Download Document

Unit-IV
Many Electron Atoms
Identical Particles
Identical Particles
Example: all electrons have the same
mass, electrical charge, magnetic
properties…
Therefore, we cannot distinguish one
electron from another.
 Quantum theory then restricts the kinds
of states for electrons

Two Families of Fundamental
Particles

Fermions
Enrico Fermi
1901-1954

Bosons
Satyendranath Bose
1894-1974
Fermions
 electrons, neutrinos, protons, quarks...
 At most one particle per quantum state
Bosons
 photons, W, Z bosons, gluons,4He
(nucleus and atom), 16O(nucleus), 85Rb
(atom),…
 Possibly many particles in the same
quantum state
Fermions versus Bosons
Quantum theory tells us that angular
momentum (amount of spin) = n (h/2)
 Experimental fact:
Fermions have n = 1, 3, 5, …
Bosons have n = 0, 2, 4, …

Many Particle Systems

Schrodinger Equation for a many particle system
H ( x1 , x2  xn )  i
 ( x1 , x2  xn )
t
with xi being the coordinate of particle i (r if 3D)
 the Hamiltonian has kinetic and potential energy
1 2
1 2
H   (

)  V ( x1 , x2  xn )
2
2
2m1 x
2m2 x
2


if only two particles and V just depends on separation
then can treat as “one” particle and use reduced mass
in QM, H does not depend on the labeling. And so if
any i  j and j i, you get the same observables or
state this as (for 2 particles)
H(1,2)=H(2,1)
Exchange Operator


Using 2 particle as example. Use 1,2 for both space coordinates
and quantum states (like spin)
the exchange operator is
P12u (1,2)  u (2,1)
P12 ( P12u (1,2))  P12u (2,1)  u (1,2)
 eigenvalues :   1

We can then define symmetric and antisymmetric states for 2 and
3 particle systems
1
1
 S  ( (1,2)   (2,1))  A  ( (1,2)   (2,1))
N
N
1
( (1,2,3)   (2,1,3)   (2,3,1)   (3,2,1)   (3,1,2)   (1,3,2))
N
1
 A  ( (1,2,3)   (2,1,3)   (2,3,1)   (3,2,1)   (3,1,2)   (1,3,2))
N
S 
Particles in a Spin State
If we have spatial wave function for 2 particles in a box which
are either symmetric or antisymmetric
 there is also the spin. assume s= ½

 s ( S z  1)  
 s ( S z  0) 
1
(    )
2
 s ( S z  1)  
 A ( S z  0) 
S=1
S=0
1
(    )
2
as Fermions need totally antisymmetric:
spatial Asymmetric + spin Symmetric (S=1)
spatial symmetric + spin Antisymmetric (S=0)
 if Boson, need totally symmetric and
so symmetric*symmetric or antisymmetric*antisymmetric

Spectroscopic Notation
n 2S + 1Lj
This code is called spectroscopic or term
symbols.
 For two electrons we have singlet states
(S = 0) and triplet states (S = 1), which
refer to the multiplicity 2S + 1.

Term Symbols

Convenient to introduce shorthand notation to label energy levels that
occurs in the LS coupling regime.

Each level is labeled by L, S and J:
◦ L = 0 => S
◦ L = 1 => P
◦ L = 2 =>D
◦ L = 3 =>F

If S = 1/2, L =1 => J = 3/2 or 1/2. This gives rise to two energy levels or
terms, 2P3/2 and 2P1/2

2S + 1 is the multiplicity. Indicates the degeneracy of the level due to
spin.
◦ If S = 0 => multiplicity is 1: singlet term.
◦ If S = 1/2 => multiplicity is 2: doublet term.
◦ If S = 1 => multiplicity is 3: triplet term.
2S+1L
J
Many-Electron Atoms






Consider two electrons outside a closed shell.
Label electrons 1 and 2.
Then
J = L1 + L2 + S1 + S2
There are two schemes, called LS coupling
and jj coupling, for combining the four angular
momenta to form J.
The decision on which scheme to use
depends on the relative strength of the
interactions.
JJ coupling predominates for heavier
elements.
LS, or Russell-Saunders, Coupling

The LS coupling scheme is used for
most atoms when the magnetic field is
weak.
L = L1 + L2
S = S1 + S2

Use upper case for
many - electron atoms.
The L and S combine to form the total
angular momentum:
J=L+S
Allowed Transitions for LS Coupling
DL = ± 1
DS = 0
DJ = 0, ± 1 (J = 0 J = 0 is
forbidden)
jj Coupling
J1 = L1 + S1
J2 = L2 + S2
J=
S Ji
Allowed Transitions for jj Coupling
Dj1 = 0,± 1
Dj2 = 0
DJ = 0, ± 1 (J = 0 J = 0 is
forbidden)
Helium Atom Spectra





One electron is presumed to be in the ground state,
the 1s state.
An electron in an upper state can have spin anti
parallel to the ground state electron (S=0, singlet
state, parahelium) or parallel to the ground state
electron (S=1, triplet state, orthohelium).
The orthohelium states are lower in energy than the
parahelium states.
Energy difference between ground state and lowest
excited is comparatively large due to tight binding of
closed shell electrons.
Lowest Ground state (13S) for triplet corresponding
to lowest singlet state is absent.
DL=±1,
DS = 0,
DJ =0, ± 1
Sodium Atom Spectrum
Selection Rule: Dl=±1
 Various Transitions:
 Principal Series (np
3s, n=3,4,5,…)
 Sharp Series (ns
3p, n=3,4,5,…)
 Diffuse Series (nd
3p, n=3,4,5,…)
 Fundamental Series (nf
3d, n=3,4,5,…)

Sodium Atom Fine Structure

Selection Rule: Dl=0, ±1