A STUDY OF TOTAL REACTION CROSS SECTIONS FOR PROTON ON SOME NUCLEI AT INTERMEDIATE ENERGIES By Saleh Marzoq Barki Al-Lugmani A thesis submitted in partial fulfillment of the requirement for the Master's Degree in Science Department of Physics Faculty of Science King Abdulaziz University Jeddah, Saudi Arabia September 2007
مستخلص هذذال لل يذذ راذذذنظ يةللذذت الرذذت لنيايذذذك لل لتذذل للصذذ لل نذذل لنخ ذذ لخ ذذذ يظ 008 م غذذ تل خذذل ت ع لذذج نذذ بل ح ذذ ث صذذن 08 يذذ ث بذذ رذذت لل لبذذ ت 12 C. تت للنةللذذت م تذذذ ت 208 Pb 90 Zr لل لسذذ ظ 40 Ca للزةك ذذ ظ لللصذذ تي ت ة ي ذس كن بل للي غظ بضهن ك ل ظ. للب ث للخضلرب ت لألخ لة مغننت م ع ت م ذ متث لذ ت للذخي ةة كنذ ت كنذ ت للخل حسخخنظ كز عت ص مب ل- ب ح للشبه ظ هلرت لل ز عت للا هلرت كيذ ت للخغ ذل للغ صذ عذل يت للغلكت لنبل ح ت للس يط بسبب مض ل ك ل ظ لنص لة للهنف حم خاه عل للغس ب رض. لل نةل لذ ت لل غ ل ذ ت ح ذل ن ت يذ ذ س كنذ بل ر يذ ص ذ نل تلذ عذ ن م اذ ل عذل مذ ن يذ ت للبل حذ ت للسذ يط عذل مذن للي يذت 08 ذ 008 م غذ تل خذل ت ع لذج. ي مذ ح يذل لل ز عذت للا هلرت خ ئش عض م كز عت صذ مب ل- ب حذ للشذبه ظ هلرذت. ب إلتذ عت صذن رضذ ت حغ ذل يت ك ل ظ له حأر ل منغ ظ ن لليايك لل لتل لليغس ب. 6
Abstract This work presents a theoretical study of the total nuclear reaction cross-section for protons at energies between 80 and 180 MeV on 12 C, 40 Ca, 90 Zr and 208 Pb nuclei. The study is made within the framework of Coulomb modified Glauber model. Recent experimental proton-nucleus total reaction cross section data are analyzed in terms of NN scattering amplitude parameters using Gambhir Patil (GP) semi-phenomenological and phenomenological densities. The downward shift in the incident proton kinetic energy due to the target Coulomb field is also taken into account. The present study shows that the Glauber model works reasonably well in the incident proton energy range 80 180 MeV and that, in general, the phenomenological density gives better results than recently proposed Gambhir-Patil semi-phenomenological density. Furthermore, we find that the Coulomb energy shift has noticeable effect on the calculated cross section, specially at lower energies. 5
CONTENTS Page ABSTRACT (ENGLISH)... ABSTRACT (ARABIC). ACKNOWLEDGEMENT CONTENTS.. LIST OF TABLES... LIST OF FIGURES... i ii iii iv vi vii Chapter 1: INTRODUCTION.... 1 1.1 Nuclear processes 2 1.2 Nuclear cross-section.. 2 1.3 Aim of the present research work... 4 1.4 Thesis organization. 5 Chapter 2: COULOMB MODIFIED GLAUBER MODEL... 6 2.1 Potential scattering.. 6 2.2 High energy approximation for potential scattering... 8 2.3 Cross sections in the high energy approximation... 12 2.4 Glauber multiple scattering theory.. 14 2.4.1 N-nucleus scattering 14 2.5 Coulomb modified Glauber model (CMGM). 21 Chapter 3: NUCLEAR DENSITY DISTRIBUTIONS.. 25 3.1 Some commonly used density distributions. 25 3.2 Semi phenomenological nucleon density distribution... Chapter 4: STUDY OF PROTON TOTAL REACTION CROSS-SECTION...... 30 4.1 Density parameters... 30 4.2 NN scattering amplitude. 32 26 0
4.3 Calculation of ζ R. 33 4.4 Results and discussion.. 35 4.5 Summary and conclusions... 42 REFERENCES... 45 APPENDIX A 47 APPENDIX B 50 APPENDIX C 51 9
LIST OF TABLES Table 4.1 The parameter values of Gambhir-Patil density... 30 Page Table 4.2 p - 12 C total reaction cross-section.... 35 40 Table 4.3 p - Ca total reaction cross-section... 35 90 Table 4.4 p - Zr total reaction cross-section... 36 Table 4.5 p - 208 Pb total reaction cross-section... 36 08
LIST OF FIGURES Fig. 2.1 Eikonal approximation.... 9 Page Fig. 2.2 N-nucleus scattering.... 15 Fig. 2.3 Schematic diagram of projectile trajectory.. 21 Fig. 4.1 p - 12 C total reaction cross-section with the Coulomb energy shift. Stars: predictions of G-P density; Open circles: predictions of the folded Yukawa density; solid circles: show experimental data of Auce et al.. 37 Fig. 4.2 Same as figure 4.1 but for p - Fig. 4.3 Same as figure 4.1 but for p - 40 Ca 90 Zr...... 37... 38 Fig. 4.4 Same as figure 4.1 but for p - Fig. 4.5 p - 208 Pb... 38 12 C total reaction cross-section. Stars: predictions of folded Yukawa density without Coulomb energy shift effect; open circles: with Coulomb energy shift effect; solid circles: show experimental data of Auce et al. 40 Fig. 4.6 Same as figure 4.5 but for p - 40 Ca... 40 Fig. 4.7 Same as figure 4.5 but for p - 90 Zr... 41 Fig. 4.8 Same as figure 4.5 but for p - 208 Pb... 41 00
Chapter 1 INTRODUCTION Nuclear reactions are used to study the properties of nuclei. Reactions that exchange energy or nucleons can be used to measure the energies of binding and excitation, quantum numbers of energy levels, and transition rates between levels. A particle accelerator, which produces a beam of high velocity charged particles (electrons, protons, alphas, or "heavy ions"), works as the source of these reactions when the accelerated particles strike a target nucleus. Nuclear reactions can also be produced in nature by high velocity cosmic ray particles, for instance in the upper atmosphere or in space. Beams of neutrons can be obtained from nuclear reactors or as secondary products when a charged particle beam knocks out weakly bound neutrons from a target nucleus. Beams of photons, mesons, muons, and neutrinos can also produce nuclear reactions. In order for nuclear reaction to occur, the nucleons in the incident particle, or projectile, must interact with the nucleons in the target. Thus the energy must be high enough to overcome the natural electromagnetic repulsion between the protons. This energy "barrier" is called the Coulomb's barrier. If the energy is below the barrier, the nuclei will bounce off each other. Early experiments by Rutherford used low-energy alpha particles from naturally radioactive material to bounce off target atoms and measure the size of the target nuclei. 02
1.1 Nuclear processes When a collimated beam of mono-energetic light particles such as protons or neutrons is shot at a target containing atoms, nuclei, etc., the projectile particles may approach so close to the target so that the two interact. And as a result of this interaction any of the following nuclear processes may occur singly or jointly: (i) The incident particle may simply be deviated from its rectilinear path so that the incoming and outgoing particles are the same. This process is known as scattering. The scattering processes may be of two types: if the incoming and outgoing particles have the same kinetic energy, the process is called elastic scattering and if the outgoing particle has lesser kinetic energy than the incident one, the process is called inelastic scattering. (ii) The incident particle be completely absorbed by the target without the emission of any particle. This process is called radiative capture. (iii) The incident and the emergent particles may be different from each other. This process is known as a nuclear reaction. However nuclear reactions themselves include a variety of special phenomena for example fission, fussion, spallations, etc. 1.2 Nuclear cross section We can determine the probability of occurrence of the nuclear process. The most naive means of expressing this probability is offered by one of the most important concepts of nuclear physics the namely cross-section. It may be visualised as the effective circular area, a target nucleus presents to the incident particles for undergoing a particular nuclear process. 03
The concept of cross-section for a nuclear process is quite different from the geometrical cross-sectional area (πr 2 ) of the nucleus. Instead of the treating the geometrical cross-sectional area as a measure of interaction probability, each target nucleus is ascribed an effective circular area ζ. The probability of a nuclear process is then just equal to the probability that the incident particle strikes within this effective area. The process becomes a certainty if the incident particle happens to strike the target within this cross-sectional area. The magnitude of the cross-section ζ, is different for different nuclear processes and for a particular nuclear process is a function of the energy of impinging particle. It has the dimensions of area and is expressed in units of barn: 1 barn = 10-28 m 2. There are a great many kinds of cross-sections which play a vital role in nuclear physics studies. Among them some are the following: (i) Total nuclear-interaction cross-section (ζ tot ): It is the effective area that a target nucleus possesses for removing particles from a collimated incident beam by all possible processes (scattering, absorption and /or nuclear reaction) involving an interaction between the target nucleus and the incident particles. The magnitude of total cross section thus defined is a function of the kinetic energy of the incident particle and it also depends upon the kind of incident particle (such as, a proton, an α-particle, etc.). (ii) Partial cross-section: The total cross-section as defined above shall be composed of the sum of several partial cross sections which represent the contributions of various distinct, independent processes by which particles can be removed from the incident beam. It is a standard practice to make distinction between (a) the scattering processes in which the outgoing particle is the same as the incident particle and (b) absorption or 04
nuclear reaction, which may be only radiative capture or may lead to the various types of nuclear reactions and in which the outgoing particle if any is of a character different from that of the incident particle. Therefore, the total cross-section may be written as (Pandya and Yadav, 1994) ζ tot = ζ sc + ζ R (1.1) where ζ sc is the scattering cross-section and ζ R is the reaction crosssection. Furthermore, scattering may be elastic or inelastic and absorption (reaction) may lead to the various nuclear reactions. Therefore, in general, total cross-section may be written as (Pandya and Yadav, 1994) ζ tot = Σ ζ partial (1.2) where the summation extends over each of the partial cross section, ζ partial referring to the individual simultaneous nuclear process, scattering, absorption, etc. 1.3 Aim of the present research work We present a study of total nuclear reaction cross-section for protons with energies between 80 and 180 MeV on 12 C, 40 Ca, 90 Zr and 208 Pb nuclei. The study has been made within the theoretical framework of the optical limit approximation to the Coulomb modified Glauber model using semi-phenomenological proton and neutron density 05
distributions as proposed by Gambhir and Patil (1986) and the measured densities as parameterized by Broglia and Winther (1981). The main aim of this work is to undertake a comparative study of the predictions of Gambhir-Patil semi-phenomenological and phenomenological densities with regard to the proton total crosssection and to demonstrate importance of the Coulomb energy shift on the calculated cross-section. 1.4 Thesis organization After the current introductory chapter, we give in chapter 2, a brief review of the potential scattering theory in the high energy approximation, the Glauber multiple scattering theory for N-nucleus scattering (for a review see also Joachain, 1975) and its modification to account for the deviation of the projectile trajectory in the Coulomb field of the target nucleus. In chapter 3, we present a brief description of the most useful models for nuclear density and their important characteristics. The results of our calculation for proton reaction cross section are presented and discussed in chapter 4. In this chapter, we also give the summary and conclusions of the present study. 06