Lectures 8-9: Multi-electron atoms Energy levels in alkali metals o Alkali atoms: in ground state, contain a set of Z - 1 completely filled subshells with a single valence electron in the next s subshell. o Alkali atom spectra o Electrons 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. o Central field approximation o Shell model o Effective potentials and screening o Experimental evidence for shell model PY3004 PY3004 Energy levels in alkali metals Hartree theory o In alkali atoms, the l degeneracy is lifted: states with the same principal quantum number n o For multi-electron atom, must consider Coulomb interactions between its Z electrons and its and different orbital quantum number l have different energies. nucleus of charge +Ze. Largest effects due to large nuclear charge. o Relative to H atom, alkali terms lie at lower energies due to increased Coulomb attraction of o Must also consider Coulomb interactions between each electron and all other electrons in nucleus. This shift increases the smaller l is. atom. Effect is weak. o For larger values of n, i.e., greater orbital radii, the terms are only slightly different from o Assume electrons are moving independently in a spherically symmetric net potential. hydrogen. o The net potential is the sum of the spherically symmetric attractive Coulomb potential due to o Also, electrons with small l are more strongly bound and their terms lie at lower energies. the nucleus and a spherically symmetric repulsive Coulomb potential which represents the average effect of the electrons and its Z - 1 colleagues. o These effects become stronger with increasing Z. o Hartree (1928) attempted to solve the time-independent Schrödinger equation for Z electrons o Non-Coulombic potential breaks degeneracy of levels with the same principal quantum in a net potential. number. o Total 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. PY3004 PY3004 Screening Central field approximation o Hartree theory results in a shell model of atomic structure, -e which includes the concept of screening. o The Hamiltonian for an N-electron atom with nuclear charge +Ze can be written: r o For example, alkali atom can be modelled as having a valence +Ze electron at a large distance from nucleus. -(Z-1)e o Moves in an electrostatic field of nucleus +Ze which is where N = Z for a neutral atom. First summation accounts for kinetic energy of electrons , screened by the (Z-1) inner electrons. This is described by the second their Coulomb interaction with the nuclues, third accounts for electron-electron effective potential Veff( r ). repulsion. o At r small, o Not possible to find exact solution to Schrodinger equation using this Hamiltonian. o Unscreened nuclear Coulomb potential. o Must use the central field approximation in which we write the Hamiltonian as: o At r large, o Nuclear charge is screened to one unit of charge. where Vcentral is the central field and Vresidual is the residual electrostatic interaction. PY3004 PY3004 Central field approximation Central field approximation o As potentials only depend on radial coordinate, can use separation of variables: o The central field approximation work in the limit where where Ri(ri) are a set of radial wave functions and Yi(!i, "i ) are a set of spherical harmonic functions. o In this case, Vresidual can be treated as a perturbation and solved later. o Following the same procedure as Lectures 3-4, we end up with three equations, one for each " =" (rˆ )" (rˆ ) " (rˆ ) polar coordinate. o By writing 1 1 2 2 K N N we end up with N separate Schrödinger equations: o Each electron will therefore have four quantum numbers: ! o l and ml: result from angular equations. with E = E + E + … + E o n: arises from solving radial equation. n and l determine the radial wave function Rnl(r ) 1 2 N and the energy of the electron. o m : Electron can either have sip up (m = +1/2) or down (m = -1/2). o Normally solved numerically, but analytic solutions can be found using the separation of s s s variables technique. o State 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 = E1 + E2 + … + EN). PY3004 PY3004 Shell model Shell model o Periodic table can be built up using this shell-filling process. Electronic configuration of first o Hartree theory predicts shell model structure, which only considers gross structure: 11 elements is listed below: 1. States are specified by four quantum numbers, n, l, ml and ms. 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. o Shell 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. o Experimental evidence for shell model proves that central approximation is appropriate. o Must apply 1. Pauli exclusion principle: Only two electrons with opposite spin can occupy an atomic orbital. i.e., no two electrons have the same four quantum numbers. 2. Hunds rule: Electrons fill each orbital in the subshell before pairing up with opposite spins. PY3004 PY3004 Shell model Shell model o Atomic shells listed in order of increasing energy. Nshell = 2(2l + 1) is the number of electrons o 4s level has lower energy than 3d level due to penetration. that can fill a shell due to the degeneracy of the ml and ms levels. Naccum is the accumulated number of electrons that can be held by atom. o Electron in 3s orbital has a probability of being found close to nucleus. Therefore experiences unshielded potential of nucleus and is more tightly bound. 3d 4s o Note, 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. PY3004 PY3004 Shell model Shell model Radial probabilities for 1s 2s 3s Radial probabilities for 3s 3p 3d 3s - red 1s - red 3p - blue 2s - blue 3d - green 3s - green PY3004 PY3004 Quantum defect Quantum defect o Alkalis are approximately one-electron atoms: filled o #(l) depends mainly on l. Values for sodium are inner shells and one valence electron. shown at right. 2 2 6 1 o Consider sodium atom: 1s 2s 2p 3s . o Can therefore estimate wavelength of a transition via o Optical spectra are determined by outermost 3s electron. The energy of each (n, l) term of the valence electron is o For sodium the D lines are 3p ! 3s transitions. Using values for #(l) from table, where #(l) is the quantum defect - allows for penetration of the inner shells by the valence electron. o Shaded region in figure near r = 0 represent the inner n = 1 and n = 2 shells. 3s and 3p penetrate the inner shells. o Much larger penetration for 3s => electron sees large => $ = 589 nm nuclear potential => lower energy. PY3004 PY3004 Shell model justification Experimental evidence for shell model o Consider sodium, which has 11 electrons. o Ionisation potentials and atomic radii: o Nucleus has a charge of +11e with 11 electrons o Ionisation potentials of noble gas elements are orbiting about it. highest within a particular period of periodic table, while those of the alkali are lowest. o From Bohr model, radii and energies of the electrons o Ionisation potential gradually increases until in their shells are n 2 13.6Z 2 shell is filled and then drops. r = a and E = " n Z 0 n n 2 o Filled shells are most stable and valence o First two electrons occupy n =1 shell. These electrons electrons occupy larger, less tightly bound orbits. 2 see full charge of +11e. => r1 = 1 /11 a0 = 0.05 Å and ! 2 2 ! E1 = -13.6 x 11 /1 ~ -1650 eV. o Noble gas atoms require large amount of energy to liberate their outermost electrons, whereas outer shell electrons of alkali metals can be o Next two electrons experience screened potential by easily liberated. 2 two electrons in n = 1 shell. Zeff =+9e => r2 = 2 /9 a0 2 2 = 0.24 Å and E2 = -13.6 x 9 /2 = -275 eV. PY3004 PY3004 Experimental evidence Experimental evidence for shell model for shell model o X-ray spectra: o Wavelength of various series of emission lines are found to obey Moseley’s law. o Enables energies of inner shells to be o For example, the K-shell lines are given by determined. o Accelerated electrons used to eject core electrons 80 keV where % accounts for the screening effect of other electrons. from inner shells. X-ray photon emitted by electrons from higher shell filling lower shell. 40 keV o Similarly, the L-shell spectra obey: o K-shell (n = 1), L-shell (n = 2), etc. Wavelength (A) o Emission lines are caused by radiative transitions after the electron beam ejects an inner shell o Same wavelength as predicted by Bohr, but now have electron. Continuum due to bremsstrahlung. and effective charge (Z - %) instead of Z. o Higher electron energies excite inner shell o %L ~ 10 and %K ~ 3. transitions. PY3004 PY3004 Experimental evidence for shell model o Assume net charge is ( Z - 1 )e.