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Gamma-ray Tracking Arrays

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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

				
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