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Lecture 8-9: Multi-electron atoms

Lecture 8-9: Multi-electron atoms. Alkali atom spectra. Central field approximation. Shell model. Effective potentials and screening. Experimental evidence for shell model. Energy levels in alkali metals.

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Lecture 8-9: Multi-electron atoms

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  1. Lecture 8-9: Multi-electron atoms • Alkali atom spectra. • Central field approximation. • Shell model. • Effective potentials and screening. • Experimental evidence for shell model. PY3P05

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

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

  4. Hartree theory • For multi-electron atom, must consider Coulomb interactions between its Z electrons and its nucleus of charge +Ze. Largest effects due to large nuclear charge. • Must also consider Coulomb interactions between each electron and all other electrons in atom. Effect is weak. • Assume electrons are moving independently in a spherically symmetric net potential. • The 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. • Hartree (1928) attempted to solve the time-independent Schrödinger equation for Z electrons in a net potential. • 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. PY3P05

  5. Screening • Hartree theory results in a shell model of atomic structure, which includes the concept of screening. • For example, alkali atom can be modelled as having a valence electron at a large distance from nucleus. • Moves in an electrostatic field of nucleus +Ze which is screened by the (Z-1)inner electrons. This is described by the effective potential Veff( r ). • At r small, Veff(r ) ~ -Ze2/r • Unscreened nuclear Coulomb potential. • At r large, Veff(r ) ~ -e2/r • Nuclear charge is screened to one unit of charge. -e r +Ze -(Z-1)e PY3P05

  6. Central field approximation • The 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. • Not possible to find exact solution to Schrodinger equation using this Hamiltonian. • Must use the central field approximation in which we write the Hamiltonian as: where Vcentralis the central field and Vresidual is the residual electrostatic interaction. PY3P05

  7. Central field approximation • The central field approximation work in the limit where • In this case, Vresidual can be treated as a perturbation and solved later. • By writing we end up with N separate Schrödinger equations: with E = E1 + E2 + … + EN • Normally solved numerically, but analytic solutions can be found using the separation of variables technique. PY3P05

  8. Central field approximation • As potentials only depend on radial coordinate, can use separation of variables: where Ri(ri) are a set of radial wave functions and Yi(i, i ) are a set of spherical harmonic functions. • Following the same procedure as Lectures 3-4, we end up with three equations, one for each polar coordinate. • Each electron will therefore have four quantum numbers: • l and ml: result from angular equations. • n: arises from solving radial equation. n and l determine the radial wave function Rnl(r ) and the energy of the electron. • ms: Electron can either have spin up (ms= +1/2)or down (ms= -1/2). • 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). PY3P05

  9. Shell model • Hartree theory predicts shell model structure, which only considers gross structure: • States are specified by four quantum numbers, n, l, ml, and ms. • Gross structure of spectrum is determined by n and l. • Each (n,l) term of the gross structure contains 2(2l + 1) degenerate levels. • 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. • Experimental evidence for shell model proves that central approximation is appropriate. PY3P05

  10. Shell model • Periodic table can be built up using this shell-filling process. Electronic configuration of first 11 elements is listed below: • Must apply • 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. • Hunds rule: Electrons fill each orbital in the subshell before pairing up with opposite spins. PY3P05

  11. Shell model • Below are atomic shells listed in order of increasing energy. Nshell = 2(2l + 1) is the number of electrons that can fill a shell due to the degeneracy of the mland mslevels. Naccumis the accumulated number of electrons that can be held by atom. • 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. PY3P05

  12. Shell model • 4s level has lower energy than 3d level due to penetration. • Electron in 3s orbital has a probability of being found close to nucleus. Therefore experiences unshielded potential of nucleus and is more tightly bound. PY3P05

  13. Shell model Radial probabilities for 4s 3d 4s - red 3d - blue Note: Movies from http://chemlinks.beloit.edu/Stars/pages/radial.htm PY3P05

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

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

  16. Quantum defect • Alkali are approximately one-electron atoms: filled inner shells and one valence electron. • Consider sodium atom: 1s2 2s2 2p6 3s1. • Optical 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. • Shaded region in figure near r = 0 represent the inner n = 1 and n = 2 shells. 3s and 3p penetrate the inner shells. • Much larger penetration for 3s => electron sees large nuclear potential => lower energy. PY3P05

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

  18. Shell model justification • Consider sodium, which has 11 electrons. • Nucleus has a charge of +11e with 11 electrons orbiting about it. • From Bohr model, radii and energies of the electrons in their shells are • First two electrons occupy n =1 shell. These electrons see full charge of +11e. => r1 = 12/11 a0 = 0.05 Å and E1 = -13.6 x 112/12 ~ -1650 eV. • Next two electrons experience screened potential by two electrons in n = 1 shell. Zeff =+9e => r2 = 22/9 a0 = 0.24 Å and E2 = -13.6 x 92/22 = -275 eV. and PY3P05

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

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

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

  22. Bohr model including screening • Assume net charge is ( Z - 1 )e. • Therefore, the potential energy is • Total energy of orbit is • Modified Bohr formula taking into account screening. • Can therefore easily show that PY3P05

  23. Shell model summary • Electrons in orbitals with large principal quantum numbers (n) will be shielded from the nucleus by inner-shell electrons. Zeff = Z - nl. • nl increases with n => Zeffdecreases with n. • nl increases with l => Zeffdecreases with l. PY3P05

  24. Shell model summary • In hydrogenic one-electron model, the energy levels of a given n are degenerate in l: • Not the case in multi-electron atoms. Orbitals with the same n quantum number have different energies for differing values of l. • As Zeff = Z - nl is a function of n and l, the l degeneracy is broken by modified potential. 3s 3p 3d 3s 3p 3d PY3P05

  25. Shell model summary • Wave functions of electrons with different l are found to have different amount of penetration into the region occupied by the 1s electrons. • This 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

  26. Shell model summary • In the case of Li, the 2s electron shows more penetration inside the first Bohr radius and is therefore lower than the 2p. • In 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. PY3P05

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