Zeeman Effect

Introduction:

The first person to study the effects of magnetic fields on the optical spectra of
atoms was Zeeman in 1896. He observed that the transition lines split when the
magnetic field is applied.

The “Zeeman effect” is the energy shift of atomic states caused by a magnetic
field. This shift is due to the coupling of the either orbital electron magnetic moment or
combined orbital and spin electron magnetic moment to the external magnetic field.

If only orbital electron magnetic moment is involved it is known as normal

Zeeman effect and if both orbital and spin magnetic moment are involved it is known
as Anomalous Zeeman effect.

In this experiment we use Cd lamp. We use red light (643.847 nm) of Cd lamp
for Normal Zeeman effect and green light (508.588nm) of Cd lamp for Anomalous
Zeeman effect. Further Zeeman splitting can be observed both in transverse and
longitudinal external magnetic field.

Objectives:

1. Quantitatively study transverse normal Zeeman effect by observing the splitting

of the rings due to magnetic field resolved by Fabry-Perot etalon using a CMOS
camera and evaluate the value of Bohr’s magneton (µB). Observe the
polarization of the rings using a polarizer
2. Observe the left circular and right circular polarized lines in anomalous normal
Zeeman effect by using quarter wave plate and polarizer.
3. Observe the transverse anomalous Zeeman effect and polarization of the rings
using a polarizer.
4. Observe the longitudinal anomalous Zeeman effect and left circular and right
circular polarized lines in anomalous normal Zeeman effect by using quarter
wave plate and polarizer
(Note. In this manual line and ring are synonymously used for spectral line)
Theory:

In quantum mechanics, a shift in the frequency and wavelength of a spectral line

implies a shift in the energy level of one or both of the states involved in the transition.
The Zeeman effect that occurs for spectral lines resulting from a transition between
singlet states is traditionally called the normal effect, while that which occurs when the
total spin of either the initial or final states, or both, is nonzero is called the anomalous
effect.
The splitting of the Cd-spectral line l = 643.8 nm into three lines, the so-called Lorentz
triplets, occurs since the Cd-atom represents a singlet system of total spin S = 0. In
the absence of a magnetic field there is only one possible 𝐷 → 𝑃 transition of 643.8
nm, as indicated by Fig. 1. In the presence of a magnetic field the associated energy
levels split into 2 L + 1 components. Radiating transitions between these components
are possible, provided that the selection rules

∆𝑀 = −1, ∆𝑀 = 0, ∆𝑀 = +1

are taken into account. In this case, therefore, there are a total of nine permitted
transitions. These nine transitions can be grouped into three groups of three transitions
each, where all transitions in a group have the same energy and hence the same
wavelength. Therefore, only three lines will be visible.

Fig.1: Splitting of lines for Cd spectral line in magnetic field and allowed
transitions: Normal Zeeman effect

The first group where ∆𝑀 = −1 gives a σ-line the light of which is polarized vertically
to the magnetic field in the transverse magnetic field. The middle group ∆𝑀 = 0 gives
a π-line. This light is polarized parallel to the direction of the field in the transverse
magnetic field. The last group where ∆𝑀 = +1 gives an σ-line the light of which is
again polarized vertically to the magnetic field in the transverse magnetic field. (Fig.2)
In the longitudinal field, we see only sigma lines (Fig.2) which are right circular and left
circular polarized (σ+ and σ-).
 Fig.2: polarization of lines in transverse and longitudinal magnetic field

In the normal Zeeman effect with the transition with 643.847 nm

the electron spins cancel each other in both the initial and final state (S=0) and the
energy of an atomic state in a magnetic field depends only on the magnetic moments
of the electron orbit. Term symbol is (2S+1LJ)

In the anomalous Zeeman effect with the transition S=1 for both initial and final states
of the transition 𝟐 𝟑𝑺1→ 𝟑𝑷2 . For this transition green light of Cd at 508.5 nm light is
used. Energy diagram splitting of the lines and allowed transitions is given in Fig. 3

There are total nine transitions in presence of magnetic field with three each for the
below selection rules.

∆𝑀 = −1, ∆𝑀 = 0, ∆𝑀 = +1

This corresponds to total 9 transitions with six sigma lines (vertically polarized) and
three pi lines (horizontally polarized in the transverse anomalous Zeeman effect and
6 sigma lines in longitudinal anomalous Zeeman effect with three left circularly
polarized and three right circularly polarized.
 Fig.3: Splitting of lines for Cd spectral line in magnetic field and allowed
transitions: anomalous Zeeman effect
So in case of LS-coupling in the anomalous Zeeman effect nine equidistant lines are
expected in this transition instead of three without spin magnetism. The polarization of
the transitions with in transversal observation is parallel to the magnetic field (here
horizontal) and the polarization of the other transitions is perpendicular to the magnetic
field.
In the case of Normal Zeeman effect, there are only two σ-lines which are termed as
‘a’ and ‘b’. Separation of them is function of transverse magnetic field (B).

𝚫𝑬 = 𝝁𝑩 𝑩 ---(1)

The difference in wave numbers of one of the lines with respect to other line of the
same order is Δk/2. For this case
∆𝒌
∆𝑬 = 𝒉𝒄 --(2)
𝟐

Combing above two equations,

∆𝒌
𝝁𝑩 = 𝒉𝒄 ---(3)
𝟐𝑩

For the σ-lines of the transversal Zeeman effect, amount of splitting increases with
increasing magnetic field strength. For a quantitative measurement of this splitting in
terms of number of wavelengths, The Fabry-Perot étalon has a resolution of
approximately 400000 is used in this experiment. That means that a wavelength
change of less than 0.002 nm can still be detected. The étalon consists of a quartz
glass plate of 3 mm thickness coated on both sides with a partially reflecting layer (90
% reflection, 10 % transmission). Refractive index of quartz at 509 nm is 1.4519 and
at 644 n is 1.4560.
𝟏 𝜹
∆𝒌 = ---(4)
𝟐𝝁𝒕 ∆

Where ‘δ’ is difference of squares of radii of different lines of same order of

interference and ‘Δ’ is the difference of squares of radii of different order. For
complete derivation of the above equations refer to the references.

Fig 4. Fabry perot etalon

Experimental arrangement:
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 to 100 mm lens in a tube, CMOS camera with
8 mm lens attached to it and connected to computer, green filter, polarizer and quarter
wave plate. Complete picture is given in the following figure.

Fig.5: complete experimental setup

In the above arrangement first 50 mm lens is to focus light on the lens attached to
tube with Fabry perot etalon. Light coming out of Etalon is focused with 300 mm lens
and brought to camera by 50 mm lens. Polarizer and quarter wave plate can be
placed in between the 300 mm lens and 50 mm lens whenever required.

Procedure:
1. Switch on the Cd lamp and wait for warm up. Identify all the optical components
given to you.
2. There two main parts of the experiment. Normal Zeeman effect and Anomalous
Zeeman effect. In each part there are two subparts: magnetic field in transverse
direction and magnetic field in longitudinal direction.
3. Get familiarized with software which captures images from CMOScamera. Use
only ‘Red gain’ in the Software for Normal Zeeman effect and ‘green’ for
Anomalous Zeeman.
4. Align the lamp, first 500 mm lens and Fabry perot tube and observe the ring
pattern coming out from etalon with a naked eye.
5. Now place the 300 mm lens and 50 mm lens along with camera and align the
entire setup, adjust the camera settings and observe the rings of first three
orders in normal Zeeman effect in the transverse field. An example picture is
given in the figure 6. (some part of the third order and may be second order is
not captured as the CMOS sensor in the camera is rectangular)
6. See that CMOS camera is equally illuminated to get the ring pattern at the
center of the screen.
7. Sometimes you may observe the pattern as Fig. 7. In Figure 7, first ring is filled
with light, try to trouble shoot by adjusting the light using given aperture by
decreasing the light and/or slightly moving the Cd lamp, finally locate the ring.
8. Sometimes you may observe the pattern as in Fig.8. Ignore the center spot and
it does not correspond to interference.
9. Observe that splitting is function of magnetic field by varying the distance
between pole pieces.
10. After confirming the ring pattern of Normal Zeeman effect with the ring pattern
up to 3 orders, take pictures using the software for each magnetic field and
save them. You need to use the saved files for estimating the radius. (here the
size of the ring (magnification) is not important as the calculation involves only
relative dimensions)
11. Place the polarizer and notice that middle line of each order is horizontally
polarized and outer lines (rings) are vertically polarized. Identify σ and 𝝅 lines.
12. Use the camera software and record the diameter of rings for first three orders
(each order three rings) for each magnetic field. Magnetic field versus pole
separation data is provided to you in the figure 10.
13. Tabulate the ring radius readings and calculate wave number using equation
𝟏 𝜹
(4) that is equation ∆𝒌 =
𝟐𝝁𝒕 ∆
 Table1. Components of each order rings are designated as ‘a, b and c’.
Here ‘a and c’ are outer σ lines and ‘b’ is π line
1st order 2nd order 3rd order
a R1,a2 R2,a2 R3,a2
b R1,b2 R2,b2 R2,b2
c R1,c2 R3,c2 R3,c2

𝟏
𝜹= 𝜹𝟏,𝒂𝒃 + 𝜹𝟏,𝒃𝒄 + 𝜹𝟐,𝒂𝒃 + 𝜹𝟐,𝒃𝒄 + 𝜹𝟑,𝒂𝒃 + 𝜹𝟑,𝒃𝒄 Where 𝜹𝒏,𝒙𝒚 = 𝑹𝟐𝒏,𝒚 − 𝑹𝟐𝒏,𝒙
𝟔

𝟏
∆= ∆𝒂𝟏 + ∆𝒃𝟏 + ∆𝒄𝟏 + ∆𝒂𝟐 + ∆𝒃𝟐 + ∆𝒄𝟐 Where, ∆𝒙𝒏 = 𝑹𝟐𝒏 𝟏,𝒙 − 𝑹𝟐𝒏,𝒙
𝟔

‘δ’ is difference of squares of radii of different lines of same order, calculate it

for both the components of ‘σ’ line and take average. ‘Δ’ is the difference of
squares of radii of different order. You may avoid some of them if they are
deviated from other values.

14. Use equation 3 to calculate Bohr magnetron(μB) by plotting necessary graph.

15. Record a picture in longitudinal normal Zeeman effect after rotating the
magnetic field and realigning the optics. In this case identify the left circularly
and right circularly polarized light using quarter wave plate and polarizer.
(quarter wave plate converts linearly polarized light to circularly polarized light
and here we use the inverse effect)
16. Bring back the magnet to transverse field, Remove Red filter and attach the
green filter to one of the lens holder.
17. Align the optics and observe 8 rings for each order (due to the insufficient
resolution of Fabry perot etalon, instead of 9 rings, we see only 8 or sometimes
7 rings.)
18. Identify the σ and π lines and record pictures for the same. Does the σ lines
are separated by π lines like in the normal Zeeman effect? They won’t.
19. Rotate the magnetic field and identify the lines left circularly polarized and right
circularly polarized σ lines in longitudinal anomalous Zeeman effect, record
pictures for the cases. Verify that all σ lines are bunched together unlike in
normal Zeeman effect in which they are separated by π line.
 Fig.6 Pattern recorded with lab setup for Normal Zeeman effect after
alignment

Fig.7. In this case first ring is filled with light instead of ring. Adjust the
intensity or use the aperture and locate the ring diameter
Fig.8. In this case there is a bright spot in the center. It does not correspond to
first order. Ignore this and take readings of diameter of rings.

Fig.9 Rings radius determination with the software (in the calculation we use
the relative radii and calibration is not necessary)
 700

600

Magnetic field
500
Magnetic field (mT)

400

300

200

100
38 40 42 44 46 48 50 52 54 56

Fig.10. pole piece separation gap(mm)

Precautions:

1. Don’t touch the lenses and filters in your finger. The single finger print will make
the things dirty and the interference fringes will be distorted.
2. Handle the filters carefully. Take the Technician help to change them.
3. While setting the optical path don’t look to the Cd-lamp in naked eye for long
time. It has a bad effect on the eye.

References:
1. For derivation of the formulas, Refer Phywe manual
https://www.nikhef.nl/~h73/kn1c/praktikum/phywe/LEP/Experim/5_1_10.pdf
2. Chapter 6, Experiments in Modern Physics, Adrian C. Melissinos and Jim
Napolitano, Academic Press.