DOCSLIB.ORG

Electronic Structure and Angular Momentum of Transition Metal Complexes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Electronic Structure and Angular Momentum of Transition Metal Complexes Electronic Structure and Angular Momentum of Transition Metal Complexes © K. S. Suslick, 2013 Angular Momentum vs. Number of Electron Spins To understand the energy of paramagnetic systems (i.e., ones with unpaired electrons), we must describe them in terms of their angular momentum. S = 0 Singlet Total Angular Momentum: S = 1/2 Doublet S = 1 Triplet J = Ms + ML S = 3/2 Quartet S = 2 Quintet Total Multiplicity: (2S+1)(2L+1) L = 0 “S term” singly orb. deg. Each “microstate” has the same L = 1 “P term” triply L and S, but different J. L = 2 “D term” pentuply L = 3 “F term” L = 4 “G term” © K. S. Suslick, 2013 1 Spin For the Total Spin of an atom or molecule the rules apply: 1. Doubly occupied orbitals do NOT contribute to the total Spin 2. Singly occupied orbitals can be occupied with either spin-up or spin-down e- 3. Unpaired e- can be coupled parallel or antiparallel, giving a total spin S 4. For a state with total spin S there are 2S+1 “components” with M = S,S-1,...,-S. Hence terms singlet, doublet, triplet, … 5. The MS quantum number is always the sum of all individual ms QNs. © K. S. Suslick, 2013 Spin Names “Russell-Saunders” Term Symbols for atoms; Irr Rep Mulliken for molecules Examples for dn configurations: Atoms Molecules doublet 2H sextet 6A triplet 3 l=1 or 3 © K. S. Suslick, 2013 2 L–S Coupling L-S Coupling = Russell-Saunders Coupling If coupling of the spin angular momentum and orbital angular momentum is relatively weak (and it usually is), then L and S remain “good” Quantum Numbers and can be treated independently of each other. Each electronic state has its own term symbol spin multiplicity 2S+1 L = 0 “S term” singly orb. deg. L = 1 “P term” triply L L = 2 “D term” pentuplicately L = 3 “F term” L = 4 “G term” (Within each term, there can be several degenerate microstates with different ML and MS.) © K. S. Suslick, 2013 L–S Coupling L = 0, 1, 2…total orbital angular momentum (“term”) ML = 0, 1, 2, L components of L (ML = ml for each e). For example, for L = 1, there are three ML values: 1, 0, -1. (analogous to l = 1 and its three ml values: 1, 0, -1) # of ML states is 2L+1 = orbital degeneracy S = total spin angular momentum Ms = S, S-1, ….-S components of S (MS = ms). For example, for S = 1, there are three Ms values, 1, 0, -1. Each electronic state has its own term symbol spin multiplicity 2S+1 L = 0 “S term” singly orb. deg. L = 1 “P term” triply L L = 2 “D term” pentuplicately L = 3 “F term” L = 4 “G term” (Within each term, there can be several degenerate microstates with different ML and MS.) © K. S. Suslick, 2013 3 Hund’s Rules 1. The ground state (GS ‘term’) has the highest spin multiplicity (S). 2. If two or more terms have the same spin multiplicity, then the GS will have the highest value of L. 3. For subshells less than half-filled (e.g., p2), lowest J is preferred; for subshells more than half-filled, highest J is preferred. Of all the states possible from degenerate orbitals, the lowest energy one will have the highest spin multiplicity (i.e., most unpaired spins). For states with the same spin multiplicity, the highest orbital degeneracy will be lowest in energy. © K. S. Suslick, 2013 The Problem: Electron-Electron Repulsion. d2 Consider as an example, 2 d electrons, one in z2 e e z2 x2-y2 g z2 x2-y2 g t2g t2g xy xz yz xy xz yz z z xz + z2 xy + z2 y y x x overlapping lobes, lobes far apart, large inter-electron repulsion small inter-electron repulsion These two electron configurations differ in energy. © K. S. Suslick, 2013 4 Microstates and Spin Orbit Coupling For a given L, the allowed values of ML and Ms are called microstates. 2N from spin (2No)! o # of microstates = up vs. down (2No –Ne)! Ne! Where No = degeneracy of orbitals in set of subshell and Ne = number of electrons e.g., for free atoms/ions, No for d orbitals = 5 © K. S. Suslick, 2013 Microstates and Spin Orbit Coupling 2N from spin (2No)! o up vs. down (2No –Ne)! Ne! Pauli X etc. etc. for 25 for 20 more © K. S. Suslick, 2013 5 Microstates and Spin Orbit Coupling Shorthand way to describe determinantal Wave Functions, e.g., for 2 electrons: mS of electron #1 mL of electron #1 2 e- in p orbitals: ml = -1, 0, +1, ms = -½ or +½ ( 1- 0+ ) mL = 1 0 -1 © K. S. Suslick, 2013 Microstates and Spin Orbit Coupling To find all the terms for a specific configuration: 3) Repeat #2 for microstates remaining with next greater ML etc. © K. S. Suslick, 2013 6 Microstates and Spin Orbit Coupling: p3 p3 examples: © K. S. Suslick, 2013 Microstates and Spin Orbit Coupling: d2 Ms mL M L 1G 3F 3P 1D 1S So, the allowed terms states for d2 are 1G, 3F, 3P, 1D, and 1S. What’s the GS?! Hund’s Rules. © K. S. Suslick, 2013 7 Hund’s Rules 1. The ground state (GS ‘term’) has the highest spin multiplicity (S). 2. If two or more terms have the same spin multiplicity, then the GS will have the highest value of L. 3. For subshells less than half-filled (e.g., p2), lowest J is preferred; for subshells more than half-filled, highest J is preferred. Of all the states possible from degenerate orbitals, the lowest energy one will have the highest spin multiplicity (i.e., most unpaired spins). For states with the same spin multiplicity, the highest orbital degeneracy will be lowest in energy. © K. S. Suslick, 2013 Microstates and Spin Orbit Coupling The allowed terms states for d2 are 1G, 3F, 3P, 1D, and 1S. What’s the GS? Hund’s Rules. G.S. is 3F. What’s the lowest excited state? Hund’s Rules DO NOT TELL YOU THAT!! (the XS energies you get from Condon-Shortley or Racah Parameters). © K. S. Suslick, 2013 8 Energy Separations of Free Ion States © K. S. Suslick, 2013 Microstates and Spin Orbit Coupling, d2 © K. S. Suslick, 2013 9 Spin Orbit Coupling © K. S. Suslick, 2013 Size of Spin Orbit Coupling For a single electron, the strength of Spin-Orbit Coupling (SOC) in a particular microstate is measured by For the whole Term State, SOC is measured by © K. S. Suslick, 2013 10 Energy Separations of Free Ion States But there is a fine structure within these states due to SOC! © K. S. Suslick, 2013 Energy Separations of Free Ion States “Lande Interval Rule” Furthermore, with L-S approximation, Spin-Orbit Coupling must preserve the “center of gravity” of the energy of the term state. (n.b., degeneracies of the J states must be counted.) © K. S. Suslick, 2013 11 Energy Separations of Free Ion States “center of gravity preserved” © K. S. Suslick, 2013 Energy Separations of Free Ion States microstates: © K. S. Suslick, 2013 12 Energy Separations of Free Ion States SOC magnitude from center of gravity L=3, S=1 J = 4, 3, 2 Degeneracy = 2J+1 © K. S. Suslick, 2013 Spin Orbit Coupling, d23F ground state SOC magnitude from center of gravity L=3, S=1, J = 4, 3, 2 Degeneracy = 2J+1 © K. S. Suslick, 2013 13 Energy Separations of Free Ion States = 9-fold degenerate 7-fold degenerate 5-fold degenerate © K. S. Suslick, 2013 Energy Separations of Free Ion States The inter-electron repulsion integrals can be broken into 3 radially and angular forms: © K. S. Suslick, 2013 14 Energy Separations of Free Ion States The inter-electron repulsion integrals can be broken into 3 radially and angular forms: (more later on Tanabe-Sugano Diagrams) © K. S. Suslick, 2013 Energy Separations of Free Ion States: d2 © K. S. Suslick, 2013 15 © K. S. Suslick, 2013 Hole Formalism Eletrons vs. holes: dn and d10-n will have the same possible term states! Their energies of interaction with the environment, however, will have opposite signs. © K. S. Suslick, 2013 16 Spin Orbit Coupling: J-J Coupling If spin-orbit coupling is very strong, then it can no longer be treated as just a perturbation. mL and ms are no longer "good" quantum numbers. Only j = mL + ms is valid. For multi-electron systems, applies only (not L and S). Applies to Lanthanides, Actinides and 3rd row metals. © K. S. Suslick, 2013 Size of Spin Orbit Coupling 120 cm–1 for Ti2+ to 830 cm–1 for Cu2+ (3d) 300 cm–1 for Y2+ to 1600 cm–1 for Pd2+(4d), 640 cm–1 for Ce3+ to 2950 cm–1 for Tb3+ (4f). © K. S. Suslick, 2013 17 Effect of Ligands on State Energies Simplest approach: “Crystal Field Theory” 1. metal d-electron Term States 2. Perturbation by point charges (i.e., ligands) 3. No consideration of other (non-d) electrons. 4. No covalency (added as an empirical correction in Ligand Field Theory) 5. Useful for optical spectra magnetism EPR Mössbauer © K. S. Suslick, 2013 Approximations in CFT kinetic coulombic inter-electron repulsion spin-orbit coupling ligand field comparable magnitudes = net pt. charge on ligand, . l = distance between ith electron and ligand, . l = ligand or crystal field © K. S. Suslick, 2013 18 Approximations in CFT 1. Free ion with SOC: 2. Weak field 3. Strong field © K. S. Suslick, 2013 Approximations in CFT So, how do we assign states in each of these conditions and how do they correlate? © K.
Load more

Recommended publications
  • 3.4 V-A Coupling of Leptons and Quarks 3.5 CKM Matrix to Describe the Quark Mixing
    Advanced Particle Physics: VI. Probing the weak interaction 3.4 V-A coupling of leptons and quarks Reminder u γ µ (1− γ 5 )u = u γ µu L = (u L + u R )γ µu L = u Lγ µu L l ν l ν l l ν l ν In V-A theory the weak interaction couples left-handed lepton/quark currents (right-handed anti-lepton/quark currents) with an universal coupling strength: 2 GF gw = 2 2 8MW Weak transition appear only inside weak-isospin doublets: Not equal to the mass eigenstate Lepton currents: Quark currents: ⎛ν ⎞ ⎛ u ⎞ 1. ⎜ e ⎟ j µ = u γ µ (1− γ 5 )u 1. ⎜ ⎟ j µ = u γ µ (1− γ 5 )u ⎜ − ⎟ eν e ν ⎜ ⎟ du d u ⎝e ⎠ ⎝d′⎠ ⎛ν ⎞ ⎛ c ⎞ 5 2. ⎜ µ ⎟ j µ = u γ µ (1− γ 5 )u 2. ⎜ ⎟ j µ = u γ µ (1− γ )u ⎜ − ⎟ µν µ ν ⎜ ⎟ sc s c ⎝ µ ⎠ ⎝s′⎠ ⎛ν ⎞ ⎛ t ⎞ µ µ 5 3. ⎜ τ ⎟ j µ = u γ µ (1− γ 5 )u 3. ⎜ ⎟ j = u γ (1− γ )u ⎜ − ⎟ τν τ ν ⎜ ⎟ bt b t ⎝τ ⎠ ⎝b′⎠ 3.5 CKM matrix to describe the quark mixing One finds that the weak eigenstates of the down type quarks entering the weak isospin doublets are not equal to the their mass/flavor eigenstates: ⎛d′⎞ ⎛V V V ⎞ ⎛d ⎞ d V u ⎜ ⎟ ⎜ ud us ub ⎟ ⎜ ⎟ ud ⎜ s′⎟ = ⎜Vcd Vcs Vcb ⎟ ⋅ ⎜ s ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ W ⎝b′⎠ ⎝Vtd Vts Vtb ⎠ ⎝b⎠ Cabibbo-Kobayashi-Maskawa mixing matrix The quark mixing is the origin of the flavor number violation of the weak interaction.
    [Show full text]
  • Strong Coupling and Quantum Molecular Plasmonics
    Strong coupling and molecular plasmonics Javier Aizpurua http://cfm.ehu.es/nanophotonics Center for Materials Physics, CSIC-UPV/EHU and Donostia International Physics Center - DIPC Donostia-San Sebastián, Basque Country AMOLF Nanophotonics School June 17-21, 2019, Science Park, Amsterdam, The Netherlands Organic molecules & light Excited molecule (=exciton on molecule) Photon Photon Adding mirrors to this interaction The photon can come back! Optical cavities to enhance light-matter interaction Optical mirrors Dielectric resonator Veff Q ∼ 1/κ Photonic crystals Plasmonic cavity Veff Q∼1/κ Coupling of plasmons and molecular excitations Plasmon-Exciton Coupling Plasmon-Vibration Coupling Absorption, Scattering, IR Absorption, Fluorescence Veff Raman Scattering Extreme plasmonic cavities Top-down Bottom-up STM ultra high-vacuum Wet Chemistry Low temperature Self-assembled monolayers (Hefei, China) (Cambridge, UK) “More interacting system” One cannot distinguish whether we have a photon or an excited molecule. The eigenstates are “hybrid” states, part molecule and part photon: “polaritons” Strong coupling! Experiments: organic molecules Organic molecules: Microcavity: D. G. Lidzey et al., Nature 395, 53 (1998) • Large dipole moments • High densities collective enhancement • Rabi splitting can be >1 eV (~30% of transition energy) • Room temperature! Surface plasmons: J. Bellessa et al., Single molecule with gap plasmon: Phys. Rev. Lett. 93, 036404 (2004) Chikkaraddy et al., Nature 535, 127 (2016) from lecture by V. Shalaev Plasmonic cavities
    [Show full text]
  • The Electron-Phonon Coupling Constant 
    The electron-phonon coupling constant Philip B. Allen Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794-3800 March 17, 2000 Tables of values of the electron-phonon coupling constants and are given for selected tr elements and comp ounds. A brief summary of the theory is also given. I. THEORETICAL INTRODUCTION Grimvall [1] has written a review of electron-phonon e ects in metals. Prominent among these e ects is sup ercon- ductivity. The BCS theory gives a relation T exp1=N 0V for the sup erconducting transition temp erature c D T in terms of the Debye temp erature .Values of T are tabulated in various places, the most complete prior to c D c the high T era b eing ref. [2]. The electron-electron interaction V consists of the attractive electron-phonon-induced c interaction minus the repulsive Coulombinteraction. The notation is used = N 0V 1 eph and the Coulomb repulsion N 0V is called , so that N 0V = , where is a \renormalized" Coulomb c repulsion, reduced in value from to =[1 + ln! =! ]. This suppression of the Coulomb repulsion is a result P D of the fact that the electron-phonon attraction is retarded in time by an amountt 1=! whereas the repulsive D screened Coulombinteraction is retarded byamuch smaller time, t 1=! where ! is the electronic plasma P P frequency. Therefore, is b ounded ab oveby1= ln! =! which for conventional metals should b e 0:2. P D Values of are known to range from 0:10 to 2:0.
    [Show full text]
  • How the Electron-Phonon Coupling Mechanism Work in Metal Superconductor
    How the electron-phonon coupling mechanism work in metal superconductor Qiankai Yao1,2 1College of Science, Henan University of Technology, Zhengzhou450001, China 2School of physics and Engineering, Zhengzhou University, Zhengzhou450001, China Abstract Superconductivity in some metals at low temperature is known to arise from an electron-phonon coupling mechanism. Such the mechanism enables an effective attraction to bind two mobile electrons together, and even form a kind of pairing system(called Cooper pair) to be physically responsible for superconductivity. But, is it possible by an analogy with the electrodynamics to describe the electron-phonon coupling as a resistivity-dependent attraction? Actually so, it will help us to explore a more operational quantum model for the formation of Cooper pair. In particularly, by the calculation of quantum state of Cooper pair, the explored model can provide a more explicit explanation for the fundamental properties of metal superconductor, and answer: 1) How the transition temperature of metal superconductor is determined? 2) Which metals can realize the superconducting transition at low temperature? PACS numbers: 74.20.Fg; 74.20.-z; 74.25.-q; 74.20.De ne is the mobile electron density, η the damping coefficient 1. Introduction that is determined by the collision time τ . In the BCS theory[1], superconductivity is attributed to a In metal environment, mobile electrons are usually phonon-mediated attraction between mobile electrons near modeled to be a kind of classical particles like gas molecules, Fermi surface(called Fermi electrons). The attraction is each of which performs a Brown-like motion and satisfies the sometimes referred to as a residual Coulomb interaction[2] that Langevin equation can glue Cooper pair together to cause superconductivity.
    [Show full text]
  • Lattice-QCD Determinations of Quark Masses and the Strong Coupling Αs
    Lattice-QCD Determinations of Quark Masses and the Strong Coupling αs Fermilab Lattice, MILC, and TUMQCD Collaborations September 1, 2020 EF Topical Groups: (check all that apply /) (EF01) EW Physics: Higgs Boson Properties and Couplings (EF02) EW Physics: Higgs Boson as a Portal to New Physics (EF03) EW Physics: Heavy Flavor and Top-quark Physics (EF04) EW Physics: Electroweak Precision Physics and Constraining New Physics (EF05) QCD and Strong Interactions: Precision QCD (EF06) QCD and Strong Interactions: Hadronic Structure and Forward QCD (EF07) QCD and Strong Interactions: Heavy Ions (EF08) BSM: Model-specific Explorations (EF09) BSM: More General Explorations (EF10) BSM: Dark Matter at Colliders Other Topical Groups: (RF01) Weak Decays of b and c Quarks (TF02) Effective Field Theory Techniques (TF05) Lattice Gauge Theory (CompF2) Theoretical Calculations and Simulation Contact Information: Andreas S. Kronfeld (Theoretical Physics Department, Fermilab) [email]: [email protected] on behalf of the Fermilab Lattice, MILC, and TUMQCD Collaborations Fermilab Lattice, MILC, and TUMQCD Collaborations: A. Bazavov, C. Bernard, N. Brambilla, C. DeTar, A.X. El-Khadra, E. Gamiz,´ Steven Gottlieb, U.M. Heller, W.I. Jay, J. Komijani, A.S. Kronfeld, J. Laiho, P.B. Mackenzie, E.T. Neil, P. Petreczky, J.N. Simone, R.L. Sugar, D. Toussaint, A. Vairo, A. Vaquero Aviles-Casco,´ J.H. Weber, R.S. Van de Water Lattice-QCD Determinations of Quark Masses and the Strong Coupling αs Quantum chromdynamics (QCD) as a stand-alone theory has 1 + nf + 1 free parameters that must be set from experimental measurements, complemented with theoretical calculations connecting the measurements to the QCD Lagrangian.
    [Show full text]
  • Feynman Diagrams Particle and Nuclear Physics
    5. Feynman Diagrams Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 5. Feynman Diagrams 1 In this section... Introduction to Feynman diagrams. Anatomy of Feynman diagrams. Allowed vertices. General rules Dr. Tina Potter 5. Feynman Diagrams 2 Feynman Diagrams The results of calculations based on a single process in Time-Ordered Perturbation Theory (sometimes called old-fashioned, OFPT) depend on the reference frame. Richard Feynman 1965 Nobel Prize The sum of all time orderings is frame independent and provides the basis for our relativistic theory of Quantum Mechanics. A Feynman diagram represents the sum of all time orderings + = −−!time −−!time −−!time Dr. Tina Potter 5. Feynman Diagrams 3 Feynman Diagrams Each Feynman diagram represents a term in the perturbation theory expansion of the matrix element for an interaction. Normally, a full matrix element contains an infinite number of Feynman diagrams. Total amplitude Mfi = M1 + M2 + M3 + ::: 2 Total rateΓ fi = 2πjM1 + M2 + M3 + :::j ρ(E) Fermi's Golden Rule But each vertex gives a factor of g, so if g is small (i.e. the perturbation is small) only need the first few. (Lowest order = fewest vertices possible) 2 4 g g g 6 p e2 1 Example: QED g = e = 4πα ∼ 0:30, α = 4π ∼ 137 Dr. Tina Potter 5. Feynman Diagrams 4 Feynman Diagrams Perturbation Theory Calculating Matrix Elements from Perturbation Theory from first principles is cumbersome { so we dont usually use it. Need to do time-ordered sums of (on mass shell) particles whose production and decay does not conserve energy and momentum. Feynman Diagrams Represent the maths of Perturbation Theory with Feynman Diagrams in a very simple way (to arbitrary order, if couplings are small enough).
    [Show full text]
  • Waves & Normal Modes
    Waves & Normal Modes Matt Jarvis February 2, 2016 Contents 1 Oscillations 2 1.0.1 Simple Harmonic Motion - revision . 2 2 Normal Modes 5 2.1 Thecoupledpendulum.............................. 6 2.1.1 TheDecouplingMethod......................... 7 2.1.2 The Matrix Method . 10 2.1.3 Initial conditions and examples . 13 2.1.4 Energy of a coupled pendulum . 15 2.2 Unequal Coupled Pendula . 18 2.3 The Horizontal Spring-Mass system . 22 2.3.1 Decouplingmethod............................ 22 2.3.2 The Matrix Method . 23 2.3.3 Energy of the horizontal spring-mass system . 25 2.3.4 Initial Condition . 26 2.4 Vertical spring-mass system . 26 2.4.1 The matrix method . 27 2.5 Interlude: Solving inhomogeneous 2nd order di↵erential equations . 28 2.6 Horizontal spring-mass system with a driving term . 31 2.7 The Forced Coupled Pendulum with a Damping Factor . 33 3 Normal modes II - towards the continuous limit 39 3.1 N-coupled oscillators . 39 3.1.1 Special cases . 40 3.1.2 General case . 42 3.1.3 N verylarge ............................... 44 3.1.4 Longitudinal Oscillations . 47 4WavesI 48 4.1 Thewaveequation ................................ 48 4.1.1 TheStretchedString........................... 48 4.2 d’Alambert’s solution to the wave equation . 50 4.2.1 Interpretation of d’Alambert’s solution . 51 4.2.2 d’Alambert’s solution with boundary conditions . 52 4.3 Solving the wave equation by separation of variables . 54 4.3.1 Negative C . 55 i 1 4.3.2 Positive C . 56 4.3.3 C=0.................................... 56 4.4 Sinusoidalwaves ................................
    [Show full text]
  • Lecture #2: Review of Spin Physics
    Lecture #2: Review of Spin Physics • Topics – Spin – The Nuclear Spin Hamiltonian – Coherences • References – Levitt, Spin Dynamics 1 Nuclear Spins • Protons (as well as electrons and neutrons) possess intrinsic angular momentum called “spin”, which gives rise to a magnetic dipole moment. Plank’s constant 1 µ = γ! 2 spin gyromagnetic ratio • In a magnetic field, the spin precesses around the applied field. z Precession frequency θ µ ω 0 ≡ γB0 B = B0zˆ Note: Some texts use ω0 = -gB0. y ! ! Energy = −µ ⋅ B = − µ Bcosθ x How does the concept of energy differ between classical and quantum physics • Question: What magnetic (and electric?)€ fields influence nuclear spins? 2 The Nuclear Spin Hamiltonian • Hˆ is the sum of different terms representing different physical interactions. ˆ ˆ ˆ ˆ H = H1 + H 2 + H 3 +! Examples: 1) interaction of spin with B0 2) interactions with dipole fields of other nuclei 3) nuclear-electron couplings • In general,€ we can think of an atomic€ nucleus as a lumpy magnet with a (possibly non-uniform) positive electric charge • The spin Hamiltonian contains terms which describe the orientation dependence of the nuclear energy The nuclear magnetic moment interacts with magnetic fields Hˆ = Hˆ elec + Hˆ mag The nuclear electric charge interacts with electric fields 3 € Electromagnetic Interactions • Magnetic interactions magnetic moment ! ! ! ! Hˆ mag ˆ B Iˆ B = −µ ⋅ = −γ" ⋅ local magnetic field quadrapole • Electric interactions monopole dipole Nuclear€ electric charge distributions can be expressed as a sum of multipole components. ! (0) ! (1) ! (2) ! C (r ) = C (r ) + C (r ) + C (r ) +" Symmetry properties: C(n)=0 for n>2I and odd interaction terms disappear Hence, for spin-½ nuclei there are no electrical ˆ elec H = 0 (for spin I€= 1/2) energy terms that depend on orientation or internal nuclear structure, and they behaves exactly like ˆ elec H ≠ 0 (for spin I >1/2) point charges! Nuclei with spin > ½ have electrical quadrupolar moments.
    [Show full text]
  • 11. the Top Quark and the Higgs Mechanism Particle and Nuclear Physics
    11. The Top Quark and the Higgs Mechanism Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 11. The Top Quark and the Higgs Mechanism 1 In this section... Focus on the most recent discoveries of fundamental particles The top quark { prediction & discovery The Higgs mechanism The Higgs discovery Dr. Tina Potter 11. The Top Quark and the Higgs Mechanism 2 Third Generation Quark Weak CC Decays 173 GeV t Cabibbo Allowed |Vtb|~1, log(mass) |Vcs|~|Vud|~0.975 Top quarks are special. Cabibbo Suppressed m(t) m(b)(> m(W )) |Vcd|~|Vus|~0.22 |V |~|V |~0.05 −25 cb ts τt ∼ 10 s ) decays b 4.8GeV before hadronisation Vtb ∼ 1 ) 1.3GeV c BR(t ! W + b)=100% s 95 MeV 2.3MeV u d 4.8 MeV Bottom quarks are also special. b quarks can only decay via the Cabbibo suppressed Wcb vertex. Vcb is very small { weak coupling! Interaction b-jet point ) τ(b) τ(u; c; d; s) b Jet initiated by b quarks look different to other jets. b quarks travel further from interaction point before decaying. b-jet traces back to a secondary vertex {\ b-tagging". Dr. Tina Potter 11. The Top Quark and the Higgs Mechanism 3 The Top Quark The Standard Model predicted the existence of the top quark 2 +3e u c t 1 −3e d s b which is required to explain a number of observations. d µ− Example: Non-observation of the decay W − 0 + − 0 + − −9 K ! µ µ B(K ! µ µ ) < 10 u=c=t νµ The top quark cancels the contributions W + from the u and c quarks.
    [Show full text]
  • Modified Gravity Theories with Non-Minimal Coupling Between Curvature and Matter Engenharia Fısica Tecnol ´Ogica
    Modified Gravity Theories with non-minimal coupling between curvature and matter Energy Conditions and Dolgov-Kawasaki Criterion Miguel Pereira Carvalho Alves de Sequeira Dissertac¸ao˜ para a Obtenc¸ao˜ do Grau de Mestre em Engenharia F´ısicaTecnologica´ J ´uri Presidente: Prof. Joao˜ Carlos Carvalho de Sa´ Seixas Orientador: Prof. Orfeu Bertolami Vogais: Prof. Alfredo Barbosa Henriques e Doutor Jorge Paramos´ May, 2009 ii Acknowledgements I would like to thank: • my supervisor, Orfeu Bertolami, not only for the guidance and support, but also for providing the opportunity to work at the edge of a compelling field such as General Relativity; • my parents for lecturing me to look ahead and for the support along these 17 years of school; • my friends for making the decompressing valve work properly. iii iv Resumo Nesta tese ´eabordado uma extens~ao`asteorias f(R) da gravita¸c~aoonde, para al´emda modifica¸c~ao no sector grav´ıticoda ac¸c~aode Einstein-Hilbert, ´eintroduzido um acoplamento n~aom´ınimo entre a curvatura e a mat´eria. Este modelo cont´emuma fenomenologia nova e interessante, principalmente no que diz respeito `atroca de energia entre os campos de mat´eria e o escalar de curvatura. Contudo, como todas as teorias de gravita¸c~aomodificada, estas s~aopass´ıveis de certas anomalias que podem, eventualmente, tornar a teoria inv´alidaou desprovida de significado f´ısico.Neste contexto, apresenta-se um estudo acerca das condi¸c~oesde energia, fundamentais para restringir os poss´ıveis tensores energia- momento, e do crit´eriode Dolgov-Kawasaki, fundamental para assegurar a estabilidade da teoria.
    [Show full text]
  • Coupling of Angular Momenta Isospin Nucleon-Nucleon Interaction
    LectureLecture 55 CouplingCoupling ofof AngularAngular MomentaMomenta IsospinIsospin NucleonNucleon --NucleonNucleon InteractionInteraction WS2012/13 : ‚Introduction to Nuclear and Particle Physics ‘, Part I I.I. AngularAngular MomentumMomentum OperatorOperator Rotation R( θθθ): in polar coordinates the point r = (r, φφφ) Polar coordinates: is transformed to r' = (r,φφφ + θθθ) by a rotation through the angle θθθ around the z-axis Define the rotated wave function as: Its value at r is determined by the value of φφφ at that point, which is transformed to r by the rotation: The shift in φφφ can also be expressed by a Taylor expansion Rotation (in exponential representation): where the operator Jz for infinitesimal rotations is the angular-momentum operator: AngularAngular MomentaMomenta Cartesian coordinates (x,y,z) Finite rotations about any of these axes may be written in the exponential representation as The cartesian form of the angular-momentum operator Jk reads ∂ℜ∂ℜ∂ℜ (θθθ ) Jˆ === −−−ih k k θθθk ===0 ∂∂∂θθθk With commutation relations (SU(2) algebra): 2 [Jk ,J ] === 0 Consider representation using the basis |jm> diagonal in both with eigenvalue ΛΛΛj =j(j+1) . CouplingCoupling ofof AngularAngular MomentaMomenta A system of two particles with angular momenta has the total angular momentum ˆ ˆ Eigenfunctions of (and J z1 and J z2 ) are: |j 1,m 1> and |j 2,m 2 > (1) A basis for the system of two particles may then be built out of the products of these states, forming the so-called uncoupled basis states (2) Such a state is an eigenstate
    [Show full text]
  • 15.8 Magnetic Interactions and Magnetic Couplings
    8. Magnetic Interactions and Magnetic Couplings. Transitions between the magnetic energy levels discussed in the previous section can be visualized as occuring through the result of magnetic torques exerted on the magnetic moment vectors of an electron spin, or equivalently, as the result of coupling of spin angular momentum to another angular momentum. We shall discuss the magnetic torques as arising from two fundamental magnetic interactions: a dipolar interaction between magnetic moments and a contact interaction between magnetic moments. These arise from some specific source of magnetic moments resulting from the motion of charged particles or spins. The coupling may occur through either a dipolar or contact interaction. In the literature the terms interaction and coupling are used more or less interchangeably. However, we use the term coupling to describe the source of the magnetic moment that causes the spin transition. The vector model allows the dipolar and contact interactions to be visualized in an analogous manner, irrespective of the source of the coupling. In addition, the vector model allows the transitions resulting from the coupling of an electron spin to any magnetic moment to be visualized in an analogous form. These features of the vector model provide a powerful and simplifying tool for discussing magnetic resonance spectroscopy and intersystem crossing through a common conceptual framework. Couplings of a magnetic field along the z axis. For example, let us consider the specific example of the intersystem crossing of a T0 state to a degernate S state or vice-versa (Figure 18, top). On the left and right of the figure the two spins are represented as "tightly" coupled to each other by electron exchange by showing their precessional cones in contactat a common point.
    [Show full text]