PY3P05 Lecture 8-9: Multi-electron atoms oAlkali atom spectra. oCentral field approximation. oShell model. oEffective potentials and screening. oExperimental evidence for shell model.

PY3P05 Energy levels in alkali metals oAlkali atoms: in ground state, contain a set of completely filled subshells with a single valence electron in the next s subshell. oElectrons in p subshells are not excited in any low-energy processes. s electron is the single optically active electron and core of filled subshells can be ignored.

PY3P05 Energy levels in alkali metals oIn alkali atoms, the l degeneracy is lifted: states with the same principal quantum number n and different orbital quantum number l have different energies. oRelative to H-atom terms, alkali terms lie at lower energies. This shift increases the smaller l is. oFor larger values of n, i.e., greater orbital radii, the terms are only slightly different from hydrogen. oAlso, electrons with small l are more strongly bound and their terms lie at lower energies. oThese effects become stronger with increasing Z. oNon-Coulombic potential breaks degeneracy of levels with the same principal quantum number.

PY3P05 Hartree theory oFor multi-electron atom, must consider Coulomb interactions between its Z electrons and its nucleus of charge +Ze. Largest effects due to large nuclear charge. oMust also consider Coulomb interactions between each electron and all other electrons in atom. Effect is weak. oAssume electrons are moving independently in a spherically symmetric net potential. oThe net potential is the sum of the spherically symmetric attractive Coulomb potential due to the nucleus and a spherically symmetric repulsive Coulomb potential which represents the average effect of the electrons and its Z - 1 colleagues. oHartree (1928) attempted to solve the time-independent Schr ö dinger equation for Z electrons in a net potential. oTotal potential of the atom can be written as the sum of a set of Z identical net potentials V( r), each depending on r of the electron only.

PY3P05Screening oHartree theory results in a shell model of atomic structure, which includes the concept of screening. oFor example, alkali atom can be modelled as having a valence electron at a large distance from nucleus. oMoves in an electrostatic field of nucleus +Ze which is screened by the (Z-1) inner electrons. This is described by the effective potential V eff ( r ). oAt r small, V eff (r ) ~ -Ze 2 /r oUnscreened nuclear Coulomb potential. oAt r large, V eff (r ) ~ -e 2 /r oNuclear charge is screened to one unit of charge. +Ze -(Z-1)e r -e

PY3P05 Central field approximation oThe Hamiltonian for an N-electron atom with nuclear charge +Ze can be written: where N = Z for a neutral atom. First summation accounts for kinetic energy of electrons, second their Coulomb interaction with the nuclues, third accounts for electron-electron repulsion. oNot possible to find exact solution to Schrodinger equation using this Hamiltonian. oMust use the central field approximation in which we write the Hamiltonian as: where V central is the central field and V residual is the residual electrostatic interaction.

PY3P05 Central field approximation oThe central field approximation work in the limit where oIn this case, V residual can be treated as a perturbation and solved later. oBy writing we end up with N separate Schrödinger equations: with E = E 1 + E 2 + … + E N oNormally solved numerically, but analytic solutions can be found using the separation of variables technique.

PY3P05 Central field approximation oAs potentials only depend on radial coordinate, can use separation of variables: where R i (r i ) are a set of radial wave functions and Y i (  i,  i ) are a set of spherical harmonic functions. oFollowing the same procedure as Lectures 3-4, we end up with three equations, one for each polar coordinate. oEach electron will therefore have four quantum numbers: ol and m l : result from angular equations. on: arises from solving radial equation. n and l determine the radial wave function R nl (r ) and the energy of the electron. om s : Electron can either have spin up (m s = +1/2) or down (m s = -1/2). oState of multi-electron atom is then found by working out the wave functions of the individual electrons and then finding the total energy of the atom (E = E 1 + E 2 + … + E N ).

PY3P05 Shell model oHartree theory predicts shell model structure, which only considers gross structure: 1.States are specified by four quantum numbers, n, l, m l, and m s. 2.Gross structure of spectrum is determined by n and l. 3.Each (n,l) term of the gross structure contains 2(2l + 1) degenerate levels. oShell model assumes that we can order energies of gross terms in a multielectron atom according to n and l. As electrons are added, electrons fill up the lowest available shell first. oExperimental evidence for shell model proves that central approximation is appropriate.

PY3P05 Shell model oPeriodic table can be built up using this shell-filling process. Electronic configuration of first 11 elements is listed below: oMust apply 1.Pauli exclusion principle: Only two electrons with opposite spin can occupy an atomic orbital. i.e., no two electrons have the same 4 quantum numbers. 2.Hunds rule: Electrons fill each orbital in the subshell before pairing up with opposite spins.

PY3P05 Shell model oBelow are atomic shells listed in order of increasing energy. N shell = 2(2l + 1) is the number of electrons that can fill a shell due to the degeneracy of the m l and m s levels. N accum is the accumulated number of electrons that can be held by atom. oNote, 19th electron occupies 4s shell rather than 3d shell. Same for 37th. Happens because energy of shell with large l may be higher than shell with higher n and lower l.

PY3P05 Shell model o4s level has lower energy than 3d level due to penetration. oElectron in 3s orbital has a probability of being found close to nucleus. Therefore experiences unshielded potential of nucleus and is more tightly bound.

PY3P05 Shell model Radial probabilities for 4s 3d 4s - red 3d - blue Note: Movies from

PY3P05 Shell model Radial probabilities for 1s 2s 3s 1s - red 2s - blue 3s - green

PY3P05 Shell model Radial probabilities for 3s 3p 3d 3s - red 3p - blue 3d - green

PY3P05 Quantum defect oAlkali are approximately one-electron atoms: filled inner shells and one valence electron. oConsider sodium atom: 1s 2 2s 2 2p 6 3s 1. oOptical spectra are determined by outermost 3s electron. The energy of each (n, l) term of the valence electron is where  (l) is the quantum defect - allows for penetration of the inner shells by the valence electron. oShaded region in figure near r = 0 represent the inner n = 1 and n = 2 shells. 3s and 3p penetrate the inner shells. oMuch larger penetration for 3s => electron sees large nuclear potential => lower energy.

PY3P05 Quantum defect o  (l) depends mainly on l. Values for sodium are shown at right. oCan therefore estimate wavelength of a transition via oFor sodium the D lines are 3p  3s transitions. Using values for  (l) from table, => = 589 nm

PY3P05 Shell model justification oConsider sodium, which has 11 electrons. oNucleus has a charge of +11e with 11 electrons orbiting about it. oFrom Bohr model, radii and energies of the electrons in their shells are oFirst two electrons occupy n =1 shell. These electrons see full charge of +11e. => r 1 = 1 2 /11 a 0 = 0.05 Å and E 1 = x 11 2 /1 2 ~ eV. oNext two electrons experience screened potential by two electrons in n = 1 shell. Z eff =+9e => r 2 = 2 2 /9 a 0 = 0.24 Å and E 2 = x 9 2 /2 2 = -275 eV. and

PY3P05 Experimental evidence for shell model oIonisation potentials and atomic radii: oIonisation potentials of noble gas elements are highest within a particular period of periodic table, while those of the alkali are lowest. oIonisation potential gradually increases until shell is filled and then drops. oFilled shells are most stable and valence electrons occupy larger, less tightly bound orbits. oNoble gas atoms require large amount of energy to liberate their outermost electrons, whereas outer shell electrons of alkali metals can be easily liberated.

PY3P05 oX-ray spectra: oEnables energies of inner shells to be determined. oAccelerated electrons used to eject core electrons from inner shells. X-ray photon emitted by electrons from higher shell filling lower shell. oK-shell (n = 1), L-shell (n = 2), etc. oEmission lines are caused by radiative transitions after the electron beam ejects an inner shell electron. oHigher electron energies excite inner shell transitions. Experimental evidence for shell model Wavelength (A) 40 keV 80 keV

PY3P05 Experimental evidence for shell model oWavelength of various series of emission lines are found to obey Moseley’s law. oFor example, the K-shell lines are given by where  accounts for the screening effect of other electrons. oSimilarly, the L-shell spectra obey: oSame wavelength as predicted by Bohr, but now have and effective charge (Z -  ) instead of Z. o  L ~ 10 and  K ~ 3.

PY3P05 Bohr model including screening oAssume net charge is ( Z - 1 )e. oTherefore, the potential energy is oTotal energy of orbit is oModified Bohr formula taking into account screening. oCan therefore easily show that

PY3P05 Shell model summary oElectrons in orbitals with large principal quantum numbers (n) will be shielded from the nucleus by inner- shell electrons. Z eff = Z -  nl. o  nl increases with n => Z eff decreases with n. o  nl increases with l => Z eff decreases with l.

PY3P05 Shell model summary oIn hydrogenic one-electron model, the energy levels of a given n are degenerate in l: oNot the case in multi-electron atoms. Orbitals with the same n quantum number have different energies for differing values of l. oAs Z eff = Z -  nl is a function of n and l, the l degeneracy is broken by modified potential. 3s3p3d 3s3p3d

PY3P05 Shell model summary oWave functions of electrons with different l are found to have different amount of penetration into the region occupied by the 1s electrons. oThis penetration of the shielding 1s electrons exposes them to more of the influence of the nucleus and causes them to be more tightly bound, lowering their associated energy states.

PY3P05 Shell model summary oIn the case of Li, the 2s electron shows more penetration inside the first Bohr radius and is therefore lower than the 2p. oIn the case of Na with two filled shells, the 3s electron penetrates the inner shielding shells more than the 3p and is significantly lower in energy.