CHAPTER 8
Atomic Physics
Multi-electron atoms
The Pauli exclusion Principle
Filling Shells
Atomic Structure and the Periodic
Table
Total Angular Momentum
Spin-Orbit Coupling
Dimitri Mendeleev (1834 – 1907)
What distinguished Mendeleev was not only genius, but a passion for the elements.
They became his personal friends; he knew every quirk and detail of their behavior.
- J. Bronowski
Prof. Rick Trebino, Georgia Tech, www.frog.gatech.edu
Multi-electron atoms
When more than one electron is involved, the
potential and the wave function are functions of
each electron’s position ri :
 4
N
V  V (r1 , r2 ,..., rN )  
i 1
  (r1 , r2 ,..., rN , t )
e
2
0
ri

N
i 1
i 1
j 1

Attraction between
nucleus and electrons
e2
4 0 ri  rj
Repulsion between
electron pairs
Solving the Schrodinger Equation exactly in this case is impossible!
But we can approximate the solution as the product of singleparticle wave functions:
(r1 , r2 ,..., rN , t )  1 (r1 , t )  2 (r2 , t )
 N (rN , t )
And we’ll approximate each i with a Hydrogen wave function.
Probability Distribution Functions  n m ( r )
Probability
densities
for some
Hydrogen
electron
states.
2
Pauli Exclusion Principle
What states do the electrons reside in? As in most cases in physics:
The electrons in an atom tend to occupy the lowest
energy levels available to them.
So they’re all in n = 1 states? Nope.
Strong absorptions are seen in multi-electron atoms from states
with large values of n, so Wolfgang Pauli proposed his famous
Exclusion Principle:
No two electrons in an atom may have the same set of
quantum numbers (n, ℓ, mℓ, ms).
It applies to all particles of half-integer spin, such as electrons,
which are called fermions.
Fermions vs. bosons
Electrons, protons, and neutrons have spin ½ and so are fermions.
Bosons have integer spin and can be in the same state. Photons
have spin 1 and so are bosons.
If states are
degenerate,
then there
can be as
many
fermions
with a given
energy as
states with
that energy.
Atomic Structure
Hydrogen:
n m ms  1 0 0  12 in the ground state.
In the absence of a magnetic field, the state ms = +½ is degenerate
with the ms = −½ state. So ms can be either +½ or −½.
Helium:
n m ms  1 0 0 12 for the first electron.
n m ms  1 0 0  12 for the second electron.
Helium’s two electrons have anti-aligned (ms = +½ and ms = −½) spins.
Lithium:
n m ms  1 0 0 12 for the first electron.
n m ms  1 0 0  12 for the second electron.
n m ms  2 0 0  12 for the third electron.
In the absence of a magnetic field, the third electron’s state ms = +½ is
degenerate with the ms = −½ state, so it can be either +½ or −½.
Electronic configuration and shells
The principle quantum number also has a letter code.
n=
1 2 3 4...
Letter =
K L M N…
n = Shells (e.g.: K shell, L shell, etc.)
nℓ = Subshells (e.g.: 1s, 2p, 3d)
The list of an atom’s
occupied energy levels
is called its electronic
configuration.
The exponent indicates
the number of electrons
in the subshell.
n
# of electrons
Hydrogen:
1s
Helium:
1s2
Lithium:
1s22s
Flourine:
1s22s22p5
etc.
ℓ
Electronic shells and subshells
Filling the shells
Filling the 2p subshell
Boron (B)
Carbon (C)
Nitrogen (N)
Oxygen (O)
Fluorine (F)
Neon (Ne)
9
10
Atomic Structure
How many electrons may be in each subshell?
For each ℓ: there are (2ℓ + 1) values of mℓ.
For each mℓ, there are two values of ms.
So there are 2(2ℓ + 1) electrons per subshell.
ℓ = 0 1 2 3 4 5 …
letter = s p d f g h …
ℓ = 0 (s state) can have two electrons.
ℓ = 1 (p state) can have six electrons, and so on.
Recall:
Electrons with higher ℓ values are on average
further from the nucleus and so are more
shielded from the nuclear charge, so electrons
with higher ℓ values lie higher in energy than
those with lower ℓ values—even if the value of n
is higher. The 4s shell fills before 3d.
Shell-filling order—a geometrical
picture
Filling the shells
Filling the shells
(metals)
Cut-off between metals
and nonmetals
Closed shells and subshells have spherically
symmetric electron probability distributions.
Elements with closed shells
have the most compact
electron distributions and so
are also the most stable and
most difficult to ionize.
They include Helium, Neon,
Argon, Krypton, Xenon, and
Radon.
Electron probability
distribution of a closed
shell or subshell
Elements with one additional
electron (beyond a closed
shell) are the largest and least
stable.
It’s easy to see why closed shells and
subshells are spherically symmetrical.
s subshell:
p subshell:
d subshell:
Subscripts
are mℓ values.
Atoms with closed shells (especially pshells) are smaller.
Atoms with closed shells (especially p-shells)
are more stable and hence difficult to ionize.
A closed d-shell can also be
quite stable for larger values of n
Additional effects: total angular momentum
The total angular-momentum quantum number for the single electron
is the sum of the orbital angular momentum and the spin. It can only
have the values:
j s
depending on their
relative directions
j and mj are also quantum numbers for single electron atoms.
The total and z-components
of angular momentum are:
J  j ( j  1)
Jz  mj
For multi-electron atoms, use vector addition of all the electrons’
orbital angular momenta and spins. Then:
L  L1  L2  ...  LN
S  S1  S2  ...  SN
  
J  LS
All of the quantities L, Lz, S, Sz, J, and Jz are quantized.
Spin-Orbit Coupling
The orbit of the electron
produces a magnetic field:
Nucleus
S
Binternal  L
But the electron spin also
produces a magnetic field
proportional to the spin.
The spin’s potential energy
in the magnetic field of the
electron orbit is:
Binternal
Vs  S  L
Parallel: repulsion
Antiparallel: attraction
This energy dependence on the interaction of the electrons’ spins
and orbital angular momenta is called spin-orbit coupling.
It yields fine structure in the energy levels and spectrum.
Total Angular Momentum
In terms of j, the selection rules for a single-electron atom become:
Δn = anything
Δmj = 0, ±1
Δℓ = ±1
Δj = 0, ±1
Hydrogen energy-level diagram for
n = 2 and n = 3 with
spin-orbit
splitting.
Similar splittings
occur for multielectron atoms.
Use capital letters
for the j value.
Hyperfine structure
The nucleus also has spin (designated by I), and its magnetic field
also contributes to the potential and yields hyperfine structure.
The total
angular
momentum,
including
nuclear spin,
is designated
by F.
Energy levels of Helium
Antiparallel
spins
Here, one
electron is
assumed to
remain in
the ground
state.
Parallel
spins
Parallel spins
means greater
separation
between
electrons
hence lower
energies.
Computation of atomic energy levels allows
us to determine the atoms present on the
sun’s surface from its
spectrum.
Line
Element
Wavelength
[nm]
b -1, 2
Mg
518.4, 517.3
c
Fe
495.8
F
H
486.1
Line
Element
Wavelength
[nm]
d
Fe
466.8
e
Fe
438.4
A -band
O2
759.4 - 762.1
f
H
434.0
B -band
O2
686.7 - 688.4
G
Fe i Ca
430.8
C
H
656.3
g
Ca
422.7
a -band
O2
627.6 - 628.7
h
H
410.2
D -1, 2
Na
589.6, 589.0
H
Ca
396.8
E
Fe
527.0
K
Ca
393.4