Download Experimental evidence for shell model

Lecture 8-9: Multi-electron atoms
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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Energy levels in alkali metals
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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Energy levels in alkali metals
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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Hartree theory
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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Screening
o
Hartree theory results in a shell model of atomic structure,
which includes the concept of screening.
-e
r
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
+Ze
-(Z-1)e
o Unscreened nuclear Coulomb potential.
o
At r large, Veff(r ) ~ -e2/r
o Nuclear charge is screened to one unit of charge.
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Central field approximation
o
The Hamiltonian for an N-electron atom with nuclear charge +Ze can be written:
Hˆ  Hˆ kinetic  Hˆ elecnucl  Hˆ elecelec
N
N
N
Ze 2
e2
 
 

2m
4

r
0 i
i1
i1
i j 4 0 rij
2
2
i
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:
N 
2

2
ˆ
H  
 i  Vcentral (ri ) Vresidual
2m


i1
where Vcentral is the central field and Vresidual is the residual electrostatic interaction.

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Central field approximation
o
The central field approximation work in the limit where
N
V
central
(ri )  Vresidual
i1
o
In this case, Vresidual can be treated as a perturbation and solved later.

o
By writing   1(rˆ1)2 (rˆ2 ) N (rˆN ) we end up with N separate Schrödinger equations:

 2 2




V
(r
)
 i ( rˆi )  E i i ( rˆi )

i
central i 
2m


with E = E1 + E2 + … + EN
o

Normally solved numerically, but analytic solutions can be found using the separation of
variables technique.
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Central field approximation
o
As potentials only depend on radial coordinate, can use separation of variables:
i (rˆi )  i (ri,i,i )  Ri (ri )i (i,i )
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).
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Shell model
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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Shell model
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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Shell model
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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Shell model
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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Shell model
Radial probabilities for 4s 3d
4s - red
3d - blue
QuickTime™ and a
Graphics decompressor
are needed to see this picture.
Note: Movies from http://chemlinks.beloit.edu/Stars/pages/radial.htm
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Shell model
Radial probabilities for 1s 2s 3s
1s - red
2s - blue
3s - green
QuickTime™ and a
Graphics decompressor
are needed to see this picture.
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Shell model
Radial probabilities for 3s 3p 3d
3s - red
3p - blue
3d - green
QuickTime™ and a
Graphics decompressor
are needed to see this picture.
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Quantum defect
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
E nl  
RH
[n   (l)]2
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.
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Quantum defect
o
(l) depends mainly on l. Values for sodium are
shown at right.
o
Can therefore estimate wavelength of a transition via


1
1
 RH 
[n   (l )]2  [n  (l )]2 


 f
f
i
i

1
o
For sodium the D lines are 3p  3s transitions. Using
values for (l) from table,



1
1
 R H 

2
2 

[3   (3s)] [3   (3p)] 
1
 1
1 
 1.0967757 10 7 


1.627 2 2.117 2 
=>  = 589 nm

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Shell model justification
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
n2
13.6Z 2
rn 
a0 and E n  
Z
n2
o
First two electrons occupy n =1 shell. These electrons
2
see full charge
 2of 2+11e. =>r1 = 1 /11 a0 = 0.05 Å and
E1 = -13.6 x 11 /1 ~ -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.
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Experimental evidence for shell model
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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Experimental evidence
for shell model
o
X-ray spectra:
o Enables energies
determined.
of
inner
shells
to
be
o Accelerated electrons used to eject core electrons
from inner shells. X-ray photon emitted by
electrons from higher shell filling lower shell.
80 keV
40 keV
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
electron.
o Higher electron energies excite inner shell
transitions.
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Experimental evidence
for shell model
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
1 1 
 (Z   K ) 2 RH  2  2 
1 n 

1
where  accounts for the screening effect of other electrons.

o
Similarly, the L-shell spectra obey:
 1 1 
 (Z   L ) 2 RH  2  2 
2 n 

1
o
Same wavelength as predicted by Bohr, but now have
and effective charge (Z - ) instead of Z.

o
L ~ 10 and K ~ 3.
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Bohr model including screening
o
Assume net charge is ( Z - 1 )e.
Vn

Z  1e 2

o
Therefore, the potential energy is
o
Total energy of orbit is E '  1 mv2  V  1 m Z  1e 2  
n
n
n


40 rn
2
2  40 n 
2
Z 1 me4


2
240  n 2 2
2
o
13.6(Z 1) 2
 En  
n2

Modified Bohr formula
taking into account screening.
o
easily show that
Can therefore
Z  1e 2
40 n 2  2
40
m( Z  1)e 2
1 1 
 (Z 1) 2 RH  2  2 
1 n 

1

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Shell model summary
o
Electrons in orbitals with large principal quantum
numbers (n) will be shielded from the nucleus by innershell electrons.
Zeff = Z - nl.
o
nl increases with n => Zeff decreases with n.
o
nl increases with l => Zeff decreases with l.
n=1
n=2
n=3
n=4
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Shell model summary
o
In hydrogenic one-electron model, the energy levels
of a given n are degenerate in l:
Z 2e 4
En  
(4 0 ) 2 2 2 n 2
o
3s
3p
3d
3s
3p
3d
Not the case in multi-electron atoms. Orbitals with
the 
same n quantum number have different energies
for differing values of l.
Z eff e 4
En  
(4 0 ) 2 2 2 n 2
2
o
As Zeff = Z - nl is a function of n and l, the l
degeneracy
is broken by modified potential.

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Shell model summary
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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Shell model summary
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.
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