
RELATIVISTIC COULOMB EXCITATION OF THE PYGMY DIPOLE AND GIANT RESONANCES A Thesis by Nathan S. Brady Submitted to the O ce of Graduate Studies of Texas A&M UniversityCommerce In partial ful llment of the requirements for the degree of MASTER OF SCIENCE December 2016 RELATIVISTIC COULOMB EXCITATION OF THE PYGMY DIPOLE AND GIANT RESONANCES A Thesis by Nathan S. Brady Approved by: Advisor: Carlos Bertulani Committee: BaoAn Li William Newton Head of Department: Matt Wood Dean of the College: Brent Donham Dean of Graduate Studies: Arlene Horne iii Copyright c 2016 Nathan S. Brady iv ABSTRACT RELATIVISTIC COULOMB EXCITATION OF THE PYGMY DIPOLE AND GIANT RESONANCES Nathan S. Brady, MS Texas A&M UniversityCommerce, 2016 Advisor: Carlos Bertulani, PhD Nuclear collisions involving heavy charged nuclei traveling at relativistic speeds can bring one or both nuclei to an excited state. In collisions where the target is excited by the electromagnetic eld of the projectile the process is called Relativistic Coulomb Excitation (RCE). Classical approximations are a good starting point for describing RCE, however, more accurate predictions require a quantum mechanical treatment. Collisions which result in RCE have a certain probability to displace a large fraction of the nuclear matter inside a nucleus. This characteristic response is intrinsic to all nuclei con taining more than a few protons and neutrons and is called a Giant Resonance (GR). Many classical and microscopic models have been used to describe this phenomenon including zero range Skyrme forces, Mean eld Approximations, and the Random Phase Approximation (RPA). As the number of neutrons become much larger than the number of protons within the nucleus the excess neutrons begin to form a skin beyond the typical charge radius. This neutron skin can be excited through RCE and cause the skin to oscillate against the strongly bound symmetric core. The strongest oscillation between the core and neutron skin is called the dipole mode and the resulting phenomenon is referred to as the Pygmy Dipole Resonance (PDR). v This thesis will look at how the PDR is in uenced by the larger strength of the GR for interactions above 100 MeV/nucleon. As the dynamics of the collision changes, the probability of exciting a PDR changes considerably. The e ects due to the coupling of the PDR to the GR indicates a need for improved theoretical studies for reactions at these collision energies. vi ACKNOWLEDGMENTS I would rst like to thank my thesis advisor, Dr. Carlos Bertulani of the Physics Department at Texas A& M UniversityCommerce (TAMUC). He was an exemplary mentor and I consider him to be a true friend. His guidance helped me through the most trying time of my career. I would also like to recognize Prof. Dr. Thomas Aumann of the Physics Department at Technische Universitat Darmstadt. Without his collaboration, the work presented here would not have been possible. My future career has been enriched and made possible with the encouragement and support of both Dr. Matt Wood, head of the TAMUC Physics Department and Dr. BaoAn Li, Regents professor of TAMUC. The classes taught by each of them continue to be an invaluable resource as I move forward with my studies. The friends I have made since joining the physics department at TAMUC have been one of the most rewarding aspects of my life. In light of that, I would also like to acknowledge my fellow colleague, o cemate, and friend, James Thomas, who also worked under Dr. Bertulani. Lastly, I would like to acknowledge my family for their unwavering support and encouragement as I pursue my passion for physics. I must also thank Sarah Cantu for putting up with me through it all. Her support has been a major factor as I worked towards nishing this thesis. Thank you,  Nathan S. Brady Contents List of Figures x Chapter 1 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2 7 COULOMB EXCITATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Underlying Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Potential Fields and Gauge Transformations . . . . . . . . . . . . . . 8 2.1.2 Lorenz Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Nuclear Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.1 HeavyIon Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.2 Classical Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Semiclassical Theory of Coulomb Scattering . . . . . . . . . . . . . . . . . . 18 Multipole Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Relativistic Coulomb Excitation . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1 Equivalent Photon Method . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.2 CoupledChannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Chapter 3 29 NUCLEAR STRUCTURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1 Giant Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.1 Classical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 vii CONTENTS viii 3.1.2 Microscopic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Pygmy Dipole Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Chapter 4 38 PRESENTATION OF FINDINGS . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Chapter 5 48 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 References 48 Vita 53 List of Figures 2.1 Interactions between fermions can be represented by a Feynman diagram. The solid lines represent fermions and the wavy line is the force carrier, i.e., the virtual photon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 The interaction picture for a particle (lower solid line) colliding with a nuclear target (shaded circle), through the exchange of a virtual photon. The products are depicted by the outgoing solid lines. . . . . . . . . . . . . . . . . . . . . 13 2.3 Scattering of an incident beam of particles by a center of force. . . . . . . . . 15 2.4 A relativistic charged projectile incident on a target with impact parameter larger than the strong interaction radius. The transfer of virtual photons is shown (wavy lines) coming from the Lorentz contracted i.e., pancake shaped, electromagnetic eld. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.5 Example cross section for the GR in Lead208. . . . . . . . . . . . . . . . . . 32 3.6 The dipole mode of the pygmy resonance in the GoldhaberTeller (GT) and SteinwedelJensen (SJ) modes. . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.7 Strength function for the E1 RPA response in 68Ni calculated with formalism described in Ref. [43]. The calculation is performed for several Skyrme inter actions, shown in the gure inset. The arrow shows the location of the pygmy resonance. (For interpretation of the references to color in this gure legend, the reader is referred to the web version of this article.) . . . . . . . . . . . . 39 ix LIST OF FIGURES x 4.8 Coulomb excitation cross section as a function of the excitation energy of 600 MeV/nucleon 68Ni projectiles incident on 197Au targets. The lled circles represent the calculations using rstorder perturbation theory, while the lled squares are the results of coupledchannel calculations. . . . . . . . . . . . . 43 4.9 Coulomb excitation cross section as a function of the excitation energy of 68Ni projectiles incident on 197Au targets at two laboratory energies. The lled circles represent the calculations using rstorder perturbation theory, while the lled squares are the results of coupledchannel calculations. . . . . . . . 45 4.10 Coulomb excitation cross sections of the PDR as a function of the bombarding energy of 68Ni projectiles incident on 197Au targets. The lled circles represent the calculations using rstorder perturbation theory, while the lled squares are the results of coupledchannel calculations. . . . . . . . . . . . . . . . . 46 1 Chapter 1 INTRODUCTION Understanding the principles by which our universe is guided and the laws that restrict its evolution have always been a large motivation for scienti c discovery. Nuclear physics is the study of phenomena which govern the interactions between matter at the femtometer scale. These interactions are built up from the residue of the strong nuclear force, described by quantum chromodynamics, and are subject to electromagnetic interactions described through quantum electrodynamics. Particle collisions are used to investigate the dynamics of interacting particles. Mea surements are made by extracting information, (e.g., mass, momentum, spin, ...) carried by the products of these collisions and inferring the original conditions which produced the observations. The probability of producing a particular event is calculated through its cross section which tells us where to expect certain products produced by said event. Statement of the Problem In peripheral collisions the distance between the two nuclei under consideration are just large enough so as not to directly interact by the strong nuclear force. The event is then dominated by Coulomb interactions, and at relativistic speeds it is better known as Relativistic Coulomb Excitation (RCE), which is a wellestablished tool used for studies in nuclear structure [1]. During an RCE event, the electric eld generated by the incident heavy ion transfers energy to the nucleons within the other nucleus exciting it to a higher energy level. At certain excitation energies the nuclear response will be ampli ed considerably and is described as the characteristic resonance of the nucleus. These characteristic resonances which dominate the low energy excitations around 8 MeV and higher for all nuclei with more than a few nucleons (i.e., protons and neutrons) are known as Giant Resonances (GR). These phenomena, rst observed by Bothe and Gentner 2 in 1937 [2], were shown to have a high probability of photoabsorption at these energies. These phenomena are well studied, see for instance [3, 4] and theories, both classical and microscopic, reproduce the observations well. An associated phenomenon which appears in nuclei with more neutrons than protons is known as the Pygmy Dipole Resonance (PDR) and is the collective vibration of the neutrons against a symmetric nuclear core. Further reading on the pygmy resonance can be found in Ref. [5] Purpose of the Study The PDR was suggested in 1987 as a possible excitation in neutronrich nuclei by Kubono, Nomura, and collaborators [7]. Theoretical support was later established by Ikeda [8] and collaborators. Experimental evidences for the existence of a collective low energy response in neutronrich nuclei, far from the valley of stability, took nearly two decades more to emerge. Initially the direct breakup of light and looselybound projectiles, such as 11Be and 11Li were suggestive of a collective nuclear response, i.e., a PDR, but was later shown to be a direct Coulomb dissociation of the weaklybound valence nucleons [9]. This suggests that the characteristics are poorly understood and further investigations, both experimentally and theoretically, are needed. The intent of this study is to investigate the dynamics of these associated phenomena produced from RCE experiments. A coupling between the PDR and the various excited vibrational modes comprising the GR is investigated. The calculation of this coupling is important to investigate since the strength and energy of the PDR would be a ected appre ciably. This in turn would a ect the calculation of the neutron skin, which is characterized by the PDR. Hypothesis The Coulomb excitation of PDR at 100 MeV/nucleon and above are investigated. As the dynamics of the collision evolve, the question of the in uence on the PDR by the larger 3 GR is addressed. The coupling of the GR to the PDR will a ect the excitation probability considerably. Di erent collision energies will a ect the amount this coupling changes the cross section of the PDR. Research Questions An important aspect of investigating the PDR is the calculation of the neutron skin which tells us more about neutron matter and how it relates to neutron stars. We will attempt to answer the following questions: How does the collision energy a ect the centroid energy and width of the PDR? How much does this change the calculation of the neutron skin and polarizability? How do the separate major modes of oscillation of the GR (i.e., Giant Dipole and Giant Quadrupole) a ect the cross section of the PDR? Is there a preference to coupling with either of the modes? Signi cance of the Study A revitalized interest in neutron matter and nuclei far from the line of stability has emerged. The ability to measure the neutron radius has historically been an experimental challenge which stems from the neutral charge of the neutrons. Since the PDR has been shown to emerge in neutronrich nuclei it has been suggested that it could be used to constrain the neutron skin thickness [6]. Accurate measurements of the neutron skin and extraction of the dipole polarizability are important for constraining the symmetry energy associated with the binding of nucleons in the nucleus. How the GR a ects the PDR is then an important question; for small changes to the observed PDR the measurement of the neutron skin will be a ected. 4 Method of Procedure Since the objective was to determine how the multipole modes of the GR a ected the PDR cross section, we utilized current predictions of both the PDR and GR's centroid energy and width. Two practical examples of reactions, 68Ni + 208Au and 68Ni + 208Pb, are used to facilitate discussions on the e ect of coupling between the GR and the PDR. De nition of Terms Nucleon. The constituent particles of nuclei, i.e., protons and neutrons, which come together to form the nucleus of an atom. Elements di er by the number of protons, Z, and isotopes, elements with the same Z, di er by the number of neutrons, N. [3, 10] Nuclei. Term which refers to the nucleus of all elements composed of nucleons. [3,10] Di erential Cross Section. Rate of scattered particles detected at some angle, d =d , where is the solid angle. Interpreted as the probability of detecting a particle within the given solid angle. [3, 10] Cross Section. An e ective \area" which an interaction can occur. [3, 10] Coulomb Interaction. Class of interactions between particles through their mutual electromagnetic elds. [11] Nuclear Excitation. The observed reaction of nuclear matter to external stimuli resulting in the jump of one or more nucleons within the nucleus into a higher energy level. The nucleus is then said to be in an excited state. [3, 10] Relativistic Coulomb Excitation. Excitation of a nucleus by the electromagnetic eld of a charged particle moving at relativistic speeds. [12] 5 Multipole Expansion. The decomposition of the angular dependence of a potential into increasingly ner segments. Multipole moments comprising the rst few terms of the expansion are de ned as the Monopole (l = 0), Dipole (l = 1), and Quadrupole (l = 2) moments. [11] Giant Resonance. A highfrequency collective nuclear response involving an appre ciable number of nucleons. Characterized by the large photoabsorption cross section observed in all nuclei with more than a few nucleons. [3, 4] Giant Dipole Resonance. The second and most prominent multipole moment of oscillation which makes up the Giant Resonance. Described as the collective vibration between the protons and neutrons. [3, 4] Giant Quadrupole Resonance. The third multipole moment of oscillation and second largest contribution to the Giant Resonance. [3, 4] Pygmy Dipole Resonance. Similar to a giant resonance, but with a smaller strength. [3, 4] Limitations Calculations utilize well known empirical data when necessary such as; the width of the resonance, location of the centroid energy, and total nuclear cross sections NN corrected for inmedium interactions. The location of the Pygmy Dipole Resonance is taken from experimental data while the width used is more consistent with theoretical calculations. This restricts our analysis of results to be in a qualitative manner. Delimitations The lowenergy resonances considered in this thesis are induced through the method of Relativistic Coulomb Excitation. Therefore, only heavyion reactions at 100 MeV/nucleon and above are considered. Since we are considering only Relativistic Coulomb Excitations, 6 we restrict ourselves to collisions at large impact parameters, i.e., no overlap of the two nuclei, so b > R1 + R2. Organization of Thesis Chapters Chapter 1 gives an outline of the study and motivations for this investigation. The problem of inducing the PDR and GR through Coulomb excitation is introduced, then a brief explanation of the coupling between the PDR and GR is given as the main focus of the thesis. Chapter 2 develops the theoretical background leading to the idea of relativistic Coulomb excitation in the context of nuclear reactions. We also show how the equivalent photon method can be used in the theory of relativistic Coulomb excitation and show how the idea of equivalent photons appears in many aspects of physics. The coupledchannels calculation, important for determining how the PDR and GR are coupled, is also introduced at the end of Chapter 2. Chapter 3 takes the ideas developed in the previous chapter and applies it to the development of nuclear physics. The giant and pygmy dipole resonances are introduced and theoretical models are brie y covered. Chapter 4 gives results on how the PDR is coupled to the GR. These results are then compared to current theoretical predictions using perturbation theory. Chapter 5 summarizes the results of the calculation and we give our conclusions. 7 Chapter 2 COULOMB EXCITATION 2.1 Underlying Theory The theory of electrodynamics, which describes the interactions between charged par ticles, plays a pivotal role in modern physics and the development of the standard model. Forgoing the highly interesting yet impertinent pre19th century studies in electrostatics, I instead choose to begin with the monumental work of Maxwell in 1873 [13, 14], which stem from numerous experimental works including the equally notable, experimental physicist, Michael Faraday. The second volume of Maxwell's treaties [14], brought together, in a mathematically coherent way, the presupposed distinct interactions of electricity and magnetism into a uni ed theory of electromagnetism with light as its propagator. In Maxwell's work a collection of di erential equations describing the propagation of the electromagnetic elds through vacuum were formulated. These equations as amended by Heaviside are, r E = 0 (2.1) r B = 0 (2.2) r E = @B @t (2.3) r B = 0J + 0 0 @E @t (2.4) setting = J = 0 for a charge free region, the vacuum equations are reproduced. In the case of electromagnetic elds in vacuum we may take the curl of both (2.3) and (2.4) and use the identity, r (r ) = r(r )r2 , which gives the well known wave equation for electromagnetic radiation. 8 @2E @t2 = c2 r2E @2B @t2 = c2 r2B with the velocity of the wave, through a nonconducting medium, denoted by c = ( 0 0)1 2 , the speed of light. Maxwell's work did away with action at a distance and cemented the idea of elds into modernday physics. The theory of electromagnetism has since been extended beyond its classical origins into a fully consistent quantum mechanical theory. 2.1.1 Potential Fields and Gauge Transformations The Maxwell equations (2.12.4) are one of the greatest strides made in physics paving the way for more modern theories. Using vector notation as above it is immediately apparent that equation (2.2) has a freedom in de ning the magnetic eld as B = r A where the transformation A ! A + r is made with (r; t) arbitrary. Since the curl of any scalar vanishes, B = r (A + r ) = r A + :0 r r , the magnetic eld remains unchanged. Applying the transformation preserves the magnetic eld, however, the electric eld changes according to equation (2.3), r E = @B @t where B = r A ) r E = @ @t (r A) = r @A @t ) r E + @A @t = 0 Since the curl of a gradient is always zero we can assign the term inside the parenthesis equal 9 to the negative of some scalar potential so that, ) E + @A @t = r ) E = r @A @t (2.5) De ning the magnetic eld as the curl of some vector eld A results in a new de nition for the electric eld given by equation (2.5). This is true for any arbitrary gauge. Now using the same transformation on equation (2.5), E = r @A @t where A ! A + r ) E = r @ @t (A + r ) = r @A @t r @ @t = r + @ @t @A @t by making a new transformation ! @ @t equation (2.5) remains unchanged. This gives the nal solutions to the partial di erential equations (2.1  2.4) as, E = r @A @t (2.6) B = r A (2.7) where A and are the magnetic vector and electric scalar potential elds respectively and are a consequence of the gauge freedom inherent of Maxwell's equations. 10 2.1.2 Lorenz Gauge Many di erent gauges have been developed to exploit this \freedom" and is known as gauge xing [11]. One such gauge transformation, which ts nicely with special relativity, is the Lorenz gauge. Named for the Danish physicist Ludvig Lorenz, it introduces a trans formation to the scalar and vector A potentials while maintaining the solutions to the Maxwell equations (2.1  2.4) as above. Beginning from equations (2.1) and (2.4) then inserting solutions (2.6) and (2.7), we get the equations of motion for a r E = 0 where E = r @A @t ) r r @A @t = 0 ) r2 + @ (r A) @t = 0 (2.8) r B = 0J + 0 0 @E @t where B = r A ) r (r A) = 0J + 1 c2 @ @t r @A @t ) r(r A) r2A = 0J 1 c2 @ @t r 1 c2 @2A @t2 ! ) r2A 1 c2 @2A @t2 ! = 0J + r r A + 1 c2 @ @t (2.9) In equation (2.9) a necessary condition in order to preserve Maxwell's equations is, r A + 1 c2 @ @t = 0 (2.10) This is known as the Lorenz condition and allows us to write the equations of motion in 11 their nal form, r2 1 c2 @2 @t2 = 0 (2.11) r2A 1 c2 @2A @t2 = 0J (2.12) where we used equation (2.10) to solve for r A and insert into equation (2.8). These are the symmetric equations of motion for a charged particle. By inspection we see a similarity between these and the wave equations for electro magnetic radiation in vacuum (i.e. = J = 0). Applying the transforms A ! A0 +r and ! 0 @ =@t to the Lorenz condition, equation (2.10) leads to, r A + 1 c2 @ @t ! r A0 + 1 c2 @ 0 @t + r2 1 c2 @2 @t2 = 0 where A0 and 0 are the original potentials. To preserve our original condition in equation (2.10) for our initial potentials A0 and 0 we require, r2 1 c2 @2 @t2 = 0 which is simply the wave equation for a scalar function (r; t). This means that under the Lorentz gauge we have an in nite number of solutions which satisfy the Maxwell equations and everything is treated as a wave. This blends nicely with the implications of special relativity and naturally works in quantum mechanics where everything is a wave. A Quantum Theory The laws which govern our universe are quantum mechanical and any theory which hopes to succeed in its attempt to describe physical reality must be consistent with quantum 12 mechanics. The theory of quantum electrodynamics (QED) is marked as the rst successful quantum eld theory, consistent with relativity. Due to the tremendous success of QED, it has become the standard measure of a theory's acceptance. In our formulation of QED the interaction between charged particles (e.g., electrons, protons, mesons, : : :) is through the exchange of photons. These photons are the mediators of the electromagnetic force. The use of Feynman diagrams, introduced by Richard Feynman in the 1940s, has become the preferred method in describing particle interactions. This pictorial method was a simple way of tracking the overwhelmingly diverse in teractions plaguing QED at the time. Following Feynman's steps provided a systematic approach in diagramming these interactions. Figure 2.1. Interactions between fermions can be represented by a Feynman diagram. The solid lines represent fermions and the wavy line is the force carrier, i.e., the virtual photon. Figure 2.1 gives the standard depiction of electron scattering, depicted as a Feynman diagram. In 2.1 the horizontal axis represents the forward ow of time while the vertical axis gives motion in 3space. Fermions are represented by solid lines while the virtual photon (wavy line) is the mediator of the electromagnetic interaction. This virtual photon connecting the two particles removes the need for action at a distance. For a more in depth discussion on QED and the use of Feynman diagrams in quantum eld theory refer to Peskin and Schroeder [15]. The development of this beautiful concept of gauge boson exchange as the force car 13 riers revolutionized theoretical nuclear/particle physics, beginning with the photon as the mediator for the Coulomb interaction. Using this concept we proceed into discussions of nuclear interactions, a manybody quantum problem. Starting with, as one might expect, a classical approximation to the process of Coulomb excitation. 2.2 Nuclear Excitation Excitation of nuclei by means of providing energy through electromagnetic interactions is known as Coulomb excitation. This situation is observed most easily in nuclear collisions where a target nucleus is perturbed by an external electromagnetic eld. This may be achieved through a number of ways, however, the excitation through scattering particles will be this works focus. 2.2.1 HeavyIon Collisions When the number of protons and neutrons involved in a nuclear collision becomes large we refer to these as heavy ion collisions. These heavy nuclear collisions present a dynamic system with a very rich collection of interesting phenomenon. In this situation a charged ion Z1 interacts with another charged particle Z2 through the exchange of a photon Figure 2.2. The interaction picture for a particle (lower solid line) colliding with a nuclear target (shaded circle), through the exchange of a virtual photon. The products are depicted by the outgoing solid lines. Figure 2.2 gives a diagram representation of two interacting nuclei Z1 and Z2 where 14 the charged particles are shown to interact through the Coulomb force, represented by a mediating particle the photon (wavy line). This idea of photon exchange will be central to our discussions later. The three outgoing particles are what results from the interaction and are the only things observed. By looking at the incoming particles and measuring the outgoing products the scientist can reconstruct what interaction took place. Several di erent ways to represent these types of interactions exist. However, the simplest method is to use Feynman diagrams. The bene t of including, implicitly, the conservation laws through vertex analysis still makes it preferable to other methods. For a brief overview of Feynman diagrams and rules see [15, 16]. 2.2.2 Classical Scattering Early investigations into nuclear physics typically involved low energy interactions between charged particles of low mass, e.g., electrons, protons, alpha particles, and a xed target nucleus. The most notable of these experiments, which revealed the nucleus as be ing composed of a dense central core surrounded by electrons, were conducted by Ernest Rutherford. In the aforementioned series of experiments, often called the Rutherford gold foil experiments, the low energy scattering of an alpha particle beam o a target of thin gold foil resulted in the detection of scattered alpha particles back towards the source. Backscattering as it's now called can be represented as a onebody elastic scattering problem which describes the trajectory of an incident charged particle as it interacts with a central force. Scattering particles is still the preferred method used to investigate nuclear structure. Since the need to probe deeper into the nucleus is more necessary, scattering particles achieves this by increasing the interaction energy and decreasing the impact parameter of the colliding nuclei. This requires the construction of larger and more powerful colliders. 15 Rutherford Rutherford scattering, as the low energy method is more commonly referred, is the classical approximation to a quantum mechanical system. In this system the interacting particles carry with them a charge, Z(1;2)e, and momentum, p(1;2), where the indices denote projectile and target, respectively. A particle is said to be scattered when its direction Figure 2.3. Scattering of an incident beam of particles by a center of force. of motion is altered. Coulomb proved that the electric force follows an inversesquare law similar to the gravitational force which implies that the orbital equations are applicable, in the classical limit. Given the charge of each particle and their relative velocity, v at in nity, we may calculate the strength of the Coulomb interaction by the Sommerfeld parameter = Z1Z2e2 ~v : (2.13) A classical approximation is appropriate when, 1. This is possible when either, Z1Z2 16 137, or, v c. The trajectory of the projectile, Z1, approaches the target, Z2, as in Figure 2.3. The collisions considered in classical scattering are not strong enough to overcome the Coulomb barrier. However, we must consider the distance of closest approach, a, between the two nuclei. From the conservation of energy, E = 1 2 mv2 0 = Z1Z2e2 a we solve for a giving the distance in terms of the charges and relative speed, a = 2Z1Z2e2 mv2 0 This is an important parameter since it gives us a known value in the projectile's trajectory. In this arbitrary collision the trajectory of the projectile is dependent on the impact parameter and its kinetic energy. The \scattering angle", , is uniquely determined for classical scattering. A scattered particle, in relation to the target, is observed passing through a cross sectional area ( ). The area scattered is dependent on the scattering angle . For a small segment of the scattering angle d multiplied over the azimuthal coordinate 2 sin the portion of solid angle is written as d = 2 sin d . Scattering strength is determined by the energy and angular momentum l. The angular momentum of a particle or planet is given as, l = b p0 = b p 2mE where p0 = mv0 is momentum and b is called the \impact parameter". The impact parameter is the distance between the centers of two scattering particles. Assuming from a classical perspective that particles at di erent impact parameters will have di erent scattering angles, the number of particles scattered are determined by the beam intensity, I = N=A, where N is the total number of incident particles passing through area A. 17 Using the di erential cross section ( ) de ned as the ux, ( ) of particles per beam intensity we have, ( ) = ( ) I (2.14) We can solve for the number of particles scattered into solid angle d by using the de nition of ux, ( ) = dn=d . The number of particles, n, incident on the target is dependent on the impact parameter as well as the scattering angle allowing the ux to be written as, ( ) = dn dA dA d where I = dn=dA, the incident beam intensity. Inserting into equation (2.14) we have, ( ) = ( ) I ) I ( ) = dn dA dA d multiplying by d on both sides and substituting in I = dn=dA we get, I ( ) d = I dA ) 2 I ( ) sin d = 2 I b db where we used the incident beam cross section A = b2 ) dA = 2 b db. The di erential cross section is now dependent on the scattering angle , solving for ( ) gives, ( ) = b sin db d (2.15) a straightforward relation between the impact parameter and the scattering angle measured. Additionally deriving the impact parameter from the eccentricity of a hyperbolic orbit equa 18 tion (2.15) becomes, ( ) = 1 4 ZZ0e2 2E !2 csc4 2 (2.16) the famous Rutherford Scattering equation. The relationship between equations (2.15) and (2.16) can be found in Goldstein's Classical Mechanics. [17] Rutherford's equation is relevant for low energy scattering where the incoming particle doesn't penetrate the targets Coulomb barrier such as the gold foil alpha particle scattering. Once the projectile and target begin to overlap, e.g., 25 MeV for +Pb, the scattering becomes inelastic and Rutherford's equation no longer applies. 2.3 Semiclassical Theory of Coulomb Scattering The validity of a semiclassical approach to the method of Coulomb excitation is well known for studies on nuclear excitation. As stated before it provides a way to study reactions without having to include the manybody nuclear forces. This is a well established method and has been extensively covered by several investigators. In the electromagnetic excitation process we describe an interaction Hamiltonian as Hint = HP + HT + V(r(t)) where the matrix HP (HT ) is the projectile(target) Hamiltonian and V(t) is a timedependent term describing the electromagnetic interaction between both projectile and target. An appropriate choice for the frame of reference (i.e., target or projectile) reduces the above interaction Hamiltonian. In our treatment of the problem we choose the frame of reference of the target and consider only stable projectiles with the ground state Hamiltonian HT . The excitation is then built from the ground state of the target. The new Hamiltonian 19 must satisfy the timedependent Schr odinger equation Hint (t) = i~ d (t) dt (2.17) Performing an expansion on the total timedependent wave function (t) in terms of the set of orthogonal eigenfunctions which form a complete basis (t) = X n an(t) neiEnt=~ (2.18) where the ground state Hamiltonian of the target nucleus satis es the eigenvalue equation HT n = En n (2.19) using the newly expanded wave function (2.18) with the interaction Hamiltonian given as Hint = HT + V(t) we obtain a new form of the timedependent Schr odinger equation (2.17) given as i~ d dt X n an(t) neiEnt=~ ! = HT + V(t) X n an(t) neiEnt=~ i~ X n _an(t) iEn ~ an(t) neiEnt=~ = X n an(t) HT + V(t) neiEnt=~ Distributing through on both sides of the equation and using the relation (2.19) we are left with the equation, i~ X n _an(t) neiEnt=~ = X n an(t)V(t) neiEnt=~ 20 multiplying both sides by m and integrating over all space we have i~ X n _an(t)eiEnt=~ mn = X n an(t) Z 1 1 d3r mV(t) n ! eiEnt=~ i~ _am(t)eiEmt=~ = X n an(t)Vmn(t)eiEnt=~ where we used R 1 1 d3r m n = mn. The matrix elements for the electromagnetic interac tion potential are given as Vmn = Z 1 1 d3r mV(t) n = hmjV (t)jni solving for the di erential time dependent amplitude of state m gives, ) _am(t) = i ~ X n hmjV (t)jni an(t)ei(EmEn)t=~ (2.20) We now have a set of m coupled equation where the sum runs over all possible excita tion states n. The solutions to which are typically found by performing a multipole expansion on the potential V(r(t)). The Multipole expansion is introduced in the next section. Equation (2.20) can be interpreted as the transition amplitude of a particle from an initial state jii to some nal state jfi with excitation ~! = Ef Ei. The excitation is taken as the Fourier component to the transition frequency of the interaction Hamiltonian so that we have, afi(t) = i ~ Z 1 1 hfjHintjii ei(EfEi)t=~dt (2.21) The term hfjHintjii is the matrix form of the interaction Hamiltonian. Integrating over a short time interval the transition amplitude is then, am(t) = i ~ Z T 0 Vmn(t)ei(EmEn)t=~dt (2.22) 21 since for short time intervals the system returns to its original state m so that an(t) = 1 for m = n else an(t) = 0. Multipole Expansion A multipole expansion of a potential is a way to approximate the e ective electro magnetic interaction between charged particles in terms of their intrinsic coordinates. In spherical polar coordinates this takes separate treatment of the radial, polar, and azimuthal coordinates, r, , and respectively. The Coulomb potential measured at r for a system of pointlike charged particles ei located at ri is given in general by the equation V (r) = X i ei jr rij (2.23) 1 jr r0j (r2 + r02 2rr0 cos )1=2 (2.24) where the primed coordinates indicate the location of charged particles and cos = cos cos 0+ sin sin 0 cos( 0) is the angle between the primed and unprimed vectors. Equation (2.24) depends on only the lengths of vectors r and r0 and the angle between them . This allows a multipole expansion to be made in terms of the mutually orthogonal Legendre polynomials of order l. 1 jr r0j = 1 r> X1 l=0 r< r> l Pl(cos ) where r< (r>) is the lesser(greater) distance between r and r0. By the addition theorem, as given in Appendix B of [18], a generalization to the geometric relation given by cos . This leads to an expression for the Legendre polynomial as a linear combination of spherical 22 harmonics, of order l, Pl(cos ) = 4 2l + 1 Xl m=l Y lm( 0; 0)Ylm( ; ) leading to a nal form for the expansion given as, 1 jr r0j = 1 r> X1 l=0 4 2l + 1 r< r> l Xl m=l Y lm( 0; 0)Ylm( ; ) Applying the above expansion, given in terms of spherical harmonics for each discrete point charge i at r0, equation (2.23) becomes, V (r) = X lm 4 2l + 1 1 rl+1Y lm( ; )M(El;m) (2.25) where M, the electric multipole moment, is given as, M(El;m) = X i eirli Ylm( i; i); m = l;l + 1; : : : ; l 1; l The sum is over each electric charge ei with position ri = (ri; i; i). Modifying the equation above, using the density operator (r) = X i ei (r ri); the multipole moment M(El;m) is now a tensor operator of rank l. Integrating over all space the operator takes on the general form, M(El;m) = Z d3r (r)rlYlm(n); n = r r (2.26) One postulate of quantum mechanics states that, for any measurable system which 23 has an associated physical observable can be represented by an operator. The multipole expansion of equation (2.23) into (2.25) puts it into a special class of quantum mechanical operators called the spherical tensor operators. 2.4 Relativistic Coulomb Excitation In the previous section we expanded the potential in terms of its multipoles and put the multipole moment in the form of an operator, a necessary step for its use in quantum mechanics. Now we must go further and put the method of Coulomb Excitation into terms that are consistent with relativity. The premise of Relativistic Coulomb Excitation (RCE) is taken from its low energy origins. As with Coulomb Excitation the fundamental assumption of RCE is that the nuclei do not interpenetrate. Reactions of this nature due to this process are short lived and enhanced due to Lorentz contraction of the eld. For the case of a projectile with energy, Elab 100 MeV/nucleon, the trajectory is nearly straight, i.e., negligible de ection, giving an interaction distance equal to the impact parameter b. We also consider only interactions involving the electric eld and the target or projectile, so that, b > RT + RP , is larger than the sum of the radii of the target and projectile, respectively. Using the same setup as with Coulomb Excitation, where we are in a target centered frame with the zaxis being the path of the projectile, we consider the case where v ' c. A particle traveling at these high energies will be Lorentz contracted in the direction of motion and can be described by the Li enardWiechert potential, (r; t) = Ze p (x b)2 + y2 + 2(z vt)2 where b is the impact parameter and the factor = (1 v2=c2)1=2 is the Lorentz factor making the component of the electric eld parallel to the direction of motion, +z, compact 24 Figure 2.4. A relativistic charged projectile incident on a target with impact parameter larger than the strong interaction radius. The transfer of virtual photons is shown (wavy lines) coming from the Lorentz contracted i.e., pancake shaped, electromagnetic eld. and pancake like. The above equation is the same electric potential eld from our discussions on Lorenz gauge. The vector potential eld used for a spinless particle following a classical trajectory is, A(r; t) = v c (r; t) where the velocity vector v = v^z. A multipole expansion can be made to the Li enard Wiechert potential in a similar way as was done in the previous section. Knowing the electric and vector potential eld equations we can easily obtain the 25 associated electric and magnetic eld equations from equations (2.6) and (2.7), Ez = Ze vt (b2 + 2v2t2)3=2 ET = Ze b (b2 + 2v2t2)3=2 BT = v c ET ; and Bz = 0 where the equations are separated by components parallel(tangent) to the direction of motion and are denoted by subscript z(T). For fast moving charged particles the interaction of the eld on the target is short. That is for 1 , by inspection of the denominator, the time is approximately, vt = b ! t = b v ' b c This shows the elds to be similar to two pulses of planepolarized radiation. In the case for radiation along the z direction, this is an exact analogy. Radiation along the T direction is only approximate due to the presence of a magnetic eld, however, this does not appreciably a ect the dynamics of the system. At speeds v = c the e ects of the Ez eld are minimal and allow us to add an additional eld, B = vEz=c. A calculation of the energy incident per unit area on the target due to an electromagnetic wave is now possible. Fourier transforming the Poynting vector S = E B, the intensity of radiation, I(!; b), is calculated with energy E = ~!. The probability of nuclear transition is then given by the integral, P(b) = Z I(!; b) (E )dE = Z n(!; b) (E ) d! ! (2.27) where (E ) is the photonuclear crosssection for photons of energy E and n(!; b) is the 26 number of equivalent photons in the electromagnetic wave at frequency ! and impact pa rameter b. The term n(!; b) in the above equation was rst introduced by Enrico Fermi [19] who proposed that the electromagnetic eld created by a charged particle can be equated to a ux of virtual photons. This idea of virtual photons mediating the electromagnetic interaction is obviously an important part to the theory of electromagnetism both classically (low energy scattering) and, as discussed above, quantum mechanically (QED). Fermi's work, published in 1924 by Zeitschrift f ur Physik and also, unconventionally, published in Italian by Nuovo Cimento [20], was extended by Weizs acer and Williams [21] to be used at relativistic velocities. This was essentially restoring the Lorentz factors in their proper places. What remains to be shown is how this Lorentz contracted electromagnetic eld inter acts with a target possessing charge. 2.4.1 Equivalent Photon Method The electromagnetic eld of a fast moving charged particle is Lorentz contracted in the direction of motion. Considering only collisions where the nuclei do not overlap, i.e., impact parameter b > RP + RT , which is the case for Coulomb excitation, the probability can be expressed in terms of the sum over all transition probabilities into a nal state If . The probability of Coulomb excitation is given as, PC(b;E ) = X f Z Pi!f (b) f (E )dE (2.28) where the sum is over all possible nal states with the density of nal states integrated over all energy transfers E = ~! = Ef Ei. The transition probabilities Pi!f are obtained from 27 the transition amplitude calculated in equation (2.21) so that, Pi!f (b) = a2 4 4 1 2Ii + 1 X Mi;Mf aif 2 Using WignerEckart theorem and respecting the orthogonality conditions of the Clebsh Gordon coe cients, which produce delta functions from the expansion coe cients between angular momentum eigenstates, the transition probability can be expressed in the form, Pi!f = Z2 1 e2 4 2a2 2 ~2 X LM B( L; Ii ! If ) (2L + 1)3 S( L;M) 2 (2.29) again where = E or M and B( L; Ii ! If ) is the reduced transition probability given as, B( L; Ii ! If ) = 1 2Ii + 1 X Mi;Mf hIiMijM( L;M)jIf ;Mf i 2 = 1 2Ii + 1 hIij M( L) jIf i 2 The transition probability given by equation (2.29) can now be plugged into equation (2.28). If we use the idea proposed by Enrico Fermi to approximate the interaction of the electric eld as a ux of virtual photons as in equation (2.27) then the probability becomes, PC(b;E ) = X L P L(b;E ) = X L Z dE E n L(b;E ) L (E ) where the sum is now over multipole states. The term n L(b;E ) is the virtual photon numbers and L is the photonuclear cross section for a given multipole. 2.4.2 CoupledChannels Coulomb Excitations are often dominated by transitions between the ground state and a continuum resonance state. Excitation into these states are treated as exact in a coupledchannels approach [22] where additional states are calculated as perturbations from 28 the ground state. In the coupledchannels approach an excitation into a continuum of states is called a doorway state, or the particular entrance taken into the continuum. The ground state is then coupled to the set of doorway states creating a coupling matrix. Amplitudes for such excitations are given as, (n)(E ) = h (E )jD(n) LMi and can be used as an expansion of the ground state potential. Integrating over all en ergy separations = E En, where E is the energy from the interaction and En is the centroid energy of the resonance n. Taking, for example, the dominant E1 excitation and summing over all magnetic quantum numbers the timedependent transition amplitude, equation (2.20), in the ground state is expressed as, _a0(t) = i ~ X M Z d h ( )jD(1) 1MihD(1) 1MjVE1;M (t)j0i a(1) ;1M(t) exp i(E1 + )t=~ = i ~ X M Z d (n)( )V (01) M (t)a(1) ;1M(t) exp i(E1 + )t=~ (2.30) where the dominant resonance amplitude is given as, d dt a(1) ;1M(t) = i ~ h (1)( ) V (01) M (t) i exp i(E1 + )t=~ (2.31) Integrating (2.31) over time and using the result in equation (2.30) the result is, da0 dt (t) = 1 ~2 X M V (01) M (t) Z d (1)( ) 2 Z t 1 dt0 h V (01) M (t0) i exp i ~ (E1 + )(t t0) a0(t0) where a(1) ;1M(t = 1) = 0 was used and the squared amplitude term (1)( ) 2 is chosen depending on the shape of the resonance. 29 Chapter 3 NUCLEAR STRUCTURE Nuclear physics is one of the largest areas of research and continues to be a signi cant area of interest. Since its conception it has contributed to many elds including the eld of astrophysics, problems in energy generation, medical technology, archaeological carbon dating, and even spawned the eld of particle physics. Early nuclear studies focused on the characterization of elements according to their constituent particles i.e., protons, electrons, and later neutrons. With the discovery of the neutron we had the nal piece to a description of the nucleus, consistent with Einstein's massenergy relation. With the neutrons discovery the energy contribution to the binding of a nucleus was nally calculable. A new look at the elements, tabulated according to the ratios of protons to neutrons, was developed. From this point of view the binding energy, which apportions energy con tributed by the constituents of the nucleus expended to the con nement of the nucleons due to the binding energy. The remaining energy constitutes the measured mass of the element. The advent of this re ned measurement to nuclear mass lead to questions apropos of the mechanism behind the con nement of neutrons and more interestingly the protons. Remembering the important relation from electrostatics, like charges repel, the protons within a nucleus, being in close proximity to one another, must feel concurrently opposing forces. These concurrent forces imposed on individual protons within the nucleus must be overcome by a much larger attractive force for a nucleus to form. This \strong force", which opposes the repulsive force of the protons, binds nucleons together and is contained within the description of the binding energy. However, its e ective range of in uence must be shorter than the electromagnetic interaction since it is not observed outside the nucleus. Hideki Yukawa, in 1935 proposed a ground breaking theory to explain the binding of nucleons by the strong force [23]. Yukawa's model of the strong force was analogous to 30 the mediating photon model of the electromagnetic force where a low mass virtual particle is exchanged between the nucleons e ectively binding them together. A straightforward calculation can be made to estimate this exchange. Using Heisenberg's uncertainty relation, E t > ~ which states that the uncertainty in time and energy must be greater than Planck's constant. Rearranging the above inequality for energy and multiplying both the top and bottom of the fraction by c the speed of light we get, E > ~c c t ~c rS where rS is the interaction distance of the strong force. Heisenberg's energy time uncertainty is valid for any \real particle", however, quantum mechanics allows us to violate this inequal ity by restating the above inequality in a contrapositive way so that the relation reads, for short time or length periods the energy must be less than or equal to some exchange energy Eex so that, Eex ~c rS An estimation of the upper bound for the lowest energy exchange particle can be calculated using some empirical data. Calculating the binding between the proton and neutron, which minimizes the in uence of the electromagnetic interaction, forming a deuteron. From ex periment the radius of the deuteron is, rD = 1:95 fm. We calculate the distance between the centers of each nucleon by subtracting the proton and neutron radii, rp = 0:8 fm and rn = 0:3 fm, from the diameter of the deuteron. Using ~c = 197 MeV fm we get an exchange energy of, Eex ~c rS ~c 2rD rp rn = 140:7 MeV 31 So the energy of an exchange particle necessary to bind a proton and neutron together must not be greater than 140 MeV. This is similar to the result found by Yukawa and corresponds to the lightest meson 0 = 135 MeV. We now see a similarity between QED and the strong force. In QED the exchange of a virtual photon mediates the electromagnetic interaction, this is analogous to result above where the exchange of a virtual pion mediates the strong interaction. However, this is not the whole story since it was later found that the pion is simply a consequence of a more fundamental theory of quarks, which are the building blocks of hadrons. It is now well known that all fundamental interactions, save for gravity, are due to some exchange of virtual quanta. The advantage to using Coulomb excitation is that we can exclude these internal interactions dominated by the strong force thereby simplifying our calculations. We now consider an important excitation where a large number of constituent nucleons are excited into a quasistable state. These excitations are highly dependent on the number of nucleons, A, implying a dependence on the binding energy. 3.1 Giant Resonance The large bulk nuclear response exhibited in nuclei with more than a few nucleons is known as a Giant Resonance (GR). The GR phenomenon has been shown to be a fundamental property of all nuclei [3,4,24] where the general features are; smooth transitions in the form, width, and centroid energy correlating to a change in the mass number A. The width is small compared to its excitation energy, and exhausts a large portion of the energyweighted sum rule. GR excitations are sensitive to several variable parameters available to the experi menter. The rst, and probably most obvious, would be the isotopes used in the reaction (e.g. p+Pb, Pb+Pb, Ni+Au, : : : ). The energy of the collision and the characteristics of the beam are then adjusted and selection of the scattering angle reveals the cross sections of interest. The GR observed varies according to the excitation energies giving a cross section 32 (E ), for excitation energy E , which can be t by a Lorentzian, (E ) = 0E2 2 (E2 E2C )2 + E2 2 (3.32) for the Giant Dipole Resonance (GDR). The centroid energy, EC, is the most probable energy of the resonance, e.g., 24 MeV for 16O and 13 MeV for 208Pb. The isotope excited into a resonance determines; the value of 0 which satis es the sum rules, and the width which varies from 4 to 8 MeV. For di erent multipole excitations, (e.g., E1, E2, M1, M2, : : :) the above parameters are adjusted accordingly. 0 50 100 150 200 250 300 350 400 450 500 550 8 10 12 14 16 18 20 22 24 26 28 Cross Section [mb] Energy [MeV] Pb208 Cross Section (Harvey) Lorentzian ’CS_Pb208_Harvey.dat’ Figure 3.5. Example cross section for the GR in Lead208. In gure 3.5 a schematic for a typical photoabsorption cross section is given. At low energies the system exhibits discrete states which are characterized by surface e ects due to 33 deformation, vibration, rotation, and a mixing of vibration and rotation. Around 8 MeV the discrete states give way to a continuum of states with the GDR, a dominant feature of the spectrum, appearing at around 8 MeV higher than where the continuum states begin. These collective excitations involve an appreciable number of nucleons and are built up from individual excitations from the ground state of the nucleus. 3.1.1 Classical models Over the years several collective models have been developed to describe the bulk response of nuclear matter. The progression of the theory from classical approximations to more sophisticated manybody models has been long running. A typical classical model for the GR phenomenon was rst developed by Goldhaber and Teller [25] and later improved by Steinwedel and Jensen [26]. Three possible explanations for the features which de ne a GR are: A restoring force between the displaced protons and neutrons exists which is inde pendent of the size of the nucleus. This possibility is discarded due to experimental evidence which clearly indicates a dependence on the number of nucleons A. A di erence between the proton and neutron densities within the nucleus. In this situation the surface of the nucleus is not a ected and the restoring force is proportional to the gradient of the di erence in density. This picture describes the SteinwedelJensen model. Two spheres comprised of either protons or neutrons oscillate against one another. This is a pure dipole mode picture with the overall density kept constant. This is the picture taken in the GoldhaberTeller model. In the GoldhaberTeller (GT) model a collection of protons, in uenced by an electric eld realized as an impinging photon, takes the nucleus to an excited state. The major feature of the GR phenomenon is of course the GDR and is described in the GT model as a 34 collection of protons moving against a collection of neutrons with their relative oscillations being opposite in phase to one another. This model allows for the collection of protons and neutrons to interpenetrate while preserving their individual incompressibility. The interpre tation as two oscillating uids will be important for our later look into the pygmy dipole resonance. The other interpretation developed by the SteinwedelJensen (SJ) model preserves the total density of the nucleus by separating the proton and neutron densities such that, A(r; t) = P (r; t) + N(r; t) the variation in density is then introduced by the addition and subtraction of a fractional density term (r; t) and requiring that the number of particles is conserved, Z d3r (r; t) = 0 A shift in the distribution of positive charge is the central idea behind both pictures of the GDR developed by GT and SJ. Each model describes the phenomenon reasonably well with the SJ giving an A dependence as A1=3 which describes the experimental data better than GT's A1=6 dependence. Each of these models takes a classical approach to this dynamic quantum system. For a more complete picture one must include quantum mechanics. Quantum mechanically describing the collective oscillations of manyparticle systems is an interesting and challenging problem in nuclear physics. In the days before computer simulations, when quantum electrodynamics was rst being developed, describing the dy namics of the wave functions for each individual nucleon inside of a nucleus was the stu of dreams. Simpli cations to the model and exploiting the symmetries and bulk properties of the system were necessary. Thus techniques such as mean eld approximations where used to reduce the degrees of freedom to a solvable problem. 35 3.1.2 Microscopic Models Although the classical models above describe the GDR reasonably well it is still neces sary to extend this into a microscopic picture. Several e ective theories exist which describe the GR on a microscopic level. Early attempts at calculations were developed based on the linear response theory [31]. Nowadays, an e ort is being undertaken to describe nuclear collective motion with more elaborate models such as the timedependent super uid local density approximation [32, 33]. Similarly, theoretical studies of the pygmy resonances have been developed based on the improvements of the hydrodynamical model [34{36], and with microscopic theories such as the random phase approximation (RPA) and its variants [37{40]. Sum Rules It is often useful to estimate the photoabsorption cross section over every transition from some initial state jii ! jfi to the possible nal states. The estimate is found by the sum rule for the system and is given in the form, S(n) i [F] = 1 2 X f (Ef Ei)n hfjFjii 2 + hijFyjfi 2 The sum rules give a tool for calculating the sum of integral cross sections for real photons over all possible nal states. 3.2 Pygmy Dipole Resonance Inherent to the majority of nuclei, excitation of the GR is a well investigated phe nomenon. As the ratio of neutrons to protons increases the distribution of protons and neutrons begins to di er. Experiments which infer the radius of the nucleus typically involve the measurement of elastically scattered particles from the target. The standard ways of measuring the radius of nuclei are easily understood through 36 classical scattering. To observe any reasonable detail of the nucleus the wavelength of the projectile must be smaller than the target. Suppose we wish to measure the radius of 208Pb. According to the radius formula, R = R0A1=3 where R0 = 1:2 1:25 fm, the radius of Lead 208 is RPb ' 7 fm, therefore the wavelength of the projectile must be 2RPb ' 14 fm which corresponds to a projectile momentum of p 89 MeV/c. Highenergy electron experiments are able to achieve beam energies from 100 MeV to 1 GeV and are a good candidate for measuring the nuclear radius. Spectroscopic analysis of only the elastically scattered electrons gives a good average estimate to the charge radius of nuclei. The electron projectiles are scattered by Coulomb interactions. This means that electron scattering tells us about the distribution of charge, i.e., protons, in the nucleus. Measurements of the distribution of all nuclear matter, which incorporates the protons and the neutrons, requires us to overcome the Coulomb barrier. Alpha particle scattering is one method where this is achieved. Using the classical scattering formula (2.16) developed by Rutherford form his famous alpha particle scattering experiments. Increasing the energy of the alpha projectiles the scattering cross section is well described by (2.16), however, at a certain energy the predictions of Rutherford's scattering formula are no longer accurate due to the e ects of the nuclear force. The results of these measurements show that the root mean square (rms) charge radius is almost the same as the rms nuclear radius di ering by less than 1 fm. Since nuclei tend to have more protons than neutrons this implies that the protons distribute themselves across the nucleus leaving more neutrons to occupy the core. When the number of neutrons become much larger than the number of protons the neutrons are then pushed out, due to their occupation pressure, beyond the rms charge radius creating a neutron rich layer. The di erence between these two radii is called the neutron skin, RSkin = Rn Rp. 37 Figure 3.6. The dipole mode of the pygmy resonance in the GoldhaberTeller (GT) and SteinwedelJensen (SJ) modes. In reactions which involve neutronrich nuclei at low excitation energies, close to the low energy tail of the GR, the excitation of the neutron skin against a symmetric nuclear core is observed. This phenomenon is known as the Pygmy Dipole Resonance (PDR) and can easily be visualized through the GT and SJ models as shown in gure 3.6. Understanding the PDR is import since it can be used as a tool to constrain the neutron skin thickness of these neutronrich nuclei. How the PDR is related to the neutron skin is through the fraction of the Energy Weighted Sum Rule (EWSR) exhausted by the PDR [27]. Measurements of the neutron skin have far reaching implications including areas of interest such as nuclear structure [28], neutron star structure [29], and heavyion collisions [30]. 38 Chapter 4 PRESENTATION OF FINDINGS (Published in Physics Letters B, Volume 757, 10 June 2016, Pages 553557 ) As research on nuclear reactions with radioactive beams became the focus over the last few decades, it became apparent that modi cations of the linear response theory predict a signi cant concentration in neutronrich nuclei at low energies of the excitation strength [41,42]. It is important to note that the amount of energy of the nuclear response exhausted by the sum rule strongly depends on how the nuclear interaction, pairing, and other physical phenomena are incorporated in the theory [37{42]. As an example, the E1 strength function numerically calculated using the public code of Ref. [43], is given as S(E) = X h jjOLj j0i 2 (E E ) (4.33) where an RPA con guration space is de ned in terms of deltafunction states and the operator OL is an electromagnetic operator. A smearing of 1 MeV of the fragmented strength function is introduced which produces a continuous distribution, shown in Fig. 4.7 for the E1 response in 68Ni. We used the option OL = jL(qr) in Eq. (4.33), where we take q = 0:1fm1 to be a representation of the momentum transfer. More details can be found in Ref. [43]. In this case, the strength function has dimensions of MeV1 and in the long wave approximation qr 1 it is proportional to the usual response for electric multipole operators. The calculation is performed for several Skyrme interactions, as shown in the gure. The arrow shows the position of the expected pygmy dipole resonance. The results presented in the literature, e.g. [37{42] show a greater response in the PDR energy range due to interactions and adaptations in the model space. There is currently no clear prediction of the exact location of the pygmy strength. It could be in the range of 712 MeV for medium mass nuclei such as Ni isotopes. The amount of the sum rule exhausted by the pygmy 39 Figure 4.7. Strength function for the E1 RPA response in 68Ni calculated with formalism described in Ref. [43]. The calculation is performed for several Skyrme interactions, shown in the gure inset. The arrow shows the location of the pygmy resonance. (For interpretation of the references to color in this gure legend, the reader is referred to the web version of this article.) resonance is also relatively unknown, although some models based on nuclear clustering can reach up to 10% of the total strength [44]. Coulomb excitation of pygmy resonances is one of the e ects overseen by the experi mental analysis of these reactions. The large excitation probability in Coulomb excitation at small impact parameters leads to a strong coupling between the pygmy and giant resonances. A manifestation of this coupling is seen as dynamical e ects such as the modi cation to transition probabilities and cross sections of the PDR. Previous observations of this coupling e ect are seen in the excitation of double giant dipole resonances (DGDR) [1, 45, 46]. The experimental observation of the DGDR is a consequence of higherorder e ects in relativis 40 tic Coulomb excitation arising due to the large excitation probabilities of giant resonances in heavy ion collisions at small impact parameters. The dynamical coupling between the usual giant resonances and the DGDR is very strong, as shown, e.g., in Ref. [22]. In the present work this dynamical coupling e ect on the excitation of the PDR is assessed using the relativistic coupled channels (RCC) equations introduced in Ref. [47]. The Smatrix for state given as, S (z; b), is obtained from the RCC equations [47] i @S (z; b) @z = X 0 h jMELj 0iS 0(z; b)ei(E0 E )z=~ ; (4.34) where is the projectile velocity andMEL is the electromagnetic operator for transitions by the electric dipole (E1) and quadrupole (E2) modes connecting states and 0. The states must satisfy the intrinsic angular momenta and parity selection rules. The ground state is denoted as j0i = jE0J0M0i and the excited states by j i = jE J M i where EJM are the intrinsic energy and angular momentum quantum numbers. The electromagnetic operators, in the longwavelength approximation, are given by [47] ME1m = r 2 3 Y1m(^ ) ZT e2 (b2 + 2z2)3=2 8>>< >>: b (if m = 1) p 2 z (if m = 0) (4.35) where is the intrinsic coordinate of the excited nucleus and Ze is the charge of the nucleus creating the interacting electromagnetic eld (in our case, the target). The electromagnetic operator for E2 transitions is [47] ME2 = r 3 10 2Y2m(^ ) ZT e2 (b2 + 2z2)5=2 8>>>>>>< >>>>>>: b2 (if = 2) 2 2bz (if = 1) p 2=3 (2 2z2 b2)z (if m = 0) (4.36) We note that the electromagnetic operator is expressed as MELm = fELm(r)OELm, where 41 OELm = LYLm(^ ) is the usual electric operator, and fELm(r) is a function of the projectile target relative position r = (b; z). Solutions to the coupled equations (4.34) are found by using S (z ! 1) = 0. For collisions at high energies and very forward angles, the cross sections for the transition j0i ! j i are given by d dE = 2 w (E) Z db b exp[2 (b)] S (z ! 1; b) 2 ; (4.37) where w (E) is the density of nal states, b is the impact parameter in the collision, and (b) is the eikonal absorption phase given by (b) = NN 4 Z dq q 1(q) 2(q)J0(qb) (4.38) where NN is the total nucleon{nucleon cross section, obtained from experiment, with medium corrections added according to Refs. [48, 49]. The Fourier transform of the ground state densities of the nuclei, i(q), is obtained from tting to electron scattering experi ments [50] for 197Au and using Hartree{Fock{Bogoliubov calculations with the SLy4 interac tion for 68Ni. A reduction of the Coulomb excitation mechanism at small impact parameters, rst introduced in Ref. [51], is used to calculate reaction cross sections relevant to excitations of the GDR and DGDR. To remain within the contexts of current experimental results we neglect e ects of nuclear excitations, and possible interferences which were subtracted in experiments [52{54]. To facilitate our discussions we consider the excitation of 68Ni on 197Au and 208Pb targets at 600 and 503 MeV/nucleon, respectively. These reactions have been experimentally investigated in Refs. [52,53]. In the rst experiment the pygmy dipole resonance was observed at EPDR ' 11 MeV with a width of PDR ' 1 MeV, for 68Ni and exhausted about 5% of the Thomas{Reiche{Kuhn (TRK) energyweighted sum rule. The excitation was identi ed through the analysis of its decay product, gamma emission. The second experiment found 42 the PDR centroid energy to be at 9.55 MeV, width a of 0.5 MeV and exhausting 2.8% of the TRK sum rule. The PDR was identi ed by measuring the neutron decay channel of the PDR. This studies focus will be on the e ects of coupling between the di erent modes of giant resonances with the PDR. Therefore, we will not take into consideration the decay channels but restrict our analysis to the calculation of the excitation function d =dE. A model is needed for including the bound and continuum discretized wavefunc tions in the matrix elements of h jMELj 0i in Eq. (4.34). To calculate the response functions, dBEL=dE = P spins w 0 h jjOELj j 0i 2 , it is appropriate to use these wavefunc tions, with obvious consideration for summing over angular momentum coe cients. To simplify calculations we assumed Lorentzian forms for the response functions dBEL=dE and assign an appropriate fraction of the sumrule based on current experimental values. We then discretized the functions into energy bins to obtain the reduced matrix elements h jjMELj j 0i 2 / E (dBEL=dE) E=E , where E = E 0 E . To keep everything in terms of real numbers a phase convention is found for the reduced matrix elements. These are then used, with proper care taken for the corresponding angular momentum coe cients, in determining the matrix elements h jMELj 0i in Eq. (4.34) (see, e.g., Ref. [46]). The Lorentzian functions are described by Eq. (3.32) with their respective centroid energies EPDR for pygmy dipole resonances and EGDR (EGQR) for the isovector (isoscalar) giant dipole (quadrupole) resonances. Their respective widths are denoted by PDR, GDR and GQR. The discretized strength function is subdivided into 35 energy bins centered around each resonance. The PDR centroid energy EPDR = 11 MeV is chosen, consistent with Refs. [52,53]. However, a full width at half maximum of 2 MeV was chosen, which is more in line with theoretical calculations [35{42] than with the experimental data [52,53]. Our choice to use the theoretical width was in essence made to better determine higherorder e ects on the modi ed tails of the PDR without changing the other aspects of the PDR appreciably. For the major mode of the GR the (isovector) 1 giant dipole resonance (GDR) was given a centroid energy and width of EGDR = 17:2 MeV and GDR = 4:5 MeV, respectively. For the 43 (isoscalar) 2+ giant quadrupole resonance (GQR) we take EGQR = 15:2 MeV and GQR = 4:5 MeV. The centroid and width for the GQR are approximations based on the systematics of GQR excitation in nickel isotopes [55] and are not experimental values. Looking at the total number of channels involved in our calculation we have, 35 3 + 35 5 + 35 5 + 1 = 456 which includes all magnetic substates plus an additional channel for the 0+ ground state. For practical purposes, to reduce computational intensity, considering only the major dynamical e ects arising from the coupling of the PDR with the GQR via the dominant E1 interaction at relativistic energies. A reduction by a factor of 2 is possible by implementing a coarser binning of the PDR and GQR states introducing only a loss of accuracy at the level of 10%. Calculations using all the channels above will converge to within 1%. Figure 4.8. Coulomb excitation cross section as a function of the excitation energy of 600 MeV/nucleon 68Ni projectiles incident on 197Au targets. The lled circles represent the calculations using rstorder perturbation theory, while the lled squares are the results of coupledchannel calculations. In Fig. 4.8 we show, for 600 MeV/nucleon 68Ni projectiles incident on 197Au targets, 44 the rstorder Coulomb excitation cross sections as a function of the excitation energy for the PDR and GQR separately. Calculations using rstorder perturbation theory are denoted by lled circles, while the lled squares represent the coupledchannel calculations. The rstorder Coulomb excitation cross sections, as shown in Ref. [?], can be obtained by means of the relation d dE = X L N L(E) E ( ) L(E) (4.39) where N L are the virtual photon numbers of multipole L and ( ) L are the real photon cross sections with multipolarity L. The virtual photon numbers include the same absorption coe cient as in Eq. (4.37) [51]. Summing over the relevant multipoles, here E1 and E2 stand for 1 and 2+ excitations, respectively. The gure shows that the coupling between these states has a visible impact on the energy dependence of the cross sections. According to the Brink{Axel hypothesis, the excitation of a giant resonance on top of any other state in a nucleus is possible [56, 57]. Therefore, the couplings are a manifestation of (PDR GQR)1, (PDR PDR)2+, (PDR GDR)2+ and (GDR GQR)1 states which we investigate, as they build up components of the PDR, GDR and GQR. Our ndings suggest the importance for the reliable extraction of the experimental strength of the PDR relative to the GDR. The dynamical calculations show that both the strength and width of the PDR are modi ed appreciably due to the coupling to the GQR. Fig. 4.8 shows separately that the population of PDR and GQR states are modi ed. The 1 states in the GDR region are a ected very weakly and so are left out of the gure. In the high energy region the GDR excitation dominates by a factor of 2{3 times that of the 2+ states. The important modi ca tions in the excitation spectrum appear from the couplings PDR$GQR$PDR by E1 elds, while couplings from PDR$GDR$PDR by E2 elds contribute very little to the 1 states in the PDR energy region. In Fig. 4.8 we notice that the tails of the PDR, and to a lesser extent those of the GQR, are appreciably modi ed. A small shift to the peaks is also seen, although barely visible for the PDR, it is evident for the GQR. It is important to also keep in mind that the strength and shape of the PDR will also be modi ed by the low energy 45 tail of the GDR. For our case considered here, a GDR strength on the order of 3.8% lies within the region of the PDR meaning that the PDR shape will only slightly be in uenced by this low energy tail. However, these e ects have been considered in the experimental analyses [52, 53]. In this work we are interested in the higherorder e ects which have been so far ignored. Figure 4.9. Coulomb excitation cross section as a function of the excitation energy of 68Ni projectiles incident on 197Au targets at two laboratory energies. The lled circles represent the calculations using rstorder perturbation theory, while the lled squares are the results of coupledchannel calculations. In Fig. 4.9 the energy region of the pygmy resonance has been singled out and we 46 plot the results of our calculations for two di erent bombarding energies: 100 MeV/nucleon and 2 GeV/nucleon, using the same notation as in Fig. 4.8. The coupling e ects change dramatically. For the lower energy interaction the in uence of the giant resonances increases appreciably for the response in the energy region of the PDR, while at the higher energy collision this e ect is much smaller and shows a slight tendency th decrease the PDR ex citation cross section. This is expected since for energies around 100 MeV/nucleon the E2 eld is dominant, with an increase to the excitation of the GQR and consequently a strong feedback to the PDR from subsequent E1 transitions. Figure 4.10. Coulomb excitation cross sections of the PDR as a function of the bombard ing energy of 68Ni projectiles incident on 197Au targets. The lled circles represent the calculations using rstorder perturbation theory, while the lled squares are the results of coupledchannel calculations. The Coulomb excitation cross sections of the PDR as a function of the bombarding energy of 68Ni projectiles incident on 197Au targets is given in Fig. 4.10. Calculations using rstorder perturbation theory are represented by the lled circles, while the lled squares 47 are the results of our coupledchannel calculations. Its apparent that at lower energies the deviation is more pronounced. At 600 MeV/nucleon the cross section for excitation of the PDR changes from 80.9 mb obtained with the virtual photon method to 92.2 mb with the coupledchannels calculation. A re ection of this e ect in the extracted PDR strength from the experimental data amounts to an appreciable change of 14%. This reduction is approximately the same amount of the strength needed to reproduce the experimental data. Calculations have also performed for 68Ni + 208Pb at 503 MeV/nucleon, corresponding to the experiment of Ref. [53]. To rstorder, the Coulomb excitation cross section for the PDR in 68Ni is found to be 58.3 mb, with the inclusion of the e ects of coupling to the giant resonances, the cross section increases to 71.2 mb, i.e., an important 18.1% correction. The dipole polarizability is de ned as D = ~c 2 2 Z dE (E) E2 (4.40) where (E) is the photoabsorption cross section. From Ref. [53] the extracted experimental value of D is 3.40 fm3 while in order to reproduce the experimental cross section with our dynamical calculations we have D = 3:16 fm3, a small but nonnegligible correction. Assuming a linear relationship between the dipole polarizability and the neutron skin [58], a reduction to the neutron skin from 0.17 fm, as reported in Ref. [53], to 0.16 fm is expected. A correction such as this lies within the experimental uncertainty of 7% for D and 0.02 fm for the neutron skin [53]. However, consideration of the coupling e ects should be taken into account in the future as more precise data becomes available, in particular, if the measurement is performed at lower bombarding energies. 48 Chapter 5 CONCLUSIONS We conclude that probabilities of giant resonances due to the large Coulomb excitation in heavy ion collisions at energies around and above 100 MeV/nucleon, the excitation of the PDR is also modi ed due to the coupling between the 1 and 2+ states. Our calculations, utilizing a Lorentzlike distribution for simplicity of the electromagnetic response and sum rules, are carried out without a detailed nuclear structure model. Future investigations carried out for nearly \abinitio" calculations based on a microscopic theory, coupled with a proper reaction mechanism, might be possible. A known alternative using individual states calculated by RPA or other microscopic models together with higher order perturbation theory, have already used in previous studies of multiphonon resonances [59]. Finally, the use of an advanced mean eld timedependent method such as that developed in Ref. [33] is also available. Deriving rather accurate dipole strength distributions from the electromagnetic excitation of the PDR is mainly of relevance to the extraction of the dipole polarizability [53], which is an important observable for constraining the symmetry energy, and is thus also important for better understanding the properties of neutronstars. A particularly important aspect for the polarizability is the lowenergy response due to the inverse weighting with energy. This opens exciting possibilities for studies of the pygmy resonance in nuclei and its use as a tool for applications in nuclear astrophysics. References [1] C.A. Bertulani and G. Baur, Phys. Rep. 163, 299 (1988). 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[52] O. Wieland, et al., Phys. Rev. Lett. 102, 092502 (2009). [53] D.M. Rossi, et al., Phys. Rev. Lett. 111, 242503 (2013). [54] D. Savran, T. Aumannnd, A. Zilges, Prog. Part. Nucl. Phys. 70, 210 (2013). [55] D.H. Youngblood, Y.W. Lui, U. Garg, R.J. Peterson, Phys. Rev. C 45, 2172 (1992). [56] D.M. Brink, PhD thesis, Oxford University, (1955). [57] P. Axel, Phys. Rev. 126, 671 (1962). [58] J. Piekarewicz, Phys. Rev. C 83, 034319 (2011). [59] V.Yu. Ponomarev, C.A. Bertulani, Phys. Rev. C 57, 3476 (1998). Vita Since graduating in 2008 from Hendrickson High School, Nathan S. Brady spent the rst two years exploring di erent career interests before discovering an obsession for physics. Nathan enrolled at The University of Nebraska at Kearney in 2010 where he pursued degrees in both physics and mathematics. There he was awarded several grants and awards including a NASA funded summer internship at the City College of New York. After graduating in 2014 Nathan enrolled at Texas A&M  Commerce to pursue a Master of Science in Physics. While attending he received several awards, including Outstanding Graduate Student and a summer Graduate Assistant Research. Nathan was also selected from a pool of candidates from around the world to participate in the TALENT workshop over the summer in Caun, France. In the Fall of 2016 Nathan was awarded the Master of Science degree and admitted into the PhD program at Texas A&M University  College Station. Permanent address: Nathan S. Brady Department of Physics and Astronomy Texas A&M UniversityCommerce P.O. Box 3011 Email: bradynsb@gmail.com 53
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Title  RELATIVISTIC COULOMB EXCITATION OF THE PYGMY DIPOLE AND GIANT RESONANCES 
Author  Brady, Nathan Scott 
Subject  Physics; Nuclear physics and radiation 
Abstract  RELATIVISTIC COULOMB EXCITATION OF THE PYGMY DIPOLE AND GIANT RESONANCES A Thesis by Nathan S. Brady Submitted to the O ce of Graduate Studies of Texas A&M UniversityCommerce In partial ful llment of the requirements for the degree of MASTER OF SCIENCE December 2016 RELATIVISTIC COULOMB EXCITATION OF THE PYGMY DIPOLE AND GIANT RESONANCES A Thesis by Nathan S. Brady Approved by: Advisor: Carlos Bertulani Committee: BaoAn Li William Newton Head of Department: Matt Wood Dean of the College: Brent Donham Dean of Graduate Studies: Arlene Horne iii Copyright c 2016 Nathan S. Brady iv ABSTRACT RELATIVISTIC COULOMB EXCITATION OF THE PYGMY DIPOLE AND GIANT RESONANCES Nathan S. Brady, MS Texas A&M UniversityCommerce, 2016 Advisor: Carlos Bertulani, PhD Nuclear collisions involving heavy charged nuclei traveling at relativistic speeds can bring one or both nuclei to an excited state. In collisions where the target is excited by the electromagnetic eld of the projectile the process is called Relativistic Coulomb Excitation (RCE). Classical approximations are a good starting point for describing RCE, however, more accurate predictions require a quantum mechanical treatment. Collisions which result in RCE have a certain probability to displace a large fraction of the nuclear matter inside a nucleus. This characteristic response is intrinsic to all nuclei con taining more than a few protons and neutrons and is called a Giant Resonance (GR). Many classical and microscopic models have been used to describe this phenomenon including zero range Skyrme forces, Mean eld Approximations, and the Random Phase Approximation (RPA). As the number of neutrons become much larger than the number of protons within the nucleus the excess neutrons begin to form a skin beyond the typical charge radius. This neutron skin can be excited through RCE and cause the skin to oscillate against the strongly bound symmetric core. The strongest oscillation between the core and neutron skin is called the dipole mode and the resulting phenomenon is referred to as the Pygmy Dipole Resonance (PDR). v This thesis will look at how the PDR is in uenced by the larger strength of the GR for interactions above 100 MeV/nucleon. As the dynamics of the collision changes, the probability of exciting a PDR changes considerably. The e ects due to the coupling of the PDR to the GR indicates a need for improved theoretical studies for reactions at these collision energies. vi ACKNOWLEDGMENTS I would rst like to thank my thesis advisor, Dr. Carlos Bertulani of the Physics Department at Texas A& M UniversityCommerce (TAMUC). He was an exemplary mentor and I consider him to be a true friend. His guidance helped me through the most trying time of my career. I would also like to recognize Prof. Dr. Thomas Aumann of the Physics Department at Technische Universitat Darmstadt. Without his collaboration, the work presented here would not have been possible. My future career has been enriched and made possible with the encouragement and support of both Dr. Matt Wood, head of the TAMUC Physics Department and Dr. BaoAn Li, Regents professor of TAMUC. The classes taught by each of them continue to be an invaluable resource as I move forward with my studies. The friends I have made since joining the physics department at TAMUC have been one of the most rewarding aspects of my life. In light of that, I would also like to acknowledge my fellow colleague, o cemate, and friend, James Thomas, who also worked under Dr. Bertulani. Lastly, I would like to acknowledge my family for their unwavering support and encouragement as I pursue my passion for physics. I must also thank Sarah Cantu for putting up with me through it all. Her support has been a major factor as I worked towards nishing this thesis. Thank you,  Nathan S. Brady Contents List of Figures x Chapter 1 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2 7 COULOMB EXCITATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Underlying Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Potential Fields and Gauge Transformations . . . . . . . . . . . . . . 8 2.1.2 Lorenz Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Nuclear Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.1 HeavyIon Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.2 Classical Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Semiclassical Theory of Coulomb Scattering . . . . . . . . . . . . . . . . . . 18 Multipole Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Relativistic Coulomb Excitation . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1 Equivalent Photon Method . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.2 CoupledChannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Chapter 3 29 NUCLEAR STRUCTURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1 Giant Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.1 Classical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 vii CONTENTS viii 3.1.2 Microscopic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Pygmy Dipole Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Chapter 4 38 PRESENTATION OF FINDINGS . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Chapter 5 48 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 References 48 Vita 53 List of Figures 2.1 Interactions between fermions can be represented by a Feynman diagram. The solid lines represent fermions and the wavy line is the force carrier, i.e., the virtual photon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 The interaction picture for a particle (lower solid line) colliding with a nuclear target (shaded circle), through the exchange of a virtual photon. The products are depicted by the outgoing solid lines. . . . . . . . . . . . . . . . . . . . . 13 2.3 Scattering of an incident beam of particles by a center of force. . . . . . . . . 15 2.4 A relativistic charged projectile incident on a target with impact parameter larger than the strong interaction radius. The transfer of virtual photons is shown (wavy lines) coming from the Lorentz contracted i.e., pancake shaped, electromagnetic eld. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.5 Example cross section for the GR in Lead208. . . . . . . . . . . . . . . . . . 32 3.6 The dipole mode of the pygmy resonance in the GoldhaberTeller (GT) and SteinwedelJensen (SJ) modes. . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.7 Strength function for the E1 RPA response in 68Ni calculated with formalism described in Ref. [43]. The calculation is performed for several Skyrme inter actions, shown in the gure inset. The arrow shows the location of the pygmy resonance. (For interpretation of the references to color in this gure legend, the reader is referred to the web version of this article.) . . . . . . . . . . . . 39 ix LIST OF FIGURES x 4.8 Coulomb excitation cross section as a function of the excitation energy of 600 MeV/nucleon 68Ni projectiles incident on 197Au targets. The lled circles represent the calculations using rstorder perturbation theory, while the lled squares are the results of coupledchannel calculations. . . . . . . . . . . . . 43 4.9 Coulomb excitation cross section as a function of the excitation energy of 68Ni projectiles incident on 197Au targets at two laboratory energies. The lled circles represent the calculations using rstorder perturbation theory, while the lled squares are the results of coupledchannel calculations. . . . . . . . 45 4.10 Coulomb excitation cross sections of the PDR as a function of the bombarding energy of 68Ni projectiles incident on 197Au targets. The lled circles represent the calculations using rstorder perturbation theory, while the lled squares are the results of coupledchannel calculations. . . . . . . . . . . . . . . . . 46 1 Chapter 1 INTRODUCTION Understanding the principles by which our universe is guided and the laws that restrict its evolution have always been a large motivation for scienti c discovery. Nuclear physics is the study of phenomena which govern the interactions between matter at the femtometer scale. These interactions are built up from the residue of the strong nuclear force, described by quantum chromodynamics, and are subject to electromagnetic interactions described through quantum electrodynamics. Particle collisions are used to investigate the dynamics of interacting particles. Mea surements are made by extracting information, (e.g., mass, momentum, spin, ...) carried by the products of these collisions and inferring the original conditions which produced the observations. The probability of producing a particular event is calculated through its cross section which tells us where to expect certain products produced by said event. Statement of the Problem In peripheral collisions the distance between the two nuclei under consideration are just large enough so as not to directly interact by the strong nuclear force. The event is then dominated by Coulomb interactions, and at relativistic speeds it is better known as Relativistic Coulomb Excitation (RCE), which is a wellestablished tool used for studies in nuclear structure [1]. During an RCE event, the electric eld generated by the incident heavy ion transfers energy to the nucleons within the other nucleus exciting it to a higher energy level. At certain excitation energies the nuclear response will be ampli ed considerably and is described as the characteristic resonance of the nucleus. These characteristic resonances which dominate the low energy excitations around 8 MeV and higher for all nuclei with more than a few nucleons (i.e., protons and neutrons) are known as Giant Resonances (GR). These phenomena, rst observed by Bothe and Gentner 2 in 1937 [2], were shown to have a high probability of photoabsorption at these energies. These phenomena are well studied, see for instance [3, 4] and theories, both classical and microscopic, reproduce the observations well. An associated phenomenon which appears in nuclei with more neutrons than protons is known as the Pygmy Dipole Resonance (PDR) and is the collective vibration of the neutrons against a symmetric nuclear core. Further reading on the pygmy resonance can be found in Ref. [5] Purpose of the Study The PDR was suggested in 1987 as a possible excitation in neutronrich nuclei by Kubono, Nomura, and collaborators [7]. Theoretical support was later established by Ikeda [8] and collaborators. Experimental evidences for the existence of a collective low energy response in neutronrich nuclei, far from the valley of stability, took nearly two decades more to emerge. Initially the direct breakup of light and looselybound projectiles, such as 11Be and 11Li were suggestive of a collective nuclear response, i.e., a PDR, but was later shown to be a direct Coulomb dissociation of the weaklybound valence nucleons [9]. This suggests that the characteristics are poorly understood and further investigations, both experimentally and theoretically, are needed. The intent of this study is to investigate the dynamics of these associated phenomena produced from RCE experiments. A coupling between the PDR and the various excited vibrational modes comprising the GR is investigated. The calculation of this coupling is important to investigate since the strength and energy of the PDR would be a ected appre ciably. This in turn would a ect the calculation of the neutron skin, which is characterized by the PDR. Hypothesis The Coulomb excitation of PDR at 100 MeV/nucleon and above are investigated. As the dynamics of the collision evolve, the question of the in uence on the PDR by the larger 3 GR is addressed. The coupling of the GR to the PDR will a ect the excitation probability considerably. Di erent collision energies will a ect the amount this coupling changes the cross section of the PDR. Research Questions An important aspect of investigating the PDR is the calculation of the neutron skin which tells us more about neutron matter and how it relates to neutron stars. We will attempt to answer the following questions: How does the collision energy a ect the centroid energy and width of the PDR? How much does this change the calculation of the neutron skin and polarizability? How do the separate major modes of oscillation of the GR (i.e., Giant Dipole and Giant Quadrupole) a ect the cross section of the PDR? Is there a preference to coupling with either of the modes? Signi cance of the Study A revitalized interest in neutron matter and nuclei far from the line of stability has emerged. The ability to measure the neutron radius has historically been an experimental challenge which stems from the neutral charge of the neutrons. Since the PDR has been shown to emerge in neutronrich nuclei it has been suggested that it could be used to constrain the neutron skin thickness [6]. Accurate measurements of the neutron skin and extraction of the dipole polarizability are important for constraining the symmetry energy associated with the binding of nucleons in the nucleus. How the GR a ects the PDR is then an important question; for small changes to the observed PDR the measurement of the neutron skin will be a ected. 4 Method of Procedure Since the objective was to determine how the multipole modes of the GR a ected the PDR cross section, we utilized current predictions of both the PDR and GR's centroid energy and width. Two practical examples of reactions, 68Ni + 208Au and 68Ni + 208Pb, are used to facilitate discussions on the e ect of coupling between the GR and the PDR. De nition of Terms Nucleon. The constituent particles of nuclei, i.e., protons and neutrons, which come together to form the nucleus of an atom. Elements di er by the number of protons, Z, and isotopes, elements with the same Z, di er by the number of neutrons, N. [3, 10] Nuclei. Term which refers to the nucleus of all elements composed of nucleons. [3,10] Di erential Cross Section. Rate of scattered particles detected at some angle, d =d , where is the solid angle. Interpreted as the probability of detecting a particle within the given solid angle. [3, 10] Cross Section. An e ective \area" which an interaction can occur. [3, 10] Coulomb Interaction. Class of interactions between particles through their mutual electromagnetic elds. [11] Nuclear Excitation. The observed reaction of nuclear matter to external stimuli resulting in the jump of one or more nucleons within the nucleus into a higher energy level. The nucleus is then said to be in an excited state. [3, 10] Relativistic Coulomb Excitation. Excitation of a nucleus by the electromagnetic eld of a charged particle moving at relativistic speeds. [12] 5 Multipole Expansion. The decomposition of the angular dependence of a potential into increasingly ner segments. Multipole moments comprising the rst few terms of the expansion are de ned as the Monopole (l = 0), Dipole (l = 1), and Quadrupole (l = 2) moments. [11] Giant Resonance. A highfrequency collective nuclear response involving an appre ciable number of nucleons. Characterized by the large photoabsorption cross section observed in all nuclei with more than a few nucleons. [3, 4] Giant Dipole Resonance. The second and most prominent multipole moment of oscillation which makes up the Giant Resonance. Described as the collective vibration between the protons and neutrons. [3, 4] Giant Quadrupole Resonance. The third multipole moment of oscillation and second largest contribution to the Giant Resonance. [3, 4] Pygmy Dipole Resonance. Similar to a giant resonance, but with a smaller strength. [3, 4] Limitations Calculations utilize well known empirical data when necessary such as; the width of the resonance, location of the centroid energy, and total nuclear cross sections NN corrected for inmedium interactions. The location of the Pygmy Dipole Resonance is taken from experimental data while the width used is more consistent with theoretical calculations. This restricts our analysis of results to be in a qualitative manner. Delimitations The lowenergy resonances considered in this thesis are induced through the method of Relativistic Coulomb Excitation. Therefore, only heavyion reactions at 100 MeV/nucleon and above are considered. Since we are considering only Relativistic Coulomb Excitations, 6 we restrict ourselves to collisions at large impact parameters, i.e., no overlap of the two nuclei, so b > R1 + R2. Organization of Thesis Chapters Chapter 1 gives an outline of the study and motivations for this investigation. The problem of inducing the PDR and GR through Coulomb excitation is introduced, then a brief explanation of the coupling between the PDR and GR is given as the main focus of the thesis. Chapter 2 develops the theoretical background leading to the idea of relativistic Coulomb excitation in the context of nuclear reactions. We also show how the equivalent photon method can be used in the theory of relativistic Coulomb excitation and show how the idea of equivalent photons appears in many aspects of physics. The coupledchannels calculation, important for determining how the PDR and GR are coupled, is also introduced at the end of Chapter 2. Chapter 3 takes the ideas developed in the previous chapter and applies it to the development of nuclear physics. The giant and pygmy dipole resonances are introduced and theoretical models are brie y covered. Chapter 4 gives results on how the PDR is coupled to the GR. These results are then compared to current theoretical predictions using perturbation theory. Chapter 5 summarizes the results of the calculation and we give our conclusions. 7 Chapter 2 COULOMB EXCITATION 2.1 Underlying Theory The theory of electrodynamics, which describes the interactions between charged par ticles, plays a pivotal role in modern physics and the development of the standard model. Forgoing the highly interesting yet impertinent pre19th century studies in electrostatics, I instead choose to begin with the monumental work of Maxwell in 1873 [13, 14], which stem from numerous experimental works including the equally notable, experimental physicist, Michael Faraday. The second volume of Maxwell's treaties [14], brought together, in a mathematically coherent way, the presupposed distinct interactions of electricity and magnetism into a uni ed theory of electromagnetism with light as its propagator. In Maxwell's work a collection of di erential equations describing the propagation of the electromagnetic elds through vacuum were formulated. These equations as amended by Heaviside are, r E = 0 (2.1) r B = 0 (2.2) r E = @B @t (2.3) r B = 0J + 0 0 @E @t (2.4) setting = J = 0 for a charge free region, the vacuum equations are reproduced. In the case of electromagnetic elds in vacuum we may take the curl of both (2.3) and (2.4) and use the identity, r (r ) = r(r )r2 , which gives the well known wave equation for electromagnetic radiation. 8 @2E @t2 = c2 r2E @2B @t2 = c2 r2B with the velocity of the wave, through a nonconducting medium, denoted by c = ( 0 0)1 2 , the speed of light. Maxwell's work did away with action at a distance and cemented the idea of elds into modernday physics. The theory of electromagnetism has since been extended beyond its classical origins into a fully consistent quantum mechanical theory. 2.1.1 Potential Fields and Gauge Transformations The Maxwell equations (2.12.4) are one of the greatest strides made in physics paving the way for more modern theories. Using vector notation as above it is immediately apparent that equation (2.2) has a freedom in de ning the magnetic eld as B = r A where the transformation A ! A + r is made with (r; t) arbitrary. Since the curl of any scalar vanishes, B = r (A + r ) = r A + :0 r r , the magnetic eld remains unchanged. Applying the transformation preserves the magnetic eld, however, the electric eld changes according to equation (2.3), r E = @B @t where B = r A ) r E = @ @t (r A) = r @A @t ) r E + @A @t = 0 Since the curl of a gradient is always zero we can assign the term inside the parenthesis equal 9 to the negative of some scalar potential so that, ) E + @A @t = r ) E = r @A @t (2.5) De ning the magnetic eld as the curl of some vector eld A results in a new de nition for the electric eld given by equation (2.5). This is true for any arbitrary gauge. Now using the same transformation on equation (2.5), E = r @A @t where A ! A + r ) E = r @ @t (A + r ) = r @A @t r @ @t = r + @ @t @A @t by making a new transformation ! @ @t equation (2.5) remains unchanged. This gives the nal solutions to the partial di erential equations (2.1  2.4) as, E = r @A @t (2.6) B = r A (2.7) where A and are the magnetic vector and electric scalar potential elds respectively and are a consequence of the gauge freedom inherent of Maxwell's equations. 10 2.1.2 Lorenz Gauge Many di erent gauges have been developed to exploit this \freedom" and is known as gauge xing [11]. One such gauge transformation, which ts nicely with special relativity, is the Lorenz gauge. Named for the Danish physicist Ludvig Lorenz, it introduces a trans formation to the scalar and vector A potentials while maintaining the solutions to the Maxwell equations (2.1  2.4) as above. Beginning from equations (2.1) and (2.4) then inserting solutions (2.6) and (2.7), we get the equations of motion for a r E = 0 where E = r @A @t ) r r @A @t = 0 ) r2 + @ (r A) @t = 0 (2.8) r B = 0J + 0 0 @E @t where B = r A ) r (r A) = 0J + 1 c2 @ @t r @A @t ) r(r A) r2A = 0J 1 c2 @ @t r 1 c2 @2A @t2 ! ) r2A 1 c2 @2A @t2 ! = 0J + r r A + 1 c2 @ @t (2.9) In equation (2.9) a necessary condition in order to preserve Maxwell's equations is, r A + 1 c2 @ @t = 0 (2.10) This is known as the Lorenz condition and allows us to write the equations of motion in 11 their nal form, r2 1 c2 @2 @t2 = 0 (2.11) r2A 1 c2 @2A @t2 = 0J (2.12) where we used equation (2.10) to solve for r A and insert into equation (2.8). These are the symmetric equations of motion for a charged particle. By inspection we see a similarity between these and the wave equations for electro magnetic radiation in vacuum (i.e. = J = 0). Applying the transforms A ! A0 +r and ! 0 @ =@t to the Lorenz condition, equation (2.10) leads to, r A + 1 c2 @ @t ! r A0 + 1 c2 @ 0 @t + r2 1 c2 @2 @t2 = 0 where A0 and 0 are the original potentials. To preserve our original condition in equation (2.10) for our initial potentials A0 and 0 we require, r2 1 c2 @2 @t2 = 0 which is simply the wave equation for a scalar function (r; t). This means that under the Lorentz gauge we have an in nite number of solutions which satisfy the Maxwell equations and everything is treated as a wave. This blends nicely with the implications of special relativity and naturally works in quantum mechanics where everything is a wave. A Quantum Theory The laws which govern our universe are quantum mechanical and any theory which hopes to succeed in its attempt to describe physical reality must be consistent with quantum 12 mechanics. The theory of quantum electrodynamics (QED) is marked as the rst successful quantum eld theory, consistent with relativity. Due to the tremendous success of QED, it has become the standard measure of a theory's acceptance. In our formulation of QED the interaction between charged particles (e.g., electrons, protons, mesons, : : :) is through the exchange of photons. These photons are the mediators of the electromagnetic force. The use of Feynman diagrams, introduced by Richard Feynman in the 1940s, has become the preferred method in describing particle interactions. This pictorial method was a simple way of tracking the overwhelmingly diverse in teractions plaguing QED at the time. Following Feynman's steps provided a systematic approach in diagramming these interactions. Figure 2.1. Interactions between fermions can be represented by a Feynman diagram. The solid lines represent fermions and the wavy line is the force carrier, i.e., the virtual photon. Figure 2.1 gives the standard depiction of electron scattering, depicted as a Feynman diagram. In 2.1 the horizontal axis represents the forward ow of time while the vertical axis gives motion in 3space. Fermions are represented by solid lines while the virtual photon (wavy line) is the mediator of the electromagnetic interaction. This virtual photon connecting the two particles removes the need for action at a distance. For a more in depth discussion on QED and the use of Feynman diagrams in quantum eld theory refer to Peskin and Schroeder [15]. The development of this beautiful concept of gauge boson exchange as the force car 13 riers revolutionized theoretical nuclear/particle physics, beginning with the photon as the mediator for the Coulomb interaction. Using this concept we proceed into discussions of nuclear interactions, a manybody quantum problem. Starting with, as one might expect, a classical approximation to the process of Coulomb excitation. 2.2 Nuclear Excitation Excitation of nuclei by means of providing energy through electromagnetic interactions is known as Coulomb excitation. This situation is observed most easily in nuclear collisions where a target nucleus is perturbed by an external electromagnetic eld. This may be achieved through a number of ways, however, the excitation through scattering particles will be this works focus. 2.2.1 HeavyIon Collisions When the number of protons and neutrons involved in a nuclear collision becomes large we refer to these as heavy ion collisions. These heavy nuclear collisions present a dynamic system with a very rich collection of interesting phenomenon. In this situation a charged ion Z1 interacts with another charged particle Z2 through the exchange of a photon Figure 2.2. The interaction picture for a particle (lower solid line) colliding with a nuclear target (shaded circle), through the exchange of a virtual photon. The products are depicted by the outgoing solid lines. Figure 2.2 gives a diagram representation of two interacting nuclei Z1 and Z2 where 14 the charged particles are shown to interact through the Coulomb force, represented by a mediating particle the photon (wavy line). This idea of photon exchange will be central to our discussions later. The three outgoing particles are what results from the interaction and are the only things observed. By looking at the incoming particles and measuring the outgoing products the scientist can reconstruct what interaction took place. Several di erent ways to represent these types of interactions exist. However, the simplest method is to use Feynman diagrams. The bene t of including, implicitly, the conservation laws through vertex analysis still makes it preferable to other methods. For a brief overview of Feynman diagrams and rules see [15, 16]. 2.2.2 Classical Scattering Early investigations into nuclear physics typically involved low energy interactions between charged particles of low mass, e.g., electrons, protons, alpha particles, and a xed target nucleus. The most notable of these experiments, which revealed the nucleus as be ing composed of a dense central core surrounded by electrons, were conducted by Ernest Rutherford. In the aforementioned series of experiments, often called the Rutherford gold foil experiments, the low energy scattering of an alpha particle beam o a target of thin gold foil resulted in the detection of scattered alpha particles back towards the source. Backscattering as it's now called can be represented as a onebody elastic scattering problem which describes the trajectory of an incident charged particle as it interacts with a central force. Scattering particles is still the preferred method used to investigate nuclear structure. Since the need to probe deeper into the nucleus is more necessary, scattering particles achieves this by increasing the interaction energy and decreasing the impact parameter of the colliding nuclei. This requires the construction of larger and more powerful colliders. 15 Rutherford Rutherford scattering, as the low energy method is more commonly referred, is the classical approximation to a quantum mechanical system. In this system the interacting particles carry with them a charge, Z(1;2)e, and momentum, p(1;2), where the indices denote projectile and target, respectively. A particle is said to be scattered when its direction Figure 2.3. Scattering of an incident beam of particles by a center of force. of motion is altered. Coulomb proved that the electric force follows an inversesquare law similar to the gravitational force which implies that the orbital equations are applicable, in the classical limit. Given the charge of each particle and their relative velocity, v at in nity, we may calculate the strength of the Coulomb interaction by the Sommerfeld parameter = Z1Z2e2 ~v : (2.13) A classical approximation is appropriate when, 1. This is possible when either, Z1Z2 16 137, or, v c. The trajectory of the projectile, Z1, approaches the target, Z2, as in Figure 2.3. The collisions considered in classical scattering are not strong enough to overcome the Coulomb barrier. However, we must consider the distance of closest approach, a, between the two nuclei. From the conservation of energy, E = 1 2 mv2 0 = Z1Z2e2 a we solve for a giving the distance in terms of the charges and relative speed, a = 2Z1Z2e2 mv2 0 This is an important parameter since it gives us a known value in the projectile's trajectory. In this arbitrary collision the trajectory of the projectile is dependent on the impact parameter and its kinetic energy. The \scattering angle", , is uniquely determined for classical scattering. A scattered particle, in relation to the target, is observed passing through a cross sectional area ( ). The area scattered is dependent on the scattering angle . For a small segment of the scattering angle d multiplied over the azimuthal coordinate 2 sin the portion of solid angle is written as d = 2 sin d . Scattering strength is determined by the energy and angular momentum l. The angular momentum of a particle or planet is given as, l = b p0 = b p 2mE where p0 = mv0 is momentum and b is called the \impact parameter". The impact parameter is the distance between the centers of two scattering particles. Assuming from a classical perspective that particles at di erent impact parameters will have di erent scattering angles, the number of particles scattered are determined by the beam intensity, I = N=A, where N is the total number of incident particles passing through area A. 17 Using the di erential cross section ( ) de ned as the ux, ( ) of particles per beam intensity we have, ( ) = ( ) I (2.14) We can solve for the number of particles scattered into solid angle d by using the de nition of ux, ( ) = dn=d . The number of particles, n, incident on the target is dependent on the impact parameter as well as the scattering angle allowing the ux to be written as, ( ) = dn dA dA d where I = dn=dA, the incident beam intensity. Inserting into equation (2.14) we have, ( ) = ( ) I ) I ( ) = dn dA dA d multiplying by d on both sides and substituting in I = dn=dA we get, I ( ) d = I dA ) 2 I ( ) sin d = 2 I b db where we used the incident beam cross section A = b2 ) dA = 2 b db. The di erential cross section is now dependent on the scattering angle , solving for ( ) gives, ( ) = b sin db d (2.15) a straightforward relation between the impact parameter and the scattering angle measured. Additionally deriving the impact parameter from the eccentricity of a hyperbolic orbit equa 18 tion (2.15) becomes, ( ) = 1 4 ZZ0e2 2E !2 csc4 2 (2.16) the famous Rutherford Scattering equation. The relationship between equations (2.15) and (2.16) can be found in Goldstein's Classical Mechanics. [17] Rutherford's equation is relevant for low energy scattering where the incoming particle doesn't penetrate the targets Coulomb barrier such as the gold foil alpha particle scattering. Once the projectile and target begin to overlap, e.g., 25 MeV for +Pb, the scattering becomes inelastic and Rutherford's equation no longer applies. 2.3 Semiclassical Theory of Coulomb Scattering The validity of a semiclassical approach to the method of Coulomb excitation is well known for studies on nuclear excitation. As stated before it provides a way to study reactions without having to include the manybody nuclear forces. This is a well established method and has been extensively covered by several investigators. In the electromagnetic excitation process we describe an interaction Hamiltonian as Hint = HP + HT + V(r(t)) where the matrix HP (HT ) is the projectile(target) Hamiltonian and V(t) is a timedependent term describing the electromagnetic interaction between both projectile and target. An appropriate choice for the frame of reference (i.e., target or projectile) reduces the above interaction Hamiltonian. In our treatment of the problem we choose the frame of reference of the target and consider only stable projectiles with the ground state Hamiltonian HT . The excitation is then built from the ground state of the target. The new Hamiltonian 19 must satisfy the timedependent Schr odinger equation Hint (t) = i~ d (t) dt (2.17) Performing an expansion on the total timedependent wave function (t) in terms of the set of orthogonal eigenfunctions which form a complete basis (t) = X n an(t) neiEnt=~ (2.18) where the ground state Hamiltonian of the target nucleus satis es the eigenvalue equation HT n = En n (2.19) using the newly expanded wave function (2.18) with the interaction Hamiltonian given as Hint = HT + V(t) we obtain a new form of the timedependent Schr odinger equation (2.17) given as i~ d dt X n an(t) neiEnt=~ ! = HT + V(t) X n an(t) neiEnt=~ i~ X n _an(t) iEn ~ an(t) neiEnt=~ = X n an(t) HT + V(t) neiEnt=~ Distributing through on both sides of the equation and using the relation (2.19) we are left with the equation, i~ X n _an(t) neiEnt=~ = X n an(t)V(t) neiEnt=~ 20 multiplying both sides by m and integrating over all space we have i~ X n _an(t)eiEnt=~ mn = X n an(t) Z 1 1 d3r mV(t) n ! eiEnt=~ i~ _am(t)eiEmt=~ = X n an(t)Vmn(t)eiEnt=~ where we used R 1 1 d3r m n = mn. The matrix elements for the electromagnetic interac tion potential are given as Vmn = Z 1 1 d3r mV(t) n = hmjV (t)jni solving for the di erential time dependent amplitude of state m gives, ) _am(t) = i ~ X n hmjV (t)jni an(t)ei(EmEn)t=~ (2.20) We now have a set of m coupled equation where the sum runs over all possible excita tion states n. The solutions to which are typically found by performing a multipole expansion on the potential V(r(t)). The Multipole expansion is introduced in the next section. Equation (2.20) can be interpreted as the transition amplitude of a particle from an initial state jii to some nal state jfi with excitation ~! = Ef Ei. The excitation is taken as the Fourier component to the transition frequency of the interaction Hamiltonian so that we have, afi(t) = i ~ Z 1 1 hfjHintjii ei(EfEi)t=~dt (2.21) The term hfjHintjii is the matrix form of the interaction Hamiltonian. Integrating over a short time interval the transition amplitude is then, am(t) = i ~ Z T 0 Vmn(t)ei(EmEn)t=~dt (2.22) 21 since for short time intervals the system returns to its original state m so that an(t) = 1 for m = n else an(t) = 0. Multipole Expansion A multipole expansion of a potential is a way to approximate the e ective electro magnetic interaction between charged particles in terms of their intrinsic coordinates. In spherical polar coordinates this takes separate treatment of the radial, polar, and azimuthal coordinates, r, , and respectively. The Coulomb potential measured at r for a system of pointlike charged particles ei located at ri is given in general by the equation V (r) = X i ei jr rij (2.23) 1 jr r0j (r2 + r02 2rr0 cos )1=2 (2.24) where the primed coordinates indicate the location of charged particles and cos = cos cos 0+ sin sin 0 cos( 0) is the angle between the primed and unprimed vectors. Equation (2.24) depends on only the lengths of vectors r and r0 and the angle between them . This allows a multipole expansion to be made in terms of the mutually orthogonal Legendre polynomials of order l. 1 jr r0j = 1 r> X1 l=0 r< r> l Pl(cos ) where r< (r>) is the lesser(greater) distance between r and r0. By the addition theorem, as given in Appendix B of [18], a generalization to the geometric relation given by cos . This leads to an expression for the Legendre polynomial as a linear combination of spherical 22 harmonics, of order l, Pl(cos ) = 4 2l + 1 Xl m=l Y lm( 0; 0)Ylm( ; ) leading to a nal form for the expansion given as, 1 jr r0j = 1 r> X1 l=0 4 2l + 1 r< r> l Xl m=l Y lm( 0; 0)Ylm( ; ) Applying the above expansion, given in terms of spherical harmonics for each discrete point charge i at r0, equation (2.23) becomes, V (r) = X lm 4 2l + 1 1 rl+1Y lm( ; )M(El;m) (2.25) where M, the electric multipole moment, is given as, M(El;m) = X i eirli Ylm( i; i); m = l;l + 1; : : : ; l 1; l The sum is over each electric charge ei with position ri = (ri; i; i). Modifying the equation above, using the density operator (r) = X i ei (r ri); the multipole moment M(El;m) is now a tensor operator of rank l. Integrating over all space the operator takes on the general form, M(El;m) = Z d3r (r)rlYlm(n); n = r r (2.26) One postulate of quantum mechanics states that, for any measurable system which 23 has an associated physical observable can be represented by an operator. The multipole expansion of equation (2.23) into (2.25) puts it into a special class of quantum mechanical operators called the spherical tensor operators. 2.4 Relativistic Coulomb Excitation In the previous section we expanded the potential in terms of its multipoles and put the multipole moment in the form of an operator, a necessary step for its use in quantum mechanics. Now we must go further and put the method of Coulomb Excitation into terms that are consistent with relativity. The premise of Relativistic Coulomb Excitation (RCE) is taken from its low energy origins. As with Coulomb Excitation the fundamental assumption of RCE is that the nuclei do not interpenetrate. Reactions of this nature due to this process are short lived and enhanced due to Lorentz contraction of the eld. For the case of a projectile with energy, Elab 100 MeV/nucleon, the trajectory is nearly straight, i.e., negligible de ection, giving an interaction distance equal to the impact parameter b. We also consider only interactions involving the electric eld and the target or projectile, so that, b > RT + RP , is larger than the sum of the radii of the target and projectile, respectively. Using the same setup as with Coulomb Excitation, where we are in a target centered frame with the zaxis being the path of the projectile, we consider the case where v ' c. A particle traveling at these high energies will be Lorentz contracted in the direction of motion and can be described by the Li enardWiechert potential, (r; t) = Ze p (x b)2 + y2 + 2(z vt)2 where b is the impact parameter and the factor = (1 v2=c2)1=2 is the Lorentz factor making the component of the electric eld parallel to the direction of motion, +z, compact 24 Figure 2.4. A relativistic charged projectile incident on a target with impact parameter larger than the strong interaction radius. The transfer of virtual photons is shown (wavy lines) coming from the Lorentz contracted i.e., pancake shaped, electromagnetic eld. and pancake like. The above equation is the same electric potential eld from our discussions on Lorenz gauge. The vector potential eld used for a spinless particle following a classical trajectory is, A(r; t) = v c (r; t) where the velocity vector v = v^z. A multipole expansion can be made to the Li enard Wiechert potential in a similar way as was done in the previous section. Knowing the electric and vector potential eld equations we can easily obtain the 25 associated electric and magnetic eld equations from equations (2.6) and (2.7), Ez = Ze vt (b2 + 2v2t2)3=2 ET = Ze b (b2 + 2v2t2)3=2 BT = v c ET ; and Bz = 0 where the equations are separated by components parallel(tangent) to the direction of motion and are denoted by subscript z(T). For fast moving charged particles the interaction of the eld on the target is short. That is for 1 , by inspection of the denominator, the time is approximately, vt = b ! t = b v ' b c This shows the elds to be similar to two pulses of planepolarized radiation. In the case for radiation along the z direction, this is an exact analogy. Radiation along the T direction is only approximate due to the presence of a magnetic eld, however, this does not appreciably a ect the dynamics of the system. At speeds v = c the e ects of the Ez eld are minimal and allow us to add an additional eld, B = vEz=c. A calculation of the energy incident per unit area on the target due to an electromagnetic wave is now possible. Fourier transforming the Poynting vector S = E B, the intensity of radiation, I(!; b), is calculated with energy E = ~!. The probability of nuclear transition is then given by the integral, P(b) = Z I(!; b) (E )dE = Z n(!; b) (E ) d! ! (2.27) where (E ) is the photonuclear crosssection for photons of energy E and n(!; b) is the 26 number of equivalent photons in the electromagnetic wave at frequency ! and impact pa rameter b. The term n(!; b) in the above equation was rst introduced by Enrico Fermi [19] who proposed that the electromagnetic eld created by a charged particle can be equated to a ux of virtual photons. This idea of virtual photons mediating the electromagnetic interaction is obviously an important part to the theory of electromagnetism both classically (low energy scattering) and, as discussed above, quantum mechanically (QED). Fermi's work, published in 1924 by Zeitschrift f ur Physik and also, unconventionally, published in Italian by Nuovo Cimento [20], was extended by Weizs acer and Williams [21] to be used at relativistic velocities. This was essentially restoring the Lorentz factors in their proper places. What remains to be shown is how this Lorentz contracted electromagnetic eld inter acts with a target possessing charge. 2.4.1 Equivalent Photon Method The electromagnetic eld of a fast moving charged particle is Lorentz contracted in the direction of motion. Considering only collisions where the nuclei do not overlap, i.e., impact parameter b > RP + RT , which is the case for Coulomb excitation, the probability can be expressed in terms of the sum over all transition probabilities into a nal state If . The probability of Coulomb excitation is given as, PC(b;E ) = X f Z Pi!f (b) f (E )dE (2.28) where the sum is over all possible nal states with the density of nal states integrated over all energy transfers E = ~! = Ef Ei. The transition probabilities Pi!f are obtained from 27 the transition amplitude calculated in equation (2.21) so that, Pi!f (b) = a2 4 4 1 2Ii + 1 X Mi;Mf aif 2 Using WignerEckart theorem and respecting the orthogonality conditions of the Clebsh Gordon coe cients, which produce delta functions from the expansion coe cients between angular momentum eigenstates, the transition probability can be expressed in the form, Pi!f = Z2 1 e2 4 2a2 2 ~2 X LM B( L; Ii ! If ) (2L + 1)3 S( L;M) 2 (2.29) again where = E or M and B( L; Ii ! If ) is the reduced transition probability given as, B( L; Ii ! If ) = 1 2Ii + 1 X Mi;Mf hIiMijM( L;M)jIf ;Mf i 2 = 1 2Ii + 1 hIij M( L) jIf i 2 The transition probability given by equation (2.29) can now be plugged into equation (2.28). If we use the idea proposed by Enrico Fermi to approximate the interaction of the electric eld as a ux of virtual photons as in equation (2.27) then the probability becomes, PC(b;E ) = X L P L(b;E ) = X L Z dE E n L(b;E ) L (E ) where the sum is now over multipole states. The term n L(b;E ) is the virtual photon numbers and L is the photonuclear cross section for a given multipole. 2.4.2 CoupledChannels Coulomb Excitations are often dominated by transitions between the ground state and a continuum resonance state. Excitation into these states are treated as exact in a coupledchannels approach [22] where additional states are calculated as perturbations from 28 the ground state. In the coupledchannels approach an excitation into a continuum of states is called a doorway state, or the particular entrance taken into the continuum. The ground state is then coupled to the set of doorway states creating a coupling matrix. Amplitudes for such excitations are given as, (n)(E ) = h (E )jD(n) LMi and can be used as an expansion of the ground state potential. Integrating over all en ergy separations = E En, where E is the energy from the interaction and En is the centroid energy of the resonance n. Taking, for example, the dominant E1 excitation and summing over all magnetic quantum numbers the timedependent transition amplitude, equation (2.20), in the ground state is expressed as, _a0(t) = i ~ X M Z d h ( )jD(1) 1MihD(1) 1MjVE1;M (t)j0i a(1) ;1M(t) exp i(E1 + )t=~ = i ~ X M Z d (n)( )V (01) M (t)a(1) ;1M(t) exp i(E1 + )t=~ (2.30) where the dominant resonance amplitude is given as, d dt a(1) ;1M(t) = i ~ h (1)( ) V (01) M (t) i exp i(E1 + )t=~ (2.31) Integrating (2.31) over time and using the result in equation (2.30) the result is, da0 dt (t) = 1 ~2 X M V (01) M (t) Z d (1)( ) 2 Z t 1 dt0 h V (01) M (t0) i exp i ~ (E1 + )(t t0) a0(t0) where a(1) ;1M(t = 1) = 0 was used and the squared amplitude term (1)( ) 2 is chosen depending on the shape of the resonance. 29 Chapter 3 NUCLEAR STRUCTURE Nuclear physics is one of the largest areas of research and continues to be a signi cant area of interest. Since its conception it has contributed to many elds including the eld of astrophysics, problems in energy generation, medical technology, archaeological carbon dating, and even spawned the eld of particle physics. Early nuclear studies focused on the characterization of elements according to their constituent particles i.e., protons, electrons, and later neutrons. With the discovery of the neutron we had the nal piece to a description of the nucleus, consistent with Einstein's massenergy relation. With the neutrons discovery the energy contribution to the binding of a nucleus was nally calculable. A new look at the elements, tabulated according to the ratios of protons to neutrons, was developed. From this point of view the binding energy, which apportions energy con tributed by the constituents of the nucleus expended to the con nement of the nucleons due to the binding energy. The remaining energy constitutes the measured mass of the element. The advent of this re ned measurement to nuclear mass lead to questions apropos of the mechanism behind the con nement of neutrons and more interestingly the protons. Remembering the important relation from electrostatics, like charges repel, the protons within a nucleus, being in close proximity to one another, must feel concurrently opposing forces. These concurrent forces imposed on individual protons within the nucleus must be overcome by a much larger attractive force for a nucleus to form. This \strong force", which opposes the repulsive force of the protons, binds nucleons together and is contained within the description of the binding energy. However, its e ective range of in uence must be shorter than the electromagnetic interaction since it is not observed outside the nucleus. Hideki Yukawa, in 1935 proposed a ground breaking theory to explain the binding of nucleons by the strong force [23]. Yukawa's model of the strong force was analogous to 30 the mediating photon model of the electromagnetic force where a low mass virtual particle is exchanged between the nucleons e ectively binding them together. A straightforward calculation can be made to estimate this exchange. Using Heisenberg's uncertainty relation, E t > ~ which states that the uncertainty in time and energy must be greater than Planck's constant. Rearranging the above inequality for energy and multiplying both the top and bottom of the fraction by c the speed of light we get, E > ~c c t ~c rS where rS is the interaction distance of the strong force. Heisenberg's energy time uncertainty is valid for any \real particle", however, quantum mechanics allows us to violate this inequal ity by restating the above inequality in a contrapositive way so that the relation reads, for short time or length periods the energy must be less than or equal to some exchange energy Eex so that, Eex ~c rS An estimation of the upper bound for the lowest energy exchange particle can be calculated using some empirical data. Calculating the binding between the proton and neutron, which minimizes the in uence of the electromagnetic interaction, forming a deuteron. From ex periment the radius of the deuteron is, rD = 1:95 fm. We calculate the distance between the centers of each nucleon by subtracting the proton and neutron radii, rp = 0:8 fm and rn = 0:3 fm, from the diameter of the deuteron. Using ~c = 197 MeV fm we get an exchange energy of, Eex ~c rS ~c 2rD rp rn = 140:7 MeV 31 So the energy of an exchange particle necessary to bind a proton and neutron together must not be greater than 140 MeV. This is similar to the result found by Yukawa and corresponds to the lightest meson 0 = 135 MeV. We now see a similarity between QED and the strong force. In QED the exchange of a virtual photon mediates the electromagnetic interaction, this is analogous to result above where the exchange of a virtual pion mediates the strong interaction. However, this is not the whole story since it was later found that the pion is simply a consequence of a more fundamental theory of quarks, which are the building blocks of hadrons. It is now well known that all fundamental interactions, save for gravity, are due to some exchange of virtual quanta. The advantage to using Coulomb excitation is that we can exclude these internal interactions dominated by the strong force thereby simplifying our calculations. We now consider an important excitation where a large number of constituent nucleons are excited into a quasistable state. These excitations are highly dependent on the number of nucleons, A, implying a dependence on the binding energy. 3.1 Giant Resonance The large bulk nuclear response exhibited in nuclei with more than a few nucleons is known as a Giant Resonance (GR). The GR phenomenon has been shown to be a fundamental property of all nuclei [3,4,24] where the general features are; smooth transitions in the form, width, and centroid energy correlating to a change in the mass number A. The width is small compared to its excitation energy, and exhausts a large portion of the energyweighted sum rule. GR excitations are sensitive to several variable parameters available to the experi menter. The rst, and probably most obvious, would be the isotopes used in the reaction (e.g. p+Pb, Pb+Pb, Ni+Au, : : : ). The energy of the collision and the characteristics of the beam are then adjusted and selection of the scattering angle reveals the cross sections of interest. The GR observed varies according to the excitation energies giving a cross section 32 (E ), for excitation energy E , which can be t by a Lorentzian, (E ) = 0E2 2 (E2 E2C )2 + E2 2 (3.32) for the Giant Dipole Resonance (GDR). The centroid energy, EC, is the most probable energy of the resonance, e.g., 24 MeV for 16O and 13 MeV for 208Pb. The isotope excited into a resonance determines; the value of 0 which satis es the sum rules, and the width which varies from 4 to 8 MeV. For di erent multipole excitations, (e.g., E1, E2, M1, M2, : : :) the above parameters are adjusted accordingly. 0 50 100 150 200 250 300 350 400 450 500 550 8 10 12 14 16 18 20 22 24 26 28 Cross Section [mb] Energy [MeV] Pb208 Cross Section (Harvey) Lorentzian ’CS_Pb208_Harvey.dat’ Figure 3.5. Example cross section for the GR in Lead208. In gure 3.5 a schematic for a typical photoabsorption cross section is given. At low energies the system exhibits discrete states which are characterized by surface e ects due to 33 deformation, vibration, rotation, and a mixing of vibration and rotation. Around 8 MeV the discrete states give way to a continuum of states with the GDR, a dominant feature of the spectrum, appearing at around 8 MeV higher than where the continuum states begin. These collective excitations involve an appreciable number of nucleons and are built up from individual excitations from the ground state of the nucleus. 3.1.1 Classical models Over the years several collective models have been developed to describe the bulk response of nuclear matter. The progression of the theory from classical approximations to more sophisticated manybody models has been long running. A typical classical model for the GR phenomenon was rst developed by Goldhaber and Teller [25] and later improved by Steinwedel and Jensen [26]. Three possible explanations for the features which de ne a GR are: A restoring force between the displaced protons and neutrons exists which is inde pendent of the size of the nucleus. This possibility is discarded due to experimental evidence which clearly indicates a dependence on the number of nucleons A. A di erence between the proton and neutron densities within the nucleus. In this situation the surface of the nucleus is not a ected and the restoring force is proportional to the gradient of the di erence in density. This picture describes the SteinwedelJensen model. Two spheres comprised of either protons or neutrons oscillate against one another. This is a pure dipole mode picture with the overall density kept constant. This is the picture taken in the GoldhaberTeller model. In the GoldhaberTeller (GT) model a collection of protons, in uenced by an electric eld realized as an impinging photon, takes the nucleus to an excited state. The major feature of the GR phenomenon is of course the GDR and is described in the GT model as a 34 collection of protons moving against a collection of neutrons with their relative oscillations being opposite in phase to one another. This model allows for the collection of protons and neutrons to interpenetrate while preserving their individual incompressibility. The interpre tation as two oscillating uids will be important for our later look into the pygmy dipole resonance. The other interpretation developed by the SteinwedelJensen (SJ) model preserves the total density of the nucleus by separating the proton and neutron densities such that, A(r; t) = P (r; t) + N(r; t) the variation in density is then introduced by the addition and subtraction of a fractional density term (r; t) and requiring that the number of particles is conserved, Z d3r (r; t) = 0 A shift in the distribution of positive charge is the central idea behind both pictures of the GDR developed by GT and SJ. Each model describes the phenomenon reasonably well with the SJ giving an A dependence as A1=3 which describes the experimental data better than GT's A1=6 dependence. Each of these models takes a classical approach to this dynamic quantum system. For a more complete picture one must include quantum mechanics. Quantum mechanically describing the collective oscillations of manyparticle systems is an interesting and challenging problem in nuclear physics. In the days before computer simulations, when quantum electrodynamics was rst being developed, describing the dy namics of the wave functions for each individual nucleon inside of a nucleus was the stu of dreams. Simpli cations to the model and exploiting the symmetries and bulk properties of the system were necessary. Thus techniques such as mean eld approximations where used to reduce the degrees of freedom to a solvable problem. 35 3.1.2 Microscopic Models Although the classical models above describe the GDR reasonably well it is still neces sary to extend this into a microscopic picture. Several e ective theories exist which describe the GR on a microscopic level. Early attempts at calculations were developed based on the linear response theory [31]. Nowadays, an e ort is being undertaken to describe nuclear collective motion with more elaborate models such as the timedependent super uid local density approximation [32, 33]. Similarly, theoretical studies of the pygmy resonances have been developed based on the improvements of the hydrodynamical model [34{36], and with microscopic theories such as the random phase approximation (RPA) and its variants [37{40]. Sum Rules It is often useful to estimate the photoabsorption cross section over every transition from some initial state jii ! jfi to the possible nal states. The estimate is found by the sum rule for the system and is given in the form, S(n) i [F] = 1 2 X f (Ef Ei)n hfjFjii 2 + hijFyjfi 2 The sum rules give a tool for calculating the sum of integral cross sections for real photons over all possible nal states. 3.2 Pygmy Dipole Resonance Inherent to the majority of nuclei, excitation of the GR is a well investigated phe nomenon. As the ratio of neutrons to protons increases the distribution of protons and neutrons begins to di er. Experiments which infer the radius of the nucleus typically involve the measurement of elastically scattered particles from the target. The standard ways of measuring the radius of nuclei are easily understood through 36 classical scattering. To observe any reasonable detail of the nucleus the wavelength of the projectile must be smaller than the target. Suppose we wish to measure the radius of 208Pb. According to the radius formula, R = R0A1=3 where R0 = 1:2 1:25 fm, the radius of Lead 208 is RPb ' 7 fm, therefore the wavelength of the projectile must be 2RPb ' 14 fm which corresponds to a projectile momentum of p 89 MeV/c. Highenergy electron experiments are able to achieve beam energies from 100 MeV to 1 GeV and are a good candidate for measuring the nuclear radius. Spectroscopic analysis of only the elastically scattered electrons gives a good average estimate to the charge radius of nuclei. The electron projectiles are scattered by Coulomb interactions. This means that electron scattering tells us about the distribution of charge, i.e., protons, in the nucleus. Measurements of the distribution of all nuclear matter, which incorporates the protons and the neutrons, requires us to overcome the Coulomb barrier. Alpha particle scattering is one method where this is achieved. Using the classical scattering formula (2.16) developed by Rutherford form his famous alpha particle scattering experiments. Increasing the energy of the alpha projectiles the scattering cross section is well described by (2.16), however, at a certain energy the predictions of Rutherford's scattering formula are no longer accurate due to the e ects of the nuclear force. The results of these measurements show that the root mean square (rms) charge radius is almost the same as the rms nuclear radius di ering by less than 1 fm. Since nuclei tend to have more protons than neutrons this implies that the protons distribute themselves across the nucleus leaving more neutrons to occupy the core. When the number of neutrons become much larger than the number of protons the neutrons are then pushed out, due to their occupation pressure, beyond the rms charge radius creating a neutron rich layer. The di erence between these two radii is called the neutron skin, RSkin = Rn Rp. 37 Figure 3.6. The dipole mode of the pygmy resonance in the GoldhaberTeller (GT) and SteinwedelJensen (SJ) modes. In reactions which involve neutronrich nuclei at low excitation energies, close to the low energy tail of the GR, the excitation of the neutron skin against a symmetric nuclear core is observed. This phenomenon is known as the Pygmy Dipole Resonance (PDR) and can easily be visualized through the GT and SJ models as shown in gure 3.6. Understanding the PDR is import since it can be used as a tool to constrain the neutron skin thickness of these neutronrich nuclei. How the PDR is related to the neutron skin is through the fraction of the Energy Weighted Sum Rule (EWSR) exhausted by the PDR [27]. Measurements of the neutron skin have far reaching implications including areas of interest such as nuclear structure [28], neutron star structure [29], and heavyion collisions [30]. 38 Chapter 4 PRESENTATION OF FINDINGS (Published in Physics Letters B, Volume 757, 10 June 2016, Pages 553557 ) As research on nuclear reactions with radioactive beams became the focus over the last few decades, it became apparent that modi cations of the linear response theory predict a signi cant concentration in neutronrich nuclei at low energies of the excitation strength [41,42]. It is important to note that the amount of energy of the nuclear response exhausted by the sum rule strongly depends on how the nuclear interaction, pairing, and other physical phenomena are incorporated in the theory [37{42]. As an example, the E1 strength function numerically calculated using the public code of Ref. [43], is given as S(E) = X h jjOLj j0i 2 (E E ) (4.33) where an RPA con guration space is de ned in terms of deltafunction states and the operator OL is an electromagnetic operator. A smearing of 1 MeV of the fragmented strength function is introduced which produces a continuous distribution, shown in Fig. 4.7 for the E1 response in 68Ni. We used the option OL = jL(qr) in Eq. (4.33), where we take q = 0:1fm1 to be a representation of the momentum transfer. More details can be found in Ref. [43]. In this case, the strength function has dimensions of MeV1 and in the long wave approximation qr 1 it is proportional to the usual response for electric multipole operators. The calculation is performed for several Skyrme interactions, as shown in the gure. The arrow shows the position of the expected pygmy dipole resonance. The results presented in the literature, e.g. [37{42] show a greater response in the PDR energy range due to interactions and adaptations in the model space. There is currently no clear prediction of the exact location of the pygmy strength. It could be in the range of 712 MeV for medium mass nuclei such as Ni isotopes. The amount of the sum rule exhausted by the pygmy 39 Figure 4.7. Strength function for the E1 RPA response in 68Ni calculated with formalism described in Ref. [43]. The calculation is performed for several Skyrme interactions, shown in the gure inset. The arrow shows the location of the pygmy resonance. (For interpretation of the references to color in this gure legend, the reader is referred to the web version of this article.) resonance is also relatively unknown, although some models based on nuclear clustering can reach up to 10% of the total strength [44]. Coulomb excitation of pygmy resonances is one of the e ects overseen by the experi mental analysis of these reactions. The large excitation probability in Coulomb excitation at small impact parameters leads to a strong coupling between the pygmy and giant resonances. A manifestation of this coupling is seen as dynamical e ects such as the modi cation to transition probabilities and cross sections of the PDR. Previous observations of this coupling e ect are seen in the excitation of double giant dipole resonances (DGDR) [1, 45, 46]. The experimental observation of the DGDR is a consequence of higherorder e ects in relativis 40 tic Coulomb excitation arising due to the large excitation probabilities of giant resonances in heavy ion collisions at small impact parameters. The dynamical coupling between the usual giant resonances and the DGDR is very strong, as shown, e.g., in Ref. [22]. In the present work this dynamical coupling e ect on the excitation of the PDR is assessed using the relativistic coupled channels (RCC) equations introduced in Ref. [47]. The Smatrix for state given as, S (z; b), is obtained from the RCC equations [47] i @S (z; b) @z = X 0 h jMELj 0iS 0(z; b)ei(E0 E )z=~ ; (4.34) where is the projectile velocity andMEL is the electromagnetic operator for transitions by the electric dipole (E1) and quadrupole (E2) modes connecting states and 0. The states must satisfy the intrinsic angular momenta and parity selection rules. The ground state is denoted as j0i = jE0J0M0i and the excited states by j i = jE J M i where EJM are the intrinsic energy and angular momentum quantum numbers. The electromagnetic operators, in the longwavelength approximation, are given by [47] ME1m = r 2 3 Y1m(^ ) ZT e2 (b2 + 2z2)3=2 8>>< >>: b (if m = 1) p 2 z (if m = 0) (4.35) where is the intrinsic coordinate of the excited nucleus and Ze is the charge of the nucleus creating the interacting electromagnetic eld (in our case, the target). The electromagnetic operator for E2 transitions is [47] ME2 = r 3 10 2Y2m(^ ) ZT e2 (b2 + 2z2)5=2 8>>>>>>< >>>>>>: b2 (if = 2) 2 2bz (if = 1) p 2=3 (2 2z2 b2)z (if m = 0) (4.36) We note that the electromagnetic operator is expressed as MELm = fELm(r)OELm, where 41 OELm = LYLm(^ ) is the usual electric operator, and fELm(r) is a function of the projectile target relative position r = (b; z). Solutions to the coupled equations (4.34) are found by using S (z ! 1) = 0. For collisions at high energies and very forward angles, the cross sections for the transition j0i ! j i are given by d dE = 2 w (E) Z db b exp[2 (b)] S (z ! 1; b) 2 ; (4.37) where w (E) is the density of nal states, b is the impact parameter in the collision, and (b) is the eikonal absorption phase given by (b) = NN 4 Z dq q 1(q) 2(q)J0(qb) (4.38) where NN is the total nucleon{nucleon cross section, obtained from experiment, with medium corrections added according to Refs. [48, 49]. The Fourier transform of the ground state densities of the nuclei, i(q), is obtained from tting to electron scattering experi ments [50] for 197Au and using Hartree{Fock{Bogoliubov calculations with the SLy4 interac tion for 68Ni. A reduction of the Coulomb excitation mechanism at small impact parameters, rst introduced in Ref. [51], is used to calculate reaction cross sections relevant to excitations of the GDR and DGDR. To remain within the contexts of current experimental results we neglect e ects of nuclear excitations, and possible interferences which were subtracted in experiments [52{54]. To facilitate our discussions we consider the excitation of 68Ni on 197Au and 208Pb targets at 600 and 503 MeV/nucleon, respectively. These reactions have been experimentally investigated in Refs. [52,53]. In the rst experiment the pygmy dipole resonance was observed at EPDR ' 11 MeV with a width of PDR ' 1 MeV, for 68Ni and exhausted about 5% of the Thomas{Reiche{Kuhn (TRK) energyweighted sum rule. The excitation was identi ed through the analysis of its decay product, gamma emission. The second experiment found 42 the PDR centroid energy to be at 9.55 MeV, width a of 0.5 MeV and exhausting 2.8% of the TRK sum rule. The PDR was identi ed by measuring the neutron decay channel of the PDR. This studies focus will be on the e ects of coupling between the di erent modes of giant resonances with the PDR. Therefore, we will not take into consideration the decay channels but restrict our analysis to the calculation of the excitation function d =dE. A model is needed for including the bound and continuum discretized wavefunc tions in the matrix elements of h jMELj 0i in Eq. (4.34). To calculate the response functions, dBEL=dE = P spins w 0 h jjOELj j 0i 2 , it is appropriate to use these wavefunc tions, with obvious consideration for summing over angular momentum coe cients. To simplify calculations we assumed Lorentzian forms for the response functions dBEL=dE and assign an appropriate fraction of the sumrule based on current experimental values. We then discretized the functions into energy bins to obtain the reduced matrix elements h jjMELj j 0i 2 / E (dBEL=dE) E=E , where E = E 0 E . To keep everything in terms of real numbers a phase convention is found for the reduced matrix elements. These are then used, with proper care taken for the corresponding angular momentum coe cients, in determining the matrix elements h jMELj 0i in Eq. (4.34) (see, e.g., Ref. [46]). The Lorentzian functions are described by Eq. (3.32) with their respective centroid energies EPDR for pygmy dipole resonances and EGDR (EGQR) for the isovector (isoscalar) giant dipole (quadrupole) resonances. Their respective widths are denoted by PDR, GDR and GQR. The discretized strength function is subdivided into 35 energy bins centered around each resonance. The PDR centroid energy EPDR = 11 MeV is chosen, consistent with Refs. [52,53]. However, a full width at half maximum of 2 MeV was chosen, which is more in line with theoretical calculations [35{42] than with the experimental data [52,53]. Our choice to use the theoretical width was in essence made to better determine higherorder e ects on the modi ed tails of the PDR without changing the other aspects of the PDR appreciably. For the major mode of the GR the (isovector) 1 giant dipole resonance (GDR) was given a centroid energy and width of EGDR = 17:2 MeV and GDR = 4:5 MeV, respectively. For the 43 (isoscalar) 2+ giant quadrupole resonance (GQR) we take EGQR = 15:2 MeV and GQR = 4:5 MeV. The centroid and width for the GQR are approximations based on the systematics of GQR excitation in nickel isotopes [55] and are not experimental values. Looking at the total number of channels involved in our calculation we have, 35 3 + 35 5 + 35 5 + 1 = 456 which includes all magnetic substates plus an additional channel for the 0+ ground state. For practical purposes, to reduce computational intensity, considering only the major dynamical e ects arising from the coupling of the PDR with the GQR via the dominant E1 interaction at relativistic energies. A reduction by a factor of 2 is possible by implementing a coarser binning of the PDR and GQR states introducing only a loss of accuracy at the level of 10%. Calculations using all the channels above will converge to within 1%. Figure 4.8. Coulomb excitation cross section as a function of the excitation energy of 600 MeV/nucleon 68Ni projectiles incident on 197Au targets. The lled circles represent the calculations using rstorder perturbation theory, while the lled squares are the results of coupledchannel calculations. In Fig. 4.8 we show, for 600 MeV/nucleon 68Ni projectiles incident on 197Au targets, 44 the rstorder Coulomb excitation cross sections as a function of the excitation energy for the PDR and GQR separately. Calculations using rstorder perturbation theory are denoted by lled circles, while the lled squares represent the coupledchannel calculations. The rstorder Coulomb excitation cross sections, as shown in Ref. [?], can be obtained by means of the relation d dE = X L N L(E) E ( ) L(E) (4.39) where N L are the virtual photon numbers of multipole L and ( ) L are the real photon cross sections with multipolarity L. The virtual photon numbers include the same absorption coe cient as in Eq. (4.37) [51]. Summing over the relevant multipoles, here E1 and E2 stand for 1 and 2+ excitations, respectively. The gure shows that the coupling between these states has a visible impact on the energy dependence of the cross sections. According to the Brink{Axel hypothesis, the excitation of a giant resonance on top of any other state in a nucleus is possible [56, 57]. Therefore, the couplings are a manifestation of (PDR GQR)1, (PDR PDR)2+, (PDR GDR)2+ and (GDR GQR)1 states which we investigate, as they build up components of the PDR, GDR and GQR. Our ndings suggest the importance for the reliable extraction of the experimental strength of the PDR relative to the GDR. The dynamical calculations show that both the strength and width of the PDR are modi ed appreciably due to the coupling to the GQR. Fig. 4.8 shows separately that the population of PDR and GQR states are modi ed. The 1 states in the GDR region are a ected very weakly and so are left out of the gure. In the high energy region the GDR excitation dominates by a factor of 2{3 times that of the 2+ states. The important modi ca tions in the excitation spectrum appear from the couplings PDR$GQR$PDR by E1 elds, while couplings from PDR$GDR$PDR by E2 elds contribute very little to the 1 states in the PDR energy region. In Fig. 4.8 we notice that the tails of the PDR, and to a lesser extent those of the GQR, are appreciably modi ed. A small shift to the peaks is also seen, although barely visible for the PDR, it is evident for the GQR. It is important to also keep in mind that the strength and shape of the PDR will also be modi ed by the low energy 45 tail of the GDR. For our case considered here, a GDR strength on the order of 3.8% lies within the region of the PDR meaning that the PDR shape will only slightly be in uenced by this low energy tail. However, these e ects have been considered in the experimental analyses [52, 53]. In this work we are interested in the higherorder e ects which have been so far ignored. Figure 4.9. Coulomb excitation cross section as a function of the excitation energy of 68Ni projectiles incident on 197Au targets at two laboratory energies. The lled circles represent the calculations using rstorder perturbation theory, while the lled squares are the results of coupledchannel calculations. In Fig. 4.9 the energy region of the pygmy resonance has been singled out and we 46 plot the results of our calculations for two di erent bombarding energies: 100 MeV/nucleon and 2 GeV/nucleon, using the same notation as in Fig. 4.8. The coupling e ects change dramatically. For the lower energy interaction the in uence of the giant resonances increases appreciably for the response in the energy region of the PDR, while at the higher energy collision this e ect is much smaller and shows a slight tendency th decrease the PDR ex citation cross section. This is expected since for energies around 100 MeV/nucleon the E2 eld is dominant, with an increase to the excitation of the GQR and consequently a strong feedback to the PDR from subsequent E1 transitions. Figure 4.10. Coulomb excitation cross sections of the PDR as a function of the bombard ing energy of 68Ni projectiles incident on 197Au targets. The lled circles represent the calculations using rstorder perturbation theory, while the lled squares are the results of coupledchannel calculations. The Coulomb excitation cross sections of the PDR as a function of the bombarding energy of 68Ni projectiles incident on 197Au targets is given in Fig. 4.10. Calculations using rstorder perturbation theory are represented by the lled circles, while the lled squares 47 are the results of our coupledchannel calculations. Its apparent that at lower energies the deviation is more pronounced. At 600 MeV/nucleon the cross section for excitation of the PDR changes from 80.9 mb obtained with the virtual photon method to 92.2 mb with the coupledchannels calculation. A re ection of this e ect in the extracted PDR strength from the experimental data amounts to an appreciable change of 14%. This reduction is approximately the same amount of the strength needed to reproduce the experimental data. Calculations have also performed for 68Ni + 208Pb at 503 MeV/nucleon, corresponding to the experiment of Ref. [53]. To rstorder, the Coulomb excitation cross section for the PDR in 68Ni is found to be 58.3 mb, with the inclusion of the e ects of coupling to the giant resonances, the cross section increases to 71.2 mb, i.e., an important 18.1% correction. The dipole polarizability is de ned as D = ~c 2 2 Z dE (E) E2 (4.40) where (E) is the photoabsorption cross section. From Ref. [53] the extracted experimental value of D is 3.40 fm3 while in order to reproduce the experimental cross section with our dynamical calculations we have D = 3:16 fm3, a small but nonnegligible correction. Assuming a linear relationship between the dipole polarizability and the neutron skin [58], a reduction to the neutron skin from 0.17 fm, as reported in Ref. [53], to 0.16 fm is expected. A correction such as this lies within the experimental uncertainty of 7% for D and 0.02 fm for the neutron skin [53]. However, consideration of the coupling e ects should be taken into account in the future as more precise data becomes available, in particular, if the measurement is performed at lower bombarding energies. 48 Chapter 5 CONCLUSIONS We conclude that probabilities of giant resonances due to the large Coulomb excitation in heavy ion collisions at energies around and above 100 MeV/nucleon, the excitation of the PDR is also modi ed due to the coupling between the 1 and 2+ states. Our calculations, utilizing a Lorentzlike distribution for simplicity of the electromagnetic response and sum rules, are carried out without a detailed nuclear structure model. Future investigations carried out for nearly \abinitio" calculations based on a microscopic theory, coupled with a proper reaction mechanism, might be possible. A known alternative using individual states calculated by RPA or other microscopic models together with higher order perturbation theory, have already used in previous studies of multiphonon resonances [59]. Finally, the use of an advanced mean eld timedependent method such as that developed in Ref. [33] is also available. Deriving rather accurate dipole strength distributions from the electromagnetic excitation of the PDR is mainly of relevance to the extraction of the dipole polarizability [53], which is an important observable for constraining the symmetry energy, and is thus also important for better understanding the properties of neutronstars. A particularly important aspect for the polarizability is the lowenergy response due to the inverse weighting with energy. This opens exciting possibilities for studies of the pygmy resonance in nuclei and its use as a tool for applications in nuclear astrophysics. References [1] C.A. Bertulani and G. Baur, Phys. Rep. 163, 299 (1988). 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[52] O. Wieland, et al., Phys. Rev. Lett. 102, 092502 (2009). [53] D.M. Rossi, et al., Phys. Rev. Lett. 111, 242503 (2013). [54] D. Savran, T. Aumannnd, A. Zilges, Prog. Part. Nucl. Phys. 70, 210 (2013). [55] D.H. Youngblood, Y.W. Lui, U. Garg, R.J. Peterson, Phys. Rev. C 45, 2172 (1992). [56] D.M. Brink, PhD thesis, Oxford University, (1955). [57] P. Axel, Phys. Rev. 126, 671 (1962). [58] J. Piekarewicz, Phys. Rev. C 83, 034319 (2011). [59] V.Yu. Ponomarev, C.A. Bertulani, Phys. Rev. C 57, 3476 (1998). Vita Since graduating in 2008 from Hendrickson High School, Nathan S. Brady spent the rst two years exploring di erent career interests before discovering an obsession for physics. Nathan enrolled at The University of Nebraska at Kearney in 2010 where he pursued degrees in both physics and mathematics. There he was awarded several grants and awards including a NASA funded summer internship at the City College of New York. After graduating in 2014 Nathan enrolled at Texas A&M  Commerce to pursue a Master of Science in Physics. While attending he received several awards, including Outstanding Graduate Student and a summer Graduate Assistant Research. Nathan was also selected from a pool of candidates from around the world to participate in the TALENT workshop over the summer in Caun, France. In the Fall of 2016 Nathan was awarded the Master of Science degree and admitted into the PhD program at Texas A&M University  College Station. Permanent address: Nathan S. Brady Department of Physics and Astronomy Texas A&M UniversityCommerce P.O. Box 3011 Email: bradynsb@gmail.com 53 
Date  2016 
Faculty Advisor  Bertulani, Carlos A 
Committee Members 
Li, BaoAn Newton, William G 
University Affiliation  Texas A&M UniversityCommerce 
Department  MSPhysics 
Degree Awarded  M.S. 
Pages  63 
Type  Text 
Format  
Language  eng 
Rights  All rights reserved. 



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