By Walter R. Johnson

ISBN-10: 3540680101

ISBN-13: 9783540680109

This e-book offers a hands-on adventure with atomic constitution calculations. fabric coated contains angular momentum equipment, the valuable box Schr?dinger and Dirac equations, Hartree-Fock and Dirac-Hartree-Fock equations, multiplet constitution, hyperfine constitution, the isotope shift, dipole and multipole transitions, easy many-body perturbation concept, configuration interplay, and correlation corrections to matrix parts. The booklet additionally comprises numerical tools for fixing the Schr?dinger and Dirac eigenvalue difficulties and the (Dirac)-Hartree-Fock equations.

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Extra info for Atomic Structure Theory: Lectures on Atomic Physics (With 21 Figures and 45 Tables)

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5 Potential Models The potential experienced by a bound atomic electron near r = 0 is dominated by the nuclear Coulomb potential, so we expect Z V (r) ≈ − , r for small r. 5 Potential Models 49 with ζ = Z − N + 1 for an N electron atom with nuclear charge Z. The transition from a nuclear potential to an ionic potential is predicted by the ThomasFermi model, for example. However, it is possible to simply approximate the potential in the intermediate region by a smooth function of r depending on several parameters that interpolates between the two extremes.

Dr Z From this equation, it follows that N (r), the number of electrons inside a sphere of radius r, is given by r d2 φ(r) dr dr2 0 r dφ = r −φ dr 0 dφ =r − φ(r) + 1 . 115) Evaluating this expression at r = R, we obtain the normalization condition 54 2 Central-Field Schr¨ odinger Equation X dφ dx =− X Z −N . 116) An iterative scheme is set up to solve the Thomas-Fermi differential equation. First, two initial values of X are guessed: X = Xa and X = Xb . 111) is integrated inward to r = 0 twice: the first time starting at x = Xa , using initial conditions φ(Xa ) = 0, dφ/dx(Xa ) = −(Z − N )/Xa Z, and the second time starting at x = Xb , using initial conditions φ(Xb ) = 0, dφ/dx(Xb ) = −(Z − N )/Xb Z.

We designate the backward difference operator by ∇: ∇f [n] = f [n] − f [n − 1] . 43) Using this notation, (1 − ∇)f[ n] = f [n − 1]. 44) .. or more generally f [n + x] = (1 − ∇)−x f [n]. 43) is to be used iteratively to determine ∇k . 45) is a general interpolation formula for equally spaced points. 45) f [n + x] = = − + − x x(x + 1) 2 x(x + 1)(x + 2) 3 ∇+ ∇ + ∇ + ··· 1! 2! 3! x(x + 1) x(x + 1)(x + 2) 1+x+ + + · · · f [n] 2! 3! 2x(x + 1) 3x(x + 1)(x + 2) + + · · · f [n − 1] x+ 2! 3! x(x + 1) 3x(x + 1)(x + 2) + + · · · f [n − 2] 2!

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Atomic Structure Theory: Lectures on Atomic Physics (With 21 Figures and 45 Tables) by Walter R. Johnson


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