Experiment: 4

Study of Zeeman Effect

Jiten Mahalik ∥ 2211051

August 27, 2024

The experiment aimed to determine the dielectric constants of different materials to gain insights into their
electrical behavior. Utilizing a parallel-plate capacitor configuration, materials including air, wood, and
styrofoam were tested. By measuring capacitance and dimensions, the dielectric constants were calculated.
The outcomes revealed significant variations in dielectric constants among the tested materials, emphasizing
their crucial role in electrical applications. This information is invaluable for electronic component design,
optimization, and materials science research, contributing to advances in electrical engineering. Though
it is not a very high-level study of dielectric materials, this study underscores the significance of dielectric
selection in various applications and enhances our understanding of the electrical behavior of diverse
materials.

1 Objectives motion. The ”normal” Zeeman effect, observed

in simple atomic systems, involves equally spaced
1. Quantitatively study transverse normal Zee- line splitting and is explained by classical physics.
man effect by observing the splitting of the However, the ”anomalous” Zeeman effect, found
rings due to magnetic field resolved by Fabry- in more complex atoms, involves unequal splitting
Perot etalon using a CMOS camera and evalu- and required the development of quantum mechan-
ate the value of Bohr’s magneton (µB). Observe ics and the introduction of electron spin for a full
the polarization of the rings using a polarizer. explanation.

2. Observe the left circular and right circular po-

larized lines in anomalous normal Zeeman ef-
fect by using quarter wave plate and polarizer. 3 Apparatus required
3. Observe the transverse anomalous Zeeman ef- 1. Experimental setup consists of Cd lamp with
fect and polarization of the rings using a po- power supply.
larizer.
2. Permanent magnet on rotating table
4. Observe the longitudinal anomalous Zeeman
effect and left circular and right circular polar- 3. Two 50 mm convex lenses
ized lines in anomalous normal Zeeman effect
by using quarter wave plate and polarizer 4. 300 mm convex lens

5. Etalon arrangement with holder for red filter

and attached to 100 mm lens in a tube.
2 Introduction
6. CMOS camera with 8 mm lens attached to it
and connected to computer.
The Zeeman effect, discovered by Dutch physicist
Pieter Zeeman in 1896, is the splitting of spectral 7. Green filter.
lines in the presence of a magnetic field. Hen-
drik Lorentz, explained this phenomenon using 8. Polarizer.
classical electrodynamics, linking it to the interac-
tion between magnetic fields and atomic electron 9. Quarter wave plate.

1
4 Theory 5.1 Magnetic Moment and Orbital An-
gular Momentum

The Zeeman effect arises from the interaction be- In an atom, electrons possess both orbital and spin
tween an external magnetic field and the magnetic angular momentum. For the normal Zeeman effect,
moments of electrons within an atom. When an we focus on the orbital angular momentum, repre-
atom is exposed to a magnetic field, the energy lev- sented by the quantum number L. The magnetic
els of its electrons are split, leading to the character- moment µL associated with this angular momen-
istic splitting of spectral lines. This phenomenon tum is given by:
is understood through quantum mechanics, where
the magnetic moments associated with the orbital
e
and spin angular momentum of electrons interact µL = − L
with the applied magnetic field, resulting in observ- 2me
able energy shifts.
where e is the electron charge, me is the electron
The Zeeman effect is broadly classified into two mass, and L is the orbital angular momentum vec-
types: the normal Zeeman effect and the anoma- tor.
lous Zeeman effect. The normal Zeeman effect
occurs for spectral lines resulting from transitions
between singlet states, where the total spin of the 5.2 Effect of an External Magnetic Field
initial and final states is zero. On the other hand,
the anomalous Zeeman effect arises when the to-
tal spin of either the initial or final states, or both, When an atom is placed in an external magnetic
is nonzero, leading to more complex splitting pat- field B, the magnetic moment µL interacts with this
terns. This interaction between both the orbital and field, leading to a potential energy given by:
spin angular momenta with the external magnetic
field often produces multiple spectral lines.
U = −µL · B
The anomalous Zeeman effect is more commonly
observed and provides deeper insights into atomic Since the magnetic moment is proportional to the
structure and the significance of electron spin. orbital angular momentum, the energy levels split
These splitting patterns not only deepen our under- depending on the orientation of L with respect to
standing of the interaction between light and matter the magnetic field.
but also serve as a crucial tool in spectroscopy and
magnetic field measurements.
5.3 Quantization and Energy Shift

The orbital angular momentum is quantized, and

its projection along the direction of the magnetic
field is given by:
5 Normal Zeeman Effect
L z = mL ℏ
The normal Zeeman effect occurs when an atom
is subjected to an external magnetic field, causing where mL is the magnetic quantum number, and ℏ
the splitting of a spectral line into multiple com- is the reduced Planck constant. The energy shift ∆E
ponents. This effect is particularly straightforward due to the magnetic field is then:
to analyze in atoms or ions with a single unpaired
electron and no spin (i.e., when the total spin quan-
tum number S = 0). The splitting occurs due to ∆E = mL µB B
the interaction between the magnetic field and the
magnetic moment associated with the orbital angu- Here, µB = 2m eℏ
e
is the Bohr magneton, and B is the
lar momentum of the electron. strength of the magnetic field. This equation shows

2
that the energy level of the electron is shifted by an
amount proportional to both the magnetic field and
the magnetic quantum number mL .

5.4 Splitting of Spectral Lines

In the absence of a magnetic field, an atomic transi-

tion between two energy levels E1 and E2 produces
a single spectral line corresponding to the energy
difference ∆E = E2 − E1 . In the presence of a mag-
netic field, each energy level E1 and E2 splits into
2L1 + 1 and 2L2 + 1 sub-levels, respectively.

The possible transitions between these sub-levels

obey the selection rule:

∆mL = 0, ±1
Figure 1: Splitting of lines for Cd spectral line in
These transitions result in the splitting of the origi- magnetic field and allowed transitions: Normal
nal spectral line into three components: Zeeman effect

• ∆mL = +1: This results in the σ+ line, shifted to 6 Anomalous Zeeman Effect
a higher frequency.
• ∆mL = −1: This results in the σ− line, shifted to The Anomalous Zeeman Effect is the splitting of
a lower frequency. spectral lines of an atom in the presence of a mag-
• ∆mL = 0: This results in the π line, which re- netic field, which cannot be explained by the classi-
mains unshifted. cal Zeeman effect alone. This phenomenon occurs
due to the interaction between the magnetic field
and the total angular momentum (including both
The frequency shift ∆ν for the σ components is orbital and spin angular momentum) of the elec-
given by: trons in the atom.

µB B
∆ν =
h 6.1 Cause of the Anomalous Zeeman Ef-
fect
where h is Planck’s constant. The π line, corre-
sponding to ∆mL = 0, has no shift because there is
In atoms, electrons possess:
no change in the projection of angular momentum.

• Orbital Angular Momentum (L): Due to their

5.5 Energy Level Diagram motion around the nucleus.

• Spin Angular Momentum (S): An intrinsic

In a simplified energy level diagram, the presence form of angular momentum.
of a magnetic field causes the original energy levels
to split into multiple sub-levels. Each of these sub-
levels corresponds to a different mL value, resulting The total angular momentum J of an electron is
in distinct energy differences between the levels. given by the vector sum:
These differences produce the observed splitting of
spectral lines.The figur is shown in below. ⃗
⃗J = ⃗L + S

3
When an atom is placed in a magnetic field B, the • σ Lines: Correspond to transitions where
interaction between the magnetic field and the mag- ∆m J = ±1. These lines are observed in the
netic dipole moments associated with both orbital perpendicular direction to the magnetic field
and spin angular momenta leads to the energy split- and are circularly polarized.
ting.
• π Lines: Correspond to transitions where
∆m J = 0. These lines are observed in the paral-
lel direction to the magnetic field and are lin-
6.2 Energy Splitting in the Anomalous early polarized.
Zeeman Effect

The interaction of the magnetic moment with the 6.4 Energy Difference and Transition
magnetic field leads to an energy shift: Lines
∆E = −⃗ ⃗
µ·B
The energy difference between two split levels is
Where µ ⃗ is the ex-
⃗ is the magnetic moment, and B given by:
ternal magnetic field. ∆E = g J µB ∆m J B
For σ transitions (∆m J = ±1):
For an electron, the magnetic moment due to both
orbital and spin angular momenta can be expressed ∆Eσ = g J µB B
as:
⃗J For π transitions (∆m J = 0):
µ⃗J = −g J µB
ℏ
∆Eπ = 0
where:

In the presence of a magnetic field, each energy

• g J is the Landé g-factor. level with total angular momentum J splits into
• µB is the Bohr magneton. 2J + 1 levels, leading to the formation of multiple
lines (σ+ , σ− , and π) in the spectrum.
• ℏ is the reduced Planck constant.

The Landé g-factor accounts for the contribution of 6.5 Visualization of the Splitting
both L and S to the total angular momentum J. It is
given by: • Without Magnetic Field: The energy levels
are degenerate, and only a single spectral line
J(J + 1) + S(S + 1) − L(L + 1) is observed.
gJ = 1 +
2J(J + 1) • With Magnetic Field: The degeneracy is lifted,
The energy shift for a particular magnetic quantum and the spectral lines split into components,
number m J is then: which can be observed as distinct lines in the
presence of the magnetic field.
∆E = g J µB m J B

Here, m J is the magnetic quantum number associ- 6.6 Energy Level Diagram
ated with J, which can take values from −J to +J.

In a simplified energy level diagram, the presence

of a magnetic field causes the original energy levels
6.3 Formation of Spectral Lines to split into multiple sub-levels. Each of these sub-
levels corresponds to a different m J value, resulting
The splitting of energy levels results in transitions in distinct energy differences between the levels.
between these levels, leading to multiple spectral These differences produce the observed splitting of
lines instead of a single line. These lines are cate- spectral lines.The figur is shown in below.
gorized as:

4
 Where ‘δ’ is difference of squares of radii of differ-
ent lines of same order of interference, ‘∆’ is the
difference of squares of radii of different order and
t is the thickness of the quartz glass.

7 Experimental Setup

Experimental setup consists of Cd lamp with power

supply, permanent magnet on rotating table, two
50 mm convex lenses, 300 mm convex lens, Etalon
arrangement with holder for red filter and attached
Figure 2: Splitting of lines for Cd spectral line in to 100 mm lens in a tube, CMOS camera with 8
magnetic field and allowed transitions: Anomalous mm lens attached to it and connected to computer,
Zeeman effect green filter, polarizer and quarter wave plate.The
figure is shown in below.
In the case of Normal Zeeman effect, there are only
two σ-lines which are termed as ‘a’ and ‘b’. Sep-
aration of them is function of transverse magnetic
field (B).

∆E = µB B (1)
Figure 3: Experimental Setup for Zeeman Effect
The difference in wave numbers of one of the lines
with respect to central line of the same order is ∆K2.
For this case:

8 Procedure
∆K
∆E = hc (2)
2 • Turn on the cadmium lamp and allow it to
Combing above two equations, warm up while identifying the optical com-
ponents.

∆K • The experiment is divided into two primary

µB = hc (3) parts: the Normal Zeeman effect and the
2B
For the σ-lines of the transversal Zeeman effect, Anomalous Zeeman effect, each with trans-
amount of splitting increases with increasing mag- verse and longitudinal magnetic field compo-
netic field strength. For a quantitative measure- nents.
ment of this splitting in terms of number of wave- • Familiarize yourself with the CMOS camera
lengths, The Fabry-Perot étalon has a resolution of software, using ’Red gain’ for the Normal Zee-
approximately 400000 is used in this experiment. man effect and ’Green gain’ for the Anomalous
That means that a wavelength change of less than Zeeman effect.
0.002 nm can still be detected. The étalon consists
of a quartz glass plate of 3 mm thickness coated • Align the lamp, the first 50 mm lens, and the
on both sides with a partially reflecting layer (90% Fabry-Perot tube to observe the ring pattern
reflection, 10% transmission). Refractive index of with the naked eye.
quartz at 509 nm is 1.4519 and at 644 nm is 1.4560.
• Ensure even illumination on the CMOS camera
to center the ring pattern on the screen.
1 δ • If the first ring appears overexposed, adjust the
∆K = (4)
2µt ∆ aperture and lamp position for better clarity.

5
 • Once the ring pattern for the Normal Zeeman B = 629 mT
effect is confirmed, capture and save images at
each magnetic field setting for later analysis.
1st order 2nd order 3rd order
Rings (Radius) (Radius) (Radius)
• Use a polarizer to observe the polarization of
lines in each order and identify the σ and π (µm) (µm) (µm)
lines. Ring a 71.29 123.92 159.34
• Record images of the first three orders (each Ring b 87.47 133.86 166.91
order with three rings) for each magnetic field Ring c 101.24 142.93 175.82
using the camera software. Use the provided
data for magnetic field versus pole separation. Table 2: Rings of various order at B = 629mT
• Tabulate the square of ring radius readings and
calculate the wave number using the equation: B = 585 mT
1 δ 1st order 2nd order 3rd order
∆K = (5)
2µt ∆
Rings (Radius) (Radius) (Radius)
• Rotate the magnetic field, realign the optics, (µm) (µm) (µm)
and capture an image of the longitudinal Nor- Ring a 70.61 124.09 158.47
mal Zeeman effect. Use a quarter wave plate
and polarizer to identify left and right circu- Ring b 86.65 132.77 166.65
larly polarized light.
Ring c 99.33 141.28 173.28
• Return to the transverse field, switch to the
green filter, and align the optics to observe Table 3: Rings of various order at B = 585mT
eight rings per order in the Anomalous Zee-
man effect.
B = 542 mT
• Rotate the magnetic field and identify left and 1st order 2nd order 3rd order
right circularly polarized σ lines in the longi-
tudinal Anomalous Zeeman effect. Record im- Rings (Radius) (Radius) (Radius)
ages and verify that all σ lines are grouped (µm) (µm) (µm)
together.
Ring a 70.56 123.73 159.4
Ring b 85.87 132.62 165.94
9 Data acquisition / Observation Ring c 97.98 140.71 170.85
Table 4: Rings of various order at B = 542mT
B = 675 mT
1st order 2nd order 3rd order B = 505 mT
Rings (Radius) (Radius) (Radius) 1st order 2nd order 3rd order
(µm) (µm) (µm)
Rings (Radius) (Radius) (Radius)
Ring a 68.58 123.24 158.25
(µm) (µm) (µm)
Ring b 86.65 133.59 166.92
Ring c 101.91 143.67 175.24
Ring a 71.65 124.05 159.38
Ring b 85.9 132.27 165.58
Table 1: Rings of various order at B = 675mT Ring c 97.1 139.5 173.71
Table 5: Rings of various order at B = 505mT

6
 B = 448 mT
1st order 2nd order 3rd order
Rings (Radius) (Radius) (Radius)
(µm) (µm) (µm)
Ring a 72.91 124.88 160.45
Ring b 84.5 132.11 165.33
Ring c 94.84 138.59 169
Table 6: Rings of various order at B = 448mT
Figure 7: Normal transverse zeeman effect σ-lines

9.1 Figures

Figure 8: Normal transverse zeeman effect π-lines

Figure 4: Normal Transverse Zeeman Effect at B =
505mT.

Figure 5: Normal Transverse Zeeman Effect at B = Figure 9: Normal longitudinal zeeman effect
629mT.

Figure 10: Normal longitudinal zeeman effect σ line

Figure 6: Normal Transverse Zeeman Effect at B =
675mT.

7
 Figure 16: Anomalous transverse zeeman effect σ-
Figure 11: Normal longitudinal zeeman effect π line
lines

Figure 12: Anomalous transverse zeeman effect Figure 17: Anomalous transverse zeeman effect π-
ines

10 Calculation

From the observation tables we will get the values

of ∆K for different magnetic field values. Now,

Figure 13: Anomalous transverse zeeman effect σ B(T) ∆K(m−1 )

line
0.675 31.52
0.629 29.48
0.585 27.13
0.542 24.51
0.505 22.64
0.488 20.22
Table 7: Values of B and ∆K for different magnetic
Figure 14: Anomalous transverse zeeman effect π field strengths
line
below is plot the ∆K vs B graph, to get its slope
which will be used further to calculate the µB .

From the above plot, we get the following things:

Slope o f the plot = (58.058 ± 3.816) m−1 T−1

(6)
Figure 15: Anomalous transverse zeeman effect Intercept o f the plot = (−7.215 ± 2.192) m−1 T−1
(7)

From the equation - 67, we get:

8
 hc
σµB = σm (12)
2
where σµB denotes the error in the µB and σm
denotes the error in the slope of the plot i.e.,
3.186 m−1 T−1 (obtained from the plot).

6.626 × 10−34 × 3 × 108

σµB = × 3.186 JT−1 (13)
2
So, we get:

σµB = 0.317 × 10−24 JT−1

Figure 18: Plot of magnetic field(B) vs ∆K

12 Discussion on deviation from

the expected values
∆K
µB = hc (8)
2B Optical Component Alignment: The precision of
hc
µB = × slope o f the plot (9) the experiment heavily depends on the accurate
2 alignment of the optical components, particularly
6.626 × 10−34 × 3 × 108 the Fabry-Perot etalon and the lenses. These com-
µB = × 58.058 JT−1 (10)
2 ponents are crucial for forming and measuring the
interference patterns (rings) that reflect the spec-
So, we get:
tral line splitting due to the Zeeman effect. Any
µb = 5.770 × 10−24 JT−1 misalignment, even by a small amount, can cause
distortions in the ring patterns, leading to incor-
So,the calculated value of the Bohr magneton is rect measurements of their diameters. Since the
5.770 × 10−24 JT−1 ,which should closely match the ring diameters directly influence the calculation of
accepted theoretical value of 9.274 × 10−24 JT−1 . the energy difference between split levels, any error
here will propagate into the final value of the Bohr
Any deviation observed can be attributed to ex- magneton.
perimental limitations such as inaccuracies in the Fabry-Perot Etalon: The etalon is sensitive to align-
alignment of optical components, calibration of the ment and spacing. If the etalon plates are not per-
magnetic field etc. fectly parallel or if the spacing is inconsistent, the
resulting interference pattern will not accurately
represent the actual wavelength shifts. This could
significantly affect the determination of the wave-
11 Error Analysis length differences, which are critical for calculating
the energy splitting and, by extension, the Bohr
magneton.
From equation-9, we get: Measurement Precision: The CMOS camera is
used to capture the interference patterns produced
by the etalon. If the camera’s resolution is insuffi-
hc cient, it may not be able to accurately resolve closely
µB = × slope o f the plot (11) spaced rings. This limitation can lead to errors in
2
measuring the ring diameters, which are essential
Let’s m= the slope of the plot. Then the error in for determining the energy levels of the split spec-
the claculation of µB will be calculated using error tral lines.
propagation formula. Image Analysis: Even with a good-resolution cam-

9
era, the software or method used to analyze the cap- 14 Source of Error
tured images must be precise. Any error in identify-
ing the exact positions of the rings or in calculating 1. Inaccurate measurement or calibration of the
their radii could result in significant inaccuracies. magnetic field strength.
Incomplete Error Analysis: The discussion sug-
gests that the calculated uncertainty in the Bohr 2. Misalignment of lenses, Fabry-Perot etalon, or
magneton value (0.317 × 10−24 JT−1 ) seems too small polarizers.
given the large deviation from the theoretical value.
This discrepancy implies that not all potential 3. Insufficient resolution of the CMOS camera
sources of error were considered in the error prop- and inaccuracies in image analysis.
agation analysis. For instance, systematic errors 4. Variations in temperature affecting the stability
like alignment issues, fluctuations in the magnetic of the magnetic field and optical components.
field, or camera resolution limitations may have
been overlooked or underestimated. 5. Non-uniform magnetic field across the sample.

15 Precaution
1. Carefully align optical components using pre-
13 Conclusion cision tools.

2. Allow the cadmium lamp to warm up fully be-

fore starting the experiment to ensure a stable
In this experiment, we successfully investigated the and consistent light source.
Zeeman effect by examining the splitting of spec-
3. Adjust the aperture and lamp position care-
tral lines in the presence of a magnetic field. Our
fully to prevent overexposure of the rings,
study effectively demonstrated both the Normal
which could lead to inaccurate measurements.
and Anomalous Zeeman effects, providing a clear
visualization of the quantum mechanical interac- 4. Ensure that the magnetic field is stable and cor-
tions between magnetic fields and atomic energy rectly calibrated before taking measurements
levels. to prevent any fluctuations that could affect
the splitting of the spectral lines.
The observed splitting of spectral lines aligned well
with the theoretical predictions for the Normal Zee-
man effect. Under the influence of a transverse
magnetic field, the spectral lines split into three 16 Reference
distinct components: one unshifted (π-line) and
two symmetrically shifted (σ-lines) in opposite di- 1. Reese, Herbert M. ”An Investigation on the
rections. This observation confirms the theoretical Zeeman Effect.” Astrophysical Journal, vol.12,
understanding of the interaction between the mag- p. 120 12 (1900): 120.
netic moment associated with the electron’s angular
momentum and the external magnetic field. 2. Zettili, N. Quantum Mechanics(2nd ed.). Ap-
proximation methods for stationary states
In contrast, the Anomalous Zeeman effect revealed
a more complex splitting pattern due to the in-
volvement of both electron spin and orbital an-
gular momentum. The uneven splitting observed
in the spectral lines, as opposed to the equally
spaced lines seen in the Normal Zeeman effect, sup-
ports the theoretical model that incorporates elec-
tron spin. The use of the quarter-wave plate and
polarizer to identify left and right circularly polar-
ized light further validated the distinct polarization
characteristics of the σ-lines.

10