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Atoms in a magnetic field
H&W Chap. 12,13
• Splitting of spectral lines due to Zeeman effect (ordinary and
anomalous) and Paschen-Back effect
• Gradient of magnetic field (B-field not homogenous): SternGerlach experiment, deflection of an atomic beam
Importance:
• Experimental evidence of the existence of the electron spin
• Powerful tool for the identification of levels (ie the assignment
of quantum numbers)
• Control motion of the atoms, eg atom trapping (will come back
to this at the end of the course)
Ordinary Zeeman effect
r
B0 // z
Magnetic energy is given by
r
= − µ j ⋅ B0
r
Vmag
r
B0 = homogenous external field
r
r r
µ j = total magnetic moment of the atom = µl + µ s
Let’s consider first the case of spin 0 states:
r
µs = 0
we are left with purely orbital magnetism
Ordinary (or “normal“) Zeeman effect
For instance Cadmium has two electrons in the outer shell, whose
spins are antiparallel
total S=0 (more on this later)
Total Hamiltonian:
H = H atom + Vmag
r
e
= H atom − µl ⋅ B0 = H atom − µl , Z B0 = H atom +
lZ B0
2m0
r
Our aim is to derive an expression for the shift in energy due to the 2nd
term. We note that the eigenfunctions ψ nlm of the atomic Hamiltonian
are also eigenfunctions of lZ (with eigenvalue hml ). Therefore we can
calculate Vmag on these eigenfunctions and we obtain:
Vmag
e
=
hml B0
2m0
Vmag = ml µ B B0
ml = −l ,−l + 1,...,+l (2l + 1 values)
This is the final result for
the ordinary Zeeman effect
(purely orbital magnetism)
Splitting into 2l+1
magnetic levels
Note that m-degeneracy is now lifted: physically, the B-field introduced
a preferred direction and H is not spherically symmetric anymore.
Example of ordinary (or “normal“) Zeeman effect: Cadmium
ml
Distance between adjacent
magnetic levels:
∆E = µ B Bo
Selection rule:
∆ml = 0, ± 1
∆ml = 0, π transition
∆ml = ±1, σ transition
Shifted Zeeman components:
∆ml
SPECTRUM
ω
 µB 
δω =   Bo
 h 
In summary, this type of splitting is observed for spin 0 states
(pure orbital magnetism).
Vmag = ml µ B Bo
Before we move on to the anomalous Zeeman effect, let’s analyse
the ordinary Zeeman effect by means of the vector model.
Vector model = classical description of the behaviour of angular
momentum vectors
0
r
torque τ = µ l × B0
r
dl
τ=
Newton’s law
dt
r
r
l
µl = − µ B
h
r
dl
µB r r
=−
l × B0
dt
h
r
l sin θ
r
l
r
µl
This is the equation of the motion for l. The resulting motion is a
precession about the direction of the B-field.
The precession frequency is known as Larmor frequency
ω L=
µB
h
B0
and it coincides with the distance between the Zeeman components
(see also tutorial problem.)
r2
From the vector model we see that l and projection lZ are constant.
This corresponds to the (already known) fact that they commute with
e
H
+
lZ B0 .
the total Hamiltonian of the system atom
2m0
Also, we note that the projection lZ , which determines
the energy
r
shift in the Zeeman effect, is the time-average of l during the
precession.
These ideas will be useful for the next topic, the anomalous Zeeman
effect.
r
So far we have considered the case of spin 0 states: µ s
r
=0
Now let’s consider the case µ s ≠ 0 . The atomic magnetic moment
is then due to a superposition of spin and orbital magnetism:
r
r
r
µ j = µl + µ s
Moreover, now we have to include also the spin-orbit coupling in the
total Hamiltonian:
2
r
p
a r r r
H=
+ V (r ) + 2 l ⋅ s − µ j ⋅ B0
424
3
2m0
h 23 1
1
14243 fine structure Vmag
gross structure
This Hamiltonian is complicated. Here we consider two extreme cases:
• weak external field B0 << Bl
• strong external field B0 >> Bl
anomalous Zeeman effect
Paschen-Back Effect
We consider the anomalous Zeeman effect first. In this case Vl , s is
much stronger than Vmag and the energy level splitting due to the
external B-field is small compared to the fine structure.
(The term “anomalous” Zeeman effect is historical, and it is actually
contradictory because it is more common than the s=0 case.)
Anomalous Zeeman effect -vector model (1)
r
r
r
Because of the spin - orbit coupling Vl ,s , s and l precess around j .
r
r
r
j r
We note that µ j is not parallel to j because the g factor
is different for orbital angular momentum and spin:
l
r
e r
r
e r
r
µ s ≅ − s ( g s ≅ 2)
µl = −
l
s
mo
2m0
r
r
Therefore µ j is not parallel to j and precesses.
r
r
The time-average of µ j is its projection along j :
r
r
e
r
gj j
(µ j ) j = −
g factor for j ,
2m0
to be determined
r
r
µs
µl
r
µj
Anomalous Zeeman effect -vector model (2)
Because of Vmag
r r
r
= − µ j ⋅ B0 , µ j also precesses around B0 .
r
However, B0 is small, so this is a slow precession
compared to the precession due to spin-orbit.
Hence it’s OK to take the time average defined above
r
and substitute it in the expression for the magnetic
µj
interaction:
r
r r
r
e
e
Vmag = −( µ j ) j ⋅ B0 =
g j j ⋅ B0 =
g j B0 jZ =
2m0
2m0
e
g j B0 hm j
=
2m0
calculate on
eigenfunctions
of jZ
Vmag = µ B g j m j B0
splitting into 2j+1 magnetic levels
(m j = − j ,− j + 1,... + j ⇒ 2 j + 1 values)
r
B0 // z
This diagram (taken from H&W) summarises the vector model
for the anomalous Zeeman effect:
The formula in the previous slide is the main result. However
we still have to derive an expression for the Lande factor g j ….
r r
j ⋅µj
r
r
−e r r
−e r r r
−e  r 2 r r
(µ j )j = r = r j ⋅ l + 2s = r j ⋅ ( j + s ) = r  j + j ⋅ s 
j
2m0 j
2m0 j
2m0 j
r r
We can write the dot product j ⋅ s as:
r
(
)
r r r r r2 1 r2 r2 r2 r2
j ⋅s = l ⋅s + s =  j − l − s + s =

2
r2
2
r2
j −l +s
2
(remember similar procedure in spin-orbit derivation). Hence:
r
(µ )
j j
r2 r2 r2

j −l +s 
−e r
=
j 1 +
r2

2m0 

2j
14442444
3
r
= g j (see def. of µ j
( )j )
Finally, we calculate g j on the set of simultaneous eigenfunctions of the
operators lˆ 2 , sˆ 2 , ˆj 2, and obtain:
j ( j + 1) − l (l + 1) + s( s + 1)
g j = 1+
2 j ( j + 1)
Lande g factor for j
j ( j + 1) − l (l + 1) + s( s + 1)
g j = 1+
2 j ( j + 1)
We get 1 for pure orbital magnetism (s=0),
2 for pure spin magnetism (l=0),
and intermediate values!
Conclusions for anomalous Zeeman effect:
• Energy splitting depends on j, l and s, hence it’s different for
different energy levels
quantum numbers can be determined
from measurements.
• Expect larger number of spectral lines, not just the triplet of the
normal Zeeman effect.
Now a few examples – anomalous Zeeman effect in Sodium D-lines:
3 p3 / 2 g j = 4 / 3
3 p1/ 2 g j = 2 / 3
3 s1/ 2 g j = 2
ANOMALOUS ZEEMAN
EFFECT OF SODIUM D-LINES
Selection rule:
∆m j = 0, ± 1
LARGE NUMBER
OF SPECTRAL LINES
Paschen Back Effect
We have so far only been dealing with WEAK magnetic fields.
If B-field applied is strong enough things become simplified
because we can neglect the spin-orbit coupling and we are left with:
Vmag
r
r r r r
e
e
= − µ j ⋅ B0 = − µl ⋅ B0 − µ s ⋅ B0 =
l z B0 +
s z B0 =
2m0
m0
r
= µ B ml B0 + 2 µ B ms B0 ⇒ Vmag = µ B (ml + 2ms ) B0
SUBSTITUTE
EIGENVALUES
Vector model:
both l and s precess independently
around the direction of the B-field.
(a) D1 and D2 in sodium
(b) Zeeman splitting
(c) Paschen Back effect
Selection rules:
∆ml = 0, ± 1
∆m s = 0
Electric dipole cannot effect
a spin flip because
it only
r
acts on ψ (r ) and not on spin.
Paschen Back effect:
triplet of spectral lines like
those of the normal Zeeman
effect
Stern-Gerlach experiment
So far we have talked about atoms in a homogenous magnetic field.
Now we are going to see what happens if the field is not homogeneous,
and in particular we will find that this affects the motion of the atoms.
In the Stern-Gerlach experiment, a collimated atomic beam travels
across a region where a magnetic field gradient is present:
z
N
B(z ) dB
dz
S
OVEN
COLLIMATED
BEAM
MAGNET
POLES
r r
As before, we can write the magnetic interaction as Vmag = − µ ⋅ B = − µ Z B
where µ is the magnetic moment of the atom. Because this magnetic
energy now depends on z, we obtain a force along z:
dVmag
dB
F =−
= µZ
dz
dz
This means that the atomic beam will be deflected. We can reasonably
assume that dB/dz doesn’t change over the length L of the magnet poles,
hence the atomic beam experiences a uniform force while inside the
magnet:
N
v
F
DEFLECTION
S
L
MAGNET
POLES
uniformly accelerated motion
parabolic trajectory
Given the force and the atom velocity, it is possible to calculate the
deflection of the beam after the magnet.
Of course the force depends on the projection of µ along B, i.e. it
r
depends on the angle α between µ and B:
r
µ
B
dB
F=µ
cosα
α
dz
We expect no preferred direction for the magnetic moment, i.e.
different atoms have different values of α. (This is often referred to as
unpolarised atomic beam.) Hence different atoms will experience a
different force.
Classically, any orientation α is permitted. Atoms with magnetic
moments perpendicular to B are not deflected, those parallel deflected
most, all intermediate values can occur. This results in a broad
distribution after the magnet:
N
v
S
PHOTOGRAPHIC
PLATE
As this experiment was first performed in 1921 (with silver atoms),
people would have expected this result, since quantum mechanics
was not yet fully developed.
However, the experiment showed two distinct peaks, in agreement
with the quantum mechanical prediction:
Ag atoms have one s electron
l=0, pure spin magnetism
r
e r
e
e h
µ s = − s ⇒ µs , z = − s z = ±
= ± µB
m0
m0
m0 2
⇒ F = µs , z
dB
dB
= ± µB
dz
dz
Due to the space quantisation of spin, the
force only takes two values, so the result is
two narrow peaks:
N
v
S
Conclusions:
• Evidence of directional quantisation
• From exp. data on deflection we can obtain values for magnetic
moment µs,z
• Observe effects due to spin magnetism (l=0 for valence electron in Ag)
• Inner shells do not contribute to the total magnetic moment
This diagram summarises the
experiment + results.
Note shape of magnet poles
to give a field gradient.