Experimental Study of the reaction - 40Zr + 124 GASP array
Page content transcription
If your browser does not render page correctly, please read the page content below
Experimental Study of the reaction 96Zr + 124Sn at 530 MeV using the 40 50 GASP array Wilmar Rodrı́guez Herrera Universidad Nacional de Colombia Facultad de Ciencias Departamento de Fı́sica Bogotá, Colombia 2014
Experimental Study of the reaction 96Zr + 124Sn at 530 MeV using the 40 50 GASP array Wilmar Rodrı́guez Herrera Master’s thesis submitted in partial fulfillment of the requirements for the degree of: Magister en Fı́sica Supervisor: Ph.D., Diego Alejandro Torres Galindo Research area: Nuclear structure Research group: Grupo de Fı́sica Nuclear de la Universidad Nacional Universidad Nacional de Colombia Facultad de Ciencias Departamento de Fı́sica Bogotá, Colombia 2014
Aknowledgments The contribution of the accelerator and target-fabrication staff at the INFN Legnaro Na- tional Laboratory is gratefully acknowledged. I would also like to thank the scientific and technical staff of Gasp and Prisma/Clara. I would like to thank all the staff of the nuclear physics group for their support along the performance of this thesis. I specially thank to professor Fernando Cristancho the director of the group whose teachings have been applied during the performance of this thesis. I specially thank to Cesar Lizarazo for the discussions of different topics of the thesis that allows me to clarify some issues. I have studied undergraduate physics as well as masters studies in physics department. The professors and administrative staff are also acknowledged for their teaching and support given. I thank to “Dirección académica” from “Universidad Nacional de Colombia” for the scholarship (Asistente Docente) that gives me the economical support which allow me to carry out my master studies. Finally the supervision of professor Diego Torres is gratefully acknowledged.
ix Abstract In this thesis an experimental study of the binary nuclear reaction 96 124 40 Zr + 50 Sn at 530 MeV using the Gasp and Prisma-Clara arrays at Legnaro National Laboratory (LNL), Legnaro, Italy is presented. The experiments populate a wealth of projectile-like and target- like binary fragments, in a large neutron-rich region below the magic number Z = 50 and at the right side of the magic number N = 50, using multinucleon-transfer reactions. The data analysis is carried out by γ-ray spectroscopy. The experimental yields of the reaction in each one of the experiments, is presented. Results on the study of the yrast and near-yrast excited states of 95 41 Nb are presented, along with a comparison of the predictions by shell model calculations. Keywords: Gamma-ray Spectroscopy, Shell Model, Neutron-Rich Nuclei, Deep Inelastic Reactions, Nuclear Structure. Resumen En este trabajo se muestra una caracterización experimental de la reacción nuclear 96 124 40 Zr+ 50 Sn a 530 MeV usando los arreglos experimentales Gasp y Prisma-Clara ubicados en el labo- ratorio nacional de Legnaro (LNL), Legnaro, Italia. En estos experimentos se poblaron una gran cantidad de fragmentos binarios de tipo proyectil y de tipo blanco en una gran área de núcleos ricos en neutrones con número de protones menores al número mágico Z = 50 y número de neutrones mayor al número mágico N = 50, usando reacciones de transferencia múltiple de nucleones. El análisis de los datos es realizado mediante espectroscopı́a de rayos γ. La producción experimental de los núcleos en cada uno de los experimentos es presen- tada. Resultados en el estudio de estados yrast y yrast-cercanos para 95 41 Nb son presentados junto con una comparación con predicciones hechas por cálculos de modelo de capas. Palabras clave: Espectroscopı́a de rayos Gamma, Modelo de Capas, Núcleos Ricos en Neutrones, Reacciones Deep Inelastic, Estructura Nuclear
x Preliminary results of the present work were presented in the conferences: XXXVI Brazilian Meeting on Nuclear Physics, Study of the Evolution of Shell Structure of Z
Contents Acknowledgments VII Abstract IX 1. Introduction 2 2. Preliminary concepts on nuclear structure 4 2.1. Chart of nuclides and the region under study . . . . . . . . . . . . . . . . . . 4 2.2. Production of neutron-rich nuclei using grazing reactions . . . . . . . . . . . 6 2.3. The nuclear shell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.1. The mean field potential . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.2. Ground state predictions . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.3. Predictions for excited states . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.4. Shell model calculations . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4. Spins and parities of excited states . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.1. Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.2. Multipolar radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 95 3. The Nb nucleus 21 4. Experimental methods 24 4.1. Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.1.1. The Prisma-Clara experiment . . . . . . . . . . . . . . . . . . . . 24 4.1.2. The Gasp experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2. Gamma-ray detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2.1. Energy resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2.2. Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5. Data analysis 34 5.1. Construction of a level scheme from Gasp experiment . . . . . . . . . . . . . 34 5.1.1. γγ coincidence matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.1.2. γγγ coincidence matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.1.3. Angular correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Contents 1 5.2. Products of the reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.2.1. The Prisma-Clara experiment . . . . . . . . . . . . . . . . . . . . 42 5.2.2. The Gasp experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6. Results 53 6.1. Products of the reaction from the Gasp and the Prisma-Clara experiments 53 6.2. Level scheme of 95 Nb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.3. Shell model calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7. Conclusions and perspectives 66 A. Appendix: Contribution to the Legnaro National Laboratory. 67 B. Appendix: Contribution to the proceedings of the XXXVI RTFNB 69 C. Appendix: Contribution to the X LASNPA proceedings 73 Bibliography 80
1. Introduction The interaction between two nucleons (protons or neutrons) mediated by the strong nuclear force, has not a complete theoretical explanation yet. The nuclear force depends not only on the relative separation of the two nucleons, but also on their intrinsic degrees of freedom. The dependence with the relative separation does not have a simple mathematical expression, moreover different attempts trying to give an analytic expression for the strong nuclear force includes around 9 terms with more than 10 parameters which have to be fitted experimen- tally, see for example Ref. . Because of this complexity of the nuclear force, different nuclei have different properties, and the characterization of a nuclear region implies an enormous task. As an example of that, in the experiments described in this thesis more than 100 nuclei were created. The number of particles in the nuclear system is not low enough to try to solve the system by use of ab-initio calculations, and it is also not large enough, for most of the nuclei, so that models do not provide a complete explanation of nuclear properties. Many models have been proposed since the discover of the nuclear force, for example the Fermi gas, the liquid drop model and the nuclear shell model. The shell model is one of the most successful, in terms of the number of correct predictions made for nuclei near the so called magic numbers. The nuclear shell model was proposed in 1949 by Eugene Paul Wigner, Maria Goeppert-Mayer and J. Hans D. Jensen, who shared the Nobel Prize in Physics for their contributions in 1963 . Currently, the nuclear shell model continues being tested experimentally in order to improve the model or to identify its limits. To succeed in this goal different nuclei, in several mass regions, must be characterized because predictions of the shell model are different for different nuclei. For instance the region approaching N ≥ 50 and Z ≈ 40 is a very interesting region for both, nuclear structure and nuclear astrophysics, due to the possibility to study shell closures and sub-closures in the neutron-rich region, and for the opportunity to increase our knowledge on nuclei in the path of the rapid neutron capture r-process nucleosynthesis, respectively. The neutron-drip line, where neutrons can no longer bind to the rest of the nucleus, is not well define by the existent nuclear model, and it is the challenging frontier that experimentalist are looking forward to reach. Recent experimental progress has been made in the theoretical side to describe the structure of neutron-rich nuclei [3, 4, 5], and large γ-ray arrays  coupled to fragment mass separators [7, 8] have provided with outstanding structural information of neutron-rich nuclei . During the last decade experimental studies of neutron-rich nuclei have been conducted
3 using deep inelastic reactions using dedicated experimental setups, such as the Prisma- Clara array at Legnaro National Laboratory, Italy. Due to the large acceptance of the Prisma magnetic spectrometer, and its use in conjunction with the high-resolution gamma- ray detector array Clara in thin target experiments, a clear identification of the sub- products of the reaction is possible. More detailed spectroscopy information can be obtained if partner thick target experiments are performed using highly efficient γ − ray arrays, such as Gasp. The latter may allow the obtention of pivotal information for a complete characterization of the nuclear states in neutron-rich nuclei. The results obtained in this work will contribute with structural information of the 95 Nb nucleus, and it is a first step toward a systematic study of isotopic chains of neutron-rich nuclei in the region. A description of the region of interest in this thesis along with an explanation of the shell model will be presented in Chapter 2. A brief description of the production of neutron- rich nuclei using grazing reactions will be also presented. Chapter 3 is a summary of the main properties of 95 Nb, which was the object under study in this thesis, as well as the latest studies carried out about 95 Nb level scheme. In Chapter 4 the experimental methods used to perform the Gasp and Prisma-Clara experiments are exposed. The data analysis performed over the data from both experiments is explained in Chapter 4. Finally in Chapter 5 the results obtained from characterization of the reaction from Gasp and Prisma-Clara experiments, along with the level scheme of 95 Nb proposed in this work, are presented.
2. Preliminary concepts on nuclear structure The atoms are the components of ordinary matter. They are formed by electrons and a nucleus with neutrons and protons inside. The electrons are bound to the atom by the Coulomb force generated between the electrons and the protons in the atomic nucleus. The atoms have an order size of ∼ 10−10 m ≡ 1 angstrom (Å). However the nucleus in the atom has a size experimentally proved to be 1.2A1/3 fm, with A the mass number. Thus the nuclear dimensions are ∼ 10−15 m ≡ 1 fm. It means that the nucleus in the atom has a size five orders of magnitude lower than the size of the complete atom. Despite the difference of sizes between the complete atom and its nucleus, most of the mass in the atom is contained in the atomic nucleus. The mass of an electron is ∼ 0.5 MeV/c2 and the mass of protons and neutrons approximately the same is ∼ 1000 MeV/c2 . For example in the case of the hydrogen atom (1 proton and 1 electron) the atomic nucleus has approximately 2000 times the mass of the electron. All these facts means that the nucleus has a very high density of ∼ 1017 Kg/m3 . Due to the Coulomb force the number of protons determines the number of electrons of an atom, and the electrons are responsible for the chemical properties of the atoms. For this reason, depending on the number of protons, the nucleus and the atom have a chemical name. Several nuclei with the same number of protons and different atomic masses can generate a bound system. These types of nuclei are called isotopes. Some isotopes are stables but most of them are unstable and decay by different ways. In the next subsection is exposed the chart of nuclides which is a tool to visualize all the nuclei, as well as the region of interest in this work. 2.1. Chart of nuclides and the region under study There are less than 300 known stable nuclei, and more than 3000 radioactive isotopes have been produced in the laboratory, so far. The way to visualize all those nuclei is to sort them in the so called “chart of nuclides”, shown in Figure 2-1, The Y-axis indicates the number of protons and the X-axis indicates the number of neutrons. Figure 2-1 also shows the neutron and proton drip lines, which indicate the limits in the number of protons or neutrons for which certain nucleus could generate bound states. While the proton drip-line has been experimentally explored during the last decades, with the use of
2.1 Chart of nuclides and the region under study 5 Figure 2-1.: Chart of nuclides. The magic numbers for protons and neutrons and different decay modes are shown as well as the proton and neutron drip lines. The region of interest in this work is also highlighted. Modified from the original at  fusion-evaporation reactions, the neutron drip-line is more difficult to access experimentally. The region of interest in this work is highlighted in Figure 2-1. Figure 2-2 shows with more detail the relevant area for this work in the chart of nuclides. In Figure 2-2 can be seen bars enclosing the magic numbers Z = 50 and N = 50. The target and the projectile are the stable isotopes of Z = 40 and Z = 50 with the highest number of neutrons. It can also be seen that the 95 Nb nucleus, that will be the subject of study in this work, is near to the N = 50 magic closed shell. In this work the region of interest corresponds to neutron-rich nuclei with A ∼ 100. These nuclei lie on the pathway of the rapid neutron capture process (r-process) , so there is also a nuclear astrophysical interest in the structure of such nuclei. The r-process is a nucleosynthesis event that occurs in core-collapse supernovae and is responsible for the creation of approximately half of the neutron-rich atomic nuclei heavier than iron. Neutron-rich nuclei decays by β − decay (n → p + e− + ν̄e ), it is, a neutron is exchanged by a proton. The r-process entails a succession of rapid neutron captures (hence the name r-process) by heavy seed nuclei and these neutrons get the nucleus faster than the β − decay occurs. Heavy elements (those with atomic numbers
6 2 Preliminary concepts on nuclear structure Figure 2-2.: Chart of nuclides in region of interest. The target, 124 50 Sn and the beam 96 Zr 40 of the reaction are located as well as the 95 Nb and the 125 In. Z > 30) are mainly synthesized by r-process and their isotopic abundances (Z > 56) are regarded as the main r-process . In this thesis an experimental study of neutron-rich nuclei in the A ∼ 100 region is performed. The nucleus 95 Nb is expected to be populated trough the reaction 2-3 and this nucleus will be study in this thesis. From the experiments analyzed in this work, it is expected that most of the nuclei below of stable nuclei shown in Figure 2-2 had been populated. This region contains isotopes with more neutrons than the stable nuclei. These nuclei are called neutron-rich. The production of nuclei is carry out colliding some nuclei against each other and in this way, different reactions can occur and produce different nuclei. Neutron-rich nuclei are usually populated by mean of grazing reactions, a type of mechanism explained in the following subsection. 2.2. Production of neutron-rich nuclei using grazing reactions Neutron-rich nuclei are difficult to produce. Currently one of the most efficient methods to populate neutron rich nuclei is using grazing reactions which could be deep inelastic and multinucleon-transfer reactions. Both type of mechanism are binary, which means that the projectile and target exchange few nucleons and the products of the reactions maintain some resemblance to the initial products. After the occurring reaction, a couple of nuclei are produced, one similar to the projectile (projectile-like) and another one similar to the target (target-like). This situation is shown in Figure 2-3 for the reaction 96 124 40 Zr + 50 Sn at Elab = 530 MeV.
2.2 Production of neutron-rich nuclei using grazing reactions 7 Figure 2-3.: Scheme of the process in a grazing reaction. The grazing angle at 530 MeV is θ = 38◦ . If the excitation energy of the ejectiles is larger than 20 MeV the reaction is called deep inelastic, due to the large amount of kinetic energy in the beam that is converted to excitation energy, otherwise the binary reaction is called multinucleon-transfer reaction. When the energy increases, the excitation energy does the same, but it has a limit imposed by the binding energy of the nucleus in the beam. The couple of products is generated in ∼ 10−22 seconds, which is too short time to be discriminated by the electronics. Experimentally nuclei already formed can be observed. It is due to the electronics time of response is ∼ 10−8 s and the typical lifetime of the excited nuclear states is ∼ 10−12 s. Grazing reactions are expected to generate more neutron-rich nuclei than other types of reactions (Inelastic or fusion-evaporation reactions). Angle with the largest cross section for the grazing reactions is called “grazing angle”. This angle is produced when the distance of maximum closest equals the sum of the radii of both nuclei implied in the reaction. The distance of closest approach is deduced in [13, 14] and is given by Zp Zt θ d= 1 + csc , (2-1) 4πǫ0 Ek 2 where Zp and Zt are the number of protons in the projectile and the target respectively. Ek is the kinetic energy of the beam. Experimentally it is found that the nuclear radius of a nucleus with A nucleons has a value given by r = 1.2 · A1/3 . So the sum of the radii of the two nuclei implied in the reaction is given by, 1/3 d = 1.2 A1/3 p + A t . (2-2) In Equation (2-2) Ap and At are the number of nucleons in the projectile and the target respectively. In this work the reaction used was, 96 40 Zr +124 50 Sn at Elab = 530 MeV. (2-3)
8 2 Preliminary concepts on nuclear structure The grazing angle for this case calculated from Equations (2-1), (2-2), (2-3) is 38◦ , as is noted in Figure 2-3. From a theoretical point of view only the transfer of a single nucleon can be explained, this due to the complexity of the nuclear force. When the number of transferred nucleons increases, the calculations get extremely complex and, for this reason, the theoretical studies of this phenomenon have not given a complete explanation. This is the case of the code “GRAZING” by G. Pollarolo . In this work the numerical code is used to simulate the total cross section for the most important yields of the reaction (2-3). The results will be shown in Chapter 6 along with a comparison of the experimental data. The nuclei generated in the reaction have excitation energies which produce a de-excitation process. In cases when this energy exceeds the bounding energy of a neutron, the nucleus will emit neutrons, this process is known as neutron emision. In the cases when the excitation energy is lower than the bound energy of a neutron, then the nucleus will decay emitting γ-rays and this γ-rays gives the information about the excited states of the nucleus. When a nucleus is close to the magic numbers in the chart of nuclides it is expected that its first excited states can be described by shell model that will be explained in the next subsection. 2.3. The nuclear shell model The nucleus is a system of A particles which interacts under the potential generated by the strong nuclear force. The hamiltonian for such system can be written as A A A X 1X X HExact = Ti + Vij (|~ ri − r~j |). (2-4) 2 j=1j6=i i=1 i=1 In Equation (2-4), Ti , is the kinetic energy of each nucleon and A is the number of nucleons. The second part in Equation (2-4) which corresponds to the potential, contains A(A − 1)/2 terms, each one corresponds to the nucleon-nucleon potential. This potential is schematically shown in Figure 2-4. At large distances the potential in Figure 2-4 is explained by the Yukawa potential which can be obtained solving the Klein-Gordon Equation for the exchange of a pion and taking the potential proportional to its wave function. At short distances the potential is repulsive. The A(A − 1)/2 terms of the second part of Equation (2-4) have the functional behavior shown in Figure 2-4. To date in the laboratory has been generated nuclei with number of nucleons, A, larger than 200. This made the calculations of Equation (2-4) a very complex problem even for a computer. Thus a model had to be developed in order to simplify the hamiltonian in Equation (2-4). The shell model was developed in 1949 by several independent works by Eugene Paul Wigner, Maria Goeppert-Mayer and J. Hans D. Jensen [16, 17]. The model consists in calculate the following approximation for the nuclear potential
2.3 The nuclear shell model 9 50 VN−N(r) Schematic VN−N (MeV) 0 −50 0.0 0.5 1.0 1.5 2.0 2.5 r (fm) Figure 2-4.: Scheme of the shape of nucleon-nucleon potential. A A A 1 XX X Vij (|~ ri − r~j |) ≈ V (ri ). (2-5) 2 j=1 i=1 i=1 Equation (2-5) replaces the interaction that acts over each nucleon due to the presence of the other ones as an interaction that depends just on the position operator, r, of each nucleon. It is assumed that the potential has a spherical symmetry. The hamiltonian proposed in this model, HSM , in this first approximation of a spherical nucleus is A X A X HSM = Ti + V (ri ). (2-6) i=1 i=1 From Equation (2-6) the following aspects have to be noted: This expression propose that nucleons inside the nucleus can be modeled as non- interacting particles and particles just interacts with a mean field potential, V (r). This potential is the same for all the nucleons and depends just on the position operator, ri , of each nucleon. The dependence of the potential results kind of counter-intuitive due to the absence of a center in the nucleus. This model had been proposed before to study the energy levels of the electrons in the atoms with several electrons. However in the atomic case was expected that the mean field potential had such a dependence because most of the interaction that acts over the electrons is central. It is due to the coulomb interaction made by the protons in the nucleus that defines the center of the atom. However this approximation also works in nuclear case.
10 2 Preliminary concepts on nuclear structure This model is coherent with the experimental data to predict excited states and g- factors among others. However the predictions are not always correct due to the fact that Equation (2-6) is an approximation to the real hamiltonian of Equation (2-4). The difference between the exact hamiltonian and the model proposed in Equation (2-6) is called the residual interaction, Hresidual . A A A 1 XX X Hresidual = Vij (|~ ri − r~j |) − V (ri ). (2-7) 2 j=1 i=1 i=1 If the model is suitable to describe the nucleus it is expected that hHresidual i ≪ hHSM i. (2-8) 2.3.1. The mean field potential The dependence of the potential, V (r), in Equation (2-6) must be coherent with experimental observations. The nuclear potential has short range and it drops quickly a few fermis away from the nucleus. This potential cannot have strong variations inside the core and in fact should be approximately constant. Taking this into account three different types of potential have been proposed being consistent with these statements. ( −V0 , if r ≤ R0 Square well =⇒ V (r) = (2-9) 0, if r > R0 " 2 # r Harmonic oscillator =⇒ V (r) = −V0 1 − (2-10) Roa −V0 Woods Saxon =⇒ V (r) = . (2-11) 1 + exp r−R 0 a The values of R0 and Roa in Equations 2-9, 2-10 and 2-11 as well as the functional shape of these potentials, are shown in Figure 2-5. The harmonic oscillator potential allows an analytical solution of the energy levels, these are given by 3 3 ǫnℓ = h̄ω0 2(n − 1) + ℓ + = h̄ω0 N + . (2-12) 2 2 Spin-orbit interaction is also present in nuclei and it is very important to understand the so called ”magic numbers”. The shell model without spin-orbit interaction does not predict all the magic numbers. The inclusion of the spin-orbit interaction in the shell model was proposed by Maria Goepert Mayer [18, 19] and can be included in the model 1 dV (r) ~ ~ ~ · S, ~ Hℓs = V0′ L · S = V0 L (2-13) r dr
2.3 The nuclear shell model 11 Figure 2-5.: Representation of harmonic oscillator, square well and Woods-Saxon potentials. where L is the angular momentum of the nucleus and S is the spin of a nucleon. There is no analytic expression for V0 in Equation (2-13). However it can be measured experimentally and its sign can be also determined. It is found that V0 ≤ 0. (2-14) Thus the hamiltonian of the shell model including spin-orbit interaction is A A h i ~ ~ X X ′ HSM = Ti + V (ri ) − |V0 | L · S . (2-15) i=1 i=1 The potential, V (r), of the Equation (2-15) can be written as ( V + |V0 | 21 (ℓ + 1), if j = ℓ − 12 V (r) = (2-16) V − |V0 | 12 ℓ, if j = ℓ + 12 . This term in the potential produces a splitting of each energy level with angular momentum ℓ 6= 0. One schematic example of the splitting generated by the spin-orbit interaction is presented in Figure 2-6. This splitting allows the shell model to predict the magic numbers that are the numbers for which some energy levels called “Shells”, of the model are full following the Pauli exclusion principle. The shells that generate the magic numbers are the ones with high gap energy between the next one.
12 2 Preliminary concepts on nuclear structure |n, J = ℓ − 1/2i |n, ℓi ∆ǫℓs |n, J = ℓ + 1/2i Figure 2-6.: Splitting of an energy level with quantic numbers n and ℓ generated by the spin-orbit interaction. The energy levels of the harmonic oscillator potential given by equation (2-12) can be written including the spin-orbit interaction as ( j = ℓ + 12 3 −ℓ ǫnℓ = h̄ω0 N + (2-17) 2 (ℓ + 1) j = ℓ − 1.2 The harmonic oscillator potential has an analytical solution, however the Woods-Saxon po- tential generates a better description of the nucleus. A modification over the harmonic osci- llator potential can be done in order to try to generate a potential similar to Woods-Saxon with an analytical solution. The modified harmonic oscillator potential is given by 1/2 1 2 2 2 M ω0 r and κµ = µ′ . HM O = h̄ω0 ρ − κh̄ω0 2ℓ · s + µ ℓ − hℓ iN with ρ = 2 h̄ (2-18) The energy levels generated by the potential of Equation (2-18) are given by ( j = ℓ + 12 3 ℓ N (N + 3) ǫN,ℓ,j = h̄ω0 N + − κ − µ′ ℓ(ℓ + 1) − (2-19) 2 −(ℓ + 1) 2 j = ℓ − 1, 2 where κ and µ′ are parameters which must be fitted experimentally and they are different for different mass regions . κ gives a measure of the strength of the spin-orbit interaction. µ′ is the parameter which gives information about the skin of the nucleus, h̄ω0 ≈ 41 · A1/3 , with A the number of nucleons. These parameters also determine the energy level scheme and the first excited states of some nuclei which can be considered to have a single-particle behavior. Figure 2-7 shows the distribution of the energy levels for the harmonic oscillator potential with and without spin-orbit interaction and also the energy levels generated by the modified harmonic oscillator potential. The energy labels in Figure 2-7 refers to the quantum numbers ℓ and J, the orbital and the total angular momenta respectively. The equivalence in angular momentum for the letters in the labels of Figure 2-7 are, s ≡ 0, p ≡ 1, d ≡ 2, f ≡ 3, g ≡ 4 and h ≡ 5. For example the level 1g9/2 refers to a level with orbital angular momentum
2.3 The nuclear shell model 13 82 1h11/2 3s 2d3/2 3s1/2 2d 1g7/2 N=4 2d5/2 κ = 0.06 1g µ’ = 0.024 50 1g9/2 2p1/2 2p 1f5/2 N=3 2p3/2 κ = 0.075 1f µ’ = 0.0263 28 1f7/2 20 1d3/2 N=2 2s + 1d 2s1/2 κ = 0.08 1d5/2 µ’ = 0.0 Harmonic -µ′ h̄ω0 ℓ2 − N (N +3) −2κh̄ω0 ℓ · s oscillator 2 Figure 2-7.: Energy levels produced by harmonic oscillator potential. At the left the levels gene- rated by a pure harmonic oscillator potential. At the middle the modification of the potential is introduced. At the right the spin-orbit interaction is added. ℓ = 4 ≡ g and total angular momentum J = 9/2. Each energy level of Figure 2-7 is called “a shell”. In each shell can be placed 2(J + 1) nucleons according with Pauli exclusion principle. Neutron-rich nuclei One of the research frontiers in nuclear structure is the experimental study of the neutron- rich nuclei, which are isotopes with larger number of neutrons than the stable nuclei. These nuclei have shown a strong variation of the κ and µ′ parameters when they are compared with the stable nuclei. For example 40 20 Ca, which is a stable nucleus, has an energy gap of 7 MeV between the shells 1d3/2 and 1f7/2 of the Figure 2-7, and on the other hand, 288 O has and energy gap of 2.5 MeV. The 28 O nucleus has 10 neutrons more than the stable isotopes of 188 O, so it is a neutron rich nucleus. Neutron-rich nuclei allow us to explore the behavior of matter with excess of neutrons, like neutron stars. Most of the nuclei generated in the experiments studied in this work are neutron-rich nuclei.
14 2 Preliminary concepts on nuclear structure The magic numbers If a nucleus has an even number of protons and neutrons its total angular momentum J is coupled to 0, because this coupling generates a lower energy state than states with other configurations. This lower energy is called “pairing energy” and it is bound energy generated when two nucleons with equal angular momenta J and opposite angular Jz -component are coupled into the same shell. When the number of protons or neutrons fills completely some shell, it is said that we have a “closed shell” in protons or neutrons. Nuclei with closed shells have bound energy larger than its neighbors due to the pairing energy. The numbers that are shown in blue in Figure 2-7 corresponds to the number of nucleons needed to fill the levels below these numbers. 20, 28, 50 and 82 are located between a couple of levels which have energy separation larger than other near levels. This energy separation means that it is more difficult to promote one nucleon in that shell to another one. These types of numbers are called “magic numbers”. Nuclei with number of protons or neutrons equal to a magic number have bound energy larger than its neighbors. For these reasons the number of stable isotopes is larger for nuclei with a magic number of protons. Magic nulcei are very well described by the shell model. There are shells in Figure 2-7 with large energy separation between them. This is the case of the 2p1/2 shell which has 40 nucleons for the close shell. For this reason 40 is known as a semi-magic number. 2.3.2. Ground state predictions Figure 2-7 can be used to make predictions about the spin and parities of the ground state. It has been proved that these predictions are in agreement with the experimental data for stable nuclei and its neighbors. As it was stated a nucleus with even number of protons and neutrons has a total angular momentum J = 0 for its ground state. If a nucleus has an even number of neutrons and an odd number of protons then the total angular momentum is given by the shell in which is located the unpaired proton. All other protons are coupled by pairs to a total angular momentum of 0. The nucleus of interest in this work is 95 Nb, with 54 neutrons and 41 protons. As the low energy state is generated when nucleons are coupled by pairs of angular momentum with Jz -component opposite, then the angular momentum J is given by the unpaired proton that can be located making the filling of the shells in Figure 2-7. In this case it is located in the shell 1g9/2 . Thus the ground state of 95 Nb is expected to have a total angular momentum J = 9/2. The parity is given by π = (−1)ℓ . (2-20) In this case ℓ = 4 ≡ g, so the parity of the ground state of 95 Nb will be positive. It is written using the typical notation in nuclear physics as J π = 9/2+ . (2-21)
2.3 The nuclear shell model 15 2.3.3. Predictions for excited states Some nucleons can be promoted to the higher shells in order to generate excited states. For these processes, however, there are some nucleons in closed shells with high bound pairing energy that are difficult to promote to other shells. For example the first excited state for 95 54 41 Nb nucleus could be generated by the promotion of the proton into the shell 1g9/2 to the higher shell 2d5/2 (see Figure 2-7) however the gap energy between these two shells is larger than, for example, the gap between the shells 2p1/2 (with two protons) and 1g9/2 . This nucleus has 4 neutrons in the 2d5/2 shell and the energy gap between this shell and the next one, 1g7/2 , is very low. Depending on the pairing energy of the two protons in the shell 2p1/2 and the pairing energy of the neutron in the 2d5/2 shell, different possible configurations are possible for the first excited state of 95 41 Nb54 nucleus. Different configurations implie that the angular momentum of the all unpaired nucleons has to be combined in order to construct the angular momentum of the excited state. 2.3.4. Shell model calculations Excited states of nuclei near magic and semi-magic numbers in the chart of nuclides are well described by shell model calculations made on the basis that excited states can be produced by promotion of nucleons between different shells in the model. These excited states are formed by “single-particle excitations”. Shell model calculations can be made to predict the energy of some excited states. These calculations are based on the fact that a nucleus with a closed shell has higher bound energy than neighbor nuclei. Some nuclei can be considered as a sum of an inert core and some valence nucleons which could be promoted to some valence orbitals to generate excited states. These concepts can be defined and illustrated with an example of the particular case of 95 41 Nb54 nucleus. Inert core; the nucleus composed by nucleons filling completely lower shells. For 95 41 Nb54 , 88 the inert core can be 38 Sr50 . Valence nucleons; nucleons in higher shells than the ones of the inert core 88 95 38 Sr50 . 41 Nb54 has 4 valence protons and 3 valence neutrons. Valence space; the energy levels available for valence nucleons. They are energy levels above the ones filled by the inert core. Neutron valence space for the 4 valence neutrons of 95 41 Nb54 is composed by the shells 2d5/2 , 1g7/2 , 3s1/2 , 2d3/2 and 1h11/2 . Proton valence space for the 3 valence protons are 2p1/2 and 1g9/2 . External orbitals; the remaining orbitals that are always empty. Figure 2-8 shows the concepts defined above for the case of 95 41 Nb54 considered as a 88 sum of the 38 Sr50 inert core plus 4 valence neutrons and 3 valence protons. A particular
16 2 Preliminary concepts on nuclear structure Neutrons Protons 82 1h11/2 82 1h11/2 2d3/2 2d3/2 External space Valence space 3s1/2 3s1/2 1g7/2 1g7/2 Valence neutrons 2d5/2 2d5/2 50 50 1g9/2 1g9/2 Valence space 2p1/2 2p1/2 1f5/2 1f5/2 Valence protons Inert core 2p3/2 2p3/2 Inert core 28 28 1f7/2 1f7/2 20 20 1d3/2 1d3/2 2s1/2 2s1/2 1d5/2 1d5/2 Figure 2-8.: Inert core, valence neutrons and protons, and valence spaces for the case of 95 Nb 41 54 nucleus. selection of the inert core and valence space must be made based on the shell model energy levels from Figure 2-7. A suitable selection of an inert core will be a nucleus with a magic number of protons and neutrons and the valence orbitals will be the higher shells. Once the inert core, valence orbitals and valence nucleons has been selected, an effective nucleon- nucleon interaction must be introduced. The success of the calculations suggest that the simple free nucleon-nucleon interaction can be regularized in the valence space. Thus there are different effective interactions for different valence spaces. Effective interactions between pair of nucleons are generated from the empirical values  which are then compared with experimental data in order to obtain better effective interactions which can describe the nuclei in some particular region. Some of the purposes of the experimental study of the excited states of the nuclei are to improve the determination of an effective interaction. The exact solution of the real interaction can be approximated by the solution of the effective
2.4 Spins and parities of excited states 17 interaction in the valence space such that Hψ = Eψ → Hef f ψef f = Eψef f , (2-22) where Hef f and ψef f are the effective halmitonian and wavefunctions in the valence space. The single particle energy levels in Figure 2-7 must be also found experimentally and they are needed to make the calculations. In this work an experimental study of the 95 Nb excited states will be presented. These data will contribute to the determination of an effective interaction in the valence space described in Figure 2-8. 2.4. Spins and parities of excited states When the nucleus decays from an excited state it emits γ-rays which have some multipora- larity. Depending on the multipolarity of the emitted γ-ray, spins and parities of the excited states can be determined. 2.4.1. Selection rules In a transition between an initial state with spin and parity Jiπi and a final state with spin π and parity Jf f , a γ-ray can be emitted with a total angular momentum jγ and parity πγ . This process is illustrated in Figure 2-9. Ei J i πi j γ πγ Eγ = Ei − Ef πf Ef Jf Figure 2-9.: Quantum numbers in a γ transition. Ei and Ef are the enegies of the initial and the π final state. Jiπi and Jf f are the spin and parity of the initial and the final state. jγ , πγ and Eγ are the angular momentum, parity and energy of the emitted γ-ray. The quantum numbers of the final state are calculated by the composition of the quantum π numbers Jf f and jγ , πγ . The angular momentum conservation is Ji = Jf + jγ . (2-23) Equation (2-23) implies an angular momentum composition which produces a selection rules on the quantum numbers jγ and Ji , |Ji − Jf | ≤ jγ ≤ Ji + Jf (2-24) |jγ − Jf | ≤ Ji ≤ jγ + Jf . (2-25)
18 2 Preliminary concepts on nuclear structure The electromagnetic decay preserves parity thus, πi = πf πγ (Xjγ ). (2-26) In Equation (2-26), jγ indicates the angular momentum of the radiation and X indicates the character of the radiation, X = E for an electric transition and X = B for a magnetic transition. Notation used in Equation (2-26) is widely used in nuclear physics, for example an E2 transition represents an electric quadrupole transition and a M1 transition represents a magnetic dipole transition, etc. The parity of the electromagnetic radiation is given by (−1)j for an electric multipole, (2-27) j+1 (−1) for a magnetic multipole. (2-28) Depending on the angular momentum of the γ-ray emitted and taking into account the section rule (2-26), the character of the radiation X can be determined. To illustrate how works the selection rules [(2-25), (2-26), (2-28)], let us consider the transition in Figure 2-10. J i πi E2 9/2+ Figure 2-10.: Transition with an emission of a E2 γ-ray to an state of spin and parity 9/2+ . The situation illustrated in Figure 2-10 is an example of a typical experimental result where the spin and parity of the ground state is known and the multipolarity character of the γ-ray emitted is measured. The objective will be to assign the spin and parity of the excited state. To do that the selection rules [(2-25), (2-26), (2-28)] must be considered. If Ji is the spin of the initial state in Figure 2-10 then the selection rule (2-25) gives |2 − 9/2| ≤ Ji ≤ 2 + 9/2 (2-29) 5/2 ≤ Ji ≤ 13/2. (2-30) The selection rule (2-27) gives the parity of the initial state in Figure 2-10. The γ-ray is of E2 type, so its parity is (−1)2 = +1, thus the parity of the initial state must be πi = (+1)(+1) = +1. (2-31) According to Equation (2-30) there are several possibilities for the spin and parity of the initial state from Figure 2-10, Jiπi = 5/2+ , 7/2+ , 9/2+ , 11/2+ , 13/2+ (2-32)
2.4 Spins and parities of excited states 19 The comparison with shell model calculations may help to determine which value given by (2-32) is the correct value. As it was stated the multipolarity character of the radiation can be measured, this will be exposed in the next subsection. 2.4.2. Multipolar radiation The γ radiation emitted by a nucleus can have either a electric or a magnetic nature. Electric and magnetic transitions are due to the redistribution of the multipole magnetic and electric moments of the nucleus, respectively. The γ-ray angular distribution depends on the multi- polarity order of the emitted radiation. This angular distribution dependence for a multipole of the order ℓ, m is given by m2 1 m(m + 1) 2 1 m(m − 1) Zℓ,m = 1− |Yℓ,m+1 | + 1− |Yℓ,m−1 |2 + |Yℓ,m |2 , (2-33) 2 ℓ(ℓ + 1) 2 ℓ(ℓ + 1) ℓ(ℓ + 1) where Yℓ,m are the spherical harmonics. For example the angular distribution of the intensity of the radiated energy by a dipole, and a quadrupole are given by Equations [(2-34), (2-35)]. The angular distribution generated by these Equations are represented in Figures [2-11, 2-12]. 1 1 3 Z1,0 (θ) = |Y1,−1 |2 + |Y1,1 |2 = |Y1,1 |2 = sin2 (θ) (2-34) 2 2 8π (a) 2D (b) 3D Figure 2-11.: Angular distribution of the emitted γ radiation of the order ℓ = 1 y m = 0. The red arrow indicates the multipole orientation. 1 1 15 Z2,0 (θ) = |Y2,1 |2 + |Y2,−1 |2 = |Y2,1 |2 = cos2 (θ) sin2 (θ) (2-35) 2 2 8π As can be seen from Figures [2-11, 2-12] the angular distribution of the energy radiated is different for different multipoles. These differences in the angular distributions allow the
20 2 Preliminary concepts on nuclear structure (a) 2D (b) 3D Figure 2-12.: Angular distribution of the emitted γ radiation of the order ℓ = 2 y m = 0. The red arrow indicates the multipole orientation. experimental determination of the multipolarity of the emitted radiation. In Chapter 5 the experimental technique utilized to determine the multipolarity of radiation will be explained and finally in Chapter 5 the results obtained for the γ-rays emitted from 95 Nb nucleus will be shown. The following subsection describes the current state of the excited states of 95 Nb measu- red by γ-ray spectroscopy. These excited states are represented in nuclear physics as a level scheme.
3. The 95Nb nucleus 95 41 Nb nucleus has a radioactive half-life of T1/2 = 35.991(6) days  and decays from the ground state via β − to the stable 95 Mo. The number of protons of 95 41 Nb is 41 protons, just one proton to the semi-magic number 40 and the number of neutrons is 54, 4 neutrons to the 50 closed shell. Due to its proximity to 88 38 Sr nucleus, which is emplyed as a standard closed-core shell , a single-particle behavior is expected. Previous experimental studies of 95 Nb nucleus have been performed using β decay , which did not populate high excited states, and also by fusion-evaporation reactions which populates high-spin states . The most recent experimental results of 95 Nb reported more than 10 different excited states with proposed spin and parity for levels close to the ground state . For the latter work data from three experiments were analyzed. The first two utilized the fusion evaporation reactions 12 Ca +82 Se at Elab = 38 MeV (3-1) 16 82 O + Se at Elab = 48 MeV. (3-2) The γ-rays produced in these reactions were detected by an array of just three Ge detectors. The low statistics generated in these experiments had to be complemented by a third experiment that made use of 16 O and 12 C contaminants from the target of the reaction 82 Se +192 Os at Elab = 470 MeV, (3-3) the γ-rays were detected using the detector array Gasp  (for specific details of the Gasp array see Chapter 4). Based on the Gasp experiment the level scheme of Figure 3-1 was proposed. In Figure 3-1 the spins and parities proposed by Bucurescu et al  are also shown. As it was mentioned in section 2.3.2, the predicted spin and parity of the 95 Nb ground state are J π = 9/2+ . (3-4) These spin and parity were measured experimentally by Rahman and Chowdhury , they found that predictions by shell-model calculations to their ground state are also correct. In the report made by Bucurescu et al.,  two problems were reported in the cons- truction of this level scheme. Firstly, the intensities of the γ-rays at each side of the energy
95 22 3 The Nb nucleus Figure 3-1.: Level scheme of 95 Nb proposed in ref 
23 level of 5643 keV were the same between the uncertainty range, like happened with the γ rays coming in and going out from the energy level of 4071 keV. Secondly, the experiment using the gasp array made use of the contaminants in the target and no the target itself. These contaminants could not be uniformly distributed which could cause difficulties in the assignment of the intensities of the γ-rays. These problems do not give confidence in the arrangement of the levels proposed in Figure 3-1, as stated in the report. The reasons presented above encourage the performance of a new experimental study of 95 the Nb nuclei, and motivates the present work. To allow that, two experiments were carried out at Legnaro National Laboratory, Legnaro, Italy. These experiments are described in the following Chapter.
4. Experimental methods 4.1. Experiments In order to study properties from nuclear states, the nucleus has to be created. To do this an accelerator must collide the nuclei in the beam with the nuclei in the target. The beam at Legnaro was initially accelerated by the Tandem and finally by the linear accelerator ALPI. As a result of the reaction, excited nuclei are generated and they decay emitting γ-rays, which will be the subject of our study. Those γ-rays will provide information about the properties of the nuclei. An array of Ge-detectors will collect information of energy and time of γ-rays emitted by the nuclei produced in the reaction. In this thesis two arrays in two different experiments: Prisma-Clara  and Gasp , were used. In Prisma-Clara experiment was utilized a thin target in order to allow the projectile- like fragments to reach the spectrometer Prisma. On the other hand a thick target was utilized for the Gasp experiment. It made the projectile-like fragments stop inside the Gasp multidetector array. A complete description of the experiments will be done in the next subsections. A summary of the experimental details of both experiments is shown in Table 4- 1. Table 4-1.: Target thickness and beam energy of the Prisma-Clara and Gasp experiments. 96 124 40 Zr + 50 Sn Prisma-Clara Gasp Target (124 2 50 Sn) thickness (mg/cm ) 0.3 8 Thickness of the backing target (mg/cm2 ) 0.04 of 12 C 40 of 208 Pb Beam energy (MeV) 530 570 a Number of working detectors 25/25 38/40 124 a The beam energy at the middle of the 50 Sn target was 530 MeV. 4.1.1. The Prisma-Clara experiment For the Prisma-Clara experiment [8, 25] the binary fragments produced in the reaction are separated in the target. The target-like products remains in its initial position, meanwhile
4.1 Experiments 25 the projectile-like fragments continue moving through Prisma which have several stages as shown in Figure 4-1. Magnetic dipole Clara detector array Magnetic quadrupole Focal Start detector plane Target detector 124 Sn Target−like Projectile−like ∆E-E detectors Beam 530 MeV 96 Zr Figure 4-1.: Prisma-Clara set-up correlating the coincidence signals at the focal plane of Prisma with the γ-ray transitions detected by CLARA. Figure 4-2.: Prisma-Clara array at the Legnaro National Laboratory. The magnetic quadrupole is used to focus the beam. The start detector and the focal plane detector gives the time of flight information which together with the length of the
26 4 Experimental methods trajectory enable us to calculate the velocity v of the beam. After the nucleus cross the magnetic dipole the beam is separated in different trajectories with a radius given by mv ρ= . (4-1) Z The incident velocity v is the same for all the nuclei on the beam, so they are separated by their charge-mass relation. When the nucleus pass trough the detectors labeled as ∆E − E in Figure 4-1 [26, 27], they loose energy depending on the width of the detector so that dE mZ 2 ∝ , (4-2) dx E where m and Z are the mass and the number of protons of the nucleus. From Equation (4-2) can be seen that the nuclei are separated by their charge, which make possible a complete identification of a nucleus. Prisma and Clara were linked at a laboratory grazing angle of 38◦ . However this link has an angular acceptance of ∆θ ∼ 12◦ and ∆φ ∼ 22◦ . Being φ the azimuthal angle with respect to the beam direction and θ the polar angle. Thus, Prisma is detecting just the nuclei produced between these angles. Besides Clara detected just the γ-rays which were in coincidence with the γ-rays emitted by the nuclei produced at these angles. This way, just the radiation produced by the nuclei produced at angles near to the grazing angle were detected. This is an important fact that will be discussed later. The Prisma-Clara experiment has the advantage of select products of the reaction at an specific angle, besides, due to Prisma magnetic spectrometer, this experimental set-up can select the radiation produced by an specific nucleus. However due to Prisma covering solid angle of 80 msr, this experimental set-up has the setback of the low yield production. To solve this problem a complementary experiment was conducted and it is called here “Gasp experiment”. 4.1.2. The Gasp experiment Gasp  is an array of 40 High-Resolution Ge-detectors, each one equipped with BGO Com- pton suppressor detectors which suppress most of the Compton events using a coincidence technique as shown in Figure 4-7. Figure 4-7 shows a Ge-detector surrounded by BGO Compton suppressor detector. If Compton event occurs in the Ge-detector it could be also detected by the high efficiency BGO detector, and this event can be suppressed. On the other hand, if an event getting the detector produces photoelectric effect, depositing all the energy of the γ-ray in the crystal, then the event does not produce a BGO detector signal, and it will be a valid event as shown in Figure 4-7. Gasp is a spherical array covering a solid angle close to 4π that has a total of 40 Ge-detectors distributed in 11 rings with the central ring hosting 8 detectors. A transversal cut of the central ring is shown in Figure 4-3.
4.1 Experiments 27 BGO Compton supressor detectors Beam 96 Zr at 574 MeV Target 124 Sn Ge Detectors 20 cm Figure 4-3.: Gasp central ring Set-up. Figure 4-4.: Gasp Set-up real image. Figure 4-3 shows also the distance between target and the position of the detectors. γ-rays from Gasp and Prisma-Clara experiments were detected using Ge-detectors su- rrounding by BGO detectors. The characteristics of such detectors will be explained in the
28 4 Experimental methods next subsections. 4.2. Gamma-ray detectors A detector is a device that is constructed with the objective of convert all the radiation that impact over it, into an electronic signal. However this is not always possible. Different detectors have been developed for different purposes. In this work just the γ-ray detectors are of interest. These detectors could be divided in three different types: Plastic: This type of detectors emits light when the radiation inside over it, but they cannot distinguish between the energy of the radiation. These detectors spend a very low time forming the signal, for this reason they are called fast detectors. Scintillators: When the radiation hit these detectors it excites the atoms and the mo- lecules in the crystal making possible the light will be emitted in the de-excitation process. This light is transmitted to the photomultiplier which convert it into a weak electric current that is amplified by an electronic system. This type of detectors has a relatively low time detection of ∼400ns (rise time of the signal after the preamplifier). On the other hand the energy resolution of these detectors is relatively low compa- red with semiconductor detectors. The most known scintillator detectors are the NaI (sodium iodide) and the BGO (bismuth germanate). Semiconductor detectors: these types of detectors need a BIAS voltage which polarizes a junction n-p in the crystal, generating a depletion zone in which a γ-ray can generate a cascade of electrons proportional to the energy of the γ-ray. This type of detectors has a very high energy resolution compared with scintillator detectors. On the other hand these detectors have a very low time of response ∼5µs. The most common detectors of this type are Ge-detectors. When a γ-ray reach a detector three different type of processes can occur, they are, Compton effect, photoelectric effect and pair production. Compton effect could occur in the electrons of the crystal. In this case the γ-ray losses energy and is also defected, this way, it could escape from the detector without loss all its energy, thus, the detector will register a count for a value of energy which is lower than the one of the initial γ-ray. Photoelectric effect could also occur. In this case the γ-ray losses all its energy inside the detector and it will generates a count in a value of energy which corresponds with the γ-ray energy. The cross section, σ, of each one of these processes depends on which it is called the attenuation coefficient µ in the following way ω µ σ= . (4-3) NA ρ
4.2 Gamma-ray detectors 29 Where ω is atomic weight, NA is the Avogrado’s number, µ is the attenuation coefficient and ρ is the density ofthe material. As it canbe 2seen from Equation (4-3) the cross section depends on the factor µρ which has units of cmg . Ge-detectors are widely used in nuclear structure experiments and for this reason is important to know howimportant is each process µ when γ-radiation interacts with germanium. Figure 4-5 shows the ρ factor of cross section of the different processes when γ-radiation interacts with germanium. 104 Compton 103 Photoelectric Pair production Total 102 µ/ρ (cm2/g) 10 1 10-1 10-2 10-3 10-4 -3 10 10-2 10-1 1 10 Energy (MeV) µ Figure 4-5.: ρ factor (proportional to the cross section) of different processes in γ-germanium interaction. When a γ-ray of energy Eγ interacts by Compton effect with an electron, the energy Eγ′ of the γ-ray after the interaction is given by Eγ Eγ Eγ′ = with ǫ = . (4-4) 1 + ǫ(1 − cos(θ)) me c2 From Equation (4-4), me represent the electron mass and c is the velocity of light. The energy, Er , registered by the detector will be the difference between the initial and final energy of the γ-ray. ǫ(1 − cos(θ)) Er = Eγ − Eγ′ = Eγ . (4-5) 1 + ǫ(1 − cos(θ)) The energy, Er , reach its maximum value when θ = 180◦ , this value is given by Equa- tion (4-6). 2ǫ Er−max = Eγ . (4-6) 1 + 2ǫ
30 4 Experimental methods Figure 4-6.: Spectrum of a 60 Co source took with a Ge-detector the Compton edge energies for the two energies of the peaks (1173 and 1332) are labeled. Because of Compton effect is present in the detection process, a typical γ spectrum of a Ge-detector is like what it is shown in Figure 4-6 for a 60 Co source which emits two γ-rays at energies of 1173 and 1332 keV. In Figure 4-6 the peak corresponds to photoelectric effect and for this reason it is called photopeak. The counts in the photopeak are located at the energy of the γ-ray that hits the detector. In this case the γ-ray leaves all its energy inside the detector. The counts in the region labeled as “Compton region” correspond to the energy that the γ-ray losses when the Compton effect takes place, it is, Er , from Equation (4-5). In this case the γ-ray does not leave all its energy in the detector, and a count is added in an undesired region of the spectrum. The edge of the “Compton region” is given by the Equation (4-6). For γ-rays at energies of 1173 and 1332 keV, as the ones in Figure 4-6, the values of Er are 963 and 1118 keV respectively. These values are located in the spectrum of Figures 4-6 and 4-8. The Compton region can be suppressed using a technique in which a γ-ray, that is de- flected by Compton effect, can be detected by another detector surrounding the Ge-detector, in the way that is shown in Figure 4-7. An incident γ-ray that is deflected by Compton effect (red line in Figure 4-7) can be detected by a BGO detector. This detector is connected in coincidence with the Ge-detector, that way, the events detected by the Ge-detector in coincidence with an event detected in the BGO detector will be suppressed from the final spectrum. The difference between a spectrum took by a Ge-detector when is used a Compton suppressor is shown in Figure 4-8. From Figure 4-8 can be seen that the Compton region has less counts when a suppressor is used. However in this last case the effect is still present. These counts could be due to a multiple scattering in the Ge-detector or it could be due to the γ-ray deflected, was not
You can also read