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UNIVERSIDAD NACIONAL DE COLOMBIA

Nuevo Metodo Hk-EOS para la

Investigacion del Continuo Nuclear:

Evaluacion de su capacidad con datos simulados y

experimentales

Edana Merchan

Trabajo presentado para optar al tıtulo deDoctor en Ciencias Fısica

Facultad de CienciasDepartamento de Fısica

Bogota, Colombia2009

http://www.unal.edu.co

http://www.ciencias.unal.edu.co

http://www.fisica.unal.edu.co)

UNIVERSIDAD NACIONAL DE COLOMBIA

New Hk-EOS Method for theInvestigation of the Nuclear Continuum:

Performance evaluation with simulated andexperimental data

Edana Merchan

Advisor:

Dr. Fernando Cristancho Mejıa

Faculty of Sciences

Bogota, Colombia2009

http://www.unal.edu.co

“Yesterday is history, tomorrow is a mystery, today is a gift.That is what it is called the present.”

Master Ogway

Abstract

The subject of this work is to develop new experimental tools in the research ofthe nuclear continuum. Presently, there is a lack of experimental methods thatpermit to obtain reliable information about nuclear structure at high excitationenergy and high angular momentum that involve predicted shape phase transitionsmaybe observed in some nuclei [1]. Considering the complexity of the experimentalmethods proposed here it is of practical interest to simulate the processes that leadto the prediction of the applicability of the Hk-EOS method.

The simulation carried out includes a complete description of the fusion-evaporationreaction processes and the interaction of the radiation with the γ detection deviceselected. It provides access to the understanding of the results of such experiments.The detection theory was studied in detail to be able to understand and describethe experimental considerations to apply the method during a possible experiment.The evaluation of the method was done for simulated data of 154Dy which allowsa detailed evaluation of the (M,E) regions that can be selected from the (k,H)pairs and the calculation of the level density parameter a for those regions. Finallyanother test was done using the experimental data of 60Ni.

Resumen

El tema del presente trabajo es desarrollar herramientas experimentales para lainvestigacion del continuo nuclear. Debido a la falta de metodos experimentalesque permiten obtener informacion acerca de la estructura nuclear a alta energıa deexcitacion y alto momento angular en donde se pueden presentar las transicionesde forma ya predichas por las teorıas para ciertos nucleos [1]. Considerando lacomplejidad de los metodos experimentales propuestos aqui es de intres practicosimular los procesos que llevan a prediccion de la aplicabiblidad del metodo Hk-EOS.

La simulacion llevada acabo incluye una descripcion completa de los procesos de lareaccion de fusion-evaporacion y la interaccion de la radiacion con el sistema dedeteccion gama seleccionado. esto a permitido una aproximacion al entendimientode los resultados de dichos experimentos. La teorıa de la deteccion fue estudiadaen detalle para poder entender y describir las consideraciones experimentales paraaplicar el metodo durante un posible experimento. La evaluacion del metodo fuerealizada para los simulados de 154Dy lo cual permite una evaluacion detallada delas regiones (M,E) que pueden ser seleccionadas de las parejas (k,H) y el calculodel parametro densidad de niveles a para esas regiones. Finalmente otra pruebafue realizada usando los datos experimetales de 60Ni.

Dedicated to my family.

Acknowledgements

Trough all these years I have learned a few things, not only about physics butabout life. I was always thinking about the future and trying to prepare and makeplans. Now I think that even if you have a dream what really matters are thedecisions you take in decisive moments. I am very glad I took the decision to makethe Ph.D. at “GFNUN” it took me exactly where I dreamt to be. It was in 2001that I began to work in the group and as an undergrad student and I always thinkabout the group as my other home, I spend more time at the “CIF” offices thatanywhere else. I am very grateful with all the people I work with there, startingof course with my advisor, Fernando Cristancho; I really admire you and I hopesome day I become as good as you are in your work, thank you very much for allthe knowledge you share with me and all the patience you had teaching me andguiding me trough all this process.

To my colleague and dear friend Pico with whom I share great times and all myscientific and not too scientific problems, thank you; you are an excellent friend,travel mate and a splendid person to work with I hope we can meet again and sharea different work environment in some exotic place. Special thanks to Zandra whohelped me in the final weeks; I doubt I could finished without your help, companyand support. Of course there are many people at the group that I am gratefulwith, but they are so many that I am just going to say that it was a pleasure tomeet and work with every person that is or was in the group with me.

This journey also brought me to many different places. I had the pleasure towork with the people of the Nuclear Structure Group at the “Lund University” inSweden. Thanks to Dirk Rudolph to allow me to have that experience of workingwith such nice people and to know the environment of the real experiments. Specialthanks to Emma and Lise-Lotte for being great friends and share with me specialmoments. Also, I had the opportunity to work at one of the greatest institutionsin nuclear physics, the “GSI” in Germany, there I met many nice people speciallythanks to Jurgen Gerl for hosting me and showing me the beautiful Germany.

The Ph.D. also brought one special person into my life, Eduardo; I hope you canunderstand how important you are to me, thank you for all the special moments,the nice words to cheer me up in the difficult moments and all the help with mywork. You were always beside me and I love you for that, you will have a specialplace in my heart for ever.

Also thanks to my best friends that maybe some of them are not close anymorebut are very important to me Cristina, Janneth (Lola), Nancy, Carolina, Claudiaand Jhonnatan.

xi

I want to say that the most important persons in my life are my family, they havealways been there for me and I love them, so thanks to my mother, my father, mysisters and brothers, specially Andres, my aunts including aunts in law and all mycousins. All of them have always supported me in many different ways and I amvery lucky to have such an amazing family.

Finally I want to thank COLCIENCIAS for the financial support of my Ph.D. andthe six months internship in Germany-Sweden. And the IAEA for the financialsupport of the three months internship in Germany. Moreover, thanks to the CIFfor the excellent work environment and the “Universidad Nacional de Colombia”for the great opportunity of being part of it as a student during all these years.

CONTENTS

Abstract v

Resumen vii

Acknowledgements xi

1 Introduction 1

2 Theoretical Background 3

2.1 Heavy-ion fusion-evaporation reactions . . . . . . . . . . . . . . . . 3

2.2 Level densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Gamma strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Nuclear structure and thermodynamics . . . . . . . . . . . . . . . . 10

3 Simulation 15

3.1 Projection Angular momentum CoupledEvaporation (Pace) . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Gamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Gamma detection device (Gasp) . . . . . . . . . . . . . . . . . . . 19

3.3.1 Gasp array simulation with Geant4 . . . . . . . . . . . . 20

3.3.2 γ-rays and neutron interaction . . . . . . . . . . . . . . . . 21

3.3.3 Spectroscopic characteristics . . . . . . . . . . . . . . . . . 22

3.3.4 Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . 24

xiii

Contents xiv

4 The Hk-EOS method 27

4.1 Primary radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 The Hk technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 The Energy-Ordered Spectra technique . . . . . . . . . . . . . . . . 30

4.4 The influence of the response function . . . . . . . . . . . . . . . . 33

5 Detection theory and experimental considerations 35

5.1 Probability theory of Hk distributions . . . . . . . . . . . . . . . . 35

5.2 Detection efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.3 Moment expansions for PN (ME; kH) . . . . . . . . . . . . . . . . 40

5.3.1 The k fold distribution PN (ME; k) . . . . . . . . . . . . . . 40

5.3.2 The total pulse height H distribution PN (ME;H) . . . . . 41

5.4 Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.5 The M → I conversion . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.6 Gasp characterization . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.7 Response functions PN (ME, kH) of Gasp . . . . . . . . . . . . . . 48

5.7.1 Construction of the entry regions . . . . . . . . . . . . . . . 50

5.7.1.1 The zeroth-order approximation . . . . . . . . . . 51

5.7.1.2 The Least-Squares Orthogonal Distance Procedureto fit (k,H) distributions . . . . . . . . . . . . . . 53

5.7.1.3 Interpolation process . . . . . . . . . . . . . . . . 55

5.8 Generation of reverse responses P ′N (kH,ME) . . . . . . . . . . . . 57

5.8.1 Iterative least-squares unfolding procedure . . . . . . . . . . 58

6 Hk-EOS method application to 154Dy 63

6.1 (k,H) distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.1.1 Influence of the Ge and BGO detectors . . . . . . . . . . . 65

6.1.2 Doppler effect influence . . . . . . . . . . . . . . . . . . . . 67

6.1.3 Reaction channels . . . . . . . . . . . . . . . . . . . . . . . 68

6.1.4 (k,H) projections and distribution moments . . . . . . . . 72

6.2 Energy Ordered Spectra (EOS) and the level density parameter a . 75

6.3 M → I conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.4 Two regions of level density parameter . . . . . . . . . . . . . . . . 79

7 Hk-EOS evaluation of experimental Gammasphere data on con-

tinuum 83

7.1 Gammasphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.2 The Gsfma138 experiment . . . . . . . . . . . . . . . . . . . . . . . 84

7.3 Data handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.4 Gammasphere in Geant4 . . . . . . . . . . . . . . . . . . . . . . . 86

7.5 Hk-EOS method for the level density parameter a calculation . . . 89

Contents xv

8 Conclusions and perspectives 93

Bibliography 95

CHAPTER 1

INTRODUCTION

Nowadays, most of the nuclear physics experiments in nuclear structure are fo-cused on the information that can be extracted when the detected γ-rays can beassociated with a very accurate energy value. Besides the γ-radiation there is ofcourse a wide variety of particles involved in the reactions, but the specific interestof this work deals just with γ-rays. The largest γ-detection arrays (Gammasphere,Gasp, Rising, etc.) use principally germanium detectors, which are well known tohave very good intrinsic energy resolution of low enery γ-rays up to 2 MeV. Thismeans that the γ-rays coming from discrete transitions, usually with low energies,are detected with a good energy resolution.

However, for the γ-rays from the nuclear continuum the detection process is differ-ent. Usually they have high transition energies (0 < Eγ < 15 MeV), and insteadof trying to detect them with high energy resolution a good efficiency is required.In this way the information that can be extracted from the statistical decay athigh energy and high spin in the nuclei is taken into account.

In this work a simulated experiment is carried out, taking advantage of high devel-oped simulation tools for the interaction of radiation with matter and other soft-ware that considers most of the physics involved in the fusion-evaporation reactionprocess. The simulation of an experimental set-up and all interactions between theproducts of the reaction and the experimental devices is undertaken. The focusin this work is to develop and test the new method, Hk-EOS, with simulated andexperimental data for the research of the nuclear continuum.

1

Chapter 1. Introduction 2

Ch. 2 will give the theoretical background in order to understand the nuclear con-tinuum starting from the experimental way to populate this region and explainingthe mathematical expressions to understand the physics involved in the nucleus.The relation between nuclear structure and thermodynamics gives the motivationof the research of new methods that allow to observe the shape phase transitionspredicted by the theory in the nucleus 154Dy.

All the computational tools used in this work are explained in Ch. 3. A fusion-evaporation reaction was simulated to populate the entry states of 154Dy using theprograms Pace2, GammaPace and Gamble which take care of different partsof the reaction. The subsequent interaction of the emitted radiation with the γdetection device is simulated by implementing the array with the Geant4 code.

The bases of the new Hk-EOS method are explained in Ch. 4 and the detectiontheory necessary to consider the experimental issues, as well as the inclusion of theresponse function and the unfolding process are explained in Ch. 5. The applicationof the method to the simulated data allows a first approach to the understandingof the results of such experiments. This is described in Ch. 6. The last chapter(Ch. 7) shows the results of the application of the method to a set of experimentaldata of 60Ni.

CHAPTER 2

THEORETICAL BACKGROUND

2.1 Heavy-ion fusion-evaporation reactions

One way to experimentally reach nuclear states at high energy and high angularmomentum is by performing heavy-ion fusion-evaporation reactions. Such kind ofexperiments are in the core of the new methods proposed in the present work andwill be described in this section.

Fusion-evaporation reactions can be analyzed as a sequence of events, starting withthe fusion of the projectile and target nucleus, until the residual nucleus reachesthe ground state. Fig. 2.1 describes all the reaction steps ( 1© - 6©). It is alsouseful to describe the reaction using two basic quantities: the mass number Aand the number of protons Z. The first stage 1© shows the projectile nucleuswith its characteristic numbers (A1, Z1) and the target nucleus with (A2, Z2). Theprojectile nucleus has an initial kinetic energy Ti and is directed to the targetnucleus which normally is at rest. In stage 2©, both nuclei fuse together when theycome into close contact forming a new system.

It takes around 10−22 s to decide the configuration of the new nucleus; one way(stage 3©) is when the new nucleus splits in two parts of more or less equal size.This is called fast fission. This reaction channel is not considered in this work.The other way illustrated in stage 4© is when the compound nucleus is formed,with characteristic numbers equal to the addition of the nucleons in the targetand the beam (A,Z) = (A1 + A2, Z1 + Z2). The lifetime of the excited states in

3

Chapter 2. Theoretical Background 4

Figure 2.1: Cartoon showing the main steps in a fusion-evaporation reaction.The sequence is explained in the text [2].

the compound nucleus is generally between 10−19 s and 10−16 s. In this stage theparticle evaporation process begins. It consists of the emission of light particleslike neutrons, protons, α-particles among others. When a given residual nucleusreaches an energy around and below the particle binding energy (stage 5©) thepreferred way for the nucleus to continue de-excitation is by emitting γ-rays untilthe ground state is reached (stage 6©). The lifetime of the excited residual nucleusis around 10−9 s.

The conservation laws dictate that the residual nucleus is formed with a highexcitation energy and high angular momentum. A brief description of the behaviorof the energy and the angular momentum in the fusion-evaporation reaction isgiven below. An extended description of the fusion-evaporation reaction can befound in Ch. 9 of Ref. [3].

Energy

If mi, mf and Ti, Tf are the masses and kinetic energies, respectively, be-fore (i) and after (f) the fusion the resulting excitation energy E∗

CN of thecompound nucleus is

E∗CN = Ti − Tf + (mi −mf )c

2. (2.1)

Usually the mass loss in the reaction is small, and the main contributionis given by the difference of kinetic energies. The projectile nucleus energy


Ti is around 4 MeV/u per nucleon and the compound nucleus has a lowerkinetic energy Tf because it is an inelastic reaction. The residual energy isthen used as excitation energy which typically has values around 40 MeV.The de-excitation process carries the compound nucleus to the ground-state.

Another important quantity is the Q-value of the reaction, defined as:

Q ≡ Tf − Ti = (mi −mf )c2 − E∗

CN . (2.2)

Even if the Q-value were small (few keV), is very important to consider thedifference between the initial and final kinetic energies in the simulation ofthe reaction.

Angular Momentum

Opposite to what happens with the excitation energy, which acquires a sin-gle value in an ideal experiment, the angular momentum of the compoundnucleus has a range of values. The angular momentum for a collision is givenby:

~L = ~r × ~p = mpvpb, (2.3)

where ~p is linear momentum vector of the projectile nucleus, with mass mp

and vp. The closest distance between the geometrical center of both nuclei,if the trajectory were a straight line, is the impact parameter b. Fig. 2.2(a)shows a scheme of a nuclear collision where the projectile nucleus has radiusRp and the target nucleus has a radius Rt. The projectile nucleus is directedtowards the target nucleus.

b

Projectile

Target

Rp

Angular Momentum

dσl

Lgr

Rt

vp

(b).(a).

Figure 2.2: (a) Scheme of a nuclear collision. The impact parameter b is thedistance between the geometrical center of both nuclei. (b) Simple represen-tation of the angular momentum distribution for a nuclear collision assuming

that the nuclei can be represented as solid spheres.

The impact parameter can be zero for a head-to-head collision to a maximumvalue for a “grazing” collision, i.e when bgr = Rp +Rt. This means that theangular momentum ranges from a minimum of zero Lmin = 0 for b = 0 toa maximum value of Lmax = mpvpbgr. An approximation of the angular


Figure 2.3: The phase space I −E in which the nuclear decaytakes place. The random emis-sion of particles populate the en-try states of a residual nucleus.From that region down, the nucleican only decay via the emission ofγ-cascades crossing the continuumat high intrinsic excitation energiestowards the yrast states in the dis-crete region. The black line repre-sents the states with the same in-

trinsic excitation energy U .

U

LevelsDiscrete

ContinuumLevels

Entry States

γ

E

I

Ex

p

yrast line

nα

momentum distribution is plotted in Fig. 2.2(b). The sharp cut of the distri-bution implies that if the nuclei are not in contact the angular momentum iszero, but in reality the values of angular momentum near the grazing limitdecrease smoothly because the edge of the nucleus is not sharp itself.

The evaporation process is also represented in Fig. 2.3 in the spin-energy (I, E)phase space. Compound nuclei have an excitation energy Ex given by Eq. 2.1, anda distribution of spin as shown in Fig. 2.2(b). The evaporation of light particles(n, p, α, etc.) allows to populate a range of entry states in the residual nucleus,which has a certain distribution in energy and angular momentum of the excitedstates. Those are represented bya a distribution from which the nucleus startsthe emission of γ-rays. For the heavy-ion fusion-evaporation reactions the entrystates are usually localized at a high energy and high spin region. The γ-decaygoes through the nuclear continuum and discrete levels, and together the yrast lineuntil the ground state is reached.

The quantum states at high energies through which this decay happens build acontinuum because the number of levels per unit energy along with their decaywidths makes it impossible to distinguish individual levels. The next section willgive a more technical description of this topic.

The energy above the ground state for a given spin, from the yrast line (seeFig. 2.3), is the intrinsic excitation energy:

U(I) = E − Eyrast(I). (2.4)


Near the yrast line at low intrinsic excitation energy U the energy levels are dis-crete, the γ-transition between these states form the discrete spectrum. The dis-crete region of most of the nuclei has been studied and characterized, and by usingthese descriptions it is possible to identify the residual nuclei.

The γ-rays coming from a small region in the entry state distribution start whatis called a cascade. All the other γ-rays following it are part of the same cascade.The number of γ-rays in the cascade is its multiplicity. The nucleus decides thefinal state of each transition and the multiplicity of the cascade. The multiplicityof the cascade will be studied in detail in Ch. 4.

2.2 Level densities

The main characteristic of the study of the nuclear continuum is that instead ofdealing with discrete energy states we face a collection of them which are betterdescribed by a level density formula, that is, an expression giving the number oflevels with given quantum numbers per energy unit. Bethe gave the first analyticaldescription based on the equidistant single particle level approximation [4]. Sincethen different approaches try to include different effects and produce correspond-ingly different formulae. In the Fermi-gas model the nucleus is regarded as anideal gas of A fermions enclosed in the fixed nuclear volume. Including the spindependence the density formula can be written as [5, p. 291]

ρ(U, I) =

√a

24

(~2

2J

)3/2

(2I + 1)exp (2

√aU)

U2, (2.5)

with J representing the moment of inertia, I the angular momentum, and U theintrinsic excitation energy of the nucleus. The relationship between energy and thethermodynamic quantity temperature T is given in second order approximationfor the Fermi-gas by

U = a(kT )2, (2.6)

where k is the Boltzmann constant and a the level density parameter. In the lastyears the approximative character of the original Bethe formula, of which Eq. (2.5)is an example, has been recognized and alternative or improved computations havebeen proposed [6] since experimental deviations of the smooth functional behavior,reflecting eventually unexpected structural effects, have been observed [7].

Experimentally, the knowledge we have on level densities comes mostly from neu-tron [8, 9] and proton resonances and more recently from the pioneering work bythe Oslo Group with light-ion reactions [7, 10–12]. This knowledge is thereforelimited to excitation energies around and below the particle binding energies and


very low spins. Parameters and dependencies deduced at these energies and spinshave been either simply implanted or extrapolated to higher energies and spins.Though these generalizations have proven to be very successful in helping otherfields (computer codes simulating the decay of heavy-ion reaction products relystrongly on such data), the fact is that there is very scarce specific experimentalinformation at such energies and spins. In particular, the level density parametera has this caveat. At present, whenever level densities are to be calculated at highenergy and spin, a value extracted from systematics (a = const×A), or from datacompilations is used.

Fig. 2.4.(a) shows a semi-empirical parametrization of a, as a function of the massnumber A [13]; the joint points are isotopes (same Z). From this parametrization itis clear that a has a complex structure, it seems to present shell structure effects.The green line in the same figure represents a linear fit of all the points. It ispossible to say that if we want to consider a linear relation between the level densityparameter and the mass number it will be a = A/8. The blue line represents theprediction with the Fermi-gas approximation of the relation as a = A/15. It meansthat even if the Fermi-gas approximation gives good values of the level density,the prediction of the level density parameter is loose.

Theoretical considerations [14–16] and experimental work [17] have already shownthat a may be lower at high excitation energy or even vary as a function of tem-perature. For example, calculations of the level density parameter for proton-richnuclei as a function of temperature have been performed [18]. Fig. 2.4.(b) plotsthe result of those calculations. It shows that for low temperatures (T = 0.6 MeV)shell effects are predominant, as the temperature increases the shell effects decreaserapidly.

(a) (b)

Figure 2.4: Variations of the level density parameter a:(a) With the massnumber A. (b) For different temperatures in the 44−58Fe isotopes [18].

Within a broader and more actual perspective, Eq. (2.6), which also applies to theensemble of electrons within a solid [19], shows the importance of the level densityparameter a since its study allows to understand the thermodynamics of a finite


quantum system composed by a very small number of fermions as it is the case ofthe nucleus, in contrast to, for example, the large number of electrons in a solid.

2.3 Gamma strengths

As mentioned in Sec. 2.1, after the evaporation of particles, the residual nucleusde-excites via γ cascades. The actual “path” chosen by a nucleus to go from apoint within the entry states to the yrast line depends on the relative probabilitiesfor γ-ray emission of the different multipolarities and electromagnetic characters.Mathematically, the most tractable theoretical description of such probabilitiesis given by the single-particle approximation [20] which states that for a givennucleus of mass A, the emission probability depends on the transition energy Eγ ,electromagnetic character X, and multipolarity ℓ:

Γsp(A,Xℓ) = C(A,Xℓ)E2ℓ+1γ . (2.7)

Later experimental research has shown that the preferred γ-decay mode from thecontinuum is the Giant Dipole Resonance (GDR) [21], whose mathematical for-mulation is given by the superposition of Lorentzians depending on the number ofnuclear deformations kdef ,

ΓGDR(E1, Eγ) =σ

3(π~c)2E4

γ

kdef∑

k=1

wkΓGk

(E2Gk

− E2γ)

2 + Γ2Gk

E2γ

, (2.8)

where σ denotes the GDR peak cross section and EGk, ΓGk

the GDR energyand width, respectively, of the vibration along the kth deformation axis. Thecontribution of each deformation to the total strength is given by the weight factorwk. In general the parameters depend on the quadrupole deformation β2 [22]. Aplot of the GDR gamma strength ΓGDR as a function of the transition energy isshown in Fig. 2.5 for a nucleus with mass number in the region A ≈ 160. When thenucleus is spherical (red line) just one term of the sum in Eq. 2.5 is needed. If thenucleus is deformed more terms appear depending on the number of deformationaxes. In the case of the gamma strength represented by the blue line, the nucleushas two deformation axes, i.e. its geometrical shape is an ellipsoid.

From the experimental point of view, historically, the same kind of experiments,(p, γ) and (n, γ), used to investigate level densities produce simultaneous informa-tion on gamma strengths. A more modern approach has been pioneered by theOslo Group [23, 24] and old topics have been brought back to life; for examplepygmy resonances [25], Giant Magnetic Dipole Resonances [26] or temperatureeffect on the GDR-strength [27].


Figure 2.5: GDR Gamma strength of a nucleus in the A ≈ 160 region. Twocases are represented, a spherical nucleus in red and a deformed nucleus with

two deformation axes in blue.

2.4 Nuclear structure and thermodynamics

Whereas the investigation on the topics mentioned in the previous sections canbe counted within the “classical” research in nuclear physics, recent theoreticaldevelopments have brought along new fields. One of them is the question aboutthe relationship between nuclear structure and thermodynamics.

The nuclear deformation of the rotational nuclei has been studied in detail when thenucleus has low excitation energy. In general the nucleus can exhibit an ellipsoidaldeformation as prolate or oblate, and more complex ones, triaxial. These types ofdeformation are parameterized with β2 for the quadrupolar deformation and γ forthe axial symmetry. Figure 2.6 shows a typical representation of the deformationparameters. When β2 = 0 and γ = 0 the nucleus is spherical. Along γ = 60 andγ = −60 the nucleus is oblate and in if the nucleus has deformation of γ = 0 orγ = −120 its shape is prolate.

It seems that similar to what happens in the discrete region, where the nuclei areobserved to have a geometrical shape, in the continuum region an assignment ofcollective deformation parameters can be done to the highly excited states in spiteof the thermalization of the degrees of freedom.

In general many nuclei have a static deformation in their ground state, i.e at zerotemperature. In such cases deformed nuclei are generated by the arrangementof nucleons out of a closed shell. When the temperature increases the thermalexcitations take away the shell effects and induce a change in the shape. Forsufficiently high temperatures (kT in the order of a few MeV) the nuclear propertiesare expected to vary smoothly with the particle numbers [26]. In conclusion theshape of a nucleus depends strongly on its temperature.


Figure 2.6: Scheme of the nucleardeformation along the β2 and γ pa-rameters. At β2 = γ = 0 the nu-cleus is spherical. For γ = 60 andγ = −60 the nucleus is oblate. Forγ = 0 and γ = −120 the nucleus

is prolate.

From the theoretical point of view, different approaches have been considered:for example Finite Temperature Hartree-Fock-Bogoljubow Cranking (FTHFBC)[28, 29], and the Landau theory of phase transitions [30]. FTHFBC is a mean fieldtheory that includes proton and neutron pair correlation fields, and also considers acranking term to describe the behavior of nuclei with angular momentum differentto zero. The clue in these theories is the inclusion in the Hamiltonian, custom-arily used for the description of discrete states, of a thermodynamical term. Thedegrees of freedom are the deformation parameters β2, γ and pairing correlationsfor neutrons ∆n and protons ∆p.

According to Goodman [28, 31–33] the competition between the quadrupole-quadru-pole interaction responsible for deformed shapes and the monopole pairing inter-action responsible for the spherical shapes determines the nuclear shape.

For nuclei in the 150 < A < 190 region calculations show that their ground statehas a static quadrupole deformation. One example is what occurs in 170Er, whichis one of the simplest nuclei that exhibit this behavior: a change of deformationwith the temperature. Fig. 2.7 shows the variation of the deformation parameter βwith temperature [31]. At zero temperature the nucleus has a prolate deformation(β = 0.33) and the deformation decreases with increasing temperature (red curve)until kT = 1.81 MeV when the nucleus becomes spherical (β = 0) and remainsspherical for higher temperatures (blue line).


Another example is what occurs in the transitional nucleus 154Dy. Phase transi-tions in this nucleus were predicted with FTHFBC, and experimental observationstried to observe its behavior [1]. A sudden change from collective to singe-particlerotations has been confirmed along the yrast line in Ref [34]. But experimentalevidence for the low and high excited states is difficult to achieve, and there arefew experimental investigations of the shape transitions above the yrast line.

Er

0

0.32

0.24

0.16

0.08

0 0.4 0.8 1.2 1.6 2.0kT (MeV)

β170

Figure 2.7: Variation of the defor-mation parameter β with the temper-

ature for 170Er. (After [31]).

Fig. 2.8 shows that in the energy-spin phasespace there are two well defined regions withdifferent nuclear shapes. With increasing en-ergy and spin the nucleus changes from a pro-late shape (collective rotation) to an oblate(single-particle rotation). Ma et al. [1] pro-posed a simple method to provide specifica-tion of the shape fluctuations in the energy-spin region, by selecting the quasicontinuumcollective E2 spectrum. The spectrum of E2transitions from excited states of a nucleusof approximately fixed deformation normallyconsists of a single broad peak. In contrast,two-peak spectra are observed when a selec-tion of E2 transitions of states from bothshape regions is measured.

0

Ene

rgy

(MeV

)

30

20

10

0 20 40 60I

Yrast Line

Dy154

PROLATE

OBLATECascade A

Cascade B

Figure 2.8: Regions of prolate andoblate deformation in the energy-spinphase space for the 154Dy. Calculated

with FTHFBC model [1].

The paths or cascades selected to obtainthe E2 transitions are represented with thecolor lines in Fig 2.8. Cascade A feeds into alow spin region of the yrast line and two E2peaks were observed, one from each phaseregion. For the cascade B the selection wasmade in such way that the cascade entersinto the yrast line at high spin, traversingonly the oblate region, in this case only theupper E2 peak is observed. According toMa et al. the two-peak spectra provide aclear signal of a deviation from constant de-formation.

However, the experimental results of 154Dyare far away from being conclusive for theentire energy-spin region. Those measure-

ments just give information from a narrow region above the yrast line, and theshape transition line is not distinguishable from the data. It is important to under-stand the approach to an experimental method to construct more appropriate tools


to the investigation of the shape transitions. The method will be implemented inthe first place for investigations of the shape fluctuations of 154Dy which promisesto be an interesting endeavor.

At finite temperature the nucleus should perform fluctuations around the mostprobable configuration. The most relevant fluctuations are those related to theshape parameters. When statistical shape fluctuations are included, the transitionline in the energy-spin phase space in Fig 2.8 is blurred. The blurring impliesthat there might not be a clear distinction between the two nearby shape regions.Fluctuations may mean that oblate states are immersed in a prolate region or viceversa, even the transition can be done trough triaxial shapes.

In addition, according to Martin and Egido [29], the consideration of the shape fluc-tuation produces an average shape, which is different of the one obtained throughthe self-consistent solution of the FTHF equations. This effect can be seen in somedetail in the plots displayed in Fig. 2.9. These figures were adapted from the theirwork. The analysis was performed using the FTHF including shape fluctuationsto describe the nuclear structure of the isotopes 152Dy, 154Dy and 156Dy at highexcitation energy and large angular momentum values.

The four plots in Fig. 2.9 are contour diagrams showing the variation of the de-formation parameters β (in blue-white scale) and γ (in red-white scale), with theintrinsic excitation energy U and the spin I. The corresponding mean values βand also γ are plotted. The mean values are obtained when the shape fluctuationsare included in the theory.

Figs. 2.9((a) and (b)) represent the self-consistent deformation parameters on the(I, U) plane. Following both plots is possible to see that at low energies and spinsthe nucleus has a prolate deformation with β=0.22 and γ=0, then with increasingenergy and spin the quadrupolar deformation slowly decreases while the triaxialparameter takes the nucleus to an oblate shape (γ = −60).

It is important to point out that unlike the well defined regions shown in Fig. 2.8where the shape transition is abrupt the FTHF theory predicts that the transitionshould be smooth and also that it is probable to include other shapes as the triaxialone.

Even more complexity is added when the deformation parameters are averaged,Figs. 2.9(c) and (d) show the averaged values of the deformation parameters β andγ when the fluctuations are considered. From there it is easy to notice that thestatistical fluctuations induce many changes in the shape distribution as a functionof the energy and spin. The only region that remains nearly unchanged is justapproximately 1 MeV near the yrast line (U = 0, T = 0). At low temperatures theresults of the mean field calculations are similar to the average shape and the shapefluctuations are small. When the temperature increases the shape fluctuation does


Figure 2.9: Contour surfaces for the variation of the deformation parametersβ and γ with the intrinsic excitation energy and the spin. The parameterobtained with the self-consistent method (right) and the average values (left)

are shown [29].

it as well, and also the differences between the average shape and the most probableshape.

The present work addresses the problem of performing measurements in the nuclearcontinuum from which reliable physical parameters can be extracted. The resultsmay be related with these phenomena. We are proposing a method to measure theshapes in different regions of the phase space, and try to determine how big thoseregions could be. However it is fundamental to perform experiments to lead to theunderstanding of the theory and to prove if it describes rightly the phenomena ofshape transitions in the nucleus.

CHAPTER 3

SIMULATION

The methods to study the nuclear continuum are complex, and not yet fully testedexperimentally, with not easy predictions about the results. It is, besides conve-nient, useful to do simulations to explore in detail all the considerations that haveto be undertaken if an experiment is carried out.

Numerical simulation allows to include many physical properties of the systemunder study, in this case the simulation of the continuum decay and the interactionof emitted radiation with the experimental set-up. It is possible to include someof the theories that explain the physical processes and array.

The computer simulation has three different stages that will be described in thenext sections: Sec.3.1 Formation of the compound nucleus; Sec.3.2 Emission ofradiation, such as particles and γ-rays; Sec.3.3 Interaction of the emitted radiationwith the detection device. Each one of these stages was simulated with a differentcode, a brief description of them is shown in the next Sections.

3.1 Projection Angular momentum CoupledEvaporation (Pace)

The first stage of the simulation deals with the formation of the compound nucleusand the evaporation of particles. The code Pace2 [35] predicts several quantitiesof practical interest like the relative cross sections for different reaction channels,

15

Chapter 3. Simulation 16

the average angular momentum and the total spectrum of emitted particles. Themethod of calculation is explained in detail in Ref. [36]. It uses Monte Carlo meth-ods to determine the decay evaporation events of the reaction using the Hauser-Feshbach formalism [37].

The probability of emitting particles and transforming the compound nucleusinto the residual nucleus is calculated using the principle of detailed balance [38],Sec.7.3. This probability can be determined if the cross section for the inverse pro-cess is known, i.e. the cross section for the production of the compound nucleusfrom a collision between the residual nucleus and the evaporated particles.

The Hauser-Feshbach formalism is necessary in order to consider the effects ofangular momentum and parity conservation on the decay modes. This is done bycalculating the decay probability of an excited nucleus with an excitation energy Eand spin I into a specific exit channel. The probability is determined by the leveldensity ρ(I, E) of the residual nucleus at (I, E) and the gamma strength Γ(Eγ)and the probabilities of particle emission.

The fusion-evaporation reaction simulated was:

Projectile︷︸︸︷3616S +

Target︷︸︸︷12250Sn ⇒

Compound nucleus︷︸︸︷15866Dy (3.1)

Calculations of the cross section of this reaction with different laboratory energiesof the projectile are shown in Fig. 3.1. The input files to obtain these results aredescribed in Ref. [39]. Here the main residual nuclei are shown for each labora-tory energies from 120 MeV to 175 MeV. Some of the residual nuclei producedin the different reaction channels are 155Dy, 154Dy, 153Dy, 152Dy, 149Gd, 150Gd,and 153Gd. For the purpose of the investigation in this work, a selection of anappropriate laboratory energy to produce and populate the high excitation energyand high spin of 154Dy was done. From laboratory energies about 130 MeV to155 MeV the cross section of 154Dy is in the order of 102 mb. The energy chosenin the simulation was Elab = 148 MeV. The most probable reaction channels atsuch energy are:

3616S +122

50 Sn

153Dy + 5n154Dy + 4n152Dy + 6n

(3.2)

Once the laboratory energy is established the following step in the simulation of thereaction process are generated by Gammapace [40, 41], which is a modification ofpace2 paying especial attention to the γ-decay in the continuum. Some featuresof practical interest were added, for example it is capable of rendering a represen-tation of the entry states of a given residual nucleus from where the maximum andminimum values of excitation energy E and spin I are taken. The corresponding


Figure 3.1: Cross section of formation of the residual nuclei in the reaction36

16S +122

50 Sn, for laboratory energies from 120 MeV to 175 MeV. For 154Dy(black full bullet points) the maximum cross section values are in the order of102 mb for laboratory energies from 130 MeV to 155 MeV. The pointed line atElab = 148 MeV represents the energy used for the reaction in the simulation

values of the minimum and maximum of energy and spin for the residual nucleus ofthe three main reaction channels with reaction energy Elab = 148 MeV are writtenin Table 3.1.

Nucleus Imin(~) Imax(~) Emin(MeV) Emax(MeV) 〈E〉(MeV)

152Dy 2 28 3 11 7154Dy 20 53 15 29 23153Dy 4 45 5 19 14

Table 3.1: Maximum and minimum values of excitation energy and spin of theentry states for the reaction channels of Eq. (3.2). The expected value 〈E〉 isthe maximum of the excitation energy distribution obtained with Gammapace.

3.2 Gamble

The code Gamble simulates, using a Monte Carlo procedure, the γ-ray emission ofresidual nuclei produced in heavy-ion fusion-evaporation reactions, i.e the emissionafter the particle evaporation stage. It contains as input a host of parameters


that allow a precise description of the quantum structure of the nucleus at highexcitation energy, high spin and finite temperature. [42]

As we saw in Sec.2.1, the recoil nucleus after the fusion-evaporation reaction pop-ulates the entry states in the energy-spin phase space. The nucleus decays sub-sequently by a cascade of γ-transitions. The first part of each cascade lies inthe nuclear continuum and is dominated by statistical transitions due to the highlevel density. The calculation of the level density is performed based on the Fermiapproximation Eq (2.5).

The entry spin distribution can be parameterized either as a Gaussian or as atriangular distribution. For the energy dependence of the entry distribution Gam-

ble allows a thermal distribution, for example a Boltzmann distribution. Theupper and lower limits, in energy and spin, of the entry states are taken fromGammapace.

The decay cascades follow their path through the continuum states and throughthe collective bands until they reach the yrast line. The experimentally knownvalues of the yrast line and several discrete states are included as input to thecode. At higher spins the yrast line is extrapolated according to the rigid rotatingliquid drop model.

To run the code it is necessary to use two programs: Gamble and Enerord. Thefirst one contains the physics involved in the γ-emission process and outputs thedata to a file similar to the event-by-event type of data obtained in an experiment.The second one is needed to analyze the Gamble output and sort the data in theappropriate way for the analysis. The most important parameters in the inputfiles with the input used in this work are described in Ref. [39].

The resulting entry states distribution for the 154Dy at a laboratory energy of148 MeV are plotted in Fig. 3.2, together with the yrast line.

3.3 Gamma detection device (Gasp)

There are several multidetector devices used to detect efficiently the γ-rays pro-duced in a heavy-ion fusion-evaporation reaction. Most of them take advantageof the High Purity Germanium detectors (Ge) capabilities to do discrete spec-troscopy. Ge detectors are well known for their high resolution, a fact that allowsto identify the level structures of the nuclei and to describe the nuclear behaviorin the discrete region of the energy-spin phase space. However, the Ge detectorhas low efficiency and it decreases with increasing gamma energies.


Figure 3.2: Entry states for 154Dy at a reaction energy of 148 MeV, the yrastline is shown.

Another type of detectors as the Bismuth Germanate (BGO) has higher efficiencybut lower resolution than the Ge detectors. A high efficiency allows to measurespecially γ-rays with high energies (Eγ > 4 MeV), typical transition energies inthe continuum. In principle the low resolution of the BGO detectors should notbe a problem (see Sec. 3.3.3).

To apply the proposed Hk-EOS method it is fundamental to select the adequatetype of array to measure the γ-radiation. The first issue is that the array mustcover more or less 4π solid angle around the target. Some multi-detector devicesthat have a 4π implementation usually do discrete spectroscopy and the totalefficiencies are low for the purpose of this work.

One good option to work with, is the Gasp [43, 44] array located at “LaboratoriNazionali di Legnaro” (LNL) in Italy. It can be arranged in mainly two ways,in “configuration I” it consists of 40 Ge detectors and 80 BGO detectors andin “configuration II” the BGO detectors are removed and the Ge detectors aremoved closer to the center to fill the spaces. As it was mentioned before the BGOdetectors of “configuration I” are necessary to apply the method proposed here.

Gasp is separated in two halves, which are put close together during the experi-ments. Fig. 3.3 (left) shows a photo of one hemisphere of Gasp in “configuration I”where it is possible to see the placement of the BGO and Ge detectors with respectto the beam line and the target. In the same figure (right) a photo of the BGO andGe detectors is shown. The Ge detector has an anti-Compton device implementedwith scintillator crystals. Since there are many different detector positions, severalrings of detectors with similar position were sorted. The assortment is tabulatedin Table 3.2. It shows the ID number of the detectors in each ring, the mean angleand the amount of detectors in each ring. There are nine BGO rings and seven Ge


Beam line

BGODetector

Target

Ge Detector+ anti−Compton

Figure 3.3: (Left) Placement of the Ge and BGO detectors in one hemisphereof the Gasp array. The target position is shown as well as the beam line. (Right)A photo of the BGO and Ge detectors, including the anti-Compton device of

the Ge detectors.

rings. The number of detectors in each ring varies from 6 to 14 for the BGO ringsand from 4 to 8 in the Ge rings.

3.3.1 Gasp array simulation with Geant4

Geant4 [45, 46] is a simulation toolkit developed at CERN in object orientedC++ language that simulates the interaction of radiation with matter. WhatmakesGeant4 so powerful is the fact that the user can add custom code describingan experiment. In that way it is possible to implement subroutines describingthe multidetector array Gasp together with the many possibilities of performingexperiments with it.

The interaction method in geant4 uses a combination of two Monte Carlo meth-ods, which involve sampling data from a distribution. This type of evaluation hasthe properties of easy sampling, quick evaluation of the selected data and a shortnumber of tries.

The original version of Gasp [47] (the simulation tool) was written by EnricoFarnea, together with the simulation of the array Agata [48]. In fact the codes for


Ring Detectors ID Angle Number of detectorsBGO detectors, 9 rings0 0-5 20.4 61 6-11 40.5 62 12-21 52.9 103 22-35 73.7 144 36-43 90.0 85 44-57 106.3 146 58-67 127.9 107 68-73 139.5 68 74-79 160.0 6

Ge detectors, 7 rings0 1-6 33.9 61 7-12 59.1 62 13-16 72.0 43 17-24 90.0 84 25-28 108.0 45 29-34 120.86 66 35-40 146.14 6

Table 3.2: Detector selection of the rings for the Doppler analysis, the anglesare an average of the different position of the detectors of each ring.

the two arrays are the same and they just have different geometry description. Bothconfigurations of the Gasp array are implemented, even some other elements asneutron detectors, the silicon array ISIS [49] and the particle detector Euclides [50]can be added. The physical description of the shell of the device is included andthe collimators as well.

The final geometry of the Gasp detectors can be observed with one of the visual-ization tools of Geant4. Fig. 3.4 shows the BGO crystals (blue hexagons), the Gedetectors and the anti-Compton devices. All detectors are arranged in a spherewith the target position in the center.

3.3.2 γ-rays and neutron interaction

When an event starts it is important to determine the range of the particle (alpha,neutron, etc.) or the radiation (γ-rays, X-rays, etc.). If the particle is chargedand there is an electromagnetic field, the propagation is done according to theequation of motion. To calculate the interaction length or mean free path of the


Target

BGOCrystals

anti−ComptonGe +

Figure 3.4: The Gasp array visualization generated with the simulation inGeant4. Each Ge detector has its anti-Compton device. The blue hexagons arethe BGO detectors. The position of the target is at the middle of the array.

particle, the macroscopic cross section is calculated for all the materials in thepath. Then an interaction point is calculated including the particle energy alongwith its lifetime.

For the γ-ray interaction some of the processes taken into account are: the photo-electric effect, the Compton scattering and the conversion into an electron-positronpair. Cross sections, mean free paths, angular distribution and final state calcula-tions are performed in each case.

The neutron interaction is simulated for those with kinetic energies up to 20 MeV,the result is a set of secondary particles which are included in the subsequentinteraction. The processes considered in this case are radiative capture, elasticscattering, fission, and inelastic scattering.

All the details of the physics involved in all different kind of processes simulatedby geant4 are described in the “Physics Reference Manual” available in its website [51].

3.3.3 Spectroscopic characteristics

Each type of detector has a set of characteristics according to the material and thedetection technique (scintillator, semi-conductor, etc.). The spectroscopic charac-teristics as detector efficiency and resolution are very important to recreate datawith similar properties as in a real experiment. To test the efficiency and reso-lution of the detectors several monoenergetic sources from a few keV to 13 MeVwere simulated. The evaluation of the absolute efficiency is plotted in Fig. 3.5.


From this figure we can see that the absolute efficiency of the set of BGO detec-tors is at least one order of magnitude above the absolute efficiency of the set ofGe detectors. This means that even for the high energy γ-rays, coming speciallyfrom the continuum, the efficiency is at least 30%.

0.001

0.01

0.1

1

0 2 4 6 8 10 12 14

Abs

olut

e E

ffici

ency

Energy (MeV)

BGO

Ge

Figure 3.5: Absolute efficiency for the complete gasp array, the efficiency ofthe BGO detectors is an order of magnitude above the Ge-detector efficiency.

With the same monoenergetic sources the dependence of the resolution with theenergy can be observed, the results are plotted in Fig. 3.6. In this case the FWHMhas to be included as:

FWHM (MeV) = a+ b√

E(MeV), (3.3)

with a = 0 and b = 4.0 (MeV)1/2 for the BGO detectors, and a = 0.648 MeVand b = 0.032 (MeV)1/2 for the Ge detectors. Opposite to what happens withthe efficiency, the resolution of the BGO detectors is lower than the resolution forthe Ge detectors. This would be a problem for discrete spectroscopy, but for thepurpose of this work the resolution effect is not decisive. This will be evaluated inSec. 6.1.1.

In Fig. 3.7 an example of the simulated spectra are plotted. They were generatedwith 105 monoenergetic γ-rays of 10 MeV. The upper spectrum is the sum ofthe recorded with the 40 Ge detectors and the lower spectrum was recorded withthe 80 BGO detectors. All the standard characteristics of real spectra can beseen here; the full-energy peak at 10 MeV, the Compton background and someother peaks coming from the conversion into electron-positron; the detection ofthe γ-rays at 511 keV and 1022 keV and also one peak at single escape peak at(10000-511) keV=9489 keV.


0.1

1

10

100

0 2 4 6 8 10 12 14

FW

HM

(M

eV)

Energy (MeV)

BGO

Ge

Figure 3.6: Array resolution, for different mono-energetic sources; it wassimulated for each detector type individually.

Figure 3.7: Spectra visualization: the upper spectrum was recorded withthe 40 Ge detectors and the lower spectrum was recorded with the 80 BGOdetectors. All the physics involved in the detection process can be observed

directly in the spectra. See text for details.

3.3.4 Doppler effect

Likewise the spectroscopy characteristics, the Doppler effect influences all the mea-surements of the γ-rays energies. The influence can be determined using the Gasp

simulation and generating the γ-sources with a velocity v. Spectra for a mo-noenergetic source emitting isotropically were collected with detectors in different


positions. Then it is possible to verify the shift induced by the Doppler effect foreach one. Fig. 3.8 shows the shift for a 1.3 MeV γ-ray detected with the Ge detec-tors. Each peak corresponds to a specific ring (See. Table 3.2) with a mean angle θand has a shift corresponding to such angle. The intensity of each peak is directlyrelated to the number of detectors in the ring. It can be easily seen from the figurethat each peak has a different energy. This is to verify the implementation of theDoppler Effect in the simulation. The relation between the angle and the energyfirst order is:

∆E = Eγv

ccos θ, (3.4)

where Eγ is the energy of the γ-ray, v the residual nucleus velocity, and c the speedof light, and a velocity of v = 0.0219 c.

Figure 3.8: Doppler shift for a γ-source of Eγ = 1300 (keV) for five differentrings. The angles are labeled for each peak. For θ < 90 the shift is to highenergy values, for θ > 90 the shift is to lower energies, in the case of 90 there

is not shift, but a broadening.

CHAPTER 4

THE Hk-EOS METHOD

The new Hk-EOS method proposed in this work is a combination between theHk technique [52], which implies to know very well the γ-ray detection process,and the Energy Ordered Spectra (EOS) [53] to extract information about thenuclear structure in the nuclear continuum. The aim of this method is to be ableexperimentally to identify the primary radiation coming from a small selection ofentry states (I, E) of the nucleus under study. The description of both parts ofthe method will be presented in this Chapter.

4.1 Primary radiation

In Sec. 2.4 it was declared that the main goal of this work is to investigate thenuclear properties at a defined region in the energy-spin phase space. The methodto analyze the nuclear behavior of a given region is by measuring the radiationcoming from it. To analyze a specific point (I, E) it is necessary to collect theradiation coming exclusively from that point, the Primary Radiation.

A scheme of a γ de-excitation process is drawn in Fig. 4.1. The sequence of γ-transitions represents any de-excitation cascade decaying from the highest statewith an excitation energy E. Each cascade has a sequence of M γ-rays emitted ina specific order (Eγ1, Eγ2, Eγ3, · · · , EγM ). The same cascade is also representedincluding the spin change in each transition.

27

Chapter 4. The Hk-EOS method 28

In both representations of Fig. 4.1 the first γ-ray is represented as a red arrow.This γ-ray is the one to be measured. The primary radiation is the collectionof γ-rays emitted directly from the states in the (I, E) region. Since the entirecascade decays very fast, (< 10−9 s), and the standard electronic time has lowerresolution, it is impossible to identify and distinguish any γ-ray, wich in particularthe primary radiation from all the gammas of each cascade.

Figure 4.1: De-excitation process of a state with a specific (I, E) value, thefirst γ-ray is shown as the red arrow.

However, it is possible to implement a method to get an approximation of theprimary radiation by using the Hk technique and the energy Ordered Spectramethod. These tools are described in the next sections.

4.2 The Hk technique

A device like Gasp allows to collect (almost) all the γ-rays emitted along the decaypaths originating from a narrow region of the excitation spin-energy map [52,54]: The instrument is able to record in an event-by-event mode nearly all thetransitions that a nucleus undergoes from the entry state to the ground state.

A cascade has two determining quantities, its energy E and its multiplicity M .The multiplicity has a direct relation with the spin I. This will be discussed atthe end of Ch. 6. The energy of each γ-ray of one cascade with multiplicity M (asin Fig. 4.1) can be added to get the total excitation energy,

M = Number of γ-rays, E =

M∑

i=1

Eγi. (4.1)


Let us suppose an ideal detection system able of registering each one of the Mγ-rays emitted in cascade with the correct energy Eγ i of each transition. The sumof the M γ-energies would reproduce the excitation energy E at which the cascadeis originated as Eq.(4.1). However, the interaction of radiation with any realdetection system brings about certain effects that hinders the exact determinationof E as well as of M .

In a real case after the detection process we will not have the number of γ-raysin the cascade but the number of fired detectors, and instead of the energies ofeach γ-ray Eγi we get the total energy pulse height of each of the fired detectorhi. This situation can be summarized as follows:

M → k, E → H =k∑

i=1

hi. (4.2)

Some of the typical γ-ray detection problems that occur in this kind of arrays andwhich can cause a wrong determination of E and M are:

3

5

4

2

1

Figure 4.2: Some detection problems in the measurement of E and M . (Ex-plained in the text)

1. If one γ-ray escapes the array, all its energy information is lost and themultiplicity is reduced by one.

2. Incomplete detection, when one γ-ray is scattered out of the array afterleaving part of its energy in just one detector. In this case only the correctenergy that the γ-ray keeps is lost, but the multiplicity is correct.

3. In the case of detector-detector scattering by Compton effect the energy issplit in (at least) two detectors and the multiplicity M is measured falsely.E might be wrong if the γ-ray escapes after the interaction with the lastdetector.


4. When two or more γ-rays arrive to one detector (pile-up), they are detectedas one. If all their energy is left in the detector, just the M value is mistaken.

5. Other sources: neutrons and high energy protons coming from the evapo-ration process are counted as γ-rays. This case can change very much themeasurement of E and M , its effect will be evaluated in Sec. 6.1.3.

All the detection devices present all the problems enumerated before, however,each array has a specific response to the γ radiation. The response function of thearray gives a final result that instead of having a collection of “true” transition γ-energies, it delivers the signals hi of the k fired detectors in the array. Symbolicallythe summation actually performed is on the right hand side of following equation,

E =M∑

i=1

Eγ i

responsefunction−→ H =

k∑

i=1

hi. (4.3)

Fortunately, methods have been developed which, after obtaining experimentallythe response function [54], run the inversion process. This process is going fromright to left in the relation (4.3), producing for every pair (k,H) not a point buta “probability cloud” in the (M,E) plane.

The issue of the deconvolution of the response function, that is the conversion

(k,H) → (M,E) (4.4)

will be described in detail in Chap. 5. One additional method that helps to obtainthe “primary radiation” will be described in the next Section.

4.3 The Energy-Ordered Spectra technique

Towards the goal of doing spectroscopy on γ-rays emitted by a state of knownenergy and spin, the Hk-technique described in the previous Section is not yetpowerful enough since the initial state of the entire cascade is known but not theγ-rays directly emitted by that state, the so called primary radiation. The idealexperimental situation is one in which the time order of the sequence of M γ-raysin a cascade can be distinguished, that is, to each γ-ray in the event an “ordinal”NT can be assigned and therefore the γ-rays in the event can be arranged as

Eγ(tNT=1), Eγ(tNT=2), ..., Eγ(tNT=M ) (4.5)

tNT=1 < tNT=2 < ... < tNT=M .


This goal is not presently reached because of the experimental impossibility ofdistinguishing a temporal order for the γ-rays in a cascade. In order to circumventthis difficulty a technique called Energy-Ordered Spectra (EOS) has been proposed[55]. It is known that the most energetic γ-rays are the most probable to be emittedfirst. Following this principle, the temporal sequence of emission is forgotten andthe ordering of the γ-rays in a cascade will be done according to their energy. TheEnergy-ordering of the cascade is carried out when the ordinal NE is associatedto each transition so that

Eγ NE=1 ≥ Eγ NE=2 ≥ · · · ≥ Eγ NE=M . (4.6)

Taking a sample of γ cascades, one can then build either Time-Ordered (TOS)or Energy-Ordered Spectra (EOS) with transitions having the same NT or NE re-spectively. The practical difference between both is that TOS are not attainableexperimentally, whereas building EOS is a very simple software task. The associ-

TOS EOS Energy ordering: Histograming:

EOS N=1

EOS N=2

EOS N=3I(co

unts

/cha

nnel

)

Eγ

N =1T

T

N =3T

N =4TN =5T

N =1

N =3

N =4E

N =2

EN =2

E

E

EN =5

Figure 4.3: Description of the time and energy ordered spectra and the con-struction of the ordered histograms.

ation of small NT values to the first instances of the decay and large ones to thelast steps of the decay is strict in the case of Time-Ordered Spectra. In particular,NT = 1 defines the primary radiation. Regarding EOS the NE = 1 spectra willcontain those transitions “largest in energy” of each cascade, the NE = 2 spectrawill contain the second largest ones, and so on.

It was shown [53] when considering γ cascades originating in an intrinsic excitationenergy stripe around the neutron binding energy, produced after a heavy ion fusion-evaporation reaction the NE=1 have striking and useful similarities to the spectrathat would be obtained if the primary radiation could be detected, NT=1. As itis shown there and suggested in Fig. 4.4, the potential usefulness of the methodlies in the fact that the intensity of NE = 1 ordered spectra depends exclusively


on the electric dipole gamma strength and the level density:

Iγ = cte · ΓGDR(E1) · ρ(Ui − Eγ). (4.7)

The expression of the level density includes also the dependences on the spin, themoment of inertia and the level density parameter as in Eq. 2.5.

Figure 4.4: The histograms represent numerical simulations taking γ-raysoriginating in an intrinsic excitation energy “stripe” between 9 and 11 MeVabove the yrast line in a typical rare earth region nucleus. (Figure taken from

Ref. [53]).

The description of the intensity of the EOS NE = 1 with these quantities allowsvarious analysis approaches:

a Quantitative determination of the level density parameter by fitting the “pri-mary radiation” theoretical function within a restricted range of the experi-mental spectrum (the high-energy side), as it is shown in Fig. 4.4.

b Most importantly, it has also been shown [56, 57] that the full spectral shapeof the Ordered Spectra is sensitive to the energy dependence of both E1gamma strength and level density. This is very important since althoughthere is not yet a method to study separately gamma strengths and leveldensities, it is however possible to distinguish which combination of the twofunctional dependencies takes place for a given nucleus in a given continuumregion by comparing experimental spectra with simulated ones.

The key fact here is that by (i) gating at (k,H) values or region of values thedefinition of (I, U) regions is achieved and, by (ii) taking the NE=1 originated inthose (k,H) values a good approximation to the corresponding “primary radiation”is obtained.


In figure 4.4 the high-energy side in both, Eγ & 6 MeV in the EOS (NE=1), and theprimary radiation spectra (NT=1) are fairly similar. This similarity can be used,e.g., to determine experimentally the level density parameter by fitting the higherenergy side of the Energy-Ordered Spectra (the only ones experimentally available)to the primary radiation theoretical function, as shown in the inset. Notice thatthe “experimental” result a = 23.3 MeV−1 would be only 16% different from the“real” value. The continuous lines are fits using the GDR gamma strength andthe Fermi Gas level density. This test case assumes ideal response function.

There are statistical tools that can describe the EOS using complete analyticalfunctions. One of them is the Order Statistics. It has been worked out previ-ously [53, 56, 58] showing that even with some deficiencies, information aboutthe nuclear continuum properties can be extracted. Moreover, the Statistics ofExtremes applied to the nuclear continuum study has been proven to be a goodmethod to describe the EOS. The present limitation of this method is to find arelation between the parameters of the distribution with the physical variablesinvolved [59].

4.4 The influence of the response function

In the final part of Sec. 4.2 the issue of the response function was postponed andit was assumed that a good deconvolution method could be used, so that thepromising analysis described in the previous Section can be done. That is, wehave been looking at spectra emitted by the nuclei as if the detectors would havenot been affected their properties: distribution, intensity, etc. The deconvolution(also called “unfolding”) performs the task of taking into account the effects ofthe γ interaction with the detectors. The energy and type of material dependenceof these effects are fairly well known, but the combination of them results in avery complex process that depends on the energy distribution of the radiation, thespecific geometry, and composition of the array of detectors.

Although there are already experimental methods designed to perform the decon-volution (e.g. Ref. [60] and references therein), the effect of the unfolding procedureon the (k,H)-gating, on the quality of the Energy-Ordered Spectra, and thereforeon the error bars of the quantities determined from them is unknown.

Since all the simulation results, as well as the experimental data, need to be relatedto the real quantities that change with the detection process, it is fundamental toimplement a good procedure to unfold the measurement effects. The unfoldingis an extensive technique which involves a complete determination of the arraycharacteristics together with the analysis of the response function for a γ-ray with


defined energy that arrives to the setup. This will be explained in detail in thenext chapter.

CHAPTER 5

DETECTION THEORY AND EXPERIMENTALCONSIDERATIONS

In order to go forward in the investigation of the applicability of the combinedmethod “Hk+ EOS” the unfolding procedure presented by Jaaskelainen et al. inRef. [54] and references there in, was implemented. Using the simulation of thedetection array, the response function and the reverse response can be calculatedfor any set of data. Despite the fact that the responses were obtained with thesimulation, the method is completely experimentally reproducible and it shouldgive a good description of the detected events in relation with the emitted ones asin a real experiment.

5.1 Probability theory of Hk distributions

First of all it is necessary to relate the experimentally observed quantities with themultiplicity distribution and have theoretical approaches to analyze the multiplic-ity measurements. The complete theoretical description of the detection processwas adopted from the work of Sarantites et al. [52], and Westerberg et al. [61].One main assumption is that the probability that each detector records a partic-ular γ-ray is the same for each detector, each γ-ray is detected with an averagedetection probability Ω and the array has a total efficiency ΩT .

One simple but important consideration is what is considered as an event in thiswork:

35

Chapter 5. Detection theory and experimental considerations 36

Event = A set of simultaneous γ-rays from a cascade withmultiplicity M and energies Eγ1

, Eγ2, · · · , EγM

.

For a given event, the response of the array is the corresponding outcome of energypulse heights hγ1

, hγ2, · · · , hγk

. The maximum information that can be extractedfrom an array with N detectors is PNk,h1,h2,...,hN

(M), the probability that eachdetector i = 1, · · · , N records a pulse height hi when only k of them are abovecertain threshold hth.

The probability theory for the detection statistics can be written by includingthe complete array response to a single γ-ray which includes all the interactionpossibilities. It is useful to include a generating function [62] together with thecounting variables:

• t1, t2, ...tN for observing the pulse heights hγ1, hγ2

, · · · , hγk.

• s1, s2, ...sN for enumerating the γ-rays that hit a given detector.

• n1, n2, ...nN the number of γ-rays triggering one detector.

In general the generating function can be written as a product of factors each onecorresponding to an individual γ-ray. This is possible because the emission of eachγ-ray is statistically independent of the emission of the others:

G(t1, t2, ...tN ) =∑

n1,n2,...,nN

Pn1,n2,...,nN× t1, t2, ...tN . (5.1)

To include a more complete description of the generating function for a single γ-ray with all its possible interactions with each detector in the array it is necessaryto take into account the detection probabilities or efficiencies. The probabilityΩi(Eγ ;hκ) that for a single γ-ray of energy Eγ only the ith detector record a pulseheight of hi, and the probability Tij(Eγ ;hκhλ) that only two detectors i and jrecord pulse heights hκ and hλ respectively. The efficiencies permit to write the


generating function including just the interaction with two detectors out of N :

G1(Eγ ; t1, t2, ...tN ) =

no response︷︸︸︷

1−N∑

i=1

κm∑

κ

Ωi(Eγ ;hκ)−N∑

ij=1i 6=j

∑

κλ

Tij(Eγ ;hκhλ)

+

only one detector︷︸︸︷

N∑

i=1

κm∑

κ

Ωi(Eγ ;hκ)thκ

i si

+

any pair of detectors︷︸︸︷

N∑

i=1i 6=j

∑

κλ

Tij(Eγ ;hκhλ)thκ

i thλ

j sisj .

(5.2)

The expression in Eq. (5.2) has three main parts. The first one includes threeterms that correspond to no response. If the total probability is equal to one, theprobability for the interaction of the γ-ray with one of two detectors is subtractedfrom the total. Another term can be added in this part of the equation to considera triple scattering and so on.

The second part of Eq. (5.2) takes into account the response of one detector. Thecounting variables t and s run over all the detectors and the maximum numberof channels in the spectrum κm respectively. In the same way the third partincludes the response of any pair of detectors. Inclusion of higher order terms isstraightforward.

Since the latter expression just includes a single γ-ray, it is possible to write thegenerating function for the cascade with multiplicity M in a more general way:

GM (Eγ ; t1, t2, ..., tN ) =

M∏

i=1

G1(Eγ ; t1, t2, ...tN ; s1, s2, .., .sN )

=∑

h1,h2,...,hNn1,n2,...,nN

P (h1, h2, ..., hN ;n1, n2, ..., nN )

× th1

1 th2

2 , ..., thN

N sn1

1 sn2

2 , ..., snN

N .

(5.3)

This expression represents the maximal description of the array response and per-mits to relate the real events and the detected signals.

For any cascade Eγ1, Eγ2

, · · · , EγM as well as its response hγ1

, hγ2, · · · , hγk

there are simpler representations in terms of the total values (M,E) and (k,H)


(introduced in Chap. 4), which are determined by the corresponding sums:

E =M∑

i=1

Eγi → (M,E), (5.4)

H =

k∑

i=1

hi → (k,H). (5.5)

In this way the information of the event is lesser, but it is more convenient forthe description of the probability distribution, in this case PN (ME, kH). Thisprobability becomes very important in the development of the unfolding procedure,it is in fact the response function of the array to a cascade.

5.2 Detection efficiency

The response function for a monoenergetic source is the simplest case of the interac-tion between a γ-ray and the detector set. The BGO and Ge detectors of Gasp areconsidered in two separate groups due to their different behavior in the detectionprocess. Fig. 5.1 shows the response function ΩT of Gasp to a set of monoenergeticγ-rays with energy Eγ = 1 MeV. The spectra show the standard characteristics ofboth types of detectors shown in Sec 3.3.3. To relate the multiplicity of a cascade

Figure 5.1: Response function ΩT for a monoenergetic source of Eγ = 1 MeV.

with the detection process, it is necessary to consider the triggering efficiencies.The single triggering can be described in terms of Ωi(Eγ ;hκ), which is the proba-bility of recording a pulse of height hκ from a γ-ray with energy Eγ in the detector


i when no other detector responds. The single triggering efficiency is defined as:

Ω(1) =

N∑

i=1

∑

κ

Ωi(Eγ ;hκ). (5.6)

For the detection of double triggering (γ − γ coincidence), the efficiency is:

Ω(2) ≡∑

ij

∑

κλ

Tij(Eγ ;hκhλ) = N(N − 1)∑

κλ

Tij(Eγ ;hκhλ). (5.7)

The previous expression of the double triggering is valid for the symmetrized model,where the N detectors of the array are taken as identical. In the symmetrizedmodel the probability generating function for the event, considering just a pair ofdetectors that record pulse heights hκ and hλ is:

G(t,N) =

M∏

i=1

(

1−N∑

κ

Ω(Eγ,i;hκ)−N(N − 1)∑

κλ

T (Eγ,i;hκ, hλ)

+ N∑

κ

Ω(Eγ,i;hκ)thκ +N(N − 1)

∑

κλ

T (Eγ,i;hκhλ)thκ+hλ

)

.

(5.8)

Any number of triggered detectors has a similar expression as Eq. (5.7) for itsefficiency. The detection efficiency for the array will be the addition of all thetriggering efficiencies:

ΩT = Ω(1) +Ω(2) +Ω(3) + · · ·+Ω(N). (5.9)

To extend the concept of detection efficiency from a single γ-ray to the detectionof the event with M γ-rays, it is necessary to introduce the concept of the foldscattering factor Fk. It was first defined in Ref. [52] as the fraction of detectionefficiency for double triggering over the total detection efficiency:

F (2) ≡ Ω(2)

ΩT, (5.10)

then it was generalized in Ref. [54] for the triple, quadruple and other orders. Thesingle triggering efficiency can be written in terms of the total detection efficiencyusing Eq. (5.9):

Ω(1) = ΩT (1−Ω(2)

ΩT− Ω(3)

ΩT− ...− Ω(N)

ΩT)

= ΩT (1− F (2) − F (3) − ...− F (N)),

(5.11)


In this way, the fold scattering factor, permits the evaluation of the triggeringdetection efficiencies from the total triggering efficiency and can be written as:

Fk ≡ F (2) + 2F (3) + 3F (4) + · · ·+ (N − 1)F (N),

=Ω(2) + 2Ω(3) + 3Ω(4) + · · ·+ (N − 1)Ω(N)

ΩT.

(5.12)

The fold scattering factor allows to calculate the average number of detectors thatfired for an input of M γ-rays when the total triggering efficiency is ΩT . In otherwords, when an event of multiplicity M and total energy E arrives to the array, itis possible to theoretically know, how many detectors k give an energy signal H.

From Eq. (5.9) and Eq. (5.12) it is possible to write the corrected total triggeringefficiency ΩT (1 + Fk) as:

ΩT (1 + Fk) = Ω(1) + 2Ω(2) + 3Ω(3) + · · ·+ (N − 1)Ω(N) (5.13)

5.3 Moment expansions for PN(ME; kH)

5.3.1 The k fold distribution PN(ME; k)

The probability of detecting an event of M γ-rays and total excitation energy Ewith and array of N detectors where k detectors are fired and the total pulse heightmeasured is H is PN (ME; kH). The generator of PN (ME; kH) is:

F (t, s) =∑

kH

PN (ME; kH)tHsk. (5.14)

This generator F (t, s) is related with the generation function G(t, n) by:

F (t, s) =N∑

n=0

sN (1− s)N−n

(N

n

)

G(t, n). (5.15)

The first moment of the probability distribution PN (ME; k), can be evaluatedfrom its generator F (t, s). If the evaluation is done over the number of γ-rays thathit the detectors, hence it gives the mean value of k as:

〈k〉 = ∂

∂sF (t, s)

∣∣∣∣t=s=1

, (5.16)


replacing Eq. (5.15) in Eq. (5.16) the mean value of k is the expressed in terms ofG(1, n):

〈k〉 = N [1−G(1, N − 1)]. (5.17)

The generating function G(1, N − 1) can be obtained from Eq. (5.8) using addi-tional terms to include triggers of greater order, and the corrected total triggeringefficiency ΩT (1 + Fk) of Eq. (5.13), as follows:

G(1, N) =

M∏

κ

(

1− ΩT (1 + Fκ)

N

)

, (5.18)

then the mean values of k can be calculated by:

〈k〉 = N

1−M∏

κ=1

[

1− ΩT (1 + Fκ)

N

]

κ

(5.19)

5.3.2 The total pulse height H distribution PN(ME;H)

In a similar way as it was done in the previous section the first moment of thevariable k in PN (ME; kH) can be calculated for H as:

〈H〉 = ∂

∂tF (t, s)

∣∣∣∣t=s=1

=

[d

dtG(t, n)

]

t=1

. (5.20)

Deriving the generating function of Eq. (5.8) over t it gives:

[d

dtG(t, n)

]

t=1

=

M∑

i=1

(

N∑

κ

Ω(Eγ,i, hκ) · hκ

+ N(N − 1)∑

κλ

T (Eγ,i;hκhλ) · (hκ + hλ)

)

.

(5.21)

If all the triggering orders are included, the terms within the parenthesis definethe spectral function ω1(Eγ). Then the expression for the mean value of H can bewritten as:

〈H〉 =M∑

i=1

ω1(Eγ,i). (5.22)


The spectral functions or order l in general are defined as:

ωl(Eγ) =N∑

κ

Ω(Eγ,i, hκ) · hlκ

+N(N − 1)∑

κλ

T (Eγ,i;hκhλ) · (hκ + hλ)l

+N(N − 1)(N − 2)∑

κλµ

T (Eγ,i;hκhλhµ) · (hκ + hλ + hµ)l + · · ·

(5.23)

It is very important to observe that for l = 0 the spectral function ω0(Eγ) is thetotal triggering efficiency ΩT of Eq. (5.9). New efficiencies can be defined for eachdetector such that they take into account the full probability for detector i of beingfired by γ-rays of any fold Ωi,F (Eγ;hκ

). Then the addition of the full probabilitiesfor all the detectors will give the total triggering efficiency:

ΩT =

N∑

i=1

∑

κ

Ωi,F (Eγ ;hκ) (5.24)

then the spectral functions can also be written in terms of the new full probabilities:

ωl(Eγ) =N∑

i=1

∑

κ

Ωi,F (Eγ ;hκ) · hlκ =

∑

κ

ΩF (Eγ ;hκ) · hlκ. (5.25)

The average total efficiency of order l for the energy Eγ is such that:

ωl ≡ Ω(l)Eγ

· Elγ . (5.26)

The generalized efficiency in first order Ω(1)Eγ

, can then be written as:

Ω(1)Eγ

≡ ω1(Eγ)/Eγ ; (5.27)

This is the fraction of γ-ray energy deposited in the entire array for an incidentγ-ray of energy Eγ .

Using the generalized efficiency it is possible to rewrite the expectation value 〈H〉in Eq. (5.22) as:

〈H〉 =M∑

κ=1

Ω(1)Eκ

Eκ (5.28)


5.4 Mappings

The measurement of an event characterized by the pair (M,E) transforms thosequantities in the pair (k,H). The mappings (M,E) → (〈k〉, 〈H〉) and (k,H) →(〈M〉, 〈E〉) permit to have a first approach of the relation between the measureddata and the corresponding event. The mappings can be obtained in the followingway:

• The mapping (M,E) → (〈k〉, 〈H〉) can be calculated using Eqs.(5.19) and(5.28).

• It is possible to calculate the reverse mapping (k,H) → (〈M〉, 〈E〉) usingan iterative inversion procedure. In order to achieve this it is necessary tomeasure the detection, generalized and corrected total triggering efficienciesexplained in Sec. 5.2 and 5.3. The i-th trial in the iteration process forcalculating the reverse mapping will be:

Mi =ln(1− k/N)

ln1− [ΩT (1− Fk)]i−1/N (5.29)

Ei =H

Ω〈Eγ〉i−1(5.30)

⇒ 〈Eγ〉 =Ei

Mi(5.31)

In the following section the procedure to obtain the response function PN (ME, kH)of the Gasp array, by using the simulation will be explained in detail. This pro-cedure considers an experimental way to obtain the probability distribution as itwas done by Jaaskelainen et al. [54].

5.5 The M → I conversion

The last aspect to be treated in the method is how to evaluate the spin value Ifrom the pair (E,M) obtained when applying the response function of the selectedpoint (k,H).

The corresponding spin is such that:

E − U = Eyrast(I), (5.32)

The value of the intrinsic excitation energy U can be extracted with the corre-sponding EOS, the maximum energy value of the spectrum is the average value of


the intrinsic excitation energy 〈U〉 = Emaxγ for the region (E, I). Actually what

can be calculated is the average of the spin 〈I〉 in the given region as:

〈E〉 − 〈U〉 = Eyrast(〈I〉), (5.33)

Fig. 5.2 shows an schema of the process of converting M to I. In Fig. 5.2(a) thefirst task is shown, the selection of U is carried out by using the EOS of the selected(M,E) region. Once the U value is obtained it it possible to find the spin, as isdepicted in Fig. 5.2(b) by reconstruct the (M,E) point in the (I, E) phase space.

(a) (b) .

EOS

EU γ I

U

(M,E)

E

Figure 5.2: To convert M in I. (a) The U value is selected from the EOS.(b) Then the (M,E) point is related to the correspondant (I, E), as is shown

in the figure.

5.6 Gasp characterization

To characterize the array the first task is to find its response to individual γ-rays,this is done by measuring the efficiencies. All the detection efficiencies dependon the γ-ray energy and can be evaluated simulating the response of the detectorarray to γ-rays of given Eγ . It is convenient to have a wide range of energiesEγ in order to have a more complete description of the array detection efficiency,specially in the high energy region Eγ > 4 MeV.

Fig. 5.3 shows the measured detection efficiencies from the simulation of the Gasp

array (configuration I). The γ-ray energies have values up to 12 MeV. The plotincludes triggering efficiencies until fifth order and the total detection efficiencyas in Eq.(5.9). In general, the single triggering efficiency (plotted in red squares)represents the higher contribution to the total detection efficiency (black circles),specially at low energies. The double triggering efficiency has a small contributionat low energies, but rises up for higher energies. The single and double detectionefficiency together represent around 90% of the total. The rest of the triggeringefficiencies are small and all of them rise linearly with the energy with different


Figure 5.3: Detection efficiency as afunction of energy for the Gasp array. Thetriggering efficiency is plotted until fifthorder. The total detection efficiency isplotted in black circles. Each order of thetriggering efficiency was fitted to an ex-pression according to Eq.(5.34), the con-tinuum lines are the fitted functions.

slopes. For orders higher than five the contribution is small and then neglected inthis work.

The energy dependence of the detection efficiencies is very strong and all of themhave a clear tendency. It is necessary to have the detection efficiency for any energyvalue. Instead of calculating it for the entire energy range, each curve is fitted toa mathematical expression. The triggering efficiencies up to fifth order were fittedwith different functions, the resulting expressions are:

Ω(1)(Eγ) = −0.16× log(Eγ) + 0.75,

Ω(2)(Eγ) = −0.32× exp

(

− Eγ

2.20

)

+ 0.32,

Ω(3)(Eγ) = 0.012× Eγ − 0.0062,

Ω(4)(Eγ) = 0.0034× Eγ − 0.0052,

Ω(5)(Eγ) = 0.00071× Eγ − 0.0013,

ΩT (Eγ) = Ω′(Eγ) + Ω′′(Eγ) + Ω′′′(Eγ) + Ω(4)(Eγ) + Ω(5)(Eγ).

(5.34)

Another quantity that characterizes the array is the corrected total triggeringefficiency ΩT (1+Fk). It was calculated using the triggering efficiencies of Eq. (5.13)and the results are plotted in Fig. (5.4). This quantity is necessary to calculatethe mean value 〈k〉 in Eq. (5.19).

It is very important to understand the differences between the pulse height recordedfor an individual detector, hκ, and the total pulse height of the array Hλ. Fig. 5.5


Figure 5.4: Total triggering efficiency corrected as is shown in Eq. (5.13),using the values of the triggering efficiency simulated up to 12 MeV.

shows the spectra recorded for monoenergetic γ-rays of Eγ = 10 MeV normalizedto the maximum of the photo-peak, for different cases:

Figure 5.5: Spectra for monoenergetic γ-rays of Eγ = 10 MeV. To observe thedifferences between the pulse height recorded for an individual detector hκ andthe total pulse height of the array Hλ, considering the Ge and BGO detectors

as two different sets.

(a) Efficiencies∑

i ΩBGOi (hκ) or probability of recording pulse heights hκ with

just one BGO detector.


Figure 5.6: Generalized efficiency as afunction of energy depending of the de-tector type Ge or BGO and the total.

(b) Efficiency of the set of BGO detectors ΩBGOH (hκ, Eγ) recording total pulse

heights HBGOλ .

(c) Efficiencies∑

i ΩGei (hκ) or probability of recording pulse heights hκ with just

one Ge detector.

(d) Efficiency of the set of Ge detectors ΩGeH (hκ, Eγ) recording total pulse heights

HGeλ .

The difference in the efficiency as well as the resolution between the BGO andGe detectors are evident in the visualization of the spectra (Fig. 5.5), but nodistinction is done in the process of determining the response function.

Finally the last quantity that characterizes the array and is possible to measureis the generalized efficiency. The distinction between the generalized efficiencymeasured with a BGO or Ge detector has to be realized. The energy deposited ina detector depends on the energy absorption mechanism. That is the reason whythe generalized efficiency depends on the detector type, Ge or BGO. Fig. 5.6 showsthe generalized efficiencies for both types of detectors in a wide range of transitionenergies Eγ . The total generalized efficiency will be:

Ω(1)Eγ

= Ω(1)Eγ(BGO) +Ω

(1)Eγ(Ge), (5.35)

The curves in Fig. 5.6 show the importance of the BGO detectors (red squares)whose generalized efficiency is around 85% of the total generalized efficiency (blacktriangles). The data obtained of generalized efficiencies for the Ge detectors (bluepoints) is lower compared to the data obtained for the BGO detectors. The dif-ference is even larger for energies below Eγ = 2 MeV. In this case there is nota simple function to fit the data across the entire energy region of interest. To


obtain the generalized efficiency for any value two regions were selected accordingto their energy:

Ω(1)Eγ

=

−0.075× Eγ + 0.81, for Eγ ≤ 2 MeV,

0.002× Eγ + 0.64, for Eγ > 2 MeV., (5.36)

This mathematical expressions will help to obtain the mean value 〈H〉 usingEq. (5.28) for any energy value.

5.7 Response functions PN(ME, kH) of Gasp

It was discussed in previous section that for an event with certain multiplicity Mand total energy E after the detection process each pair (M,E) is transformedinto a (k,H) pair. This transformation depends on the response function of thearray PN (ME, kH),

(M,E) ⇒ PN (ME, kH) ⇒ (k,H). (5.37)

An example of the process to experimentally obtain a distribution probabilityis described with the two plots of Fig. 5.7. All the energies from now on willbe given in MeV. A selection of an event with a given energy and multiplicity(M,E) = (5, 5) is plotted as a black square in Fig. 5.7(a), the events laying in aregion with ∆E = 0.5 MeV and the same M value. The distribution probabilityobtained with the simulation after the detection process is plotted in Fig. 5.7(b).The distribution shows a probability cloud with some specific characteristics. Thecharacterization of the probability cloud will be discussed later.

Figure 5.7: (a) Event with (M,E) = (5, 5). (b) Its distribution probabilityPN (M = 5, E = 5; kH) plotted in a (k,H) map.

The generation of the complete set of probability distributions for each possible(M,E) pair to construct the response function PN (ME, kH) is a delicate process.


It was implemented as an experimental procedure in Ref. [54]. In this work thedata were generated by the Gasp simulation as follows:

1. It is necessary to determine which is the response function for events withinany set of (M,E). This can be done by measuring monoenergetic γ-sourcesof all the γ-energies Eγ and sort the measured γ-rays in cascades with all themultiplicity M values. Experimentally it is not possible to do this, becausethere are not enough monoenergetic γ-sources to describe all the possible en-ergies. Also if one of these sources is measured with the array the backgroundradiation adds wrong information to the measured response function.

An alternative to monoenergetic γ-sources is to use 2-γ sources. With the2-γ sources and a multi-detector array it is possible to measure both γ-rays and build several events with the energies for the γ-transitions and anymultiplicity M . The exact procedure is explained below.

2. A set of five 2-γ sources were simulated, the nuclides and energies of bothγ-rays are registered in Table 5.1.

nuclide Eγ1(MeV) Eγ2

(MeV)75Se 0.126 0.265207Bi 1.064 0.57060Co 1.173 1.33288Y 0.898 1.83624Na 2.754 1.369

Table 5.1: γ-ray energies for the γ-ray sources used to find the detectionefficiencies.

3. A 2-γ source is placed inside the detector array. Both γ-rays are emittedsimultaneusly. Then a gate selection is performed, when one of the γ-raysis completely detected in just one detector. This ensures that the eventsrecorded in the other detectors correspond to the companion of the γ-raygated. The interaction of the second γ-ray is recorded, its detection presentsall the characteristics described in Sec. 4.2: It could escape the array, itcould have an incomplete detection, or any other interaction. The numberof detectors fired for each γ-ray is ki each one with an energy pulse heighthi.

4. Taking a group of γ-rays from the 2-γ sources with the same energy Eγ ispossible to recreate an entire cascade. The number of γ-rays selected givesthe multiplicity M of the cascade, and the resulting pulse heights for eachγ-ray hi will be added to obtain the total pulse height H. The k value is setby all the detectors that were fired by the individual γ-rays ki.


5. Cascades of multiplicity between M = 2 to M = 50 were generated with the10 γ-energies from the five 2-γ sources in Table 5.1. The (M,E) cascades willhave total values (M,M · Eγ) and they generate the corresponding (k,H)distribution probability.

Contrary to the multiplicity which allows the selection of any value, in the caseof energy it is possible to select just a reduced set of values because the numberof different transition energies Eγ in the Table 5.1 is only 10. The distributionprobability of those pairs (M,E) no obtainable with the γ-rays from the sources,are interpolated with the two nearest distributions. A detailed explanation of thiswill be given in Sec. 5.7.1.

M

E

0 10 20 30 40 50

10

20

30

40

50

Figure 5.8: Limits of the interpola-tion due to the lowest and highest val-ues of the γ-rays in the 2-γ source, forevents in the red region is not possibleto interpolate probability distributions.

There are some limitations in the distribu-tions that can be generated by the interpola-tion due to the maximum and minimum en-ergy values of the γ-rays of the 2-γ sources.For example for M = 10 the distributionwith minimum total excitation energy valuethat can be generated has:

E = 136 keV×M ≈ 1.4 MeV,

because Eγ = 136 keV is the lower energy ofthe γ-rays from the 2-γ sources. The samehappens with the highest γ-ray, to createan event with M = 10 it will have a totalexcitation energy maximum of:

E = 2754 keV×M ≈ 27 MeV.

The same happens for all the multiplicities. Fig. 5.8 shows the region (in blue)where all the values of (M,E) were generated, and the region for the values of(M,E) which no distributions can be generated (in red). This limitations are notcrucial for this work because the region of interest is located at high excitationenergy and high multipolarities.

5.7.1 Construction of the entry regions

To generate the response function PN (ME, kH) for all the (M,E) pairs it is nec-essary to interpolate the (k,H) distributions obtained with the 2-γ sources. Eachprobability distribution can be represented as a surface of a three-dimensional dis-tribution (x, y, x) = (k,H, PN (ME, kH)). To describe the surface a set of contours


are generated using Gnuplot [63]. The automatic algorithm generates individ-ual segments making up a contour line, determined by subdividing the surface intosmall triangles and computing the contours assuming a linear variation on each tri-angle generating continuous strings of segments. Each contour is an equipotential,all the points in the contour have approximately the same intensity.

Fig. 5.9(a) shows a distribution with three contours, some non-integer values ofk result from the contour generation algorithm but it does not mean that thosevalues have physical interpretation. Later during the unfolding procedure it isnecessary to replicate the distribution probability for each (M,E) pair. For thispurpose each contour is fitted to an ellipse with a set of five parameters: Theminor axis a, the major axis b, the center coordinates (kc, Hc) and the pose angleθ (See Fig. 5.9(b)). In this work nine contours are usually used to describe the

6 8

10

10 8 6 4 2 0 0 2 4

6 8

10

10 8 6 4 2 0 0 2 4

(b). Contour fitting to an ellipse.(a). Contour selection.

k

H(M

eV)

k

H(M

eV)

ab

(kc, Hc)θ

Figure 5.9: (a) Generation of three contours for the probability distributionPN (M = 5, E = 5; kH). (b) Contour fitting to an ellipse with parameters

(a, b, kc, Hc, θ).

distribution probabilities for each (M,E). The most inner contour represents anintensity of 90%, the successive external contours will have 80%, 70% and so on.Finally the most external contour represents 10% of the total intensity.

Two different ways are used to characterize these distributions. The first method isvery simple and is named here as “The zeroth-order approximation”. The secondone implies a more accurate way to fit the distributions and uses a least squaresprocedure.

5.7.1.1 The zeroth-order approximation

The zeroth order approximation uses five parameters (kc, Hc, a, b, θ) of the ellipsefitting as it is shown in Fig. 5.9(b), the initial parameters are obtained in thefollowing way:


1. (kc, Hc): The center of the ellipse is obtained just by taking the mean of the(k,H) points position in a given contour.

kc =

m∑

i=0

kim, and Hc =

m∑

i=0

Hi

m, (5.38)

where m is the number of points in the contour.

2. θ: The pose angle is calculated by fitting all the contour points to a straightline, it is important to do this after the ellipse is centered in (kc, Hc). Thenthe points are rotated to the angle subtended between the line and the k axisto find the length of the axes.

3. a: Minor axis length. It is calculated when the ellipse is centered at (kc, Hc) =(0, 0) and the ellipse has no pose angle (θ = 0). Once the ellipse is in theright position the maximum km value is taken as the minor axis.

4. b: Major axis length. Also calculated when the ellipse is centered at (kc, Hc) =(0, 0) and the ellipse has no pose angle (θ = 0). In this case when the ellipseis in the right position the maximum Hm value is taken as the major axis.

Figure 5.10: Using the zeroth order approximation to fit some contours. Thefit data are transformed to arbitrary coordinates (x, y) just for the implemen-

tation process.

Some tests were carried out in order to probe the accuracy of the program. Fig. 5.10shows some cases, from a well positioned ellipse to a real contour, extracted fromtheGasp simulation. The ellipses are transformed into arbitrary coordinates (x, y)


to fit the data points. Plots (a) and (b) in Fig. 5.10 show fits of points distributedon a more or less well defined ellipse. Fig. 5.10(a) is centered with pose angle zero.This is the simplest ellipse tested. Fig. 5.10(b) is tilted a small angle, but alsocentered. The points in Fig. 5.10 (c) and (d) come from two different contours ofsimulated (k,H) distributions selected randomly.

The fitting algorithm approximation is good enough if just one contour is consid-ered. The ellipse does not fit exactly the data points but it clearly follows thedata. This procedure was tested for different contours of different distributions.Then it is important to focus on having a good description of the contours withthe ellipses and to see if they also describe the (k,H) distributions.

In most of the cases the ellipse describes very well the contour, however sometimeswhen various contours are considered together, the fitting ellipses cross each otheras in Fig. 5.11(a), in that case it can not be possible to assume that those ellipsesrepresent the distribution. This problem can be easily solved if just five contoursare considered. The fitted distribution in Fig. 5.11(b) have the five contours andthey describe very well the distribution. The five fitting ellipses satisfactorilyrepresent the nine contours. This is acceptable since the main interest is to describeaverage values in the distribution, not the exact local values.

(a) (b)

Figure 5.11: Distributions fitted with the zeroth order approximation: (a) For(M,E) = (20, 10), using 10 contours. Some of the ellipses cross each other. (b)For (M,E) = (20, 11), using just five contours. The ellipse crossing is reduced

and still there is a good description of the distribution.

5.7.1.2 The Least-Squares Orthogonal Distance Procedure to fit (k,H)distributions

A more sophisticated procedure to fit the (k,H) distributions is by using allthe points in each contour to fit the ellipse. One method is described as the


Least-Square Orthogonal Distance Procedure (LSOP) with distance fitting algo-rithm [64]. In this procedure the squares sum of the error distances is minimized,the distances are defined with the orthogonal, or shortest, distances from the pointto the ellipse. They are described with the same five parameters used for the ze-roth order approximation (kc, Hc, a, b, θ). Fig. 5.12 shows a pictorial description

Figure 5.12: Ellipse centered in (xc, yc) with axis lengths a and b and poseangle θ. The data point (xi, yi) is connected with the ellipse by a segment (blueline) which is the shortest distance to the ellipse. The orthogonal contacting

point is (x′

i, y′

i).

of the ellipse and the parameters used in the fitting procedure. The process startsin the following way:

1. Transform the points (ki, Hi) into a new coordinate system (xi, yi) where thepoint (0, 0) will be in the ellipse center (xc, yc). Then the ellipse is rotatedand positioned with a pose angle θ equal to zero. This reduces the numberof parameters to just two, the axes lengths a and b.

2. Locate the nearest corresponding point of the data to the ellipse, i.e. theorthogonal contacting point (x′

i, y′i) on the ellipse. It must simultaneously

satisfy:

(a) The equation of the ellipse in a standard position:

k2

a2+

H2

b2= 1, (5.39)

(b) The connecting line between the two points (blue line in Fig 5.12), whichis perpendicular to the ellipse.

3. To evaluate the position of the data points the zeroth order approximationfit was used for the initial parameters.


4. If the data points have a value close to zero the numerical method is unstable.It is necessary to ensure that all the points have both of its componentsdifferent to zero (xi, yi) 6= (0, 0). Also each contour must have at least5 different points because there are five parameters. Usually for the innercontours it is necessary to create some extra points. This is done by includingintermediate points between each pair of contiguous points.

5. Then it is possible to apply the generalized Newton method [64] to get thebest fit of the data points to the ellipse.

6. Finally, to calculate the error distances vector in order to evaluate the fitcertainty, it is necessary to use the backward transformation to the originalcoordinate system (k,H).

(a) (b)

Figure 5.13: Fitting of the distributions using the LSOP. (a). For (M,E) =(5, 6), the ellipse fit with 10 contours represents very well the distribution.(b). For high energy and high spin values (M,E) = (50, 50) the fit does notrepresent the distribution. In this cases the zeroth order approximation must

be used.

Fig. 5.13 shows two examples of the application of the LSOP for some distributions.Fig. 5.13(a) is the distribution probability for (M,E) = (5, 6), the procedure fitsvery well each contour and also the distribution. The result is not so good fordistribution probabilities with higher excitation energy and high spin values. Forexample for (M,E) = (50, 50) in Fig. 5.13(b). This type of distributions presentseveral problems because the number of events is very low and the contours are notwell defined, the LSOP ends up with a fit of each contour which does not representthe distribution. In such cases, it turns out that the zeroth order approximationperforms correctly the fit.

5.7.1.3 Interpolation process

Before the interpolation of the probability distributions is performed it is very im-portant to normalize each distribution to the number of events. Once the original


3

4

5

6

7

8

9

10

11

12

12 13 14 15 16 17 18 19 20 21 22 23

H(M

eV

)

k

Figure 5.14: In blue, interpolation of the (M,E)=(20,10) distribution withthe two red ones. The upper one at (M,E)=(20,11) and the lower one at

(M,E)=(20,6).

distributions (obtained with the 2-γ sources) are fitted the interpolation processcan be completed. During this process a selection of the nearest two sets of (M,E)values, one above and the other below the point to interpolate with the same M ,is done.

Both distributions must have the same number of contours because the interpola-tion is done contour by contour. Then the new set of ellipses are transformed intoa surface with the corresponding equipotential contours. Fig. 5.14 shows in bluethe distribution of (M,E)=(20,10) interpolated with the two red distributions, theupper one at (M,E)=(20,11) and the lower one at (M,E)=(20,6).

The process is carried out many times for each (M,E), and all the possible distri-bution probabilities PN (ME, kH) are produced. Fig 5.15(a) shows various distri-butions obtained by interpolation for several (M,E), with equal numerical valuesof M and E. Fig 5.15(b) and (c) shows the projections on the H and k axes.

5.8 Generation of reverse responses P ′N(kH,ME)

The reverse responses will give the probabilities of each (M,E) pair to contributeto a given (k,H) pair, i.e., an event or cascade detected with certain (k,H) thatwould come from a particular event (M,E) but with certain probability given bythe reverse response P ′

N (kH,ME). With the response functions PN (ME, kH)the reverse responses P ′

N (kH,ME) can now be calculated. To achieve that, theanalysis is done by looking in all the response function matrices for the events witha specific (k,H) value and a new distribution is built up with these events andtheir corresponding weights.


(a).

(c).

(b).

Figure 5.15: (a). Some PN (EM,Hk) distributions obtained by interpolationfor different pairs of (k,H). And the respective projections: (b) in the H axis,

and (c) k axis.

The new reverse response functions have to be normalized to the number of eventsin the distribution. One important fact to pay attention during the normalizationprocess is that the number of cascades in each response function is different. Thisis because cascades of different multiplicity were artificially created with the samenumber of γ-rays (107) in the Gasp simulation. Fig. 5.16 shows a sequence ofthe number of cascades generated with the same number of γ-rays. For M = 2it is possible to construct 107/2 cascades, for M = 3 it is possible to construct107/3 cascades. Then the number of cascades with the same M that is possible toconstruct will be given by 107/M . This quantity is the number used to normalizethe distributions. The reverse responses obtained for some pairs (k,H) are plottedin Fig. 5.17. These distributions represent the probability that one cascade with(k,H) comes from a given cascade with values of total excitation energy andmultiplicity (E,M).


Figure 5.16: Process of grouping 107 simulated events in cascades with mul-tiplicity M . The number of cascades that is possible to arrange with the same

M is 107/M .

(a).

(c).

(b).

Figure 5.17: (a). Some reverse responses P ′

N (kH,ME) distributions, and therespective projections: (b) in the E axis, and (c) in the M axis. For pairswith (k,H) = (5, 5), (10, 10), (20, 20), (25, 25) and (30, 30). These distributions

describe the region in (M,E) which populates a given (k,H).


5.8.1 Iterative least-squares unfolding procedure

With all the possible responses PN (ME, kH) and reverse responses P ′N (kH,ME)

for all different set of cascades the unfolding procedure can be started followingJaaskelainen et al. [54]. To deconvolute a given measured event the followingequation system has to be solved:

Q(k,H) =∑

M,E

PN (ME, kH)R(M,E) (5.40)

wereQ(k,H) is the measured ”experimental” population for a given (k,H). R(M,E)is the deduced population distribution for a given (M,E), the real distribution ofevents from the reaction. The relation between Q(H, k) and R(M,E) is given bythe reverse response functions P ′

N (ME, kH), the distribution in the (M,E) spacethat contributes to a given point (k,H).

1. Initial estimate of the entry population R(E,M).

The method of solving Eq. (5.40) uses an iterative least squares unfolding.As in every iterative process, an initial estimation of the function to fitis necessary. To obtain this the mapping transformation T = ((M,E) →(〈k〉, 〈H〉)) is used. In this case R(E,M) is calculated as the initial estimationusing the reverse mapping T−1 = ((k,H) → (〈M〉, 〈E〉)) (see Sec. 5.4) toshift the measured population Q(k,H).

Every event measured with a pair Q(kκ, Hλ) is implanted to the correspond-ing pair (Mi, Ej) using the reverse mapping T−1. A graphic example of suchtransformation is shown in Fig. 5.18. This process is done point by point:

(Mi, Ej) = T−1(kκ, Hλ)

R1(Mi, Ej) = Q(kκ, Hλ)(5.41)

The transformation T−1 is performed according the reverse mapping usingfor the first trial values of M1 = k and E = H/0.75. Then the original valuesof Q(kκ, Hλ) are given to their corresponding values R1(Mi, Ej). After thisprocess some of the R1(Mi, Ej) values could be empty due to the distributionof the reverse mapping. In this case each point in the new distribution willbe spread to its neighboring points using a bi-variate Gaussian distribution.

2. First fit.

The first fit F1(k,H) to Q(k,H) is calculated using the initial estimate calledR1(Mi, Ej) in Eq. (5.40). This is, applying the response functions for every


Figure 5.18: Transformation of an event measured as Q(k,H) to an event ofthe initial estimate R1(M,E) via the reverse response T

−1.

(M,E) to the initial estimate R1(M,E):

F1(k,H) =∑

M,E

PN (ME, kH)R1(M,E) (5.42)

This step is the mathematical equivalent to measure the data with the Gasp

array, because the response functions are specific for this array.

3. Improvement of the estimated intensity.

The first fit and the initial estimate are very important to start the iterationprocess. Now is necessary to improve R1(M,E). Estimation of how well it isfitted is done by comparing the distributions point by point and calculatingits χ2. But before, an improvement of the estimated intensity can be added.To improve the initial estimate the ratio matrix between it and the first fitis calculated point by point as:

B1(M,E) =R1(M,E)

F1(k,H)=

R1(M,E)

R′1(k,H)

, (5.43)

Then, it is essential to transform the first fit F1(k,H) into a new distributionR′

1(M,E) using the reverse mapping in the same way as it was done to obtainthe first estimate. With the transformation completed the ratio will give ascaling factor, which will be used in the second estimation. The reverseresponse functions P ′(kH,ME) should be modified point by point by therelative probabilities of each (M,E). Since all distributions are normalized


to the total number of events the improved function is:

P ′1(kH,ME) =

R1(M,E)P ′(kH,ME)∑

M,E R1(M,E)= R1(M,E)P ′(kH,ME) (5.44)

To use in an appropriate way the new reverse response functions it is re-quired to construct a coordinate transformation table for each (k,H). Thecorresponding P ′

1(kH,ME) distribution is taken to evaluate the expectationvalues (〈M〉, 〈E〉). Table 5.2 shows a schematic organization of such a table.

(k1, H1) (〈M〉, 〈E〉)11(k2, H1) (〈M〉, 〈E〉)21

... ...(ki, Hj) (〈M〉, 〈E〉)ij

... ...(kimax

, Hjmax) (〈M〉, 〈E〉)imax,jmax

Table 5.2: Schematic table of the coordinate transformation obtained fromP ′

1(kH,ME).

4. Second estimate.

The second estimate R2(M,E) is calculated using the scaling factor and thecoordinate transformation with the new reverse responses P ′

1(kH,ME) inthe following way:

R2(M,E) = B1(ME)Q′(M,E) (5.45)

where

Q(k,H) → Q′(M,E) is transformed via P ′1(kH,ME) (5.46)

for each new estimate we use the response function to check the result, as wedid before with the first estimate. Generating the second fit F2(k,H) withthe response function means solving:

F2(k,H) =∑

M,E

P (ME, kH)R2(M,E) (5.47)

5. χ2 calculation.

To control the fit process the chi-square χ22 is calculated as:

χ22 =

∑

ij

[F2(ki, Hj)−Q(ki, Hj)]2, (5.48)


depending on the χ22 value a second iteration of the entire procedure is re-

quired, starting with the re-calculation of the scale factor. With the firstiteration the value of the chi-square obtained is χ2

2 = 0.0026. Both themeasured population Q(k,H) and the resulting distributions for the secondestimate R2(M,E) are plotted in Fig. 5.19.

(a). (b).

Figure 5.19: (a) Measured valuesQ(k,H) of 154Dy using the response functionof Gasp. (b) Second estimate R2(M,E) of the “real” values (M,E).

The difference between the final fit F2(k,H) and the measured data Q(k,H)can be quantified also with their expectation values and standard deviations,tabulated in Table. 5.3. The distribution of the F2(k,H) obtained with theunfolding up to second order is narrower than the measured data Q(k,H).The standard deviation is lower in one unit in both k andH. The expectationvalues of both distributions are very close. Considering that the unfolding isa very complex process the differences are small.

(〈k〉, 〈H〉) σ(k,H)Q(k,H) (〈21〉, 〈15〉) (4,3)F2(k,H) (〈23〉, 〈16〉) (3,2)

(〈M〉, 〈E〉) σ(M,E)R2(M,E) (〈26〉, 〈18〉) (4,3)

Table 5.3: Expectation values for the measured data. The second fit and theresulting second estimate with their standard deviations.

The distribution R2(M,E) represents the entry states from where the values(k,H) were measured. The intensity of the resulting distribution is verysmall but it is a good estimate of the entry states.

CHAPTER 6

Hk-EOS METHOD APPLICATION TO 154Dy

All the tools that permit to apply the new combined Hk-EOS method were dis-cussed in the previous Chapters, and will be used in this Chapter to obtain in-formation about the nuclear continuum of 154Dy. The interest of evaluating thismethod for the nucleus 154Dy was addressed before in Sec. 2.4. This process wasdone by a code set developed for this work, detail information about the imple-mentation and use can be founded in Ref. [39].

One advantage of having a simulation of the detection process is the possibility ofdetermining how large are the (I, E) regions being populated by one pair (k,H) inthe distribution probability (Sec. 6.1), and if individual EOS can be extracted fromthose regions (Sec. 6.2). Another advantage is to be able to fit the level densityparameter a from the measured EOS, then compare it to the original value set inthe simulation.

In this chapter an evaluation is done of how good this technique is to allow theextraction of information from the nuclear continuum. Together with the analysisof the influence of some experimental issues as: the detector type, the reactionprocess, the channel selection, and the effects of the neutron emission.

6.1 (k,H) distributions

The (k,H) distributions were constructed from the γ-rays, measured with theGasp simulation, of the cascades coming from the 154Dy entry states, simulated

63

Chapter 6. The Hk-EOS technique for 154Dy 64

for the reaction 122Sn(36S,4n)154Dy with Pace2-Gamble (Chap. 3). The entrystates accuracy depends on the set of parameters included in the simulation. Thefirst approach is to consider 154Dy with a constant value of density parameter forall the (I, E) region. Some of the parameters used to generate the entry states are:

• Level density

The level density ρ is calculated, using the Fermi-gas model:

ρ(U, I) =

√a

24

(~2

2J

)3/2

(2I + 1)exp (2

√aU)

(U + 3

2kT)2 , (6.1)

Where the level density parameter a is written in terms of the mass numberA, and surface effects are taken into account [22, Eq.(4.5)]:

a =A

14.61(1 + 4A−1/3) = 18.4 MeV−1. (6.2)

The thermodynamical temperature of the Fermi gas is calculated as:

kT =

√

U

a+

9

16a2+

3

4a= 0.66 MeV. (6.3)

The intrinsic excitation energy U = 7 MeV is used, because it should bearound the neutron binding energy.

• Gamma strength

The E1 gamma strength is calculated with the GDR gamma strength usingEq. (2.5) for deformed nuclei:

ΓGDR(E1, Eγ) =σ

3(π~c)2E4

γ

2∑

k=1

wkΓk

(E2k − E2

γ)2 + Γ2

kE2γ

, (6.4)

using the empirical parameters for the peaks energies Ek and widths Γk, andweights, wk, of Table 6.1 taken from Ref. [22, Table 1].

k ωk Ek (MeV) Γk (MeV)

1 0.4 78 ·A−1/3 − 3 · 0.6 2.5

2 0.3 78 ·A−1/3 + 3 · 0.4 5

Table 6.1: Empirical parameters used in Gamble for the gamma strength ofa deformed nucleus. (Taken from [22, Table 1]).


• Structure

The value Q0, for calculating the quadrupolar moment of each transition,was obtained by taking the mean value from those measured by Azgui etal. [65], Q0 = 4.67e · b. The transitional Qt is then calculated as:

Qt = Q0

(

1 + dQ0

[

R(0, 1)− 1

2

])

, (6.5)

where dQ0 = 1 e · b and R(0, 1) is a rectangular distribution of randomnumbers in the interval (0, 1) to generate all the possible Qt values.

Figure 6.1: (a) Entry states of 154Dy generated with Pace2-Gamble. (b).The distribution probability in (k,H) for the entry states measured with the

Gasp simulation.

The entry states generated are plotted in Fig. 6.1(a), together with the yrastvalues used to delimit the states accessible to the system. The entry states of154Dy are distributed between 20 < I < 50 values of spin. The maximum ofthe distribution has an intrinsic excitation energy about U = 7 MeV and thedistribution spreads around the constant intrinsic excitation energy line (blackline). The total distribution probability of the (k,H) values of the cascades thatdecay from the entry states is plotted in Fig. 6.1(b). The maximum probability isgiven for a value (k,H) = (22, 15).

The (k,H) distribution is affected experimentally by many factors, as detectorproperties, efficiency and resolution, and Doppler effect. Also the evaporated par-ticles may affect the distribution. A detailed analysis of the distribution probabilitywill be undertaken in the following sections.


6.1.1 Influence of the Ge and BGO detectors

It is important to analyze how much of the (k,H) distribution is detected byeach type of detectors. For this purpose the γ-rays were discriminated duringthe analysis process whether they were detected by the Ge or BGO detectors.Fig. 6.2(a) shows the distribution probability in (k,H) for the entry states of 154Dy,for each detector type. The set of 40 Ge detectors generates the small distributionwith values of energy and spin between 0 and 5. Due to the low efficiency of theGe detectors, the number of detectors fired by one cascade is always very low andthe amount of energy lost is very high, around 90%.

Fig. 6.2(a) also shows the distribution probability measured with the 80 BGO de-tectors plotted with the red contours, which can be compared to the distributionprobability obtained with the total array (blue contours). Both of them are dis-tributed more or less around the same values. Of course the slight shift to lowervalues is because the Ge are not considered. Even though, the small quantitiesof energy collected by the Ge detectors moves up the total distribution in energyabout 2 MeV and in k about 3 units. This means that if the energy deposited inthe Ge detectors and the number of Ge detectors fired by the γ-rays of the eventis included in the total (k,H) distribution, then the total energy E of each eventcould be measured more exactly. However, the effect of having a larger k does notmean a more accurate measurement of M .

(a) (b) .

Figure 6.2: (a) (k,H) distributions obtained with each type of detectors Geand BGO, and the distribution obtained with the total array. The contributionof the Ge detector is small but it is important to have the better measurementof the total energy of the event. (b) Measured (k,H) distributions with the

BGO detectors and different resolutions regions. See the text for details.

It was already established that the BGO detectors have a low energy resolution,then it is interesting to evaluate the influence of the detector resolution in the mea-surement of the distribution (k,H). Since the simulation includes a parameter to


set the FWHM of each type of detector (See Sec.3.3.3), it is possible to change theresolution of the BGO detectors, by decreasing the parameter b in the expression:

FWHM = b√

Eγ(keV ), (6.6)

to measure the γ-rays with a higher resolution. Fig 6.2(b) shows two (k,H) distri-butions obtained with different b parameter: The one that was measured with lowresolution detectors (b = 4 MeV−1) and the other measured with higher resolutiondetectors (b = 0.1 MeV−1). The main effect of modifying the resolution is thatthe ellipse becomes thinner for a lower value of the resolution.

6.1.2 Doppler effect influence

Another experimental issue is the shift of the γ-rays energy due to the Dopplereffect. The shift in energy is given by Eq. (3.4). In this case v = 0.0219 c. Thenfor a given γ-ray of energy Eγ the energy shift will be:

∆E = 0.0219 · Eγ cos θ. (6.7)

Usually the energies have to be corrected by subtracting the shifts to every de-tected γ-ray. To check how the Doppler effect influences the (k,H) probabilitydistribution it is a good idea to analyze first the energy spectrum.

With the simulation is possible to set the velocity of the residual nuclei to differentvalues. This option was used to generate spectra with two residual velocities v = 0and v = 0.0219 c. Setting the residual velocity to zero permits to have a gammaspectrum without shift.

Fig 6.3 shows a comparison of the spectra measured with the BGO and Ge de-tectors with two residual velocities, labeled “(No-Doppler)” in the case of v = 0and “(Doppler)” when v = 2.19%c. The differences due to the energy shift areappreciable only in the low energy region and specially for the Ge detected spec-tra. An additional shift of Eγ = 100 keV is artificially added in the “(Doppler)”spectra to emphasize those differences. Despite the extra shift the differences areappreciable only for Eγ < 1 MeV. In the region of high energies Eγ > 1 MeVthere is practically no difference, this region in the spectra is formed mainly withthe γ-rays coming from the nuclear continuum. It means that for those γ-rays theDoppler effect can not be clearly identified, i.e. in that region the spectral shapeis the same with and without Doppler shift.


Figure 6.3: Spectra of the detection with the BGO and Ge detectors. Witha residual nucleus with velocity v = 0.0219 c (Doppler) and considering theresidual nucleus at rest v = 0 (No-Doppler). To observe the effect of the energyshift the (Doppler) spectra were shifted manually 100 keV. To observe thateven with this large shift there is no difference in the high energy part of the

spectra.

6.1.3 Reaction channels

As it was explained in Sec. 3.1 the entry states of 154Dy were populated using thereaction:

122Sn(36S,4n)154Dy, (6.8)

and the three main reaction channels are:

3616S →122

50 Sn

153Dy + 5n154Dy + 4n152Dy + 6n

(6.9)

In the simulation it is possible to restrict the entire process to the desired reactionchannel, in this case for 154Dy. But to recreate a better approach to the realreaction the two other channels were considered as well as the emission of neutrons.

First, to generate the entry states for each residual nucleus it is necessary toconsider the excitation energy of the compound nucleus. For a beam energy ofElab = 148 MeV, the compound nucleus has an excitation energy of E∗

NC =64 MeV. Some of this energy is invested in the emission of particles, and thisprocess leaves the residual nucleus with an energy E. For the reaction and decaychannels considered here, the emitted particles are only neutrons. The neutronenergy distribution plotted in Fig. 6.4 gives the probability of the emission of aneutron with energy En. This distribution is generated with Pace2 and the datapoints were fitted to a Maxwell-Boltzmann distribution. The figure shows that the


most probable kinetic energy for the evaporated neutrons is 1 MeV. The equationthat was fitted to the neutron evaporation distribution is:

Pneutrons =ct

σ3/2

√

En(MeV)− µ× exp

(−(En(MeV)− µ)

σ

)

(6.10)

where the parameters fitted were σ = 1.36 MeV and µ = 0.35 MeV. This equationwas used to generate the energies of the neutrons in the simulation.

Figure 6.4: Energy distribution of emitted neutrons from 158Dy in the reaction(See Eq. (6.8)), generated by Pace2. The data points were fitted to a Maxwell-

Boltzmann distribution.

In addition, for every neutron emission part of the energy is invested in separatingthe neutron from the rest of the nucleus, for that the binding energies Bn haveto be calculated for each emission. Table 6.2 shows some values of Bn for thereaction taken from Ref. [66]. The Btotal

n values of the table corresponds to thetotal amount of energy that have to be invested to extract 4, 5 or 6 neutrons fromthe compound nucleus in each reaction channel.

From To Bn(MeV) Btotaln (MeV) (E∗

CN −Btotaln )(MeV)

158Dy 157Dy 9.05157Dy 156Dy 6.96156Dy 155Dy 9.44155Dy 154Dy 6.83 32.30 31.79154Dy 153Dy 9.32 41.61 22.47153Dy 152Dy 7.09 48.71 15.38

Table 6.2: Neutron binding energies required to extract neutrons from someof the isotopes of 158Dy. The values Btotal

n correspond to the total amount ofenergy necessary to extract 4, 5 and 6 neutrons from the compound nucleus [66]

If the energy Btotaln is subtracted from the compound nucleus excitation energy

E∗CN , this energy difference, minus the E value of the corresponding cascade, will


Figure 6.5: Entry states of 154Dy,153Dy and 152Dy generated byPace2-Gamble. The blue line rep-resents the (E∗

NC − Btotal

n ) limit foreach channel reaction. Yrast lines foreach nucleus were plotted to have areference of the entry states distribu-

tion.

be distributed among the respective number of evaporated neutrons following thedistribution of Eq. 6.10. It is important to extract the energies of the emittedneutrons in each cascade to consider the effect in the experiment.

Calculating the binding energy Btotaln for each reaction channel and subtracting it

from the excitation energy E∗NC of the compound nucleus helps to generate the

cascades from the entry states with the appropriate values. The value (E∗NC −

Btotaln ) limits the energy-spin phase space, and the resulting entry states for each

reaction channel is plotted in Fig. 6.5. The entry states for each Dy isotopeproduced in the reaction populates different regions in the (I, E) phase space.

The plot in Fig. 6.5(a) shows that the entry states of 152Dy are located very closeto the yrast states. The maximum total energy E is around 12 MeV and the


highest spin values are about 29. The entry states for 153Dy plotted in Fig. 6.5(b)have energy values up to 20 MeV and values of spin up to 45. The distribution ofentry states of 154Dy shown in Fig. 6.5(c) has values of energy up to 25 MeV andthe spin values are distributed at values from I = 20 to I = 50.

With the simulation of the individual channels it is possible to observe the specificregion where the entry states are distributed. However a careful discrimination ofthe evaporation channel has to be done. Such discrimination can be done by gatingin the characteristic discrete lines. Then the selection of the events in coincidencewith a given γ-ray with an energy of one of the discrete transitions is the simplestway to do it experimentally.

Until here there is not any consideration of the effect of neutrons in the measure-ment, but they can generate pulse heights in the BGO and Ge detectors, if they arenot discriminated during the pulse recollection. Then the neutron energy signalgenerated by a neutron interacting with the detectors is mistakenly interpretedas a γ-ray signal and it induces a big change in the measured (k,H) values. Toillustrate this Fig. 6.6 shows in red contours a distribution probability for 154Dyincluding the interaction of the four neutrons emitted in the evaporation processthat leads to the 154Dy residual nucleus. Then the pulse heights and the num-ber of detectors fired generated for all the neutrons recorded were added togetherwith the pulse heights and detectors fired by the γ-rays in the cascade to buildthe (k,H) distribution probability. In the same plot the blue contours representthe (k,H) distribution considering just γ-ray interactions. It is plotted here tocompare the size of the distributions while considering neutrons. The discrimina-

Figure 6.6: Distribution probability in (k,H) for 154Dy in red contours, con-sidering the possible detection with the BGO and Ge detectors of the four neu-trons emitted in each decay process with the energy distribution of Eq. 6.10.The blue contours represent the (k,H) distribution considering just γ-ray in-

teractions.

tion of γ-rays or neutrons can be done by pulse shape discrimination. The aimof constructing the (k,H) distribution including neutron signals is to observe the


possible implications. Now that most of the experimental influences in the con-struction of the distribution was studied, the analysis of the information that canbe extracted from it will be undertaken in the next subsection.

6.1.4 (k,H) projections and distribution moments

The (k,H) distribution considered in this part of the work was built for 154Dywith the signals measured with the Ge and BGO detectors of the Gasp arrayfor the γ-rays of the cascade generated from the entry states (I, E) shown inFig. 6.1(a). Each pair (k,H) of the distribution probability has a well definedcorresponding pair (I, E). The simulation permits to select a small region of(k,H), with ∆H = 0.5 MeV and k fixed, and observe the region where theycome from. Fig. 6.7 shows the distribution probability and the position of threepoints used to extract (I, E) regions: A=(15,10), B=(22,15) and C=(28,19). Theselection was done trying to cover a wide range in k and H, where the statisticshas a maximum. In fact the point B has the maximum value of the distributionprobability.

Figure 6.7: Distribution probability in (k,H) for 154Dy. The small regionslabeled as A=(15,10), B=(22,15) and C=(28,19) are used to select the (I, E)

distributions that populate them with ∆H = 0.5 MeV and k fixed.

Fig. 6.8 shows the distribution of states with values (I, E) that are populated bypoint B in Fig. 6.7. All the events with the same I value were added to constructthe spin projection and then they are normalized to the maximum value. Toconstruct the E projection a bin of 1 MeV was selected to add all the events thatbelong in the same bin, then the projection is normalized to the maximum value.The projections give a characterization of the distribution structure and permitsto evaluate their distribution moments of the (I, E) regions.

The same procedure of extracting (I, E) regions and constructing its projectionsis done for the points A and C. The set of projections are compared in Fig. 6.9.


Figure 6.8: Distribution of the pairs (I, E) that were measured with a pair(k,H) = (22, 15) with ∆H = 0.5MeV . Together with the projections in energy

and spin normalized to the maximum value.

Each distribution has a region that can be differentiated from the others and themaxima are well defined in each case. The distributions of the projections in Iand E do not follow a Gaussian distribution, and the functional shape and othercharacteristics of the projections has to be analyzed using the moments of thedistribution.

(a) (b) .

Figure 6.9: (a) Total projection in I and (b) total projections in E for theselected (k,H) regions A, B and C shown in Fig. 6.7. Normalized to their

maximum value in order to make a better comparison.

A quantitative analysis of the distribution moments will give information on howthe (I, E) distribution was measured with certain (k,H) pair. In order to have awide selection additional points were taken over the black line in Fig. 6.7 for eachk, including the points labeled A, B and C.


(a) .

(b) .

Figure 6.10: (a) Distribution moments up to fourth order of the I projectionsfor different values of k. (b) Distribution moments up to fourth order of E

projections for different values of H.

The resulting values of the moments up to fourth order are plotted in Fig 6.10,where the mean values are plotted in the inset. The variance, skewness and kur-tosis are plotted together. The plot in Fig 6.10(a) has the values obtained fromthe projections in E for different values of M . The relation between the meanvalues of the energy projections, which define the value to which the distributionis centered, and the selected H is almost linear. On the other hand, the variancegives information on how much the distribution is spread. The maximum varianceis 3 MeV and it decreases as H increases. It means that the regions that populatehigh values of H are narrower than the regions that populate low values of H. Theskewness and kurtosis for the energy projections are more or less constant whichmeans that the shape of all the distributions are similar, except in the width.

Fig 6.10(b) is plotted in the same way as Fig 6.10(a), but in this case the momentscorrespond to the projections on I for different values of k. The relation between


the mean values of the projections and k is not linear. It seems to follow twolinear segments separated at k = 24. The variance for the I projections havethe maximum values of 5 for k = 17. After k = 17 the variance diminisheswith increasing k up to 3. This is important because as for the variance of theenergy projections, small variance means better defined regions. The skewness haspositive values up to k = 18, then the skewness is negative. This shows that inmost of the cases all the (k,H) pairs have contributions from almost any value ofI but with different proportions.

After the selection and characterization of the (I, E) regions, it is important toextract some useful information, for instance it is possible to get the γ-ray spectraof such regions. This is the next step to do for the application of the Hk-EOSmethod and will be shown in the next Section.

6.2 Energy Ordered Spectra (EOS) and the leveldensity parameter a

As it was observed in the last section it is possible to select well defined regionsin the (I, E) phase space from a specific point (k,H). It is important to check ifthe EOS that can be constructed with the events in each (k,H) point generatesdifferent spectra. From the cascades that build each (k,H) pair the EOS can beconstructed. Fig. 6.11 shows some EOS built from a different selection of (k,H)pairs. Each plot represents respectively:

(a) The selection of (k,H) pairs was done for H constant and varying k by one.The EOS were built for the pairs (21,15), (22,15) and (23,15). The minimumenergy of the spectra is around 1 MeV, but the maximum energies vary withk. For small k the maximum energy becomes smaller.

(b) Fixing k and building EOS for different values of H for the pairs (22,14),(22,15) and (22,16). The spectra show that the differences of the minimumenergy in each case are more noticeable than having H constant. Also themaximum energies differ significantly. For H = 14 MeV the maximum energyis around Eγ = 4.5 MeV, for H = 15 MeV the maximum energy is aroundEγ = 5.5 MeV and forH = 16 MeV a maximum energy around Eγ = 6.5 MeV.


(a). (b). .

Figure 6.11: EOS constructed from different (k,H) regions. (a) With kfixed and consecutive values of H. (b) Fixing the H value and varying k

with consecutive values.

The EOS can be fitted using the expression of level density of Eq. 2.5 and theGDR gamma strength ΓGDR(Eγ) Eq. (2.5) to:

Iγ(Eγ) ∝ΓGDR(E1, Eγ)× ρ(U, I),

∝ σ

3(π~c)2E4

γ

kdef∑

k=1

wkΓGk

(E2Gk

− E2γ)

2 + Γ2Gk

E2γ

×(2I + 1)

√a exp

(

2√

a(U − Eγ))

((U − Eγ) +

34T)2 ,

(6.11)

Observe that the expression of Eq. (6.11) is used to describe the Time OrderedSpectra (TOS). As it was shown in Sec. 4.3 the TOS are not experimentally ac-cessible, however, the EOS and TOS are practically the same at high γ-energies.Then the fit must be done for the high energy part of the spectrum. Since theexact starting energy value from where the TOS and EOS become the same isunknown, then a limiting fitting process was done using a threshold Eth at low en-ergies. Such threshold is increased and the fit performed with the different energyvalues.

The fit was performed to the EOS obtained with the events of the point B, (k,H) =(22, 15), corrected in efficiency. It is necessary to know the average values of theintrinsic excitation energy U and I where the event comes from. These values canbe determined from the response function PN (kH,ME) as it was shown in Chap. 4.Finally to get the (I, E) values the process explained in Sec. (5.5) was followed.For the pair (k,H) = (22, 15) the mean values of the projections give (〈M〉, 〈E〉) =(25, 17.6), and the maximum value of Eγ for the EOS gives an intrinsic excitation


energy of 〈U〉 = 6 MeV. Then the selected region is characterized by the values:

(〈I〉, 〈E〉) = (44, 17.6) (6.12)

Some of the thresholds and level density parameters obtained with several fitsperformed to the EOS, selection (k,H) = (22, 15), are listed in Table 6.3. Fromthe values of a obtained with this process, it is possible to observe that betweenthe energy thresholds Eth = 3.25 MeV and Eth = 3.50 MeV there is a suddenchange of the a value.

Eth (MeV) a (MeV−1)

2.00 24.6(2)2.50 24.7(3)3.00 24.4(5)3.25 24.3(6)3.50 18.4(5)3.75 18.4(7)4.00 18.4(8)

Table 6.3: Values of level density parameter a obtained by fitting the EOSwith different threshold values Eth al low energies.

Figure 6.12: EOS spectrum fitted with different intrinsic excitation energyvalues extracted from the simulation.

Fig. 6.12 shows the EOS spectra for (k,H) = (22, 15) and two of the fits withdifferent thresholds. The fitting curves are plotted in dots for the energy valueslower than the selected threshold. The region of the curve fitted to the spectrais drawn in solid lines. The blue curve is one fit with Eth = 3.00 MeV which


gives a value of a = 24.4(7) MeV−1, and the red curve represents the fit withEth = 4.00 MeV which gives a value of a = 18.4(7) MeV−1. The errors are theasymptotic standard errors calculated during the fit process.

However, any fit could have another error induced by the fluctuations in the EOS,specially if the Eth is high. For example, for the Eth = 4.00 MeV (see Fig. 6.13). Itis possible to draw two other functions with a = 27.0 MeV−1 that follows the highpart of the fluctuations, and the other one with a = 15.0 MeV−1 that follows thelower part of the fluctuations. This problem induces a big error in the calculationof the level density parameter and can be reduced by increasing the statistics ofthe EOS.

Figure 6.13: EOS spectrum fitted with different intrinsic excitation energyvalues extracted from the simulation.

Since the fit only describes the high energy region of the spectrum, then it ispossible to say that the level density parameter for the region (k,H) = (22, 15) is

a = 18.4(7)MeV−1. (6.13)

This result is in agreement with the value set in the simulation as Eq.(6.2) shows.

6.3 M → I conversion

A consistency test on the transformation between M and I can be done by usingGamble. It allows to consider the small region in (I, E) selected with I fixed andE with ∆E = 0.5 MeV, and produce a set of cascades coming from there, payingespecial attention to the given spin values. Then it is possible to count the number


of γ-rays in each cascade and compare it with the total spin I. Table 6.4 shows fora fixed value of spin I, the excitation energy of the yrast state of Eyrast(I) and thetotal excitation energy E = Eyrast(I) + U at such spin value. Also the selectionif made in energy where [EL;EU ] are the lower and upper values. This gives theaverage multiplicity 〈M〉 obtained by counting the number of γ-transitions comingfrom the selection.

Hk-EOS Gamble

I Eyrast(I) E (MeV) [EL;EU ] (MeV) 〈M〉40 10.4 16.4 [16; 17] 22.841 11.1 17.1 [17; 18] 23.542 11.1 17.1 [17; 18] 23.943 11.8 17.8 [17; 18] 24.544 11.9 17.9 [17; 18] 24.545 12.5 18.5 [18; 19] 24.9

Table 6.4: Selection of a decay box [EL;EU ] at spin I, a calculation of 〈M〉for the selected region performed by Gamble.

This calculation leads to conclude that the results for spins 43 ≤ I ≤ 45 areconsistent with the value of M obtained with the response function.

6.4 Two regions of level density parameter

Until now the simulation deals only with one constant value of the level densityparameter a for all the energy-spin phase space. As it was shown in Sec. 2.4the nuclear structure depends on the temperature. In particular the level den-sity parameter could change for different excitation energies and spin values. Thedeformation calculations performed on 154Dy in Ref. [1] show two different defor-mation regions in E − I (See Fig. 2.8). However there is not a precise descriptionof the shape transition line or the structure parameters in both regions in order toperform an accurate simulation.

What is relevant now is to determine if it is possible to distinguish both regionsby using the Hk-EOS method. For this aim, values of the level density parameterat low intrinsic excitation energies (U < 10 MeV) in both regions were assigneddepending on the spin. For the low spin region the nucleus is prolate and thelevel density parameter is taken as a1 = 22.1 MeV−1. For higher spin values(I = 40), where the nucleus is oblate, the level density parameter is taken asa2 = 18.4 MeV−1. The change between both regions is represented using a Fermidistribution to get a smooth transition of the level density parameter between bothregions. Fig. 6.14 shows the dependence of the level density parameter with the


spin, the transition line was placed at I = 40 as suggested by Ref. [1] for low valuesof the intrinsic excitation energy.

Figure 6.14: Level density parameter as a function of spin. Two differentvalues of a were chosen to represent different structure regions in 154Dy. Thetransition line was represented using a Fermi distribution in order to have a

smooth transition.

It is important to point out that the values of the level density parameters used inthis section were chosen in order to test the analysis method and do no representthe real structure of the 154Dy. The value a2 was chosen to be 20% higher thana1 the value used in the first part of the analysis.

With this characteristic of the level density parameter the (E, I) entry states weregenerated as well as the corresponding (k,H) distribution. These distribution arestatistically equal to those shown in Fig. 6.1. In the same way as before it ispossible to select specific (k,H) points as it was shown in Fig. 6.7 and extract theEnergy Ordered Spectra (EOS), the same points A,B, and C were used.

In this case the main interest is to compare the EOS from the points marked asA=(15,10) and C=(28,19). The events measured with these specific values areexpected to come from the two different regions simulated, i.e. the events in Ashould reflect a prolate structure and the events in C should reflect an oblatenucleus, by having different level density parameter. To be able to observe thisdifferences the statistics was improved to 5 × 106 events in order to get the EOSwith better definition. The reduction of the fluctuations at high Eγ leads to abetter fit of the level density parameter.

The EOS corresponding to the regions A and C were obtained and plotted inFig. 6.15, together with the intensity function Iγ(Eγ) to fit the level density pa-rameter for each curve using Eq. (6.11). In both cases the energy threshold wasset to Eth=4 MeV. One value of the level density parameter was obtained for each


Figure 6.15: Energy ordered spectra extracted from two regions in the (k,H)phase space. A=(15,10) and C=(28,19). The intensity curves are fitted to thelevel density parameter correspondent to the two regions with different nuclear

structures in the 154Dy.

region, its value can be assigned to a small region in the (E, I) phase space usingthe response function for the Gasp array.

Table 6.5 summarizes for each region: the values corresponding to the (k,H)selection, the expected value of the intrinsic excitation energy 〈U〉 obtained withthe maximum values of Eγ in the EOS, the level density parameter a, the expectedvalues (〈M〉, 〈E〉) obtained after applying the response function, and the finalexpected values of the two regions (〈I〉, 〈E〉).

(k,H) 〈U〉 a MeV−1 (〈M〉, 〈E〉) (〈I〉, 〈E〉)A (15, 10) 5.5 22.1(9) (11, 16) (30, 16)C (28, 19) 6.3 18.4(7) (22, 31) (50, 30)

Table 6.5: The regions labeled A and C, the level density parameter a fittedfrom the EOS, and their expected values in (M,E) and (I, E).

The final expected values (〈I〉, 〈E〉) extracted after the application of the Hk-EOSmethod show that each point represents two different regions each one of themwith its own level density parameter.

CHAPTER 7

Hk-EOS EVALUATION OF EXPERIMENTALGAMMASPHERE DATA ON CONTINUUM

The following is an evaluation of the quality of the data obtained with Gammas-

phere. Although the set-up was not optimized for Hk-EOS studies, it is howeverimportant to evaluate the possibilities of the method with real data, and at thesame time to identify instrumental origins of the possible difficulties.

The first approach to the application of the method Hk-EOS to experimental datawill be carried out for 60Ni, from an experiment performed at Argonne NationalLaboratory using the Gammasphere array. This experiment was done to investigatethe nuclear structure at high excitation energy and high spin of nucleus in theregion A = 60. The Hk distributions and EOS spectra were constructed withthe 60Ni data, and the reverse response functions were extracted from a Geant4

simulation of Gammasphere.

7.1 Gammasphere

The Gammasphere [67, 68] multi-detector array is located at Argonne NationalLaboratory. It consists of a spherical shell of up to 110 large volume HPGe de-tectors, each enclosed in a BGO Compton-suppression shield. Figure 7.1 shows aside view of the set-up (top) and a scheme of the Ge, BGO, and neutron detectorsarrangement (bottom). The Ge detectors are hexagonal and each one has sevenBGO detectors, six around the Ge crystal and one on the back.

83

Chapter 7. Evaluation of Gammasphere data on continuum 84

The detectors are positioned towards the center of the array where the target isplaced. BGO detectors usually are used to distinguish when a γ-ray interacts withone Ge detector and then by Compton effect is scattered to one of the BGO’sthat surrounds the Ge crystal. A heavymet shielding is placed in front of theBGO detectors to avoid γ-rays to arrive directly to the detector. In order to usethe BGO detectors not as Compton suppressors but as individual detectors, theheavymet is removed and the electronics is changed to enable the data read-out ofthe individual BGO crystals.

GammaSphere

BGO&Gedetectors detectors

Neutron

Figure 7.1: Top: Gammasphere array side view. Bottom: Arrangement ofthe BGO, Ge, and neutron detectors. (Pictures taken from [69]).

7.2 The Gsfma138 experiment

The experiment Gsfma138 was performed at Argonne National Laboratory, tostudy the structure of nuclei in the mass region A = 60 by a fusion-evaporationreaction. The experiment was performed using the Gammasphere array togetherwith charged-particle detectors, neutron detectors and the Fragment Mass Ana-lyzer (FMA). The reaction used was:

28Si(36Ar, 4p)60Ni (7.1)


with a beam energy of 134 MeV. The compound nucleus in the reaction is 64Gewhich evaporates four protons and populates the high energy and high spin entrystates of 60Ni. It is very important to identify all the reaction products, and thedetection of the evaporated neutron, protons and α-particle will provide a goodselection of the residual nuclei.

The experimental setup consisted of 77 of the 110 Ge detectors including the cor-responding BGO detectors (539), the charged-particle detector is a combination ofthe LuWuSiA [70] and two Microball [71] rings. It also includes 30 neutron detectorsof the Neutron Shell replacing the most forward Ge-detectors in Gammasphere.To identify even the weak reaction channels the FMA can be used to separate theresidual nuclei by their mass-charge ratio. The heavymets were removed to collectthe signals of the BGO detectors independently of the signals collected with theGe detectors.

7.3 Data handling

All the details of the data format and the calibration methods for the γ-detectionwith the Ge detectors, as well as the time alignment, Doppler, kinematic andangular corrections are explained in Refs. [72, 73]. The selection of the EOS andthe generation of the kH matrices of 60Ni with the data of the Gsfma138 experimentto be analyzed here were done by L.G. Sarmiento [74]. The calibration of LuWuSiA

is explained in [70].

Figure 7.2: ∆E-E map for the 4-proton selection.

The selection of 60Ni events was done considering that the evaporation particlesin this reaction channel are four protons, 4p. Then to extract the data of 60Ni a


gate in the number of protons detected with the LuWuSIA particle detector wasset up. Fig. 7.2 shows the two dimensional spectrum generated with the E∆E andEE signals collected by the telescope system of LuWuSIA [72]. In the figure twobanana structures can be clearly differentiated, the upper one at higher energiesare the events corresponding to an α detection and the one placed at lower energiesare the events that correspond to a proton detection. All the γ-signals recorded incoincidence with a four proton detection were stored and the total (k,H) matrixconstructed.

7.4 Gammasphere in Geant4

In order to be able to unfold the measured (k,H) distribution it is necessary to havea complete knowledge of the detection system. This can be done by simulatingthe Gammasphere array in Geant4. The simulation is a first approach to thereal device, it was developed taking especial care of the geometry of some of thedetectors used in the Gsfma138 experiment. The 80 Ge with the corresponding560 BGO detectors and 25 neutron detectors were considered [75].

BGOdetectors

DetectorsGe Neutron

Shell

To the FMABeam direction

Figure 7.3: Visualization of the geometry of GammaSphere and Neutron Shellfor the experiment Gsfma138, simulated in Geant4 [75].


Fig. 7.3 shows the geometry generated by the simulation. The detectors weresimulated only with the sensitive regions, the crystals in the case of the BGO andGe detectors and the scintillation liquid for the neutron detectors.

(a) Back view (b) Side viewGe Detectors

BGO detectors

Neutron detector

(c)

Figure 7.4: Views of the individual detectors in the Gammasphere simula-tion. (a) Back view of one Ge-BGO cell. For each Ge detector there are sevenBGO detectors, six of the BGO detectors surround the Ge detector, the sev-enth BGO detector is placed behind the Ge detector. (b) Side view, the BGOdetectors were simulated as wedges. (c) The neutron detectors were simulated

as a hexagonal polyhedra.

Fig. 7.4 shows some views of the individual Ge, BGO and neutron detectors in theGammasphere simulation. In Fig. 7.4(a) the back view of one of the 110 Ge-BGOcells is shown. For each Ge detector there are seven BGO detectors, six of the BGOdetectors surround the Ge detector, the seventh BGO detector is placed behindthe Ge detector. In Fig. 7.4(b) is possible to see that the BGO detectors weresimulated as wedges, in reality the shape of the BGO detectors is more complex,but the wedge shape that was simulated can be a good first approximation. InFig. 7.4(a) one of the neutron detectors is shown, they were simulated as hexagonalpolyhedra.

Besides the geometry, all the physics implemented by Geant4 was included toGammaSphere. The physics process that involve the generation of γ-rays andthe matter-radiation interaction as well as the generation of spectra for all the γdetectors (80 Ge + 560 BGO). Then is possible to characterize the array by itstotal efficiency and resolution. The results of the measurement of such quantitieswere used to generate the reverse responses of the array.

Some spectra were generated to show the behavior of the detection process. Fig 7.5shows the total spectra simulated for an energy Eγ = 3 MeV, Fig 7.5(a) showathe measurement of γ-rays with all the Ge detectors and Fig 7.5(b) with all theBGO detectors. In both spectra the photoelectric peak and the double and single-escape peak can be clearly seen, as well as the peak at Eγ=511 keV. The Compton


background in the Ge spectrum is very low, the opposite happens in the BGOspectra where the Compton background is very high.

Figure 7.5: Spectra generated with all the Gammasphere simulation for mo-noenergetic source of Eγ = 3 MeV. (a) Spectra measured with the Ge detectors.

(b) Spectra measured with the BGO detectors.

The efficiency of the Ge detectors is well known from previous works [72], but theabsolute efficiency of the BGO detectors has to be derived from the simulation be-cause there is no previous data available. The process of determining the efficiencyof the BGO detectors has a main difficulty, the dispersion by Compton effect isvery high, specially at low energies Eγ < 0.5 MeV. Also the double and single-escape peaks, generated by the pair creation effect, are very intense compared withthe photopeak.

Then to generate the spectra for several monoenergetic sources the absolute effi-ciency can be calculated. Two different ways of performing the efficiency calibra-tion were done, in each case the Ge and BGO detectors were calibrated indepen-dently and then added to have the total efficiency. Then the EOS spectra can bedisplayed in what is going to be called “cases”:

I. The spectrum with no efficiency corrections.

II. Using the number of detected events in photoelectric, single-escape and double-escape peaks. This was done considering that any of those events can berecorded as the most energetic of a given cascade. Fig. 7.6(a) shows the Ge,BGO and total efficiency for this case.

III. Calculating the efficiency just with the events in the photo-peak. Fig. 7.6(b)shows the Ge, BGO and total efficiency for this case.


(a) (b) .

Figure 7.6: Absolute efficiency of theGammasphere-Neutron shell array,for the Ge, BGO detectors and the total, calculated in two different ways. (a)Using the events in the photoelectric, single escape and double escape peaks.

(b) Using only the events in photo-peak.

7.5 Hk-EOS method for the level density param-eter a calculation

The entry states of 60Ni were also simulated with Gamble in order to have areference of what is observed in the measured distributions. Both the simulatedentry states and the (k,H) are plotted in Fig. 7.7. Fig. 7.7(a) shows that thedistribution in (I, E) is spread between the spin values 5 ≤ I ≤ 17 and is near theyrast line. The measured population in (k,H) is shown in Fig. 7.7(b). This wasgenerated with the data of the Gsfma138 by gating in the four protons reactionchannel to obtain 60Ni data. Notice that the measured distributions are spreadat low energies and spin. This type of effect depends on the response functionof the detector array. In the measured distribution probability the black squarerepresents a region with (k,H) = (8, 9) and ∆H = 0.5 MeV an k fixed. Usingthe reverse response function this point corresponds to a distribution with meanvalues (〈I〉, 〈E〉) = (12, 15).

The maximum value of the probability distribution was selected to extract theEOS. Then it is necessary to correct the spectrum in the array efficiency and fitit to calculate the level density parameter. The EOS is shown in Fig. 7.8(a) foreach of the cases mentioned above. When the spectrum is corrected in efficiencyit changes its slope at high energies. The slopes were calculated to observe the


(a) (b) .

Figure 7.7: (a) Entry states of 60Ni simulated with Gamble, the yrast line isplotted to have a reference. (b) The measured distribution probability (k,H),obtained with the data from the Gsfma138 experiment. The point (8, 9) is the

maximum value of the distribution.

behavior of the spectra at high energies and the results are:

sI = 1.87 counts/MeV,

sII = 2.07 counts/MeV,

sIII = 2.13 counts/MeV,

(7.2)

The slopes are different in each case. This shows that the spectrum is very sen-sitive to the efficiency calibration and a wrong calibration could generate a badcalculation of the level density parameter. Another problem is that the EOS in-cludes several peaks at low energy Eγ < 3 MeV, forming peaks. Fig. 7.8(b) showsthe EOSIII , plotted to observe the peaks formed. For example, energies 1.33 MeV,1.46 MeV and 1.76 MeV, correspond to discrete transitions of 60Ni.

The issues mentioned above induce to a wrong calculation of the level densityparameter a. A detailed study of the possible solutions to this problems mustbe undertaken in order to get a good result in the level density parameter fittingprocess.


(a) (b) .

Figure 7.8: (a) EOS with different cases for the efficiency correction, the blacklines are the slopes of the high energy part of the spectra, the values in each casesI = 1.87 counts/MeV, sII = 2.07 counts/MeV and sIII = 2.13 counts/MeV.(b) The EOS in the Case III, were it is possible to distinguish peaks from the

discrete transitions of 60Ni.

CHAPTER 8

CONCLUSIONS AND PERSPECTIVES

The Hk-EOS method was presented. All the considerations related to the detec-tion theory, several experimental issues and the unfolding of the response functiontogether with the generation of EOS were implemented for the investigation ofthe nuclear continuum. The simulation carried out for 154Dy includes a detaileddescription of the reaction and detection processes. The analysis of the influenceof the detector type, the Doppler effect and the neutrons of the reaction was per-formed.

The influence of the resolution and efficiency of the BGO and Ge detectors of theGasp array was considered. It was shown that even if a good efficiency is necessary,the fact of having narrower distributions in energy, with higher resolution, couldbe a good improvement in the method. In the near future it would be possibleto consider new devices as Agata or Greta, which could be a good option toperform this type of experiments.

It was shown that the evaluation of the simulated (I, E) events extracted fromseveral (k,H) points of the probability distribution, permits the selection of rathersmall and well defined regions. Therefore it is possible to analyze the properties ofnuclear states in such region by generating the EOS. Since all the information thatcan be extracted from EOS should be assigned to the (I, E) region, the capacity ofdistinguishing EOS for each (k,H) will make possible to assign nuclear propertiesto a small region in the (I, E) phase space. The limitations could be generated bythe method used to deconvolute the (k,H) points using the array response.

93

Chapter 8. Conclusions and perspectives 94

The response function of the Gasp array as well as the reverse response were cal-culated. The values (M,E) obtained by unfolding the values of the selected (k,H)regions, together with the consideration that for the given EOS the maximum en-ergy Emax

γ is the average intrinsic excitation energy 〈U〉 of the region, give thevalues used to fit the EOS with the product of the GDR gamma strength and theFermi-gas level density to get the level density parameter.

The level density parameter for the region (k,H) = (22, 15) of the 154Dy distri-bution probability was calculated by fitting the EOS spectrum corrected in ef-ficiency. A low energy threshold must be set during the fitting process becausethe region where the EOS is equal to the TOS is at high energies. The result ofa = 18.4(7) MeV−1 with a threshold Eγ = 3.5 MeV is in agreement with the leveldensity parameter set in Gamble (a = 18.4 MeV−1). However factors as a lowstatistics could introduce large error bars.

Additionally, it was shown that even when two different structure regions areconsidered in the E − I phase space, it is possible to extract for each region thecorresponding level density parameter. An example was carried out by choosinga = 22.1 Mev−1 for the proposed prolate region and a = 18.4 Mev−1 for theoblate region in 154Dy. EOS for both regions were extracted and the level densityparameters were fitted and the simulated values were recovered.

To use the 60Ni data from the Gsfma138 experiment on Gammasphere and testthe Hk-EOS method, the response and reverse response functions were calculatedfrom the simulation of Gammasphere with Geant4. The efficiency of the arraywas calculated. It was shown that an insufficient determination of the efficiencycould induce a wrong evaluation of the level density parameter. The distributionprobability (k,H) was generated from the 60Ni data as well as the EOS for theselected region (k,H) = (8, 9). However, the level density parameter was notcalculated because some correction has to be done to the data. First the correctcalibration of the array efficiency has to be performed. Also it was shown thatthe EOS for 60Ni have a big contribution of the discrete transitions. This must betaken into account for the fitting of the EOS.

The Hk-EOS method proved to have good perspectives. However, some othertests have to be performed: Reduce the (I, E) region by subtracting the events inthe near regions. The inclusion of regions with different level density parametersin the (I, E) phase space of 154Dy and generate EOS with more statistics. Itis important to be able of carrying out the experiment and compare it with thesimulation results.

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