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PHYS 490/891 – Winter 2010 Nuclear and Particle Physics Lecture Notes Wolfgang Rau Queen’s University 1 PHYS 490/891 – Winter 2009 L01 1. Introduction 1.1 From Elements to Particles The question of what the world around us consists of is as old as the history of the human mind. Many old cultures classified the material world in a small number of elements. The ancient Greeks e.g. considered Earth, Air, Fire and Water as the basic elements. Sometimes additional non-material elements were considered like Idea, Aether (Ether) or Light. Air wet hot Figure 1: Four Elements in ancient Greece Water Fire cold dry Earth With the advancement of science it was realised that there are quite a few more distinguishable elements. Nowadays the term elements usually refers to the roughly 100 chemical elements. Group → 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Period ↓ 1 2 3 4 5 6 7 Lanthanides Actinides Figure 2: Table of chemical elements (http://en.wikipedia.org/wiki/Chemical_element) What was earlier considered the four elements could now be viewed as the states of matter (Earth → solid; Water → liquid; Air → gas; and perhaps Fire → plasma). Even though suspected since a long time we learned only at the beginning of the 20th century for sure that matter is not continuous but rather composed of smallest quanta, the atoms. Atoms were considered as small (~ 10 m) solid balls of the respective substance. Figure 3: Matter composed of atoms in different states solid liquid gas 2 PHYS 490/891 – Winter 2009 L01 The history of the atom as atomos (from Greek: indivisible) did not last very long: almost at the same time when the existence of atoms was confirmed the investigation of cathode rays and radioactivity showed that atoms could emit constituent (sub-atomic) particles. Based on the work of Thomson, Rutherford, Bohr and others a model of the atom was developed which involved a tiny (~10 m) positively charged nucleus and an atomic shell made out of electrons. Figure 4: Bohr atom with nucleus and orbiting electrons (not to scale) This opened the door to study the atom as such, creating the field of atomic physics. But at the same time it became clear that radioactivity involved changes in the atomic nuclei. So the nuclei themselves are composed objects. The lighter elements have mostly masses which are roughly integer multiples of the mass of the hydrogen atom and so might just be composed of such. But this is not true for all elements. It was realised that there were different types of atoms for certain elements with the same chemical properties but different masses (called isotopes). The explanation of isotopism followed in the 1930s when Chadwick discovered the neutron. With this, the material components of the matter surrounding us were identified: all matter is made of atoms; atoms consist of nuclei and electrons while nuclei are composed of protons (the nuclei of hydrogen atoms) and neutrons. Protons and neutrons are called nucleons. Still missing was the answer to the question what it was that held the nuclei together, overcoming the strong repulsive Coulomb force which pushes the protons apart. Also in another aspect this was not the end of the story: in the attempt to better understand radioactivity – which had led to the conclusion that nuclei are not fundamental but composed objects – the cosmic radiation was discovered which in turn held new surprises in the form of additional particles which were not part of the usual matter. So beyond atomic physics there was now nuclear physics, trying to understand the rich phenomenology of atomic nuclei, and the new field of particle physics. These three branches of physics are closely connected. The two main tools used in all three branches to investigate the respective objects are spectroscopy – investigating the radiation emitted when a system transitions from an excited state to the ground state (or to another excited state) – and the scattering of probes (high energetic particles) off the object of interest, which is governed by the de Broglie relation between momentum and wavelength: the smaller the object of interest 3 PHYS 490/891 – Winter 2009 L01 is the shorter a wavelength and consequently higher momentum of the probe is required. Rutherford’s famous experiment which showed the existence of nuclei was of the latter type while one of the basic inputs for Bohr’s model of the atom was the Balmer formula which is based on spectroscopy. Gamma radiation from radioactive nuclei is the exact equivalent to the emission lines from atoms, telling us about excited states in nuclei, while Rutherford-like experiments were used to investigate the structure of nuclei showing that protons and neutrons are themselves again composed of smaller components. Scattering experiments beyond Rutherford became possible with the advancement of the technology of accelerating particles which then served as probes with sufficient energy to study the small structures. The same technology is the basis of particle physics where by means of Einstein’s relation between energy and mass new particles with high masses can be produced if enough energy is available from the colliding primary particles. A large variety of new particles has been found in this type of experiments. Many of these new particles are compound particles (similar to the nucleons) and again from the decay of excited states we can learn about their components and interactions. Today we know of two different types of particles: the quarks, which are the constituents of the nucleons and other compound particles, but are never observed as free particles, and the leptons, to which e.g. the electron and the neutrino belong. There are six of each type (plus their anti-particles) and they can be grouped into three generations of families with recurring properties, seen as indication by some that we are looking at composed instead of elementary objects once more. The subject of this course is Nuclear and Particle Physics, but we may from time to time refer back to Atomic Physics and use analogies to facilitate our understanding of the new phenomena. 1.2 Scales and Units All three branches of physics mentioned in the last section have their own typical length and energy scales. Lengths are measured in units derived from the meter, while the usual unit for energies is the electron-volt ( V, the energy a particle with the charge of an electron gains in a potential of 1 V. 1 V 1.602 10 J The relevant length scales for Atomic Physics is the size of the atom which is in the order of 10 m 1 Angstrom (or Ångström, ). The typical energies for atomic processes are from a few V up to several tens of k V. The length scale for Nuclear Physics is the size of an atomic nucleus (order of 10 m ). A very important quantity is the cross section which is related to the probability for a given interaction to occur. To 4 PHYS 490/891 – Winter 2009 L01 first order this can be visualized as the geometrical cross section of the involved interaction partners. The typical unit in use is the barn (b): 1b 10 m . However the cross section depends very strongly on the type of interaction and the energy and can range from less than 10 b (fb) to thousands of barns. The energies for nuclear processes are in the order of M V (10 V; typical binding energy for a nucleon in a nucleus is 7 8 M V). The length scale for Particle Physics is at 10 m (1 femto meter, fm also called fermi) and below. The size of a proton is roughly 1 fm. The size of elementary particles (like quarks or the electron) is not known. We only know that it is less than about 10 m without indication that they have any finite size at all. They might as well be point-like. To produce new particles the necessary energy is given by Einstein’s formula ⁄ Consequently the masses of particles are usually measured in V/ . Masses of known elementary particles range from below 1 V/ for neutrinos up to ~170 G V/c for the top quark. Typical energies are in the G V (10 V) range; the highest energies produced at particle accelerators to date are about 1 T V (10 V) for protons. For the description of scattering processes the momentum is needed which is usually measured in V/c. In many cases the interactions in particle physics involve highly relativistic particles for which the relationship between energy and momentum is the same as for photons: . The momentum is also relevant to determine the wavelength of a given particle. According to de Broglie the wavelength is given by ⁄ with the Planck constant . An easy-to-remember form of this constant is as reduced Planck constant multiplied by the speed of light: ⁄2 197.3 MeV fm 200 MeV fm. To resolve structures inside the atomic nucleus (~ 10 fm) one needs particles with a wavelength in the same range which corresponds to a momentum of ~20 M V/ or an energy of ~20 M V if electrons are used, which are highly relativistic in this energy range (note that this is very close to the typical nuclear binding energies of 7 8 M V mentioned above). For the typical size of an atom (10 m 10 fm) we find a corresponding energy of the order of k V which is in the above discussed range of typical atomic energies. For theoretical calculations particle physicists often use the so called natural units where 1. In those units masses and momenta are measured in V (or M V, G V) instead of V/c or V/c. However it is easy to get confused in this system so we will mostly account for factors and (or ) explicitly. 5 PHYS 490/891 – Winter 2009 L02 1.3 Relativistic Theories and the four Forces Schrödinger’s Equation is derived from the non-relativistic relation between energy and momentum: (1.1) where is the potential energy (but technically should also include the rest energy of the respective particle). The second ingredient is the observation that for a plain wave we can determine energy and momentum by derivation with respect to time and space. Using complex plain waves exp we find: and (1.2) So we can write the above equation (1.1) in terms of operators: with (1.3) where we call the Hamilton Operator or Hamiltonian. We now switch to the correct relativistic energy momentum relation (we consider here only the case without potential energy) m c (1.4) If we want to write this in terms of operators in the same general form as in (1.3) we find a Hamiltonian of the form · ̂ (1.5) with the momentum operator ̂ . When Dirac first wrote this equation down for relativistic electrons he found that and cannot be just numbers but in the simplest are case 4×4 matrices. Accordingly, the solution is a vector of wave functions with four components: x, t x, t x, t (1.6) x, t x, t Two of the wave functions correspond to the positive energy solution of equation (1.4) and represent the two spin states ( 1⁄2). The other two wave functions correspond to the negative energy solution. Investigating these solutions for an electron it turns out these states behave exactly like a particle with positive charge, but otherwise the same properties as the electron. This led to the prediction that for each particle there should be an antiparticle with the opposite charge. This was verified for the case of the electron by the discovery of the positron (the anti-electron) in 1933 and for all other particles as well since then. In general, antiparticles are symbolized by a bar (e.g. the proton is symbolised by a while we use for the anti-proton). 6 PHYS 490/891 – Winter 2009 L02 This solution is completely symmetric and it is arbitrary which of the solutions we correlate with particles and which one with anti-particles. The only reason to call the electron a particle and the positron an anti- particle is based on the fact that we know electrons longer than positrons. In non-relativistic quantum mechanics we usually discuss particles in certain potentials. However the potentials themselves are not part of the theory, they are added ad-hoc. A consistent physical theory has to include a description of these potentials. Theories which deliver such a description are called field theories since they describe fields in addition to particles. The best possible description of course should include relativistic effects and should be consistent with quantum mechanics, which leads to relativistic quantum field theories. It turns out that in these theories we find a new phenomenon: not only the particles are discrete and the states they are in, but also the fields are quantised. The continuous and quasi-static potential has to be replaced by quanta which are continuously exchanged between the interacting particles. Each type of interaction has its own types of quanta which are exchanged. For the electro-magnetic interaction it turns out that the exchange particle is the photon. Beyond the electro-magnetic force there are three other known forces. Gravity is important at large scales, but is almost always negligible when dealing with small objects such as elementary particles. The force that holds nuclei together, overcoming the strong Coulomb repulsion of the protons, is a consequence of the strong force which holds to quarks inside the nucleons together, similar to the chemical forces (like the van-der-Waals force which is a consequence of the incomplete cancellation of the electric forces inside the atoms or molecules). The strong force has only a short range and the respective exchange particles are called gluons. The strong force acts between particles that carry the “charge” of the strong force, which comes in three variations (as opposed to the electrical charge which only comes in two different varieties, referred to as positive and negative) and is usually referred to as color in analogy to the three colors that produce white light (red, green, and blue). Be aware that this has nothing to do with an actual color! For anti-particles we find the respective anti- colors (anti-red, anti-green, and anti-blue). Quarks, as well as the gluons themselves carry a strong charge, but leptons do not. The fourth force is the weak force which is e.g. responsible for the radioactive beta decay. The exchange particles for this force are the W , the W and the Z . All three are very heavy and can only exist for a very short time, which makes this force very short range and weak. All particles participate in the weak interaction. While all the particles discussed in the previous section (leptons and quarks) are fermions, the exchange particles are all bosons. 7 PHYS 490/891 – Winter 2009 L03 1.4 Symmetries Symmetries play an important role in physics since they are always correlated with conservation laws. The most important conservation laws (energy, momentum, angular momentum) are correlated to continuous symmetries: conservation of energy follows from the fact that a process independent of the point in time at which it occurs; the corresponding transformation is . Conservation of momentum is a consequence of the fact that it does not matter where in space a process occurs (transformation: and the fact that the orientation in space does not matter leads to the conservation of angular momentum ( ). All these conservation laws are additive (i.e. the sum of the energies, momenta or angular momenta in a system is conserved). In addition to these continuous symmetries there is a set of discrete symmetries which are relevant for all fundamental interactions. The most important are Parity, Charge Conjugation, and Time Reversal. Parity refers to spatial reflection. We introduce the parity operator which produces the transformation: . If we apply this operator to a wave function we find: (1.7) So if is an eigenfunction of the eigenvalues can only be 1. If the potential in Schrödinger’s equation is symmetric: , then the Hamiltonian in (1.3) is symmetric as well and the solution consequently can be expressed in terms of eigenfunctions of the parity operator. We therefore find that free particles (this implies 0 which is obviously symmetric) have an intrinsic parity which can be positive or negative ( or ). For a spherically symmetric problem the solutions are usually given in polar coordinates , and with sin cos , sin sin and cos . (1.8) They can be expressed as a product of a function of the radius and a generic set of functions of the angles, the spherical harmonics , . The parity operator does not change but affects the angles: and . (1.9) From the definition of the spherical harmonics follows: , , 1 , (1.10) and consequently , 1 , 1 (1.11) From this we can see that if a situation is described by a set of wave functions the parity of the combined wave function is the product of the individual parities. A detailed analysis of the relativistic 8 PHYS 490/891 – Winter 2009 L03 wave equation discussed above shows that the product of the intrinsic parities of a particle and its antiparticle is negative. Since the parity of a single particle cannot be measured, positive parity is arbitrarily assigned to particles and negative parity to antiparticles. Charge conjugation (usually symbolised by the charge conjugation operator ) is the operation which transforms a particle into its antiparticle. In this operation not only the charge is changed but also a number of other basic properties (e.g. the magnetic moment or quantum numbers like the lepton number which we will discuss later). Like the parity operator the charge conjugation operator has eigenvalues 1. A charged particle can obviously not be an eigenstate of this operator but we can construct eigenstates by combining particles and antiparticles appropriately. The eigenvalue is then again determined by the angular momentum (in this case including the spin). The third important discrete symmetry is time reversal. Applied to a given situation it reverses all processes in time. However it is slightly different from the other two discussed discrete symmetries: if is the time reversal operator we have . (1.12) So this operator does not just reverse the time but it also transforms the wave function into its complex conjugate. Most interactions conserve these symmetries. If we have e.g. as initial state an excited state of an atom and as final state the same atom in the ground state plus a photon which is emitted in this process then the parity of initial and final state will be the same. Consequently there are some transitions in atomic physics which are no allowed because they would violate the parity conservation law. However there are interactions which do not conserve parity. As far as we know the combination of all three of these symmetries (usually referred to as ) is always conserved. 1.5 Interactions In a very condensed form we can write down particle interactions like chemical reactions. The decay of a neutron can e.g. be written as: (1.13) All quantities which are conserved in an interaction have to be the same on both sides of this equation. Like in a mathematical equation we can shift terms from one side to the other if we make sure that all conservation laws are obeyed. In a mathematical equation we have to invert the terms or switch their signs depending on the operation we perform. Here we have to replace particles by antiparticles. We may e.g. take away the electron on the right side of (1.13) and compensate for this by adding an anti-electron (or positron) on the left side: (1.14) 9 PHYS 490/891 – Winter 2009 L03 The basic physics which describes both these processes (1.13) and (1.14) is the same. Richard Feynman invented a very intuitive way to write down particle interactions in a graphical form, the so-called Feynman diagrams. In addition to the initial and final state which we find in the above format the Feynman diagrams also include the exchange particle. One possible Feynman diagram for process (1.14) is shown in Fig. 5. Figure 5: Feynman diagram for process (1.14). Time is going from left to right; the exchange particle for this process is a boson. Note that the arrows for antiparticles (positron and antineutrino in this case) point backwards in time. The could be a going from the upper vertex to the lower one or a going the other way. There are slight variations in the conventions for drawing Feynman diagrams, the most obvious being that the flow of time might go in a different direction (e.g. bottom to top) but this is usually easily recognizable. We will use straight lines for particles and anti-particles (with arrows forward and backward in time respectively), dashed lines for and , wavy lines for photons and spirals for gluons. The points where lines meet are called vertices and are of special importance in a Feynman diagram since this is where things happen. For most purposes we will only deal with Feynman diagrams with three-line-vertices. There are rules how to convert such diagrams into a calculation for the probability for the respective process to happen, but in this course we will use these diagrams mostly just as a convenient visualisation and for order-of-magnitude estimates. All conservation laws have to be obeyed at each vertex. However, as we know from Heisenberg’s Uncertainty Principle, certain quantities may not be well defined. If e.g. the time between the two interactions in Fig. 5 (the two vertices) is very short, the energy transferred at each vertex is not very well defined. In our example the exchange particle is very heavy, so the incoming positron cannot possibly decay into a neutrino and a and still conserve energy. However, for a short time we can ‘borrow’ energy to produce the exchange particle. Another way to think of this is saying the exchange particle is emitted with negative energy. At the other vertex where the exchange particle is absorbed we have to pay back the ‘debt’ so that in the total process energy is conserved. From this perspective we can understand why the weak force with its heavy exchange particles has only a short range: if the interaction partner is too far away the exchange particle cannot reach it in time to pay back the energy debt and has to be re-absorbed by the emitter, so no interaction happens.
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