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Zeeman Effect
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Division of Physics & Applied Physics
PH2198/PAP218 Physics Laboratory IIa
Zeeman Effect
Risk of Electrical ShockEnsure all wiring are secure before
turning on power supply
Hot SurfacesAllow the light to cool before handling
1. Historical background
In 1862 Michael Faraday wanted to find out whether magnetic fields had an influence on the
spectral lines emitted from sodium vapor in a Bunsen burner flame. He used the most powerful
magnet and the best prism spectroscope available at that time, but failed to see anything. Three
decades later Pieter Zeeman in Leiden (Netherlands) attempted to do the same experiment. Hewas able to use stronger magnets, and instead of prisms (which rely on dispersion for separating
the wavelength) he used gratings fabricated by Rowland at Johns Hopkins University (USA).
In 1896 Zeeman discovered that a strong magnetic field is able to split the lines of sodium intotwo or more lines, which he could detect using the Rowland grating. Zeeman proposed an
explanation of his discovery based on Hendrik Antoon Lorentz’ idea that “in all bodies small
electrically charged particles of definite mass are present”. We consider, as shown in the figure,an electrically charged particle of charge –e and mass m circling around a nucleus in an orbit of
radius R with a velocity v.
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Fig 1. An orbiting electron
The magnitude of the centripetal force on such an electron is given by
Rm R
mvF
S
22
ω == (1)
Where ω=v/R is the angular velocity. If a magnetic field acts along the z-axis, the corresponding
Lorentz force on the electron is given by FL=-evB and is pointing radially outwards (remember
the right-hand rule). Thus, in the simplest approximation the new force on electron is F=mω2
R
±evB, where the ± sign takes into account that the electron may orbit in the opposite direction as
well. Since we require that the radius of the orbit is almost constant and that Eq. (1) is still valid,we are led to conclude that the frequency ω must change a small amount ∆ω, such that
evB Rm Rm ±=∆+22)( ω ω ω (2)
We assume that this frequency change is very small, such that second order terms (∆ω2
) can be
neglected. Then it is easy to see from Eq. (2) that
m
eB
2±=∆ω (3)
In this way Zeeman could explain that the magnetic field splits the spectroscopic line into threecomponents by an amount linearly proportional to the applied magnetic field. Zeeman andLorentz won the Nobel prize of Physics in 1902 for their studies of these systems.
Today the Zeeman effect is used to determine the spectral properties of gases and solid with highaccuracies. It is also used in laser cooled condensates to control the magnetic moments of the
atoms. Thus, the Zeeman effect is still important, and it is therefore of importance to all
physicists to have a basic understanding of this effect.
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2. Quantum theory of Zeeman effect
It is clear that the simple explanation provided by Zeeman and Lorentz (Eqs. 1-3) is only an
order of magnitude calculation which relies on classical physics. It can therefore not give an
entirely correct picture of the situation. Instead, we must resort to quantum theory to understand
the underlying physical mechanisms. In modern quantum terminology we say that the Zeeman
effect is the breaking of the degeneracy in atomic levels due to the interaction between the
magnetic moment of the atoms and an external magnetic field .
An external magnetic field will interact with the magnetic dipole moment of an atom which
results is
BU ⋅−= µ θ )( (4)
The magnetic dipole moment associated with the orbital angular momentum is given by
Lme
e
orbital2
−= µ (5)
For magnetic field in the z-direction, B=B0z this gives
Bm
em B L
m
eU
e
l z
e 22
h== (6)
Considering the quantization of angular momentum, this gives equally spaced energy levels
displaced from the zero field level by
Bm Bm
em E Bl
e
l µ ==∆2
h(7)
where µB
= 9.2740154x10-24
J/T is the Bohr magneton. This displacement of the energy levels
gives the uniformly spaced multiplet splitting of the spectral lines which is called the Zeeman
effect.
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Fig 2. Splitting of the spectral lines due to Zeeman effect
The magnetic field also interacts with the electron spin magnetic moment, so it contributes to the
Zeeman effect in many cases. The electron spin had not been discovered at the time of Zeeman'soriginal experiments, so the cases where it contributed were considered to be anomalous. The
term "anomalous Zeeman effect" has persisted for the cases where spin contributes. In general,
both orbital and spin moments are involved, and the Zeeman interaction takes the form
Bmg BS Lm
e E j B L
e
µ =⋅+=∆ )2(2
rr
(8)
The factor of two multiplying the electron spin angular momentum comes from the fact that it is
twice as effective in producing magnetic moment. This factor is called the spin g-factor or
gyromagnetic ratio. The evaluation of the scalar product between the angular momenta and the
magnetic field here is complicated by the fact that the S and L vectors are both precessing aroundthe magnetic field and are not in general in the same direction. The persistent early
spectroscopists worked out a way to calculate the effect of the directions. The resulting
geometric factor gL
in the final expression above is called the Lande g factor. It allowed them to
express the resultant splittings of the spectral lines in terms of the z-component of the totalangular momentum, m
j.
The above treatment of the Zeeman effect describes the phenomenon when the magnetic fields
are small enough that the orbital and spin angular momenta can be considered to be coupled. Forextremely strong magnetic fields this coupling is broken and another approach must be taken.The strong field effect is called the Paschen-Back effect.
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3. Fabry-Perot Interferometer
The Fabry-Perot étalon consists of two parallel flat glass plates coated on the inner surface with
partially reflecting surface. An incoming ray at an angle θ with the horizontal will be split into
many rays. The condition for constructive interference occur when
θ η λ cos2 t n = (9)
where η is the reflective index and t is the thickness of the étalon.
Fig 3. Constructive interference produced by an etalon
The emerging parallel rays are brought to focus using a convex lens into a series of bright rings.
The radius of the rings are given by
nnn f f r θ θ ≈= tan (10)
According to equation (9),
)2
sin21(
coscos2
2 n
o
non
n
nt
n
θ
θ θ λ
η
−=
==
)2
1(2
n
onθ
−≈ (11)
Since n0
is not a generally not a whole number and n1
< n0, we have n
1= n
0− ε, where n
1is the
closest integer to n0
and 0 < ε < 1. Thus, the p-th ring of the pattern, measured from center out is
np
= (n0
- ε) - (p -1). Combining it with equations (10) and (11), the radius of the p ring can be
expressed as
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)1(2 2
ε +−= pn
f r
o
p (12)
Hence the difference in the square of the radii of adjacent rings is a constant,
o
p pn
f r r
222
1
2=−
+ (13)
When the spectral lines split, they will have fractional orders at the center εa
and εb, subscript a
and b denotes the two components of the split,
aaa
a
a nvt nt
,1,1 22
−=−=λ
ε
bbb
b
b nvt nt ,1,1 22 −=−=
λ ε (14)
The difference in wave numbers of the two components is
t v ba
2
ε ε −=∆ (15)
Using equations (12) and (13), we get
pr r
r
p p
p−
−=
+
+
22
1
2
1ε (16)
Applying the above equation (16) to the components a and b and substitute them into (15) will
yield the difference in wave number to be
)(2
12
,
2
,1
2
,1
2
,
2
,1
2
,1
b pb p
b p
a pa p
a p
r r
r
r r
r
t v
−−
−=∆
+
+
+
+
(17)
Verify for yourself that
2
,
2
,1
,1,1
a pa p
p p
b
p p
a r r −=∆=∆+
++
(18)
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4. Experiment
The setup of the apparatus is shown in Figure 4. The red filter is not inserted into the Fabry-Perot
etalon during the initial setup. The coils of the electromagnet are connected in parallel and via an
ammeter connected to the variable power supply of up to 20VDC, 12A. A capacitor of 22,000 µF
is then connected parallel to the power supply to smoothen the DC-voltage. (You may use adifferent lens for L2 to acquire a different magnification)
Fig 4. Arrangement of the optical components
Calibration of the magnetic field will be required. Remove the cadmium lamp. Insert the
teslameter between the electromagnetic poles. Increase the current coil to 4A. Record themagnetic field between the electromagnetic poles. Increase the current and repeat until 10A. Plot
the calibration graph of magnetic field against current.
!!! !!! !!!DO NOT maintain large currents through the
electromagnetic coils for extended periods of time.Reset the current to zero when not in use.
!!! !!! !!!
Get a life picture by opening the Motic Images Plus software followed by File Capture
Window. Insert the red filter to pick out the 643.8 nm line of the cadmium spectrum. Fine-tunethe positions of the optics components and the image settings until a satisfactory series of sharp
rings are obtained in the life picture window.
Increase the coil current to about 4A and observe the splitting of the rings. Capture the imageusing the Capture icon. Increase the current and repeat until 10A.
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5. Measurement and Evaluation
Once the pictures are collected, the radius of each ring can be measured by selecting clicking
Measure Circle or Circle (3 Points). Obtain the best fit circle and the program will
automatically calculate the radius of the circle. Repeat for as many of the rings as possible. The
radii of the components a and b of the rings are used to calculate
2
,1
2
,
1,1,
a pa p
p p
b
p p
a r r −
−−−=∆=∆
2
,
2
, b pa p
p r r −=δ (19)
The difference of the wave numbers can be calculated by the equation
∆=∆
t v
2
1(20)
where t (= 3mm) is the spacing of the Fabry-Perot étalon, and the mean values ∆ and δ are
calculated in the following way
)(2
1 1,1,
1
−−
=
∆+∆∑=∆npnp
b
npnp
a
n
pn
pn
pnδ δ
2
12
1
=
∑= (21)
Since the central line split symmetrically, the change in energy of radiating electrons is given by
2
vhc E
∆=∆ (22)
and this change in energy ∆E is proportional to the magnetic flux density B by a factor µB,
B E B µ =∆ (23)
Determine the Bohr magneton, µB.
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6. Questions
1. Does the position of the analyzer affect the results of the experiment?
2. If the coils and light source are turned 90°, how will the results change? Describe the
changes if any.