3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fund of Mat Sci: Bonding – Lecture 7ALPHABET SOUPPhotograph of a bowl of alphabet soup removed for copyright reasons.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Homework for Wed Oct 5• Study: 21.5, 21.6• Read: 17.2 (Stern-Gerlach)• Non-graded PS3 will be posted today3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Last time: 1. Hamiltonian in a spherical potential:2. all commute with each other3. We can find common eigenfunctionslabeled with the 3 quantum numbers n, l, m222221ˆ()22eedd LHrVrmr dr dr mr⎛⎞=− + +⎜⎟⎝⎠h2,, andˆˆ ˆzHL L() () (, )nlm nl lmrRrYψϑϕ=r3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Three Quantum Numbers• Principal quantum number n (energy, accidental degeneracy)• Angular momentum quantum number ll = 0,1,…,n-1 (a.k.a. s, p, d… orbitals)• Magnetic quantum number mm = -l,-l+1,…,l-1,l()()22 2 222200 13.6058 eV 1 Ry8neZ Z ZEan n nπε=− =− =−ˆH ↔ˆzL ↔2ˆL ↔3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)221100 3 4R10000.20.41 2 3 4r2R220r2R210r2R221r2R230r2R231r2R232rrrrrrrrr-0.20000.20.40.040.080.120.62rrr6 102 6 10000.10.050.050.152 6 10000000002 6 10R20R21R30R31R320.1448812121616448812121616004 8 12 16r0 4 8 12 16-0.10.20.40.040.08-0.040.040.080.020.040.40.800.40.8Radial functions Rnl(r) and radial distribution functions r2R2nl(r) foratomic hydrogen. The unit of length is aµ = (m/µ) a0, where a0 is thefirst Bohr radius.0.15Figure by MIT OCW.Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)The Full Alphabet SoupCourtesy of David Manthey. Used with permission. Source: http://www.orbitals.com/orb/orbtable.htm3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Orbital levels in hydrogenoid atomsOrbital Energy (kJ / mol)0-82-146-328-13131s4s2s3s4p3p2p4d3d4fHydrogenFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Balmer lines in a hot starCourtesy of the Department of Physics and Astronomy at the University of Tennessee. Used with permission.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Emission and absorption linesCourtesy of the Department of Physics and Astronomy at the University of Tennessee. Used with permission.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)XPS in Condensed MatterDiagram of Moon composition as seen in X-rays removed for copyright reasons.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Composition AnalysisImages of X-Ray element maps of mine waste soil particles removed for copyright reasons.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Orbital levels in multi-electron atomsOrbital Energy (kJ / mol)0-82-146-328-13130-82-146-328-13131s1s4s4s2s2s3s3s4p4p3p3p2p 2p4d4d3d3d4f4fHydrogen Multielectron AtomsFigure by MIT OCW.(I) “Centripetal” repulsion3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)21-1-21 2 3 4 5 6Vcentripetal(r)1010r(m)Veff(r)VCoulomb(r)1018v(r) (J)Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)(II) Screening3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)(III) Screeningˆˆˆ ˆˆˆˆ(2) )(2BzzzBHH LSBHLSBµµ++→+ + ⋅=rh hGoudsmit and UhlenbeckStern-Gerlach ExperimentImage courtesy of Theresa Knott.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Spin• Dirac derived the relativistic extension of Schrödinger’s equation; for a free particle he found two independent solutions for a given energy• There is an operator (spin S) that commutes with the Hamiltonian and that can only have two eigenvalues• In a magnetic field, the spin combines with the angular momentum, and they couple via ˆˆˆ ˆ(2)BHH LSBµ→+ + ⋅rh3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Spin Eigenvalues/Eigenfunctions•Norm (s integer → bosons, half-integer → fermions) • Z-axis projection (electron is a fermion with s=1/2)• Spin-orbital: product of the “space” wavefunction and the “spin” wavefunction()22ˆ1ˆ2spin spinz spi n spinSssSΨ=
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