Walter R. Johnson Atomic Structure Theory

Transcription

1 Walter R. Johnson Atomic Structure Theory

2 Walter R. Johnson Atomic Structure Theory Lectures on Atomic Physics With 21 Figures and 45 Tables 123

3 Professor Dr. Walter R. Johnson University of Notre Dame Department of Physics Nieuwland Science Hall Notre Dame, IN, USA Library of Congress Control Number: ISBN Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera ready by author Production: LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig Cover: estudio Calamar Steinen SPIN /3100/YL Printedonacid-freepaper

4 To Sally for her assistance, encouragement, and patience with this project.

5 Preface This is a set of lecture notes prepared for a course on Atomic Physics for second-year graduate students, first given at the University of Notre Dame during the spring semester of My aim in this course was to provide opportunities for hands-on practice in the calculation of atomic wave functions and energies. The lectures started with a review of angular momentum theory including the formal rules for manipulating angular momentum operators, a discussion of orbital and spin angular momentum, Clebsch-Gordan coefficients and threej symbols. This material should have been familiar to the students from firstyear Quantum Mechanics. More advanced material on angular momentum needed in atomic structure calculations followed, including an introduction to graphical methods, irreducible tensor operators, spherical spinors and vector spherical harmonics. The lectures on angular momentum were followed by an extended discussion of the central-field Schrödinger equation. The Schrödinger equation was reduced to a radial differential equation and analytical solutions for Coulomb wave functions were obtained. Again, this reduction should have been familiar from first-year Quantum Mechanics. This preliminary material was followed by an introduction to the finite difference methods used to solve the radial Schrödinger equation. A subroutine to find eigenfunctions and eigenvalues of the Schrödinger equation was developed. This routine was used together with parametric potentials to obtain wave functions and energies for alkali atoms. The Thomas-Fermi theory was introduced and used to obtain approximate electron screening potentials. Next, the Dirac equation was considered. The bound-state Dirac equation was reduced to radial form and Dirac-Coulomb wave functions were determined analytically. Numerical solutions to the radial Dirac equation were considered and a subroutine to obtain the eigenvalues and eigenfunctions of the radial Dirac equation was developed. In the third part of the course, many-electron wave functions were considered and the ground state of a two-electron atom was determined variationally. This was followed by a discussion of Slater-determinant wave functions and a

6 VIII Preface derivation of the Hartree-Fock (HF) equations for closed-shell atoms. Numerical methods for solving the HF equations were described. The HF equations for atoms with one electron beyond closed shells were derived and a code was developed to solve the HF equations for the closed-shell case and for the case of a single valence electron. Finally, the Dirac-Fock equations were derived and discussed. The fourth section of the material began with a discussion of secondquantization. This approach was used to study a number of structure problems in first-order perturbation theory, including excited states of two-electron atoms, excited states of atoms with one or two electrons beyond closed shells and particle-hole states. Relativistic fine-structure effects were considered using the no-pair Hamiltonian. A rather complete discussion of the magneticdipole and electric-quadrupole hyperfine structure from the relativistic point of view was given, and nonrelativistic limiting forms were worked out in the Pauli approximation. fortran subroutines to solve the radial Schródinger equation and the Hartree-Fock equations were handed out to be used in connection with weekly homework assignments. Some of these assigned exercises required the student to write or use fortran codes to determine atomic energy levels or wave functions. Other exercises required the student to write maple or mathematica routines to generate formulas for wave functions or matrix elements. Additionally, more standard pencil and paper exercises on Atomic Physics were assigned. The second time that this course was taught, material on electromagnetic transitions was included and a chapter on many-body methods was started; the third time through, additional sections on many-body methods were added. The fourth time that this course was taught, material on hyperfine structure was moved to a separate chapter and a discussion of the isotope shift was included. The chapter on many-body methods was considerably extended and a separate chapter on many-body methods for matrix elements was added. In order to squeeze all of this material into a one semester three credit hour course, it was necessary to skip some of the material on numerical methods. However, I am confident that the entire book could be covered in a four credit hour course. Notre Dame, Indiana, September 2006 Walter R. Johnson

7 Contents 1 Angular Momentum Orbital Angular Momentum - Spherical Harmonics QuantumMechanicsofAngularMomentum Spherical Coordinates - Spherical Harmonics SpinAngularMomentum Spin 1/2andSpinors Infinitesimal Rotations of Vector Fields Spin1andVectors Clebsch-GordanCoefficients Wigner3jSymbols Irreducible Tensor Operators Graphical Representation - Basic rules Spinor and Vector Spherical Harmonics Spherical Spinors Vector Spherical Harmonics Problems Central-Field Schrödinger Equation Radial SchrödingerEquation CoulombWaveFunctions NumericalSolutiontotheRadialEquation Adams Method (adams) Starting the Outward Integration (outsch) Starting the Inward Integration (insch) Eigenvalue Problem (master) Quadrature Rules (rint) PotentialModels ParametricPotentials Thomas-FermiPotential SeparationofVariablesforDiracEquation RadialDiracEquationforaCoulombField... 56

8 X Contents 2.8 NumericalSolutiontoDiracEquation Outward and Inward Integrations (adams, outdir, indir) Eigenvalue Problem for Dirac Equation (master) ExamplesusingParametricPotentials Problems Self-Consistent Fields Two-ElectronSystems HFEquationsforClosed-ShellAtoms NumericalSolutiontotheHFEquations Starting Approximation (hart) Refining the Solution (nrhf) AtomswithOneValenceElectron Dirac-FockEquations Problems Atomic Multiplets Second-Quantization jSymbols Two-ElectronAtoms AtomswithOneorTwoValenceElectrons Particle-Hole Excited States jSymbols Relativity and Fine Structure He-likeIons AtomswithTwoValenceElectrons Particle-Hole States Problems Hyperfine Interaction & Isotope Shift HyperfineStructure AtomswithOneValenceElectron PauliApproximation Isotope Shift NormalandSpecificMassShifts CalculationsoftheSMS AngularDecomposition ApplicationtoOne-ElectronAtoms FieldShift Problems...155

9 Contents XI 6 Radiative Transitions ReviewofClassicalElectromagnetism ElectromagneticPotentials ElectromagneticPlaneWaves QuantizedElectromagneticField Eigenstates of N i InteractionHamiltonian Time-DependentPerturbationTheory TransitionMatrixElements GaugeInvariance Electric-DipoleTransitions Magnetic-Dipole and Electric-Quadrupole Transitions Nonrelativistic Many-Body Amplitudes TheoryofMultipoleTransitions Problems Introduction to MBPT Closed-ShellAtoms AngularMomentumReduction Example:Second-OrderEnergyinHelium B-SplineBasisSets Hartree-FockEquationandB-splines B-splineBasisfortheDiracEquation Application:HeliumCorrelationEnergy AtomswithOneValenceElectron Second-OrderEnergy AngularMomentumDecomposition Quasi-ParticleEquationandBruecknerOrbitals Monovalent Negative Ions Relativistic Calculations BreitInteraction AngularReductionoftheBreitInteraction Coulomb-BreitMany-ElectronHamiltonian Closed-ShellEnergies OneValenceElectron CICalculations Relativistic CI Calculations MBPTforDivalentAtomsandIons Two-ParticleModelSpaces First-OrderPerturbationTheory Second-OrderPerturbationTheory AngularMomentumReduction Problems...234

10 XII Contents 8 MBPT for Matrix Elements Second-OrderCorrections AngularReduction Random-PhaseApproximation GaugeIndependenceoftheRPA TheRPAforHyperfineConstants Third-OrderMatrixElements Matrix Elements of Two-Particle Operators Two-Particle Operators: Closed-Shell Atoms Two-Particle Operators: One Valence Electron Atoms CICalculationsforTwo-ElectronAtoms E1TransitionsinHe Second-OrderMatrixElementsinDivalentAtoms SummaryRemarks Problems Solutions References Index...307