Andrew Spencer Jamieson. A Thesis. Presented to. The University of Guelph. In partial fulfilment of requirements. for the degree of.

Transcription

1 The nuclear ucture of 11 Cd studied through the 111 Cd( d,p 11 Cd single neutron transfer reaction by Andrew Spencer Jamieson A Thesis Presented to The University of Guelph In partial fulfilment of requirements for the degree of Master of Science in Physics Guelph, Ontario, Canada c Andrew Spencer Jamieson, January, 14

2 ABSTRACT The nuclear ucture of 11 Cd studied through the 111 Cd( d,p 11 Cd single neutron transfer reaction Andrew Spencer Jamieson University of Guelph, 14 Advisors: Professor Paul E. Garrett The cadmium isotopes have been cited as excellent examples of vibrational nuclei for decades, with multi-phonon quadrupole, quadrupole-octupole, and mixed-symmetry states proposed. A large amount of spectroscopic data has been obtained from a variety of experiments, recently focused on γ-ray studies. In the present work, the single particle ucture of 11 Cd has been investigated using the 111 Cd( d,p 11 Cd reaction. The high energy resolution investigation was carried out using a MeV beam of polarized deuterons obtained from the Maier-Leibnitz Laboratory at Garching, Germany. The reaction ejectiles were momentum analyzed using a Q3D magnetic spectrometer, and 19 levels have been identified up to 4. MeV of excitation energy. Spin-parity has been assigned to each analyzed level, and angular diibutions for the reaction cross sections and analyzing powers were obtained. Optical model calculations have been performed, and the calculated angular diibutions were compared with the experimental cross sections and analyzing powers. Many additional levels have been observed compared with the previous (d,p study performed with 8 MeV deuterons, including ongly populated 5 and 6 states. The former was previously assigned as a member of the quadrupole-octupole quintuplet, based on a ongly enhanced B(E value to the 3 state, but is now reassigned as being predominately s 1/ h 11/ two-quasineutron configuration.

3 Acknowledgements A number of people deserve my sincerest gratitude for helping me complete this work. Above all, I would like to thank my parents, Elaine Ruddock and Lindsay Jamieson, for their support and encouragement throughout my life. My supervisor professor Paul Garrett has provided me with invaluable opportunities and guidance over theyears. Ithankhimforhispatienceandthesupporthehasofferedme. Several individuals provided me with insights and assistance on this project. I would like to thank Dr. Ian Thompson for his help with the code FRESCO, and Dr. Filomena Nunes for bringing the Adiabatic Distorted Wave Approximation to my attention. CollaborationwithDr. KyleLeachwasalso immensely helpful tome, andiwould like to thank him for uggling with me to sort out various confusions over conventions. I would like to thank Dr. Chandana Sumithrarachchi for his assistance compiling various programs when I began this project. It has also been an honour to work with Dr. Carl Svensson. Finally I would like to thank my friends and colleagues in the nuclear physics group at the University of Guelph including: Laura Bianco, Vinzenz Bildstein, Michelle Boudreau, Sophie Chagnon-Lessard, Greg Demand, Alejandra Diaz Varela, Ryan Dunlop, Paul Finlay, Baharak Hadinia, Badamsambuu Jigmeddorj, Alex Laffoley, Andrew MacLean, Julian Michetti-Wilson, Andrew Phillips, Allison Radich, Evan Rand, and James Wong. iii

4 Contents Acknowledgements iii 1 Introduction Collective ucture in vibrational nuclei Intruder states Single nucleon transfer reactions Distorted Wave Born Approximation DWBA for (d, p reactions The optical model potential The adiabatic approximation Experiment 7.1 The Maier-Leibnitz Laboratory (MLL The Ion source and tandem Van de Graaff accelerator The Q3D Energy loss of charged particles in a gas and proportional counters Momentum analysis and particle identification Beam particle normalization and dead time corrections Off-line data analysis iv

5 3 Analysis Peak fitting Energy calibrations Cross sections and analyzing powers Elastic scattering data DWBA and ADWA calculations Results Angular diibutions Spectroscopic engths Evaluation of the level scheme kev level kev levels kev level kev level kev level, unobserved kev levels kev level kev level kev levels kev level kev levels kev level kev levels kev levels kev level v

6 kev levels kev levels kev levels kev levels kev level kev levels kev level kev level Sum rules Conclusion 99 Bibliography 11 Appendix A: Angular Diibutions 15 vi

7 List of Tables.1 Values from the high-momentum bite runs, polarization up Values from the high-momentum bite runs, polarization down Values from the low-momentum bite runs, polarization up Values from the low-momentum bite runs, polarization down Values from the elastic runs, polarization up Values from the elastic runs, polarization up Deuteron optical model parameters Proton optical model parameters Deuteron adiabatic optical model parameters High-momentum bite spectroscopic engths and spin-parities Low-momentum bite spectroscopic engths and spin-parities Sums of engths from l=-5 transfers Strengths from 111 Cd(d,p 11 Cd and 111 Cd(d,t 11 Cd vii

8 List of Figures 1.1 Partial level scheme for a quadrupole-octupole vibrational system Partial level scheme of assigned phonon states in 11 Cd Coordinates involved in the 111 Cd(d,p 11 Cd reaction Nilsson states for neutrons above N= Nilsson states for neutrons above N= Wood-Saxon form volume potential well Wood-Saxon derivative surface potential well Layout of the Maier-Leibnitz Laboratory Schematic diagram of the ion source Schematic diagram of a tandem Van de Graaff accelerator Accelerator and Q3D Geometry of the opening to the Q3D Schematic diagram of the Q3D magnetic spectrometer Schematic diagram of the ip-cathode detector Ungated and gated histograms of energy losses Ungated and gated histograms of energy losses Histograms of the total integrated current on anode wires viii

9 3.1 Sample of fit data from FitPic High-momentum bite energy calibration curve Low-momentum bite energy calibration curve Lab frame and centre-of-mass frame kinematics Deuteron elastic scattering angular diibutions Proton elastic scattering angular diibutions Sample FRESCO input Single particle calculations for even parity states Single particle calculations for odd parity states Single particle calculations for a 3 + state Mixed single particle calculation for a 3 + state Sample spectra of observed 11 Cd levels Angular diibutions for E = -.6 MeV Angular diibutions for E = MeV Angular diibutions for E = MeV Angular diibutions for E = MeV Angular diibutions for E = MeV Angular diibutions for E = MeV Sums of ength from each j-transfer value ix

10 Chapter 1 Introduction Cadmium isotopes, and 11 Cd in particular, have been described for decades as excellent examples of nuclei that exhibit collective vibrational excitations. Many of the low lying levels of 11 Cd have been interpreted as quadrupole, octupole, and coupled quadrupole-octupole vibrations of the nuclear surface [1]. Some of the evidence in favour of this interpretation will be discussed below. The focus of the current work is to determine whether high resolution spectroscopic data from the ( d,p reaction is consistent with the vibrational interpretation of low lying levels in 11 Cd. Before discussing the details and results of the single neutron transfer experiment, a brief overview will be given in this chapter of nuclear ucture, direct nuclear reactions, and how these reactions are used to investigate nuclear ucture. Describing the ucture of a given nucleus requires solving a quantum many-body problem for a certain number of interacting protons and neutrons. The interaction between nucleons is a residual force originating from the ong nuclear force that binds quarks together. The nucleon-nucleon potential arises from the more fundamental ong nuclear potential in an analogous way to how interatomic forces originate from the Coulomb potential that binds electrons to an atomic 1

11 nucleus. However, unlike electrodynamics, the ong nuclear interaction is not perturbative at low energies, and it is not yet possible to calculate the nucleon-nucleon interaction directly from quantum chromodynamics. Even if such a calculation were possible, the size of the model space for all but the light-mass nuclei is too large to make calculations tractable. A phenomenological model is used to describe single particle states within a nucleus. This model is called the shell model, although there are various realizations of the shell model, each based on a different nucleon-nucleon potential. In general, shell model potentials must reproduce the observed features of nucleon-nucleon interactions, including the observation that they are ong only within a short range, (on the order of 1 15 m, of the nuclear centre of mass, and that they involve a spin-orbit coupling. Shell model states are labelled by total angular momentum, parity, and a principle quantum number associated with the energy of a state. The nuclear shell model has been used to successfully predict a variety of uctural features in nuclei, including the nucleon magic numbers at which maxima in the separation energy for protons and neutrons are observed. The shell model also provides very accurate predictions for the ucture of light nuclei and nuclei near closed shells. For heavier nuclei with a large number of valence nucleons, the shell model approach is impractical. As the number of nucleons increases, the number of states accessible for configuration mixing renders shell model calculations intractable. This is especially the case in even-even nuclei (with even numbers of both protons and neutrons, where pairing correlations dramatically increases the size of the model space. Instead of taking the single particle perspective, collective models of nuclear ucture can be used to describe heavy nuclei. The collective model, in contrast to

12 the shell model, focuses on the behaviour of the nucleus as a whole. One of the earliest and most influential models of collective nuclear ucture was the liquid-drop model. In this model, the nuclear medium is treated as a continuous fluid, which is incompressible at low energies. A nucleus is described as a spherical droplet of this fluid. Collective excitations of the nucleus result in either static or dynamic deformations of the spherical nuclear shape. For instance, a collective excitation in the nucleus can result in the size of the nucleus increasing along one axis while becoming smaller along the other two axes (a prolate deformation, or one axis can becoming smaller than the other two (an oblate shape. If deformation occurs, the spherical symmetry is broken and rotational degrees of freedom become available. Also, a superposition of deformation states can occur, which is a situation referred to as shape coexistence. On the other hand, since the nuclear medium is bound together by an attractive force, it possesses surface tension on which quantized, vibrational modes of excitation can occur. Each of these examples of collective excitation can occur with signature observable consequences for the energy spacings, spin-parities, lifetimes, and branching ratios of excited states. The collective model is thus a useful tool for the interpretation of spectroscopic data in nuclear physics. 1.1 Collective ucture in vibrational nuclei Since the current work focuses on a nucleus that has been regarded as an excellent example of vibrational ucture, the particular signature observables of a vibrational system will be discussed here. The details presented here, and below in the section 3

13 on intruder states follows the presentation found in Nuclear Structure from a Simple Perspective, by Richard F. Casten []. Vibrational excitations in a nucleus can be described by treating the nuclear radius as if it oscillates in a time-dependant way around some average value. The oscillation can be expanded as a series of spherical harmonics: λ R(t = R av + a λµ (ty λµ (θ,φ (1.1 λ= µ= λ where the time-dependant ucture coefficients, a λµ (t dictate the excitation energies of different modes of oscillation. The indices, λ and µ label the orbital angular momentum and one component of the orbital angular momentum vector for an oscillation mode. A mode carrying a given value of orbital angular momentum is referred to as a λ -pole phonon. The first mode is a monopole phonon, which represents purely radial oscillations. Since the nuclear medium is incompressible to a good approximation at low excitation energies, this mode is not relevant for the current work. The second mode is a dipole phonon, which corresponds to an excited state in which the protons oscillate completely out of phase with the neutrons, and the centre-of-mass of each type of nucleon oscillates sinusoidally around the nuclear centre-of-mass. The dipole vibrational excitation is also only relevant at much higher excitation energies than those discussed in the current work The quadrupole and octupole phonons, with λ = and λ = 3 respectively, are the next two vibrational states of excitation in the multipole expansion. As a result of the pairing interaction between nucleons, which tend to couple together to zero angular momentum, the ground state of an even-even nucleus is a + state. The first excited 4

14 4 hω } quadrupole-octupole multi-phonon quintuplet 3 hω 3 } one octupole phonon three quadrupole phonons } hω } two quadrupole phonons 1 hω + } one quadrupole phonon + Figure 1.1: Partial level scheme for a quadrupole-octupole vibrational system, with single and multi-phonon quadrupole and octupole modes, and a mixed quadrupoleoctupole excitation state is a + state, which is interpreted in the vibrational model as a quadrupole phonon excitation. If another quadrupole phonon were excited within the nucleus, a set of degenerate states would occur at twice the energy of the single quadrupole phonon level, with spin-parities: +, +, and 4 +. These degenerate states are simply all possible results of adding the angular momenta of two indistinguishable spin- particles. With an additional quadrupole phonon, there would be a set of degenerate states with excitation energies at three times the one quadrupole phonon level, with spin-parities: +, +, 3 +, 4 +, and 6 +. It is possible to continue on this way, adding single quadrupole phonons to the system, working out the addition of angular momentum for n quadrupole phonons to obtain a set of states at n times the single quadrupole phonon excitation energy. Thus a purely harmonic quadrupole vibrational spectrum would contain evenly spaced excitation levels with a signature set of spin-parity values at each level. Consider the situation in which a single octupole phonon state occurs at the same energy as that of three quadrupole phonons. This situation is shown in Figure

15 4 hω } quadrupole-octupole multi-phonon quintuplet 3 hω 3 5 } one octupole phonon } three quadrupole phonons hω two quadrupole 131 } phonons 1 hω } one quadrupole phonon + Figure1.: Partial level scheme of assigned phonon states in 11 Cd for low lying states below 7 kev, all energies are given in kev, only those states that have previously been interpreted as phonon states are shown Note that the single octupole phonon energy need not be related in this way, or in any other particular way, to the single quadrupole phonon energy. The 3 phonon can couple with a single quadrupole phonon, resulting in a set of states at four times the single quadrupole phonon level, with spin-parities: 1,, 3, 4, and 5. This set of states is referred to as the quadrupole-octupole quintuplet, and is a signature of a multi-phonon, quadrupole-octupole vibrational system [3]. Some of the low lying states of 11 Cd exhibit the energy spacing, and spin-parities that are characteristic of a quadrupole-octupole vibrational system. Comparing Figure 1.1 with Figure 1., which displays the levels of 11 Cd that have previously been assigned as vibrational states, there is a remarkable resemblance between the two level schemes. However, some states have been omitted from the partial level scheme presented in Figure 1., including the 14 kev + level, the 1469 kev + level, and the 1871 kev 4 + level. These omitted states require an interpretation that cannot be given by the vibrational model alone. 6

16 1. Intruder states One of the prominent features of the nuclear interaction is nucleon-nucleon pairing. Protons and neutrons often form j = pairs within the nucleus, in p-p, n-n and p- n configurations. The proton-neutron interaction is particularly important for the understanding of nuclear shape deformations and intruder states. An intruder state can be loosely understood as an excitation that appears to have a significantly lower energy than one might have naively expected. These types of excitations are prevalent in nuclei near closed shells. It requires a considerable amount of energy to promote valence nucleons near a shell gap into states above the shell gap. However, for isotopes such as 11 Cd, the binding energy associated with pairing interactions can lower the energy of these excitations significantly. In 11 Cd there are 48 protons, which is two protons away from the Z=5 shell closure. In this case we can consider 11 Cd to consist of two proton holes that can interact with valence neutrons. Since there are N = 64 neutrons, this leaves 14 valence neutrons that occupy the N=5-8 shell. By promoting a pair of protons across the shell gap, this leaves behind four proton holes below the Z=5 shell closure, but also means that there are now two proton particles within the Z=5-8 shell, which can interact with the valence neutrons there. In essence, in this excited state there are now six valence protons (two particles, four holes, that can interact with the valence neutrons. The binding energy gained from interactions or correlations between these protons and the valence neutrons greatly lowers the energy of the excitation, such that it lies much lower in the level scheme than might have naively been predicted from the shell model. The pairing interaction of valence nucleons, as well as the number of accessible states available for valence nucleon pairs to occupy, can explain why there are a 7

17 number of low lying intruder states interspersed among the vibrational excitations of 11 Cd. However, this characterization of intruder states suggests that E transitions between intruder states and vibrational states should be forbidden. Observation of such forbidden transitions can be explained by mixing between intruder configurations and phonon levels. Depending on how close the unperturbed levels are in energy before mixing, the amount of mixing could be quite large. 1.3 Single nucleon transfer reactions A deformed nuclear state can be described using the Nilsson model. The wave function of a deformed nucleus is described within the Nilsson model as a superposition of states from the spherically symmetric shell model. The amplitudes for the shell model states that make up a Nilsson model wave function depend on the nuclear deformation parameters, such as the static quadrupole and octupole deformations. In general, any excited nuclear wave function will involve configuration mixing between multiple Nilsson model states []. In order to probe these nuclear wave functions, it is desirable to selectively populate nuclear states in a way that only samples a certain component of the wave function. Under the right experimental conditions, transfer reactions provide an excellent tool for selectively populating nuclear states in a way that probes specific wave function components. The experimental conditions that make it possible to selectively probe components of a nuclear wave function are ones in which direct nuclear transfer reactions are favoured. A nuclear reaction is considered to be direct if the configuration of the final state nucleus does not greatly differ from the configuration of the initial nucleus. In a direct reaction, one or two nucleons might change their 8

18 state, leaving the rest of the nucleons in a configuration that is similar to their initial state. The direct reaction happens as a single-step process, without intermediate configurations occurring between the initial and final state. A compound nuclear reaction, on the other hand, involves a final state in which many nucleons have altered their configurations, which can happen in a multi-step process with one or more intermediate states. Although the concepts of direct and compound reactions are idealizations, and any physical reaction will be both direct and compound to some extent, it is still possible to create experimental conditions under which direct reactions are favoured. For instance, a single nuclear transfer reaction can happen in a number of different ways. For the (d,p reaction to occur, a deuteron must approach the field of the target nucleus. When the neutron is transferred from the deuteron, the proton could also inelastically scatter off of the resulting nucleus, or Coulomb excitation could occur. However, both of these processes involve two steps: transfer and then inelastic scattering, transfer and then Coulomb excitation. In a direct (d,p reaction, the only thing that happens is the transfer. The cross sections for the direct and compound parts of the reaction fluctuate with scattering angle. However, with the target mass and beam energy used in this experiment, the direct reaction channels are favoured over compound processes. There is dependence on both the target mass and the deuteron energy, but typical (d,p reactions are performed at energies of around 1 to MeV for nuclei with A > 1 [4]. 9

19 1.4 Distorted Wave Born Approximation In quantum mechanical scattering theory, information about observables is stored within the S-matrix. The S-matrix is given by the overlap of incoming and outgoing states: S αβ = β OUT α IN (1. In the formalism of quantum scattering theory, the Hamiltonian is separated into a free part and an interacting part. The free Hamiltonian gives rise to free particle states, which are wave packets formed from appropriately normalized plane waves. The S-matrix can be defined as the inner product of free particle states with a scattering operator: β OUT α IN = φ OUT S φ IN (1.3 The only physical outgoing states that have non-zero overlap with incoming states are the ones that conserve energy, E i = E f. One possibility is that the outgoing state is identical to the incoming one, so the S-matrix can be shown to have the form: S k,k = δ 3 ( k k iπt k,k (1.4 where the delta-function accounts for events in which no interaction occurs, projectiles simply pass though the target without scattering. The matrix T k,k is called the transition matrix, and contains all information about events in which the projectile interacts with the target. The transition matrix can be expressed as the inner product of an incoming scattering particle wave function and an outgoing plane wave with the 1

20 interaction potential [5]: T k,k = φ k V ψ (+ k (1.5 Since the time scale and distance scale separating the initial state from the moment of interaction are much larger than the time scale and distance scale involved in the interaction event, the incoming state may be treated as coming from asymptotically far away in the asymptotic past. Similarly, the out-going states may be treated as going asymptotically far away into the asymptotic future from the interaction. The wave functions that satisfy Schrödinger s equations with the full Hamiltonian are assumed to have the asymptotic form of an incoming plane wave plus an outgoing spherical wave [4]: ψ (+ k (r = A (e i k r +f(θ eirk r (1.6 where A is the normalization and f(θ is the scattering amplitude. Using the Lippmann-Schwinger equations, it can be shown that the transition matrix is proportional to the scattering amplitudes (the constant of proportionality depends on how the plane waves are normalized [5]. Here we will adopt a normalization convention that makes the transition matrix equal to the scattering amplitudes. If the interaction potential is weak, the scattering particle wave function can be expanded in a perturbative series. The first-order approximation to the scattering wave functions are the free-particle plane waves, this approximation is called the first Born approximation. At this order, the scattering amplitudes are: f(θ = φ k V φ k (1.7 11

21 Since nuclear interactions are not weak at small distance scales, the first Born approximation is not generally a good description of elastic and inelastic scattering events. A better approximation can be obtained by separating the interaction potential into two parts, one that involves onger, nuclear and Coulomb interactions, V, and one that involves weaker interactions that can be dealt with perturbatively, W. In this case, the ong interaction potential is included with the free Hamiltonian, which is now the nonperturbative part of the Hamiltonian. The separation of the Hamiltonian in a perturbative and a nonperturbative component gives rise to distorted wave functions satisfying the Schrödinger equation: (E k K V χ (± k = (1.8 where E k is the energy of the particle with wave vector k, K is the kinetic energy operator and V is the nonperturbative part of the potential. The transition matrix, expressed in terms of the distorted plane waves is [5]: T k,k = χ ( V φk k + χ ( W ψ (+ k k (1.9 The ong potential, V, is chosen to describe elastic scattering, so that information about inelastic events is given by the second term of the above expression. The analog of the first Born approximation is to replace the scattering wave function in the transition matrix with a distorted plane wave, so that the inelastic part of the transition matrix becomes: Tk,k inel = χ ( W χ (+ k k (1.1 1

22 This is called the Distorted Wave Born Approximation (DWBA. This approximation is valid if the dominant contribution to the transition matrix is from elastic scattering events, so that non-elastic events may be treated perturbatively. In order to compute the transition matrix element for a nucleon being transferred into a given single particle state, the distorted waves are calculated under a partial wave expansion. The nuclear potentials are short-ranged and this makes the partial wave expansion converge very quickly, since larger values of angular momentum correspond to larger impact parameters. Projectiles with large impact parameters are not influenced ongly by the nuclear interactions that result in the transfer, and so a large number of terms in the partial wave expansion is unnecessary. 1.5 DWBA for (d, p reactions For DWBA calculations involving a general scattering process, A(α,βB, the initial state is a product of the internal wave functions of the target and projectile, and the distorted plane wave for the projectile external wave function. The internal wave functions are obtained from the binding potentials of the target and the projectile respectively. The distorted plane wave is obtain by solving Equation 1.8 with a potential V α, which describes the interaction between the incoming projectile and the target. Similarly, the final state is a product of the internal wave functions of the residual and the ejectile, and the outgoing distorted wave [6]. The distorted wave is obtained by solving Equation 1.8 with the potential V β, which describes the interaction between the residual and the ejectile. 13

23 For (d,p reactions, the system may be treated as containing a binding potential for the neutron within the deuteron, a binding potential for the transferred neutron in the residual nucleus and a core-core interaction between the proton and the target nucleus. The Hamiltonian is [5]: H = K R +K r +V d ( r+v t ( r +V cc ( r c (1.11 where R is the relative positionof the deuteron andtarget nucleus and r is the relative position of the neutron and proton that comprise the deuteron. Since the sum of initial kinetic energies is equal to the sum of final kinetic energies, the Hamiltonian may also be equivalently written: H = K R +K r +V d ( r+v t ( r +V cc ( r c (1.1 where R is the relative position of the outgoing proton and residual nucleus and r is the relative position of the residual nucleus core and the transferred neutron. The coordinates involved in the transfer are shown in Figure 1.3. Due to recoil effects during the transfer, the position vectors R and R are not equal [4]. These two ways of writing the Hamiltonian are called the prior and post form respectively. In order to perform DWBA calculations, the interaction must be written in terms of a potential, V(R, which describes elastic scattering and a potential U( R, r, which can be treated perturbatively and is responsible for the transfer events. By writing the initial bound-state deuteron Hamiltonian as H d = K r + V d ( r, the prior form Hamiltonian is: H = K R +V(R+H d +U( R, r (

24 n r r R R 111 Cd p r c Figure 1.3: Coordinates involved in the 111 Cd(d,p 11 Cd reaction, unprimed coordinates refer to positions prior to the reaction, and primed coordinates refer to positions after the reaction By comparing Equation 1.14 with Equation 1.13, we can identify the perturbative potential: U( R, r = V t ( r+v cc ( r c V(R (1.14 The first termintheabove potential is abinding potential, while theother two are collectively referred to as the remnant. The two terms of the remnant are of similar order of magnitude, and are sometimes neglected to simplify the transition matrix calculation. However, by making a zero-range approximation, the transition matrix calculation is greatly simplified. Under the zero-range approximation, the product of the perturbative potential and the internal deuteron wave function is proportional to a Dirac delta-function [4]: φ d ( ru( R, r = D δ 3 ( r (1.15 The factor, D, is calculated based on models of the deuteron. The standard value is D = MeV fm 3 [5]. Investigations of zero range and finite range calculations 15

25 have demonated that the differences between the two are negligible at the beam energy of this experiment [7]. The quantity that we are interested in extracting using the DWBA calculation is the spectroscopic ength. Spectroscopic engths are defined in the current work to be the reduced matrix element of the operator which adds a single neutron to the l j single particle orbital: S lj = Ψ A+1 J f a lj J i +1 Ψ A J i (1.16 where the factor of J i +1 is conventional, and J i is the spin of the initial nucleus. With this definition, spectroscopic engths for single neutron ipping reactions have the same form as ones from pick-up reactions. In general, the neutron is transferred into a superposition of available single particle states. The single particle states are given by the shell model. There are 64 neutrons in 11 Cd, which is between the 5 neutron and 8 neutron shell gaps. For 11 Cd, the available states that are predominantly occupied by the transferred neutron are the 3s1, d3, d5, 1g7, and 1h11, which are below the 8 neutron shell gap shown in Figure 1.4. There are also contributions from states above the N=8 shell gap. These contributions become more important at higher excitation energies, above 3 MeV. The states above the N=8 shell gap include 3p1, 3p3, f5, f7, 1h9, and 1i13, shown in Figure 1.5. The measured differential cross section is a sum over contributions from these states: dσ = lj ( J i +1 dσ J f +1 S lj lj,f RESCO (

26 where (lj denotes the amount of transferred orbital and total angular momentum, J i and J f are the total angular momenta of the initial and final states respectively, and FRESCO is the program that was used to calculated the single particle cross sections. With this definition, the spectroscopic engths do not depend on the spins of the initial and final states. The coefficients, S lj, are the spectroscopic engths for each of the single particle states. These engths are obtained by scaling the angular diibutions from DWBA calculations to the measured angular diibution of the differential cross section. The spectroscopic ength is a measure of how ong the transfer into a given single particle state is. It is typical for one single particle state to dominate the transfer into a given level, so it is common to report spectroscopic engths for only these dominant single particle states [4]. Summing all of the ength from a single particle orbital from both a single neutron ipping reaction and a single neutron pick-up reaction on the same target nucleus should add up to the degeneracy of the single particle level [8]: S lj + S lj = j +1 (1.18 pick up ipping It should be noted that another convention for spectroscopic engths is frequently used in the literature. Sometimes the engths are defined: S lj = Ψ A+1 J f a lj J +1 Ψ A J i (

27 where the factor of J in the denominator is the spin of the heavier nucleus. With this convention, the spectroscopic engths for ipping reactions are related to those for pick-up reactions by: S ipping = J f +1 J i +1 S pick up (1. The sum rules using this convention are: pick up S lj + ipping J f +1 J i +1 S lj = j +1 (1.1 FRESCO includes the factor of (J f +1/(J i +1 in the single particle cross sections it outputs. 1.6 The optical model potential In order to perform the DWBA calculations for the single neutron transfer reactions, the interaction potentials are required. There are five interactions involved: the binding potential of the initial deuteron, the incoming deuteron interacting with the target nucleus, the outgoing proton interacting with the residual nucleus after the transfer, the binding potential for the transferred neutron, and the core-core interaction. In order for the transfer to occur at all, the interaction potentials must be complex valued. If the potential were real, only elastic scattering would occur. The imaginary part of the potential serves to remove current from the elastic scattering reaction channel [5]. 18

28 å ä ã âá å à æ âá å à ç æá àß Þ Ö Õ Õ ÔÓ Ö Õ Ó Ö Õ Ô Ó ù ÒÑ Ð Ö Ð ù Ó àß Þ àß ã ÒÑ Ð ÒÑ Ð ÒÑ ø ï ðñòóô õ ôö Æ È Â Ã Ä ÅÆÇ ± ² «É ÊËÌÍ Í ÎÏ» ¼½¾ ½À Á èéêëìíêî úûüýþüúÿ Þßàáâããä ª å æ ç è é ååê e fghi j jk øùúûüüýþ &'(&*(+ ƒ c ^^_`ab bd ˆ Š q r O PQR S TUV l m n opq Ý u v w x yxz Ÿ Ÿ Ž Ï ØÙÚÛ Ð Ñ ÒÓÔÑ Ü Ý Õ B ±² 9:; <= ;>? G HIJ K LMN \ [] W WXYZ[ ÁÂÃÄÅÆÃÇ ƒ ˆ ÈÉÊ Ë ÌÍ Ê Î º»¼½¾ À ëìíîïëë𠪫³ µ ¹º s tuvw w xy z{ }~zz Ö Ø Ù Ú ÛÚÜ œ ž Ÿ Š Œ Ž š ÿ, -. /,. 1 š œ ñ ò ó ôõ õ ö ³ µ µ ¹! "# z{ } žz / { } ~ (*+(,*- $$% JKLMNJOP CDEFGGHI lmnopqrs XYZ[\ZZ] defghijk t u v w x yxz ^_`ab``c Q R S T U QVW :5; < = AB!" #$%& ' Figure 1.4: Nilsson states for neutrons near the N=5 and N=8 shell gaps [9] A 19

29 W V U TS W Z Y TS Ž Ž h g f fe ~ } Œ Š Ž h b g ie Š Œ Š W Y Ì S P RQ U y RQ P RQ X Œ Š dc b {z y Œ Š dc b {z y {z } Ÿ š œ ž!" #$ Ü ÝÞßà á Üâ CDEFGEEH j klmn o jp ìíîïðñòó ¹º» ¼ ºº½ I JKLM N KO ƒ `_a [\]^_ [ š ã é ã ä å æçè Ž ê Ø Ù Ú ÛÜ Ý ¾ ÀÁ  ÃÄ ˆ ±²³ µ C A % &'( <=>?@AAB * +, ôõö øøùú '(*+,- ÆÇÈÉÆÊËÌ ±²³ K LMN O MPQ ÎÏÐÍÑÒÓ ù úûüý ¼ ± ½¾ Ô!"#Õ þ Í ùÿ ²³ µ ¹º» Ô "& é ë u tv å æ ç èéê cd` e f o pqrst ``ab wxyz{yw ˆ Š Œ Ž œ žÿ Y^_ Y YZ [ \] RSTUVTWX Õ Ö Ï Ð Ñ ÒÓÔ ª««ÀÁÂÃÄÁÅ ëìíîïëðñ < =>?@< A B ò óôõö òø É Ê Ë ÌÍÉ Ë Î ÅÆÇÈÍÅÎË qrstuvwx ÅÆÇÈÉÊÅË Ö ØÙÖÚØÛ ª«vwx y z{ v g hijkl m ln Ã Ä Å Æ Ã ÇÅÈ ˆ â ä Þ ß à áâã œ ž Ÿ C D E - F G./1 HHI } ~ 34 }. : ; /1 5 š : 3 4!" ¼ ½ ¾ # $%"& À ¾Á µ ¹ º» } ~ } ƒ Š Œ Ž ŠŒ û ü ý þÿÿ ðíîïòjòó 56789:7; ª««./ : ;<=>??@A BCDEFGHI Y Z [ \]^ ^_ T UV W VX RRS JKL M NOP Q `ab c d eef g h i j k lim Figure 1.5: Nilsson states for neutrons near the N=8 and N=16 shell gaps [9] B ' ( * +,+- n o p qrs t u!" #$%" & ƒ/1.. 4

30 Since the nuclear interactions involved in the transfer are not theoretically determined, a phenomenological potential, called the optical model potential, is used. The optical model divides the nuclear interaction into three parts: interactions with the nuclear volume, interactions that dominate at the nuclear surface, and a spin-orbit interaction that also occurs at the nuclear surface. The volume interaction is described by a Wood-Saxon form, which involves a well-depth, a radius that cuts off the interaction at radial distances outside of the nuclear surface, and a diffuseness [1]. The Wood-Saxon form, plotted with typical deuteron parameters in Figure 1.6, resembles a square potential well with tapered walls. This form qualitatively captures the short-range nature of the nuclear interaction. To obtain a surface-dominant interaction, the derivative of a Wood-Saxon form is used. A plot of the surface potential is also shown in Figure 1.7 with typical parameters for a deuteron. The optical model potential has the form [4]: V(r = V VR f(r;r R,a R iv VI f(r;r I,a I +i4v SI a I df dr (r;r I,a I +V SO λ π r df dr (r;r SO,a SO l s+v C (r (1. where V C (r is the Coulomb potential, λ π is the pion Compton wavelength, λ π, and f(r;r,a is the Wood-Saxon form: f(r;r,a = df dr (r;r,a = 1 a 1 1+exp( r R (1.3 a exp( r R a (1+exp( r R a (1.4 1

31 potential well-depth (MeV potential well-depth (MeV radial distance (fm Figure 1.6: Wood-Saxon volume potential well, with well-depth V=-91.6 MeV, radius R=5.6 fm and diffuseness a=.75 fm 5 1 radial distance (fm Figure 1.7: Wood-Saxon surface potential well, with well-depth V=48.7 MeV, radius R=6.4 fm and diffuseness a=.87 fm Each part of the optical model involves a well-depth, a radius and a diffuseness. Since there are three parts of the optical model, each with a real and imaginary part, the optical model generally involves 19 parameters, which includes a radius for the Coulomb potential. These parameters are fixed by fitting a large amount of elastic scattering data, performing aχ minimization tooptimize theparameters. Ingeneral, the real part of the surface potential does not contribute and the imaginary part of the spin-orbit potential is negligible except at high beam energies [11]. It is typically the case that the imaginary parts of the volume and surface potentials have the same radius and diffuseness, although sometimes the real and imaginary parts of the volume potential are given the same radius and diffuseness and the surface parameters are left independent [1, 13, 14]. The number of relevant parameters is reduced to eleven. In order to optimize the parameters, each one is written as a function of nuclear mass, nuclear charge and beam energy. Different optical model potentials are obtained

32 by assuming different dependencies of the parameters on these properties. Global optical model parameter sets are obtained by optimizing the parameters over a large range of nuclear masses, nuclear charges and projectile beam energies. The task of measuring spectroscopic engths requires the evaluation of transition matrix elements using the DWBA with a standard set of global optical model parameters (OMPs. One of the advantages of using global OMPs is that, since the phenomenological potential is derived from elastic scattering data, the choice of which parameter set to use can be justified based on elastic scattering. By collecting elastic scattering data for deuterons on 111 Cd, the global deuteron OMPs that reproduce the experimental data best are the parameters that should be used to extract spectroscopic engths. The situation is similar for the interaction potential between the outgoing proton and the residual nucleus, and the core-core interaction. The global optical model potentials fix three of the five interaction potentials involved in the DWBA calculations. The other two are the deuteron binding potential and the final state binding potential. It is standard to give the deuteron binding potential a Gaussian form, and fix the well-depth so that it reproduces the known deuteron binding energy. Similarly, the final state binding potential is given a Wood- Saxon form, fixing the well-depth to the known neutron separation energy [15]. The radius and diffuseness parameters for the two binding potentials are given standard values, which are known to give good reproduction of experimental data; the results of the calculations are not very sensitive to these parameters [4]. It is possible to extrapolate global optical model parameter sets to negative energies. To do this the dispersive optical model is used, which associates an energy dispersion relation with the optical model parameters [16]. However, there is currently no justification for 3

33 extrapolating the spin-orbit part of the potential to bound states. The dispersive optical model was not used for the analysis in this work. 1.7 The adiabatic approximation There are some disadvantages from using a deuteron optical model potential to obtain the distorted wave in the incoming channel of the (d,p reaction. The deuteron optical model potential neglects the composite ucture of the deuteron, and thus neglects the deuteron break-up within the potential of the target nucleus. The interactions that result in the break-up are taken to be implicitly described within the global deuteron OMPs, from which the incoming distorted wave is calculated. However, the global deuteron OMP is obtained by fitting elastic scattering data. The elastic scattering data includes only asymptotic components of the wave function, whereas the mechanisms that result in the deuteron break-up are short-ranged interactions, which occur at distance scales smaller than the scale of the deuteron binding potential [5]. It is not clear that these break-up mechanisms can be well described with a phenomenological model that is based on elastic scattering. An alternative reaction model exists for the (d,p reaction that requires only minor changes to the DWBA calculation. By making an adiabatic approximation, the potential in the incoming channel can be approximated by the sum of two global optical model potentials, one for the proton and one for the neutron [17]. If the incoming wave function for the deuteron is written as the product of the deuteron internal, ground state wave function and a distorted wave function, φ (rχ(r, this wave function is only required for smaller values of r than the deuteron binding 4

34 potential radius, which is about 1 to fm. In this case, the zero-range approximation greatly simplifies the calculation of the incoming distorted wave, since the wave function is only required at r= under this approximation. The Schrödinger equation for the distorted wave becomes [5]: ( KR +V ad (R (E ǫ χ(r = (1.5 where V ad is the adiabatic optical model potential and ǫ is the deuteron binding energy. The adiabatic potential is: V ad (R = V p (R+V n (R (1.6 which is just the sum of the proton and neutron optical model potentials evaluated at the distance of the deuteron s centre-of-mass from the target nucleus, at half the energy of the deuteron. The adiabatic approximation assumes that the energy per particle of the deuteron beam is much greater than the energy associated with the relative motion of the protonandneutronwithinthedeuteron. ThisapproximationisvalidforE d >1MeV [17]. The calculation scheme, called the Adiabatic Distorted Wave Approximation (ADWA, was developed by Johnson and Soper in order to address poor reproduction of spectroscopic data from (d,p experiments, using DWBA with deuteron global OMPs at beam energies of around MeV [17]. One benefit of this alternative approach to DWBA calculations is that there are global OMPs that have been defined to work for both protons and neutrons, so the calculations can be carried out using only one phenomenological potential. However, the relationship between the adiabatic potential and the standard deuteron 5

35 potential is not well understood. Since deuteron elastic scattering data is no longer the basis for the phenomenological potential under the ADWA, the adiabatic approximation does not reproduce elastic scattering data well. In fact, the adiabatic approximation neglects the asymptotic form of the deuteron wave function, which contains the information about elastic scattering. Good reproduction of elastic scattering data cannot be appealed to for justifying the choice between different OMPs under ADWA. 6

36 Chapter Experiment.1 The Maier-Leibnitz Laboratory (MLL The Maier-Leibnitz Laboratory is located in Garching, Germany. The facility is jointly run by the Ludwig Maximillian University of Munich and the Technical University of Munich. The experimental data that was analyzed for the current work was obtained at the MLL, using the tandem Van de Graaff accelerator. A polarized beam of deuterons was produced at a beam energy MeV per deuteron. The deuterons were extracted from a Stern-Gerlach polarized ion source, with 8% polarization achieved. A layout of the laboratory is shown in Figure.1. The deuteron beam was incident on a target of 111 Cd. The target manufacturer reported the target thickness to be approximately 15 µg cm. The target was manufactured without a backing. Reaction ejectiles from deuteron scattering and transfer reactions with the target were detected and momentum analyzed using the Q3D magnetic spectrometer. A cathode-ip detector was used for particle identification and momentum analysis. 7

37 Data were collected for the single neutron transfer reaction, 111 Cd(d,p 11 Cd, and elastic scattering of deuterons off of 111 Cd. Elastic scattering data were collected at angles of the outgoing particle with respect to the beam axis from 15 to 1 in 5 increments. Spectra for the transfer reaction were collected over two different energy ranges at each angle. The lower excitation energy range (which will be referred to as the high-momentum bite included the ground state of 11 Cd and went up to about 4 kev. The higher excitation energy range (the low-momentum bite went from kev to 4 kev. The transfer data were collected at angles from 1 to 5 in 5 increments for both momentum bites, with additional data at an angle of 6 taken for the low-momentum bite. Two data sets were taken at each angle and momentum bite, one for each beam polarization.. The Ion source and tandem Van de Graaff accelerator At the front end of the accelerator, polarized ions were prepared by a Stern- Gerlach polarized ion source, which was developed by Hertenberger et al., and is described in [19]. The ion source produces ions in two stages. First, the atoms of the deuterium gas are singly ionized within ECR-plasma. This single electron ipping is accomplished with an efficiency of only a few percent. During the second stage, the deuterium ions are passed through a cesium vapour, where the deuterium ions pick up two electrons, resulting in negatively charged ions. The double charge pickup has an efficiency of about 3%. This two-step ionization process is much more efficient than a one-step single charge exchange process, and produces beams of around 1µA. 8

38 Figure.1: Layout of the Maier-Leibnitz Laboratory, adapted from [18] 9

39 Figure.: A schematic diagram of the ion source, adapted from [19] Atoms that are extracted into the ionization unit are polarized using permanent sextuple magnet FeNdB, with maximum magnetic field ength of about 1.4 T. In order to maintain polarization throughout the charge exchange process, the ionization occurs within a ong magnetic field, which is of the order of the hyperfine interaction ength. Polarizations of around 8% are achieved this way. Once polarized, the ions are accelerated towards the tandem Van de Graaff accelerator. At the centre of the accelerator there is a large electric conductor attached to a rubber belt. The belt transfers charge from the conducting plate, which electrostatically attracts the negative ions from the source. A beam line feeds ions from the source into the accelerator. At the centre of the conducting plate there is a ipper foil that the ions pass through after the first stage of acceleration. The ipper foil ips the deuterium ions of their two electrons, leaving only the deuterons on the far side of the foil. These deuterons are electrostatically repelled by the conducting plate, and so they undergo 3

40 Figure.3: A schematic diagram of a tandem Van de Graaff accelerator, adapted from [], note that a gas ipping channel is shown in this diagram, whereas a ipping foil was used in the accelerator at the MLL a second stage of acceleration, out of the accelerator. With the conducting plate charged to 11 MV, the deuterons exiting the accelerator tank have a kinetic energy of MeV, having accelerated through 11 MV twice..3 The Q3D The Q3D (Quadrupole-three-Dipole is a magnetic spectrometer that provides high-resolution spectroscopic data. The spectrometer can be rotated relative to the beamaxis, andiscentredontheaxisofthetargetinthemiddleofthetargetchamber. The rotation allows the Q3D to measure angular diibutions of spectroscopic data. A circular iron track surrounds the Q3D and a motor controls a wheel that attaches the Q3D to the iron track. The wheel rolls along the track, swiveling the Q3D around the target chamber. The track is etched with marks that label the angle of the Q3D 31

41 Figure.4: Accelerator and Q3D, schematically outlining the path ions follow through the MLL beam line, from the ion source to the Q3D focal plane detector, image is from [1] relative to the beam axis. Using the Q3D and these rotation mechanisms, angular diibutions of differential cross sections with respect to solid angle were measured. Reaction ejectiles produced in the target chamber are accepted into the Q3D through a small opening, which has adjustable slits to control the amount of current going into the detector. The positions of the vertical and horizontal slits determines the amount of solid angle covered by the detector during a run. Figure.5 shows the geometry of the opening to the Q3D. The total area of the Q3D opening is given by the area of the rectangle of sides X Y minus the area of the four corners that are blocked by the slits. X and Y are the distances between the centre of the opening and the horizontal and 3

42 Figure.5: Geometry of the opening to the Q3D, shaded area is the solid angle coverage of the Q3D, diamond shape is the Q3D aperture with width A = 31.5 mm and width B = mm vertical slits respectively. If A and B are half the height and width of the opening respectively, the total area of the opening is: ( Area = 4 X Y X A+ A Y ( Y B+ B X B A (.1 The slit settings are read by a micrometer, but they have to be corrected for the horizontal and vertical offsets. If D is the distance from the target to the opening, the solid angle coverage is given by: Ω = 1 ( [4 X Y X A+ A Y ( Y B + B X ] D B A (. 33

43 The horizontal slit settings also affect the angular acceptance of the Q3D. The effect of the finite angular acceptance on angular diibutions of differential cross sections is most noticeable when there is rapid change in the cross section with respect to angle. For instance, if there is a sharp minimum in the cross section at a specific angle, the finite angular acceptance of the Q3D will soften this minimum. This effect can be seen in the l = transfers, which are shown in the next two chapters. Such rapid rates of change in the differential cross sections are seen in the elastic scattering data at low angles. However, the horizontal slit settings are smallest here, due to the high rates, and so the angular acceptance is not very large for these data sets. The angular acceptance is: ( x θ =tan 1 D (.3 The range of angle covered by the Q3D is within θ ± θ. The distance from the centre of the target chamber to the opening of the Q3D is D = mm. The Q3D is composed of three dipole magnets, a quadrupole magnet and a multipole magnet. Figure.6 shows a diagram of the Q3D. The initial quadrupole magnet that particle s pass through is used for focusing purposes, to obtain an optimal resolution for the particular reaction ejectile of interest. The multipole magnetic was not used in this experiment. Charged particles that pass through the dipole magnetics are subject to a constant magnetic field that is perpendicular to their velocity. The trajectory of the charged particles passing through the dipole magnetic is circular, with a radius given by: ρ = p me qb = qb (.4 34

44 !"#" $!%$!"#"&% '(# *%+!#," *%#%'#, -(,#"'.% / 1 %$%,34 Figure.6: Schematic diagram of the Q3D magnetic spectrometer, adapted from [], showing the target chamber, the quadrupole and dipole magnetic fields, and the focal plane, note that the multipole magnetic was not used in this experiment where p is the particle s momentum, q is the particle s charge, and B is the ength of the magnetic field. By adjusting the magnetic field ength of the three dipole magnets, the momentum range covered by the detector can be controlled. The dipole field ength is chosen such that a particle of a certain type and with a certain energy will end up at the centre of the detector. This magnetic field ength can be calculated from the mass and charge of the particle of interest, and the radius associated with the centre of the detector. The Q3D has a focal plane that particles pass through after they have been steered through the dipole magnets. The position of particles on the focal plane is determined by their momentum and the magnetic settings of the Q3D. Since the radius of a particle s trajectory through a dipole magnet is proportional to the particle s momentum, higher momentum particles pass through the outer end of the 35

45 focal plane. A cathode-ip detector is placed at the focal plane for the purposes of particle identification and position measurement. Due to the curvature of the focal plane (the detector is not similarly curved, and cannot be since the curvature depends on the Q3D magnetic settings, only a limited portion of the focal plane is useful for momentum analysis without having to do sophisticating ray-tracing calculations. Also, the calibration relating a particle s position on the focal plane to the detected particle s kinetic energy is non-linear..4 Energy loss of charged particles in a gas and proportional counters A heavy charged particle moving through a gas will lose energy by ionizing the atoms it passes. The rate of energy loss with respect to distance travelled through a medium is called the stopping power of the medium. The stopping power can be calculated by treating each ionization event as Coulomb scattering of the heavy charged particle off of a stationary electron [3]. If we treat the interaction classically, the amount of momentum transferred is: p = Z pe 4πǫ bv (.5 For a charged projectile with atomic number Z p, velocity v, and impact parameter b. The amount of energy lost over a small distance can be calculated by considering a cylindrical ring of thickness db, which the charged particle interacts within over a 36

46 distance dx. The amount of energy lost by the charged particle is [3]: de = 4πZ azp ρ ( e db dx (.6 m e v 4πǫ b where ρ is the density of the gas. Since the amount of energy lost is inversely proportional to the square of the impact parameter, integrating this expression gives a formula for the stopping power of a gas in terms of the minimum and maximum amounts of energy loss that a charged particle can undergo travelling small distances through the gas: de dx = πz azpρ ( e ln m e v 4πǫ ( ǫmax ǫ min (.7 The maximum possible energy transfer is ǫ max = Mv, where M is the mass of the charged particle. The minimum energy loss is a property of the material, ǫ min = I, called the mean excitation potential. A typical approximation for the mean excitation potential is given by I = (1eVZ a. The following expression for the stopping power of a gas is the non-relativistic version of the Bethe-Bloch formula: de dx = πz azp ρ ( e Mv ln( m e v 4πǫ I (.8 Asthechargedparticletravels throughthegas, itcreatesionpairs. Thenumber of ion pairs created is proportional to the amount of kinetic energy lost to ionization. By integrating the current from the ion pairs, the total ionized charge can be determined, which is proportional to the energy lost by the charged particle in the gas. A gas chamber that measures energy loss in this way is called a proportional counter. There are two proportional counters in the detector used in this experiment, which measured 37

47 the energy losses of reaction ejectiles at two different stages within the detector [4]. The dependence of energy loss on both the Z p and the mass of the reaction ejectiles enables their identification..5 Momentum analysis and particle identification Upon entering the detector, particles pass through a kapton entrance foil. There is an isobutane filled gas chamber and a single anode wire between the entrance foil and the first cathode foil, which is made of mylar. Between the first cathode foil and the second foil, which is a segmented kapton cathode foil, there is a chamber also filled with isobutane at 5 mbar, with two anode wires [5]. The anode wires are positioned so that one is vertically above the other, and both run the length of the detector. The cathode ip foil is segmented into 7 ips, each 3 mm in width, with.5 mm gaps between them [6]. After crossing the cathode-ip foil, particles are stopped by a plastic scintillator. Charged particles ionize the isobutane when they enter the detector. Electrons from the ionization move towards one of the two anode wires, producing a charge avalanche. The charge avalanche induces a positive charge on the segments of the cathode-ip foil that are nearest to the position of the event. The amount of charge induced on each segment is monitored by individual preamplifiers and current integrating modules. In order for an event to be recorded, at least three adjacent segments must have an induced charge above the minimum threshold value set for the detector. A single event can have no more than seven adjacent segments charged above the threshold [5]. A histogram is obtained from the induced charge diibution on the cathode ip foil for every event. During online analysis, the 38

48 Figure.7: Schematic diagram of the focal plane ip-cathode detector, adapted from [6], ions enter through the opening on the left, and pass through the two gas chamber proportional counters before being stopped by the plastic scintillator spectra are obtained by a centre-of-gravity calculation from the charge diibution. However, this rough determination of the event position yields spectra that show the periodic ucture of the cathode ip foil. For offline analysis, the charge diibution histograms are fit with a Gaussian diibution, which yields spectra that are not affected by the ip detector s ucture. The two anode wires were used to monitor the detector s vertical position throughout the experiment. If the detector height is not correctly adjusted, particles might miss the plastic scintillator behind the gas chambers, and no rest energy measurement would be obtained for the event. The number of events on the two wires should be roughly equal during each run, for a properly centred beam and detector. It was sometimes the case that the amount of charge deposited on each wire would become unbalanced, as momentum settings and detection angle were changed, or the beam drifted. A large imbalance between the two anode wires 39