
RELATIVISTIC CORRECTIONS IN HEAVY ION ELASTIC SCATTERING A Thesis By George Phillip Robinson Submitted to the O ce of Graduate Studies Of Texas A&M UniversityCommerce In ful llment of the requirements For the degree of MASTERS OF SCIENCE August 2017 RELATIVISTIC CORRECTIONS IN HEAVY ION ELASTIC SCATTERING A Thesis By George Phillip Robinson Approved by: Adviser: Carlos Bertulani Committee: BaoAn Li William Newton Head of Department: Matt Wood Dean of the College: Brent Donham Dean of Graduate Studies: Mary Beth Sampson iii Copyright c 2017 George Phillip Robinson iv ABSTRACT RELATIVISTIC CORRECTIONS IN HEAVY ION ELASTIC SCATTERING George Phillip Robinson, MS Texas A&M UniversityCommerce, 2017 Advisor: Carlos Bertulani, PhD Relativistic e ects of heavy ion scattering were investigated at interme diate collision energies, at or above about 50 MeV/n. Two methods for evaluating these e ects were compared for their validity. The rst method involves a full account of the retardation of the Coulomb potential by solu tion of the covariant equations of motion for charged particles. The second method involved the expansion of the e ective Lagrangian, including the elec tromagnetic Darwin Lagrangian, in orders of (v/c). This study allowed for the determination of the degree of involvement of e ects such as relativistic magnetic interactions, kinematic corrections, and relativistic mass increase in the motion of the heavy charge particles. It was shown that the numeric solutions of the coupled di erential equations presented were not necessary as the analytic formulations su ciently describe all of the scattering parameters needed for nuclear experimentation. v ACKNOWLEDGMENTS: Dr. Carlos Bertulani Ravinder Kumar Michael Hartos vi TABLE OF CONTENTS LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vii INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii CHAPTER 1. DARWIN LAGRANGIAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Electrodynamics with Retardation . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 LienardWiechert Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Darwin Lagrangiian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. RUTHERFORD SCATTERING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3. THE INCLUSION OF RETARDATION IN COULOMB SCATTER ING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4. SOLUTION OF THE ELECTROMAGNETIC TWOBODY SCAT TERING WITH AN EFFECTIVE LAGRANGIAN EXPANSION 23 5. RUNGE KUTTA NUMERICAL METHOD . . . . . . . . . . . . . . . . . . . . . 28 5.1 Basic Runge Kutta Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28 5.2 Adaptive Step Size Runge Kutta Method . . . . . . . . . . . . . . . . . . 30 6. RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.2 Analytic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41 6.3 Analysis of the Distance of Closest Approach . . . . . . . . . . . . . . 44 7. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 vii LIST OF FIGURES 1. LienardWiechert Potential for a particle moving through an arbitrary path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Simple 2D scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3. Classical scattering by a central potential . . . . . . . . . . . . . . . . . . . . . . . 15 4. 208Pb+208 Pb collision at 100MeV n . Scattering angle percent di erence vs. impact parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5. 208Pb +208 Pb collision at b = Rp + Rt. Scattering angle percent di erence vs. bombarding energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6. 208Pb+208Pb collision at 100MeV n . Di erential scattering cross section percent di erence vs. scattering angle. . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7. 208Pb +208 Pb collision at b = Rp + Rt. Di erential scattering cross section percent di erence vs. bombarding energy . . . . . . . . . . . . . . . .39 8. 208Pb +208 Pb collision at b = Rp + Rt. Di erential scattering cross section vs. bombarding energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 9. 17O+208 Pb collision at 100MeV n . Di erential scattering cross section percent di erence vs. scattering angle . . . . . . . . . . . . . . . . . . . . . . . . . . .43 10. Multiple di erent collisions at b = Rp + Rt. Distance of closest approach percent di erence vs. bombarding energy . . . . . . . . . . . . . 46 viii Introduction In modern radioactive beam facilities, Coulomb excitation is used to probe phenomenon such as dipole polarizeability, pygmy dipole resonance, neutron skin thickness, equations of state of nuclear material, and general nuclear structure. This is done because Coulomb interactions are well understood. Experimentally, it is presumed that Coulomb scattering dominates in small angle scattering especially in the nonheadon elastic scattering of heavy ions. Many of the beam experiments are performed at appreciably high kinetic energies. Due to this, it is crucial that relativistic e ects be accounted for in order to properly address relativistic kinematics, any number of possible internal excitations, and any other relativistic reaction dynamics. We sought to investigate the best method to account for relativistic e ects in elastic Coulomb scattering of heavy ions. To do this, we explored two di erent proposed methods for assessing relativistic scattering. The rst involved a solution to the covariant equations of motion for a charged particle moving in the electric and magnetic eld of another. The second used an expansion of the e ective Lagrangian to determine the extent of the relativistic velocities. Both numeric and analytic solutions were given using each method, and both were compared for validity. First however, it is necessary to discuss some background material. 1 Chapter 1 DARWIN LAGRANGIAN 1.1 Electrodynamics with Retardation In classical electrodynamics, the Lagrangian for multiple charged particles can be written as follows: L = T U (1) where T is the kinetic energy, and U is the potential energy. S = Z mcds = Z Tdt (2) where S is the classical action and ds = cdt q 1 v2 c2 . Therefore: T = mc2 r 1 v2 c2 (3) while the potential follows easily from the Force(F = q[E + (v B)]) as: U = q 2 V (r) A(r) v c (4) 2 (these terms are de ned later when we discuss Maxwell's equations, where they are more relevant) bringing us to the nal Lagrangian: L = X n " mnc2 r 1 v2n c2 + qn 2 vn c A(rn) V (rn) # (5) The Darwin Lagrangian is a term based on the expansion of the classical electrodynamic's Lagrangian above for two relativistic charged particles interacting in free space. It was originally derived in 1920 by Charles Darwin by expansion of the LienardWiechert potential (we di er from the original derivation, as explained below). The term accounts for the reaction of one particle to the magnetic eld created by the other. The Darwin Lagrangian follows from the expansion to the order of v2=c2 , but higher orders are required to account for retardation e ects. To nd this Lagrangian term, we begin with Maxwells equations: r E = 0 r B = 0 r E = @B @t r B = 0J + 0 0 @E @t (6) 3 We take the curl of the third Maxwell's equation: r (r E) = r @B @t r(r E) r2E = @(r B) @t r 0 r2E = @ @t 0J + 0 0 @E @t (7) making the electric eld wave equation with the inclusion of source terms: 2E = r 0 + 0 @J @t (8) where 2 = r2 1 c2 @2 @t2 . We can now integrate the Green's function (9) and the sources to nd the electric eld: G(r; t; r0; t0) = (t t0) 4 jr r0j (9) E(r; t) = Z d3r0dtG(r; t; r0; t0) r0 (r0; t0) 0 + 0 @J(r0; t0) @t0 (10) Through integration by parts, and the unique properties of derivatives of the Greens function(rG = r0G), we have the following emerge: E(r; t) = rV @A @t (11) where the scalar potential and the vector potential are respectively de ned 4 by: V (r; t) = 1 4 0 Z (r0; tr) R d3r0 (12) A = 0 4 Z J(r0; tr) R d3r0 (13) Similar manipulations of the magnetic eld term from Maxwell's equations, and following the same procedure just performed for E leads to: B(r; t) = r A (14) In this case, both the E and B eld equations include the term for the retarded time, tr = t R c where R is the position term, jr r0j from the Green's function, and c is the speed of light. Additionally, and J are the charge and current distributions respectively. It is then evident that the E eld di ers from the electrostatic case by the addition of the time derivative of the vector potential term, while the B eld seems to remain the same as the magnetostatic case. The B eld term does change however, due to the inclusion of the position dependence of the retarded time (as well as R) when the curl is evaluated. This leads to multiple terms through a chain rule, clearly making it di erent from the magnetostatic case. We will discuss the Lagrangian that arises from this derivation, but this is a convenient place to stop and discuss the LienardWiechert Potential that we use later. So we digress shortly, and will return to this topic after. 5 1.2 LienardWiechert Potential Our goal in this section is to nd the potential at some location from a charged particle moving through an arbitrary path. So we begin, as we do, with the de nition of terms represented in Figure 1. Figure 1: LienardWiechert Potential for a particle moving through an arbi trary path As shown above, a particle of charge q is moving through an arbitrary path traced by rq at a velocity vq(t). Additionally, we de ne, the same as above, a vector from the particle to the point P which we will simplify by calling it R. We can also quite easily de ne a trivial charge density (r; t) = q (r rq(t)) and a current J(r; t) = qvq (r rq(t)). Although we previously derived terms for V (r; t) and A(r; t) only in terms of of a volumetric integral, we will revert back to the time dependent, more generic 6 versions because we now would like to explicitly solve the time integral rather than take it for granted. We begin with the Scalar Potential: V (r; t) = 1 4 0 Z d3r0dt0 (r 0; t0) (t t0 R c ) R = q 4 0 Z d3r0dt0 (r 0 rq(t0)) (t t0 R c ) R = q 4 0 Z dt0 (t t0 Rq(t) c ) Rq(t) (15) where Rq(t) = jr rq(t)j, which it should be obvious from this notation that Rq is time dependent while the original R was independent of time. So to solve the integral, we must rst examine some unique properties of delta functions: [f(t)] = X i (t ti) jf0(ti)j (16) In this, f(t) is any arbitrary function of t, and atf(ti) = 0 while f0(ti) 6= 0 and in this case, f(t0) = t t0 Rq(t0) c . Therefore, t0 0 = t Rq(t0) c and f(t0 0) = 0, but t0 is quite obviously the retarded time tr. We also need the derivative of f: f0(t0) = Rq(t0) vq(t0)) Rq(t0)c 1. Now returning to the integral: V (r; t) = q 4 0 Z dt0 (t tr Rq(t) c ) Rq(t) = q 4 0 Z dt0 (t tr) (1 Rq(t0) Rq(t0) vq(t0) c )Rq(t) V (r; t) = q 4 0 " 1 Rq Rq vq c # retarded time (17) 7 Similarly, the vector potential A can be washed through the same steps to obtain the following: A(r; t) = 0q 4 " vq Rq Rq vq c # retarded time (18) These two equations for the scalar and vector potentials are the LienardWiechert Potentials, but it is no longer necessary to use the q subscript, so we can simplify by writing both as follows: V (r; t) = q 4 0 " 1 R(1 ^R ) # retarded time A(r; t) = V (r; t) c retarded time (19) 1.3 Darwin Lagrangian Prior to the LienardWiechert Potential explanation, we came up with a term for the E and B, and from these we can write the Lagrangian for the relativistic interaction of a particle of charge q within an electromagnetic eld as: L = qrV + q c (v A) (20) where v is the velocity of the charged particle. Applying the Coulomb Gauge, (r A = 0), and removing 1 c2 @2A @t2 8 because A is of order v2 c2 , we can write the following from Maxwells laws: r2A 1 c r @V @t = 4 c Jt (21) where Jt is the transverse current created by the movement of the second charge. This indicates that the currents divergence is zero throughout. It should be noted that the original derivation done by Darwin did not use this method, however, this method, as performed by Jackson [2], leads to an exact Coulomb potential V and transfers the approximation to the vector potential, A. which becomes: A(r) = Z d3r0 jr r0j 1 c J(r0) 1 4 c r0V (@r0) @t (22) We can write J(r0) for n particles as qnvn 3(r0 rn) and V as qn jr0r0 nj A(r) = Z d3r0 jr r0j qnvn c 3(r0 rn) qn 4 c r0 vn (r0 rn) jr0 rnj3 (23) Then by integrating by parts, and removing the surface at in nity, while also utilizing the property of greens functions where r0 r: A(r) = qnvn c jr rnj + qn 4 c Z d3x0 vn (r0 rn) jr0 rnj3 r0 1 jr r0j (24) A(r) = qnvn c jr rnj qn 4 c r Z d3(r0 rn) vn (r0 rn) jr0 rnj3 1 r0 r (25) 9 Finally, by evaluating this nontrivial integral, we reach: An = qn 2cr vn + rvn r r (26) where r = jvrj. If we multiply qv c to take into account the second (or incidentally, any number by including a subscript to account for the nth) charged particle, and correct the original free particle Lagrangian by saying that mc2 1 + mc2 1 2mv2[1 + 1 4 v2 c2 ] We arrive at the nalized Darwin Lagrangian: X i6=j 1 2 miv2 i + 1 8c2miv4 i + qiqj rij 1 2 + 1 4c2 (vi vj + (vi r^ij)(vj r^ij)) (27) where we have used a di erent index here merely for convenience, and to avoid confusing subscripts. This Lagrangian is useful when radiation can be neglected, but was originally derived under the assumption of nonrelativistic speeds, although that assumption should not be necessary because radiation is due to acceleration. A Lagrangian that both neglects radiation and the assumption of low speeds was proposed by H. Ess en [3] and leads to a succinct interaction Lagrangian for the interaction of two particles as: L12 = g v2 c2 e1e2 r21 v2 c2 e1e2 r21 (28) where g(x) 1 1+ p 1x , r21 = jr2 r1j, and e is the particle's charge. The 10 term g can be expended in terms of v2 c2 and at the limit of v ! c, g(x) = 1, and the related force is zero. This method allows for all retardation dependencies to be accounted for, which eliminates the assumption of low velocities as required by the original Darwin derivation. 11 Chapter 2 RUTHERFORD SCATTERING In May of 1911, Ernst Rutherford published his ndings from a scattering experiment in which alpha particles were red at a thin sheet of gold foil [4]. The atomic models of the time did not conform with the ndings of his experiment, as a high percentage of alpha particles were unexpectedly backscattered at angles greater than 90o. It was this conclusion that led to our current understanding of the ultradense positively charged center to atoms, which we now know as the nucleus, surrounded by a large volume, negatively charged space occupied by the electrons. The following are the derivations associated with Rutherford's Scattering model. Scattering being integral to this subject, it seems prudent to discuss it in some detail. We begin with a few de nitions in classical scattering: An incoming particle, the projectile, is incident on another particle, the target, which we will assume has a very large mass as compared to the incident one so we can neglect recoil of the target. We will discuss removing this assumption later, but for now we will use it to discuss the simpler case. The incoming particle approaches with an impact parameter, b, which we de ne as the perpendicular distance from the direction of motion of the incoming projectile, and the center of the target. The incident particle will have an initial velocity v0. After interacting with the target, the projectile 12 emerges at an angle , as measured from the projectile's original direction of motion, and it will have a velocity denoted v1 for its velocity as t ! 1. Figure 2: Simple 2D Scattering To obtain Rutherford's Scattering equation, we must begin with some assumptions, (in addition to having the mass of the target be much larger than that of the projectile): rst, we assume that the only interaction between the projectile and the target is the Coulomb repulsion; this implies azimuthal symmetry. We will also only consider a single projectile and only a single target. And lastly, (and most importantly for this paper) we will consider only nonrelativistic speeds (for now). So from these assumptions and the above de nitions, we can now begin to solve for a relationship between the impact parameter b and the scattering angle . (these being the best adjustable and measurable parameters respectively for experimentalists in the lab setting). We begin 13 with the Coulomb Force: F = Z1Z2e2 4 0r2 (29) where Ze is the charge of the particle, and r is the vector that traces the projectile's motion. We then consider the conservation of angular momentum. We know that because angular momentum is conserved, then the angular momentum itself must be equal to some constant, C: jLj jr pj = jmrvj sin = C (30) Where L is the angular momentum, p is the linear momentum, and is the angle between r and v. We must then consider the initial angular momentum, where the projectile has not yet experienced any de ection from the Coulomb repulsion. At this time, sin = b r , so L0 = mv0b = C. So we now have a term for the constant C. Combining the Coulomb Force (29) with our understanding of angular momentum, we can write r2 = bv0( d dt )1 (31) Then by writing the force in Cartesian coordinates. Fx = F cos = Z1Z2e2 4 0r2 cos = Z1Z2e2 4 0bv0 d dt cos = mdvx dt Fy = F sin = Z1Z2e2 4 0r2 sin = Z1Z2e2 4 0bv0 d dt sin = mdvy dt (32) 14 solving this integral for vi : vx( ) = Z1Z2e2 4 0mbv0 sin + v0 vy( ) = Z1Z2e2 4 0mbv0 (cos + 1) for 0 (33) Now we take a look at the conservation of energy, which tells us that the magnitude of the initial and nal energy must be equal (v0 = v1) and at t ! 1, = , so (focusing only on the ydirection): Z1Z2e2 4 0mbv0 (cos( ) + 1) = v0 sin Z1Z2e2 4 0mbv0 (cos( ) + 1) = v0 sin Z1Z2e2 4 0mbv0 = v0 sin cos + 1 Z1Z2e2 4 0mbv0 = v0 tan 2 (34) one nal rearrangement brings us to b as a function of . b( ) = Z1Z2e2 4 0v2 0mtan( 2 ) (35) Now, we take into account the more appropriate three dimensional version of classical scattering with a beam of incoming projectiles rather than only a single projectile. The projectile incident in a crosssectional area element d = 2 bdb at the cylindrically symmetric impact parameter b 15 will scatter to a correlating solid angle element d = 2 sin d . From these, we can de ne a so called di erential scattering crosssection, d d : Figure 3: Classical scattering by a central potential We can say that the number of incoming particles that pass through this di erential cross section d with an intensity, I, must be equal to the number of particles that scatter through the di erential solid angle d : I 2 bdb = I d d d (36) and plugging in what we know about d and d we can rearrange to nd 16 an expression for the di erential scattering cross section in terms of b and : d d = b sin db d (37) Finally, we can plug in b( ) (35) and its derivative with respect to into (37) to obtain a term for the di erential cross section solely in terms of . db d = Z1Z2e2 16 0K0 sin2( 2 ) (38) d d = Z1Z2e2 8 0K0 2 1 2 sin tan 2 sin (39) where K0 is the initial kinetic energy, K0 = 1 2mv2 0. Then by simply using the trigonometric identity: sin x = 2 sin x 2 cos x 2 , we arrive at last, at the elegant formula: d d = Z1Z2e2 16 0K0 2 1 sin4 2 (40) 17 Chapter 3 THE INCLUSION OF RETARDATION IN COULOMB SCATTERING In 1987, R. Matzdorf and G. So published a paper describing the collisions of simirelativistic heavy ions [5]. In which, they derived a set of coupled equations for the motion for relativistic charged particles. Their method does intentionally neglect radiative corrections. We will use their method in conjunction with a method outlined in a paper written by C. E. Aguiar et al. [6] which is discussed in the following section. Matzdorf begins with the covariant equations of motion that describe a charged particle moving in the external electromagnetic eld of another charged particle. Assuming the projectile is moving in the ^x direction, the covariant equation of motion for the two particle system moving under the in uence of their own electric elds and magnetic elds: dp d = q c F U (41) where eld strength tensor, F can be written as: dp d = q c 0 BBBBBBB@ 0 E1 E2 E3 E1 0 B3 B2 E2 B3 0 B1 E3 B2 B1 0 1 CCCCCCCA U (42) where p is the 4momentum and U is the 4velocity. is the proper time. 18 The E and B elds are then derived from the Lienard Wiechert potentials (19). Then plugging those back into (42), and produce three relevant equations for the velocities which can be expressed as: 4 0 BBBB@ 2 + u21 u1u2 u1u3 u1u2 2 + u22 u2u3 u1u3 u2u3 2 + u23 1 CCCCA u_ = q(p) m(p) 0 c 0 BBBB@ E(t) 1 B(t) 3 u2 + B(t) 2 u3 E(t) 2 + B(t) 3 u1 B(t) 1 u3 E(t) 3 B(t) 2 u1 + B(t) 1 u2 1 CCCCA (43) This method produces a fourth equation 4(u; u_ ) = q(p) m(p) 0 E(t) 1 u1 E(t) 2 u2 E(t) 3 u3 , but it provides no additional information, so it is disregarded. We denote u = v c as the normalized velocity to the speed of light, and = (1 2) 1 2 . The subscript 1, 2, and 3 denote the direction component as in (x1; x2; x3). In addition, p and t refer to the projectile and target respectively (although the distinction is nonessential because the reference frame can simply be switched to that of the motionless projectile and moving target and all remains the same). To solve for u_ , a mathematically straightforward matrix manipulation is performed. Although it is straightforward, it is also fairly lengthy, so it is not shown here. By con ning the scattering plane to the xy plane, and solving for u_ we can reduce the 3 3 matrix to a 2 2 matrix involving only u1 and u2 terms. Using the target's frame of reference, the scalar and vector potentials 19 can be expressed as: A = 0 V = q R (44) Performing a Lorentz transformation into the laboratory reference frame, we nd E and B can be expressed as: E =q n 2 [1 2 + (n )2] 3 2 R2 B = E (45) Then plugging in the nonvanishing terms, after the restriction to two dimensions, E1, E2, and B3, can be expressed as: E1 =q n1 2 [1 2 + (n )2] 3 2 R2 E2 =q n2 2 [1 2 + (n )2] 3 2 R2 B3 = 1E2 2E1 where 2 = 2 1 + 2 2 (46) Lastly, Matzdorf restricts the Coulomb eld to be purely classical (i.e. E = q n R2 and B = 0 ) and, due to its essentially negligible contribution, on the order of only a single percent di erence at speeds less than 0:99c, the magnetic eld's relativistic contribution is omitted. This leads to the nalized equation for the normalized accelerations 20 in each direction. u_ 1 = q(p)q(t) m(p) 0 3 ( 2 + u22 )n1 u1u2n2 R2[( 2 + u21 )( 2 + u22 ) u21 u22 ] u_ 2 = q(p)q(t) m(p) 0 3 ( 2 + u21 )n2 u1u2n1 R2[( 2 + u21 )( 2 + u22 ) u21 u22 ] (47) where R is the magnitude of the radius vector considering the target at time t, and n represents the unit vector in its respective direction. These equations, (43) and (47), are only soluble numerically as they are obviously interdependent upon one another. The numerical methods are discussed in Chapter 5 and the solutions are discussed in Chapter 6. In addition to this, the scattering angle can be solved if the target begins at the origin at rest at t = 1, and the projectile moves originally in the ^x direction, then the solution for the scattering angle can be found by use of the solution to (47) and: (t ! 1) = arctan u2 u1 = arctan dy(t) dx(t) (48) The next step, and the ultimate goal of all collision calculations, would be to use these equations to nd a relationship between the impact parameter b and the scattering angle . This is exactly what we do. Following a method presented by Sommerfeld [7], where they describe the orbital motion of a relativistic electron, and simply change signs to account for the positively charged projectile we wish to use. We then transform the 21 nonrelativistic relationship for the angle, which we denote as , to its relativistic counterpart, . In this formalism, 0 is simply equal to the arccos of the inverse of the eccentricity. This eccentricity is a function of the nonrelativistic angular momentum, and because the goal is to transform everything into the relativistic domain, it was therefore necessary to also rewrite the nonrelativistic angular momentum, L0 in terms of the relativistic angular momentum, L = L0 . This led to the description of 0 as: 0 = 1 q 1 (ZpZte2)2 c2L2 arccos 0 BB@ ZpZte2 r Lv1 1 + ZpZte2 Lv1 2 1 CCA (49) For succinctness, the simpli cation, k(b) = ZpZte2 Lv1 = d 2b p 1 2 is made. The parameter d is de ned as the collision diameter as: d = 2ZpZte2 m(p) 0 v21 . This leads to the condensation: 0 = 1 p 1 k(b)2 2 arccos k(b) p 1 + k(b)2 ! (50) A conversion from 0 into , = 2 arccot(k(b)) p 1 k(b)2 2 (51) and plugging this into a rearranged (37) where db d = dk d db dk . This produces 22 the ultimate goal of this endeavor: d d = b2 sin [1 + k(b)2][1 k(b)2 2] 2[1 + k(b)2]k(b)2 2 2 2k(b) p 1 k(b)2 2 (52) It is this, equation (52), and the equations of normalized velocity components, equations (47), that are numerically evaluated in [5] and by ourselves. Matzdorf additionally claims an accuracy of 104 in a collision of a proton with a Uranium atom. Although it was passed over without comment earlier, the magnetic eld contribution, or more appropriately, the lack thereof, should be discussed. Matzdorf showed that the magnetic eld does not contribute heavily enough to the motion of the relativistic motion of charged particles to be considered on the range from 0:1c to 0:99c. This in and of itself is not exceptionally interesting, but it has intriguing consequences. The relativistic mass correction must then be quite signi cant. They reported this speci cally for Xe + U collisions. 23 Chapter 4 SOLUTION OF THE ELECTROMAGNETIC TWOBODY SCATTERING WITH AN EFFECTIVE LAGRANGIAN EXPANSION Similar to the Matzdorf paper, C. E. Aguiar, A. N. F. Aleixo, and C. A. Bertulani [6], sought to nd an equation to describe the motion of charged particles in the relativistic domain interacting with one another through the Coulomb eld. Unlike Matzdorf, Aguiar et al. attempted to solve the problem beginning with the Darwin Lagrange, or more accurately, an expansion of the e ective Lagrangian including the Darwin Lagrange. The derivation of the Darwin Lagrangian was discussed in Section 1.3. We begin with the full Lagrangian written as: L = L(0) + L(2) (53) where L(0) is the classical, or zerothorder Lagrangian and is simply described as: L(0) = v2 2 Z1Z2e2 r (54) and L(2) is the secondorder Lagrangian correction, or the Darwin term, which is written in terms of c2 as: L(2) = 4v4 8c2 1 m31 + 1 m32 2Z1Z2e2 2m1m2c2r (v2 + v2 r ) (55) In both (54) and (55), refers to the reduced mass, and in (55), 24 vr = v r=r. It is clear here that there is a missing third and higherorders in the Lagrangian. The thirdorder term describes the dipole radiation emission. Aguiar explains that this term is usually obtained through the AbrahamLorentz Formalism which is quite mathematically intensive and will not be discussed here. From this formalism, only cases that have equal charge to mass ratios lead to nonrunaway solutions, and more speci cally, the damping force vanishes entirely. With this assumption, we are able to exclude radiative emissions altogether, and therefore not include a L(3) term which is the third order term of v c . By using a standard Hamiltonian, we are able to write a term for the velocity (dr=dt) and the force (dp=dt) to the order of c2, as: dr dt = p p2 2c2 1 m31 + 1 m32 p + Z1Z2e2 m1m2c2r h p + pr r r i (56) dp dt Z1Z2e2 r3 r + Z1Z2e2 2m1m2c2r2 h (p2 + 3p2r ) r r 2prp i (57) where p is the canonical momentum and can be expressed as: p = v + 4v2 2c2 1 m31 + 1 m32 v 2Z1Z2e2 m1m2c2r h v + vr r r i (58) Next, we would like to consider the symmetric system where the charges and the masses of the projectile and target are equal proportions. This simpli cation allows for a fourthorder correction to the Lagrangian to 25 be tacked onto (53) which is given by: L(4) = mv6 512c4 + Z2e2 16c4r 1 8 (v4 3v4 r + 2v2 rv2) + Z2e2 mr (3v2 r v2) + 4Z4e4 m2r2 where: L =L(0) + L(2) + L(4) (59) In addition, the equations of motion for the canonical momentum, the velocity, and the force can be derived in the same manner as above, where: dr dt = 2 p2 m2c2 9p4 4m4c4 p m + Z2e2 m2c2r 1 p2 2m2c2 p + 1 + 3p2r 2m2c2 pr r r + Z4e4 m3c4r2p (60) and, dp dt = Z2e2 r3 r + Z2e2 2m2c2r2 p2 + 3p2r p4 4m2c2 + 15p4r 4m2c2 r r Z2e2pr m2c2r2 1 + 3p2r 2m2c2 p + Z4e4p2 m3c4r4 r 3Z6e6 4m2c4r5 r (61) where the canonical momentum is written as: p = 1 2 1 + v2 8c2 + 3v4 128c4 mv Z2e2 4c2r 1 v2 8c2 v2 r 8c2 + Z2e2 2mc2r v + 1 v2 8c2 + 3v2 r 8c2 3Z2e2 2mc2r vr r r (62) From here, Aguiar performs the numerical analysis of equations of 26 motion not assuming equal charge to mass ratios, equations (56) and (57). We preform our own analysis of this result in Chapter 6. We would then like to consider the more simple case of a light particle scattering o a much heavier particle. In this case, because recoil can be neglected (in addition to keeping with the assumption from above: not allowing radiation emission), the retardation e ects vanish. This is due to the immobile larger particle, considered to have in nite mass as compared to the other. This causes the eld of the heavy particle to also be static. Relativity cannot be entirely ignored however, because of the increase in the mass of the lighter particle due to relativity. We begin our consideration of this case by stating the scattering angle is given by: = 2 p 2 2 arctan p 2 2 (63) where is merely the abbreviation: = vL Z1Z2e2 (64) where L, as before, is the angular momentum. As discussed in Chapter 2, in elastic collisions, angular momentum is conserved. Thus the angular momentum must be equal to the linear momentum multiplied by the impact parameter, L = pb. From this logic, is a function of L and therefore a function of b. Because we ultimately would like to solve the di erential cross section, which relates b and , this leads us to expand , as 27 a function of , to the second order of explicitly in addition to a term representing the consolidated higher order terms, O( 4). This expansion is then plugged into (37) where , as a function of the impact parameter, replaces b in the following way: d d = Z1Z2e2 2mv2 sin2( =2) 2 [1 h( ) 2 + O( 4)] (65) where h( ) = 1 + 1 2 [1 + ( ) cot ] tan2 2 and in this case, m is the relativistic mass. The last step performed, was to express (65) in terms of the kinetic energy, K. d d = Z1Z2e2 4K sin2( =2) 2 1 + g( ) K mc2 + O K2 (mc2)2 (66) where g( ) = 3 2h( ). It is evident that the rst term in the brackets is the classical Rutherford scattering formula. The term that follows is Aguiar's proposed relativistic correction to the di erential cross section. This nal method proposed by Aguiar, although bulky, can be solved analytically without the need for numeric computational methods. Should the analytic solutions presented in either method above prove to be su ciently close to the according numeric solution, then the time costly numeric solution would not be necessary. This is discussed for both methods in Chapter 6, Results. Before we discuss the results of our investigation, we need to review the numerical method we employed to solve the numeric solutions presented. 28 Chapter 5 RUNGE KUTTA NUMERICAL METHOD 5.1 Basic Runge Kutta Method The last bit of background necessary to discuss is an extremely useful numerical method, called the Runge Kutta method. This method is used to solve ordinary rst order di erential equations of the form: dy dx = f(x; y) (67) If we attempt to solve this di erential equation over some small interval of i ! i + 1 (what is considered small depends entirely on the function) we would proceed by: dy =f(x; y)dx Z yi+1 yi dy = Z xi+1 xi f(x; y)dx yi+1 y1 = Z xi+1 xi f(x; y)dx yi+1 =yi + Z xi+1 xi f(x; y)dx (68) under the assumption of a nonprecipitous function, or at least relatively smooth over the required interval, the integral in the last term can be reduced to the slope of the function, multiplied by the horizontal 29 displacement or the step interval, h, reducing the equation to: yi+1 = yi + h (69) This is the form of the simplest Runge Kutta scheme, and is of the rst order. The second order is solved using something that is very similar to (69), but the slope is modi ed to be = (a1k1 + a2k2) where a1 and a2 are constants, and k1 = f(xi; yi) and k2 = f(xi + p1h; yi + q11k1h) where p1 and q11 are also just constants. We would like to nd out what all of these constants are, and to do this we use a Taylor expansion of (69) about the point yi including the new , where yi+1 = yi + f(xi; yi)h + f0(xi;yi) 2! h2 + : : : and what this does, is it allows us to apply some constraints to the mess of constants we have in the following way: a1 + a2 = 1 a2 p1 = 1 2 a2 q11 = 1 2 (70) From here, we employ a classic solution to solve the remaining portions of (70) called the Midpoint method where we arti cially set a2 = 1 2 . This then solves the remainder of the constants where a1 = 1 2 , p1 = 1, and q11 = 1. There are three broadly adopted methods, Heun's Method where you set a2 = 1 2 , the Midpoint Method mentioned above, and the Ralston's Method where you set a2 = 2 3 . These are all valid methods, 30 but the Midpoint method speci cally involves using the derivative at the starting point, and using that to approximate the derivative at the midpoint. This slope at the midpoint is then used in a straight line approximation from the original location to nd the new position location after a step h. As an example, the midpoint method we will use from here on. It then follows that (69) becomes: yi+1 = yi + k2 (71) where k1 = hf(xi; yi) and k2 = hf(xi + 1 2h; yi + 1 2k1) 5.2 Adaptive Step Size Runge Kutta Method We would now like to consider an adaptive solution that tries to take into account the inherent error associated with approximating the solutions of di erential equations. In these methods, it seems obvious that as the step size decreases, accuracy increases. The natural response is then to have h ! 0. This has a problem however, in that as h gets smaller, the computation time increases, so it is not realistic to have h = 0, because that would then require in nite computation time. In most cases however, we do not need the exact answer; we merely need an answer that is su ciently close to the exact answer, within some acceptable margin of error. Additionally, error can be reduced by using higher order methods. The last problem we seek to address is that of uneven error that occurs as a curve 31 transitions from relatively smooth to steep (or vise versa). If the local error, or the instantaneous error associated with each point, was uniform, that would greatly improve any approximation. To achieve this, we introduce what is known as an adaptive step size Runge Kutta method. One of the major tenets of this method is error reduction, so we require a way to estimate the error at each step. The way this is done, is by using two di erent methods at the same time. The solution that better matches the exact, or known value is assumed to be very close to the exact value, so the method that produced the less accurate values has a local error equal to the absolute di erence between the two values. For an example, using di erent methods, say: Method 1: y1(x + h) Method 2: y2(x + h) Local Error: jy1(x + h) y2(x + h)j (72) At this point, if the local error is far below some preset tolerance (usually set by the computation power available to you), then the approximation for this point is more accurate than necessary, meaning it is wasting valuable computation time, and we can double the step size. If the error is far above that tolerance, then we can half the step size to improve the accuracy in this region. Lastly, if the error approximates the tolerance, then the step size is acceptable for this point, and we move on to the next 32 step. An example of di erent methods would simply be to use two methods of di erent orders, where the higher order method can be assumed to be the more accurate method, and the lower order method to be assumed to be less accurate. Two such methods could be Runge Kutta second order approximation (71) and the Runge Kutta third order approximation (not discussed here). In this example, the third order Runge Kutta is considered to be essentially exact, and the error is calculated by nding the di erence between the value of the third order approximation and the second order. 33 Chapter 6 RESULTS 6.1 Numerical Results We seek to investigate Matzdorf's [5] solutions to the motion of relativistic, heavy, charged particles interacting, equations (47) and (43), and compare those to Aguiar's [6] solutions, equation (59). We solved these equations numerically using the adaptive step size control Runge Kutta method discussed in Section 5.2. We began by using the same initial conditions as [5], where the target particle is taken to be motionless at the origin at time, t = 1. The impact parameter, y(t = 1) = b where the projectile moves towards the target in the ^x direction; the total trajectory distance is con ned to 80; 000 fm for our calculations. At time t = +1 the projectile scatters to an angle , with a velocity v1 (which is equal to v1 from conservation of momentum). These parameters are all nearly identical to the classical scattering parameters discussed in Chapter 2 for reference. The impact parameter is varied in increments of b = 0:1 fm from 60 fm to the sum of the atomic radii of the target and projectile, Rt + Rp. To begin with, the scattering angles obtained using Methods [5] and [6] are checked against well known classical, nonrelativistic Rutherford scattering angle (c), [8]: (c) = 2 arctan qpqt v2b (73) 34 In Figure 4 and Figure 5, a 208Pb +208 Pb collision is shown with the relative percent di erence between each method, with the classical de ection angle presented as a percent di erence: % = [5]or[6] (c) (c) 100 (74) plotted against the impact parameter in Figure 4 at 100 MeV per nucleon, and in Figure 5, plotted against the bombarding energy, Elab in MeV per nucleon at a grazing impact parameter equal to the sum of the atomic radii of the target and projectile. 35 0 10 20 30 40 50 60 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 208Pb+208Pb E lab =100 MeV/n (Θ  Θ NR ) / Θ NR [%] b (fm) Figure 4: 208Pb +208 Pb collision at 100MeV n . Scattering angle percent di er ence vs. impact parameter. 36 0 50 100 150 200 250 0 5 10 15 20 208Pb + 208Pb b = R P + R T (Θ  Θ NR ) / Θ NR [%] E lab (MeV/n) Figure 5: 208Pb +208 Pb collision at b = Rp + Rt. Scattering angle percent di erence vs. bombarding energy. The dashed line represents the Aguiar method to order v c 4 , allowed to this order because of the equal charge to mass ratio as shown in [5], and the solid line represents the Matzdorf method. From Figure 4, it is evident that the Matzdorf method compares more closely to the nonrelativistic scattering angle than the method of Lagrangian expansion by Aguiar. Because of the known, small contribution from the magnetic eld, the smaller deviation from the nonrelativistic angle in Matzdorf's method can be attributed to their noninclusion of the magnetic terms, while the 37 Darwin Lagrangian relies heavily on the magnetic eld. Figure 4 also shows that the scattering angle % di erence increases as b increases, but the percent caps at about 7% for Aguiar's method; and caps at about 5% for Matzdorf's method. This cap in the percentage is an intuitive result because at larger impact parameters, the defection angle decreases for both methods as well as the Rutherford classical scattering at approximately the same rate after small impact parameters. Therefore the percent di erence should cap asymptotically as b ! 1. In Figure 5 it is clear that both methods increasingly deviate from the Rutherford classical scattering angle as the bombarding energy increases. This is perfectly reasonable to expect, as retardation and relativistic corrections cause substantial deviations from the classical solutions as we approach the relativistic domain. Once again, this gure shows that the Aguiar method exceeds the percent di erence in the scattering angle as compared to that of the Matzdorf method. This is most likely due to Aguiar's consideration of relativistic mass but reduced consideration of retardation associated with this method. Next, a comparison is made to the classical di erential cross section, d d (c) displayed in the same method as above in the form of a percent di erence between the two methods and the classical cross section. These are displayed in Figure 6 and Figure 7 where the former shows a 208Pb +208 Pb collision at 100 MeV per nucleon plotted against the scattering angle, while the latter shows the same Lead collisions plotted 38 against the laboratory energy at a grazing impact parameter. 2 4 6 8 10 12 9 10 11 12 13 14 15 208Pb+208Pb E lab = 100 MeV/n (dσ  dσ NR ) / dσ NR [%] Θ (Deg) Figure 6: 208Pb +208 Pb collision at 100MeV n . Di erential scattering cross section percent di erence vs. scattering angle. 39 0 50 100 150 200 250 0 10 20 30 40 50 208Pb + 208Pb b = R P + R T (dσ  dσ NR ) / dσ NR [%] E lab (MeV/n) Figure 7: 208Pb+208 Pb collision at b = Rp +Rt. Di erential scattering cross section percent di erence vs. bombarding energy. It is clear from Figure 6 that the two methods' solutions for the di erential scattering cross section are quite far apart from one another when they are plotted against the scattering angle ( 4:5%). Once again, the dashed line represents Aguiar's method, and the solid line represents Matzdorf's method. It should be noted that each method becomes more like the classical scattering as the scattering angle is increased. This is can be accounted for by a close look at the classical di erential scattering cross section. The di erential scattering cross section, being proportional to the 40 probability that the particle is scattered to a di erential solid angle from a di erential cross sectional area, should decrease very rapidly as the scattering angle increases. More succinctly, you would expect fewer de ections to high angles, and more de ections at low angles. However, at higher energies, you would expect smaller de ection angles for the aggregate, regardless of the impact parameter, but it would still drop o rapidly. We would therefore expect the higher classical di erential scattering cross sections for higher scattering angles than that of the relativist methods, but both should approach zero as the de ection angle increases. This should manifest itself as a negative slope on the percent di erence vs. angle plot. As is evident by the data from Figure 6, the curve does indeed drop o as the scattering angle increases, which is exactly what we should expect. Similar to Figure 5, Figure 7 shows that both methods increasingly depart from the classical di erential cross section as the laboratory energy is raised. As before, Matzdorf's method does not deviate from the classical di erential as much as Aguiar's method for both Figure 6 and 7. For all the numeric solutions presented for symmetric collisions, both methods are in close proximity to one another, and agree well with the classical regime at low energies. Matzdorf's solution is far more close to the classical Rutherford solution in all of the gures presented. 41 6.2 Analytic Results As both methods are presented with two possible solutions: a more involved solution that requires a numerical method approach (the previous section), and a less involved analytic solution, it seems prudent to also compare these more time e cient methods to see if they hold muster. First we examine the di erential scattering cross section produced by each method for the same symmetric collision of 208Pb +208 Pb at a grazing angle b = Rp + Rt. This is a pure solution graphic, not a comparison to the classical di erential scattering cross section in the form of a percent as before. 42 0 50 100 150 200 250 103 104 105 106 208Pb + 208Pb b = R P + R T dσ / dΩ ( fm2 / sr ) E lab (MeV/n) Figure 8: 208Pb+208 Pb collision at b = Rp +Rt. Di erential scattering cross section vs. bombarding energy. Figure 8 shows that both methods agree very closely at low energies and separate only slightly at higher energies. It should be noted that the vertical axis, which displays the di erential scattering cross section, is logarithmic. This can make large di erences seem smaller, but the two methods are in relatively good agreement. The analytic formulas proposed by both papers are valid for light particles scattering o a heavy target. In Figure 9 the di erential scattering cross section percent di erence is plotted against the scattering angle for 43 the asymmetric collision of 17O +208 Pb. Due to the asymmetry, Aguiar's method is only valid up to order of v c 2 . 0 2 4 6 8 10 12 8.5 9.0 9.5 10.0 10.5 11.0 11.5 17O + 208Pb E lab = 100 MeV/n (dσ  dσ NR ) / dσ NR [%] Θ (Deg.) Figure 9: 17O+208Pb collision at 100MeV n . Di erential scattering cross section percent di erence vs. scattering angle. Figure 9 shows three di erent lines. The dashed line represents Aguiar's analytic solution (65), while the solid line represents Aguiar's exact, numeric solution up to the order of v c 2 . These are both in excellent agreement with one another. The dotted line represents Matzdorf's analytic equations (51). Not shown in this graphic is the Matzdorf numeric exact result, which would lie ontop of the dotted line, and would be 44 indistinguishable from Matzdorf's solutions at this resolution; it agrees within 0:1% of that of the exact result. It is clear from Figure 9 that the two methods are in much better agreement with one another for light on heavy particle collisions. Not only do the analytic methods presented by Aguiar and Matzdorf agree well with one another, they also agree quite well with the exact result of the numeric solutions, the consequences of which are discussed in Chapter 7. 6.3 Analysis of the Distance of Closest Approach We rst start by de ning half the distance of closest approach in a head on nonrelativistic collision, a0 , which is a useful scattering parameter characterized by: a0 = qpqt v2 (75) To begin a study of the distance of closest approach, it is convenient to rst employ the use of a helpful parameterization common to orbital motion problems: x =a0(cosh ! + ) y =a0 p 2 1 sinh ! z =0 (76) where ! is the angular frequency and is the eccentricity which is related to the scattering angle by = sin1( 2 ). Then if we wish to express this 45 parameterization in terms of the radial distance r = p x2 + y2 + z2, then we arrive at r = a0( cosh ! + 1). Then by solving v = dr dt = dr d! d! dt and rearranging for t, we attain the useful time parameterization: t = a0 v ( sinh ! + !) (77) Next we move this parameterization into the relativistic domain by replacing all instances of a0 with a where a = a0 . We can nally introduce the relativistic distance of closest approach bc associated with the impact parameter by: bc = a + p a2 + b2 (78) With this, we calculate the impact parameter by solving Matzdorf's equations (51) and the de nition of k(b) for the distance of closest approach, labeled bAnalyt and compare that to the relativistic distance of closest approach in a similar manner as the prior gures, in the form of a percent di erence. These are displayed in Figure 10. 46 50 100 150 200 250 0.01 0.1 1 0.01 0.1 1 ( b c  bAnalyt ) / b c [%] E lab (MeV/n) p + 208Pb 17O + 208Pb 40Ca + 208Pb 208Pb + 208Pb Figure 10: Multiple di erent collisions at b = Rp + Rt. Distance of closest approach percent di erence vs. bombarding energy. The solid line represents a p +208 Pb collision (p is a single proton), the dashed line represents a 17O +208 Pb collision, the dotted line represents a 40Ca +208 Pb collision, and the dasheddotted line represents the symmetric 208Pb +208 Pb collision; all of which were evaluated at a grazing impact parameter. The accuracy of these collisions is excellent even for the severely asymmetric case. Equation (78) therefore, produces a result very close to the \exact" result, all at or below the accuracy of 1% or less, further showing the validity of Matzdorf's methods. 47 Chapter 7 CONCLUSIONS In this paper, we have reviewed methods for solving for the relativistic motion of heavyonheavy and lightonheavy ion collisions. These collisions were evaluated over a spread of impact parameters, bombardment energies, and scattering angles; and from these the di erential scattering cross section and the relativistic distance of closest approach were calculated and plotted. Through a thorough investigation of the relativistic e ects manifested in the form of the magnetic interactions included in the Coulomb scattering, the retardation of the Coulomb electric potential, and the change in mass due to relativistic speeds, we have reached a a few important interpretations from our results. The solutions to the scattering angle and di erential scattering cross section presented by Matzdorf [5] as a solution to the covariant equations of motion are shown to be markedly better than the method of Lagrangian expansion by orders of v c outlined by Aguiar et al. [6]. The covariant equation solution accounts for retardation very well, while the Lagrangian expansion does not. The e ects of retardation being quite large in relativistic motion, this is something that needs to be considered in high energy elastic scattering. In addition to this, the inclusion of magnetic interaction terms proves to be unnecessary, as its contributions are negligible to the motion of relativistic ions. This could be 48 further reason for the superiority of the Matzdorf method, as the Lagrangian expansion includes the Darwin Lagrangian, which is heavily in uenced by magnetic terms. Further, the Matzdorf's analytic solutions were found to be in good enough agreement with the corresponding numeric solutions of equations (43) and (47). The di erential scattering cross section has been shown to be best described using equation (52). The scattering angle is solved most e ciently using equation (51). Lastly, the relativistic distance of closest approach, which is described in equation (78), in addition to the time dependence parameterization of the trajectory, given by equations (76), are shown to agree very well with one another. In summation, the solution of the covariant equation of a charged particle moving in the external electromagnetic eld of another charged particle method for accounting for the elastic scattering of heavy ions is more e ective than the method of Lagrangian expansion. In addition, the analytic formulation following the covariant equation solution method is su cient for all circumstances tested. Therefore, the more computationally involved numerical method is not necessary even for the severely asymmetric cases. 49 REFERENCES [1] Darwin C. G., Philos. Mag Ser. 6, 39, p. 537, 1920. [2] Jackson J. D., 1999, Classical Electrodynamics, 3rd edition (John Wiley & Sons, New York). [3] Ess en H., 2005, Eur. J. Phys., 26, p. 279. [4] Rutherford E., May 1911, The Scattering of and Particles by Matter and the Structure of the Atom, Philosophical Magazine. Series 6, 21. [5] Matzdorf R., So G.,Mehler G., 1987, Elastic Collisions of Heavy Ions at Intermediate Energies, Zeitschrift Physik D Atoms, Molecules and Clusters, 6, Number 1, pp. 512. [6] Aguiar C. E., Aleixo A. N. F., Bertulani C. A., November 1990, Elastic Coulomb Scattering of Heavy Ions at Intermediate Energies, American Physical Society Physics Review C, 42, iss. 5. [7] Sommerfeld A., 1978, Atombau und Spektrallinien. Frankfurt: Harri Deutsch. [8] Goldstein H., Poole C., Safko J., 2002, Classical Mechanics, 3rd edition (Addison Wesley), p. 110 50 VITA George Phillip Robinson attended Angelo State University in San Angelo, Texas where he received his Bachelors of Science in Physics in May of 2015. George further continued his education obtaining a Masters of Science in Physics from Texas A&M UniversityCommerce in Commerce, Texas in August of 2017.
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Title  Relativistic Corrections in Heavy Ion Elastic Scattering 
Author  Robinson, George Robinson 
Subject  Nuclear physics and radiation 
Abstract  RELATIVISTIC CORRECTIONS IN HEAVY ION ELASTIC SCATTERING A Thesis By George Phillip Robinson Submitted to the O ce of Graduate Studies Of Texas A&M UniversityCommerce In ful llment of the requirements For the degree of MASTERS OF SCIENCE August 2017 RELATIVISTIC CORRECTIONS IN HEAVY ION ELASTIC SCATTERING A Thesis By George Phillip Robinson Approved by: Adviser: Carlos Bertulani Committee: BaoAn Li William Newton Head of Department: Matt Wood Dean of the College: Brent Donham Dean of Graduate Studies: Mary Beth Sampson iii Copyright c 2017 George Phillip Robinson iv ABSTRACT RELATIVISTIC CORRECTIONS IN HEAVY ION ELASTIC SCATTERING George Phillip Robinson, MS Texas A&M UniversityCommerce, 2017 Advisor: Carlos Bertulani, PhD Relativistic e ects of heavy ion scattering were investigated at interme diate collision energies, at or above about 50 MeV/n. Two methods for evaluating these e ects were compared for their validity. The rst method involves a full account of the retardation of the Coulomb potential by solu tion of the covariant equations of motion for charged particles. The second method involved the expansion of the e ective Lagrangian, including the elec tromagnetic Darwin Lagrangian, in orders of (v/c). This study allowed for the determination of the degree of involvement of e ects such as relativistic magnetic interactions, kinematic corrections, and relativistic mass increase in the motion of the heavy charge particles. It was shown that the numeric solutions of the coupled di erential equations presented were not necessary as the analytic formulations su ciently describe all of the scattering parameters needed for nuclear experimentation. v ACKNOWLEDGMENTS: Dr. Carlos Bertulani Ravinder Kumar Michael Hartos vi TABLE OF CONTENTS LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vii INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii CHAPTER 1. DARWIN LAGRANGIAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Electrodynamics with Retardation . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 LienardWiechert Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Darwin Lagrangiian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. RUTHERFORD SCATTERING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3. THE INCLUSION OF RETARDATION IN COULOMB SCATTER ING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4. SOLUTION OF THE ELECTROMAGNETIC TWOBODY SCAT TERING WITH AN EFFECTIVE LAGRANGIAN EXPANSION 23 5. RUNGE KUTTA NUMERICAL METHOD . . . . . . . . . . . . . . . . . . . . . 28 5.1 Basic Runge Kutta Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28 5.2 Adaptive Step Size Runge Kutta Method . . . . . . . . . . . . . . . . . . 30 6. RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.2 Analytic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41 6.3 Analysis of the Distance of Closest Approach . . . . . . . . . . . . . . 44 7. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 vii LIST OF FIGURES 1. LienardWiechert Potential for a particle moving through an arbitrary path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Simple 2D scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3. Classical scattering by a central potential . . . . . . . . . . . . . . . . . . . . . . . 15 4. 208Pb+208 Pb collision at 100MeV n . Scattering angle percent di erence vs. impact parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5. 208Pb +208 Pb collision at b = Rp + Rt. Scattering angle percent di erence vs. bombarding energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6. 208Pb+208Pb collision at 100MeV n . Di erential scattering cross section percent di erence vs. scattering angle. . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7. 208Pb +208 Pb collision at b = Rp + Rt. Di erential scattering cross section percent di erence vs. bombarding energy . . . . . . . . . . . . . . . .39 8. 208Pb +208 Pb collision at b = Rp + Rt. Di erential scattering cross section vs. bombarding energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 9. 17O+208 Pb collision at 100MeV n . Di erential scattering cross section percent di erence vs. scattering angle . . . . . . . . . . . . . . . . . . . . . . . . . . .43 10. Multiple di erent collisions at b = Rp + Rt. Distance of closest approach percent di erence vs. bombarding energy . . . . . . . . . . . . . 46 viii Introduction In modern radioactive beam facilities, Coulomb excitation is used to probe phenomenon such as dipole polarizeability, pygmy dipole resonance, neutron skin thickness, equations of state of nuclear material, and general nuclear structure. This is done because Coulomb interactions are well understood. Experimentally, it is presumed that Coulomb scattering dominates in small angle scattering especially in the nonheadon elastic scattering of heavy ions. Many of the beam experiments are performed at appreciably high kinetic energies. Due to this, it is crucial that relativistic e ects be accounted for in order to properly address relativistic kinematics, any number of possible internal excitations, and any other relativistic reaction dynamics. We sought to investigate the best method to account for relativistic e ects in elastic Coulomb scattering of heavy ions. To do this, we explored two di erent proposed methods for assessing relativistic scattering. The rst involved a solution to the covariant equations of motion for a charged particle moving in the electric and magnetic eld of another. The second used an expansion of the e ective Lagrangian to determine the extent of the relativistic velocities. Both numeric and analytic solutions were given using each method, and both were compared for validity. First however, it is necessary to discuss some background material. 1 Chapter 1 DARWIN LAGRANGIAN 1.1 Electrodynamics with Retardation In classical electrodynamics, the Lagrangian for multiple charged particles can be written as follows: L = T U (1) where T is the kinetic energy, and U is the potential energy. S = Z mcds = Z Tdt (2) where S is the classical action and ds = cdt q 1 v2 c2 . Therefore: T = mc2 r 1 v2 c2 (3) while the potential follows easily from the Force(F = q[E + (v B)]) as: U = q 2 V (r) A(r) v c (4) 2 (these terms are de ned later when we discuss Maxwell's equations, where they are more relevant) bringing us to the nal Lagrangian: L = X n " mnc2 r 1 v2n c2 + qn 2 vn c A(rn) V (rn) # (5) The Darwin Lagrangian is a term based on the expansion of the classical electrodynamic's Lagrangian above for two relativistic charged particles interacting in free space. It was originally derived in 1920 by Charles Darwin by expansion of the LienardWiechert potential (we di er from the original derivation, as explained below). The term accounts for the reaction of one particle to the magnetic eld created by the other. The Darwin Lagrangian follows from the expansion to the order of v2=c2 , but higher orders are required to account for retardation e ects. To nd this Lagrangian term, we begin with Maxwells equations: r E = 0 r B = 0 r E = @B @t r B = 0J + 0 0 @E @t (6) 3 We take the curl of the third Maxwell's equation: r (r E) = r @B @t r(r E) r2E = @(r B) @t r 0 r2E = @ @t 0J + 0 0 @E @t (7) making the electric eld wave equation with the inclusion of source terms: 2E = r 0 + 0 @J @t (8) where 2 = r2 1 c2 @2 @t2 . We can now integrate the Green's function (9) and the sources to nd the electric eld: G(r; t; r0; t0) = (t t0) 4 jr r0j (9) E(r; t) = Z d3r0dtG(r; t; r0; t0) r0 (r0; t0) 0 + 0 @J(r0; t0) @t0 (10) Through integration by parts, and the unique properties of derivatives of the Greens function(rG = r0G), we have the following emerge: E(r; t) = rV @A @t (11) where the scalar potential and the vector potential are respectively de ned 4 by: V (r; t) = 1 4 0 Z (r0; tr) R d3r0 (12) A = 0 4 Z J(r0; tr) R d3r0 (13) Similar manipulations of the magnetic eld term from Maxwell's equations, and following the same procedure just performed for E leads to: B(r; t) = r A (14) In this case, both the E and B eld equations include the term for the retarded time, tr = t R c where R is the position term, jr r0j from the Green's function, and c is the speed of light. Additionally, and J are the charge and current distributions respectively. It is then evident that the E eld di ers from the electrostatic case by the addition of the time derivative of the vector potential term, while the B eld seems to remain the same as the magnetostatic case. The B eld term does change however, due to the inclusion of the position dependence of the retarded time (as well as R) when the curl is evaluated. This leads to multiple terms through a chain rule, clearly making it di erent from the magnetostatic case. We will discuss the Lagrangian that arises from this derivation, but this is a convenient place to stop and discuss the LienardWiechert Potential that we use later. So we digress shortly, and will return to this topic after. 5 1.2 LienardWiechert Potential Our goal in this section is to nd the potential at some location from a charged particle moving through an arbitrary path. So we begin, as we do, with the de nition of terms represented in Figure 1. Figure 1: LienardWiechert Potential for a particle moving through an arbi trary path As shown above, a particle of charge q is moving through an arbitrary path traced by rq at a velocity vq(t). Additionally, we de ne, the same as above, a vector from the particle to the point P which we will simplify by calling it R. We can also quite easily de ne a trivial charge density (r; t) = q (r rq(t)) and a current J(r; t) = qvq (r rq(t)). Although we previously derived terms for V (r; t) and A(r; t) only in terms of of a volumetric integral, we will revert back to the time dependent, more generic 6 versions because we now would like to explicitly solve the time integral rather than take it for granted. We begin with the Scalar Potential: V (r; t) = 1 4 0 Z d3r0dt0 (r 0; t0) (t t0 R c ) R = q 4 0 Z d3r0dt0 (r 0 rq(t0)) (t t0 R c ) R = q 4 0 Z dt0 (t t0 Rq(t) c ) Rq(t) (15) where Rq(t) = jr rq(t)j, which it should be obvious from this notation that Rq is time dependent while the original R was independent of time. So to solve the integral, we must rst examine some unique properties of delta functions: [f(t)] = X i (t ti) jf0(ti)j (16) In this, f(t) is any arbitrary function of t, and atf(ti) = 0 while f0(ti) 6= 0 and in this case, f(t0) = t t0 Rq(t0) c . Therefore, t0 0 = t Rq(t0) c and f(t0 0) = 0, but t0 is quite obviously the retarded time tr. We also need the derivative of f: f0(t0) = Rq(t0) vq(t0)) Rq(t0)c 1. Now returning to the integral: V (r; t) = q 4 0 Z dt0 (t tr Rq(t) c ) Rq(t) = q 4 0 Z dt0 (t tr) (1 Rq(t0) Rq(t0) vq(t0) c )Rq(t) V (r; t) = q 4 0 " 1 Rq Rq vq c # retarded time (17) 7 Similarly, the vector potential A can be washed through the same steps to obtain the following: A(r; t) = 0q 4 " vq Rq Rq vq c # retarded time (18) These two equations for the scalar and vector potentials are the LienardWiechert Potentials, but it is no longer necessary to use the q subscript, so we can simplify by writing both as follows: V (r; t) = q 4 0 " 1 R(1 ^R ) # retarded time A(r; t) = V (r; t) c retarded time (19) 1.3 Darwin Lagrangian Prior to the LienardWiechert Potential explanation, we came up with a term for the E and B, and from these we can write the Lagrangian for the relativistic interaction of a particle of charge q within an electromagnetic eld as: L = qrV + q c (v A) (20) where v is the velocity of the charged particle. Applying the Coulomb Gauge, (r A = 0), and removing 1 c2 @2A @t2 8 because A is of order v2 c2 , we can write the following from Maxwells laws: r2A 1 c r @V @t = 4 c Jt (21) where Jt is the transverse current created by the movement of the second charge. This indicates that the currents divergence is zero throughout. It should be noted that the original derivation done by Darwin did not use this method, however, this method, as performed by Jackson [2], leads to an exact Coulomb potential V and transfers the approximation to the vector potential, A. which becomes: A(r) = Z d3r0 jr r0j 1 c J(r0) 1 4 c r0V (@r0) @t (22) We can write J(r0) for n particles as qnvn 3(r0 rn) and V as qn jr0r0 nj A(r) = Z d3r0 jr r0j qnvn c 3(r0 rn) qn 4 c r0 vn (r0 rn) jr0 rnj3 (23) Then by integrating by parts, and removing the surface at in nity, while also utilizing the property of greens functions where r0 r: A(r) = qnvn c jr rnj + qn 4 c Z d3x0 vn (r0 rn) jr0 rnj3 r0 1 jr r0j (24) A(r) = qnvn c jr rnj qn 4 c r Z d3(r0 rn) vn (r0 rn) jr0 rnj3 1 r0 r (25) 9 Finally, by evaluating this nontrivial integral, we reach: An = qn 2cr vn + rvn r r (26) where r = jvrj. If we multiply qv c to take into account the second (or incidentally, any number by including a subscript to account for the nth) charged particle, and correct the original free particle Lagrangian by saying that mc2 1 + mc2 1 2mv2[1 + 1 4 v2 c2 ] We arrive at the nalized Darwin Lagrangian: X i6=j 1 2 miv2 i + 1 8c2miv4 i + qiqj rij 1 2 + 1 4c2 (vi vj + (vi r^ij)(vj r^ij)) (27) where we have used a di erent index here merely for convenience, and to avoid confusing subscripts. This Lagrangian is useful when radiation can be neglected, but was originally derived under the assumption of nonrelativistic speeds, although that assumption should not be necessary because radiation is due to acceleration. A Lagrangian that both neglects radiation and the assumption of low speeds was proposed by H. Ess en [3] and leads to a succinct interaction Lagrangian for the interaction of two particles as: L12 = g v2 c2 e1e2 r21 v2 c2 e1e2 r21 (28) where g(x) 1 1+ p 1x , r21 = jr2 r1j, and e is the particle's charge. The 10 term g can be expended in terms of v2 c2 and at the limit of v ! c, g(x) = 1, and the related force is zero. This method allows for all retardation dependencies to be accounted for, which eliminates the assumption of low velocities as required by the original Darwin derivation. 11 Chapter 2 RUTHERFORD SCATTERING In May of 1911, Ernst Rutherford published his ndings from a scattering experiment in which alpha particles were red at a thin sheet of gold foil [4]. The atomic models of the time did not conform with the ndings of his experiment, as a high percentage of alpha particles were unexpectedly backscattered at angles greater than 90o. It was this conclusion that led to our current understanding of the ultradense positively charged center to atoms, which we now know as the nucleus, surrounded by a large volume, negatively charged space occupied by the electrons. The following are the derivations associated with Rutherford's Scattering model. Scattering being integral to this subject, it seems prudent to discuss it in some detail. We begin with a few de nitions in classical scattering: An incoming particle, the projectile, is incident on another particle, the target, which we will assume has a very large mass as compared to the incident one so we can neglect recoil of the target. We will discuss removing this assumption later, but for now we will use it to discuss the simpler case. The incoming particle approaches with an impact parameter, b, which we de ne as the perpendicular distance from the direction of motion of the incoming projectile, and the center of the target. The incident particle will have an initial velocity v0. After interacting with the target, the projectile 12 emerges at an angle , as measured from the projectile's original direction of motion, and it will have a velocity denoted v1 for its velocity as t ! 1. Figure 2: Simple 2D Scattering To obtain Rutherford's Scattering equation, we must begin with some assumptions, (in addition to having the mass of the target be much larger than that of the projectile): rst, we assume that the only interaction between the projectile and the target is the Coulomb repulsion; this implies azimuthal symmetry. We will also only consider a single projectile and only a single target. And lastly, (and most importantly for this paper) we will consider only nonrelativistic speeds (for now). So from these assumptions and the above de nitions, we can now begin to solve for a relationship between the impact parameter b and the scattering angle . (these being the best adjustable and measurable parameters respectively for experimentalists in the lab setting). We begin 13 with the Coulomb Force: F = Z1Z2e2 4 0r2 (29) where Ze is the charge of the particle, and r is the vector that traces the projectile's motion. We then consider the conservation of angular momentum. We know that because angular momentum is conserved, then the angular momentum itself must be equal to some constant, C: jLj jr pj = jmrvj sin = C (30) Where L is the angular momentum, p is the linear momentum, and is the angle between r and v. We must then consider the initial angular momentum, where the projectile has not yet experienced any de ection from the Coulomb repulsion. At this time, sin = b r , so L0 = mv0b = C. So we now have a term for the constant C. Combining the Coulomb Force (29) with our understanding of angular momentum, we can write r2 = bv0( d dt )1 (31) Then by writing the force in Cartesian coordinates. Fx = F cos = Z1Z2e2 4 0r2 cos = Z1Z2e2 4 0bv0 d dt cos = mdvx dt Fy = F sin = Z1Z2e2 4 0r2 sin = Z1Z2e2 4 0bv0 d dt sin = mdvy dt (32) 14 solving this integral for vi : vx( ) = Z1Z2e2 4 0mbv0 sin + v0 vy( ) = Z1Z2e2 4 0mbv0 (cos + 1) for 0 (33) Now we take a look at the conservation of energy, which tells us that the magnitude of the initial and nal energy must be equal (v0 = v1) and at t ! 1, = , so (focusing only on the ydirection): Z1Z2e2 4 0mbv0 (cos( ) + 1) = v0 sin Z1Z2e2 4 0mbv0 (cos( ) + 1) = v0 sin Z1Z2e2 4 0mbv0 = v0 sin cos + 1 Z1Z2e2 4 0mbv0 = v0 tan 2 (34) one nal rearrangement brings us to b as a function of . b( ) = Z1Z2e2 4 0v2 0mtan( 2 ) (35) Now, we take into account the more appropriate three dimensional version of classical scattering with a beam of incoming projectiles rather than only a single projectile. The projectile incident in a crosssectional area element d = 2 bdb at the cylindrically symmetric impact parameter b 15 will scatter to a correlating solid angle element d = 2 sin d . From these, we can de ne a so called di erential scattering crosssection, d d : Figure 3: Classical scattering by a central potential We can say that the number of incoming particles that pass through this di erential cross section d with an intensity, I, must be equal to the number of particles that scatter through the di erential solid angle d : I 2 bdb = I d d d (36) and plugging in what we know about d and d we can rearrange to nd 16 an expression for the di erential scattering cross section in terms of b and : d d = b sin db d (37) Finally, we can plug in b( ) (35) and its derivative with respect to into (37) to obtain a term for the di erential cross section solely in terms of . db d = Z1Z2e2 16 0K0 sin2( 2 ) (38) d d = Z1Z2e2 8 0K0 2 1 2 sin tan 2 sin (39) where K0 is the initial kinetic energy, K0 = 1 2mv2 0. Then by simply using the trigonometric identity: sin x = 2 sin x 2 cos x 2 , we arrive at last, at the elegant formula: d d = Z1Z2e2 16 0K0 2 1 sin4 2 (40) 17 Chapter 3 THE INCLUSION OF RETARDATION IN COULOMB SCATTERING In 1987, R. Matzdorf and G. So published a paper describing the collisions of simirelativistic heavy ions [5]. In which, they derived a set of coupled equations for the motion for relativistic charged particles. Their method does intentionally neglect radiative corrections. We will use their method in conjunction with a method outlined in a paper written by C. E. Aguiar et al. [6] which is discussed in the following section. Matzdorf begins with the covariant equations of motion that describe a charged particle moving in the external electromagnetic eld of another charged particle. Assuming the projectile is moving in the ^x direction, the covariant equation of motion for the two particle system moving under the in uence of their own electric elds and magnetic elds: dp d = q c F U (41) where eld strength tensor, F can be written as: dp d = q c 0 BBBBBBB@ 0 E1 E2 E3 E1 0 B3 B2 E2 B3 0 B1 E3 B2 B1 0 1 CCCCCCCA U (42) where p is the 4momentum and U is the 4velocity. is the proper time. 18 The E and B elds are then derived from the Lienard Wiechert potentials (19). Then plugging those back into (42), and produce three relevant equations for the velocities which can be expressed as: 4 0 BBBB@ 2 + u21 u1u2 u1u3 u1u2 2 + u22 u2u3 u1u3 u2u3 2 + u23 1 CCCCA u_ = q(p) m(p) 0 c 0 BBBB@ E(t) 1 B(t) 3 u2 + B(t) 2 u3 E(t) 2 + B(t) 3 u1 B(t) 1 u3 E(t) 3 B(t) 2 u1 + B(t) 1 u2 1 CCCCA (43) This method produces a fourth equation 4(u; u_ ) = q(p) m(p) 0 E(t) 1 u1 E(t) 2 u2 E(t) 3 u3 , but it provides no additional information, so it is disregarded. We denote u = v c as the normalized velocity to the speed of light, and = (1 2) 1 2 . The subscript 1, 2, and 3 denote the direction component as in (x1; x2; x3). In addition, p and t refer to the projectile and target respectively (although the distinction is nonessential because the reference frame can simply be switched to that of the motionless projectile and moving target and all remains the same). To solve for u_ , a mathematically straightforward matrix manipulation is performed. Although it is straightforward, it is also fairly lengthy, so it is not shown here. By con ning the scattering plane to the xy plane, and solving for u_ we can reduce the 3 3 matrix to a 2 2 matrix involving only u1 and u2 terms. Using the target's frame of reference, the scalar and vector potentials 19 can be expressed as: A = 0 V = q R (44) Performing a Lorentz transformation into the laboratory reference frame, we nd E and B can be expressed as: E =q n 2 [1 2 + (n )2] 3 2 R2 B = E (45) Then plugging in the nonvanishing terms, after the restriction to two dimensions, E1, E2, and B3, can be expressed as: E1 =q n1 2 [1 2 + (n )2] 3 2 R2 E2 =q n2 2 [1 2 + (n )2] 3 2 R2 B3 = 1E2 2E1 where 2 = 2 1 + 2 2 (46) Lastly, Matzdorf restricts the Coulomb eld to be purely classical (i.e. E = q n R2 and B = 0 ) and, due to its essentially negligible contribution, on the order of only a single percent di erence at speeds less than 0:99c, the magnetic eld's relativistic contribution is omitted. This leads to the nalized equation for the normalized accelerations 20 in each direction. u_ 1 = q(p)q(t) m(p) 0 3 ( 2 + u22 )n1 u1u2n2 R2[( 2 + u21 )( 2 + u22 ) u21 u22 ] u_ 2 = q(p)q(t) m(p) 0 3 ( 2 + u21 )n2 u1u2n1 R2[( 2 + u21 )( 2 + u22 ) u21 u22 ] (47) where R is the magnitude of the radius vector considering the target at time t, and n represents the unit vector in its respective direction. These equations, (43) and (47), are only soluble numerically as they are obviously interdependent upon one another. The numerical methods are discussed in Chapter 5 and the solutions are discussed in Chapter 6. In addition to this, the scattering angle can be solved if the target begins at the origin at rest at t = 1, and the projectile moves originally in the ^x direction, then the solution for the scattering angle can be found by use of the solution to (47) and: (t ! 1) = arctan u2 u1 = arctan dy(t) dx(t) (48) The next step, and the ultimate goal of all collision calculations, would be to use these equations to nd a relationship between the impact parameter b and the scattering angle . This is exactly what we do. Following a method presented by Sommerfeld [7], where they describe the orbital motion of a relativistic electron, and simply change signs to account for the positively charged projectile we wish to use. We then transform the 21 nonrelativistic relationship for the angle, which we denote as , to its relativistic counterpart, . In this formalism, 0 is simply equal to the arccos of the inverse of the eccentricity. This eccentricity is a function of the nonrelativistic angular momentum, and because the goal is to transform everything into the relativistic domain, it was therefore necessary to also rewrite the nonrelativistic angular momentum, L0 in terms of the relativistic angular momentum, L = L0 . This led to the description of 0 as: 0 = 1 q 1 (ZpZte2)2 c2L2 arccos 0 BB@ ZpZte2 r Lv1 1 + ZpZte2 Lv1 2 1 CCA (49) For succinctness, the simpli cation, k(b) = ZpZte2 Lv1 = d 2b p 1 2 is made. The parameter d is de ned as the collision diameter as: d = 2ZpZte2 m(p) 0 v21 . This leads to the condensation: 0 = 1 p 1 k(b)2 2 arccos k(b) p 1 + k(b)2 ! (50) A conversion from 0 into , = 2 arccot(k(b)) p 1 k(b)2 2 (51) and plugging this into a rearranged (37) where db d = dk d db dk . This produces 22 the ultimate goal of this endeavor: d d = b2 sin [1 + k(b)2][1 k(b)2 2] 2[1 + k(b)2]k(b)2 2 2 2k(b) p 1 k(b)2 2 (52) It is this, equation (52), and the equations of normalized velocity components, equations (47), that are numerically evaluated in [5] and by ourselves. Matzdorf additionally claims an accuracy of 104 in a collision of a proton with a Uranium atom. Although it was passed over without comment earlier, the magnetic eld contribution, or more appropriately, the lack thereof, should be discussed. Matzdorf showed that the magnetic eld does not contribute heavily enough to the motion of the relativistic motion of charged particles to be considered on the range from 0:1c to 0:99c. This in and of itself is not exceptionally interesting, but it has intriguing consequences. The relativistic mass correction must then be quite signi cant. They reported this speci cally for Xe + U collisions. 23 Chapter 4 SOLUTION OF THE ELECTROMAGNETIC TWOBODY SCATTERING WITH AN EFFECTIVE LAGRANGIAN EXPANSION Similar to the Matzdorf paper, C. E. Aguiar, A. N. F. Aleixo, and C. A. Bertulani [6], sought to nd an equation to describe the motion of charged particles in the relativistic domain interacting with one another through the Coulomb eld. Unlike Matzdorf, Aguiar et al. attempted to solve the problem beginning with the Darwin Lagrange, or more accurately, an expansion of the e ective Lagrangian including the Darwin Lagrange. The derivation of the Darwin Lagrangian was discussed in Section 1.3. We begin with the full Lagrangian written as: L = L(0) + L(2) (53) where L(0) is the classical, or zerothorder Lagrangian and is simply described as: L(0) = v2 2 Z1Z2e2 r (54) and L(2) is the secondorder Lagrangian correction, or the Darwin term, which is written in terms of c2 as: L(2) = 4v4 8c2 1 m31 + 1 m32 2Z1Z2e2 2m1m2c2r (v2 + v2 r ) (55) In both (54) and (55), refers to the reduced mass, and in (55), 24 vr = v r=r. It is clear here that there is a missing third and higherorders in the Lagrangian. The thirdorder term describes the dipole radiation emission. Aguiar explains that this term is usually obtained through the AbrahamLorentz Formalism which is quite mathematically intensive and will not be discussed here. From this formalism, only cases that have equal charge to mass ratios lead to nonrunaway solutions, and more speci cally, the damping force vanishes entirely. With this assumption, we are able to exclude radiative emissions altogether, and therefore not include a L(3) term which is the third order term of v c . By using a standard Hamiltonian, we are able to write a term for the velocity (dr=dt) and the force (dp=dt) to the order of c2, as: dr dt = p p2 2c2 1 m31 + 1 m32 p + Z1Z2e2 m1m2c2r h p + pr r r i (56) dp dt Z1Z2e2 r3 r + Z1Z2e2 2m1m2c2r2 h (p2 + 3p2r ) r r 2prp i (57) where p is the canonical momentum and can be expressed as: p = v + 4v2 2c2 1 m31 + 1 m32 v 2Z1Z2e2 m1m2c2r h v + vr r r i (58) Next, we would like to consider the symmetric system where the charges and the masses of the projectile and target are equal proportions. This simpli cation allows for a fourthorder correction to the Lagrangian to 25 be tacked onto (53) which is given by: L(4) = mv6 512c4 + Z2e2 16c4r 1 8 (v4 3v4 r + 2v2 rv2) + Z2e2 mr (3v2 r v2) + 4Z4e4 m2r2 where: L =L(0) + L(2) + L(4) (59) In addition, the equations of motion for the canonical momentum, the velocity, and the force can be derived in the same manner as above, where: dr dt = 2 p2 m2c2 9p4 4m4c4 p m + Z2e2 m2c2r 1 p2 2m2c2 p + 1 + 3p2r 2m2c2 pr r r + Z4e4 m3c4r2p (60) and, dp dt = Z2e2 r3 r + Z2e2 2m2c2r2 p2 + 3p2r p4 4m2c2 + 15p4r 4m2c2 r r Z2e2pr m2c2r2 1 + 3p2r 2m2c2 p + Z4e4p2 m3c4r4 r 3Z6e6 4m2c4r5 r (61) where the canonical momentum is written as: p = 1 2 1 + v2 8c2 + 3v4 128c4 mv Z2e2 4c2r 1 v2 8c2 v2 r 8c2 + Z2e2 2mc2r v + 1 v2 8c2 + 3v2 r 8c2 3Z2e2 2mc2r vr r r (62) From here, Aguiar performs the numerical analysis of equations of 26 motion not assuming equal charge to mass ratios, equations (56) and (57). We preform our own analysis of this result in Chapter 6. We would then like to consider the more simple case of a light particle scattering o a much heavier particle. In this case, because recoil can be neglected (in addition to keeping with the assumption from above: not allowing radiation emission), the retardation e ects vanish. This is due to the immobile larger particle, considered to have in nite mass as compared to the other. This causes the eld of the heavy particle to also be static. Relativity cannot be entirely ignored however, because of the increase in the mass of the lighter particle due to relativity. We begin our consideration of this case by stating the scattering angle is given by: = 2 p 2 2 arctan p 2 2 (63) where is merely the abbreviation: = vL Z1Z2e2 (64) where L, as before, is the angular momentum. As discussed in Chapter 2, in elastic collisions, angular momentum is conserved. Thus the angular momentum must be equal to the linear momentum multiplied by the impact parameter, L = pb. From this logic, is a function of L and therefore a function of b. Because we ultimately would like to solve the di erential cross section, which relates b and , this leads us to expand , as 27 a function of , to the second order of explicitly in addition to a term representing the consolidated higher order terms, O( 4). This expansion is then plugged into (37) where , as a function of the impact parameter, replaces b in the following way: d d = Z1Z2e2 2mv2 sin2( =2) 2 [1 h( ) 2 + O( 4)] (65) where h( ) = 1 + 1 2 [1 + ( ) cot ] tan2 2 and in this case, m is the relativistic mass. The last step performed, was to express (65) in terms of the kinetic energy, K. d d = Z1Z2e2 4K sin2( =2) 2 1 + g( ) K mc2 + O K2 (mc2)2 (66) where g( ) = 3 2h( ). It is evident that the rst term in the brackets is the classical Rutherford scattering formula. The term that follows is Aguiar's proposed relativistic correction to the di erential cross section. This nal method proposed by Aguiar, although bulky, can be solved analytically without the need for numeric computational methods. Should the analytic solutions presented in either method above prove to be su ciently close to the according numeric solution, then the time costly numeric solution would not be necessary. This is discussed for both methods in Chapter 6, Results. Before we discuss the results of our investigation, we need to review the numerical method we employed to solve the numeric solutions presented. 28 Chapter 5 RUNGE KUTTA NUMERICAL METHOD 5.1 Basic Runge Kutta Method The last bit of background necessary to discuss is an extremely useful numerical method, called the Runge Kutta method. This method is used to solve ordinary rst order di erential equations of the form: dy dx = f(x; y) (67) If we attempt to solve this di erential equation over some small interval of i ! i + 1 (what is considered small depends entirely on the function) we would proceed by: dy =f(x; y)dx Z yi+1 yi dy = Z xi+1 xi f(x; y)dx yi+1 y1 = Z xi+1 xi f(x; y)dx yi+1 =yi + Z xi+1 xi f(x; y)dx (68) under the assumption of a nonprecipitous function, or at least relatively smooth over the required interval, the integral in the last term can be reduced to the slope of the function, multiplied by the horizontal 29 displacement or the step interval, h, reducing the equation to: yi+1 = yi + h (69) This is the form of the simplest Runge Kutta scheme, and is of the rst order. The second order is solved using something that is very similar to (69), but the slope is modi ed to be = (a1k1 + a2k2) where a1 and a2 are constants, and k1 = f(xi; yi) and k2 = f(xi + p1h; yi + q11k1h) where p1 and q11 are also just constants. We would like to nd out what all of these constants are, and to do this we use a Taylor expansion of (69) about the point yi including the new , where yi+1 = yi + f(xi; yi)h + f0(xi;yi) 2! h2 + : : : and what this does, is it allows us to apply some constraints to the mess of constants we have in the following way: a1 + a2 = 1 a2 p1 = 1 2 a2 q11 = 1 2 (70) From here, we employ a classic solution to solve the remaining portions of (70) called the Midpoint method where we arti cially set a2 = 1 2 . This then solves the remainder of the constants where a1 = 1 2 , p1 = 1, and q11 = 1. There are three broadly adopted methods, Heun's Method where you set a2 = 1 2 , the Midpoint Method mentioned above, and the Ralston's Method where you set a2 = 2 3 . These are all valid methods, 30 but the Midpoint method speci cally involves using the derivative at the starting point, and using that to approximate the derivative at the midpoint. This slope at the midpoint is then used in a straight line approximation from the original location to nd the new position location after a step h. As an example, the midpoint method we will use from here on. It then follows that (69) becomes: yi+1 = yi + k2 (71) where k1 = hf(xi; yi) and k2 = hf(xi + 1 2h; yi + 1 2k1) 5.2 Adaptive Step Size Runge Kutta Method We would now like to consider an adaptive solution that tries to take into account the inherent error associated with approximating the solutions of di erential equations. In these methods, it seems obvious that as the step size decreases, accuracy increases. The natural response is then to have h ! 0. This has a problem however, in that as h gets smaller, the computation time increases, so it is not realistic to have h = 0, because that would then require in nite computation time. In most cases however, we do not need the exact answer; we merely need an answer that is su ciently close to the exact answer, within some acceptable margin of error. Additionally, error can be reduced by using higher order methods. The last problem we seek to address is that of uneven error that occurs as a curve 31 transitions from relatively smooth to steep (or vise versa). If the local error, or the instantaneous error associated with each point, was uniform, that would greatly improve any approximation. To achieve this, we introduce what is known as an adaptive step size Runge Kutta method. One of the major tenets of this method is error reduction, so we require a way to estimate the error at each step. The way this is done, is by using two di erent methods at the same time. The solution that better matches the exact, or known value is assumed to be very close to the exact value, so the method that produced the less accurate values has a local error equal to the absolute di erence between the two values. For an example, using di erent methods, say: Method 1: y1(x + h) Method 2: y2(x + h) Local Error: jy1(x + h) y2(x + h)j (72) At this point, if the local error is far below some preset tolerance (usually set by the computation power available to you), then the approximation for this point is more accurate than necessary, meaning it is wasting valuable computation time, and we can double the step size. If the error is far above that tolerance, then we can half the step size to improve the accuracy in this region. Lastly, if the error approximates the tolerance, then the step size is acceptable for this point, and we move on to the next 32 step. An example of di erent methods would simply be to use two methods of di erent orders, where the higher order method can be assumed to be the more accurate method, and the lower order method to be assumed to be less accurate. Two such methods could be Runge Kutta second order approximation (71) and the Runge Kutta third order approximation (not discussed here). In this example, the third order Runge Kutta is considered to be essentially exact, and the error is calculated by nding the di erence between the value of the third order approximation and the second order. 33 Chapter 6 RESULTS 6.1 Numerical Results We seek to investigate Matzdorf's [5] solutions to the motion of relativistic, heavy, charged particles interacting, equations (47) and (43), and compare those to Aguiar's [6] solutions, equation (59). We solved these equations numerically using the adaptive step size control Runge Kutta method discussed in Section 5.2. We began by using the same initial conditions as [5], where the target particle is taken to be motionless at the origin at time, t = 1. The impact parameter, y(t = 1) = b where the projectile moves towards the target in the ^x direction; the total trajectory distance is con ned to 80; 000 fm for our calculations. At time t = +1 the projectile scatters to an angle , with a velocity v1 (which is equal to v1 from conservation of momentum). These parameters are all nearly identical to the classical scattering parameters discussed in Chapter 2 for reference. The impact parameter is varied in increments of b = 0:1 fm from 60 fm to the sum of the atomic radii of the target and projectile, Rt + Rp. To begin with, the scattering angles obtained using Methods [5] and [6] are checked against well known classical, nonrelativistic Rutherford scattering angle (c), [8]: (c) = 2 arctan qpqt v2b (73) 34 In Figure 4 and Figure 5, a 208Pb +208 Pb collision is shown with the relative percent di erence between each method, with the classical de ection angle presented as a percent di erence: % = [5]or[6] (c) (c) 100 (74) plotted against the impact parameter in Figure 4 at 100 MeV per nucleon, and in Figure 5, plotted against the bombarding energy, Elab in MeV per nucleon at a grazing impact parameter equal to the sum of the atomic radii of the target and projectile. 35 0 10 20 30 40 50 60 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 208Pb+208Pb E lab =100 MeV/n (Θ  Θ NR ) / Θ NR [%] b (fm) Figure 4: 208Pb +208 Pb collision at 100MeV n . Scattering angle percent di er ence vs. impact parameter. 36 0 50 100 150 200 250 0 5 10 15 20 208Pb + 208Pb b = R P + R T (Θ  Θ NR ) / Θ NR [%] E lab (MeV/n) Figure 5: 208Pb +208 Pb collision at b = Rp + Rt. Scattering angle percent di erence vs. bombarding energy. The dashed line represents the Aguiar method to order v c 4 , allowed to this order because of the equal charge to mass ratio as shown in [5], and the solid line represents the Matzdorf method. From Figure 4, it is evident that the Matzdorf method compares more closely to the nonrelativistic scattering angle than the method of Lagrangian expansion by Aguiar. Because of the known, small contribution from the magnetic eld, the smaller deviation from the nonrelativistic angle in Matzdorf's method can be attributed to their noninclusion of the magnetic terms, while the 37 Darwin Lagrangian relies heavily on the magnetic eld. Figure 4 also shows that the scattering angle % di erence increases as b increases, but the percent caps at about 7% for Aguiar's method; and caps at about 5% for Matzdorf's method. This cap in the percentage is an intuitive result because at larger impact parameters, the defection angle decreases for both methods as well as the Rutherford classical scattering at approximately the same rate after small impact parameters. Therefore the percent di erence should cap asymptotically as b ! 1. In Figure 5 it is clear that both methods increasingly deviate from the Rutherford classical scattering angle as the bombarding energy increases. This is perfectly reasonable to expect, as retardation and relativistic corrections cause substantial deviations from the classical solutions as we approach the relativistic domain. Once again, this gure shows that the Aguiar method exceeds the percent di erence in the scattering angle as compared to that of the Matzdorf method. This is most likely due to Aguiar's consideration of relativistic mass but reduced consideration of retardation associated with this method. Next, a comparison is made to the classical di erential cross section, d d (c) displayed in the same method as above in the form of a percent di erence between the two methods and the classical cross section. These are displayed in Figure 6 and Figure 7 where the former shows a 208Pb +208 Pb collision at 100 MeV per nucleon plotted against the scattering angle, while the latter shows the same Lead collisions plotted 38 against the laboratory energy at a grazing impact parameter. 2 4 6 8 10 12 9 10 11 12 13 14 15 208Pb+208Pb E lab = 100 MeV/n (dσ  dσ NR ) / dσ NR [%] Θ (Deg) Figure 6: 208Pb +208 Pb collision at 100MeV n . Di erential scattering cross section percent di erence vs. scattering angle. 39 0 50 100 150 200 250 0 10 20 30 40 50 208Pb + 208Pb b = R P + R T (dσ  dσ NR ) / dσ NR [%] E lab (MeV/n) Figure 7: 208Pb+208 Pb collision at b = Rp +Rt. Di erential scattering cross section percent di erence vs. bombarding energy. It is clear from Figure 6 that the two methods' solutions for the di erential scattering cross section are quite far apart from one another when they are plotted against the scattering angle ( 4:5%). Once again, the dashed line represents Aguiar's method, and the solid line represents Matzdorf's method. It should be noted that each method becomes more like the classical scattering as the scattering angle is increased. This is can be accounted for by a close look at the classical di erential scattering cross section. The di erential scattering cross section, being proportional to the 40 probability that the particle is scattered to a di erential solid angle from a di erential cross sectional area, should decrease very rapidly as the scattering angle increases. More succinctly, you would expect fewer de ections to high angles, and more de ections at low angles. However, at higher energies, you would expect smaller de ection angles for the aggregate, regardless of the impact parameter, but it would still drop o rapidly. We would therefore expect the higher classical di erential scattering cross sections for higher scattering angles than that of the relativist methods, but both should approach zero as the de ection angle increases. This should manifest itself as a negative slope on the percent di erence vs. angle plot. As is evident by the data from Figure 6, the curve does indeed drop o as the scattering angle increases, which is exactly what we should expect. Similar to Figure 5, Figure 7 shows that both methods increasingly depart from the classical di erential cross section as the laboratory energy is raised. As before, Matzdorf's method does not deviate from the classical di erential as much as Aguiar's method for both Figure 6 and 7. For all the numeric solutions presented for symmetric collisions, both methods are in close proximity to one another, and agree well with the classical regime at low energies. Matzdorf's solution is far more close to the classical Rutherford solution in all of the gures presented. 41 6.2 Analytic Results As both methods are presented with two possible solutions: a more involved solution that requires a numerical method approach (the previous section), and a less involved analytic solution, it seems prudent to also compare these more time e cient methods to see if they hold muster. First we examine the di erential scattering cross section produced by each method for the same symmetric collision of 208Pb +208 Pb at a grazing angle b = Rp + Rt. This is a pure solution graphic, not a comparison to the classical di erential scattering cross section in the form of a percent as before. 42 0 50 100 150 200 250 103 104 105 106 208Pb + 208Pb b = R P + R T dσ / dΩ ( fm2 / sr ) E lab (MeV/n) Figure 8: 208Pb+208 Pb collision at b = Rp +Rt. Di erential scattering cross section vs. bombarding energy. Figure 8 shows that both methods agree very closely at low energies and separate only slightly at higher energies. It should be noted that the vertical axis, which displays the di erential scattering cross section, is logarithmic. This can make large di erences seem smaller, but the two methods are in relatively good agreement. The analytic formulas proposed by both papers are valid for light particles scattering o a heavy target. In Figure 9 the di erential scattering cross section percent di erence is plotted against the scattering angle for 43 the asymmetric collision of 17O +208 Pb. Due to the asymmetry, Aguiar's method is only valid up to order of v c 2 . 0 2 4 6 8 10 12 8.5 9.0 9.5 10.0 10.5 11.0 11.5 17O + 208Pb E lab = 100 MeV/n (dσ  dσ NR ) / dσ NR [%] Θ (Deg.) Figure 9: 17O+208Pb collision at 100MeV n . Di erential scattering cross section percent di erence vs. scattering angle. Figure 9 shows three di erent lines. The dashed line represents Aguiar's analytic solution (65), while the solid line represents Aguiar's exact, numeric solution up to the order of v c 2 . These are both in excellent agreement with one another. The dotted line represents Matzdorf's analytic equations (51). Not shown in this graphic is the Matzdorf numeric exact result, which would lie ontop of the dotted line, and would be 44 indistinguishable from Matzdorf's solutions at this resolution; it agrees within 0:1% of that of the exact result. It is clear from Figure 9 that the two methods are in much better agreement with one another for light on heavy particle collisions. Not only do the analytic methods presented by Aguiar and Matzdorf agree well with one another, they also agree quite well with the exact result of the numeric solutions, the consequences of which are discussed in Chapter 7. 6.3 Analysis of the Distance of Closest Approach We rst start by de ning half the distance of closest approach in a head on nonrelativistic collision, a0 , which is a useful scattering parameter characterized by: a0 = qpqt v2 (75) To begin a study of the distance of closest approach, it is convenient to rst employ the use of a helpful parameterization common to orbital motion problems: x =a0(cosh ! + ) y =a0 p 2 1 sinh ! z =0 (76) where ! is the angular frequency and is the eccentricity which is related to the scattering angle by = sin1( 2 ). Then if we wish to express this 45 parameterization in terms of the radial distance r = p x2 + y2 + z2, then we arrive at r = a0( cosh ! + 1). Then by solving v = dr dt = dr d! d! dt and rearranging for t, we attain the useful time parameterization: t = a0 v ( sinh ! + !) (77) Next we move this parameterization into the relativistic domain by replacing all instances of a0 with a where a = a0 . We can nally introduce the relativistic distance of closest approach bc associated with the impact parameter by: bc = a + p a2 + b2 (78) With this, we calculate the impact parameter by solving Matzdorf's equations (51) and the de nition of k(b) for the distance of closest approach, labeled bAnalyt and compare that to the relativistic distance of closest approach in a similar manner as the prior gures, in the form of a percent di erence. These are displayed in Figure 10. 46 50 100 150 200 250 0.01 0.1 1 0.01 0.1 1 ( b c  bAnalyt ) / b c [%] E lab (MeV/n) p + 208Pb 17O + 208Pb 40Ca + 208Pb 208Pb + 208Pb Figure 10: Multiple di erent collisions at b = Rp + Rt. Distance of closest approach percent di erence vs. bombarding energy. The solid line represents a p +208 Pb collision (p is a single proton), the dashed line represents a 17O +208 Pb collision, the dotted line represents a 40Ca +208 Pb collision, and the dasheddotted line represents the symmetric 208Pb +208 Pb collision; all of which were evaluated at a grazing impact parameter. The accuracy of these collisions is excellent even for the severely asymmetric case. Equation (78) therefore, produces a result very close to the \exact" result, all at or below the accuracy of 1% or less, further showing the validity of Matzdorf's methods. 47 Chapter 7 CONCLUSIONS In this paper, we have reviewed methods for solving for the relativistic motion of heavyonheavy and lightonheavy ion collisions. These collisions were evaluated over a spread of impact parameters, bombardment energies, and scattering angles; and from these the di erential scattering cross section and the relativistic distance of closest approach were calculated and plotted. Through a thorough investigation of the relativistic e ects manifested in the form of the magnetic interactions included in the Coulomb scattering, the retardation of the Coulomb electric potential, and the change in mass due to relativistic speeds, we have reached a a few important interpretations from our results. The solutions to the scattering angle and di erential scattering cross section presented by Matzdorf [5] as a solution to the covariant equations of motion are shown to be markedly better than the method of Lagrangian expansion by orders of v c outlined by Aguiar et al. [6]. The covariant equation solution accounts for retardation very well, while the Lagrangian expansion does not. The e ects of retardation being quite large in relativistic motion, this is something that needs to be considered in high energy elastic scattering. In addition to this, the inclusion of magnetic interaction terms proves to be unnecessary, as its contributions are negligible to the motion of relativistic ions. This could be 48 further reason for the superiority of the Matzdorf method, as the Lagrangian expansion includes the Darwin Lagrangian, which is heavily in uenced by magnetic terms. Further, the Matzdorf's analytic solutions were found to be in good enough agreement with the corresponding numeric solutions of equations (43) and (47). The di erential scattering cross section has been shown to be best described using equation (52). The scattering angle is solved most e ciently using equation (51). Lastly, the relativistic distance of closest approach, which is described in equation (78), in addition to the time dependence parameterization of the trajectory, given by equations (76), are shown to agree very well with one another. In summation, the solution of the covariant equation of a charged particle moving in the external electromagnetic eld of another charged particle method for accounting for the elastic scattering of heavy ions is more e ective than the method of Lagrangian expansion. In addition, the analytic formulation following the covariant equation solution method is su cient for all circumstances tested. Therefore, the more computationally involved numerical method is not necessary even for the severely asymmetric cases. 49 REFERENCES [1] Darwin C. G., Philos. Mag Ser. 6, 39, p. 537, 1920. [2] Jackson J. D., 1999, Classical Electrodynamics, 3rd edition (John Wiley & Sons, New York). [3] Ess en H., 2005, Eur. J. Phys., 26, p. 279. [4] Rutherford E., May 1911, The Scattering of and Particles by Matter and the Structure of the Atom, Philosophical Magazine. Series 6, 21. [5] Matzdorf R., So G.,Mehler G., 1987, Elastic Collisions of Heavy Ions at Intermediate Energies, Zeitschrift Physik D Atoms, Molecules and Clusters, 6, Number 1, pp. 512. [6] Aguiar C. E., Aleixo A. N. F., Bertulani C. A., November 1990, Elastic Coulomb Scattering of Heavy Ions at Intermediate Energies, American Physical Society Physics Review C, 42, iss. 5. [7] Sommerfeld A., 1978, Atombau und Spektrallinien. Frankfurt: Harri Deutsch. [8] Goldstein H., Poole C., Safko J., 2002, Classical Mechanics, 3rd edition (Addison Wesley), p. 110 50 VITA George Phillip Robinson attended Angelo State University in San Angelo, Texas where he received his Bachelors of Science in Physics in May of 2015. George further continued his education obtaining a Masters of Science in Physics from Texas A&M UniversityCommerce in Commerce, Texas in August of 2017. 
Date  2017 
Faculty Advisor  Bertulani, Carlos A 
Committee Members 
Li, BaoAn Newton, William 
University Affiliation  Texas A&M UniversityCommerce 
Department  MSPhysics 
Degree Awarded  M.S. 
Pages  58 
Type  Text 
Format  
Language  eng 
Rights  All rights reserved. 



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