DOCSLIB.ORG

Majorana Fermions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Majorana Fermions Majorana Fermions QFFF MSc Dissertation Aran Sivaguru September 2012 Advisor: Dr. A. Rajantie Contents 1 Introduction 1 2 Particles and Anti-Particles 2 2.1 The Dirac Equation . 2 2.1.1 Starting from Schr¨odinger . 2 2.1.2 Enter Dirac . 3 2.2 Field Theory and Anti-Matter . 5 2.2.1 The Dirac Field . 5 2.2.2 Neutral Particles . 7 3 Neutrinos 10 3.1 Lepton Number Conservation . 10 3.2 Massive Neutrinos and the Standard Model . 12 3.2.1 Neutrino Oscillations . 12 3.2.2 The Seesaw Mechanism . 14 3.3 Neutrinoless Double Beta Decay . 17 3.3.1 In Search of 0νββ ............................. 17 3.3.2 Theory vs. Experiment . 18 4 Condensed Matter Physics 19 4.1 Electrons and Holes . 19 4.2 Superconductors . 20 4.2.1 Cooper Pairs . 20 4.2.2 Vortices . 21 4.3 Non-Abelian Symmetries . 22 4.3.1 Discovering the Majorana Fermion? . 23 4.3.2 Quantum Computing . 24 5 Discussion 25 5.0.3 Supersymmetry . 25 5.0.4 Dark Matter . 25 5.1 Conclusion . 26 1 INTRODUCTION 1 Introduction In 1937 Ettore Majorana suggested that chargeless fermions could be described by a real wave equation, leading to the possibility that this particle would be identical to its anti- particle[1]. These Majorana fermions seemed a mathematical quirk without any physical basis. If fermions and anti-fermions were indeed indistinguishable, they could co-exist with- out annihilating one another - not something that we seem to see in nature. However, recent experiments have indicated the existence of these particles within condensed matter systems. If further data bears this out, it could lead to myriad applications in fields as diverse as quantum computing and cosmology[2]. With its grounding in quantum field theory, Majorana fermions provide a useful insight into the utility of field theories, and the interplay between theoretical physics and its exper- imental cousin - with a 75 year-old theoretical prediction only recently gaining experimental credence. This dissertation will discuss some of the physics and applications of Majorana fermions, looking at the implications of their discovery. Firstly, this report will look at some of the basic concepts in field theory that underpin the theoretical aspects of these particles. As Majorana fermions could be integral to the description of massive neutrinos - an exten- sion of the Standard Model, the next section will be devoted to this. Then, as recent results in condensed matter physics have shown glimpses of these particles, a qualitative look at Majorana fermions in this branch of research will be sought. Finally, the last section will briefly discuss some further applications of Majorana fermions and its utility in theoretical and experimental physics. 1 2 PARTICLES AND ANTI-PARTICLES 2 Particles and Anti-Particles 2.1 The Dirac Equation Quantum field theory provides a convenient mathematical framework to view particles and their anti-particles. So, as with all discussions surrounding quantum field theory, it is prob- ably best to start with looking at the Dirac equation[3]. This equation was Dirac's way 1 of formulating a description of elementary spin- 2 particles that was consistent with both quantum mechanics and special relativity. 2.1.1 Starting from Schr¨odinger Dirac's famous equation was motivated by Schr¨odinger'sequation for a free particle h2 @ − r2 = i (2.1) 2m ~@t The left hand side of this equation gives the non-relativistic kinetic energy. However, rel- ativity treats space and time on the same footing. Therefore a relativistic treatment of Schr¨odinger'sequation must have differential equations of the same order with respect to space and time. We can try to do this by starting with the relativistic invariance of 4- momentum E2 − p2 = m2c2 (2.2) c2 ^ @ If we express the energy and momentum as operators i.e. E = i~ @t andp ^ = −i~r, we can get the Klein-Gordon equation1 @2 − + r2 = m2 (2.3) @t2 1From here on c = ~ = 1 2 2 PARTICLES AND ANTI-PARTICLES The wavefunction is now a relativistic scalar and the space and time derivatives are both second order. However, although equation 2.3 is a valid wave equation, it is not a good relativistic generalisation of Schr¨odinger'sequation. This is because the initial values of and @ @t can be chosen freely, and as a result the probability density is no longer positive definite. This leaves open the possibility of negative probabilities - which is somewhat worrisome. 2.1.2 Enter Dirac So making the connection between Schr¨odinger'sequation and relativity was not so straight- forward. Equation 2.3 gives a relativistic wave equation, but with the possibility of negative probabilities. Dirac wondered if instead of getting a second order equation in space and time (as in the Klein-Gordon equation), he could find a first order equation that would be positive definite. He started by trying to take the square root of the wave equation @2 @ @ @ @ 2 r2 − = A + B + C + iD (2.4) @t2 @x @y @z @t @2 @2 when the right hand side of this equation is multiplied out, the cross terms ( @x@y , @x@z etc) cancel if the anti-commutators2 of the coefficients vanish fA; Bg = fC; Dg = fA; Dg = fB; Cg = ::: = 0 (2.5) and they each square to give the identity 2 2 2 2 A = B = C = D = 14×4 (2.6) From this Dirac realised that to get a generalisation of quantum mechanics that was com- patible with relativity you need to construct matrices that are at least 4 × 4 - leading to a 2with fx; yg = xy + yx being the usual anti-commutation relation. 3 2 PARTICLES AND ANTI-PARTICLES wavefunction that has 4 components, known as a spinor. Using equations 2.3 and 2.4 we can write @ @ @ @ A + B + C + iD = X (2.7) @x @y @z @t with X still to be determined. Applying X to both sides we have @2 r2 − = X2 (2.8) @t2 by setting X = m we have a first order equation in space and time @ @ @ @ A + B + C + iD − m = 0 (2.9) @x @y @z @t setting A = iβα1, B = iβα2, C = iβα3 and D = β @ @ @ @ iβ α + α + α + − m = m (2.10) 1 @x 2 @y 3 @z @t inserting r @ iβ α · r + = m (2.11) @t From 2.6 it follows that β2 = 1 and β = β−1 and therefore we get @ iα · r + i = mβ (2.12) @t Then, if we insert the momentum operatorp ^ = −ir, we have the original form of the Dirac equation @ i = (mβ + α · p^) (2.13) @t 4 2 PARTICLES AND ANTI-PARTICLES By defining the gamma matrices γµ γ0 = β (2.14) i γ = βαi (2.15) we get the more usual form of Dirac's equation µ (iγ @µ − m) = 0 (2.16) with being the four component spinor. 2.2 Field Theory and Anti-Matter 1 So equation 2.16 allows us to describe relativistic spin- 2 particles, but what does this tell us about particles and their anti-particle partners? And what does this have to do with the Majorana fermion? 2.2.1 The Dirac Field When building a quantum filed theory you can start with the Lagrangian for your fields then canonically quantise the Lagrangian by promoting the fields to operators. The Lagrangian of the free Dirac field is given by µ L = (iγ @µ − m) (2.17) 5 2 PARTICLES AND ANTI-PARTICLES with ≡ yγ0. It is easy to see that when this Lagrangian is varied we get back the Dirac equation, which is just what we want. The gamma matrices γµ follow the Clifford Algebra fγµ; γνg = γµγν + γνγµ (2.18) = 2ηµν (2.19) Dirac found gamma matrices that were complex. From the form of equation 2.17 this means that the spinor field must also be complex. This makes sense from the point of view of field theory as a complex field would create particles and annihilate anti-particles while its complex conjugate would create anti-particles and annihilate particles. s s y s More explicitly we can expand in terms of creation and annihilation operators ap, ap ,bp, s y bp Z d3p 1 X (x) = (as us(p)e−ip·x + bs yvs(p)eip·x) (2.20) (2π)3 p p p 2Ep s Z d3p 1 X (x) = (bs vs(p)e−ip·x + as yus(p)eip·x) (2.21) (2π)3 p p p 2Ep s With the creation and annihilation operators obeying the normal anti-commutation relations r s y r s y 3 (3) rs fap; ap g = fbp; bp g = (2π) δ (p − q)δ (2.22) s y s y The explanation for these two types of operators is that ap creates a fermion and bp creates an anti-fermion, both with energy Ep and momentum p. Therefore, one-particle states are 6 2 PARTICLES AND ANTI-PARTICLES created by operating on the vacuum state j0i − p s y j e ; p; si ≡ 2Epap j0i (2.23) + p s y j e ; p; si ≡ 2Epbp j0i (2.24) Here, e− is the electron and e+ is its anti-particle partner, the positron. These anti-particles have the same mass as their particle partners but with opposite electric charge. When pairs meet they annihilate each other to produce photons, thus conserving total charge. Field theory gives a neat way to describe particles and their anti-particles, each being created by its own operator. However, as anti-particles are usually defined as having the same mass but opposite charge from their partners, what does this mean for neutral particles? 2.2.2 Neutral Particles It is possible for neutrally charged particles to have a distinct anti-particle.
Load more

Recommended publications
  • Supersymmetric Dark Matter
    Supersymmetric dark matter G. Bélanger LAPTH- Annecy Plan | Dark matter : motivation | Introduction to supersymmetry | MSSM | Properties of neutralino | Status of LSP in various SUSY models | Other DM candidates z SUSY z Non-SUSY | DM : signals, direct detection, LHC Dark matter: a WIMP? | Strong evidence that DM dominates over visible matter. Data from rotation curves, clusters, supernovae, CMB all point to large DM component | DM a new particle? | SM is incomplete : arbitrary parameters, hierarchy problem z DM likely to be related to physics at weak scale, new physics at the weak scale can also solve EWSB z Stable particle protect by symmetry z Many solutions – supersymmetry is one best motivated alternative to SM | NP at electroweak scale could also explain baryonic asymetry in the universe Relic density of wimps | In early universe WIMPs are present in large number and they are in thermal equilibrium | As the universe expanded and cooled their density is reduced Freeze-out through pair annihilation | Eventually density is too low for annihilation process to keep up with expansion rate z Freeze-out temperature | LSP decouples from standard model particles, density depends only on expansion rate of the universe | Relic density | A relic density in agreement with present measurements (Ωh2 ~0.1) requires typical weak interactions cross-section Coannihilation | If M(NLSP)~M(LSP) then maintains thermal equilibrium between NLSP-LSP even after SUSY particles decouple from standard ones | Relic density then depends on rate for all processes
    [Show full text]
  • Effective Dirac Equations in Honeycomb Structures
    Effective Dirac equations in honeycomb structures Effective Dirac equations in honeycomb structures Young Researchers Seminar, CERMICS, Ecole des Ponts ParisTech William Borrelli CEREMADE, Universit´eParis Dauphine 11 April 2018 It is self-adjoint on L2(R2; C2) and the spectrum is given by σ(D0) = R; σ(D) = (−∞; −m] [ [m; +1) The domain of the operator and form domain are H1(R2; C2) and 1 2 2 H 2 (R ; C ), respectively. Remark The negative spectrum is associated with antiparticles, in relativistic theories. Effective Dirac equations in honeycomb structures Dirac in 2D The 2D Dirac operator The 2D Dirac operator is defined as D = D0 +mσ3 = −i(σ1@1 + σ2@2) + mσ3: (1) where σk are the Pauli matrices and m ≥ 0 is the mass of the particle. It acts on C2-valued spinors. The domain of the operator and form domain are H1(R2; C2) and 1 2 2 H 2 (R ; C ), respectively. Remark The negative spectrum is associated with antiparticles, in relativistic theories. Effective Dirac equations in honeycomb structures Dirac in 2D The 2D Dirac operator The 2D Dirac operator is defined as D = D0 +mσ3 = −i(σ1@1 + σ2@2) + mσ3: (1) where σk are the Pauli matrices and m ≥ 0 is the mass of the particle. It acts on C2-valued spinors. It is self-adjoint on L2(R2; C2) and the spectrum is given by σ(D0) = R; σ(D) = (−∞; −m] [ [m; +1) Effective Dirac equations in honeycomb structures Dirac in 2D The 2D Dirac operator The 2D Dirac operator is defined as D = D0 +mσ3 = −i(σ1@1 + σ2@2) + mσ3: (1) where σk are the Pauli matrices and m ≥ 0 is the mass of the particle.
    [Show full text]
  • Majorana Fermions in Condensed Matter Physics: the 1D Nanowire Case
    Majorana Fermions in Condensed Matter Physics: The 1D Nanowire Case Philip Haupt, Hirsh Kamakari, Edward Thoeng, Aswin Vishnuradhan Department of Physics and Astronomy, University of British Columbia, Vancouver, B.C., V6T 1Z1, Canada (Dated: November 24, 2018) Majorana fermions are fermions that are their own antiparticles. Although they remain elusive as elementary particles (how they were originally proposed), they have rapidly gained interest in condensed matter physics as emergent quasiparticles in certain systems like topological supercon- ductors. In this article, we briefly review the necessary theory and discuss the \recipe" to create Majorana particles. We then consider existing experimental realisations and their methodologies. I. MOTIVATION A). Kitaev used a simplified quantum wire model to show Ettore Majorana, in 1937, postulated the existence of how Majorana modes might manifest as an emergent an elementary particle which is its own antiparticle, so phenomena, which we will now discuss. Consider 1- called Majorana fermions [1]. It is predicted that the neu- dimensional tight binding chain with spinless fermions trinos are one such elementary particle, which is yet to and p-orbital hopping. The use of unphysical spinless be detected via extremely rare neutrino-less double beta- fermions calls into question the validity of the model, decay. The research on Majorana fermions in the past but, as has been subsequently realised, in the presence few years, however, have gained momentum in the com- of strong spin orbit coupling it is possible for electrons pletely different field of condensed matter physics. Arti- to be approximated as spinless in the presence of spin- ficially engineered low-dimensional nanostructures which orbit coupling as well as a Zeeman field [9].
    [Show full text]
  • $ Z $ Boson Mediated Dark Matter Beyond the Effective Theory
    MCTP-16-27 FERMILAB-PUB-16-534-T Z boson mediated dark matter beyond the effective theory John Kearney Theoretical Physics Department, Fermi National Accelerator Laboratory, Batavia, IL 60510 USA Nicholas Orlofsky and Aaron Pierce Michigan Center for Theoretical Physics (MCTP) Department of Physics, University of Michigan, Ann Arbor, MI 48109 (Dated: April 11, 2017) Direct detection bounds are beginning to constrain a very simple model of weakly interacting dark matter|a Majorana fermion with a coupling to the Z boson. In a particularly straightforward gauge-invariant realization, this coupling is introduced via a higher-dimensional operator. While attractive in its simplicity, this model generically induces a large ρ parameter. An ultraviolet completion that avoids an overly large contribution to ρ is the singlet-doublet model. We revisit this model, focusing on the Higgs blind spot region of parameter space where spin-independent interactions are absent. This model successfully reproduces dark matter with direct detection mediated by the Z boson, but whose cosmology may depend on additional couplings and states. Future direct detection experiments should effectively probe a significant portion of this parameter space, aside from a small coannihilating region. As such, Z-mediated thermal dark matter as realized in the singlet-doublet model represents an interesting target for future searches. I. INTRODUCTION Z, and their spin-dependent (SD) couplings, which at tree level arise from exchange of the Z. The latest bounds on SI scattering arise from PandaX Weakly interacting massive particles (WIMPs) re- [1] and LUX [2]. DM that interacts with the Z main an attractive thermal dark matter (DM) can- boson via vectorial couplings, g (¯χγ χ)Zµ, is very didate.
    [Show full text]
  • Quantum Statistics: Is There an Effective Fermion Repulsion Or Boson Attraction? W
    Quantum statistics: Is there an effective fermion repulsion or boson attraction? W. J. Mullin and G. Blaylock Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003 ͑Received 13 February 2003; accepted 16 May 2003͒ Physicists often claim that there is an effective repulsion between fermions, implied by the Pauli principle, and a corresponding effective attraction between bosons. We examine the origins and validity of such exchange force ideas and the areas where they are highly misleading. We propose that explanations of quantum statistics should avoid the idea of an effective force completely, and replace it with more appropriate physical insights, some of which are suggested here. © 2003 American Association of Physics Teachers. ͓DOI: 10.1119/1.1590658͔ ␺ ͒ϭ ͒ Ϫ␣ Ϫ ϩ ͒2 I. INTRODUCTION ͑x1 ,x2 ,t C͕f ͑x1 ,x2 exp͓ ͑x1 vt a Ϫ␤͑x ϩvtϪa͒2͔Ϫ f ͑x ,x ͒ The Pauli principle states that no two fermions can have 2 2 1 ϫ Ϫ␣ Ϫ ϩ ͒2Ϫ␤ ϩ Ϫ ͒2 the same quantum numbers. The origin of this law is the exp͓ ͑x2 vt a ͑x1 vt a ͔͖, required antisymmetry of the multi-fermion wavefunction. ͑1͒ Most physicists have heard or read a shorthand way of ex- pressing the Pauli principle, which says something analogous where x1 and x2 are the particle coordinates, f (x1 ,x2) ϭ ͓ Ϫ ប͔ to fermions being ‘‘antisocial’’ and bosons ‘‘gregarious.’’ Of- exp imv(x1 x2)/ , C is a time-dependent factor, and the ten this intuitive approach involves the statement that there is packet width parameters ␣ and ␤ are unequal.
    [Show full text]
  • Compton Scattering from the Deuteron at Low Energies
    SE0200264 LUNFD6-NFFR-1018 Compton Scattering from the Deuteron at Low Energies Magnus Lundin Department of Physics Lund University 2002 W' •sii" Compton spridning från deuteronen vid låga energier (populärvetenskaplig sammanfattning på svenska) Vid Compton spridning sprids fotonen elastiskt, dvs. utan att förlora energi, mot en annan partikel eller kärna. Kärnorna som användes i detta försök består av en proton och en neutron (sk. deuterium, eller tungt väte). Kärnorna bestrålades med fotoner av kända energier och de spridda fo- tonerna detekterades mha. stora Nal-detektorer som var placerade i olika vinklar runt strålmålet. Försöket utfördes under 8 veckor och genom att räkna antalet fotoner som kärnorna bestålades med och antalet spridda fo- toner i de olika detektorerna, kan sannolikheten för att en foton skall spridas bestämmas. Denna sannolikhet jämfördes med en teoretisk modell som beskriver sannolikheten för att en foton skall spridas elastiskt mot en deuterium- kärna. Eftersom protonen och neutronen består av kvarkar, vilka har en elektrisk laddning, kommer dessa att sträckas ut då de utsätts för ett elek- triskt fält (fotonen), dvs. de polariseras. Värdet (sannolikheten) som den teoretiska modellen ger, beror på polariserbarheten hos protonen och neu- tronen i deuterium. Genom att beräkna sannolikheten för fotonspridning för olika värden av polariserbarheterna, kan man se vilket värde som ger bäst överensstämmelse mellan modellen och experimentella data. Det är speciellt neutronens polariserbarhet som är av intresse, och denna kunde bestämmas i detta arbete. Organization Document name LUND UNIVERSITY DOCTORAL DISSERTATION Department of Physics Date of issue 2002.04.29 Division of Nuclear Physics Box 118 Sponsoring organization SE-22100 Lund Sweden Author (s) Magnus Lundin Title and subtitle Compton Scattering from the Deuteron at Low Energies Abstract A series of three Compton scattering experiments on deuterium have been performed at the high-resolution tagged-photon facility MAX-lab located in Lund, Sweden.
    [Show full text]
  • Ettore Majorana: Genius and Mystery
    «ETTORE MAJORANA» FOUNDATION AND CENTRE FOR SCIENTIFIC CULTURE TO PAY A PERMANENT TRIBUTE TO GALILEO GALILEI, FOUNDER OF MODERN SCIENCE AND TO ENRICO FERMI, THE "ITALIAN NAVIGATOR", FATHER OF THE WEAK FORCES ETTORE MAJORANA CENTENARY ETTORE MAJORANA: GENIUS AND MYSTERY Antonino Zichichi ETTORE MAJORANA: GENIUS AND MYSTERY Antonino Zichichi ABSTRACT The geniality of Ettore Majorana is discussed in the framework of the crucial problems being investigated at the time of his activity. These problems are projected to our present days, where the number of space-time dimensions is no longer four and where the unification of the fundamental forces needs the Majorana particle: neutral, with spin ½ and identical to its antiparticle. The mystery of the way Majorana disappeared is restricted to few testimonies, while his geniality is open to all eminent physicists of the XXth century, who had the privilege of knowing him, directly or indirectly. 3 44444444444444444444444444444444444 ETTORE MAJORANA: GENIUS AND MYSTERY Antonino Zichichi CONTENTS 1 LEONARDO SCIASCIA’S IDEA 5 2 ENRICO FERMI: FEW OTHERS IN THE WORLD COULD MATCH MAJORANA’S DEEP UNDERSTANDING OF THE PHYSICS OF THE TIME 7 3 RECOLLECTIONS BY ROBERT OPPENHEIMER 19 4 THE DISCOVERY OF THE NEUTRON – RECOLLECTIONS BY EMILIO SEGRÉ AND GIANCARLO WICK 21 5 THE MAJORANA ‘NEUTRINOS’ – RECOLLECTIONS BY BRUNO PONTECORVO – THE MAJORANA DISCOVERY ON THE DIRAC γ- MATRICES 23 6 THE FIRST COURSE OF THE SUBNUCLEAR PHYSICS SCHOOL (1963): JOHN BELL ON THE DIRAC AND MAJORANA NEUTRINOS 45 7 THE FIRST STEP TO RELATIVISTICALLY DESCRIBE PARTICLES WITH ARBITRARY SPIN 47 8 THE CENTENNIAL OF THE BIRTH OF A GENIUS – A HOMAGE BY THE INTERNATIONAL SCIENTIFIC COMMUNITY 53 REFERENCES 61 4 44444444444444444444444444444444444 Ettore Majorana’s photograph taken from his university card dated 3rd November 1923.
    [Show full text]
  • A Generalization of the One-Dimensional Boson-Fermion Duality Through the Path-Integral Formalsim
    A Generalization of the One-Dimensional Boson-Fermion Duality Through the Path-Integral Formalism Satoshi Ohya Institute of Quantum Science, Nihon University, Kanda-Surugadai 1-8-14, Chiyoda, Tokyo 101-8308, Japan [email protected] (Dated: May 11, 2021) Abstract We study boson-fermion dualities in one-dimensional many-body problems of identical parti- cles interacting only through two-body contacts. By using the path-integral formalism as well as the configuration-space approach to indistinguishable particles, we find a generalization of the boson-fermion duality between the Lieb-Liniger model and the Cheon-Shigehara model. We present an explicit construction of n-boson and n-fermion models which are dual to each other and characterized by n−1 distinct (coordinate-dependent) coupling constants. These models enjoy the spectral equivalence, the boson-fermion mapping, and the strong-weak duality. We also discuss a scale-invariant generalization of the boson-fermion duality. arXiv:2105.04288v1 [quant-ph] 10 May 2021 1 1 Introduction Inhisseminalpaper[1] in 1960, Girardeau proved the one-to-one correspondence—the duality—between one-dimensional spinless bosons and fermions with hard-core interparticle interactions. By using this duality, he presented a celebrated example of the spectral equivalence between impenetrable bosons and free fermions. Since then, the one-dimensional boson-fermion duality has been a testing ground for studying strongly-interacting many-body problems, especially in the field of integrable models. So far there have been proposed several generalizations of the Girardeau’s finding, the most promi- nent of which was given by Cheon and Shigehara in 1998 [2]: they discovered the fermionic dual of the Lieb-Liniger model [3] by using the generalized pointlike interactions.
    [Show full text]
  • 7. Gamma and X-Ray Interactions in Matter
    Photon interactions in matter Gamma- and X-Ray • Compton effect • Photoelectric effect Interactions in Matter • Pair production • Rayleigh (coherent) scattering Chapter 7 • Photonuclear interactions F.A. Attix, Introduction to Radiological Kinematics Physics and Radiation Dosimetry Interaction cross sections Energy-transfer cross sections Mass attenuation coefficients 1 2 Compton interaction A.H. Compton • Inelastic photon scattering by an electron • Arthur Holly Compton (September 10, 1892 – March 15, 1962) • Main assumption: the electron struck by the • Received Nobel prize in physics 1927 for incoming photon is unbound and stationary his discovery of the Compton effect – The largest contribution from binding is under • Was a key figure in the Manhattan Project, condition of high Z, low energy and creation of first nuclear reactor, which went critical in December 1942 – Under these conditions photoelectric effect is dominant Born and buried in • Consider two aspects: kinematics and cross Wooster, OH http://en.wikipedia.org/wiki/Arthur_Compton sections http://www.findagrave.com/cgi-bin/fg.cgi?page=gr&GRid=22551 3 4 Compton interaction: Kinematics Compton interaction: Kinematics • An earlier theory of -ray scattering by Thomson, based on observations only at low energies, predicted that the scattered photon should always have the same energy as the incident one, regardless of h or • The failure of the Thomson theory to describe high-energy photon scattering necessitated the • Inelastic collision • After the collision the electron departs
    [Show full text]
  • Realization of the Low Background Neutrino Detector Double Chooz: from the Development of a High-Purity Liquid & Gas Handling Concept to first Neutrino Data
    Realization of the low background neutrino detector Double Chooz: From the development of a high-purity liquid & gas handling concept to first neutrino data Dissertation of Patrick Pfahler TECHNISCHE UNIVERSITAT¨ MUNCHEN¨ Physik Department Lehrstuhl f¨urexperimentelle Astroteilchenphysik / E15 Univ.-Prof. Dr. Lothar Oberauer Realization of the low background neutrino detector Double Chooz: From the development of high-purity liquid- & gas handling concept to first neutrino data Dipl. Phys. (Univ.) Patrick Pfahler Vollst¨andigerAbdruck der von der Fakult¨atf¨urPhysik der Technischen Universit¨atM¨unchen zur Erlangung des akademischen Grades eines Doktors des Naturwissenschaften (Dr. rer. nat) genehmigten Dissertation. Vorsitzender: Univ.-Prof. Dr. Alejandro Ibarra Pr¨uferder Dissertation: 1. Univ.-Prof. Dr. Lothar Oberauer 2. Priv.-Doz. Dr. Andreas Ulrich Die Dissertation wurde am 3.12.2012 bei der Technischen Universit¨atM¨unchen eingereicht und durch die Fakult¨atf¨urPhysik am 17.12.2012 angenommen. 2 Contents Contents i Introduction 1 I The Neutrino Disappearance Experiment Double Chooz 5 1 Neutrino Oscillation and Flavor Mixing 6 1.1 PMNS Matrix . 6 1.2 Flavor Mixing and Neutrino Oscillations . 7 1.2.1 Survival Probability of Reactor Neutrinos . 9 1.2.2 Neutrino Masses and Mass Hierarchy . 12 2 Reactor Neutrinos 14 2.1 Neutrino Production in Nuclear Power Cores . 14 2.2 Energy Spectrum of Reactor neutrinos . 15 2.3 Neutrino Flux Approximation . 16 3 The Double Chooz Experiment 19 3.1 The Double Chooz Collaboration . 19 3.2 Experimental Site: Commercial Nuclear Power Plant in Chooz . 20 3.3 Physics Program and Experimental Concept . 21 3.4 Signal . 23 3.4.1 The Inverse Beta Decay (IBD) .
    [Show full text]
  • 2 Lecture 1: Spinors, Their Properties and Spinor Prodcuts
    2 Lecture 1: spinors, their properties and spinor prodcuts Consider a theory of a single massless Dirac fermion . The Lagrangian is = ¯ i@ˆ . (2.1) L ⇣ ⌘ The Dirac equation is i@ˆ =0, (2.2) which, in momentum space becomes pUˆ (p)=0, pVˆ (p)=0, (2.3) depending on whether we take positive-energy(particle) or negative-energy (anti-particle) solutions of the Dirac equation. Therefore, in the massless case no di↵erence appears in equations for paprticles and anti-particles. Finding one solution is therefore sufficient. The algebra is simplified if we take γ matrices in Weyl repreentation where µ µ 0 σ γ = µ . (2.4) " σ¯ 0 # and σµ =(1,~σ) andσ ¯µ =(1, ~ ). The Pauli matrices are − 01 0 i 10 σ = ,σ= − ,σ= . (2.5) 1 10 2 i 0 3 0 1 " # " # " − # The matrix γ5 is taken to be 10 γ5 = − . (2.6) " 01# We can use the matrix γ5 to construct projection operators on to upper and lower parts of the four-component spinors U and V . The projection operators are 1 γ 1+γ Pˆ = − 5 , Pˆ = 5 . (2.7) L 2 R 2 Let us write u (p) U(p)= L , (2.8) uR(p) ! where uL(p) and uR(p) are two-component spinors. Since µ 0 pµσ pˆ = µ , (2.9) " pµσ¯ 0(p) # andpU ˆ (p) = 0, the two-component spinors satisfy the following (Weyl) equations µ µ pµσ uR(p)=0,pµσ¯ uL(p)=0. (2.10) –3– Suppose that we have a left handed spinor uL(p) that satisfies the Weyl equation.
    [Show full text]
  • The Bound States of Dirac Equation with a Scalar Potential
    THE BOUND STATES OF DIRAC EQUATION WITH A SCALAR POTENTIAL BY VATSAL DWIVEDI THESIS Submitted in partial fulfillment of the requirements for the degree of Master of Science in Applied Mathematics in the Graduate College of the University of Illinois at Urbana-Champaign, 2015 Urbana, Illinois Adviser: Professor Jared Bronski Abstract We study the bound states of the 1 + 1 dimensional Dirac equation with a scalar potential, which can also be interpreted as a position dependent \mass", analytically as well as numerically. We derive a Pr¨ufer-like representation for the Dirac equation, which can be used to derive a condition for the existence of bound states in terms of the fixed point of the nonlinear Pr¨uferequation for the angle variable. Another condition was derived by interpreting the Dirac equation as a Hamiltonian flow on R4 and a shooting argument for the induced flow on the space of Lagrangian planes of R4, following a similar calculation by Jones (Ergodic Theor Dyn Syst, 8 (1988) 119-138). The two conditions are shown to be equivalent, and used to compute the bound states analytically and numerically, as well as to derive a Calogero-like upper bound on the number of bound states. The analytic computations are also compared to the bound states computed using techniques from supersymmetric quantum mechanics. ii Acknowledgments In the eternity that is the grad school, this project has been what one might call an impulsive endeavor. In the 6 months from its inception to its conclusion, it has, without doubt, greatly benefited from quite a few individuals, as well as entities, around me, to whom I owe my sincere regards and gratitude.
    [Show full text]