Download zeeman effect

The Zeeman Effect
o Atoms in magnetic fields:
o Normal Zeeman effect
o Anomalous Zeeman effect
Zeeman Effect
o
First reported by Zeeman in 1896. Interpreted by
Lorentz.
o
Interaction between atoms and field can be
classified into two regimes:
o Weak fields: Zeeman effect, either normal
or anomalous.
o Strong fields: Paschen-Back effect.
o
Normal Zeeman effect agrees with the classical
theory of Lorentz. Anomalous effect depends on
electron spin, and is purely quantum mechanical.
Norman Zeeman effect
o
Observed in atoms with no spin.
o
Total spin of an N-electron atom is Sˆ   sˆi
N
i1
o
Filled shells have no net spin, so only consider valence electrons. Since electrons have spin
1/2, not possible to obtain S = 0 from atoms with odd number of valence electrons.

o
Even number of electrons can produce S = 0 state (e.g., for two valence electrons, S = 0 or 1).
o
All ground states of Group II (divalent atoms) have ns2 configurations => always have S = 0
as two electrons align with their spins antiparallel.
o
Magnetic moment of an atom with no spin will be due entirely to orbital motion:
ˆ 


B ˆ
L
Norman Zeeman effect
o
Interaction energy between magnetic moment and a uniform magnetic field is:
ˆ  Bˆ
E  
o
Assume B is only in the z-direction:

o
0 
 
Bˆ  0 
 
Bz 
The interaction energy of the atom is therefore,

E  z Bz  B Bz ml
where ml is the orbital magnetic quantum number. This equation implies that B splits the
degeneracy of the ml states evenly.

Norman Zeeman effect transitions
o
But what transitions occur? Must consider selections rules for ml: ml = 0, ±1.
o
Consider transitions between two Zeeman-split atomic levels. Allowed transition frequencies
are therefore,
h   h  0   B Bz
ml  1
h  h 0
ml  0
ml  1
h   h  0   B Bz
o
Emitted photons also have
 a polarization, depending

on which transition they result from.
Norman Zeeman effect transitions
o
Longitudinal Zeeman effect: Observing along magnetic field, photons must propagate in zdirection.
o Light waves are transverse, and so only x and y polarizations are possible.
o The z-component (ml = 0) is therefore absent and only observe ml = ± 1.
o Termed -components and are circularly polarized.
o
Transverse Zeeman effect: When observed at right angles to the field, all three lines are
present.
o ml = 0 are linearly polarized || to the field.
o ml = ±1 transitions are linearly polarized at right angles to field.
Norman Zeeman effect transitions
o
Last two columns of table below refer to the
polarizations observed in the longitudinal and
transverse directions.
o
The direction of circular polarization in the
longitudinal observations is defined relative to B.
o
Interpretation proposed by Lorentz (1896)
(ml=-1 )

(ml=0 )
+
(ml=+1 )
Anomalous Zeeman effect
o
Discovered by Thomas Preston in Dublin in 1897.
o
Occurs in atoms with non-zero spin => atoms with odd number of electrons.
o
In LS-coupling, the spin-orbit interaction couples the spin and orbital angular momenta to
give a total angular momentum according to
Jˆ  Lˆ  Sˆ
o
In an applied B-field, J precesses about B at the Larmor
frequency.

o
L and S precess more rapidly about J to due to spin-orbit
interaction. Spin-orbit effect therefore stronger.
Anomalous Zeeman effect
o
Interaction energy of atom is equal to sum of interactions of spin and orbital magnetic
moments with B-field:
E  z Bz
 (zorbital  zspin )Bz

 Lˆ z  gsSˆ z B Bz
where gs= 2, and the < … > is the expectation value. The normal Zeeman effect is obtained
by setting Sˆz  0 and Lˆ z  ml .

o
In the case of precessing atomic magnetic in figure on last slide, neither Sz nor Lz are constant.
 Jˆz  m
Only
j is well defined.
o
Must therefore project L and S onto J and project onto
 z-axis =>
Jˆ
Jˆ B
ˆ   | Lˆ | cos1

 2 | Sˆ | cos 2
ˆ
ˆ
|J |

|J |
Anomalous Zeeman effect
o
The angles 1 and 2 can be calculated from the scalar products of the respective vectors:
Lˆ  Jˆ | L || J | cos1
Sˆ  Jˆ | S || J | cos
2
which implies that
o
ˆ 

Lˆ  Jˆ
Sˆ  Jˆ B ˆ

2
J
| Jˆ |2
| Jˆ |2

Now, using Sˆ  Jˆ  Lˆ implies that
(1)
Sˆ  Sˆ  (Jˆ  Lˆ )  (Jˆ  Lˆ )  Jˆ  Jˆ  Lˆ  Lˆ  2Lˆ  Jˆ

ˆ
therefore L  Jˆ  (Jˆ  Jˆ  Lˆ  Lˆ  Sˆ  Sˆ ) /2


so that

j( j  1)  l(l  1)  s(s  1) 2 /2
Lˆ  Jˆ


j( j  1) 2
| Jˆ |2

o
Similarly,


 j( j  1)  l(l  1)  s(s  1)
2 j( j  1)
Sˆ  Jˆ  (Jˆ  Jˆ  Sˆ  Sˆ  Lˆ  Lˆ ) /2

and
Sˆ  Jˆ
 j( j  1)  s(s  1)  l(l  1)

2 j( j  1)
| Jˆ |2
Anomalous Zeeman effect
o
We can therefore write Eqn. 1 as
 j( j  1)  l(l  1)  s(s  1)
ˆ  

2 j( j  1)

o
This can be written in the form
ˆ  g j

2
 j( j  1)  s(s  1)  l(l  1)B Jˆ
2 j( j  1)


B ˆ
J

where gJ is the Lande g-factor given by

o
This implies that
g j  1
j( j  1)  s(s  1)  l(l  1)
2 j( j  1)
z  g j B m j
and hence the interactionenergy with the B-field is

o
E  z Bz  g j B Bz m j
Classical theory predicts that gj = 1. Departure from this due to spin in quantum picture.

Anomalous Zeeman effect spectra
o
Spectra can be understood by applying the selection rules for J and mj:
j  0,1
m j  0,1
o
Polarizations of the transitions follow the
 as for normal Zeeman effect.
same patterns
o
For example, consider the Na D-lines
at right produced by 3p  3s transition.