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o  Alkali atom spectra.

o  Central field approximation.

o  Shell model.

o  Effective potentials and screening.

o  Experimental evidence for shell model.

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o  Alkali atoms: in ground state, contain a set of completely filled subshells with a single valence electron in the next s subshell.

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.

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o  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.

o  Relative to H-atom terms, alkali terms lie at lower energies. This shift increases the smaller l is.

o  For larger values of n, i.e., greater orbital radii, the terms are only slightly different from hydrogen.

o  Also, electrons with small l are more strongly bound and their terms lie at lower energies.

o  These effects become stronger with increasing Z.

o  Non-Coulombic potential breaks degeneracy of levels with the same principal quantum number.

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o  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.

o  Must also consider Coulomb interactions between each electron and all other electrons in atom. Effect is weak.

o  Assume electrons are moving independently in a spherically symmetric net potential.

o  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.

o  Hartree (1928) attempted to solve the time-independent Schrödinger equation for Z electrons in a net potential.

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.

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o  Hartree theory results in a shell model of atomic structure, which includes the concept of screening.

o  For example, alkali atom can be modelled as having a valence electron at a large distance from nucleus.

o  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 ).

o  At r small, Veff(r ) ~ -Ze2/r

o  Unscreened nuclear Coulomb potential.

o  At r large, Veff(r ) ~ -e2/r

o  Nuclear charge is screened to one unit of charge.

+Ze

-(Z-1)e

r

-e

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o  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.

o  Not possible to find exact solution to Schrodinger equation using this Hamiltonian.

o  Must use the central field approximation in which we write the Hamiltonian as:

where Vcentral is the central field and Vresidual is the residual electrostatic interaction.

!

ˆ H = ˆ H kinetic + ˆ H elec"nucl + ˆ H elec"elec

= "!2

2m# i

2 "Ze2

4$%0rii=1

N

&i=1

N

& +e2

4$%0riji> j

N

&

!

ˆ H = "!2

2m# i

2 "Vcentral (ri)$

% &

'

( )

i=1

N

* + Vresidual

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o  The central field approximation work in the limit where

o  In this case, Vresidual can be treated as a perturbation and solved later.

o  By writing we end up with N separate Schrödinger equations:

with E = E1 + E2 + … + EN

o  Normally solved numerically, but analytic solutions can be found using the separation of variables technique.

!

Vcentral (ri)i=1

N

" >> Vresidual

!

" ="1(ˆ r 1)"2(ˆ r 2)…"N (ˆ r N )

!

"!2

2m# i

2 "Vcentral (ri)$

% &

'

( ) * i( ˆ r i) = Ei* i( ˆ r i)

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o  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.

o  Following the same procedure as Lectures 3-4, we end up with three equations, one for each polar coordinate.

o  Each electron will therefore have four quantum numbers: o  l and ml: result from angular equations. o  n: arises from solving radial equation. n and l determine the radial wave function Rnl(r )

and the energy of the electron. o  ms: Electron can either have spin up (ms = +1/2) or down (ms = -1/2).

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).

!

"i(ˆ r i) =" i(ri,#i,$i) = Ri(ri)%i(#i,$i)

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o  Hartree theory predicts shell model structure, which only considers gross structure:

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.

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o  Periodic table can be built up using this shell-filling process. Electronic configuration of first 11 elements is listed below:

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 4 quantum numbers. 2.  Hunds rule: Electrons fill each orbital in the subshell before pairing up with opposite spins.

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o  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 ml and ms levels. Naccum is the accumulated number of electrons that can be held by atom.

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.

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o  4s level has lower energy than 3d level due to penetration.

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.

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Radial probabilities for 4s 3d

4s - red 3d - blue

Note: Movies from http://chemlinks.beloit.edu/Stars/pages/radial.htm

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Radial probabilities for 1s 2s 3s

1s - red 2s - blue 3s - green

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Radial probabilities for 3s 3p 3d 3s - red 3p - blue 3d - green

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o  Alkali are approximately one-electron atoms: filled inner shells and one valence electron.

o  Consider sodium atom: 1s2 2s2 2p6 3s1.

o  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.

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 nuclear potential => lower energy.

!

Enl = "RH

[n "#(l)]2

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o  #(l) depends mainly on l. Values for sodium are shown at right.

o  Can therefore estimate wavelength of a transition via

o  For sodium the D lines are 3p ! 3s transitions. Using values for #(l) from table,

=> $ = 589 nm

!

1"

= RH1

[n f #$(l f )]2 #

1[ni #$(li)]

2

%

& ' '

(

) * *

!

1"

= RH1

[3#$(3s)]2#

1[3#$(3p)]2

%

& '

(

) *

=1.0967757 +107 11.6272

#1

2.1172%

& '

(

) *

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o  Consider sodium, which has 11 electrons.

o  Nucleus has a charge of +11e with 11 electrons orbiting about it.

o  From Bohr model, radii and energies of the electrons in their shells are

o  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.

o  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.

!

rn =n2

Za0

!

En = "13.6Z 2

n2and

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o  Ionisation potentials and atomic radii:

o  Ionisation potentials of noble gas elements are highest within a particular period of periodic table, while those of the alkali are lowest.

o  Ionisation potential gradually increases until shell is filled and then drops.

o  Filled shells are most stable and valence electrons occupy larger, less tightly bound orbits.

o  Noble gas atoms require large amount of energy to liberate their outermost electrons, whereas outer shell electrons of alkali metals can be easily liberated.

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o  X-ray spectra:

o  Enables energies of inner shells to be determined.

o  Accelerated electrons used to eject core electrons from inner shells. X-ray photon emitted by electrons from higher shell filling lower shell.

o  K-shell (n = 1), L-shell (n = 2), etc.

o  Emission lines are caused by radiative transitions after the electron beam ejects an inner shell electron.

o  Higher electron energies excite inner shell transitions.

Wavelength (A)

40 keV

80 keV

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o  Wavelength of various series of emission lines are found to obey Moseley’s law.

o  For example, the K-shell lines are given by

where % accounts for the screening effect of other electrons.

o  Similarly, the L-shell spectra obey:

o  Same wavelength as predicted by Bohr, but now have and effective charge (Z - %) instead of Z.

o  %L ~ 10 and %K ~ 3.

!

1"

= (Z #$K )2RH

112#1n2

%

& '

(

) *

!

1"

= (Z #$ L )2RH

122#1n2

%

& '

(

) *

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o  Assume net charge is ( Z - 1 )e.

o  Therefore, the potential energy is

o  Total energy of orbit is

o  Modified Bohr formula taking into account screening.

o  Can therefore easily show that

( )n

n reZV

0

2

41

!"#

#=

( ) ( )

2

220

0

22

0

22'

)1(44

141

21

21

eZmneZ

neZmVmvE nnn

!

!!""

#

$%%&

' !=+=

!! ()()

()

!

= "Z "1( )2me4

2 4#$0( )2n2!2

!

"En = #13.6(Z #1)2

n2

!

1"

= (Z #1)2RH112#1n2

$

% &

'

( )

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o  Electrons in orbitals with large principal quantum numbers (n) will be shielded from the nucleus by inner-shell electrons.

Zeff = Z - %nl.

o  %nl increases with n => Zeff decreases with n.

o  %nl increases with l => Zeff decreases with l.

n=1n=2n=3n=4

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o  In hydrogenic one-electron model, the energy levels of a given n are degenerate in l:

o  Not the case in multi-electron atoms. Orbitals with the same n quantum number have different energies for differing values of l.

o  As Zeff = Z - %nl is a function of n and l, the l degeneracy is broken by modified potential.

!

En = "Z 2µe4

(4#$0 )22!2n2 3s 3p 3d

!

En = "Zeff

2µe4

(4#$0 )22!2n2

3s 3p 3d

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o  Wave functions of electrons with different l are found to have different amount of penetration into the region occupied by the 1s electrons.

o  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.

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o  In the case of Li, the 2s electron shows more penetration inside the first Bohr radius and is therefore lower than the 2p.

o  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.