Honors Physics IIIa
Lecture 14:
Periodic table, Complex Atoms:
L-S coupling & Zeeman Effect
http://www.physics.rutgers.edu/ugrad/273a
Weida Wu
Multi-Electron Atoms

Atoms with 2 or more electrons have
a new feature:
Electrons are indistinguishable!
 There is no way to tell them apart!


Any measurable quantity (probability,
expectation value, etc.) must not
depend on which electron is labeled
1, 2, etc.
A(1)
B(2)
2
Pauli Exclusion Principle (1925)
No two electrons in an atom may have the same set
of quantum numbers, e.g.: n, , mℓ, ms.
Examples:
 Ground state of helium (2 electrons):
 Electron #1: n=1, =0, mℓ=0, ms=+1/2 (or 1/2)
 Electron #2: n=1, =0, mℓ=0, ms=-1/2 (or
+1/2)
 Ground state of lithium (3 electrons):
 Electrons 1 and 2 are like helium.
 Electron #3 n=2, =0, mℓ=0, ms=+1/2 or -1/2

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Shells and Subshells
Each value of n corresponds to a shell:
n:
1
2
3
4
…
Shell:
K
L
M N …
Each n and l constitute a subshell:
l:
0
1
2
3
4
…
s
p
d
f
g
…
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Population of Shells
Shell n
l
ml
ms
Sub-shell Max. Sub-shell
population
Max. Shell
population
K
1
0
0
±1/2
1s
2
2
L
2
0
1
0
0, ± 1
±1/2
±1/2
2s
2p
2
6
8
0
1
2
0
0, ± 1
0, ± 1,
±2
±1/2
±1/2
±1/2
3s
3p
3d
2
6
10
M
3
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For each l, there are (2l+1) values of ml and 2 values of ms,
so the maximum subshell population is 2(2l+1)
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Maximum Shell Population


For each n, l = 0, 1, … (n-1)
So the maximum shell population is:
6
Question
A certain subshell can have a maximum of 14 electrons.
Which of the following could that subshell be?
 Recall:
The maximum subshell population is 2(2l+1)
l:
0
1
2
3
4
…
s
p
d
f
g
…
A. 3p
B. 3d
C. 3f
D. None of the above

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Question
A certain subshell can have a maximum of 14 electrons.
Which of the following could that subshell be?
 Recall:
The maximum subshell population is 2(2l+1)
l:
0
1
2
3
4
…
s
p
d
f
g
…
A. 3p
2(2l+1)=14  l =3  f
B. 3d
But we must have n>l, so 3f is impossible!
C. 3f
Recall that l = 0, 1, 2, …, (n-1)
D. None of the above

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Multi-Electron Atoms



Chemical properties of atoms are
(mostly) determined by the outermost
electrons.
Outer electrons do not feel the full
nuclear charge, because inner electrons
partially shield the nucleus.
 This is called shielding or
screening.
Also radii of complex atoms are within
a factor of two of hydrogen!
 The inner electrons get pulled in
closer to the nucleus.
rn ,Z
n2
 a0
Z
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Ionization Energy
Atomic number Z: # of
protons in the nucleus






The ionization energy: the energy necessary to remove an electron from the
neutral atom.
Ionization energy of hydrogen: 13.6eV
Other atoms: 5-25 eV
So within a factor of two or so of hydrogen.
It is a minimum for the atoms (alkali metals) which have a single electron
outside a closed shell.
It generally increases across a row on the periodic table.
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 Maximum for the noble gases which have closed shells.
Question
What is the maximum number of electrons that can
populate a 3d subshell?
 Recall:
The maximum subshell population is 2(2l+1)
l:
0
1
2
3
4
…
s
p
d
f
g
…
A. 5
B. 6
C. 10
D. 14
E. None of the above

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Question
What is the maximum number of electrons that can
populate a 3d subshell?
 Recall:
The maximum subshell population is 2(2l+1)
l:
0
1
2
3
4
…
s
p
d
f
g
…
A. 5
“d” means ℓ=2.
B. 6
So mℓ can be -2, -1, 0, 1, 2 (5 possibilities)
And there are two electron spin possibilities.
C. 10
So … 5×2=10.
D. 14
E. None of the above

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Energy of Complex Atoms


Hydrogen atom:
 In Schrodinger theory, H atom exhibits l-degeneracy
 In Dirac theory: energy depends on n and j
For complex atoms:
 For a given n, the energy increases with l
 Mainly a consequence of screening.
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

Charge of nucleus = Ze
Potential energy of outer electron is:

Large l means large orbital angular
momentum.
 L=mvr, so orbit is rather circular, implying
lots of screening. So Zeffective ~ 1.

Small l means L is small, so orbit is elongated.
 “outer” electron goes through nucleus 
not much screening
 So Zeffective ~ Z of nucleus

For small l , Zeffective is large and energy is
more negative.
Energy increases with l .

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In terms of wavefunctions
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Build-up of Periodic Table
Element



Electrons fill subshells in
order of increasing energy:
 1s 2s 2p 3s 3p 4s 3d 4p 5s
4d 5p 6s 4f 5d 6p 7s ….
Note that the l dependence of
energy becomes more and
more important as Z increases.
Each element is described by
giving its electron
configuration , i.e. the
population of each subshell.
Population
(sub)shell full Inert gas
Alkali metal; electron in
2s mostly determines
chemistry of Li
Another inert gas
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Periodic table
Mendeleev
(1834–1907)
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Hund’s Rules


When electrons fill a subshell, do they follow any
special ordering of ml and ms?
For example:
ml
ms
ml
ms
1
+1/2
1
+1/2
1
-1/2
-1
+1/2
0
+1/2
0
+1/2
0
-1/2
1
-1/2
-1
+1/2
-1
-1/2
-1
-1/2
0
-1/2
OR
OR…
Turns out this is preferred. Why?
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Hund’s Rules

They follow Hund’s Rules:
 First priority is to maximize spin angular
momentum, so parallel spins are preferred. Then
electrons will be far apart because of exchange
forces and


Second priority is to maximize orbital angular
momentum.
Pauli Exclusion Principle is the underlying reason for
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the stability of matter!
Electron Configurations

They follow Pauli’s
exclusion principle and
Hund’s rules
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Complex Atoms


There could be several outer electrons, with orbital angular
momenta li and spin angular momenta si .
There are two ways to get grand total angular momentum J.
1.
LS coupling: Most atoms observe LS coupling.
2.
So li precess about L, si precess about S and then L and S
precess about J
jj coupling: Large Z-atoms (Z>~80) observe jj coupling.
We will only deal with LS coupling.
“Term symbol”:
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