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Electrons in Atoms • Atomic Units (A.U.) --> Standard Usage

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Electrons in Atoms • Atomic Units (A.U.) --> Standard Usage Electrons in atoms • Atomic units (a.u.) --> standard usage – electron mass m e unit of mass – elementary charge e unit of charge – length and time such that numerical values of and 4 ⇥ 0 are unity – then atomic unit of length Bohr radius 2 4⇥0 11 a.u. (length) = a0 = 2 5.29177 10− m e me ⇥ × a0 17 – and time a.u. (time) = 2.41888 10− s αc ⇥ × e2 1 – where α = is the fine structure constant 4⇤⇥0c ≈ 137.036 2 – energy unit = Hartree EH = 2 27.2114 eV mea0 ≈ QMPT 540 Hamiltonian in a.u. • Most of atomic physics can be understood on the basis of N 2 N N pi Z 1 1 HN = + + Vmag 2 − ri 2 ri rj i=1 i=1 i=j | | | − | • for most applications V V eff = ζ s mag ⇥ so i i · i i • Relativistic description required for heavier atoms – binding sizable fraction of electron rest mass – binding of lowest s state generates high-momentum components • Sensible calculations up to Kr without Vmag • Shell structure well established QMPT 540 Shell structure • Simulate with N N H0 = H0(i) i=1 2 • with pi Z H0(i)= + U(ri) 2 − ri • even without auxiliary potential ⇒ shells (2 + 1) (2s + 1) – hydrogen-like: ∗ degeneracy Z2 ε = – but n − 2 n 2 does not give correct shell structure (2,10,28... – degeneracy must be lifted – how? QMPT 540 Effect of other electrons in neutral atoms • Consider effect of electrons in closed shells for neutral Na • large distances: nuclear charge screened to 1 • close to the nucleus: electron “sees” all 11 protons 0 • approximately: !5 !10 V(r) [a.u.] Sodium !15 !20 012345 r [a.u.] • lifts H-like degeneracy: ε2s < ε2p ε3s < ε3p < ε3d • “Far away” orbits: still hydrogen-like! 523-14 Example: Na • Fill the lowest shells • Use schematic potential H n⇤m m = ε n⇤m m 0 | s n | s • Ground state: fill lowest orbits according to Pauli 300m , 211 1 , 211 1 ,...,100 1 , 100 1 | s 2 − 2 2 − 2 i⌘ Φ (Na) | 0 i • Excited states? 523-14 Closed-shell atoms • Neon • Ground state Φ (Ne) = 211 1 , 211 1 ,...,100 1 , 100 1 | 0 i | 2 − 2 2 − 2 i • Excited states 1 n` (2p)− = a† a Φ (Ne) | i n` 2p | 0 i • operators: see later • Note the H-like states • Splitting? • Basic shell structure of atoms understood ⇒ IPM 523-14 Periodic table HF F-RPA Σ(2) 523-14 Level sequence (approximately) 523-14 Helium • IPM description is very successful • Closed-shell configuration 1s2 • Reaction more complicated than for Hydrogen • DWIA (distorted wave impulse approximation) N 1 N 2 S = dp Ψ − a Ψ n | p | 0 ⇥ ⇥ agreement with IPM! → 1 Phys. Rev. A8, 2494 (1973) 523-14 Other closed-shell atoms • Spectroscopic factor becomes less than 1 • Neon 2 p removal: S = 0.92 with two fragments each 0.04 • IPM not the whole story: fragmentation of sp strength • Summed strength: like IPM • IPM wave functions still excellent • Example: Argon 3 p S = 0.95 a.u 3 2.0 Argon − • Rest in 3 small fragments 1.6 3p 1.2 0.8 0.4 0.0 Differential cross section (10 0123 p (a.u.) 523-14 Argon spectroscopic factors • s strength also in the continuum: Ar++ + e • note vertical scale • red bars: 3s fragments exhibit substantial fragmentation 8% 523-14 Fragmentation in atoms • ~All the strength remains below (above) the Fermi energy in closed-shell atoms • Fragmentation can be interpreted in terms of mixing between a ΦN α | 0 • and N a a a† Φ β ⇥ ⇤ | 0 • with the same “global” quantum numbers • Example: Argon ground state ΦN = (3s)2(3p)6(2s)2(2p)6(1s)2 | 0 | + 1 N 2 5 2 6 2 • Ar ground state (3p)− = a Φ = (3s) (3p) (2s) (2p) (1s) | 3p | 0 | 1 N 1 6 2 6 2 • excited state (3s)− = a Φ = (3s) (3p) (2s) (2p) (1s) | 3s | 0 | 2 N 1 2 4 2 6 2 • also (3p)− 4s = a a a† Φ = (4s) (3s) (3p) (2s) (2p) (1s) | 3p 3p 4s | 0 | 2 N 1 2 4 2 6 2 • and (3p)− nd = a3pa3pand† Φ0 = (nd) (3s) (3p) (2s) (2p) (1s) | | | 523-14 .
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