MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.5.80 Lecture #5 Fall, 2008 Page 1 Lecture #5: Alkali and Many e– Atomic Spectra What am I doing spending all of this time on ATOMS? One of the goals of spectroscopy is to recover what is unique about the system by using what isuniversal and general as a road map. Form and flavor of electronic structure models. * patterns for assignment of spectra* predictions of unobserved states and properties* ways to estimate size and shape of orbitals* quantitative reconstruction of V(r) from spectrum* zero-order pictures for describing dynamics. 0 ( ) time evolving non-eigenstateoΨ (Q, t)= ∑ ai(t)ψi Qexpressed in terms of basis states, ψi i ψi ↔ ∑αiψi0 eigenstate expressed in terms of basis states, ψio Construct interesting Ψ(Q, 0) and predict its time evolution. How do we know n, , and Z for 1e– spectrum? pattern * convergence ℜZ2/n2 * fine structure Z4/[n3 ( + 1)( + 1/2)]* hyperfine?* selection rules for electric dipole transition redundancy * exactly repeated intervals in two series Alkalis SCF → mathematical definition of 1e– orbitals Scaling generalized ⎫ interpretive Z → Zn ⎪eff ⎪ intuitive n → n* = n − δ ⎬⎪ diagnostic ⎪⎭ systematic probe core, valence, Rydbergcore-penetrating, non-penetrating Ontogeny recapitulates phylogeny [Mulliken] What happens when you throw an e– at a closed shell ion? - intuition - Quantum mechanical wavepacket calculation.5.80 Lecture #5 Fall, 2008 Page 2 V (r) = − Zeff (r)e2 + 2( + 1) -dependent effective potentialr r2 # of radial nodes ? 2s vs 2p?spacing of radial nodes? λ = h/p p(r) = [(E–V(r))2µ]1/2 (+1)effect of r2 on En, on rn? V(r) r ≠ 0 1e2/r Zeff(r)e2 r Real curve is more attractive than Z = 1 curve  = 0 Ze2/r V=0(r)/eV = – 14.4Zr/Å5.80 Lecture #5 ask for these and other effects. Why? Shell model Fall, 2008 Page 3 En’s are lowered for n? for n*? nodes are closer together (same node count?)inner part of Rn becomes more compact not a simpler/Z scaling of ψ2s 2p1s break 1e– atom degeneracy 3s 3p3d outer part of Rn same as Z = 1 at that value of E that corresponds to n* valence region K shell L shell z Zeff(r) ZCORE core valence Rydberg r (filled) low-n* orbitals are exclusively inside corevalence orbital penetrates inside core → HOAO → n*0 penetrating low- δ > 0 Rydberg orbitals non-penetrating high- see ZCORE = integer δ = 0 δ=0 > δp > δd > δf ≈ 05.80 Lecture #5 Fall, 2008 Page 4 eff Zn core orbitals - XPS spectrumcore part of valence orbitalsproperties like spin-orbit core part of penetrating Rydberg orbitals and hyperfine Zeff > ZCORE and δ ≠ 0 and independent of n Z = ZCORE * outer part of penetrating orbitals n* = n – δ * outer part of valence orbitalsOR * all of non-penetrating orbitals ZION(integer) ontogeny recapitulates phylogeny  Rydberg Series * Replicated inner lobes* n*–3/2 amplitude scale factor e– ↔ core energy exchange e– scattered off core πδ phase shift (with respect to H+ + e–)inter-channel interactions, due to 1/rij, with core excited states simple picture follows in order to understand δ systematics and ψn* recapitulation. Eigenstatesin this energyregion V(r) asymptotic limit core levels very far apart andnot following constant-ZRydberg equation “Rydberg” levels closetogether, converging, andscalable with constant Z What do we know about this kind of potential? Are all eigenfunctions pictorially related?How do E levels tell us about form of V(r)?5.80 Lecture #5 Fall, 2008 Page 5 Why is δ n-independent? Boundary condition: Rn(r) → 0 at r → ∞ (so phase at outer turning point, r>, must be just right toprevent blow-up at r → ∞) Outside “core”, Zn (r) have identical asymptotic form (except for theeff (r) → ZCORE (integer) thus all Rnpossibility of a phase shift), because the V(r) → –ZCORE/r. Inside “core” — all of the extra phase accumulates because Zn All ψn* in a specificeff (r) > ZCORE. Rydberg series (channel) exit core with same phase. Must splice (universal) Coulomb long rangewavefunction onto (-specific) core wavefunction. small rangeof KE enormous average KE n = 3 n = 2 r> n = 1 V(r) core: inside n = 1 regionKE is enormous KE does not vary significantly with n*de Broglie λ’s (nodal structure) insidethe core are independent of n*along each series. e – exits core with same phase,independent of n*. amplitude inside core ~ n*–3/2 period ∝ n–3 SEMI-CLASSICAL Harmonic oscillator period is T = ν1 = 2π ω Quantum Mechanical period is ⎡En+1 − En−1 ⎤−1 → n *−3 ⎣⎢ 2h ⎦⎥ ∆t inside core independent of n* ∆t T ∝ n–3 probability inside core (amplitude inside core ∝ n–3/2)5.80 Lecture #5 Fall, 2008 Page 6 Quantum Defect Theory throw e– at M+ ion set of δ’s tells us about Zeff(r) from extra phase accumulated inside core region. different ’s tell us different depths of penetration — partial wave analysis. [complementary inside-core information from spin-orbit and hfs ↔ Zeff] What if e– hits a core e– and scatters it out? Perturbation of Rydberg series member n2L by a core excited state? Usually it costs too much topromote a core e–. Except for 3d104s1 (Cu, Ag, Au) (but not for a Rydberg series converging to anelectronically excited state of the ion.) Doubly excited states — Rydberg series built on a core hole. e.g. Na [1s2 2s2 2p5]3s n* 2,4 + 1  are the possible states 2P CORE  – 1 Spectrum gets very complicated at high E! Autoionization: eject e– matrix elements of 1/rij between ionization continuation and doubly excited state. Crucial differences between hydrogenic and alkali-like spectra * loss of degeneracy between n2Lj=+1/2 and n2(L + 1)j(different shielding/core-penetration of s,p, d, f…) * loss of simple analytic f(quantum numbers) for all radial properties. Retain empirically correctedscaling relationships. Retain ability to estimate sizes.(n-independence of δ’s means that a quantitative theory exists) * possibility of core-excited states (core no longer 1S closed shell)possibility of core e– ↔valence e– energy transfer “autoionization”,