LIFETIME MEASUREMENTS OF - INFN LIFETIME MEASUREMENTS OF EXCITED STATES IN THE 133SB NUCLEUS WITH FAST-TIMING TECHNIQUES Relatore: Prof.ssaSilviaLeoni Correlatori: Dott. SimoneBottoni

Università degli studi di Milano

Facoltà di Scienze e TecnologieCorso di laurea magistrale in Fisica

LIFETIME MEASUREMENTS OFEXCITED STATES IN THE 133SB NUCLEUS

WITH FAST-TIMING TECHNIQUES

Relatore:Prof.ssa Silvia Leoni

Correlatori:Dott. Simone BottoniDott. Łukasz Iskra

Tesi di laurea di:Annachiara Filippini

Matr.n. 914950

Anno Accademico 2018-2019

ContentsIntroduction 2

1 Physics background 31.1 The nuclear shell model . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Isomeric states and decay probabilities . . . . . . . . . . . . . . . . . . . 71.3 Study of the 133Sb isotope . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 The experiment 132.1 The LOHENGRIN mass spectrometer . . . . . . . . . . . . . . . . . . . 132.2 Fission reactions and their role in exotic nuclei production . . . . . . . . 162.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Characteristics of γ-rays detectors . . . . . . . . . . . . . . . . . . 19

3 Lifetime measurements 223.1 The fast timing method . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Time spectra analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 The slope fit method . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2 The convolution fit method . . . . . . . . . . . . . . . . . . . . . 243.2.3 The centroid shift method . . . . . . . . . . . . . . . . . . . . . . 24

4 Detectors calibration 314.1 HPGe calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 LaBr3 calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 152Eu lifetimes analysis 41

6 Data Analysis 516.1 Detectors calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2 Mass selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.3 Identification of contaminants . . . . . . . . . . . . . . . . . . . . . . . . 596.4 133Sb lifetimes analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7 Comparison with theoretical calculations 757.1 The Hybrid Configuration Mixing model . . . . . . . . . . . . . . . . . . 757.2 133Sb level scheme and wave function composition . . . . . . . . . . . . . 767.3 Reduced transition probabilities . . . . . . . . . . . . . . . . . . . . . . . 78

Conclusion 79

Acknowledgements 80

1

IntroductionAtomic nuclei are complex systems where the mutual interactions between nucleons aredriven by three fundamental forces: strong, electromagnetic and weak. Even thoughthe experimental knowledge has been extended significantly in the last decades, the fulldescription of all the effects occurring inside the nucleus is still incomplete. Neutronor proton rich nuclei, which lie far from the path of stability in the nuclear chart, areparticularly important to develop new theoretical models and to study how the nuclearstructures evolve. The doubly-magic 132Sn isotope is one of the most important nucleiaway from the stability valley, which is also considered the best shell closure of the nuclearchart. Nuclei in the mass region around 132Sn are suitable to investigate structurescoming from the couplings between core excitation and a valence nucleon. In this thesis,an analysis of 133Sb data will be performed. The structure of this nucleus is characterizedby the couplings of collective and non-collective excitations of the doubly magic 132Sncore and a single valence proton; the study of its properties is of crucial importanceto give an accurate description of the fundamental excitations around shell closures innuclear models.A good way to test the properties of nuclei is to measure the lifetimes of the states ofinterest using γ-spectroscopy techniques with fast scintillators that permit high-precisionmeasurements. For this purpose, a new experiment has been conducted at the InstitutLaue Langevin (ILL) in Grenoble (France), where the 133Sb isotope has been producedin a neutron-induced fission reaction of a 235U target. Subsequently, fission fragmentshave been selected in the mass spectrometer LOHENGRIN, that provides clean isotopicseparation according to their mass, energy and charge state. The states of interesthave been populated during the decay of the long-lived 21/2+ isomer of 133Sb, which issituated at 4526(30) keV excitation energy and has a half-life of 16.6 µs. The experimentalresults will represent an important benchmark for the development of a new theoreticalapproach, named the Hybrid Configuration Mixing (HCM) model, aimed at describingodd-mass systems around doubly-magic closed-shell nuclei based on couplings betweensingle particles and both collective and non-collective excitations of the core.The present work is organized into seven chapters; in the first one, information aboutthe physics background will be given. In the second chapter, an explanation of thepresent experiment will be provided, describing the experimental setup and some generalproperties of radiation detectors, as well as the LOHENGRIN mass spectrometer. Inthe third chapter the fast timing technique for lifetime measurements will be brieflyexplained, in particular the Generalized Centroid Difference (GCD) method. The fourthchapter is dedicated to the calibration of HPGe and LaBr3(Ce) detectors; the fifth isabout lifetime analysis based on the 152Eu source, which will serve as an example forthe application of the GCD method, while the sixth chapter concerns the 133Sb dataanalysis. The last chapter will include a comparison with the theoretical model as wellas final conclusions and perspectives.

2

1 Physics background

1.1 The nuclear shell model

The nuclear shell model is based on a mean autoconsistent potential created by thenucleons themselves, and permits to describe properties of the nucleus such as excitationenergies or level spins [1].Nuclei having a magic number of protons or neutrons are characterized by, e.g., highnucleon separation energy and high energy of the first excited state. These attributesmanifest themselves periodically and, in analogy to the periodic properties of atoms,suggest the shell structure of the nucleus.Some orbitals are grouped together, and such a group is called a shell; the energy spacingbetween two shells is called a shell gap. The occurrence of magic numbers is connectedwith the existence of those energy gaps between states or group of states.In the nucleus we assume that the motion of a single nucleon is governed by a potentialcaused by the other nucleons. In the first approximation, the harmonic potential can bechosen. Assuming N nucleon per shell and calling n the principal quantum number,

N =∑

(n+ 1)(n+ 2) (1)

Since it can reproduce just the first three magic numbers, one can take into considerationthe Woods-Saxon potential. In the case of spherical nuclei, it is represented in Figure 1and has the following shape:

V (r) = − U0

1 + er−Ra

(2)

where R is the nuclear radius, a is a parameter that defines the nuclear surface and iscalled diffuseness (its value lies around 0.5 - 0.6 fm), and U0 is the potential depth thatis typically around -50 MeV.

The Woods-Saxon potential does not completely reproduce the whole sequence ofmagic numbers. Since nucleon interaction strongly depends on spin, a spin-orbit couplingterm is introduced [1]:

VS.O.(r) = VLS(r)~l · ~s (3)

where ~l is the particle angular momentum and ~s its spin.

3

Figure 1: Schematic representation of the Woods-Saxon potential in comparison withthe harmonic one [2].

The total angular momentum ~j is given by the combination ~j = ~l + ~s, thus modify-ing the level energies:

j = l +1

2→ Enj = Enl + ~ELS

l

2(4)

j = l − 1

2→ Enj = Enl − ~ELS

l + 1

2(5)

where ~ELS ∼ −20A−23 MeV is the mean value of VLS(r) (A is the mass number). This

spin-orbit coupling successfully reproduces the shell structure and the observed magicnumbers [3], as it can be seen in Figure 2, in comparison with predictions based on theprevious potentials.

4

Figure 2: Energy levels predicted by the shell model obtained assuming an harmonicpotential (left), a Woods-Saxon potential (center) and including the spin orbit couplingterm (right). The maximum number of nucleons in the shell are given in bold. On theright, the occupation numbers of the various levels are shown [4].

5

Around shell closures, states arise from the coupling of a particle (or a hole) andexcitations of a doubly magic core (phonons in particular), like in case of 133Sb, whichcan be seen as the coupling of a 132Sn core with a proton. The coupling between aphonon and a particle generates states with angular momentum J given by

|λ− s| ≤ J ≤ |λ+ s| (6)

where s is the particle spin and λ the phonon multipolarity. Electromagnetic transitionprobabilities are connected to the core’s properties and to the strength of the coupling[5]. Figure 3 shows the experimental excitation spectrum of the doubly magic 132Snnucleus.

Figure 3: The energy level scheme of the 132Sn nucleus [6].

6

1.2 Isomeric states and decay probabilities

Prediction of the presence of isomeric states is a good test for nuclear structure modelsand their selection rules. An isomer is a metastable state of a nucleus, where protonsand neutrons occupy specific orbitals.Isomers find themselves in an excited state with an half-life significantly longer than the"ordinary" ones. They decay through the emission of γ-rays or conversion electrons (thishappens when the energy of nuclear de-excitation is not emitted as a photon, but it isused to eject one of the inner electrons).In general, the decay of a nuclear state is a stochastic process, where an excited statepasses to a lower-energy one following an exponential law [7]. If N is the relative popu-lation of the isomer, the decay rate is given by

dN(t)

dt= −λN(t) (7)

where λ is the decay constant.By solving this differential equation, the decay law is derived:

N(t) = N0e−λt (8)

The inverse of λ gives the time constant τ . The half-life is the time taken for half of theisomeric yield decay, and it can be expressed as T1/2 = τ ln2 [8].Excited nuclear states may decay via several branches with different probabilities. Inthis case, only the total decay constant can be experimentally observed as the sum of allthe involved ones:

n∑i=1

λγi =n∑i=1

1

τi(9)

The decay constant is directly related to the wave function of the initial and final statesthrough the Fermi’s Golden Rule [4]:

λ =2π

h|mfi|2ρ(E) (10)

where ρ(E) is the density of final states and mfi stands for the transition matrix elementof the initial and final states, and it’s defined as

mfi = 〈Ψf |m|Ψi〉 (11)

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In an electromagnetic transition σL (σ denotes the electric or magnetic character and Lthe multipolarity), the value of λ depends on the spin and the parity Jπ of the twostates and on their energy difference. Therefore, the reduced transition probabilityB(σL, J1 → J2) is generally used to describe the decay of an excited state and canbe expressed by the equation [9]:

B(σL, J1 → J2) =∑µM2

|〈J1M2|m(σL, µ)|J1µ1〉|2 = (2J1 + 1)−1|〈J2||m(σL)||J1〉|2 (12)

where −J ≤M ≤ J and −L ≤ µ ≤ L are the magnetic substates.The Clebsch-Gordan coefficients relate the reduced transition matrix element 〈J2||m(σL)||J1〉to the transition matrix element mfi described above. For a γ-ray transition with a givenmultipolarity, it’s possible to relate the reduced transition probabilities B(σL) to the to-tal decay probability T , for both electric transitions

T (E1) = 1.59 · 1015E3γB(E1) (13)

T (E2) = 1.22 · 109E5γB(E2) (14)

T (E3) = 5.67 · 102E7γB(E3) (15)

T (E4) = 1.69 · 10−4E9γB(E4) (16)

and magnetic transitions

T (M1) = 1.76 · 1013E3γB(M1) (17)

T (M2) = 1.35 · 107E5γB(M2) (18)

T (M3) = 6.28 · 100E7γB(M3) (19)

T (M4) = 1.87 · 10−6E9γB(M4) (20)

in which Eγ is given in MeV, B(EL) in e2(fm)2L and B(ML) in ( eh2Mc

)2(fm)2L−2.The measured lifetime τ of a certain level can be related to a quantity called partiallifetime τγ of a particular γ decay:

τγ(σL) = τItot

Iγ(σL)(21)

where Iγ is the intensity of the γ transition and Itot is the sum of the intensities of allthe γ’s that depopulate the level. The quantity T is the inverse of the lifetime.Transition rates are often measured in Weisskopf units, based on single particle transi-tion strength. In this way, it’s possible to compare the transition rates between nucleiin different mass regions.

8

The following relations between these two units hold:

BW (EL) =1.22L

4π

( 3

L+ 3

)2

A2L3 e2(fm)2L (22)

BW (ML) =10

π1.22L−2

( 3

L+ 3

)2

A2L−2

3eh

2Mc(fm)2L−2 (23)

Finally, angular momentum coupling determines the following selection rules for electro-magnetic transitions decay:

|Ji − Jf | ≤ ∆L ≤ Ji + Jf (24)

∆L > 0 (25)

∆π =

(−1)L if σ = E

(−1)L+1 if σ = M(26)

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1.3 Study of the 133Sb isotope

The excitation spectrum of the 133Sb isotope has been studied and extended followingseveral experiments, in the last decades. The first identification of the isomeric 21/2+

state at E = 4526 keV dates back to 1978, as reported in Ref. [10], while the half-lifeT1/2 = 16.6 µs has been measured by Urban in Ref. [11]. However, up to the recentexperimental campaigns at ILL (Grenoble), there was no information about the lifetimeof the states in the 133Sb nucleus through which the isomer decays. The campaign namedEXILL (performed at ILL in 2012, 2013) allowed the investigation of the structure ofexotic nuclei produced in neutron-induced reactions [12]. The ILL’s reactor (see Figure4) produces the world most intense continuous neutron beam, with I = 1.5 ·1015 n s−1

cm−2, and makes use of thermal (Ekin ∼ 0.025 eV) or cold (Ekin < 0.025 eV) neutrons.

Figure 4: The High Flux Reactor situated at ILL.

The 133Sb isotope has been produced during the fission of 235U and 241Pu targets[13]. The levels in the nucleus of interest were populated up to high excitation energies(around 6 MeV) and high spin values. The 21/2+, 16.6 µs isomeric state in 133Sb decaysvia a main cascade of five γ transitions: an unknown isomeric one with Eγ < 30 keVfollowed by 62, 162, 1510 and 2792 keV γ-rays feeding the 7/2+ ground state [14, 15].Coincidences between prompt (within 200 ns from the fission event) and delayed γ-rayswere measured with a highly efficient HPGe array. For the first time, high-spin excitedstates above the 21/2+, 16.6 µs isomer were observed (up to the 25/2+ state). The searchfor higher spin structures resulted in the location of three new energy levels, which arepresented in Figure 5.

10

Figure 5: Experimental level scheme of 133Sb: the already known decays are labeled inblack, while the newly identified transitions above the isomer are in red [14].

In the second part of the EXILL experiment, the lifetime analysis of the 13/2+ and15/2+ states has been performed with the fast-timing technique by employing a compactsetup of LaBr3(Ce) scintillators; this method permits the measurement of half-lives in theps range. As it will be explained in the next chapters, from the time difference betweenthe centroids of the time distributions it’s possible to deduce the lifetime of a given state(see Figure 6). An average of the half-life values obtained from 235U and 241Pu data givesT1/2 = 31(8) ps and T1/2 < 20 ps for the 13/2+ and 15/2+ states, respectively.

11

Figure 6: Time distributions used in the analysis of the 13/2+ (a) and 15/2+ (b) statesfor the 235U data set, associated with the γ-ray cascades shown on the right [14].

The lifetime analysis gave a large difference in the magnitude of the B(M1) reducedtransition probability for the 15/2+ → 13/2+ and the 13/2+ → 11/2+ transitions. Thisdifference suggests some change in the nature of particle-core excitations in states of133Sb with increasing spin, going from a collective to a non-collective character [14].In the EXILL campaign, the background induced by the fission reaction was very high.For this reason, the experiment analyzed in this thesis was performed, with the aim ofimproving the lifetime measurement precision, particularly in the case of the 13/2+ state,and of obtaining a definite value for the 15/2+ one.

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2 The experimentThe experiment on 133Sb, analyzed in this thesis work, was performed in October 2018at the Institut Laue-Langevin (ILL) in Grenoble. It made use of the LOHENGRIN spec-trometer which can separate mass fragments coming from neutron-induced fission of the235U target. In this chapter the experimental setups used in the conducted measurementare presented.

2.1 The LOHENGRIN mass spectrometer

The ILL’s reactor produces the most intense continuous neutron flux in the world (witha thermal power up to 58.3 MW). Neutrons are transported to the different experimentalareas via neutron guides. Figure 7 gives a representation of their position and the variousexperimental areas. LOHENGRIN (PN1) is a recoil mass spectrometer for the study ofthe properties of exotic isotopes produced during fission processes, allowing to select themass, kinetic energy and charge distribution with very high resolution [16].

Figure 7: Plan of the ILL experimental areas [16]. The red arrow on the left showswhere the LOHENGRIN mass spectrometer is installed.

13

At LOHENGRIN, a thermal neutron flux of 5.3 ·1014 n cm−2s−1 induces the fission ofa target (with typical thickness of 200-400 µg/cm2) placed near the core of the reactor.The fission fragments then pass through a thin Ni layer, so that they have a statisticallydistributed ionic charge, and a collimator selects those emitted toward the beam tube.Their typical energy is about 1 MeV per nucleon. The main magnet and the electrostaticdeflector produce a magnetic and electric field, respectively, thanks to which fragmentseparation is achieved: according to their A/q and E/q ratios (where A is the mass, E thekinetic energy and q the ionic charge), the fragments are split onto different parabolasin such a way that only those with the selected values of the ratio pass the magnet(Figure 8). The magnet is perpendicular to the ions’ momentum, and deflects themin the horizontal direction. The electrostatic deflector is positioned just after the mainmagnet, and it’s designed to make ions travel along a circular line of equal electric fieldstrength E. Deflection here happens in the vertical direction.

Figure 8: Schematic drawing of the LOHENGRIN mass spectrometer [16]. The toppanel shows the deflection in the x-direction induced by the main magnet, while thebottom panel shows the deflection in the y-direction induced by the electrostatic deflec-tor. The red line represents the path of focused fission fragments that finally reach theionization chamber, where they are stopped.

14

Defining as rmag and rcon the radii of the trajectories on which the ions are respec-tively forced by the main magnet and the condenser, the equations followed by the ionstravelling at velocity v in these fields are

qv ~B =Av2

rmag(27)

q ~E =Av2

rcon(28)

The electric and magnetic field are adjusted in such a way that A, q and E of thefragments correspond to the radii of curvature that allow them to pass through thewhole spectrometer. It can be demonstrated that the focusing conditions are

A

q=

1

χ

~B2

U(29)

E

q= ΦU (30)

with parameters

Φ =rcon2d

(31)

χ =2Φ

r2mag

(32)

Here U is the electrostatic potential and d the distance between the condenser’s plates,that define the electric field in its interior [17].For spectroscopy studies of neutron rich nuclei, different ancillary detectors can be in-stalled at the LOHENGRIN’s focal plane. In the case of our setup, there are two ioniza-tion chambers, two HPGe clover detectors and four LaBr3 scintillators. In general, twotypes of γ-spectroscopy experiments can be performed at LOHENGRIN: β decay studiesand µs isomer spectroscopy, as in the case treated in this thesis, where the lifetime ofthe intermediate states in the nucleus of interest are measured by using the coincidencebetween the γ-rays recorded in the LaBr3 scintillators. The levels of interest are typicallypopulated from an isomer with half-life longer than 500 ns, which is the time sufficientfor the fission fragment to travel through the separator.

15

2.2 Fission reactions and their role in exotic nuclei production

Exotic nuclei lie far from the β-stability valley, and they’re very important in the con-text of nuclear structure research: they need other production methods than standardtransfer reactions with stable beams, as would be suitable for non-stable isotopes close tostability. For example, neutron- and proton-rich nuclei can be produced through nuclearfragmentation, while another way of investigating neutron rich species may be throughthe study of fission products.During nuclear fission, a compound nucleus is elongated until it splits in two parts,together with the emission of two or three neutrons. The fission fragments have highexcitation energy and high spin values. This kind of process is schematically illustratedin Figure 9.

Figure 9: Schematic illustration of the different stages leading to a binary fission inducedby neutrons [17].

16

The distribution of fission products is in general asymmetric and highly depends onthe target material, as it can be seen in Figure 10. While the heavier fragment always hasthe highest yield around mass A = 130, the lighter one is more sensitive to the reactionsystem: its maximum production moves toward lower masses when a less neutron-richtarget is employed.

Figure 10: Fission yields for different targets [18].

The fission process leads to very neutron-rich products, becoming a good tool for thestudy of exotic nuclei. Since in this type of reaction hundreds of different isotopes canbe populated, one then needs to identify and select the particular isotopes of interest byseparating them according to the mass A and the nuclear charge z. This can be doneby using a combination of electric and magnetic fields, like in the LOHENGRIN massspectrometer.

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2.3 Experimental setup

The fast-timing measurements analyzed in this thesis have been performed using a com-pact setup of four LaBr3 scintillators. This type of detector has an energy resolution,which is much higher than in other scintillating materials (∼3% at 662 keV, against acorresponding value of ∼7% with a NaI:Tl crystal [19], [20]). Moreover, the LaBr3 de-tectors have a very good time resolution, which makes them useful in measuring nuclearstates lifetimes. Figure 11 shows the geometry of a four LaBr3 timing system.

Figure 11: Experimental geometry of a timing system composed of four LaBr3 detectors[21].

In our experimental setup there are also two HPGe clovers, each consisting of fourHPGe semiconductor detectors, thus reaching a solid angle coverage much larger thanthat for a single crystal detector. They have been used for the study of the coincidencesbetween the γ rays of the products separated with the LOHENGRIN spectrometer. Insemiconductors, the information carriers are electron-hole pairs produced by charged ion-izing particles along their path into the medium [22]. Since the necessary energy for thecreation of such pairs is about one order of magnitude lower compared to that requiredby scintillators, the energy resolution of semiconductor detectors is much better. Highpurity Ge detectors are typically cooled down to 77 K through a liquid nitrogen coolingsystem, in order to reduce the thermally-induced leakage current.

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HPGe detectors have a worse time response in comparison with scintillators becauseof the charge collection time, which is slower than the de-excitation process in scintil-lating materials [23]. The setup used in this thesis experiment was meant to combinethe high resolving power of germanium detectors and the good timing properties of thescintillators for the detection of the γ-rays.

2.3.1 Characteristics of γ-rays detectors

Energy resolution

The energy resolution R refers to the detector’s ability to resolve incident photons withsimilar energy values. It is defined as

R =FWHM

H0

, (33)

where FWHM refers to the energy distribution (Gaussan curve) and H0 is the peakcentroid position. Detectors with small R values are very useful to help distinguishingsignals from photons of very close energy values. Given a number N of charge carriersand assuming that their formation process in the detector is poissonian, one shouldexpect a standard deviation of

√N ; in the limit of large N , the shape of the pulse height

distribution tends to a Gaussian (see Figure 12), with FWHM = 2.35σ.

Figure 12: Illustration of a Gaussian distribution, with indicated FWHM and standarddeviation σ.

19

Since the response of many detectors is linear, the peak centroid value can be writtenas H0 = KN , so that its standard deviation becomes σ = K

√N , where K is a propor-

tionality factor. It can be seen that, in this case, the resolution only depends on thenumber of charge carriers, and it improves as N increases:

R|Poisson =FWHM

H0

=2.35K

√N

KN=

2.35√N

(34)

Since charge carriers production cannot actually be described via Poisson statistics, theFano factor F has been introduced, which is defined by the ratio of the observed variancein N to the Poisson predicted variance:

F =σ2

(K√N)2

(35)

This factor is much smaller than unity for semiconductor detectors, while it’s close tothe unitary value in the case of scintillators. The resolution becomes

R =2.35K

√N√F

KN= 2.35

√F

N(36)

A detector’s energy resolution depends on many factors, such as the fluctuations in thenumber of produced charge carriers, electronic fluctuations caused by external condi-tions, electronic noise, etc. Consequently, the global resolution is affected by all of thesecontributions:

FWHMtot =√F · FWHM2

statistical + FWHM2noise + FWHM2

drifts (37)

Detector’s efficiency

Another fundamental characteristic of radiation detectors is the efficiency, which variesaccording to the energies of the considered transitions. This property represents theprobability that a photon entering the detector produces a measurable signal, so it allowsto understand how much emitted radiation is being measured. A distinction has to bemade for absolute and intrisic efficiency [22]:

εabs =γ’s detected

γ’s emitted by the source(38)

εintr =γ’s detected

γ’s that effectively entered the detector(39)

20

The following relation holds:

εabs =Ω

4πεintr (40)

The proportionality factor corresponds to the portion of solid angle covered by the de-tector.

Timing response

The time response is an important detector property to form the signal after the radiationhas arrived, and it has a strong influence on the detector’s capability of resolving thesignals arrival times.During the period of duration of a signal, a second event should not be accepted becauseit would pile up on the first: the minimum time separation between two distinct eventsis the detector dead time, that obviously limits the count rate at which a detector canoperate [22].

21

3 Lifetime measurementsPrecise lifetime measurements of excited states are essential to verify and develop nuclearstructure models, since the lifetime τ and the decay constant λ = 1/τ depend on theconfiguration of the wave functions of the involved states. A way to achieve this goal isto use the electronic fast timing method, which is the technique that has been employedin this thesis.

3.1 The fast timing method

The technique of electronic fast timing consists in determining the lifetime of a nuclearstate through the time difference between two measured signals (start and stop): one isnamed feeder and it is associated to the γ ray populating the state of interest, the otheris called decay and it is related to the γ ray depopulating the state.The setup in the simple case of two detectors is schematically depicted in Figure 13:the signals are sent through a constant fraction discriminator (CFD) which generates alogic time signal, then a time to amplitude converter (TAC) measures the time differencebetween the start and stop input. The two detector energy signals, E1 and E2, are alsorecorded [17].

Figure 13: Fast timing setup for a two detector system: start and stop are connectedto constant fraction discriminators (CFD) to produce logic timing signals. The timedifference measurement is obtained by using a time-to-amplitude converter (TAC) [17].

22

A fast timing setup is characterized by the Prompt Response Function (PRF) P (t).Two coincident signals are prompt if their time difference can’t be resolved with the usedsetup: using LaBr3 scintillators, events with a delay of about 5 ps can be consideredas prompt. Their time spectrum is the PRF of the setup, which can be approximatedby a Gaussian distribution and its FWHM contains all timing uncertainties from thescintillator and the photomultiplier tubes, electronics and setup geometry.In general, the time distributions show an asymmetric shape made by the convolutionof the energy dependent PRF with an exponential decay [21, 24, 25]:

D(t) = nλ

∫ t

−∞P (t′)e−λ(t−t′)dt′, (41)

n being a normalization factor.

3.2 Time spectra analysis

Assuming two transitions in delayed coincidence, detector 1 starts the TAC and is gatedon the transition of energy E1 that feeds the state of interest, while detector 2 is gatedon the decay transition of energy E2 and stops the TAC (Figure 13). The lifetime of thestate can be obtained by studying the asymmetry of the time spectrum which results ina tail (slope) consisting of delayed events.

3.2.1 The slope fit method

If τ is larger than the time resolution of the setup, then the time distribution becomesasymmetric: the visible tail corresponds to the exponential decay. The lifetime can beobtained from the pronounced slope of the delayed time distribution D(t):

ln[D(t)] ∼ λt (42)

In fact, by fitting the straight line observed on a semi-logarithmic plot of D(t) outsidethe region of the PRF, one gets [21]

d

dtln[D(t)] = 1/τ (43)

This method, which is independent of the shape of the PRF, can be applied if there areenough data points in the region where P (t)

D(t) 1.

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3.2.2 The convolution fit method

If the conditions to apply the slope method are not fulfilled but the time distributionis still asymmetric, the lifetime can be extracted by fitting the time spectrum with theconvolution function given in Eq. 41 (the shape of the PRF must be known to calculatesuch a convolution, but this condition is not always satisfied). This procedure can giveconsistent results with other kinds of methods, considering lifetimes not much shorterthan the time resolution.

3.2.3 The centroid shift method

This method is used for the determination of lifetimes which are smaller than the setuptime resolution, like in the case of the states in the 133Sb isotope analyzed in this thesis.For this purpose, the time distribution C(D(t)) can be defined as [21, 24, 25]:

C(D) =< t >=

∫tD(t)∫D(t)

(44)

The lifetime can be determined by measuring the difference of the centroids of the delayedtime distribution and the prompt time distribution, corresponding to the respectiveenergy gates:

τ = C(D)− C(P ) (45)

In the following, two specific applications of these technique are discussed.

The mirror symmetric centroid difference method

The prompt centroid is dependent on the time response, i.e. the time walk character-istics of both detectors in the fast timing setup. As a consequence, the main issue ofthe previous method is the prompt curve calibration: the two-detectors system has acombined time walk that depends on the energy of the two gates, but in γ-γ experimentsthese gates change according to the state that one wants to measure. In the mirrorsymmetric centroid difference (MSCD) method, the difference ∆C of the two centroidsC(D)stop and C(D)start is calculated (see Figure 14), so that the asymmetry in the timingcharacteristics is canceled:

∆C = C(D)stop − C(D)start (46)

24

Figure 14: The MSCD method for a two detectors system [21]. The two time spectraof a certain γfeeder − γdecay cascade are obtained by gating on the decay transition withthe start detector (Cstart) and with the stop detector (Cstop); the feeding transition isdetected by the second detector. The example refers to the case of the 1213-244 keV γ-γcascade of the 152Sm nucleus.

In Ref. [24] it is shown that, when the decay transition is gated on the stop detec-tor, the centroid curve C(D)stop is shifted by +τ from the corresponding prompt curveC(P )stop; if the decay transition is gated on the start detector, then C(D)start is shiftedby -τ from the prompt curve. Defining ∆Eγ = Efeeder − Edecay, the centroid differencecan be expressed as

∆C(∆Eγ)decay = C(D)stop − C(D)start (47)

The subscript ’decay’ indicates that the reference timing signal of the setup is providedby the decay transition; the subscript ’start’ (’stop’) means that the transition gives thestart (stop) timing signal.

25

Remembering that, from the discussion on the centroid shift method, τ = C(P ) −C(D), it follows that

∆C(∆Eγ)decay = CP (∆Eγ)stop + τ − (Cp(∆Eγ)start − τ) = PRD(∆Eγ)decay + 2τ (48)

As it can be seen in Figure 15, there is a +2τ shift between the linearly combined centroiddifference curve and the corresponding PRD (Prompt Response Difference), which is thecentroid difference of a prompt coincidence.

Figure 15: The obtained energy-dependent PRD curve when a decay transition is usedas the reference energy gate of the setup [24]. When it crosses the zero-point, the centroiddifference at ∆Eγ = 0 corresponds to 2τ .

26

Taking into consideration the feeding transition as the reference energy gate, anidentical prompt curve C(P )stop is obtained by gating the reference feeding transition onthe stop detector, but the centroid curve C(D)stop is shifted towards shorter time values.The centroid difference for a reference feeding transition is then defined as

∆C(∆Eγ)feeder = C(D)start − C(D)stop

= CP (∆Eγ)start + τ − (CP (∆Eγ)stop − τ)

= PRD(∆Eγ)feeder + 2τ

(49)

Therefore,

∆C(∆Eγ)decay = ∆C(∆Eγ)feeder (50)

PRD(∆Eγ)decay = PRD(∆Eγ)feeder (51)

Since the centroid difference depends on the energies of the two gates and on which isfeeder or decay, a reference transition is introduced: its choice fixes the character andthe energy Eref of one gate. Considering a reference energy gate corresponding to theenergy of the feeding transition, the PRD can be written as

PRD(−∆Eγ)ref=Efeeder= CP (Eγ − ref)start − CP (Eγ − ref)stop

= −(CP (Eγ − ref)stop − CP (Eγ − ref)start),(52)

where ∆Eγ = Eγ − ref . The following symmetries (responsible for the name of themethod itself) can be found:

PRD(∆Eγ)ref.=Edecay= −PRD(−(∆Eγ)ref.=Efeeder

) (53)

∆C(∆Eγ)decay = −∆C(−∆Eγ)feeder (54)

and accordingly

PRD(∆Eγ = 0) = 0 (55)

|∆C(∆Eγ = 0)| = 2τ (56)

27

These mirror symmetric relations imply that for a reference energy, the value of thecentroid difference and of the PRD is independent on the energy of the feeder and thedecay. For a given reference energy as the decay, the obtained PRD curve can be trans-formed into the reference feeder of the same energy; its mirror symmetry makes thedetermination of the PRD for any energy combination possible using:

PRD(Efeeder − Edecay) = PRD(Efeeder)− PRD(Edecay) (57)

The centroid difference ∆C(E)ref of a prompt coincidence is the prompt response dif-ference PRD(E)ref for a reference transition with energy Eref .For a delayed cascade connecting a level with lifetime τ , the following relations hold [21]:

2τ = ∆C(Efeeder)decay − PRD(Efeeder)decay (58)

2τ = ∆C(Edecay)feeder − PRD(Edecay)feeder (59)

To extract lifetimes from the spectra, the PRD must be calibrated using full energy peaksfrom prompt cascades. A possible fit for the PRD, as proposed in [24], can be given by:

PRD(Eγ)ref =a√

b+ Eγ+ cEγ + d (60)

It provides the uncertainty for the MSCD method only (assuming no background con-tributions from time spectra).

The generalized centroid difference method

In case of a setup with a number N of detectors, a way to evaluate fast timing datais given by the generalized centroid difference (GCD) method, which is an extension ofthe MSCD technique for multi-detector systems. The PRDij for any detector-detectorcombination (with i 6= j ∈ N) is

PRDij = Cpi + Cp

j (61)

There is a number N(N − 1)/2 of possible PRDij combinations [21]: their linear combi-nation gives

N−1∑i=1

N∑j>i

PRDij = (N − 1)N∑i=1

Cpi (62)

28

By linearly combining the N(N − 1)/2 independent centroid differences ∆Cij, one gets:

N−1∑i=1

N∑j>i

∆Cij =N−1∑i=1

N∑j>i

PRDij +N(N − 1)

22τ

= (N − 1)N∑i=1

Cpi +N(N − 1)τ

(63)

Instead of evaluating N(N − 1)/2 PRDij, one can evaluate the centroid of the sum ofall combinations, determine a mean PRD and then extract the lifetimes by calculatingthe shift of the mean centroid difference with respect to the mean PRD. This is done as[21]:

∆C(Eγ) = PRD(Eγ) + 2τ, (64)

with

∆C =2∑N−1

i=1

∑Nj>i ∆Cij

N(N − 1), (65)

PRD(Eγ) =2

N

N∑i=1

Cpi (66)

All the combinations can be measured with a common stop fast timing circuit (which isfavoured over the common start to reduce dead time): detector i starts TAC i, which isstopped by all detectors with index number j > i. Then, all combinations are recordedwith N − 1 TACij instead of N(N − 1)/2 (see Figure 16).

29

Figure 16: Common stop timing circuit for N detectors with (N - 1) TACij [17]. EachTAC is started by an individual detector i and stopped by detectors with higher indexnumbers j > i.

30

4 Detectors calibrationThe experimental timing setup used in this thesis work is composed of four LaBr3 andtwo HPGe clover detectors, each clover consisting of four germanium crystals. There arealso two ionization chambers, which give us information about the mass of the fissionfragments.The first step in the calibration of a detector is to find a relation between the channelsand the energy of the γ-rays emitted by a nucleus. For this purpose, the most intensetransitions from a 152Eu source have been used for both HPGe and LaBr3.

4.1 HPGe calibration

The initial task in the analysis procedure is to build the events based on the raw datacollected during the experiment. This conversion reduces the original data size by at leastone order of magnitude by grouping coincident hits together into events that fulfill certainspecific requirements. The obtained HPGe spectra after event building are displayedusing the SOCOv2 software package [26] and are shown in Figure 17.

Figure 17: Spectra of the uncalibrated single Ge crystals corresponding to the transitionsin the 152Eu source.

31

The strongest peaks in each detector can be integrated and it’s possible to determinethe channel corresponding to the peak’s centroid position. Subsequently, by plottingthe transition energies as a function of these channels, the corresponding calibrationcoefficients can be obtained. In the case of HPGe detectors, this assignment has beendone according to a linear fit: Figures 18 and 19 report a single HPGe spectrum and therelative calibration line, respectively.

Figure 18: Uncalibrated spectrum of a 152Eu source, considering the first HPGe detectorin the first clover (C10). The labels above the peaks represent their energies (in keV).

32

Figure 19: Linear calibration of one HPGe crystal (C10). The energies of the peaksare plotted as a function of the position of the centroids, and their fit gives the reportedequation.

33

Table 1 reports the channels and the corresponding energies of the γ’s emitted bythe 152Eu source in the case of a single HPGe crystal, while Table 2 lists the calibrationcoefficients for all eight crystals.

Channel Energy (keV)672.9 121.81353.0 244.71842.5 344.32273.6 411.22454.9 444.04307.1 778.94796.8 867.45331.3 964.16149.6 1112.17786.0 1408.1

Table 1: On the second column the literature energy values of the strongest transitionsin the 152Eu source are given. On the first one, the corresponding channels in the C10HPGe crystal are reported.

Type of detector Linear calibrationC10 y = 9.6·10−2 + 0.18xC11 y = 6.0·10−2 + 0.19xC12 y = 0.20 + 0.19xC13 y = 7.7·10−2 + 0.19xC20 y = 4.0·10−3 + 0.17xC21 y = 0.18 + 0.20xC22 y = 7.5·10−2 + 0.20xC23 y = 0.13 + 0.19x

Table 2: Calibration coefficients for the HPGe detectors in the two clovers used in thepresent experiment.

34

4.2 LaBr3 calibration

The same calibration procedure has been applied also to the four LaBr3 scintillators:Figures 20 and 21 show, respectively, the spectra of all uncalibrated detectors and thatof a single one (with energy labels). From Figure 21, one can notice the presence of a 40keV peak, which comes from X-rays emission.

Figure 20: Spectra of uncalibrated LaBr3 detectors corresponding to the transitions inthe 152Eu source.

35

Figure 21: Uncalibrated single spectrum of a 152Eu source, considering the first LaBr3

detector (LaBr3 [0]). The labels above the peaks represent their expected energies (keV).

Channel Energy (keV)375.8 121.8766.1 244.71083.6 344.31296.5 411.11404.8 444.02474.7 778.92758.8 867.43070.0 964.14474.0 1408.0

Table 3: On the second column the literature energy values of the strongest transitions inthe 152Eu isotope are given, and on the first one the corresponding channels are reportedin the case of the first LaBr3 detector. The 1112 keV peak has not been consideredbecause in LaBr3 detectors it appears as the sum of two peaks (see Figure 21).

36

The energy resolution of the LaBr3 scintillators is worse than that of HPGe detectors,and depends on the energy of the transition. Therefore, it’s important to calculate thiseffect to have control on the distinction between two close-lying peaks. This has beendone by fitting the energy resolution as a function of the literature values of the transitionenergies. The result of such analysis is presented in Figure 22.

Figure 22: Energy resolution of the four LaBr3 detectors. Error bars are not visible inthe figure, having the size of the points.

The energy response of LaBr3 detectors shows a non-linear character, therefore a thirddegree polynomial was needed to fit the peak energies as a function of the channels, asit is shown in Figure 23 for a single detector. The calibration coefficients of all the fourLaBr3 detectors are summarized in Table 4. We note that the second and third ordercalibration coefficients are very small, however, they become relevant in the high energyregion.

37

Figure 23: Third-degree polynomial calibration of the LaBr3 [0] detector.

Type of detector Third degree polynomial calibrationLaBr3 [0] y = 2.88 + 0.31x - 2.8·10−6x2 + 4.7·10−10x3

LaBr3 [1] y = 2.62 + 0.36x - 4.5·10−6x2 + 5.7·10−10x3

LaBr3 [2] y = 2.94 + 0.35x - 3.9·10−6x2 + 6.5·10−10x3

LaBr3 [3] y = 3.00 + 0.33x - 2.9·10−6x2 + 6.3·10−10x3

Table 4: Calibration coefficients for the four LaBr3 detectors.

38

In the case of LaBr3 detectors, an effect that is often observed is a drift of the detectorsignal over time, which stems from changes of the signal voltage induced by temperaturefluctuations over time. This implies that any calibration might be correct for a fewruns and would induce a systematic error if used on other runs: this error increases theFWHM of the peaks and thus reduces the resolution, as it can be seen in panel a) ofFigure 24.SOCOv2 provides a method for compensating these shifts by using the single spectra togenerate appropriate shifts for correction: the basic idea behind the shift tracking is tocalculate the numeric differential of that spectrum and to find the maxima by searchingfor zero-crossings in the differential. In order to reduce the noise, each single spectrumis first filtered out through the Hann function [26]:

w(n) =1

2

(1− cos

( 2πn

N − 1

))(67)

where 0 ≤ n ≤ N − 1. By choosing the adequate filter window size N , the zero-crossingsin the differentiated spectra show a constant correlation to the true peak position andtherefore may be used to create polynomial fits that map each drifted peak back to theoriginal position in the reference spectrum. In panel b) of Figure 24, it can be seen howthe four spectra appear very well aligned after including the shift files generated by theshift-tracker command in SOCOv2.

In Table 5 the peak energies obtained from the 152Eu source can be seen: in the firstcolumn before the shift-tracker procedure, in the second after. The values after applyingthe shift-tracker agree with the literature values of the transition energies in the 152Euisotope.

Before shift-tracker After shift-tracker124.0 121.6248.1 244.4349.3 344.0417.2 410.9451.8 443.7793.2 778.1883.6 866.6982.5 964.11424.1 1408.3

Table 5: Comparison between the obtained peak energies (keV) before and after theimplementation of the shift-tracker procedure in SOCOv2.

39

Figure 24: Comparison of the obtained LaBr3 spectra in the two cases: a) the shiftshave not been corrected; b) the shifts have been taken into account. Both spectra arezoomed around the same region, for comparison purposes.

40

5 152Eu lifetimes analysisThe 152Eu nucleus has two possible decay modes, as it is shown in Figure 25. It decayseither to an excited state in the 152Sm isotope, or to an excited level in the 152Gd; therespective ground states are then reached through a series of γ cascades [27].

Figure 25: Decay scheme of the 152Eu nucleus. The energy (in keV) of each stateis reported on the left, while the vertical numbers represent the energies of the γ-raysdepopulating the corresponding levels. The half-lives of the excited states are writtenon the right.

From the 152Eu spectrum, one can distinguish the two different decay chains by se-lecting a gate, which means selecting a certain energy interval around one peak andusing it as the reference energy state in the GCD method. The result is a coincidencefast-timing matrix, in which the gated peak is in coincidence with the transitions whichbelong to the same decay cascade.In this chapter, the lifetimes of the states in the decay path of the 152Eu isotope will beextracted in order to validate the fast timing technique. The first example is the statein 152Sm at 366 keV excitation energy.

41

In this case, the 867 keV γ-ray coming from the excited 1234 keV state feeds the 366 keVone, with an half-life of 57.7(6) ps. If a gate on the 245 keV peak (the decay) is selectedin the LaBr3 detector, the other Sm peaks - at 122 and 867 keV - are clearly visible inthe output fast-timing matrices, which are called start and stop. The difference in theoutput matrices is whether the γ-ray recorded in the LaBr3 detector was the start or thestop signal for the TAC [26]. By selecting the 867 keV peak as a feeding transition (seepanel (b) of Figure 26), the TAC projection can be obtained as shown on Figure 27.

Figure 26: a) The 152Eu spectrum measured by LaBr3 detectors is reported and theselected gate on the decay transition is highlighted; b) the projection of the coincidencematrix is shown before setting a gate on the feeding transition of 867 keV.

42

Figure 27: TAC projection of the start matrix of 152Sm, in a logarithmic scale, for the867 - 245 keV transitions configuration.

43

The same procedure is repeated with the stop matrix, and the resulting spectrum isrepresented in Figure 28 together with the one obtained from the start matrix. At thispoint, the centroid difference ∆C of the transition of interest can be extracted.

Figure 28: TAC projections of the start and stop matrices for the 867 - 245 keVtransitions configuration of 152Sm, in logarithmic scale, from which the centroid difference∆C is extracted.

44

In order to calculate the lifetime τ of the state of interest, it’s necessary to know thePRD values of the decay and the feeder: the PRD curve is reported in Figure 29. Oncethe PRD values are obtained, one can calculate the lifetime through the relation

2τ = ∆C − PRD(Efeeder − Edecay) (68)

wherePRD(Efeeder − Edecay) = PRD(Efeeder)− PRD(Edecay) (69)

Figure 29: PRD curve as a function of energy. The red curve corresponds to the bestfit, while the dashed lines give the PRD associated to the fit uncertainties.

The fitting curve for the PRD is given by Eq. 60, with final parameters:

• a = 5608.48 ± 528.5;

• b = 112.208 ± 16.75;

• c = 0.097026 ± 0.006253;

• d = -295.328 ± 21.37.

45

The reported uncertainties are represented by the two dashed lines in Figure 29.In this way the errors on the PRD corresponding to decay and feeder transitions have

been deduced. The parameters determining the upper and lower curve are obtained bysumming or subtracting the errors to the previous values of a, b, c and d (see Table 6).

Upper curve Lower curvea 6136.98 5079.98b 128.958 95.458c 0.103279 0.090773d -273.958 -316.698

Table 6: Parameters for the determination of the dashed curves in Figure 29.

The PRD values of the feeder and the decay will be different, according to whichcurve we consider. They are reported in Table 7 in each case.

Energy (keV) PRDup (ps) PRD (ps) PRDlow (ps)245 68.70 25.19 -19.14867 10.05 31.98 -74.25

Table 7: In the second column, the PRD values obtained with the upper curve inFigure 29 are reported; those obtained with the middle and lower curve are reported inthe third and fourth column, respectively. They all refer to the energy specified in thefirst column.

For both dashed curves in Figure 29 the absolute error on the obtained PRD valuehas been calculated with respect to the one received using the red curve, resulting inwhat follows (Table 8):

Energy (keV) PRDup (ps) PRDlow (ps)245 68.70 ± 43.51 -19.14 ± 44.33867 10.05 ± 42.02 -74.25 ± 42.27

Table 8: Absolute errors on the PRD values obtained with the two dashed curves inFigure 29.

Since the PRD values obtained with the red curve in Figure 29 can be consideredas the mean of the ones obtained with the dashed lines, it’s possible to deduce theiruncertainties through error propagation. They are reported in Table 9, together withthe other necessary measured values to calculate the lifetime of the transition of interest.

46

Since the TAC spectra are given in channels (where one channel corresponds to 10 ps),the values from Table 9 have been multiplied accordingly.

Cstart (ch) 3990.91 ± 0.37Cstop (ch) 4002.89 ± 0.37

PRDfeeder (ps) -31.98 ± 29.80PRDdecay (ps) 25.19 ± 31.06

Table 9: Measured values for the lifetime determination of the 867 - 245 keV transitionscombination of 152Sm.

Since the half-life of the state of interest can be defined as

T 12

= τ · ln2, (70)

using the values in Table 9, a (88.5 ± 21.5) ps lifetime can be extracted, leading to a(61.3 ± 14.9) ps half-life. The obtained results are in agreement with the best literature(57.7 ± 0.6) ps value.This procedure can be repeated in the case of the second state in the 152Sm nucleus at122 keV excitation energy (see Figure 25). In this case, the 245 - 122 transitions havebeen taken into account: by first gating on the 122 keV peak and then projecting thematrices around the 245 keV peak, the respective TAC projections are shown in Figure30. The exponential trend (which is linear in the logarithmic scale) is clearly visible.

With the parameters given in Table 10, the (1.98 ± 0.02) ns lifetime can be calculatedgiving T1/2 = (1.37 ± 0.02) ns. This result is also in agreement with the (1.403 ± 0.011)ns value reported in the literature database.


PRDfeeder (ps) 25.19 ± 31.06PRDdecay (ps) 82.98 ± 30.82

Table 10: Measured values for the lifetime determination of the 245 - 122 keV transitionscombination of 152Sm.

Since τ is larger than the setup timing resolution, one could also apply the slope fitmethod: the coefficent of the linear equation obtained by fitting the start TAC projectionin Figure 30 corresponds to the decay constant λ, which results equal to 4.7·10−4 ns−1.This corresponds to a 2.1 ns lifetime and a 1.5 ns half-life, in agreement with the previousresult; by doing the same on the stop TAC projection, similar values are obtained.

47

Figure 30: TAC projections of the two start and stop matrices for the 245 - 122 keVtransitions combination in the 152Sm isotope.

Agreement is also found for the 779 - 344 keV transitions combination in the 152Gdisotope. The resulting lifetime and half-life are (49.3 ± 21.4) ps and (34.1 ± 14.9) ps,respectively, using the values given in Table 11.


PRDfeeder (ps) -31.88 ± 29.83PRDdecay (ps) 0.63 ± 30.73

Table 11: Measured values for the lifetime determination of the 779 - 344 keV transitionscombination in the case of 152Gd.

48

The other considered transitions combination in 152Gd is the 679 - 411 pair. In Figure31 the corresponding TAC projections are shown. By referring to panel a), the calculatedparameters for lifetime determination shown in Table 12 give a (30.0 ± 21.3) ps lifetimeand a (20.8 ± 14.8) ps half-life, to be compared with the literature (7.3 ± 0.4) ps half-life.

Figure 31: TAC projections of the two start and stop matrices for the 679 - 411 keVtransitions combination in the 152Gd isotope: in panel a) there is no exclusion of detectorscombinations, while in panel b) such exclusion has been enabled.

49


PRDfeeder (ps) -30.06 ± 29.80PRDdecay (ps) -10.26 ± 30.50

Table 12: Measured values for lifetime determination of the 679 - 411 keV transitionscombination in 152Gd, without exclusion of the undesired detectors combinations.

This discrepancy may be due to an inter-detector Compton scattering: in fact, thedetectors are usually positioned as near as possible to the target (and thus to each other)to achieve the maximum solid angle coverage. This vicinity between detectors leads tointer-detector Compton scattering, so it is desirable to exclude those events containingan unwanted combination of neighbouring detectors. This can be done in SOCOv2,by excluding detectors combinations that should be ignored during the analysis. As aresult, the TAC projection presented in panel b) of Figure 31 can be obtained, where theamount of events is reduced: therefore, excluding combinations of neighbouring detectorsis advised when the statistics is high enough, and for this reason this option hasn’t beenimplemented in the analysis of 133Sb data.The new parameters for lifetime determination are summarized in Table 13, giving thelifetime and half-life values of the state of interest: (12.6 ± 21.4) ps and (8.7 ± 14.9) ps,respectively.


PRDfeeder (ps) -30.06 ± 29.80PRDdecay (ps) -10.26 ± 30.50

Table 13: Measured values for the lifetime determination of the 679 - 411 keV transitionscombination in 152Gd, with the exclusion of the undesired detectors combinations.

It’s worth highlighting that the errors have been calculated according to the errorpropagation theory, knowing the uncertainties on the PRD and on the centroid channels(which are given by the software that has been used to analyze spectra and fast-timingmatrices).

50

6 Data AnalysisIn this chapter, the analysis of 133Sb data will be presented. First, detector calibrationswill be discussed along with the identification of all the γ-ray peaks. Second, the selec-tivity of the LOHENGRIN setup on the isomeric decay will be outlined, followed by thelifetime analysis here performed.

6.1 Detectors calibration

The same calibration obtained for 152Eu events has been used also for 133Sb events, inthe case of HPGe detectors. The sum of all the calibrated HPGe spectra is reported inFigure 32.For what concerns LaBr3 detectors, a different event-based calibration was needed; thiswas necessary in order to account for non-negligible gain shifts occurred during theexperiment. The sum of all the four LaBr3 detectors is reported in Figure 33, and Table14 lists the new calibration coefficients for LaBr3 detectors.

Type of detector Third degree polynomial calibrationLaBr3 [0] y = 4.85 + 0.32x + 1.7·10−6x2 - 3.6·10−10x3

LaBr3 [1] y = 4.86 + 0.35x + 2.5·10−6x2 - 8.6·10−10x3

LaBr3 [2] y = 5.00 + 0.36x + 2.3·10−6x2 - 6.7·10−10x3

LaBr3 [3] y = 4.97 + 0.33x + 2.6·10−6x2 - 4.5·10−10x3

Table 14: New calibration coefficients for the four LaBr3 detectors.

In both Figure 32 and Figure 33, the γ rays corresponding to the isomeric decay of the21/2+, 16.6 µs state in 133Sb are marked in bold.

51

Figure 32: Sum of the eight calibrated HPGe detectors, where all the peaks have beenidentified. The bold ones refer to the 21/2+, 16.6 µs isomeric decay of 133Sb.

52

Figure 33: Sum of the four calibrated LaBr3 detectors, with the identification of thevarious peaks. The X-rays 40 keV peak is still visible, and it was also observed in thecase of the 152Eu source.

53

6.2 Mass selection

The experimental setup also includes an ionization chamber (IC) divided in two sections,placed at the focal plane of the spectrometer. These are detectors that collect thecharges created by direct ionization within the gas contained in their interior, through theapplication of an electric field. In Figure 34, it is possible to see that two different massesproduced in the fission of 235U were transported to the focal plane of LOHENGRIN. Thisis due to their similar energy and A/q, which are selected by the optical elements of thespectrometer (see Section 2.1).

Figure 34: a) The two separated IC spectra; b) same as a), but after a re-calibrationthrough the linear relation IC2 = 2.15·IC1 + 1.9·102.

54

The two masses were identified by looking at the γ rays measured in coincidence withthem: they turned out to be A = 133 and A = 93, and the two main produced nucleiare 133Sb and 93Rb (see discussion in Section 6.3).In order to apply a proper gate on masses to select the events of interest, a linearcalibration was performed with the purpose of overlapping the signals coming from thetwo IC sections:

IC2 = A · IC1 +B (71)

The angular coefficient A and the intercept B are calculated considering the position ofthe centroids of the two distributions in both IC spectra, and taking those from IC1 asy and those from IC2 as x.Once the two IC spectra are aligned (see panel b) of Figure 34), they can be summedand the ratio of the counts in the two peaks can be calculated, which turns out to be ∼8.3. This factor is used to normalize the γ spectra measured in coincidence with A = 93and A = 133, as depicted in panel b) of Figure 35.

By subtracting the two normalized spectra, only the four peaks corresponding to thedecay of the 21/2+, 16.6 µs isomer in 133Sb remain (see Figures 36 and 37).

55

Figure 35: LaBr3 spectra gated on A = 133 and A = 93 in panel a) and b), respectively.In panel b), the LaBr3 spectrum has been multiplied by a 8.3 factor and it is possible tosee that the peaks corresponding to the 133Sb isomeric transition (marked in red in thepanel above) have disappeared. The 168.5 keV line from 93Y is instead visible.

56

Figure 36: Mass-gated HPGe spectra and the result of their subtraction (in blue). Theremaining peaks are those relative to the considered isomeric transition in 133Sb.

57

Figure 37: Same as Figure 36, but considering the two mass-gated LaBr3 spectra (i.e.,the two spectra in Figure 35).

58

6.3 Identification of contaminants

Along with the γ-ray decay of 133Sb, other lines were observed in the present measure-ment, as can be seen in Figures 36 and 37. These lines are listed in Table 15, and theymainly come from the β decay of 133I, 133Te (which are produced by the β decay of 133Sb)and of other isotopes produced during the neutron-induced fission of the 235U target, like93Rb and 93Y, whose partial level schemes are reported in Figure 38.

Figure 38: Level schemes of contaminant transitions from a) 93Rb and b) 93Y.

Furthermore, some random lines from the natural background are also present: 40Kemits a γ ray at 1461 keV energy, 214Pb at 352 keV and 208Tl at 2615 keV.

59

Energy (keV) Isotope Decay mode

61.7162.51510.32791.7

133Sb Isomeric transition

334.3817.8836.91096.2

133Te β decay into 133I

74.181.695150.8169213.5261.6312.1407.6444.9602.1647.5710.4724786.9844.4864912.7978.31333.21458.9

133I β decay into 133Xe

Energy (keV) Isotope Decay mode

529.61596.2

93Rb β decay into 93Y

168.5590.2875.7

93Y β decay into 93Zr

609.31120.31377.71385.31683.91729.61764.522042447

214Bi Background

1460.8 40K Background

351.5 214Pb Background

2614.5 208Tl Background

Table 15: List of all the peaks that have been identified in the HPGe spectrum, includingthose coming from natural background.

60

6.4 133Sb lifetimes analysis

As anticipated in the previous Section, four transitions have been observed in 133Sb,following the decay of the 21/2+ isomer, as shown in Figure 39.

Figure 39: Partial level scheme of 133Sb following the decay of the 21/2+, 16.6 µs isomer.

These four γ’s are emitted in coincidence: for example, if a gate is set on the 1510keV transition on the LaBr3 scintillators, the other transitions at 62 and 162 keV in 133Sbwill be clearly visible (Figure 40).

Similarly, Figure 41 shows the γ coincidences when the gate is set on the 2792 keVtransition using HPGe detectors.

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Figure 40: a) The 133Sb spectrum from LaBr3 detectors is reported and the gate on the1510 keV transition is highlighted; b) the coincidence spectrum measured with LaBr3

detectors, with a gate on the 1510 keV line.

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Figure 41: 133Sb spectrum from the HPGe detectors with a gate on the 2792 keVtransition, showing the other transitions depopulating the 21/2+ isomer.

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The purpose of this thesis is to use fast-timing techniques to determine the lifetimesof the two 13/2+ and 15/2+ states, fed by the 21/2+, 16.6 µs isomer in 133Sb.The procedure is exactly the same as described in Chapter 5 for 152Eu: for each state, agate is put on both the feeder and decay transition; then, the start and stop fast-timingmatrices are produced and projected on the time axis, from which one can extract thecentroid position. In order to select only the events of interest, LaBr3 spectra gated onA = 133 have been taken into account.First of all, the lifetime analysis has been performed without considering backgroundand mass contributions. Then, in order to extract a more precise lifetime value, thesecorrections have been taken into consideration and, in both cases, two background regionshave been selected (one on the left and one on the right of the peak) in such a way thattheir sum gives the amplitude of the energy gate itself (see Figure 42).

Figure 42: Selection of the two background regions around the 162 keV peak, when thegate is set on the 1510 keV transition.

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Time spectra are then obtained by projecting the (Tγ, Eγ) matrix, considering thebackground energy intervals defined above: the result is the total background spectrumfor the combination of interest, which can be subtracted from the original peak timespectrum, as it can be seen in Figure 43.

Figure 43: Time spectra obtained by projecting the (Tγ, Eγ) matrix on the TAC axisand considering the energy intervals shown in Figure 42.

After this background subtraction, the mass selection is performed. In the LaBr3

spectrum gated on A = 93, one selects the same energy gate as in LaBr3 spectrum gatedon A = 133 and projects the cut on the TAC axis; after a proper normalization, it canbe subtracted from the other A = 133 gated spectrum. All the TAC projections ofthe matrices are shown from Figure 44 to Figure 49, while the values used for lifetimemeasurements are shown in Tables 16-21, reporting the PRD for the transitions of interest(see Chapter 5) and the centroid position in the start and stop time matrices.

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Figure 44: TAC projections of the two start and stop matrices for the 13/2+ state in133Sb, gating on the decay transition at 1510 keV. a) No corrections are included; b) thespectra have been corrected for the background; c) mass correction has been performed.

13/2+ No corrections Background correction Mass correctionCstart (ch) 4003.34 ± 0.35 4003.89 ± 0.37 4004.80 ± 0.41Cstop (ch) 3997.75 ± 0.35 3996.88 ± 0.36 3996.35 ± 0.38

PRDfeeder (ps) 59.08 ± 31.10PRDdecay (ps) -9.57 ± 30.56

Table 16: Measured values for the lifetime determination of the 13/2+ state, gating onthe decay transition at 1510 keV.

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Figure 45: TAC projection of the two start and stop matrices for the 13/2+ state in133Sb, gating on the feeder transition at 162 keV.



Table 17: Measured values for the lifetime determination of the 13/2+ state, gating onthe feeder transition at 162 keV.

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Figure 46: TAC projections of the two start and stop matrices for the 15/2+ state in133Sb, gating on the decay transition at 162 keV. a) No corrections are included; b) thespectra have been corrected for the background; c) mass correction has been performed.



Table 18: Measured values for the lifetime determination of the 15/2+ state, gating onthe decay transition at 162 keV.

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Table 19: Measured values for the lifetime determination of the 15/2+ state, gating onthe feeder transition at 62 keV.

Figure 47: TAC projections of the two start and stop matrices for the 15/2+ state in133Sb, gating on the feeder transition at 62 keV. a) No corrections are included; b) thespectra have been corrected for the background; c) mass correction has been performed.

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13/2+ + 15/2+ No corrections Background correction Mass correctionCstart (ch) 4013.65 ± 1.00 4017.21 ± 1.20 4020.80 ± 1.60Cstop (ch) 3995.64 ± 0.98 3994.50 ± 1.10 3992.30 ± 1.40


Table 20: Measured values for the lifetime determination of the sum of the two 15/2+

and 13/2+ states, gating on the decay transition at 1510 keV.

Figure 48: TAC projections of the two start and stop matrices for the sum of thetwo 13/2+ and 15/2+ state in 133Sb, gating on the decay transition at 1510 keV. a) Nocorrections are included; b) the spectra have been corrected for the background; c) masscorrection has been performed.

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13/2+ + 15/2+ No corrections Background correction Mass correctionCstart (ch) 3994.04 ± 0.92 3993.00 ± 1.00 3991.60 ± 1.40Cstop (ch) 4013.32 ± 0.95 4014.80 ± 1.10 4018.00 ± 1.40


Table 21: Measured values for the lifetime determination of the sum of the two 15/2+

and 13/2+ states, gating on the feeder transition at 62 keV.

Figure 49: TAC projections of the two start and stop matrices for the sum of thetwo 13/2+ and 15/2+ states in 133Sb, gating on the feeder transition at 62 keV. a) Nocorrections are included; b) the spectra have been corrected for the background; c) masscorrection has been performed.

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Table 22 reports the values of the measured lifetimes τ of the states of interest. Themean τ is also reported, by averaging on the obtained lifetime values in the two directand reverse configurations (i.e. when the gate is set on the decay or feeder transition,respectively).

13/2+ τ (ps), 1510-162 τ (ps), 162-1510 Mean τ (ps)

No corrections -6.4 ± 21.8 -8.4 ± 21.8 -7.4 ± 15.4

Background correction 0.7 ± 21.8 2.7 ± 21.8 1.7 ± 15.4

Mass correction 7.9 ± 21.8 8.7 ± 21.8 8.3 ± 15.4

15/2+ τ (ps), 162-62 τ (ps), 62-162 Mean τ (ps)

No corrections 55.2 ± 21.4 57.2 ± 21.4 56.2 ± 15.1


Mass correction 42.0 ± 21.4 50.7 ± 21.4 46.4 ± 15.1

13/2+ + 15/2+ τ (ps), 1510-62 τ (ps), 62-1510 Mean τ (ps)

No corrections 17.5 ± 21.2 23.8 ± 21.2 20.6 ± 15.0


Mass correction 69.9 ± 21.2 59.4 ± 21.2 64.7 ± 15.0

Table 22: Measured lifetimes of the states of interest with or without corrections. Inthe second column the lifetime τ has been calculated with the gate on the decay, whilein the third one the gate is on the feeder. In the fourth column, the value of τ is givenby an average of the previous two.

If no correction is made for the 13/2+ state, a negative lifetime is achieved regardlesswhether the gate is set on the feeder or the decay transition; after subtracting the back-ground and imposing the mass gate, the lifetime values increase to positive values. Whenthe gate is set on the decay, the lifetime reaches ∼ 0.7 ps after background subtractionand it increases again after mass correction; if the gate is on the feeder, τ increases to apositive value after the first correction and then it takes almost the same absolute valueof the original result after the second one, and the same thing happens to the mean valueof τ .The 15/2+ state behaves differently: when the gate is set on the decay transition at 162keV, τ is about 55 ps and becomes ∼ 13% longer after background correction, but thendecreases to 42 ps after mass correction. When the gate is on the feeder, τ drops by ∼11% with respect to the original value after background and mass subtraction; the meanτ doesn’t change much after the first correction, and then decreases by ∼ 17% compared

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to the non-corrected τ after taking the mass contribution into account.A third case has been analyzed, i.e. the 1510 - 62 cascade. When the gate is set on the62 keV transition, τ increases by a factor ∼ 2.3 after background correction and by afactor of ∼ 4 after mass correction; in the reverse configuration, τ increases by ∼ 50%after the subtraction of background and by a factor ∼ 2.5 after mass correction. Themean lifetime gets a factor ∼ 1.9 and ∼ 3.1 longer after the two corrections, respectively.Since the 1510 - 62 cascade can be seen as the 1510 - 162 one followed by the 162 - 62cascade, it should be expected that the resulting lifetime values are compatible with thesum of the ones obtained separately in those two cases. In the last three rows of Table22, it is possible to see that the calculated lifetimes are compatible with the sum of theprevious ones, either with or without corrections. The sum of the mean values of τ (inthe fourth column) also appears to be consistent within the errors, which are calculatedaccording to error propagation and are underestimated because the uncertainty contri-bution introduced by the gate hasn’t been considered.Taking into account the decay branchings from the two 15/2+ and 13/2+ states, B(M1)values were extracted for the 15/2+ → 13/2+ and 13/2+ → 11/2+ transitions throughthe equation

B(M1) =1

1.76 · 1013E3γT (M1)

(72)

T is the inverse of the partial lifetime τγ, that is related to the measured lifetime τ ofthe particular γ decay:

τγ(M1) = τItot

Iγ(M1)(73)

Iγ is the intensity of the γ transition and Itot is the sum of the intensities of all theγ’s depopulating the level. The γ’s that depopulate the two 15/2+ and 13/2+ levelshave energies of 162 keV and 110 keV, with relative intensities Iγ = 100 and Iγ = 0.8,respectively. The branchings can be seen in Figure 50, and Table 23 report the measuredB(M1) values.

.

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Figure 50: Partial level scheme of 133Sb, in which the decay branchings of interest areshown together with the relative intensities of the γ decays.

13/2+ → 11/2+ B(M1) (W.u.), B(M1) (W.u.), Mean B(M1)

1510-162 162-1510 (W.u)

No corrections (-2.8 ± 9.5)·10−2 (-2.1 ± 5.5)·10−2 (-2.4 ± 5.0)·10−2


Mass correction (2.3 ± 6.3)·10−2 (2.1 ± 5.2)·10−2 (2.2 ± 4.0)·10−2

15/2+ → 13/2+ B(M1) (W.u.), B(M1) (W.u.), Mean B(M1)

162-62 62-162 (W.u.)

No corrections (12.4 ± 4.8)·10−2 (11.9 ± 4.5)·10−2 (12.2 ± 3.3)·10−2

Background correction (10.9 ± 3.7)·10−2 (13.5 ± 5.7)·10−2 (12.1 ± 3.2)·10−2

Mass correction (16.3 ± 8.3)·10−2 (13.5 ± 5.7)·10−2 (14.7 ± 4.8)·10−2

Table 23: Reduced transition probabilities, calculated from the values in Table 22 forthe 13/2+ and 15/2+ states.

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7 Comparison with theoretical calculationsThe preliminary results on the structure of 133Sb (which can be seen as the nucleus132Sn plus a proton) are interpreted by the Hybrid Configuration Mixing model, aimedat describing odd-mass systems around doubly-magic nuclei based on couplings withvarious types of core excitations (both collective and non-collective), using a Skyrmeeffective interaction.

7.1 The Hybrid Configuration Mixing model

The Hybrid Configuration Mixing (HCM) model is a new microscopic model developedby the group of Milano that aims at the description of low-lying states in odd nuclei madeof a magic core plus an extra nucleon, like 133Sb. The calculation is self consistent, sincethe description of single-particle states and core excitations is based on a Hartree-Fock(HF) description and Random Phase Approximation (RPA) calculations performed withthe same Skyrme interaction [28].One begins from a basis of states that are either pure particle states outside a core, orparticles coupled with a core excitation emerging from RPA. The particle states havethe usual quantum numbers n, l, j and m, that will be shorthanded in the form jm;the associated energies are denoted with ε, while a and a† stand for the annihilation andcreation operators of fermionic states, respectively. For each spin and parity JπM , thecore excitations will be labelled with an index N and the parity label will be dropped;the associated energies, annihilation, and creation boson operators are denoted by hω,Γ and Γ†, respectively. The basis states are

|jm〉 = a†jm|0〉 (74)

where |0〉 represents the even-even core, or

|[j′m′ ⊗NJ ]jm〉 =∑m′M

〈j′m′JM |jm〉a†j′m′Γ†JM |0〉 (75)

For each j, the Hamiltonian H = H0 +V is diagonalized. H0 represents the unperturbedHF + RPA part and V their coupling, and they are expressed as

H0 =∑jm

εja†jmajm +

∑NJM

hωNJΓ†NJMΓNJM (76)

V =∑jmj′m′

∑NJM

h(jm; j′m′, NJM)ajm[a†j′m′ ⊗ Γ†NJ ]jm (77)

where h are the coupling matrix elements, which can be calculated once the structureof the RPA states is known [29]. This Hamiltonian can be diagonalized separately in

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different Hilbert subspaces with good angular momentum j and parity π.Calling N the overlap matrix between the states in the space in which H is solved, thecorrection for the non-orthonormality of the basis formed by those states made up withone particle and one core excitation is taken into account by solving the generalizedeigenvalue problem (H − EλN)|λ〉 = 0 [30].

7.2 133Sb level scheme and wave function composition

In Figure 51, the experimental level scheme of 133Sb is compared with theoretical cal-culations performed with the SkX Skyrme interaction. As it can be seen, the modelreproduces rather well the energies of the observed states.

Figure 51: Comparison between theoretical (left) and experimental (right) level schemeof 133Sb.

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The wave function compositions of each state are reported in Table 24. It can benoticed that the 7/2+ ground state and the lowest 5/2+, 3/2+ and 11/2− states havemainly a single-particle character, with a valence proton occupying a certain orbital (g7/2,d5/2, d3/2 and h11/2, respectively). On the other hand, the wave function of the 11/2+

state is dominated by the coupling between a g7/2 proton and the collective 2+1 state in

132Sn. Moreover, the 13/2+ and the 15/2+ states have a quite mixed wave function, whilethe highest spin states 17/2+ and 21/2+ are dominated by a valence proton coupled tonon-collective core excitations.

Table 24: Wave function composition of positive-parity states in 133Sb, as calculated bythe HCM model. Only contributions larger than 10% are reported.

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7.3 Reduced transition probabilities

The experimental B(M1) values for the 15/2+ → 13/2+ and 13/2+ → 11/2+ transitionsobtained in this case were also compared with theoretical predictions performed with theHCM model. The latter are 2.919·10−4 W.u. and 1.691·10−4 W.u., respectively. Despitethe difference on the absolute values between the experimental and theoretical results(from Table 23, the measured reduced transition probabilities are (14.7 ± 4.8)·10−2 W.u.for the 15/2+ → 13/2+ transition and (2.2 ± 4.0)·10−2 W.u for 13/2+ → 11/2+), theratio between B(M1; 15/2+ → 13/2+) and B(M1; 13/2+ → 11/2+) is qualitativelyreproduced. This is ∼ 2 in the case of the HCM model predictions and ∼ 7 for theexperimental values.

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ConclusionIn this thesis, a detailed study of lifetimes in the picosecond range of excited states inthe 133Sb nucleus was performed, with specific emphasis on the 13/2+ and 15/2+ statesabove 4 MeV. The experiment was performed at the ILL laboratory in Grenoble, exploit-ing the neutron-induced fission of 235U. The 133Sb nucleus produced in the fission processwas selected by the LOHENGRIN mass spectrometer and transported to its focal plane,where the γ-ray decay from the 21/2+, 16.6 µs isomer was measured by using a mixedsetup comprising two HPGe detectors and four LaBr3 scintillators.A very clean selection on the transitions of interest was achieved by means of mass-γ-γ coincidences, with the mass being reconstructed from the ions detected by the ion-ization chamber of the spectrometer. Lifetime measurements were performed throughfast-timing techniques, using LaBr3 detectors and different background-subtraction con-ditions.The analysis here presented provided τ = (8.3 ± 15.4) ps and τ = (46.4 ± 15.1) ps forthe 13/2+ and 15/2+ states, respectively. The reduced transitions probabilities deducedfrom them are B(M1; 15/2+ → 13/2+) = (14.7 ± 4.8)·10−2 W.u. and B(M1; 13/2+

→ 11/2+) = (2.2 ± 4.0)·10−2 W.u. respectively.Experimental results have been compared with theoretical calculations performed withthe Hybrid Configuration Mixing (HCM) model developed by the group of Milano. Thismicroscopic, self-consistent approach was specifically designed to describe odd-mass nu-clei in the vicinity of closed shells, using Hartree-Fock and Random Phase Approximationcalculations and a Hamiltonian of Skyrme type.A good agreement between the experimental and theoretical level scheme of 133Sb wasachieved. The model suggests that low-lying states have a nearly pure single-proton na-ture, whereas those at higher excitation energy are dominated by couplings with 132Sn,collective and non-collective core excitations. Moreover, the model predicts a ratio be-tween B(M1; 15/2+ → 13/2+) and B(M1; 13/2+ → 11/2+) of ∼ 2, which is in qualita-tively agreement with the experimental value of ∼ 7. This work represents an importantbenchmark for the HCM model and validates its predictive power for odd-mass nucleiclose to doubly-magic isotopes, which are typically difficult to describe in terms of clas-sical shell model calculations.

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AcknowledgementsIl primo ringraziamento va alla professoressa Silvia Leoni, per avermi dato l’opportunitàdi svolgere questo lavoro di tesi e per il suo costante supporto.Grazie a Simone per l’aiuto e la disponibilità che ha dimostrato, oltre che per la sua sim-patia; Łukasz, dzięki za pomoc i zachętę, które mi dałes w tym okresie (i za cierpliwośćkiedy próbowałam mówić po polsku).Ringraziamenti speciali vanno poi ai miei amici di sempre e a quelli che mi hanno accom-pagnato in questi anni di studio, in particolare Veronica, Valentina, Arianna e Francesca.Grazie a Marta e Chiara, amiche e compagne di tesi, con cui ho condiviso gioie, dolori ecibo.Alice: meriti un ringraziamento speciale, perché sei colei che più mi ha supportato inquesti due anni. Il nostro legame va al di là di qualsiasi cosa e io sono orgogliosa di fareparte della tua vita.I ringraziamenti più sentiti e più importanti vanno però alla mia famiglia, soprattutto amamma e papà che mi supportano e sopportano da sempre. Grazie per tutto quello cheavete fatto e per la serenità che siete sempre riusciti a garantirmi.

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Documents

LIFETIME MEASUREMENTS OF - INFN LIFETIME MEASUREMENTS OF EXCITED STATES IN THE 133SB NUCLEUS WITH FAST-TIMING TECHNIQUES Relatore: Prof.ssaSilviaLeoni Correlatori: Dott. SimoneBottoni