neutrons                                              Protons

                            82       1h11/2                       82      1h11/2
                                     2d3/2                                2d3/2 External space
 Valence space                       3s1/2                                3s1/2
                                     1g7/2                                1g7/2
 Valence neutrons                    2d5/2                                2d5/2

                            50                                   50
                                    1g9/2                                 1g9/2
                                                                                  Valence space
                                    2p1/2                                 2p1/2
                                    1f5/2                                 1f5/2 Valence protons
 Inert core                         2p3/2                                 2p3/2
                                                                                Inert core
                            28                                   28
                                    1f7/2                                 1f7/2

                            20                                   20
                                     1d3/2                                1d3/2
                                     2s1/2                                2s1/2
                                     1d5/2                                1d5/2

Figure 2-8.: Inert core, valence neutrons and protons, and valence spaces for the case of   95 Nb
                                                                                            41    54
             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 → Hef f ψef f = Eψef f ,                                                                 (2-22)
where Hef f and ψef f 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 95 Nb 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 Jiπi and a final state with spin
              π
and parity Jf f , a γ-ray can be emitted with a total angular momentum jγ and parity πγ .
This process is illustrated in Figure 2-9.

                                Ei              J i πi
                                                                     j γ πγ

                                                                     Eγ = Ei − Ef
                                                     πf
                                Ef              Jf

Figure 2-9.: Quantum numbers in a γ transition. Ei and Ef are the enegies of the initial and the
                                    π
             final state. Jiπi and Jf 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
         π
numbers Jf 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 = πf πγ (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)
       j+1
(−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.

                                            J i πi
                                                                  E2

                                            9/2+

  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 Ji
is 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,
Jiπi = 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

                                                                           m2
                                                      
        1       m(m + 1)             2  1      m(m − 1)
Zℓ,m =      1−              |Yℓ,m+1 | +     1−             |Yℓ,m−1 |2 +          |Yℓ,m |2 , (2-33)
        2        ℓ(ℓ + 1)               2       ℓ(ℓ + 1)                ℓ(ℓ + 1)

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].
          1           1                       3
Z1,0 (θ) = |Y1,−1 |2 + |Y1,1 |2 = |Y1,1 |2 =    sin2 (θ)                                   (2-34)
          2           2                      8π

                    (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.

          1          1                       15
Z2,0 (θ) = |Y2,1 |2 + |Y2,−1 |2 = |Y2,1 |2 =    cos2 (θ) sin2 (θ)                          (2-35)
          2          2                       8π

    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 95 Nb nucleus will
be shown.
    The following subsection describes the current state of the excited states of 95 Nb measu-
red by γ-ray spectroscopy. These excited states are represented in nuclear physics as a level
scheme.
3. The 95Nb nucleus
95
41 Nb  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 95 Mo. The number of protons of 95   41 Nb 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 88  38 Sr nucleus, which is emplyed as a standard
closed-core shell [5], a single-particle behavior is expected.
     Previous experimental studies of 95 Nb 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 95 Nb 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

12
     Ca +82 Se at Elab = 38 MeV                                                            (3-1)
16       82
     O + 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 16 O and 12 C contaminants from the target of the reaction
82
     Se +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 95 Nb ground
state are

J π = 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
95
22                                                                 3 The        Nb nucleus

     Figure 3-1.: Level scheme of   95 Nb   proposed in ref [23]
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
     95
the Nb 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.
                                                 96          124
                                                 40 Zr   +    50 Sn
                                                                      Prisma-Clara       Gasp
              Target (124                     2
                       50 Sn) thickness (mg/cm )                            0.3             8
           Thickness of the backing target (mg/cm2 )                    0.04 of 12 C   40 of 208 Pb
                     Beam energy (MeV)                                      530           570 a
                 Number of working detectors                               25/25          38/40
                                            124
 a
     The beam energy at the middle of the    50 Sn   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.

                                                             Magnetic
                                                              dipole

Clara detector
    array                                       Magnetic
                                                quadrupole
                                                                         Focal
                             Start detector                              plane
          Target                                                        detector
          124 Sn

     Target−like    Projectile−like                                      ∆E-E
                                                                         detectors

                     Beam
                   530 MeV   96 Zr

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
      mv
ρ=       .                                                                               (4-1)
      Z
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   mZ 2
   ∝      ,                                                                              (4-2)
dx    E

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

                                                           BGO Compton
                                                        supressor detectors
                                     Beam
                            96 Zr   at 574 MeV
                                        Target
                                        124 Sn
                                                              Ge Detectors

                                       20 cm

                           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

         
   ω     µ
σ=          .                                                                             (4-3)
   NA    ρ
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 ofthe
                           material. As it canbe 2seen
                                                      from Equation (4-3) the cross section
depends on the factor µρ which has units of cmg . Ge-detectors are widely used in nuclear
structure experiments and for this reason is important to know howimportant
                                                                            is each process
                                                                    µ
when γ-radiation interacts with germanium. Figure 4-5 shows the ρ factor of cross section
of the different processes when γ-radiation interacts with germanium.

                                   104
                                                                  Compton
                                   103                       Photoelectric
                                                            Pair production
                                                                      Total
                                   102
                     µ/ρ (cm2/g)

                                    10

                                     1

                                   10-1

                                   10-2

                                   10-3

                                   10-4 -3
                                      10       10-2       10-1         1      10
                                                      Energy (MeV)
                
                µ
Figure 4-5.:    ρ   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 Eγ′
of the γ-ray after the interaction is given by

               Eγ                   Eγ
Eγ′ =                     with ǫ =       .                                                    (4-4)
        1 + ǫ(1 − cos(θ))          me c2

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.

                                ǫ(1 − cos(θ))
Er = Eγ − Eγ′ = Eγ                              .                                             (4-5)
                              1 + ǫ(1 − cos(θ))

The energy, Er , reach its maximum value when θ = 180◦ , this value is given by Equa-
tion (4-6).

                   2ǫ
Er−max = Eγ             .                                                                     (4-6)
                 1 + 2ǫ
30                                                                   4 Experimental methods

Figure 4-6.: Spectrum of a 60 Co 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 60 Co 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
You can also read