University of Texas at El Paso DigitalCommons@UTEP Open Access Theses & Dissertations 2011-01-01 Equation Of State Of A Dense And Magnetized Fermion System Israel Portillo Vazquez University of Texas at El Paso, [email protected] Follow this and additional works at: https://digitalcommons.utep.edu/open_etd Part of the Physics Commons Recommended Citation Portillo Vazquez, Israel, "Equation Of State Of A Dense And Magnetized Fermion System" (2011). Open Access Theses & Dissertations. 2365. https://digitalcommons.utep.edu/open_etd/2365 This is brought to you for free and open access by DigitalCommons@UTEP. It has been accepted for inclusion in Open Access Theses & Dissertations by an authorized administrator of DigitalCommons@UTEP. For more information, please contact [email protected]. EQUATION OF STATE OF A DENSE AND MAGNETIZED FERMION SYSTEM ISRAEL PORTILLO VAZQUEZ Department of Physics APPROVED: Efrain J. Ferrer , Ph.D., Chair Vivian Incera, Ph.D. Mohamed A. Khamsi, Ph.D. Benjamin C. Flores, Ph.D. Acting Dean of the Graduate School Copyright © by Israel Portillo Vazquez 2011 EQUATION OF STATE OF A DENSE AND MAGNETIZED FERMION SYSTEM by ISRAEL PORTILLO VAZQUEZ, MS THESIS Presented to the Faculty of the Graduate School of The University of Texas at El Paso in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE Department of Physics THE UNIVERSITY OF TEXAS AT EL PASO August 2011 Acknowledgements First, I would like to acknowledge the advice and guidance of my mentor Dr. Efrain Ferrer. During the time I have been working with him, the way in which I think about nature has changed considerably. He introduced me to the wonderful world of theoretical physics by making me part of his research group. I am grateful to him for his great teaching, both inside and outside the classroom. I thank him for this encouragement to continue studying physics, and for his financial assistance. He is my scientific role model. I would also like to thank Dr. Jorge Lopez for giving me the opportunity to come to University of Texas at El Paso (UTEP). I acknowledge my professors Dr. Ramon Ravelo and Dr. Murat Durandurdu. Every one of them showed me different aspects of physics, which were of great value for my development as a scientist. I would like to thank Dr. Vivian Incera, Chairman and Professor of the Physics Department. Her character, motivation, and great scientific spirit were of great influence in the decisions I have made. I am indeed grateful to the members of my family. Even though they do not know what I was doing at UTEP, my family in Santa Barbara (Israel, Estela, Isabel, and Aurelia) trusts it should be something good. My family in El Paso (Roberto, Rosalia, Yanci, and Masai) showed me that there are no barriers when we have the enough courage to face every obstacle in life. They make me a member of the family and influenced me greatly during my work. I am indebted to my many of my classmates, secretaries, professors, relatives, and friends, who played important roles in the developed of my life during the last two years. I also thanks to UTEP, the Department of Physics for giving me this opportunity to develop my research. I thank the members of my committee for taking the time for reviewing and working on my thesis: Dr. Efrain J. Ferrer, Dr. Vivian Incera, and Mohamed A. Khamsi. Israel Portillo Vazquez University of Texas at El Paso July 2011 iv Table of Contents Acknowledgements ................................................................................................. iv Table of Contents ..................................................................................................... v List of Tables ........................................................................................................ vii List of Figures ...................................................................................................... viii Chapter 1: Introduction ............................................................................................ 1 1.1 Antecedents ............................................................................................ 4 1.2 Objectives of the Thesis ......................................................................... 6 1.3 Outline ................................................................................................... 7 Chapter 2: Astrophysical Background ..................................................................... 9 2.1 Nucleosynthesis ..................................................................................... 9 2.2 Stellar Evolution .................................................................................. 14 2.3 Neutron Stars ....................................................................................... 26 Chapter 3: Theoretical Background ....................................................................... 39 3.1 Notation ............................................................................................... 39 3.2 Special Theory of Relativity & General Theory of Relativity ............ 42 3.3 Lagrangian and Hamiltonian Formalism ............................................. 48 3.4 Electromagnetic Field .......................................................................... 51 3.5 Conservation Laws and the Energy-Momentum Tensor ..................... 53 3.7 Thermodynamic Approach .................................................................. 55 3.8 Motion of a Charged Particle in a Magnetic Field .............................. 56 Chapter 4: Compact Star’s Magnetic Field Estimates ........................................... 61 4.1 Self-Bound Compact Stars .................................................................. 62 2.2 Gravitationally-Bound Compact Stars ................................................. 63 Chapter 5: Equations of State: Functional Method Approach ............................... 67 5.1 Energy-Momentum Tensor .................................................................. 67 5.2 Energy and Pressures ........................................................................... 72 5.3 Energy-Momentum Tensor Covariant Structure ................................. 77 5.4 Thermodynamical Potential of a Magnetized Fermion System .......... 78 v Chapter 6: Equations of State: Numerical Results ................................................ 83 Chapter 7: Concluding ........................................................................................... 89 References .............................................................................................................. 91 Vita..… .................................................................................................................. 97 vi List of Tables Table 2.1: Evolutionary stage of a 25 M⊙ star. ......................................................................................... 24 Table 2.2: Summary of properties of the known quarks. .......................................................................... 35 vii List of Figures Figure 2.1: Energy-density of the Universe. .............................................................................................. 11 Figure 2.2: Nuclear Binding Energy. ......................................................................................................... 13 Figure 2.3: Proton-proton chain mechanism. ............................................................................................ 14 Figure 2.4: Motion of the atoms in the interstellar cloud. (a) The atoms are attracted by gravity, (b) the collision of the atoms take them away. ...................................................................................................... 16 Figure 2.5: Hydrostatic equilibrium. ......................................................................................................... 17 Figure 2.6: Star’s layers of the super red giant. ......................................................................................... 18 Figure 2.7: Star’s Evolution from protostar to white dwarfs. .................................................................... 21 Figure 2.8: Concentric-fusing-layer of a star. ............................................................................................ 22 Figure 2.9: Structure of a Neutron Star. .................................................................................................... 32 Figure 2.10: Structure of a neutron and a proton. ...................................................................................... 34 Figure 2.11: Proposed phase diagram for QCD. ........................................................................................ 36 Figure 3.1: Moving charge in a uniform and constant magnetic field. (a) particle without acceleration in the x3 directions, and (b) particle with acceleration in the x3 directions. ................................................... 58 Figure 3.2: Landau levels. ......................................................................................................................... 59 Figure 4.1: (a) The mass density coefficient a vs. the density parameter n. (b) The core magnetic field H0 vs. the field coefficient b. .......................................................................................................................... 64 Figure 5.1: Symmetry braking by the magnetic field B: (a) Symmetry O(3), (b) Symmetry O(2). .......... 69 Figure 6.1: System energy density vs. magnetic field for μ=400 MeV. (a) System