Introduction to Particle Physics

Introduction to Plasma Physics: A graduate level course Richard Fitzpatrick 1 Associate Professor of Physics The University of Texas at Austin 1 In association with R.D. Hazeltine and F.L. Waelbroeck. Contents 1 Introduction 5 1.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 What is plasma? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 A brief history of plasma physics . . . . . . . . . . . . . . . . . . . 7 1.4 Basic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 The plasma frequency . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Debye shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.7 The plasma parameter . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.8 Collisionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.9 Magnetized plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.10 Plasma beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Charged particle motion 20 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Motion in uniform fields . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Method of averaging . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Guiding centre motion . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5 Magnetic drifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.6 Invariance of the magnetic moment . . . . . . . . . . . . . . . . . 31 2.7 Poincar ´ e invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.8 Adiabatic invariants . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.9 Magnetic mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.10 The Van Allen radiation belts . . . . . . . . . . . . . . . . . . . . . 37 2.11 The ring current . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.12 The second adiabatic invariant . . . . . . . . . . . . . . . . . . . . 46 2.13 The third adiabatic invariant . . . . . . . . . . . . . . . . . . . . . 48 2.14 Motion in oscillating fields . . . . . . . . . . . . . . . . . . . . . . . 49 3 Plasma fluid theory 53 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Moments of the distribution function . . . . . . . . . . . . . . . . . 56 3.3 Moments of the collision operator . . . . . . . . . . . . . . . . . . . 58 3.4 Moments of the kinetic equation . . . . . . . . . . . . . . . . . . . 61 2 3.5 Fluid equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.6 Entropy production . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.7 Fluid closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.8 The Braginskii equations . . . . . . . . . . . . . . . . . . . . . . . . 72 3.9 Normalization of the Braginskii equations . . . . . . . . . . . . . . 85 3.10 The cold-plasma equations . . . . . . . . . . . . . . . . . . . . . . 93 3.11 The MHD equations . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.12 The drift equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.13 Closure in collisionless magnetized plasmas . . . . . . . . . . . . . 100 4 Waves in cold plasmas 105 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.2 Plane waves in a homogeneous plasma . . . . . . . . . . . . . . . . 105 4.3 The cold-plasma dielectric permittivity . . . . . . . . . . . . . . . . 107 4.4 The cold-plasma dispersion relation . . . . . . . . . . . . . . . . . 110 4.5 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.6 Cutoff and resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.7 Waves in an unmagnetized plasma . . . . . . . . . . . . . . . . . . 114 4.8 Low-frequency wave propagation in a magnetized plasma . . . . Introduction to Particle Physics Introduction to Particle Physics Bởi: OpenStaxCollege Part of the Large Hadron Collider at CERN, on the border of Switzerland and France The LHC is a particle accelerator, designed to study fundamental particles (credit: Image Editor, Flickr) Following ideas remarkably similar to those of the ancient Greeks, we continue to look for smaller and smaller structures in nature, hoping ultimately to find and understand the most fundamental building blocks that exist Atomic physics deals with the smallest units of elements and compounds In its study, we have found a relatively small number of atoms with systematic properties that explained a tremendous range of phenomena Nuclear physics is concerned with the nuclei of atoms and their substructures Here, a smaller number of components—the proton and neutron—make up all nuclei Exploring the systematic behavior of their interactions has revealed even more about matter, 1/2 Introduction to Particle Physics forces, and energy Particle physics deals with the substructures of atoms and nuclei and is particularly aimed at finding those truly fundamental particles that have no further substructure Just as in atomic and nuclear physics, we have found a complex array of particles and properties with systematic characteristics analogous to the periodic table and the chart of nuclides An underlying structure is apparent, and there is some reason to think that we are finding particles that have no substructure Of course, we have been in similar situations before For example, atoms were once thought to be the ultimate substructure Perhaps we will find deeper and deeper structures and never come to an ultimate substructure We may never really know, as indicated in [link] The properties of matter are based on substructures called molecules and atoms Atoms have the substructure of a nucleus with orbiting electrons, the interactions of which explain atomic properties Protons and neutrons, the interactions of which explain the stability and abundance of elements, form the substructure of nuclei Protons and neutrons are not fundamental—they are composed of quarks Like electrons and a few other particles, quarks may be the fundamental building blocks of all there is, lacking any further substructure But the story is not complete, because quarks and electrons may have substructure smaller than details that are presently observable This chapter covers the basics of particle physics as we know it today An amazing convergence of topics is evolving in particle physics We find that some particles are intimately related to forces, and that nature on the smallest scale may have its greatest influence on the large-scale character of the universe It is an adventure exceeding the best science fiction because it is not only fantastic, it is real Summary • Particle physics is the study of and the quest for those truly fundamental particles having no substructure 2/2 An Introduction to GEOMETRICAL PHYSICS R. Aldrovandi & J.G. Pereira Instituto de F´ısica Te´orica State Univers ity of S˜ao Paulo – UNESP S˜ao Paulo — Brazil To our parents Nice, Dina, Jos´e and Tito i ii PREAMBLE: SPACE AND GEOMETRY What stuff’tis made of, whereof it is born, I am to learn. Merchant of Venice The simplest geometrical setting used — consciously or not — by physi- cists in their everyday work is the 3-dimensional euclidean space E 3 . It con- sists of the set R 3 of ordered triples of real numbers such as p = (p 1 , p 2 , p 3 ), q = (q 1 , q 2 , q 3 ), etc, and is endowed with a very special characteristic, a metric defined by the distance function d(p, q) =  3  i=1 (p i − q i ) 2  1/2 . It is the space of ordinary human experience and the starting point of our geometric intuition. Studied for two-and-a-half millenia, it has been the object of celebrated controversies, the most famous concerning the minimum number of properties necessary to define it completely. From Aristotle to Newton, through Galileo and Descartes, the very word space has been reserved to E 3 . Only in the 19-th century has it become clear that other, different spaces could be thought of, and mathematicians have since greatly amused themselves by inventing all kinds of them. For physi- cists, the age-long debate shifted to another question: how can we recognize, amongst such innumerable possible spaces, that real space chosen by Nature as the stage-set of its processes? For example, suppose the space of our ev- eryday experience consists of the same set R 3 of triples above, but with a different distance function, such as d(p, q) = 3  i=1 |p i − q i |. This would define a different metric space, in principle as good as that given above. Were it only a matter of principle, it would be as good as iii iv any other space given by any distance function with R 3 as set point. It so happens, however, that Nature has chosen the former and not the latter space for us to live in. To know which one is the real space is not a simple question of principle — something else is needed. What else? The answer may seem rather trivial in the case of our home space, though less so in other spaces singled out by Nature in the many different situations which are objects of physical study. It was given by Riemann in his famous Inaugural Address 1 : “ those properties which distinguish Space from other con- ceivable triply extended quantities can only be deduced from expe- rience.” Thus, from experience! It is experiment which tells us in which space we actually live in. When we measure distances we find them to be independent of the direction of the straight lines joining the points. And this isotropy property rules out the second proposed distance function, while admitting the metric of the euclidean space. In reality, Riemann’s statement implies an epistemological limitation: it will never be possible to ascertain exactly which space is the real one. Other isotropic distance functions are, in principle, admissible and more experi- ments are necessary to decide between them. In Riemann’s time already other geometries were known (those found by Lobachevsky and Boliyai) that could be as similar to the euclidean geometry as we might wish in the re- stricted regions experience is confined to. In honesty, all we can say is that E 3 , as a model for our ambient space, is strongly favored by present day experimental evidence in scales ranging from (say) human dimensions down to about 10 −15 cm. Our knowledge on smaller scales is limited by our ca- pacity to probe them. For larger scales, according to General Relativity, the validity of this model depends on the presence and strength of gravitational fields: E 3 is good only as long as gravitational fields are very weak. “ These data are — like all data — not logically necessary, but only of empirical [...]... programming language Programming languages can be divided into two major categories: low-level languages designed to work with the given hardware, and high-level languages that are not related to any specific hardware Simple machine languages and assembly languages were the only ones available before the development of high-level languages A machine language is typically in binary form and is designed to work... computing and global computing is elucidated in Koniges (2000), Foster and Kesselman (2003), and Abbas (2004) 1.1 Computation and science Modern societies are not the only ones to rely on computation Ancient societies also had to deal with quantifying their knowledge and events It is interesting to see how the ancient societies developed their knowledge of numbers and calculations with different means and tools... of programming and debugging They are more advanced than machine languages because they have adopted symbolic addresses But they are still related to a certain architecture and wiring of the system A translating device called an assembler is needed to convert an assembly code into a native machine code before a computer can recognize the instructions Machine languages and assembly languages do not... discussion on the Fortran language and its applications, see Edgar (1992) The newest version of Fortran, known as Fortran 90, has absorbed many important features for parallel computing Fortran 90 has many extensions over the standard Fortran 77 Most of these extensions are established based on the extensions already adopted by computer manufacturers to enhance their computer performance Efficient compilers... programming languages that are used in scientific computing The longest-running candidate is Fortran (Formula translation), which was introduced in 1957 as one of the earliest high-level languages and is still one of the primary languages in computational science Of course, the Fortran language has evolved from its very early version, known as Fortran 66, to Fortran 77, which has been the most popular language... replace human beings in this regard and the quest for a better understanding of Nature will go on no matter how difficult the journey is Computers will certainly help to make that journey more colorful and pleasant 1.3 Computer algorithms and languages Before we can use a computer to solve a specific problem, we must instruct the computer to follow certain procedures and to carry out the desired computational. .. protocols and environments under various software packages, which we will leave to the readers to discover and explore 1.3 Computer algorithms and languages The other popular programming language for scientific computing is the C programming language Most system programmers and software developers prefer to use C in developing system and application software because of its high flexibility (Kernighan and... (Kernighan and Pike, 1984) now used on almost all workstations and supercomputers was initially written in C In the last 50 years of computer history, many programming languages have appeared and then disappeared for one reason or another Several languages have made significant impact on how computing tasks are achieved today Examples include Cobol, Algol, Pascal, and Ada Another object-oriented language... speed of the motorcycle is 67 m/s, the air density is ρ = 1.2 kg/m3 , the combined mass of the motorcycle and the person is 250 kg, and the coefficient c is 1, find the tilting angle of the taking-off ramp that can produce the longest range 1.9 One way to calculate π is by randomly throwing a dart into the unit square defined by x ∈ [0, 1] and y ∈ [0, 1] in the x y plane The chance of the dart landing inside... efficient approach to learning computational physics is .. .Introduction to Particle Physics forces, and energy Particle physics deals with the substructures of atoms and nuclei and is particularly aimed at finding those truly fundamental particles... covers the basics of particle physics as we know it today An amazing convergence of topics is evolving in particle physics We find that some particles are intimately related to forces, and that... there is some reason to think that we are finding particles that have no substructure Of course, we have been in similar situations before For example, atoms were once thought to be the ultimate