Department of Physical Electronics
Faculty of Science, Masaryk University, Brno
Physical laboratory 3
Task E Zeeman effect
Once a man was walking on a street in Copenhagen (it was W. Pauli) with his face so gloomy
that an unknown older lady worriedly asked him after the reason for his anxiety. “ But, madam,”
he answered, “ I cannot understand the anomalous Zeeman effect.” After its discovery, the Zeeman
effect was indeed quite a riddle suggesting something about the atomic structure, but it was
yet challenging to solve this enigma. The spectral lines of atoms were splitting into a different
number of components. The change in the energy of the radiation was directly proportional to
the magnetic induction. However, the proportionality constant was sometimes an integer multiple
and sometimes a rational multiple of a different constant, the so-called Bohr magneton. Different
components exhibited different polarisation, as well as their numbers were different. This was
moreover dependent on the direction from which the observation was made. Today, the change of
the energy of atoms in a magnetic field discovered in the Zeeman effect is used not only in optical
spectroscopy. It also forms the basis of electron paramagnetic resonance, polarisation spectroscopy,
different laser tuning and cooling using the adiabatic demagnetisation of paramagnetics.
Tasks
1. Verify the behaviour of the Fabry-Perot interferometer. Show that the measured radii of
different interference rings of a single wavelength correspond to a suitable formula from the
manual.
2. Determine the value of the Bohr magneton using the shift of the wavenumbers in the normal
Zeeman effect. Turn the magnetic field on only after getting clearance from the teacher!
3. Determine which components of the split spectral lines are emitted in the direction perpendicular
to the magnetic induction direction and which are emitted in the direction parallel
to the magnetic induction direction. Measure the polarisation of all the different line components
in both directions mentioned above.
Think before the laboratory:
• How do you construct a telescope from two converging lenses with focal lengths of 5 cm and
30 cm?
• Why do energy levels of electrons change in a magnetic field?
• How do you determine the energy difference of two photons of two close spectral lines from
the interference pattern provided by the Fabry-Perot interferometer?
Physical laboratory 3 manuals (version May 10, 2023) 2
Figure 1: Energy level diagram of cadmium.
• How do you measure the radiation if it has circular polarisation and only a linear polarisation
filter and a quarter-wave plate is available? You can find some fundamental facts about
quarter-wave plates in the addendum to this manual.
Theory of Zeeman effect
The Zeeman effect is the name for the splitting of spectral lines emitted by atoms in magnetic
fields. This splitting occurs due to the energy changes of the individual atomic energy levels
caused by the external magnetic field. Pieter Zeeman first described this effect in 1896, and it
brought in new insights into electron energy levels in the atom and about the quantum nature
of momentum projection in a selected direction. When observing radiation emitted by an atom
transition between two singlet states, we will see the so-called normal Zeeman effect. Singlet states
are states where the spins of all the electrons compensate, and the total spin of the atom is zero.
The normal Zeeman effect is caused only by the interaction of the orbital magnetic momentum
of the atom with the external magnetic field. If the total spin of the atom is non-zero, the effect
of the spin in the external magnetic field also has to be counted towards the energy of the atom.
When observing spectral lines of transitions between states where at least one is not a singlet, we
will observe a more complicated line splitting - the so-called anomalous Zeeman effect. We will
not take into account any possible atomic nucleus’ spin in this task.
We will observe the Zeeman effect on selected spectral lines of cadmium. The electron configuration
of cadmium in the ground state is (Kr) 4d10 5s2, and it is a singlet. The ground state
and the excited states of cadmium with the corresponding energies are shown in figure 1. The
Physical laboratory 3 manuals (version May 10, 2023) 3
cadmium states are denoted by something like 2 3S1. In such a case, the letter “S” denotes the total
orbital momentum of the atom represented by the quantum number L. “S” specifically denotes
a state with L = 0, “P” would denote a state with L = 1, “D” with L = 2, “F” with L = 3 etc.
The superscript “3” in the upper left is called the multiplicity, and it can be calculated from the
quantum number S showing the magnitude of the total electron spin of the atom as 2S + 1. Singlet
states carry the multiplicity of 1. The subscript “1” on the bottom right shows the quantum
number of the total momentum of the atom J. It can have the values of L + S, L + S − 1 etc. up
to |L − S|. The leading number “2” is connected to the principal quantum number of the excited
electron – electron in its ground state is denoted by the number “1”. Therefore, an electron in the
2 3S1 state is excited to the 6s level (the ground state has no electrons higher than 5s).
In a magnetic field, the cadmium energy levels split and their energy shifts according to the
projection of the atomic magnetic moment in the external magnetic field direction. When we
describe the motion of a single electron in an atom using a wave function and the laws of quantum
mechanics, we find out that it can exist only in the states described by an integer value of the socalled
orbital angular momentum (or azimuthal) quantum number l. In these states the electron
orbital momentum l1 magnitude is |l1| = l(l + 1). This single electron would have the magnetic
moment due to its orbital motion (at the moment not considering spin) of
µ1 = −
e
2me
l1,
where e denotes the elementary charge and me the electron mass. The magnetic moment magnitude
will be
|µ1| =
e
2me
l(l + 1) = µB l(l + 1),
where
µB =
e
2me
≈ 9,274.10−24
Am2
(1)
is called the Bohr magneton. The projection of the magnetic moment µ1 in a selected direction
can only have the values of µB.ml, where ml is an arbitrary integer in the interval of −l to +l.
Similarly for a multielectron system with the total orbital momentum of
Ltot =
i
li
we can write
|Ltot| = L(L + 1) (2)
|µL| = µB L(L + 1), (3)
where µL is the orbital magnetic momentum.
The total atomic momentum Jtot is in the case of the LS bond a sum of the total orbital and
total spin (Stot) momentum
Jtot = Ltot + Stot. (4)
We again describe its magnitude by the quantum number J as:
|Jtot| = |Ltot + Stot| = J(J + 1). (5)
The momentum Jtot is connected to the magnetic momentum µJ . The magnitude of the mean
magnetic momentum can be described as
µ = gJ µB J(J + 1), (6)
where gJ is the Landé g-factor.
Physical laboratory 3 manuals (version May 10, 2023) 4
B=0 B>0 m
−1 0
σ σπ
+1
1
2
J
−2
−1
0
1
2
−1
0
1
L=2
S=0
L=1
S=0
m =J∆
3 D
1
2 P
1
Figure 2: Splitting of the 3 1D2 and 2 1P1 cadmium
states in a magnetic field. The arrows
denote electric dipole transitions.
2
1
−1
0
1
−1
0
1
L=0
S=1
L=1
S=1
2 S
3
3
2 P
∆m =J −1
π σ
2
−2
−2
0
2
−3
0
3
3/2
−3/2
σ
0 +1
2
3
1
4
5
6
7
8
9
B=0 B>0 m m gJ J J
Figure 3: Splitting of the 2 3S1 and 2 3P2 cadmium
states in a magnetic field. The arrows
denote electric dipole transitions.
Normal Zeeman effect
The normal Zeeman effect occurs for transitions between singlet states, where the total electron
spin Stot is zero. The total magnetic momentum of the atom can then be written simply as
µJ = µL, J = L a gJ = 1. The projection of the total magnetic momentum in the magnetic
field direction can take on values of mJ · gJ · µB, where the magnetic quantum number mJ can
be an arbitrary integer between −J and +J. The energy of the interaction between the magnetic
momentum µJ and the external magnetic field with the induction B has the magnitude of
EmJ = −mJ gJ µB B. (7)
In this laboratory, we will observe the normal Zeeman effect on the cadmium 3 1D2 → 2 1P1
transition corresponding to the wavelength of 643.847 nm. In the initial state J = L = 2 and the
magnetic quantum number mJ1 can take on values of -2, -1, 0, 1, 2. In the final state J = L = 1
and the magnetic quantum number mJ1 can take on values of -1, 0, 1. As the Landé g-factor
gJ = 1 in both states, the energy difference between two neighbouring energy sub-levels in both
states is µB · B. The corresponding energy level splitting is shown in figure 2. The energy of the
photon emitted in such a transition changes by
EmJ1 − EmJ2 = (mJ2 − mJ1) µB B (8)
As the dipole moment transition selection rules allow only the ∆mJ difference of -1, 0, 1, the
observed line will split into three lines in a magnetic field. The centre line with ∆mJ = 0 is
denoted as π, and its wavelength does not change in the magnetic field. The side-lines with
∆mJ = ±1 are denoted σ, and their photons have their energy shifted by ±µB B from the central
π line. All of these lines are emitted in the direction perpendicular to the magnetic induction.
However, only some of them are emitted in the direction parallel to the magnetic field. Not only
the number of the lines but also their polarisation is dependent on the emission direction. The
polarisation can change all the way from linear to circular. Measuring the line polarisation and
determining which lines are emitted into which directions is one of the tasks of this laboratory.
Although the changes in the polarisation and the different number of lines in different directions
might seem confusing at first glance, both can be understood even in the classical conception of
an electron oscillating in a magnetic field.
Physical laboratory 3 manuals (version May 10, 2023) 5
Anomalous Zeeman effect
The situation is more complicated in the case of the anomalous Zeeman effect. This is due to the
existence of a non-zero total electron spin Stot =
i
si. Its magnitude and magnetic momentum
are described by the quantum number S as
|Stot| = S(S + 1) (9)
|µS| = gs µB S(S + 1), (10)
where gs = 2.0023 is called the gyromagnetic factor. We obtain the total momentum of the
atom as the sum of its orbital and spin momenta according to (4) and (5). The total magnetic
momentum of the atom µJ = µL + µS is not parallel to the total momentum of the atom Jtot
due to the existence of the gyromagnetic factor of 2.0023. We are, however, interested only in the
mean value of the magnetic momentum. This value is at the same time the projection of the µJ
vector in the direction of the Jtot vector. We can calculate its value from geometric considerations
µ =
3J(J + 1) + S(S + 1) − L(L + 1)
2 J(J + 1)
µB
valid in the gs ≈ 2 approximation. From the definition of the Landé g-factor
µ = gJ µB J(J + 1)
we obtain
gJ = 1 +
J(J + 1) + S(S + 1) − L(L + 1)
2J(J + 1)
. (11)
As the projection of µ into the direction of the magnetic field can take on values of mJ · gJ · µB
with the magnetic quantum number mJ = J, J − 1, . . . , −J, we once again obtain formula (7)
for the description of the change of the energy of the atom in a magnetic field.
In this laboratory, we will study the anomalous Zeeman effect on the 2 3S1 → 2 3P2 cadmium
transition with the emission line with the wavelength of 508.588 nm. The initial state is characterised
by quantum numbers L = 0, S = 1 a J = 1 and the Landé g-factor gJ1 = 2 and the final
state by L = 1, S = 1, J = 2 and gJ2 = 3/2. The magnetic quantum number mJ can have the
values of -1, 0, 1 in the initial state and -2, -1, 0, 1, 2 in the final state. The diagram of the
split atomic levels of the 2 3S1 and 2 3P2 states together with dipole transitions allowed by the
transition rule ∆mJ = -1, 0, 1 is shown in figure 3. It can be seen that the observed line will
split into nine components (three π and six σ). The energy shift of the different components is
described by
EmJ1 − EmJ2 = (mJ2 gJ2 − mJ1 gJ1) µB B (12)
The properties of the individual lines can be deduced from the following table.
Number 1 2 3 4 5 6 7 8 9
∆mJ 1 1 1 0 0 0 -1 -1 -1
mJ2 gJ2 − mJ1 gJ1 -2 -3/2 -1 -1/2 0 1/2 1 3/2 2
Fabry-Perot interferometer
As the individual line components are very close in energy and wavelength, a high-resolution
device is needed for their analysis. Therefore a Fabry-Perot interferometer is used. In this case, it
is a 3 mm silica plate covered on both sides by a thin film reflecting 90 % of the incident light. The
Fabry-Perot interferometer is often used in spectroscopy, telecommunications, and laser technology
for wavelength measuring or filtering a specific wavelength. For example, it is used for the choice of
different wavelengths on the recipient’s side of a wave multiplex, it forms the basis of interference
Physical laboratory 3 manuals (version May 10, 2023) 6
d
α
Θ
b
a
Figure 4: Beam paths in the Fabry-Perot interferometer.
The optical path difference of two
neighbouring beams is 2na − b.
f tg(Z )
ZΘm
Θm
f
Figure 5: Interference pattern displaying by a
converging lens.
standards and some dichroic filters, it is also used to choose a single-mode from all the modes
of the resonance chamber of a single-mode laser. Owing to its high resolution, the Fabry-Perot
interferometer allows for the measurement of narrow spectral lines. The way how to measure the
difference of the close spectral lines is described in the following paragraphs.
The path of the individual rays in the Fabry-Perot interferometer is shown in figure 4. For
the greatest intensity of the radiation passing through the FP interferometer, the phase difference
of the individual rays needs to be integer times 2π. Because the optical path difference of two
neighbouring rays is 2nd cos α, the constructive interference condition can be written as
mλ = 2nd cos αm, (13)
where m is an arbitrary integer, λ is the wavelength of the light, 2 = 1 + 1, n the refractive
index of silicate glass (1.4519 for the wavelength of 509 nm and 1.4560 for 644 nm) and d the
thickness of the interferometer. As the condition (13) is fulfilled for only certain angles αm, the
monochromatic radiation after passing the interferometer will form a pattern consisting of light
and dark circles. Let us denote
M =
2nd
λ
.
It can be seen that the maxima of interference are formed for m ≤ M. We can rewrite (13) for
small angles αm as
m = M cos αm ≈ M 1 −
α2
m
2
and for the angles fulfilling the constructive interference condition we obtain
αm ≈
2 (M − m)
M
(14)
The light is refracted after exiting the interferometer, and the constructively interfering rays
exit at the angle Θm. The interference pattern is then Z-times magnified by a telescope and then
displayed by a converging lens with the focal distance f onto a chip of a CCD camera (see fig. 5),
where it forms light rings with radii
rm = f tg(ZΘm) ≈ fZΘm ≈ fZ nαm (15)
We can get the final formula or the radii of the interference rings from the last to equations. Let
us observe rays propagating at a small angle Θm. Their m is, however, a large number that is
difficult to determine exactly. It is therefore practical to use the index of p = mmax − m + 1
Physical laboratory 3 manuals (version May 10, 2023) 7
numbering the rings from the smallest one to larger ones (mmax is the largest index m fulfilling
the m ≤ M condition). Combining (14) and (15) using new indexing we obtain
r2
p = 2(fZn)2
1 −
mmax + 1
M
+
p
M
(16)
We can measure the distance between two close spectral lines by comparing the difference of
the interference ring radii of these lines with the difference of the neighbouring rings of a single
spectral line. To do this we rewrite (16) using wavenumber ˜λ = 1/λ as
r2
p = 2(fZn)2
1 −
mmax + 1
2nd˜λ
+
p
2nd˜λ
and for the difference of the squares of the radii of two spectral lines (ra a rb) with close wavenumbers
˜λa a ˜λb we obtain
r2
b,p − r2
a,p = 2(fZn)2 mmax + 1 − p
2nd
1
˜λa
−
1
˜λb
(17)
Next we calculate the difference of the squares of radii of two neighbouring rings of a single spectral
line
r2
a,p+1 − r2
a,p = 2(fZn)2 1
2nd˜λa
(18)
and by dividing (17) by (18) we obtain
r2
b,p − r2
a,p
r2
a,p+1 − r2
a,p
= (mmax+1−p)
˜λb − ˜λa
˜λb
We can easily measure the left side of the equation. As mmax ≈ M 1 a ˜λa ≈ ˜λb, we can rewrite
the previous equation as
r2
b,p − r2
a,p
r2
a,p+1 − r2
a,p
= 2nd (˜λb − ˜λa). (19)
This formula enables us to easily measure the difference of the wavenumbers of two close spectral
lines.
Experimental setup
The cadmium lamp and the magnets are placed on a rotating table (see fig. 6). The lamp overheats
if the magnetic field is turned on for too long. Therefore, the magnetic field needs to be applied
for the shortest time possible! The radiation is exiting from the lamp through the gap between the
electromagnets and through a hole in the magnet holder, enabling to observe the emission lines
from a direction perpendicular to and parallel with the vector of the magnetic induction. The
electromagnets can be powered by a DC current of up to 10 A. The relation between the current
going through the electromagnets and the magnetic induction magnitude in the gap between the
electromagnets can be found in a table uploaded to the study materials in the information system.
You can use the following excerpt as a rough guide:
I [A] 2.97 5.05 6.06 7.1 8.06 8.96 9.95
B [mT] 188 319 381 444 500 549 594
An iris used for observing the light emitted in the direction perpendicular to the magnetic field
is placed behind the rotating table. The hole in the magnetic field holder serves as the iris when
observing the light emitted in the direction parallel to the magnetic field. Next, a converging lens
Physical laboratory 3 manuals (version May 10, 2023) 8
with a focal length of 5 cm is placed. It creates light with nearly parallel beams that are entering
the Fabry-Perot interferometer.
The student builds the optical path between the interferometer and a camera. The interference
pattern needs to be magnified by a telescope. You can use two converging lenses with focal lengths
of 5 cm and 30 cm to build the telescope. A CCD camera captures the final image at the end of
the optical path. This image is being sent to a computer. Moreover, a polarising filter and
a quarter-wave plate, used for the determination of the line polarisation, can be placed in the
optical path. As cadmium emits many different wavelengths, the resulting interference pattern is
unclear. Therefore, we need to filter out only the measured optical line. Use the red filter placed
into a slot at the end of the Fabry-Perot interferometer to observe the 644 nm for the normal
Zeeman effect. Use the green interference filter that can be mounted on any lens to observe the
anomalous Zeeman effect.
Experimental procedure
• Turn the table with the magnets and the cadmium lamp to allow the light emitted perpendicularly
to the magnetic field to go through the optical path. The iris should be placed 5 cm
in front of the converging lens in the optical path. While the magnetic field is turned off,
place the two converging lenses behind the Fabry-Perot interferometer. The lenses with the
focal lengths of 5 cm and 30 cm should form a magnifying telescope making the interference
clearly visible on the CCD. Cover the optical path from the exterior lighting for consecutive
measurements.
• Image as many interference circles of the red (643.847 nm) or green (508.588 nm) cadmium
line as possible. Block the other lines by using the red or the green interference filter. Verify
that the interference ring radii are in accordance with the theoretical expectations shown in
section 2. It is advisable to use the line showing the higher number of interference rings.
• You will be using the magnetic field in the following tasks. As the cadmium lamp overheats
in such strong magnetic fields, it is necessary to measure the following tasks swiftly. Do not
set the current going through the electromagnets above 10 A.
• While observing the light emitted perpendicularly to the magnetic field, image the interference
patterns of the studied line at different currents through the electromagnets. You
will measure the spectral line splitting and calculate the Bohr magneton from these images.
Determine the Bohr magneton from the red line only. Determine the polarisation of all the
components of the split spectral line. You have the linear polarisation filter and a quarterwave
plate available for this task. Do this for the normal Zeeman effect of the red line as
well as the anomalous Zeeman effect of the green line. Turn the magnetic field off while not
acquiring images!
interferometr kamera
CCD
elektromagnet
Cd lampa
clona
irisová
filtr
polarizační
Fabry−Perotův
otočný stolek
electromagnet
Cd lamp
rotating
table
iris
Fabry-Perot
interferometer
polarising
filter
CCD
camera
Figure 6: Diagram of the experimental system.
Physical laboratory 3 manuals (version May 10, 2023) 9
21 4 5 6 7 83
9 10
Figure 7: Experimental system: 1 - Cadmium lamp power source, 2 - Electromagnet power source,
3 - Cadmium lamp with electromagnets, 4 - Iris, 5 - Fabry-Perot interferometer, 6 - Lenses, 7 Polarising
filter, 8 - CCD camera, 9 - Interference filter, 10 - Quarter-wave plate.
• When the magnetic field is turned off, rotate the table so that the emission going through
the hole in the magnet holder is aimed at the optical path. You do not need to use the iris
now. Check if the interference pattern is still visible on the screen.
• Determine which components of both lines (643.847 nm and 508.588 nm) are emitted in the
direction parallel to the magnetic field and what is their polarisation.
• You can use “Motic images” to evaluate the acquired interference patterns. The images are
saved by “file – capture window – capture” and the radii of the interference rings can be
directly measured by “measure – circle” or “circle (3 points)”. Measure three line triplets for
each of 10 different magnetic fields. Compare the calculated value of the Bohr magneton
with the tabulated value (1).
Addendum: Waveplates
Waveplates are simple devices made of optically anisotropic materials, i.e. materials with the
refraction index depending on the polarisation of the light. The waveplate will slow the radiation
polarised in the y axis compared to the radiation polarised in the x axis by a phase difference δ
given by the refractive indexes of the phase plate material for the two polarisations (ny − nx), the
plate thickness (df ) and the wavelength of the radiation as
δ =
2π(ny − nx)df
λ
.
For example, a waveplate made of mica having refractive indexes of 1.594 a 1.599 for 633 nm
radiation would cause a phase shift by π having a thickness of only 63 µm. The phase shift δ
of many materials can be affected, e.g. by an electric field. This is used in optoelectronics and
imaging technology.
Physical laboratory 3 manuals (version May 10, 2023) 10
The x and y axes are called the fast axis and the slow axis of the waveplate. We can describe
the radiation by the Jones vector with its components being complex amplitudes of projection of
the wave’s electric field into these axes. The waveplate is characterised by the Jones matrix
F =
1 0
0 e−iδ
Nothing interesting happens if the incident radiation is polarised in the direction of the principal
axes of the plate. The situation is more interesting if the radiation is polarised in a general
direction or has a circular polarisation. Practically we use two main waveplate types:
• A quarter-wave plate shifts the phase by δ = π/2. A linearly polarised light with its
polarisation axis being turned by 45◦ compared to the principal axes of the plate changes to
a circularly polarised, while previously circularly polarised light becomes linearly polarised.
• A half-wave plate with δ = π rotates the plane of linearly polarised light, changes rightcircularly
polarised light into left-circularly polarised light and vice versa.