Level density and γ-ray strength function in 87 Kr

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The Oslo Method in Inverse Kinematics

Level density and γ-ray strength function in ⁸⁷ Kr

Vetle Wegner Ingeberg

Thesis submitted for the degree of Master of Science

60 credits

Faculty of mathematics and natural sciences

UNIVERSITY OF OSLO

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The Oslo Method in Inverse Kinematics

Level density and γ-ray strength function in ⁸⁷ Kr

Vetle Wegner Ingeberg

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The Oslo Method in Inverse Kinematics http://www.duo.uio.no/

Printed: Reprosentralen, University of Oslo

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Abstract

The first ever investigation of the application of the Oslo Method in inverse kinematics experiments has been performed by analysis of an experiment where a

86Krbeam at300MeV was used on deuterated polyethylene and polystyrene targets at iThemba LABS to induce d(⁸⁶Kr,p)⁸⁷Kr reactions. The experiment was done at the AFRODITE array featuring two large volume (3.5×8”)LaBr3(Ce) detectors, eightCLOVERdetectors and two particle telescopes each consist- ing of two segmented silicon detectors. Challenges only seen in inverse kinematics such as Doppler shift of γ-ray energies, large angular dependencies, etc. have been solved. As part of this master thesis a new software code for analyzing inverse kinematics experiments with the Oslo Method was developed.

Theγ-ray strength function and the level density of⁸⁷Krwere extracted and a large enhancement of theγ-ray strength function at low energy was found. The experimental level density is consistent with a constant temperature model for the level density.

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Acknowledgements

I would like to thank my supervisors Sunniva Siem and Mathis Wiedeking for excellent help and guidance. Thanks to the Nuclear Physics Group at the Uni- versity of Oslo for providing a fantastic work environment, the passion that you all have for the field is truly inspiring. Thanks to everyone who helped set up the experiment and everyone who sacrificed their weekends to take shifts.

Takk til mamma og pappa som alltid har støttet meg og alltid stiller opp. Takk til Anders, Torgeir, Sigve, Wilhelm, Henrik, Anne-Marthe, Jørgen og Jon Vegard for fantastiske luncher og alt annet tull og tøys. Takk til familien min, venner, Fysikkforeningen, Fysisk Fagutvalg og miljøet p ˚a Lillefy.

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List of Figures

1.1 Nucleonic energy levels . . . 2

1.2 Illustration of the evolution of the level spacing with increasing excitation energy . . . 4

1.3 Theγ-ray strength function of^56,57Fe . . . 4

1.4 Effects of low energy enhancements on neutron capture cross sections . . . 5

2.1 Two-body scattering in laboratory frame . . . 12

2.2 Two-body scattering in center-of-mass frame . . . 13

3.1 Floor plan of iThemba LABS . . . 18

3.2 Picture of the AFRODITE array . . . 19

3.3 Picture of the particle telescopes . . . 21

3.4 Illustration of the particle telescopes . . . 22

3.5 Burned C2D4targets. . . 23

4.1 γ-ray calibration spectra . . . 25

4.2 Silicon strip calibration spectra . . . 27

4.3 Observed drift in particle telescope . . . 28

4.4 ∆E - E coincidence spectra . . . 30

4.5 Time spectrum of aLaBr3(Ce)detector. . . 31

4.6 Time spectrum of aCLOVERdetector. . . 32

4.7 Illustration of a two-body reaction experiment . . . 34

4.8 Excitation energy . . . 36

4.9 Singles excitation spectrum for⁸⁷Kr . . . 37

4.10 Recoil velocity of⁸⁷Kr . . . 39

4.11 Experimental excitation of⁸⁷Krfor proton-γcoincidences . . . . 40

4.12 γ-ray spectrum for proton coincidences . . . 41

4.13 Particle-γcoincidence matrices . . . 42

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LIST OF FIGURES

5.1 Compton scattering . . . 44

5.2 Response spectrum . . . 45

5.3 Interpolation of response function . . . 46

5.4 The response matrix ofLaBr3(Ce) . . . 47

5.5 The response matrix ofCLOVER. . . 47

5.6 Simulation of theLaBr3(Ce)response function for the iThemba LABS setup with4.438MeVγ-rays . . . 48

5.7 Simulation ofLaBr3(Ce)response function in the CACTUS array with4.438MeVγ-rays . . . 49

5.8 The unfolded particle-γcoincidence spectra . . . 53

5.9 Illustration ofγ-ray cascade . . . 54

5.10 The first generation spectra of thed(⁸⁶Kr,p)⁸⁷Krreaction . . . . 57

5.11 Experimental level density of⁸⁷Kr . . . 60

5.12 Experimentalγ-ray strength function of⁸⁷Kr . . . 62

5.13 Comparison of GSF for⁸⁷Krwith otherKrisotopes . . . 63

6.1 Excitation energy of⁸⁷Kras a function of proton scattering angle and energy . . . 65

6.2 Excitation energy of ⁸⁶Kr as a function of deuteron scattering angle and energy . . . 66

6.3 Experimental angle-particle energy correlation matrix for proton events. . . 67

6.4 Experimental angle-particle energy correlation matrix for proton events. . . 68

6.5 The∆E - E spectra at different angles . . . 69

6.6 Deuteron breakup . . . 70

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List of Tables

3.1 List ofγ-detectors and their mounting angle . . . 19 4.1 Theγ-ray energies used in the calibration of theγ-ray detectors. 26 4.2 Theα-particle energies used in the calibration of the silicon strips

of the particle telescopes. . . 26 4.3 Total number of particle-γcoincidences . . . 39 5.1 Neutron resonance spacing for n+⁸⁶Kras reported from vari-

ous sources. . . 59

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Chapter 1

Introduction

Nuclear physics is the study of the atomic nuclei and its properties. The nucleus consists of protons and neutrons, together called nucleons. The protons and neutrons consist of three quarks and are held together by the gluon which is the carrier of the strong force. According to the exchange force model the nucleons are kept together inside the nucleus by the short range nuclear force which is mediated by the exchange of light mesons between the nucleons. This model explains many of the fundamental properties of the interactions between the nucleons but is fairly complicated, making calculations with many nucleons dif- ficult. Several additional models have been adopted to describe various properties of nuclei such as mass, energy states, spin, parity, etc. These models range from macroscopic models such as the liquid drop model to microscopic models such as the shell model.

In the liquid-drop model the nucleus is viewed as a droplet where the mass is given by the number of nucleons and the ratio of protons and neutrons. It is extremely successful at describing the overall properties and trends in the masses of the known stable nuclei and the process of fission, but fails at ex- plaining why nuclei with particular neutron and/or proton numbers are more stable than others or why odd-odd proton and neutron numbers are consis- tently less stable than those with even-even numbers. Nor does it describe microscopic properties such as the nucleus ground state spin, parity and the discrete excited states. These properties can be predicted by the nuclear shell model which explains the nucleus as nucleons trapped in a potential similar to the finite three dimensional square well with slopes resembling that of the har- monic oscillator. This results in the nucleons having to be placed in different energy levels of finite capacity as illustrated in Figure 1.1. At some particular energy levels there are major gaps in energy to the next level. When one of

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CHAPTER 1. INTRODUCTION

Figure 1.1: Nucleonic energy levels in which protons and neutrons are placed.

Major gaps between the levels marks the nuclear shells that cause added stability when filled. Figure is taken from [1].

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these levels is filled the nucleus is said to have reached a shell closure and these particular proton or neutron numbers are the ”magic numbers” (2,8,20, 28, etc.) where the nucleus is particularly strongly bound.

Moving nucleons from filled levels to unfilled levels creates the excited states of the nucleus. Such excited states are unstable and will decay back to a stable state. The mode of decay is γ-decay back down to the ground state of the nucleus, provided that the excitation energy is less than the threshold for emission of particles. The rate of decay from one excited state to another of lower energy is determined by the electromagnetic properties of the states and the spins of the states. Many nucleons can contribute to an excited state of the nucleus, making predictions of states and transitions complex and compu- tationally challenging.

As the excitation energy increases the number of possible stats will increase exponentially and the energy spacing between the individual states will de- crease. At a certain excitation energy the spacing between the states will be- come so small that the widths of the states will start to overlap. This energy marks the onset of the continuum region of excitation energy. At the low end of the excitation energy scale there are only a few excited states and these are well separated. This region is said to be the discrete region. Between the discrete region and the continuum region is the so called quasi-continuum¹, see illustration in Figure 1.2. In this region the nucleus is best described by average properties such as the average number of states per unit energy (level density) and the average electromagnetic properties (γ-ray strength function).

One of the most successful methods at measuring these properties is the Oslo Method [3] which enables the simultaneous extraction of both the level density and the γ-ray strength function from particle - γ coincidence experiments. About a decade ago this technique revealed an unexpected increase in theγ-ray strength for lowγ-energies,Eγ ≤3MeV, in^56,57Feas reported by [4]. The experiment was later repeated withLaBr3(Ce) detectors by [5] and resulted in theγ-ray strength function seen in Figure 1.3. This experiment also revealed that the low energy enhancement was of dipole radiative nature. This enhancement has later been observed in many nuclei in the light and medium mass region ranging fromTitoLa[6–11]. The existence of the enhancement has later been confirmed by a model independent method [12]. Theoretical explanations of the origin of the feature have recently been presented and ex-

1The definition of the quasi-continuum is somewhat elusive and is not well defined in the litera- ture. It might be generalized as the region of excitation where the properties of the states can be described by their average.

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Figure 1.2: Illustration of how the level spacing evolves with increasing excitation energyEx. HereΓ is the width of the states andD is the level spacing.

Figure taken from [2].

Figure 1.3: Theγ-ray strength function for^56,57Fe. Figure taken from [5].

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Figure 1.4: The average neutron capture cross section calculated with low energy enhancements divided by calculations without. Figure taken from [15].

plains it as either of dipole magnetic nature [13] or dipole electric nature [14], but the true nature of the enhancement has yet to be determined experimentally.

The level density and the γ-ray strength function are parameters that are especially important in calculations of nuclear reaction cross sections as they provide all the probability of transitions between energy levels by emission ofγ- rays. Of the reaction cross-sections, the cross section for neutron capture is of particular interest as it has applications both in reactor physics and in nuclear astrophysics. Currently these cross sections have been fairly well documented for most stable nuclei, but are not very well known for unstable nuclei. This represents a large source of uncertainty for models predicting the production and formation of elements at sites of high neutron flux (r-process). In 2010 it was shown by [15] that the inclusion of the newly discovered enhancements in theγ-ray strength function can increase the neutron capture cross section by a factor of about 10² for some neutron-rich nuclei, see Figure 1.4. Such an increase in the capture cross section has been shown to have a significant impact on calculated abundances in the r-process resulting in an average of up to43%change [16]. Measuringγ-ray strength functions and level densities of neutron-rich nuclei will help put constraints on the neutron capture rates.

Currently the Oslo method has mostly been applied to experimental data from traditional experiments where light particles (p,d,³He,α) have been accelerated onto targets containing suitable isotopes for the formation of the nucleus of interest. The targets used in these experiments need to be reasonably stable both chemically and radioactively in order for them to persist thought the length of the experiment, thus limiting the experimentally accessible nuclei to only those close to the valley of stability and close to chemical stability. Re-

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cently the method has been extended to nuclei where the quasi-continuum are populated byβ-decay [17] which has been a great step towards studying more neutron-rich nuclei. This method, known as the β-Oslo method, does have some limitations as it requires that the quasi-continuum is populated by theβ- decay, thus requiring the energy released by the decay to be large. Another more general way of reaching unstable nuclei is to have inverse kinematics experiments, where heavy nuclei are accelerated towards targets containing high amounts of light nuclei such as protons or deuterons. Such an ”inverse Oslo”

method will be much more powerful than the traditional experimental setup as it would only be limited by the beams available. The first ever attempt at such an experiment with the Oslo Method is presented in this thesis.

Moving to inverse kinematics experiments poses new challenges that are absent in the traditional experimental setup. As inverse kinematics involve pro- jectiles with much larger energies the kinematics requires more rigorous treatment due to the increased angular sensitivity. The excited nucleus will have a high velocity after the reaction, resulting in significant Doppler shifts of the observedγ-ray energies. There are also technical challenges since there was no software available for data handling and sorting the data as required for the Oslo method, thus new code had to be developed (≈4300lines of code). This thesis will resolve these challenges and provide a proof of concept by analysing data from a inverse kinematics experiment where a⁸⁶Krbeam at300MeV has been accelerated onto targets of deuterated polyethylene and polystyrene.

The thesis is structured such that chapter 2 will present some theoretical concepts needed in the analysis of the experiment while the experimental setup is presented in chapter 3 and the analysis of the resulting data is given in chapter 4. The extraction of the level density and theγ-ray strength function will be presented in chapter 5. Chapter 6 will discuss some of the experimental difficulties found related to the use of inverse kinematics as well as some of the results obtained from the experiment. The thesis will be summarized and a future outlook is given in chapter 7.

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Chapter 2

Theory

2.1 Statistical nuclear properties

The region of excitation energy in the nucleus between the discrete region (where states have well-defined quantum numbers) and the continuum region (where states over-lap each other) is defined as the quasi-continuum. The region is characterized by levels that are spaced so close together that it is impractical to treat them as individual levels and is best described by the average properties of the nucleus, rather than the individual levels. Such properties include the level density andγ-ray strength function which are the topic of this thesis.

2.1.1 Nuclear level density

The nuclear level density (NLD) is defined as the number of quantum levels per unit of excitation energy and is the inverse of the average energy spacing between the states. To accurately describe the NLD of a certain nucleus one would have to solve the Schr ¨odinger equation with the appropriate potential for the nucleus and find all the excited states, which can be a tedious and diffi- cult process if at all possible. There are models that avoid these problems by relying on thermodynamical approximations for the nucleus. The most com- mon approximation is to model the nucleus as a Fermi gas. The first Fermi gas model was introduced by H. Bethe in 1936 and relies on a description of the nucleons as non-interacting fermions inside a potential. The original model states that the NLD for an excitation energyE can be described by [18]

ρ(E) =

√π 12

exp(2√ aE)

a^1/4E^5/4 (2.1)

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CHAPTER 2. THEORY

where the level-density parameterais given by a= π

6(gp+gn)

Where the parametersgpandgnare the single-particle level density parameter for protons and neutrons, respectively.

The formula was later revised by A. Gilbert and A. G. W. Cameron in 1965 [19]:

ρ(U) =

√π 12

exp(2√ aU a^1/4U^5/4

√1

2πσ (2.2)

where they introduced the back-shifted energyU =E−∆p−∆nwhere∆pand

∆n are the paring energies for protons and neutrons, respectively. They also introduced the spin cut-off parameterσas

σ²=g m²

T (2.3)

whereg =gp+gn, m²

≈0.146A^2/3 is the mean-square magnetic quantum number for single-particle states and the nuclear temperature:

T = rU

a

A large study of the systematics of the NLD done by T. von Egidy and D.

Bucurescu in 2005 [20] lead to further modifications to the NLD:

ρ(E) = exp(2√

E−E1) 12√

2σa^1/4(E−E1)^5/4 (2.4) and

σ²=IT. (2.5)

The rigid-body value for the nuclear moment of inertia is I= 2

5 m0r²₀

~² A^5/3

wherem0is the nucleon mass andr0is the nuclear radius. The temperature is given by

T = 1 +p

1 + 4a(E−E1)

2a (2.6)

The level-density parameter a and the energy shift E1 are in [20] treated as free parameters to be fitted to experimental data.

Another model of the NLD is the constant temperature model and is expressed as [19]

ρ(E) = 1

Te^E−E^T ⁰ (2.7)

where E is the excitation energy and the free parameter E0 and T are the energy shift and the constant nuclear temperature, respectively. The energy shift and nuclear temperature have to be fitted to measurements of the NLD.

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2.2. KINEMATICS OF INDUCED NUCLEAR REACTIONS

2.1.2 The γ-ray strength function

γ-ray strength functions describes the average electromagnetic properties of excited nuclei and are closely connected to the process of radiative decay and photo-absorption and govern the transitions between states in the quasi- continuum. Theγ-ray strength function for electromagnetic characterX and multipolarityLcan be formulated as [21]

fXL(Eγ) = hΓγli DlEγ^2L+1

(2.8) wherehΓγliis the average radiative width andDlis the resonance spacing for l-wave resonances (s,p, etc.) and can be measured in neutron/proton average resonance capture experiments. Equation 2.8 is often referred to as the

”downward” strength since it is related to theγ decay of states. The opposite reaction, photo-excitation, is often referred to as the ”upward” strength function and is determined by the average photo-absorption cross sectionhσXL(Eγ)i when summed over all possible spins of final states

fXL(Eγ) = 1 (2L+ 1)π²~²c²

hσXL(Eγ)i Eγ^2L⁻¹

(2.9) and is, according to Fermi’s golden rule and the principal of detailed balance, equal to Equation 2.8, provided that the same states are populated in both cases.

The γ-ray strength function can also be related to the γ-ray transmission coefficient:

T_XL(Eγ) = 2πE_γ^2L+1fXL(Eγ) (2.10)

2.2 Kinematics of induced nuclear reactions

When planning and analyzing inverse kinematics experiments there are a few key aspects that need to be considered. The particles involved are electri- cally charged and will interact and slow down as they travel through matter.

The kinematics of two body reactions is needed to understand the processes that takes place in an event and are important in understanding the excitation energy received by an residual nucleus. As the energies involved in inverse kinematics are high there is also a need to consider the Doppler shift ofγ-rays emitted by the excited nucleus.

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CHAPTER 2. THEORY

2.2.1 Interaction between charged particles and matter

The passage of charged particles in matter is mainly govern by its inelastic collisions with electrons and elastic scattering off nuclei. These processes cause an overall slowing down of the particle and can contribute to a deflection of the particle trajectory. The slowing down of the particle is usually expressed as the loss in energy per unit length, also known as the stopping power.

The first model for stopping power was developed by Bohr and was the result of a classical treatment of inelastic collisions with the electrons present in the material [22]. Bohr’s result showed that the change in energy per unit length was

−dE

dx = 4πz²e⁴ mev² Nelog

γ²mv³ ze²ν¯

(2.11)

where the incident particle has the chargeze, massm, velocityv and energy E. The material has a electron density ofNe and the bound atomic electrons of the material have a mean frequency ofν¯. The factorγis the Lorentz factor

γ= 1 p1−^v_c

wherecis the speed of light.

The Bohr formula gives reasonable results when compared to experimental values, but doesn’t incorporate any quantum effects. The first calculations that accounted for quantum mechanics was done by Bethe, Bloch and several others which resulted in the following formula known as the Bethe-Bloch formula [22]:

−dE

dx = 2πNar²_emec²ρZ A

z² β²



2 ln





2mec²γ²β² I

q

1 + 2^m_M^eγ+_M^m²^e2



−2β²



 (2.12)

where

2πNar_e²mec²= 0.1535 MeVcm²/g

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re: Classical electron radius =2.817×10⁻¹³cm ρ: Density of absorbing material

me: Electron mass

z: Charge of incident particle in units ofe Na: Avogadro’s number =6.022×10²³mol⁻¹

β=v/c vis the velocity of the incident particle,cis the speed of light I: Mean excitation potential

Z: Atomic number of the absorbing material A: Atomic weight of the absorbing material M: Mass of the incident particle

The Bethe-Bloch equation shows that the stopping power of a particular material can be approximated as

−dE dx =z²f

T M

(2.13) wherez is the charge of the incident particle, T its kinetic energy and M its mass. The function f is specific for the material. This leads to the following scaling of the stopping power

−E2

dx(T) =−z₂² z₁²

E1

dx M1

M2

T

(2.14) making it possible to extract the stopping power experienced by a charged particle from the known stopping power of another.

The stopping power for compounds materials has to be done by averaging the stopping power of each of the elements in the material,

1 ρ

dE dx = 1

Am

X

i

aiAi

ρi

dEi

dx (2.15)

whereai is the number of atoms of elementiin the molecule,Aiis its atomic weight of the element and _ρ¹

i

dEi

dx is the stopping power in units of mass thickness. Am = P

aiAi is the total weight of the molecule. This summing rule is known as Bragg’s rule and gives a good approximate value to the actual stopping power of the compound [22].

As both the Bethe-Bloch formula and the Bohr formula are approximate models it is necessary to rely on tabulated values for high accuracy calculations. Sources of such tabulated values can be evaluated databases such as PSTAR [23] or software such as SRIM [24].

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CHAPTER 2. THEORY

m ₂ m 1

m ₃

m ₄

✓

₃

✓

₄

p

4

p

3

p

2

Figure 2.1: Illustration of a two-body scattering in the laboratory frame. The incident projectile with massm2has momentump2and is incident on the stationary target particle with massm1. After the reaction there are two new particles, a fragment particle with mass m3 and momentum p3 and a residual particle with massm4and momentump4. The fragment and the residual are scattered at anglesθ3andθ4, respectively.

2.2.2 Kinematics of nuclear two-body reactions

Having an understanding of the two-body scattering reaction is instrumental for analyzing data from nuclear two-body reaction experiments. The equa- tions governing such reactions have been derived numerous times by count- less sources, but will be derived once more in order to get a notation suitable for this thesis. The derivation presented is heavily inspired by the derivation done by [25].

First, consider a particle with mass m2 and kinetic energy T2 incident on a stationary particle with massm1. When the two particles meet they interact and are transformed such that after the reaction there are new particles with massm3andm4. Introducing the notation

m1(m2, m3)m4 (2.16)

where the particle with massm1is denoted as the target,m2is the projectile, m3 is the fragment andm4 is the residual. The process is illustrated in Fig- ure 2.1. The relativistic four-momentum vectors of the projectile and the target are

P₁= (m1,0,0,0) P₂= (T2+m2,

q

T₂²+ 2m2T2,0,0)

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m ₂ m ₁

m ₃

m ₄

pCM

p⁰_CM

✓CM

✓_CM

Figure 2.2: Illustration of a two-body scattering as seen in the center-of-mass frame. The projectile and the target particle have the same amount of momen- tumpCM but have opposite directions. The fragment and the residual particle are scattered in an angleθCM.

The square of the sum of the four-momentum vectors is

s= (P₁+P₂)²= (m1+m2+T2)²−T_k²−2m2T2= (m1+m2)²+ 2m1T2 (2.17) and is Lorentz invariant. The reaction is easiest handled in the center-of-mass frame, that is the frame of reference where both particles have equal momentum but have opposite direction, thus the total sum of momentum vanishes.

The reaction viewed from the CM frame is illustrated in Figure 2.2. The four- momentum vectors of the target and the projectile in the CM frame is:

P⁰₁= ( q

m²₁+p²_CM,−pCM,0,0) P⁰₂= (q

m²₂+p²_CM, pCM,0,0)

The magnitude of the CM momentum,pCM is easily determined sincesis invariant:

pCM=

r(s−(m²₁+m²₂))²−4m²₁m²₂

4s =m1

s T₂²+ 2m2T2

(m1+m2)²+ 2m1T2

(2.18) The Lorentz transform that takesP₁from the laboratory frame to the CM frame is given as

P⁰₁= (γm1,−γβm1,0,0) = (q

m²₁+p²_CM,−pCM,0,0)

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CHAPTER 2. THEORY

Whereγ= √¹

1−β² andβ= ^v_c,vis the velocity of the CM frame. The rapidityχ is defined asβ = tanhχand thusγ= coshχandβγ= sinhχ.coshχ+sinhχ= e^χallows for the determination ofχ:

χ= ln pCM+p

m²₁+p²_CM m1

!

= ln T2+m1+m2+p

T₂²+ 2m2T2

p(m1+m2)²+ 2m1T2

!

(2.19) when adding togetherp

m²₁+p²_CM andpCM.

In CM frame the four-momentum of the two particles after the interaction is:

P⁰₃= ( q

m²₃+p⁰²_CM, p⁰_CMcosθCM, p⁰_CMsinθCM,0) P⁰₄= (q

m²₄+p⁰_CM² ,−p⁰_CMcosθCM,−p⁰_CMsinθCM,0)

Where the fragment and residual momentum has some angleθCMwith respect to the axis that the CM frame is moving along.

Sinces= (P⁰₃+P⁰₄)²it is easy to find the magnitude of the momenta of the fragment and the residual in the CM frame:

p⁰_CM=

r(s−(m²₃+m²₄))²−4m²₃m²₄

4s (2.20)

Lorentz transforming back to the laboratory frame of reference gives the observed energy and momentum for the fragment and residual

E3=q

p⁰²_CM+m²₃coshχ+p⁰_CMcosθCMsinhχ p3cosθ3=p⁰_CMcosθCMcoshχ+

q

p⁰_CM² +m²₃sinhχ p3sinθ3=p⁰_CMsinθCM

(2.21)

E4=q

p⁰²_CM+m²₄coshχ−p⁰_CMcosθCMsinhχ p4cosθ4=−p⁰_CMcosθCMcoshχ+

q

p⁰_CM² +m²₄sinhχ p4sinθ4=−p⁰_CMsinθCM

(2.22)

Solving for cosθCM andsinθCM and adding them together squared gives the laboratory momenta as a function of the laboratory angles

p3=

pp⁰²_CM+m²₃cosθ3sinhχ±coshχq

p⁰²_CM−m²₃sin²θ3sinh²χ

1 + sin²θ3sinh²χ (2.23) p4=

pp⁰²_CM+m²₄cosθ4sinhχ±coshχq

p⁰²_CM−m²₄sin²θ4sinh²χ

1 + sin²θ4sinh²χ (2.24)

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Since the momentum cannot be imaginary, the maximum laboratory angle of deflection is given by

θi≤arcsin

p⁰_CM misinhχ

(2.25) fori= 3andi= 4.

Excitation energy

Assuming now that the residual particle can obtain some energyExthat is not kinetic, but an internal excitation. This will lead to an increase in the mass of the particle such thatm4→m4+Ex. Solving

√s=q

m²₃+p⁰²_CM+q

(m4+Ex)²+p⁰²_CM

with respect toExgives

Ex= r

s−2√ s(

q

m²₃+p²₃coshχ−p3cosθ3sinhχ) +m²₃−m4

= (m1+m2)

1 + 2m1−m3−T3

(m1+m2)² T2−2T3+m3

m1+m2

+2

p(T₂²+m2T2)(T₃²+m3T3)

(m1+m2)² cosθ3+ m²₃ (m1+m2)²

!^1/2

−m4

(2.26)

where the CM energy of the fragment,p

m²₃+p⁰_CM² , has been replace by the Lorentz transform from the laboratory energy:

q

m²₃+p⁰_CM² = q

m²₃+p²₃coshχ−p3cosθ3sinhχ

=(T2+m1+m2)(T3+m3)−p

(T₂²+ 2m2T2)(T₃²+ 2m3T3) cosθ3

p(m1+m2)²+ 2m1T2

Assuming that the fragment scattering angle is measured to beθ3±∆θ3, the fragment kinetic energy is measured asT3±∆T3 and the incident energy is known to beT2±∆T2, then the uncertainty in the excitation is given as:

(∆Ex)²= 1 (Ex+m4)²

(T₂²+ 2m2T2)(T3+ 2m3T3) sinθ3(∆θ3)²

+(m1−(T3−m3) + (T2+m2)

sT₃²+ 2m3T3

T₂²+ 2m2T2cosθ3)²(∆T2)²

+((T3+m3) s

T₂²+ 2m2T2

T₃²+ 2m3T3

cosθ3−(T2+m2+m3))²(∆T3)²

#

(2.27)

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CHAPTER 2. THEORY

Trajectory reconstruction

If the momentum and the scattering angle of the fragment is known, then the CM frame energy and momentum is given by

q

m²₃+p⁰_CM² = q

m²₃+p²₃coshχ−p3cosθ3sinhχ p⁰_CMcosθCM=p3cosθ3coshχ−

q

m²₃+p²₃sinhχ p⁰_CMsinθCM=p3sinθ3

When combined with (2.22) and thatp

m²₄+p⁰²_CM =√ s−p

m²₃+p⁰²_CM, giving the energy and momentum for the residual:

E4=√

scoshχ− q

m²₃+p²₃ p4cosθ4=√

ssinhχ−p3cosθ3

p4sinθ4=−p3sinθ3

(2.28)

This result can be extended to spherical coordinates by adding an additional axis:

E4=√

scoshχ− q

m²₃+p²₃ p4sinθ4cosφ4=√

ssinhχ−p3sinθ3cosφ3

p4sinθ4sinφ4=−p3sinθ3sinφ3

p4cosθ4=−p3cosθ3

(2.29)

2.2.3 Doppler correction

A photon emitted in the rest frame of a particle moving with a velocity v in the laboratory frame will appear to have a shifted energy when measured in the laboratory frame. If the rest frame energy of the photon is E_γ⁰, then the laboratory frame energy of the photon is [26]

Eγ = E_γ⁰

γ(1−βcosθ) (2.30)

where γ = √ ¹

1−(v/c)² with c equal to the speed of light and θ is the angle between the trajectory of the photon and the particle as viewed in the laboratory frame.

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Chapter 3

Experimental setup

3.1 Introduction

The data analyzed in this thesis was collected at iThemba LABS (Laboratory for Accelerator Based Sciences)¹during approximately160hours of beam time at the end of April and the start of May 2015. During the week days the accelerator facilities at iThemba are used to provide beams for treatment of can- cer patients and for radio nuclide production, while on the weekends providing beam for various research purposes.

3.2 Beam facilities

The⁸⁶Krbeam was provided by a two step acceleration process. Initially the

86Krions were accelerated by a solid pole injector cyclotron (SPC2) to an initial energy and injected in a larger separated sector cyclotron (SSC) that provided the bulk of acceleration up to approximately300MeV. The beam provided was pulsed with a frequency of approximately42.5MHz and had a beam cur- rent/intensity of approximately0.5to1.5enA during the experiment and had a charge state of+12. An overview of the layout of the facilities at iThemba LABS can be seen in Figure 3.1.

1iThemba LABS, PO Box 722, Somerset West 7129, South Africa

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CHAPTER 3. EXPERIMENTAL SETUP

Figure 3.1: Floor plan of iThemba LABS [27].

3.3 AFRODITE array

The facility used in this experiment was theAFRODITE (AFRican Omnipur- pose Detector for Innovative Techniques and Experiments) array. The array featured a total of eight high-purity germanium (HPGe) detectors (CLOVER) and two large volume (3.5×8 inch)LaBr3(Ce) detectors provided by the Uni- versity of Oslo for this experiment. Each of theCLOVER detectors consisted of four 50×70 mmHPGecrystals inside a Bismuth Germanate scintillator (BGO) shield with a3.5cm long collimator. The target chamber had a diameter of23 cm and the distance from the center of the target to the detectors was21cm and14cm for theCLOVER and theLaBr3(Ce) detectors, respectively. The thickness of the target chamber walls at the points where theγ-ray detectors were pointing was 6 mm. The mounting distances are assumed to have an uncertainty of0.5cm, making the total solid angle of theLaBr3(Ce) detectors 4.7±0.3%of4πand13.7±1.3%of4πfor theCLOVER detectors. The total solid angle covered byγ-ray detectors was18.4±1.35%of4π.

The mounting position of the detectors with respect to the target placed at the origin is given in Table 3.1. Here the anglesθandφare the spherical angles when the beam is assumed to be along they-axis. The angleϕis the angle with respect to the beam axis. The angle between the trajectory of aγ-ray and the beam axis will have an uncertainty of≈ ±17^◦ and≈ ±13^◦ when incident upon theLaBr3(Ce)detectors andCLOVERdetectors, respectively. Figure 3.2 depicts theAFRODITEarray when closed.

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3.3. AFRODITE ARRAY

Figure 3.2: The AFRODITE array with eight CLOVER detectors and two LaBr3(Ce)detectors.

Detector: θ:[°] φ:[°] ϕ:[°] Channel id:

CLOVER1 90 0 90 64-67 CLOVER2 45 0 90 72-75 CLOVER3 135 180 90 96-99 CLOVER4 45 180 90 104-107 CLOVER5 45 270 135 128-131 CLOVER6 90 225 135 136-139 CLOVER7 90 315 135 160-163 CLOVER8 90 180 90 168-171 LaBr3(Ce)1 90 135 45 520 LaBr3(Ce)2 45 90 45 521

Table 3.1: List ofγ-detectors and their mounting angle. It is assumed that the target is at the origin and that the beam moves along they-axis. θis the polar angle, φ is the azimuthal angle and ϕis the angle with respect to the beam axis.

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3.4 Particle telescopes

The inside of the target chamber was fitted with two particle telescopes mounted with their centers at±45° with respect to the beam axis. Both telescopes consisted of a thin and a thick silicon double sided strip detector (DSSD) in a∆E - E configuration. Telescope A at+45^◦ had a284 µm thick∆E detector and 996µmthick E detector. Telescope B at−45^◦had a500µmthick∆E detector and982µmthick E detector. Both telescopes were fitted with52 mg/cm²thick Tafoils in front of the∆E detectors to shield the detectors from heavy particles (A>4). The telescopes mounted in the scattering chamber without their Ta shielding is depicted in Figure 3.3. Each of the DSSD detectors had a total of 16vertical and16horizontal strips that can be read out individually. All of the strips of the thin∆E front detectors were read out, while only the vertical strips were read out from the thick E back detector. The signals from the strips in the telescopes were amplified using Mesytec MPR-32 charge sensitive preampli- fiers. The width of the strips was3 mm and the center of the telescopes was mounted at a distance of6cm from the target. Each of the telescopes covered scattering angles from24.5° up to67° and together covered a total of8.8%of 4π. The horizontal and the vertical strips of the telescopes formed a16×16 pixel matrix. The pixels were numerated (i, j) wherei is the number of the vertical strips andjis the number of the horizontal strips. The strips of the detectors were numerated such that the vertical strip furthermost from the beam axis was chosen to be stripi= 1and the horizontal down-most was chosen as j= 1, see Figure 3.4 for a more detailed sketch of the numeration of the pixels.

The position relative to the target of each pixel is found by:

xi= r0−(i−17/2)w

√2 (3.1)

yi= r0+ (i−17/2)w

√2 (3.2)

zj = (j−17/2)w (3.3)

wherer0= 6cm is the distance from the target to the center of the telescopes andw= 0.3mm is the width of the strips. The angle of a pixel(i, j)is then

θij = cos⁻¹





yi

qx²_i +y²_i +z_j²



 (3.4)

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3.5. DATA ACQUISITION

Figure 3.3: Particle telescopes mounted in the target chamber without Ta shields.

when the beam is along they-axis. The maximum scattering angle covered by pixel(i, j)is

θ_ij^max= cos⁻¹

















yi−₂^w^√₂ q

x²_i+y_i²+z_i²+(xi−yi)^√^w

2−ziw+^w₂², j≤8

y_i−₂^w^√₂ qx²_i+y_i²+z_i²+(xi−yi)^√^w

2+ziw+^w₂², j >8







(3.5)

and the minimum

θ^min_ij = cos⁻¹

















yi+ ^w

2√ 2

q

x²_i+y_i²+z_i²+(yi−xi)^√^w₂−ziw+^3w₄², j≤8

yi+ ^w

2√ 2

q

x²_i+y_i²+z_i²+(yi−xi)^√^w₂−ziw+^3w₄², j >8







(3.6)

3.5 Data acquisition

The data acquisition (DAQ) system used at iThemba LABS is a fully digital system using a total of eleven XIA Pixie-16 digital gamma finder (DGF) cards. The DGF cards sample the input signal at a frequency of100MHz with a resolution of14bits. The samples are then processed by a field programmable gate array (FPGA) and a digital signal processor (DSP) that determines if there is a pulse from the detector and the height of the pulse. The parameters used by the internal algorithm of the DGF cards has to be optimized to the pulses emitted

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Figure 3.4: Illustration of the right hand side particle telescope. Only the label- ing of the pixels at the corners of the telescope is shown. The second telescope is mirrored along the dotted line.

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3.6. TARGETS

Figure 3.5: Burned C₂D₄targets.

by each of the detectors connected. The recorded pulse height is combined with a timestamp and the channel number, and is read out from the XIA cards and saved in the TDR data format²to a file for further offline analysis. An incre- ment of one unit in the timestamp is equal to10ns resulting in an uncertainty in time of±5ns. The system recorded continuously all pulses larger than the threshold set for each particular detector. Pulses from a HPGe crystals in coincidence with pulses from the BGO shield around the sameCLOVERdetector were discarded as part of the Compton suppression system of theCLOVER detectors.

3.6 Targets

The targets used in the experiments were of enriched polyethylene (C2D4) with 99% deutrons and polystyrene (C8H3D5). The polyethylene targets were be- tween110 µg/cm² and500 µg/cm² thick and the polystyrene targets ranged from1.0 to2.5 mg/cm². Picture of targets mounted on the target ladder can be seen in Figure 3.5. The polyethylene targets were produced at Argonne National Laboratory while the polystyrene targets were produced at iThemba LABS. Targets had to be changed several times during the experiments as they melted while in beam, with the thinner targets breaking more frequently than the thicker.

2See http://npg.dl.ac.uk/MIDAS/DataAcq/GreatFormat_3.1.html for a description of the data format.

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Chapter 4

Data analysis

A large part of the work done in this thesis was the development of a new sorting software able to work with the TDR format used by iThemba LABS. This work resulted in TDRreader and is available at GitHub¹ and partially shown in section A.2. The software implements everything described in this chapter with the exception of the calculations of the excitation energy coefficients (section 4.6) which is implemented in a separate software.

4.1 Calibration

Several calibration runs with radioactive sources (¹⁵²Eu, ⁵⁶Co) were done to map the channel number of the ADC to theγ-ray energy. The energy of a ADC channel was determined by

E(ch) =s+g·ch (4.1)

wherechis the ADC value recorded by the XIA DGF card,sis the shift andg is the gain. The gain and shift were found by a least square fit to the peaks in the calibration spectra.

The γ-ray detectors were calibrated using a newly produced ⁵⁶Co source which contained a substantial amount of⁵²Mnand⁴⁸V. An uncalibrated spectrum of the source can be seen in the upper panel in Figure 4.1 and the calibrated spectrum in the lower panel. The peaks used in the fit of Equation 4.1 are listed in Table 4.1 and are marked with arrows in the upper panel of Fig- ure 4.1.

1https://github.com/vetlewi/Master

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4.1. CALIBRATION

ADC channel

0 200 400 600 800 1000 1200 1400 1600 1800

Counts

0 100 200 300 400 500 600 700 800 900 1000

103

×

Energy [keV]

200 400 600 800 1000 1200 1400

Counts

0 100 200 300 400 500 600

103

×

Figure 4.1: The calibration spectrum for aCLOVERdetector. The upper panel is the uncalibrated spectrum and the lower is the calibrated spectrum. The arrows in the upper panel indicate the peaks used in the calibration, see Ta- ble 4.1.

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CHAPTER 4. DATA ANALYSIS

Mean energy: Origin:

511keV Annihilation

744keV Decay radiation of⁵²Mn 936keV Decay radiation of⁵²Mn 1434keV Decay readiaton of⁵²Mn

Table 4.1: Theγ-ray energies used in the calibration of theγ-ray detectors.

The silicon strips of the particle telescopes were calibrated using a ²²⁶Ra α-emitting source. The resulting uncalibrated spectrum can be seen in the upper panel of Figure 4.2. Theα-particle energies used in the fit of Equation 4.1 are tabulated in Table 4.2 and are marked by arrows in the upper panel of Fig- ure 4.2. The fit resulted in the spectrum shown in the lower panel of Figure 4.2.

Mean energy: Origin:

4784keV Decay radiation of²²⁶Ra 5489keV Decay radiation of²²²Rn 6002keV Decay radiation of²¹⁸Po 7687keV Decay radiation of²¹⁴Po

Table 4.2: Theα-particle energies used in the calibration of the silicon strips of the particle telescopes.

Calibration runs with γ sources were done before and after each week- end of beam time, while the silicon strips in the particle telescopes were only calibrated at the start and at the end of the experiment. As can be seen in Figure 4.3, there were some drift observed in the calibration of the particle telescopes with the largest being less than4%. For theγ-detectors there were no noticeable changes in the calibration for the duration of the experiment.

4.2 Determining events

The data output by the DAQ system was a continuous stream of data words for each of the signals acquired by the XIA cards and did not contain any type of indicator of events. Each of the data words featured a number identifying the source of the signal, the size of the signal and a timestamp indicating the time of arrival.

The determination of events was done in software with the following condi- tions, separate for each particle telescope:

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4.2. DETERMINING EVENTS

ADC channel

1200 1400 1600 1800 2000 2200 2400 2600 2800

Counts

0 100 200 300 400 500 600 700 800 900 1000

Energy [keV]

4000 4500 5000 5500 6000 6500 7000 7500 8000

Counts

0 100 200 300 400 500

Figure 4.2: The calibration spectra for a silicon strip detector. The upper panel is the uncalibrated spectrum and the lower is the calibrated spectrum. The arrows in the upper panel indicates theα-particle energies used in the calibration, see Table 4.2.

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ADC channel

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000

Counts

0 50 100 150 200 250 300

Start of exp.

End of exp.

Figure 4.3: Largest observed drift in a single silicon strip in the particle telescopes.

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4.3. PARTICLE IDENTIFICATION

1. One strip in one of theE detectors is hit at time t0. Additional signals from theEdetector had to be separated by at least350ns.

2. One of the vertical strips in the ∆E detector is hit at time t such that

|t−t0| ≤ 60ns. Additional signals from vertical strips arriving at timet are allowed as long as|t−t0|>60ns, but are not considered as a part of the event.

3. One of the horizontal strips in the∆E detector is hit at timet such that

|t−t0| ≤60ns. Additional signals from horizontal strips arriving at timet are allowed as long as|t−t0|>60ns, but are not considered as a part of the event.

4. Signals from theγ-ray detectors arriving at timethad to be within−350≤ t−t0≤1500ns.

Two consecutive particle events within350ns were allowed provided that the two events did not originate from the same particle telescope and were separated at least by50ns.

4.3 Particle identification

The reaction taking place in an event was determined by the type of particle recorded in the particle telescope. The telescopes allowed for this by having two silicon detectors in each telescope. The energy deposited in the two detectors are governed by Equation 2.12 where the energy loss per unit length is dependent on both the mass and the charge of the particle. The range of a particle in the detector will therefor differ depending on the mass and charge of the particle. A particle with initial energyEk will deposit some of its energy in the∆E detector and the rest in the E detector, provided thatEkis large enough to penetrate the∆E detector. The ratio of energy deposited in the∆E detector versus the amount deposited in the E detector will change with the mass, charge and initial kinetic energy of the particle. The result is that the energy deposited in the ∆E detector as a function of the energy deposited in the E detector will create distinct banana-shaped curves for different particles. Such a plot can be seen in Figure 4.4, where one can easily distinguish between protons and deuterons.

The selection of reaction was done by requiring that the particles had to deposit its energy such that it followed the curve for the particle emitted in the

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Figure 4.4: ∆E - E coincidence matrix used to identify particles. Protons and deuterons fall within distinct curves allowing for the distinction between them.

The upper curve corresponds to deuterons and the lower curve are for protons.

reaction. This was implemented by

gl(EE)≤E∆E≤gu(EE) (4.2) whereEE is the energy deposited in the E detector andE∆E the energy deposited in the ∆E detector. gl(EE) and gu(EE) are polynomials that follow the lower part of the curve and the upper part of the curve, respectively. The portion of the curves that ”moves back” is known as punch-through and is due to particles with large energies that penetrate both the∆E detector and the E detector, thus depositing only parts of its energy in the telescope. Events with energies corresponding to punch-through were discarded as the full energy of the particle in these events cannot be determined.

4.4 Coincidence gating

Time spectra were created by letting the time of the signal be relative to the timestamp of the back detector strip that acted as the trigger of the event. γ- rays in coincidence with the particle then formed a prompt peak in the time spectra corresponding toγ-rays in coincidence with the particle. A typical time

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4.4. COINCIDENCE GATING

0 20 40 60 80 100 120 140

(Ce) energy [keV]

LaBr3

0 1000 2000 3000 4000 5000 6000

Time [ns]

−300

−200

−100 0 100 200 300

Figure 4.5: Time spectrum of aLaBr3(Ce)detector.

- energy spectrum forγ-rays in coincidence with protons is shown in Figure 4.5 and Figure 4.6 forLaBr3(Ce)andCLOVERdetectors, respectively. As can be seen clearly in Figure 4.5 and Figure 4.6 there were in fact two prompt peaks.

This is due to an error in the way that the XIA DGF cards handles the timing of the signals and is entirely artificial. Gates were set on both of the prompt peaks to get theγ-rays in coincidence. An additional gate was placed far from the two prompt peaks and the γ-ray energies arriving within this gate were subtracted from the created spectra later in the analysis. There is some energy dependent walk in the prompt time peaks that can be countered by mapping the timestamps down with an energy dependent function, but has been omitted in this analysis.

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0 2 4 6 8 10 12 14

CLOVER energy [keV]

0 1000 2000 3000 4000 5000 6000

Time [ns]

−200 0 200 400 600 800 1000 1200 1400

Figure 4.6: Time spectrum of aCLOVERdetector.

Level density and γ-ray strength function in 87 Kr

The Oslo Method in Inverse Kinematics