Download Alkali Elements Alkali Elements: Excited States

Alkali Elements
The alkali elements are the elements in column one of the periodic table.
H (Hydrogen), Li (Lithium), Na (Sodium), K (potassium), Rb (rubidium) …
These elements are characterized by a single electron in the outermost s-subshell.
This ns1 electron is the first electron in a new shell.
Configuration
Hydrogen:
Lithium:
Sodium:
Potassium:
1s1
2
1s 2s1
1s22s22p63s1
2
2
1s 2s 2p63s23p64s1
Abbreviated Configuration
1s1
[He]2s1
[Ne]3s1
[Ar]4s1
“Nobel gas” core electrons
“Optical” electron.
Spherically symmetric
Weakly bonded.
Strongly bonded.
Easy to excite and
Shields nuclear charge.
ionize.
Do not participate in
low energy excitation
processes.
Alkali Elements: Excited States
We consider now excited
states of the alkali elements in
which the optically active
electron is excited into higher
subshells.
e.g. for Sodium (Na):
1s22s22p63s1
1s22s22p63p1
1s22s22p64s1
1s22s22p63d1
…
2
1s 2s22p66f1
…
For large n, (i.e. the optical
electron is in “far-out” shells),
the nuclear shielding is
effectively complete. Z(r)=1
and the energies correspond to
that of hydrogen.
Figure constructed from
optical line spectra.
Alkali Elements: Spin-Orbit Coupling
Close inspection of the optical transitions between these states reveals “multiplet”
lines in the spectrum. Each state is split into two due to spin-orbit coupling of
the optically active electron.
Reminder: Spin-orbit coupling is the interaction of the magnetic moment due the
electrons orbital motion with the magnetic moment due to electron spin.
The spin orbit coupling energy is determined by the spin quantum number s, the
orbital angular momentum number l, and the total angular momentum quantum
number j.
A constant for a given
subshell (n,l).
s is always 1/2 for one electron. l is given by the subshell that the optical electron
is in. j determines how spin and orbital angular momentum are oriented relative to
each other.
e.g. Sodium Doublet
Emission: 1s22s22p63p1
!
1s22s22p63s1
Work out the spin coupling for the optical electron in both
configurations:
3p1:
"
l=1, s=1/2
"
Term symbol:
j = 3/2, 1/2
2P , 2P
3/2
1/2
2P :
3/2
<ESO> = K [ 15/4 - 2 - 3/4 ] = +K
2P :
1/2
<ESO> = K [ 3/4 - 2 - 3/4 ] = -2K
3s1:
"
l=0, s=1/2
"
Term symbol:
2S :
1/2
3p1
2P
3/2
2P
1/2
j = 1/2
2S
1/2
<ESO> = K [ 3/4 - 0 - 3/4 ] = 0
3s1
2S
1/2
Spin Orbit : Alkali Element Dependence
Energy splitting into 2P1/2 and 2P3/2 as a function of the element. Spin orbit
splitting increases massively with Z [which in turn affects V(r) and the
integral over dV(r)/dr in the spin-orbit formula].
Element
Li
Na
K
Rb
Cs
Subshell
2p1
3p1
4p1
5p1
6p1
Spin-orbit
Splitting
(10-4 eV)
0.42
21
72
295
687
Atoms with several optically active electrons
i.e. atoms with several electrons in partially filled subshells.
e.g. carbon (ground state):
e.g. nitrogen (ground state):
e.g. beryllium (excited state):
e.g. uranium (ground state):
1s22s22p2
1s22s22p3
1s22s12p1
[Rn]7s26d15f3
Most of the energetics of these atoms is well described by the Hartree model;
however, in detail (e.g. in high-resolution spectroscopy), spin-orbit coupling and
the residual coulomb interaction are important.
Residual Coulomb Interaction: The Coulomb interaction that is not captured by the
effective Hartree potential (remember: the Hartree method is an approximation in
that it replaces the instantaneous, pairwise interaction of moving electrons with an
averaged interaction potential !). The residual Coulomb interaction is the difference
between the instantaneous Coulomb interaction between electrons in reality and the
averaged Coulomb interaction in Hartree theory.
Spin-Orbit Coupling: As before, the magnetic interaction between orbital- and spinmagnetic moments. However, now we have multiple electrons to consider (each with
an orbital and an spin moment).
Coupling of Angular Momenta
Residual Coulomb Interaction:
In a partially filled shell, the potential is no longer spherical
symmetric, this has an effect on the relative motion of two
optically active electrons in a partially filled shell.
Tendency (i.e. energetic preference) to align orbital and spin
angular momenta, so as to maximize total orbital angular
momentum L’ and total spin momentum S’.
LS-Coupling (elements with small Z; spin-orbit interaction << residual coulomb):
First couple all single electron L together into L’, and all the single electron S into
S’. Then couple L’ and S’ to get total angular momentum J’.
JJ-Coupling (elements with large Z; spin-orbit interaction >> residual coulomb):
For every electron, couple L and S to get J; then couple the single-electron J
together into total angular momentum J’
e.g. Helium (two optical electrons)
s’=0
1s12p1
1s12s1
Helium Ground State: 1s2
l’=1
s’=1
j’=1
1P
1
3P
2 3
P1
3
j’=2,1,0 P0
s’=0
j’=0
s’=1
j’=1
s’=0
j’=0
l’=0
l’=0
1S
0
3S
1
1S
0