experimental study of the reaction 96zr + 124sn at … · experimental study of the reaction 96...

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

Dedicated

to my parents

Whose unconditional support has allowed me to

reach this point.

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)


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.


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)


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.


|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


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.


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)


2.3.3. 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


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)


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


(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

22 3 The 95Nb nucleus

Figure 3-1.: Level scheme of 95Nb 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

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


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)


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


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


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.


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.


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.


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.


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


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.


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


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


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

60 6 Results

Figure 6-7.: Angular correlations for different γ−rays of the level scheme 6-6.

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-


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].


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.

A. Appendix: Contribution to the

Legnaro National Laboratory.(Annual Report 2011)

68 A Appendix: Contribution to the Legnaro National Laboratory.

B. Appendix: Contribution to the

proceedings of the XXXVI RTFNB(XXXVI Reuniao de Trabalho sobre Fısica Nuclear no Brasil)

70 B Appendix: Contribution to the proceedings of the XXXVI RTFNB

72 B Appendix: Contribution to the proceedings of the XXXVI RTFNB

C. Appendix: Contribution to the X

LASNPA proceedings(Latin American Symposium on nuclear physics and applications)

74 C Appendix: Contribution to the X LASNPA proceedings

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