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Gamma-ray Tracking Arrays Sara Norain Billal (sarab@aims.ac.za) African Institute for Mathematical Sciences (AIMS) Supervised by Prof. David Aschman University of Cape Town Co. supervisor Dr.Simon Mullins iThemba LABS May 24, 2007 Abstract The objective of this essay is to determine the basic nuclear structures of nuclei and to review some models that focus on diﬀerent aspects of nuclear behaviour such as the liquid drop model and the nuclear shell model. An instrument of major importance for these studies is a high eﬃciency gamma-ray tracking spectrometer capable of disentangling the structure of exotic nuclei. Methods of tracking gamma-rays such as those of clusterization and backtracking methods are also discussed. i Contents Abstract i List of Figures iv 1 Gamma-rays 1 1.1 The importance of gamma-ray spectroscopy . . . . . . . . . . . . . . . . . . . 1 1.1.1 The interaction of γ radiation with matter . . . . . . . . . . . . . . . . 1 1.1.2 Photoelectric eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.3 Compton eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.4 Pair production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Charge carriers in semiconductors . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Intrinsic charge carriers . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Doped semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.4 The p-n junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Semiconductor detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Forward bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2 Reverse-bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.3 The p-n junction detector . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.4 Germanium detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.5 Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.6 Energy resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.7 Compton suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Nuclear structure 11 2.1 Nuclear models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 The liquid drop model . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.2 Volume binding energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.3 Surface term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 ii 2.1.4 Coulomb energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.5 Asymmetry Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.6 Pairing energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.7 The nuclear shell model . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.8 The Woods-Saxon potential with a spin-orbit coupling . . . . . . . . . . 14 2.1.9 The spin-orbit potential . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.10 Deformed nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.11 Collective motion of nuclei . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.12 Vibrational states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.13 Rotational states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.14 Superdeformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Gamma-ray tracking methods 20 3.0.15 The clusterisation method . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.0.16 The backtracking method . . . . . . . . . . . . . . . . . . . . . . . . . 23 4 Physics with AGATA 24 5 Conclusion 25 Bibliography 27 iii List of Figures 1.1 Photon mass-attenuation coeﬃcients for aluminium and lead as a function of the photon energy. Dashed lines show the separate contributions due to the photoelectric eﬀect, Compton scattering and pair production[4]. . . . . . . . . . 2 1.2 Geometry of Compton scattering.[9] . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The depletion zone and the reverse-bias eﬀect[9]. . . . . . . . . . . . . . . . . . 7 1.4 Typical pulse height spectrum of a γ-ray[8]. . . . . . . . . . . . . . . . . . . . . 8 1.5 Spectrum of γ-rays emitted following β-decay of 60 Co to 60 Ni. The lower spec- trum was obtained with an unsuppressed Ge detector, the upper one with the Ge detector inside an escape suppression shield[10]. . . . . . . . . . . . . . . . . . 10 2.1 Relative contributions to the binding energy per nucleon showing the various terms in the semi-empirical mass formula (SEMF).[10] . . . . . . . . . . . . . . . . . 13 2.2 Sequences of bound single-particle states calculated for diﬀerent forms of the nuclear shell-model potential. The number of protons (and neutrons) allowed in each state is indicated in the parentheses and the numbers enclosed in circles indicate magic numbers corresponding to the closed shells.[4] . . . . . . . . . . 15 2.3 The phonon and multi-phonon excitations in spherical and deformed nuclei[9]. . 18 2.4 Superdeformed and heyperdeformed nuclei[9] . . . . . . . . . . . . . . . . . . . 19 3.1 Annihilation generates two γ-ray tracks originating from the interaction point[10]. 20 3.2 Clusterisation method[7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3 The total peak eﬃciency of ”cluster-tracking” reconstructed data is shown as a function of assumed position resolution and γ-ray multiplicity[7] . . . . . . . . . 23 iv 1. Gamma-rays 1.1 The importance of gamma-ray spectroscopy Gamma spectroscopy is a method of study of nuclear structure. A gamma spectrometer will de- termine the energy and the count rate of gamma-rays emitted by radioactive substances. Gamma spectroscopy is an extremely important method. Most radioactive sources produce gamma-rays of various energies and intensities. When these emissions are collected and analyzed with a gamma spectroscopy system, a gamma energy spectrum can be produced[8]. 1.1.1 The interaction of γ radiation with matter An excited nucleus may lose energy by emitting a γ-ray photon, whose energy Eγ is equal to the energy diﬀerence of ∆E between the initial and ﬁnal nuclear states. Gamma-ray energies cover a wide range up to tens of MeV but, typically, are of the order of 1 MeV. The wavelength of a 1-MeV γ-ray is about 1240 fm. In principle the initial state could decay to any state at a lower energy but, in reality, the prob- ability of a particular transition is very dependent upon the quantum numbers of the states and the transition energy ( E)[1]. If a ﬂux of gamma-ray passes through matter, the number emerg- ing decreases exponentially with the thickness of the absorber. Hence we have a relationship analogous to the fundamental decay law given by: N = N0 e−µx , (1.1) where µ is the total absorbtion coeﬃcient and x is the thickness. There are three primary processes by which γ-rays interact with matter. These are the photo- electric eﬀect, Compton scattering and pair production. Their relative importance is illustrated in Fig 1.1 which plots, as a function of photon energy Eγ, the total mass attenuation coeﬃcient µ for aluminium and lead. This is the measure of the probability of the γ-ray interacting in the material, and depends on the atomic number Z of the atoms of the absorbing media and varies strongly with γ-ray energy Eγ . By comparing the two sets of curves for aluminium and lead, we ﬁnd that each eﬀect has its own characteristic energy and Z dependence. The contribution from the Compton scattering shows a very weak dependence on Z while the contribution from the photoelectric eﬀect and pair production are relatively more important in lead (Z=82) than in aluminium (Z=13)[4]. 1 Section 1.1. The importance of gamma-ray spectroscopy Page 2 Figure 1.1: Photon mass-attenuation coeﬃcients for aluminium and lead as a function of the photon energy. Dashed lines show the separate contributions due to the photoelectric eﬀect, Compton scattering and pair production[4]. 1.1.2 Photoelectric eﬀect The photoelectric eﬀect involves the absorption of a photon by an atomic electron which is, as a result, knocked out of the atom. This photo-electron emerges with kinetic energy given by[4] T = Eγ − Be , (1.2) where Be is the binding energy of the electron. The atom that has lost the electron may be de-excited by releasing other, less tightly bound electrons. Electrons emitted by this process are called Auger electrons. Alternatively an electron from a higher shell may ﬁll the vacancy in the inner shell with an emission of a characteristic X-ray photon . The photoelectric mass attenuation coeﬃcient, shown in Fig 1.1 varies strongly with Eγ and Z. A sharp rise at low energy for lead is due to the contribution of the innermost, K-shell electrons which in lead have a binding energy BK of about 90 keV. When the photon’s energy is below 90 keV, these electrons do not contribute to the photoelectric eﬀect because there is not enough energy to eject them from the atom. The discontinuity in the energy dependence of µ is known as the K edge. Note that the relative contribution of the two K electrons is much greater than that of all the rest of the electrons in the lead atom put together. The need to conserve both energy and momentum forbids the photoelectric eﬀect from occurring on a free electron. The smaller the electron’s binding energy relative to Eγ the more it appears like a free electron and the smaller will be the probability of the photoelectric absorption. An Section 1.1. The importance of gamma-ray spectroscopy Page 3 approximate expression, illustrating these dependencies, gives for the photoelectric cross section: 3.5 σpe ∝ Z 5 /Eγ . (1.3) 1.1.3 Compton eﬀect This is an electromagnetic interaction where a photon with incident energy Eγ scatters from an atomic electron that may be regarded as free , as shown in Fig 1.2. The result is a recoiling electron with a kinetic energy T which depends on the photon’s scattering angle θ, and a photon with lower energy Eγ . Using relativistic kinematics and energy conservation, the kinetic energy of the electron is given by[4]: T = Eγ − Eγ = E − mc2 , (1.4) From conservation of momentum pγ = pγ + p, (1.5) where pγ and pγ are the momentum of the incident and scattered photon, p is the momentum of the electron and E is the total energy of the recoil electron including its mass energy mc2 where c is the speed of light. Using the cosine rule, we can write: p2 c2 = Eγ + (Eγ )2 − 2Eγ Eγ cos θ = E 2 − m2 c4 , 2 (1.6) where we have substituted for pγ and pγ and used the relation E 2 = p2 c2 + m2 c4 . By using equations 1.6 and 1.4 we will get: Eγ − Eγ + mc2 = E. (1.7) By squaring the formula we get: 2 2 Eγ + Eγ − 2Eγ Eγ + m2 c4 + 2mc2 Eγ − 2mc2 Eγ = E 2 , (1.8) Eγ Eγ (1 − cos θ) + mc2 Eγ = mc2 Eγ , (1.9) then mc2 Eγ Eγ = . (1.10) Eγ (1 − cos θ) + mc2 Finally, we obtain an expression for the scattered photon energy: Eγ Eγ = Eγ . (1.11) 1+ ( mc2 )(1 − cos θ) Section 1.2. Semiconductors Page 4 Figure 1.2: Geometry of Compton scattering.[9] ‘ 1.1.4 Pair production Pair production refers to the creation of an elementary particle and its antiparticle from a photon. In this process the entire photon energy is converted in the ﬁeld of an atom into the creation of an electron-positron pair with a total kinetic energy given by T− + T+ = Eγ + 2mc2 . e (1.12) Pair production requires the presence of a heavy body in order to conserve both energy and momentum. There is an energy threshold of 2mc2 = 1.022 MeV and the pair production cross e section does not become important until Eγ exceeds several MeV. 1.2 Semiconductors Semiconductors’ intrinsic electrical properties are very often permanently modiﬁed by introducing impurities, in a process known as doping. Semiconductors are very similar to insulators. The two categories of solids diﬀer primarily in that insulators have larger band gaps than the energies that electrons must aquire to be free to ﬂow. In semiconductors at room temperature, just as in insulators, very few electrons gain enough thermal energy to leap the band gap, which is necessary for conduction. For this reason, pure semiconductors and insulators, in the absence of applied ﬁelds, have roughly similar electrical properties. The smaller band gaps of semiconductors, however, allow for many other means besides temperature to control their electrical properties. 1.2.1 Charge carriers in semiconductors In the lowest energy state of a semiconductor the electrons in the valence band all participate in the covalent bonding between the lattice atoms. Both silicon and germanium have four valence Section 1.2. Semiconductors Page 5 electrons. In a semiconductor the electric current arises from two sources: the movement of free electrons in the conduction band and the movement of the holes in the valence band. 1.2.2 Intrinsic charge carriers Under stable conditions, an equilibrium concentration of electron-hole pairs is established. If ni is the concentration of electrons (or equally holes) and T is the temperature, then Eg 3 Eg ni = Nc N v exp − = AT 2 exp − , (1.13) 2kT 2kT where k is Boltzmann constant, Nc and Nv are the number of states of the conduction band and the valence band respectively, Eg is the energy gap at 0 K. Typical values for ni are of the order of 2.5 × 1013 cm−13 for germanium and 1.5 × 1010 cm−3 for silicon at T=300 K. This should be put into perspective, by noting that they are of the order of 1022 atoms/cm3 in these materials. This means that only 1 in 109 germanium atoms is ionised and 1 in 1012 in Si. Despite the large exponents, the concentrations are very low. 1.2.3 Doped semiconductors In pure semiconductor crystals, the balance of the number of electrons and holes can be changed by producing a small amount of impurity atoms having one more or one less valence electron in their outer shell. These impurities integrate themselves into the crystal lattice to create what are called doped or extrinsic semiconductors. If the impurity consits of one less valence electron there will not be enough electrons to ﬁll in the valence band. There is thus an excess of holes. These impurities perturb the band structure by creating an additional energy state close to the valence band. This excess of holes increases the normal concentration of the holes, so the holes become the majority charge carriers. Such materials are referred to as p-types semiconductor. If the impurity consits of one more valence electron, the electrons in the ground state ﬁll up the valence band which contains just enough room for four valence electrons per atom. Since the impurity has ﬁve valence electrons, an extra electron is left which does not ﬁt into this band. This electron resides in a discrete energy level created in the energy gap. This level is close to the conduction band where it will enhance the conductivity of the semiconductor. In such material the current is mainly due to the movement of electrons. These doped semiconductors are called n-type semiconductor . The product of the number of holes and electrons is given by Eg np = n2 = AT −3 exp i − . (1.14) KT Section 1.3. Semiconductor detectors Page 6 1.2.4 The p-n junction A pn-junction is formed by the juxtaposition of a p-type and n-type semiconductor. The formation of a pn-junction creates a special zone around the interface between the two materials. Because of the diﬀerence in the concentration of the electrons and holes between the two materials, a number of holes move towards the n-region and a similar number of electrons move towards the p-region. The electrons ﬁll up holes in the p-region while the holes capture electrons in the n-region. Due to this process a non-conducting layer called the depletion zone occurs as shown in Fig 1.3. Any hole entering the zone will be swept out by the electric ﬁeld. This characteristic of the depletion zone is particularly attractive for radiation detection. Ionizing radiation entering this zone will liberate electron-hole pairs which are then swept out by the electric ﬁeld. If electrical contacts are placed on either end of the junction device, a current signal proportional to the ionization will then be detected. 1.3 Semiconductor detectors 1.3.1 Forward bias Forward bias occurs when the p-type block is at a higher potential. With this set-up, the holes in the p-type region and the electrons in the n-type region are pushed towards the junction. This reduces the depletion zone. Eventually the non-conducting depletion zone becomes small as the charge carriers can tunnel across the barrier. This causes an electric current to ﬂow. 1.3.2 Reverse-bias Connecting the p-type region to a negative potential, produces the reverse-bias eﬀect. The ’holes’ in the p-type region are pulled away from the junction, causing the width of the non-conducting depletion zone to increase. The electrons will be pulled away from the junction. This eﬀectively increases the potential barrier and greatly increases the electrical resistance against the ﬂow of charge carriers. For this reason there will be minimal electric current across the junction. 1.3.3 The p-n junction detector This type of detector is a diode, which is formed at the boundary between two diﬀerent types of semiconductors. When a junction is made between two types of semiconductors, electrons from the n-type migrate across the boundary and combine with holes in the p-type material, creating a zone of intrinsically much purer (compensated) material depleted of either weakly-bound electrons or trapping sites. The movement of charges sets up a bias voltage across the junction. The depleted zone is the region of the detector sensitive to radiation. Electrons and holes created by the radiation move under the inﬂuence of the electric ﬁeld across the region and form a current Section 1.3. Semiconductor detectors Page 7 Figure 1.3: The depletion zone and the reverse-bias eﬀect[9]. pulse. The depleted zone can be made deeper by applying an external voltage V in the reverse bias direction. Reverse bias increases the electric ﬁeld in the depletion zone and, therefore, also increases the rate and eﬃciency with which the holes and electrons are collected at the electrodes. Thermal excitation is much more likely in germanium than in silicon because of the narrow energy band gap (0.75 eV) and, for this reason, germanium detectors are usually cooled. 1.3.4 Germanium detectors For gamma-ray detection, germanium is preferred over silicon because of its much higher atomic number (ZSi =14 ,ZGe =32). The photoelectric cross section is thus about 60 times greater in germanium than silicon. Germanium however, must be operated at low temperature because of its smaller band gap. High-Purity Germanium detectors (HpGe) produce high energy resolution. Table 1.1. Some physical properties of Silicon and Germanium: Silicon Germanuim Atomic number Z 14 32 Atomic weight A 28.1 72.6 2 Density [g/cm ] 2.33 5.32 Energy gap (300 K) [eV] 1.1 0.7 Energy gap (0 K) [eV] 1.21 0.785 Electronic mobility (300K) [cm/Vs] 1350 3900 Hole mobility (300K) [cm/Vs] 480 1900 Section 1.3. Semiconductor detectors Page 8 Figure 1.4: Typical pulse height spectrum of a γ-ray[8]. 1.3.5 Energy spectrum Energy spectrum is proportional to the probability distribution function of the energy deposited by events and expementaly is measured by simply plotting the number of events verses the energy. A plateau continuum at lower pulse heights shown in Fig 1.4 is due to the Compton scattering with the gamma-ray escaping undetected. The upper limit to the plateau is known as the Compton edge and its energy is easily obtained from Equations 1.4 and 1.11. The maximum energy of the recoil electron is given by: 2 2Eγ T (1800 ) = Eγ − Eγ (180) = (1.15) (mc2 + 2Eγ ) The full-energy peak is a sharp peak in the energy spectrum corresponding to events in which the gamma-ray transfers all its energy Eγ to the electron in the crystal, either in a single photoelectric convertion or by Compton scattering several times before undergoing photoelectric absorption. 1.3.6 Energy resolution Energy resolution is measured in terms of the full width at half maximum (FWHM) of the full- energy peak and is expressed either in units of energy or as a percentage of the peak energy. The peak is often ﬁtted to a Gaussian distribution of the form exp[ (E−Eγ ) ] where Eγ is the peak 2σ 2 energy and σ is the standard deviation of the distribution[4]. The FWHM for a Gaussian distribution is approximately equal to 2.35σ. The resolution of a Section 1.3. Semiconductor detectors Page 9 semiconductor (Ge) detector is far superior to that of NaI scintillation detector. One reason for this is the diﬀerence in the statistical variation on the critical number of the events which determines the signal in each case. Taking the average energy to create an e-h pair in Ge crystal to be 3eV, we ﬁnd that a 660 keV γ ray excites n =220 000 pairs; the standard deviation σ which is the square root of the n is about 470. Using this , we estimate the FWHM to be 2.35 × 470 × 3eV=3.3 keV or about 0.5% of the γ-ray energy. Fig 1.4 shows the Compton edge. 1.3.7 Compton suppression Compton scattering means that many γ-rays which enter the Ge detector will not deposit their full energy. In order to reduce the contribution of scattered γ-rays the Ge detector can be surrounded by a scintillator detector. The two detectors are operated in anti-coincidence, which means that if an event occurs at the same time in both detectors, then the event is rejected. The majority of escape suppression shields, as they are known, use bismuth germinate B4 G3 O12 (BGO). The reason for this choice is that BGO has excellent timing properties, which is desirable for coincidence work, and high density (7.3 g/cm3 ), so that a relatively small amount of material is needed in order to stop fully the scattered photons. This fact becomes important when we wish to closely pack many detectors around a target to form an array. An example of the eﬀect of Compton suppression is shown in Fig 1.5. Fig1.5 shows a spectrum of emitted γ-rays following the β − decay of 60 Co to 60 Ni. The most important features are labelled. The Compton edge corresponds to the maximum energy loss when θ = π in the Compton scattering equation 1.8. The photopeak is of particular interest to the γ-ray spectroscopist. This corresponds to the situation when the incident γ-ray deposits its full energy in the Ge detector. An important quantity is the peak-to-total ratio, which is the ratio of the number of counts in the photopeak to the total number of counts in the spectrum, above an energy of around 100 keV. It can be seen from Fig 1.5 that escape suppression improves this ratio considerably, since the number of counts in the Compton continuum region is reduced, whilst the number of counts in the photopeak is not aﬀected appreciably. It is preferable that the peak-to-total ratio is as high as possible. Section 1.3. Semiconductor detectors Page 10 Figure 1.5: Spectrum of γ-rays emitted following β-decay of 60 Co to 60 Ni. The lower spectrum was obtained with an unsuppressed Ge detector, the upper one with the Ge detector inside an escape suppression shield[10]. 2. Nuclear structure 2.1 Nuclear models Diﬀerent models focus on diﬀerent aspects of nuclear behaviour such as the ground state prop- erties, excited state energies and nuclear reactions at diﬀerent energies. 2.1.1 The liquid drop model The liquid drop model was formulated by Niels Bohr. The model treats the nucleus as a spherical drop of incompressible nuclear ﬂuid bound together by the strong force. Although the nuclear force is strong , it acts over a relatively small distance compared to the nuclear radius. Therefore, a nucleon will only interact via the strong force with a few neighbouring nucleons. This property is called saturation. The liquid drop model calculates the interaction force between the nucleons as a sum of the repulsive coulomb force between the protons of the nucleus and the attractive surface tension on the surface of the liquid drop model. 2.1.2 Volume binding energy A nucleon in the nuclear interior interacts, on average, with a ﬁxed number of neighbouring nucleons within the short range of the nuclear force. The binding energy per nucleon, therefore will be constant and the total contribution to the nuclear binding energy will be proportional to A. The binding energy of the nucleus will be proportional to the number of nucleons Evolume (A, Z) = aν A, (2.1) where aν is a parameter called the volume term coeﬃcient. 2.1.3 Surface term Nucleons close to or at the surface of the nucleus are surrounded by fewer nucleons than those at the centre and will therefore experience less attraction. Thus a term proportional to the surface area must be subtracted from the volume term, which will reduce the binding energy of the nucleus. The binding energy for the surface will be: 2 Esurface (A, Z) = −as A 3 , (2.2) 1 where as is a parameter, since R∼ A 3 . 11 Section 2.1. Nuclear models Page 12 2.1.4 Coulomb energy The electric repulsion between each pair of protons in a nucleus contributes toward decreasing its binding energy. The nucleus has a total charge Ze distributed uniformly. The electrical potential Q2 energy of uniformly charged sphere , radius R is equal to 3 4πε0 R . Then the Coulomb energy 5 term is given by: Z2 ECoulomb (A, Z) = −ac , (2.3) R where ac is a parameter. 2.1.5 Asymmetry Energy This term expresses the charge-symmetric nature of the nucleon-nucleon force. In the absence of the coulomb force, the most stable nuclei would have equal numbers of neutrons and protons. The form of the symmetry term follows from the Pauli principle and the fact that the eﬀective force in the nucleus is stronger between unlike nucleons (n-p) than between identical nucleons (n-n or p-p). Easymmetry (A, Z) = −aA (N − Z)2 A−1 . (2.4) where aA is a parameter. 2.1.6 Pairing energy The ﬁnal term in the SEMF reﬂects the tendency for like nucleons to form spin-zero pairs in the same spatial state. When coupled, they spend more time being closer together within the range of the nuclear force then when they occupy diﬀerent orbitals. The pairing contribution is positive if I N and Z are both even and the nucleons are all coupled to form a spin-zero pairs, and is negative if both N and Z are odd. It is zero if either N or Z is odd (A odd)[6] The pairing correction takes the form: +δ even-even Epair (A, Z) = 0 even-odd . (2.5) −δ odd-odd The Bethe-Weizsacker formula for a liquid drop model [3] which shows the total contribution to the binding energy takes the following form: 2 Z2 BE (A, Z) = aν A − as A + ac 3 + aA (N − Z)2 A−1 ± δ. (2.6) R BE(A,Z) is the binding energy in MeV. Fig 2.1 shows the relative contributions to the binding energy per nucleon and the various terms in the semi-empirical mass formula. Table 2.1.6 shows a set of the values of the coeﬃcients: Section 2.1. Nuclear models Page 13 Figure 2.1: Relative contributions to the binding energy per nucleon showing the various terms in the semi-empirical mass formula (SEMF).[10] Table 2.1: M eV av 15.75 as 17.8 ac 0.711 aA 23.7 δ 11.18 Section 2.1. Nuclear models Page 14 2.1.7 The nuclear shell model The nucleus is a quantum-mechanical object and it’s structure of the nucleus is constrained by the Pauli principle. The shell model is partly analogous to the atomic shell model which describes the arrangement of electrons in an atom, in that a ﬁlled shell results in greater stability. There are certain magic numbers of nucleons [6]: Z=2,8,20,36,54,86. N=2,8,20,28,50,82 and 126. which are more tightly bound than an average for other particle. Atomic nuclei consisting of such magic numbers of nucleons have higher average binding energy per nucleon based upon predictions such as the semi-empirical mass formula shown in equation 2.6. Noble gases represent particularly stable and inactive electron conﬁguration. The corresponding ”magic” numbers are: Z=2,10,28,36,54,86. The shell model tries to identify the causes for the observed features for nuclei with magic numbers. The diﬀerent single particle states (orbits) are obtained by solving the time-independent o Schr¨dinger equation for the diﬀerent potential given in equation 2.8. A magic number occurs when there is a large energy gap between the last ﬁlled level and the next unoccupied one. The magic numbers predicted by the diﬀerent wells are indicated in Fig 2.2. The ﬁrst two wells correspond to experimental values but neither of these wells account for the higher ones. 2.1.8 The Woods-Saxon potential with a spin-orbit coupling The average potential energy V experienced by a nucleon in a nucleus is generated by the interac- tions with the other nucleons. The nuclear potential energy experienced by nucleon in a nucleus is often parametrised in the following way[2]: −V0 V (r) = ) (2.7) 1 + exp[ r−R ] a which is known as the Woods-Saxon form of the potential. 1 where R=1.2×A 3 fm, a=0.6fm, V0 o The Schr¨dinger equation can be solved in three dimensions using this potential, and the allowed energy states are shown in Fig 2.2 . o The time-independent Schr¨dinger equation in three dimensions is given by: 2 − ψ + V ψ = Eψ. (2.8) 2m Section 2.1. Nuclear models Page 15 Figure 2.2: Sequences of bound single-particle states calculated for diﬀerent forms of the nuclear shell-model potential. The number of protons (and neutrons) allowed in each state is indicated in the parentheses and the numbers enclosed in circles indicate magic numbers corresponding to the closed shells.[4] Section 2.1. Nuclear models Page 16 The states in Fig 2.2 are labelled s,p,d,f,g, etc. Successive l states in a nucleus are labelled sequentially in order of increasing energy. For each value of l, there are 2l+1 substates cor- responding to the allowed orientations of the angular momentum along a given direction. For spherical nuclei, the potentail wells are spherically symmetric and in this case, substates of a given l all have the same energy, and each l state is said to consist of 2l+1 degenerate states. Nucleons occupy states in ascending order of energy according to the exclusion principle. Thus, each l state can accommodate up to 2(2l+1) nucleons of each type. A magic number occurs when there is a large energy gap between the last ﬁlled level and the next unoccupied one. The magic numbers predicted by the diﬀerent wells are indicated in Fig 2.2. 2.1.9 The spin-orbit potential In 1949 Mayer, Haxel, Jensen and Suess showed that the proper spacing of levels with observed shell closing could be obtained by adding a spin-orbit term to the nuclear potential. The simple shell model works for three magic numbers 2,8,20 but not for the remaining numbers, so it was necessary to include the spin-orbit coupling which further splits the nl states[4]. The spin-orbit (SO) potential has the form -Vso (r)I.s where I and s are the orbital and spin angular momenta, respectively of a nucleon moving in the nuclear well. Nucleons with diﬀerent values of total angular momentum j = I + s will have diﬀerent energies. A nucleon has a spin quantum number s= 1 , which means that each I state can have a total 2 angular momentum quantum number j=l ± 2 (except l=0 for which only j= 1 is allowed)[2]. 1 2 The energy levels split into two corresponding to the j=l + 1 and j=l − 1 . The energy levels with 2 2 j=l − 1 are raised and the energy levels with j=l + 2 are lowered. 2 1 2.1.10 Deformed nuclei The liquid drop model indicates that since for a given volume, a spherical shape has a minimum surface area. Any deformation from spherical shape is determined by the competition between the coulomb and surface energies. Therefore a stable deformation will occur when the coulomb exceeds the surface energy. In light nuclei the surface energy is more powerful and in this case there will be resistance to deformation. In the case of nuclei with a large number of A, the coulomb energy decreases and leads to a stable deformation rather than an equilibrium spherical shape. It was established that many nuclei with N and Z values between magic numbers are permanently deformed in their ground state. The deformation arises because of the way valence nucleons arrange themselves in an unﬁlled shell. Nucleons ﬁlling a shell-model state with a given l value will tend to group into substates with similar values of projection number m because this maximises the binding energy from the nucleon-nucleon attraction Once these substates are full, further nucleons will go into other substates which have the greatest Section 2.1. Nuclear models Page 17 overlap with the ﬁlled ones. Also, certain substances of higher levels may be favoured by the deformed potential and ﬁll preferentially, thus adding to the deformation. The tendency to drive the nucleus into a non-spherical shape is greatest when the shell is about half ﬁlled. Beyond this point, additional nucleons are constrained to enter the remaining, unﬁlled substance in the shell. A closed shell resists becoming deformed and it is only when both proton and neutron shells are partially ﬁlled that we ﬁnd permanently deformed nuclei. Deformed stable nuclei are found throughout the periodic table and are most common in the mass regions 150<A<190. and A>230. 2.1.11 Collective motion of nuclei A collective nuclear model assumes to be a drop of incompressible nuclear ﬂuid but slightly compressible at high energy. If we consider the even-even nuclei, such a nucleus is expected to have spherical symmetry in its ground state. However, this spherical nucleus will be deformable and excited states are to be expected in which the nucleus oscillates about its spherical shape. 2.1.12 Vibrational states The liquid-drop model predicts that the j=l − 1 nucleus will be spherical in its ground state. 2 Any deformation of the shape from equilibrium increases the surface-energy term. For small deformations near the minimum the shape is parabolic. It is possible for the nucleus to vibrate about its equilibrium and exist in quantum states. Fig 2.3 shows the phonon and multi-phonon excitations in spherical and deformed nuclei[2]. 2.1.13 Rotational states Collective rotational motion can only be observed in nuclei with a non-spherical shape. A nucleus shaped like an ellipsoid can rotate about one of its equal axis, but not about the third (symmetry) axis. This is related to the fact that, in quantum mechanics, a wave function representing a perfectly spherically symmetric system has no preferred direction in space and a rotation does not lead to any observable change. Only if there is a deviation from spherical symmetry can a rotation be detected. The magnitude of the rotational angular momentum is given by Iω where I is the eﬀective moment of inertia and ω is the angular frequency of the rotation. The expression for rotational energy is given by: 1 (Iω)2 I(I + 1) 2 E(I) = Iω 2 = = (2.9) 2 2I 2I where I is the quantum angular momentum. Section 2.1. Nuclear models Page 18 Figure 2.3: The phonon and multi-phonon excitations in spherical and deformed nuclei[9]. 2.1.14 Superdeformation A super deformed nucleus is a nucleus that is very far from spherical, forming an ellipsoid with axis in ratios of approximately 2:1. Normal deformation is approximately 1.3:1. In the spherical potential well, magic numbers occur because the shell model states group together to form shells with large energy intervals between them. A ﬁlled shell forms a stable structure because considerable energy is required to break a nucleon away from the closed shell and move it across the energy gap to the next one[2]. The ﬁrst superdeformed states to be observed were the ﬁssion isomers. The nucleus can be described by the liquid drop model. The drop’s energy as a function of deformation is at a minimum for zero deformation, due to the surface tension term. In the deformed well, energy gaps occur at N and Z that are quit diﬀerent to the magic numbers for a spherical well. Magic numbers in the 2:1 deformed well occur at N , Z values that correspond to nuclei with partially ﬁlled spherical shells and which may be deformed at the ground state. Such nuclei are called Superdeformed nuclei. Section 2.1. Nuclear models Page 19 Figure 2.4: Superdeformed and heyperdeformed nuclei[9] Superdeformed nuclei’s rotational frequencies are extremely large; for an energy of 0.6 MeV the corresponding rotational frequency is 1022 Hz. The single particle motion and collective motion, mainly rotational together determine the characteristics of the superdeformed nuclei bands. Fig 2.4 shows the diﬀerent types of deformation. 3. Gamma-ray tracking methods In nuclear and particle physics, charged particles are tracked using their continuous ionisation in a position sensitive detector. For gamma-rays, the situation is completely diﬀerent since their interaction probabilities follow a statistical law and are much lower, generally resulting in a few scattered interaction points that can be separated by large distances. Therefore, the scattering path of a gamma-ray in the detector volume can not be easily deduced[7]. Most gamma-rays in the energy range around 1 MeV, which interact with a Ge detector, will gen- erally Compton scatter several times before ﬁnally photo absorbtion or escape takes place. That is why gamma-ray tracking requires powerful algorithms that take into account the physical char- acteristics of the gamma-ray interaction in the detector, ie. Compton-scattering, pair production, and photoelectric eﬀect. Fig3.1 shows the annihilation photons generating two γ-rays. Figure 3.1: Annihilation generates two γ-ray tracks originating from the interaction point[10]. To apply the Compton scattering law the information from the γ-ray tracking detectors on the individual interactions positions and the respective energy depositions as well as the total inte- grated energy deposition must be used. The development of the γ-ray tracking algorithms relies primarily on simulated Monte Carlos. In the clusterisation method a preliminary identiﬁcations of clusters of interaction points is followed by a comparison of all possible scattering angles within a cluster against the Compton-scattering formula. The second approach, called backtracking starts from points likely to be the last interaction and goes back, step by step, to the origin of the incident γ-ray. Compton scattering is the most important eﬀect in the energy range of interest and is the only mechanism that allows real tracking to be performed. The scattering path of electrons in the 20 Page 21 MeV range is of the order 1 mm so that any practical detector will see it as an energy release point very close to the scattering vertex and not the track. A position-sensitive detector provides both the value of the energy released at the interaction points and the 3-dimension coordinates of the scattering position. Low energy γ-rays (below 150 keV) usually are absorbed directly by photoelectric eﬀect and hence mostly detected as single points. There is actually no safe way to decide whether an isolated low-energy interaction point corresponds to transition of the same energy or is the result of a Compton-scattered and partly escaped higher energy γ-ray. In germanium, pair production becomes an important detection mechanism for the γ-rays above a few MeV and it overcomes Compton scattering at 9 MeV. Given the energy range of our interest and the energy dependence of the pair-production cross section, we need to consider in practice only the case where the pair is produced at the ﬁrst interacton. The total kinetic energy of the electron-positron pair (Eγ -2mc2 ) is shared by the two partners. However, as both particles are in the MeV range, they are stopped in close vicinity to the pair production point. Being close to each other, they are normally seen by the detector as one individual energy deposit. The slowed-down positron binds to an atomic electron and forms a positronium atom that rapidly annihilates emitting two collinear 551 keV γ-rays as shown in Fig 3.1. These will either escape or be absorbed in some other part of the detector. In case of full absorption the detection pattern of the experimental points has an energy coore- sponding to the total detected energy minus 1022 keV, while the other points are from the two 511 keV γ-rays. 3.0.15 The clusterisation method For a γ-ray scattering in a large Ge detector, the interaction points tend to conﬁne themselves within a rather limited volume. This eﬀect is due to the slight forward peaking of the Compton scattering cross section as given in equation 3.4. For an incident photon of energy γ, the diﬀerential cross section is given by the Klein-Nishina: Eaf ter = hν (3.1) Ebef ore = hν0 (3.2) 2 3 dσ 1 2 ν ν 2 ν = re − sin θ + , (3.3) dΩ 2 ν0 ν0 ν0 dσ 2 = 0.5re (P (Eγ , θ) − P (Eγ , θ)2 sin2 (θ) + P (Eγ , θ)3 ). (3.4) dΩ Page 22 where θ is the scattering angle; re is the classical electron radius; me is the mass of an electron; and P (Eγ ,θ) is the ratio of photon energy after to that before the collision given in equation 1.8. Figure 3.2: Clusterisation method[7] The clusterisation method identiﬁes clusters of the interaction points from individual γ-ray’s. The energy of a gamma-ray is equal to the sum of the energies of its interaction points within a cluster. Out of the clusters that have been formed, some will correspond to all the interaction points of one fully absorbed γ-ray (”good” clusters) and others will not (”bad” clusters). Bad clusters can arise when two good clusters or parts of them are treated as one, or when a good cluster is misidentiﬁed as being two. Page 23 3.0.16 The backtracking method Low-energy γ-rays (below 150 keV) usually are absorbed directly by photoelectric eﬀect and hence mostly detected as single points. There is actually no safe way to decide whether an isolated low energy interaction point corresponds to a transition of the same energy or is the result of a Compton-scattered and partly escaped higher energy γ-ray[11]. The second γ-ray tracking method called the backscattering method is based on the observation that the energy deposition of the ﬁnal photoelectric interaction after scattering usually falls into a narrow energy band. Here the photo and Compton spectra of the energy depositions in all the individual interactions of the gamma-rays within the Ge detector are observed, considering that in most cases they interact by several Compton scatterings before photo absorbtion ﬁnally takes place. The idea is to look for the ﬁnal interaction and then reconstruct a track onto the original emission point. This method allows in principle, to disentangle the interaction points of two γ-rays entering the detector very close to one another. The FWHM of the distrubution of gamma-rays is inversely proportional to the energy of the interaction point. As a further feature, points that are closer to each other are packed together to an average energy position. Fig 3.3 shows the total peak eﬃciency of ”cluster tracking” reconstruction as a function of assumed position resolution and gamma-ray multiplicity. The position resolution produces good data with just a γ-ray in the event. It becomes an important factor already at multiplicity 2, because the packed points can now belong to diﬀerent transitions. Figure 3.3: The total peak eﬃciency of ”cluster-tracking” reconstructed data is shown as a function of assumed position resolution and γ-ray multiplicity[7] 4. Physics with AGATA AGATA, the ﬁrst complete 4π gamma-ray spectrometer which is proposed in Europe and not yet built from germanium (Ge) detectors, is based on the novel technique of gamma-ray tracking. AGATA will be an instrument of major importance for nuclear structure studies at the very limits of nuclear stability, capable of measuring gamma-radiation in a very large energy range (from a few tens of keV up to 10 MeV and more), with the largest possible eﬃciency and with a very good spectral response. AGATA will be several orders of magnitude more powerful than all current and near-future gamma-ray spectrometers. AGATA is a programme that develops, a new generation of position-sensitive high-purity germanium detectors. Spectroscopy of strongly deformed-nuclei from super to hy- perdeformation. Enormous advances in understanding the behaviour of the atomic nucleus when subjected to extremes of angular momentum, have resulted from the high-eﬃciency Compton suppressed multi-detector arrays such as Euroball and Gammasphere. Experimentalists have been able to probe the ﬁne structure of the nuclei created in rapidly spinning states. Compton suppressed multidetector HpGe gamma-ray arrays gave the experimental capability to discover and study superdeformed nuclei. Under special conditions, atomic nuclei can possess a very elongated superdeformed shape at high angular momenta. In heavy nuclei the formation of a hyperdeformed (HD) shape may happen at low spins, just before ﬁssion. In medium-heavy nuclei, HD shapes are expected however, to occur near the largest angular momenta which a nucleus could sustain preceding ﬁssion. The ﬁrst search for the existence of HD shapes at high spins was made in the γ-energy correlations spectra of 152 Dy . In both cases a (heavy-ion,proton-xn-γ) nuclear reaction was used. In both cases a heavy ion induced incomplete ﬁssion reaction (HI,p-xn) was used from which it was conjectured that charged-particle emission from the tip of HD nuclei may be enhanced relative to that from normal deformed nuclei. If so, tagging on charged particles may help much in the preferential selection of gammas from decay of HD nuclei. What more can now be done? A gamma-ray tracking device such as AGATA or GRETA which will be a national instrument, movable between several major accelerators in the US, made from highly segmented HpGe will have more increased eﬃciency for detecting gamma-ray energies and direction in events with high gamma multiplicity. This will enable nuclear physicists to study phenomenas where modes of excitation of the nucleus are only weekly populated, for example hyperdeformation, or even exotic nuclear shapes such as a tetrahedral or octrahedral. 24 5. Conclusion The nucleus is a complex system which displays remarkably regular and often simple excitation modes. In the liquid drop model, the nucleons are imagined to interact strongly with each other, like the molecules in a drop of liquid. It permits us to correlate many facts about nuclear masses and binding energies. It also provides a useful model for understanding nuclear ﬁssion. Collective rotations of nuclei can only be observed in nuclei with a non-spherical shape. Any deformation from a spherical shape is determined by the competition between the Coulomb and surface energies. If the surface energy dominates the Coulomb energy, nuclei preserve their spherical shape. The single-shell model shows evidence which suggests that nuclei with certain numbers of neutrons and protons are particularly stable. Experimental values of the binding energy per nucleon deviate most notably from the semi-empirical mass formula curve for certain values of N and Z. Diﬀerent features of the nucleus can only be simultaneously observed by a new generation of spectrometers from close-packed arrangements of gamma-ray detectors and resembling a 4π shell of large segmented germanium crystals. Gamma-ray detector systems play a key role across a wide range of sciences. The development of gamma-ray detection systems is capable of tracking the location and energy deposition at every gamma-ray interaction point in the detector, representing a major advance in detector technology. The clusterisation method identiﬁes clusters of the interaction points from individual gamma-ray’s origins, between points within one cluster and spatial separation of points in the cluster. The clusterisation method identiﬁes clusters of the interaction points from individual γ-ray’s. The energy of a gamma-ray is equal to the sum of the energies of its interaction points within a cluster. Out of the clusters that have been formed, some will correspond to all the interaction points of one fully absorbed γ-ray (”good” clusters) and others will not (”bad” clusters). Bad clusters can arise when two good clusters or parts of them are treated as one, or when a good cluster is misidentiﬁed as being two. The backscattering method is based on the observation that the energy deposition of the ﬁnal photoelectric interaction after scattering usually falls into a narrow energy band. The idea of the backtracking method is to look for the ﬁnal interaction and then reconstruct a track back to the original emission point. The aim of the gamma-ray tracking is to disentangle the interaction points i.e. to reconstruct individual photon trajectories and to output the energies, incident and scattering directions. 25 Acknowledgements I would like to dedicate this work to my wonderful parents Yanawil Tilar and Mirghani Bilal and to my brothers Nasir, Mohammed, Bilal and my sister Lona. I would like to express my sincere gratitude to my supervisors professor David Aschman and Dr. Simon Muller for their encouragement and patience throughout the duration of this essay. I would also like to thank Dr. Sam for reading my essay and oﬀering valuable advice, helpful guidance and constant assistance. I would also like to thank Anahita for her continuous support throughout the year. I am also gratetful to Irvin and Alfred who have made my days at AIMS so wonderful and cheerful. I wish to extend my gratitude to Prof. Neil Turok and Prof Fritz Hahne and all the tutors and lecturers. 26 Bibliography [1] W.E.Burcham, Elements of Nuclear Physics, 1981. o [2] Sven G¨sta Nilsson and Ingemar Ragnarsson, Shapes and Shells in Nuclear Structure, 1995. [3] B.R.Martin, Nuclear Models, 1996. [4] John Lilley, Nuclear Physics, 2001. [5] B.R.Martin, Nuclear and Particle Physics, 2006. [6] R .J. Blin-Stoyle, 1991. Nuclear and Particle Physics. [7] D.Bazzocco, B.Cresswell, J.Gerl, and W.Korten. pages 26–58. Advanced Gamma-ray Track- ing Array, 2001. [8] http://en.wikipedia.org. [9] http://hyperphysic.phy-astr gsu.edu/quantum/comptint.html. [10] http://images.google.co.za. [11] http://www w2k.gsi.de/agata/setup.htm. 27